GENERALIZED OPTION PRICING MODELS: NUMERICAL SOLUTIONS AND THE PRICING OF A NEW LIFE INSURANCE CONTRACT by EDUARDO SAUL SCHWARTZ B.Eng., School of Engineering, University of Chile, 1963 M. Sc., U n i v e r s i t y of B r i t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Faculty of Commerce and Business Administration We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1975 In presenting this thesis an advanced degree at the L i b r a r y in p a r t i a l the U n i v e r s i t y of s h a l l make it freely I f u r t h e r agree t h a t p e r m i s s i o n for of the requirements B r i t i s h Columbia, available for reference for e x t e n s i v e copying o f I agree and this for that study. thesis s c h o l a r l y purposes may be granted by the Head o f my Department or by h i s of fulfilment representatives. It this thesis for financial written C cyT^vc^v^O^-- University of B r i t i s h 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date gain s h a l l permission. Department of The is understood Columbia that not copying or p u b l i c a t i o n be allowed without my ABSTRACT The option p r i c i n g model developed by Black and Scholes and extended by Merton gives r i s e to p a r t i a l d i f f e r e n t i a l governing the value of an option. equations When the underlying stock pays no dividends - and i n some very r e s t r i c t i v e cases when i t does -, a closed form solution to the p a r t i a l d i f f e r e n t i a l equation subject to the appropriate boundary condition, has been obtained. But, i n more relevant cases such as the one i n which the stock pays d i s c r e t e dividends, no closed form solution has been found. This study shows how to solve those equations by numerical methods. The equations i n question are l i n e a r parabolic p a r t i a l equations. differential Although s o l u t i o n procedures are w e l l known i n Science and Mathematics, they have not been treated extensively i n the finance l i t e r a t u r e . approximating The numerical procedure used consists i n p a r t i a l d e r i v a t i v e s by f i n i t e d i f f e r e n c e s to obtain expressions which can be handled by the computer. A general numerical solution to the p a r t i a l d i f f e r e n t i a l equation governing the value of an option on a stock which pays d i s c r e t e dividends, i s developed i n this d i s s e r t a t i o n . In a d d i t i o n , the optimal strategies f o r exercising American options are derived. For a s u f f i c i e n t l y large value of the stock c l e a r l y larger than the exercise p r i c e , i t may pay to exercise the American option at dividend payment dates. This study shows how to determine the " c r i t i c a l stock p r i c e " above which i t w i l l pay to exercise the option. T r a d i t i o n a l l y l i f e insurance contracts have provided for benefits fixed i n terms of the l o c a l unit of account. The prolonged period of i n f l a t i o n during the l a s t three decades has served to diminish the usefulness of such usually long-term currency denominated contracts. As a r e s u l t , l i f e insurance companies have increasingly moved toward the study and issuance of equity based products the benefits of which depend upon the performance of some reference p o r t f o l i o of common stock. This d i s s e r t a t i o n deals with the p r i c i n g of equity-linked l i f e insurance contracts with asset value guarantee. In these contracts the benefits depend upon the performance of a p o r t f o l i o of common stock. The insurance company, however, guarantees a minimum l e v e l of benefits (payable on death or maturity) regardless of the investment performance of the reference p o r t f o l i o . In an equity based l i f e insurance p o l i c y with t h i s type of provision the insurance company assumes part of the investment r i s k as w e l l as the mortality risk. When the contract becomes a claim (either because of death of maturity), the policyholder i s e n t i t l e d to receive an amount equal either to the value of the reference p o r t f o l i o at that time or to the guaranteed sum, whichever i s the greater. This study shows how the benefits of the contract can be viewed i n terms of options and i t obtains the p a r t i a l d i f f e r e n t i a l equations governing the value of these options, which can also be solved by the above V mentioned numerical methods. It i s shown that the premium payments can be interpreted as negative dividend payments on the reference p o r t f o l i o . Another important problem i n r e l a t i o n to these p o l i c i e s i s the determination of the appropriate investment p o l i c i e s f o r insurance companies to enable them to hedge against the major r i s k s associated with the provision of the guarantee. I t i s shown how the equilibrium option p r i c i n g model determines the optimal investment strategy to be followed by insurance companies. The need f o r an appropriate model for p r i c i n g equitylinked l i f e insurance p o l i c i e s with asset value guarantee, has long been apparent i n the a c t u a r i a l l i t e r a t u r e . The model developed i n t h i s study gives normative rules for p r i c i n g such contracts. Moreover, the prices determined by these rules are equilibrium prices, that i s , they would p r e v a i l i n a p e r f e c t l y competitive market and would have the property that i f the insurance company were to charge them and were to follow the investment policy determined by the model, i t w i l l bear no r i s k and w i l l make neither p r o f i t nor l o s s . It i s the hope of the author of t h i s study that the model w i l l f i n d p r a c t i c a l applications i n the l i f e insurance industry. vi TABLE OF CONTENTS Page CHAPTER 1. 2. INTRODUCTION 1 1.1 1-2 1.3 Preamble The Valuation of Options on Dividend Paying Stocks.... The P r i c i n g of a New L i f e Insurance Contract 1 2 4 1.4 Outline of the Thesis 7 THE PRICING OF OPTIONS 10 2.1 Introduction 2.2 S t a t i s t i c a l Approach to Option Valuation 2.3 Expected Value Approach to Option Valuation 2.4 Market Equilibrium Approach to Option Valuation 2.5 Merton s Derivation of the B-S Formula 2.6 Extension of the Model 2.7 More Recent Developments 2.8 Conclusions PROCEDURES FOR PRICING OPTIONS ON STOCKS WITH CONTINUOUS DIVIDEND PAYMENTS 10 12 15 17 21 26 30 33 ? 3. 3.1 3.2 3.3 3.4 4. 35 35 36 37 Introduction Introduction to P a r t i a l D i f f e r e n t i a l Equations Difference Equations The P a r t i a l D i f f e r e n t i a l Equation Governing the Value of an Option 3.5 Truncation Error 3.6 S t a b i l i t y and Convergence 3.7 Boundary Conditions 3.8 System of Linear Equations 3.9 Solution Algorithm 3.10 Numerical Example 3.11 Conclusions 40 42 42 43 44 47 48 55 OPTION STRATEGIES FOR EXERCISING OPTIONS ON STOCKS WITH DISCRETE DIVIDENDS 57 4.1 4.2 4.3 4.4 Introduction Theory Solution Algorithm A p p l i c a t i o n to the ATT Warrant: S e n s i t i v i t y A n a l y s i s . 4.4.1 Dividends and C r i t i c a l Stock Prices 4.4.2 Variance Rates and C r i t i c a l Stock Prices 4.4.3 Interest Rates and C r i t i c a l Stock Prices 4.4.4 ATT Theoretical Warrant Prices 4.4.5 Comparing Warrant Values 57 58 64 67 68 69 72 72 79 vii Chapter 4.5 5. 5.11 5.12 5.13 5.14 I n t r o d u c t i o n to L i f e Insurance E q u i t y Based L i f e I n s u r a n c e A l t e r n a t i v e Approaches t o P r i c i n g B r i e f Survey o f t h e A c t u a r i a l L i t e r a t u r e The Guarantee as an O p t i o n V a l u i n g the O p t i o n Component of an Insurance C o n t r a c t . The S i n g l e Premium C o n t r a c t The Constant Continuous Premium C o n t r a c t The P e r i o d i c Premium C o n t r a c t The C a l l O p t i o n , The Put Option-and t h e P r i c i n g o f Investment R i s k The Measurement o f M o r t a l i t y Mortality Risk The E q u i l i b r i u m V a l u e o f the C o n t r a c t : Investment R i s k and M o r t a l i t y R i s k Conclusions THE EQUILIBRIUM PRICING OF ASSET VALUE GUARANTEES ON PERIODIC PREMIUM EQUITY LINKED L I F E INSURANCE POLICIES: APPLICATIONS 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 7. Conclusions 83 THE EQUILIBRIUM PRICING OF ASSET VALUE GUARANTEES ON EQUITY LINKED L I F E INSURANCE POLICIES: THEORY 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 6. page Introduction S o l u t i o n A l g o r i t h m f o r t h e P e r i o d i c Premium C o n t r a c t . The V a r i a n c e Rate o f t h e TSE I n d u s t r i a l Index Put Premium o r " R i s k Premium" Numerical Example Number 1 Numerical Example Number 2 N u m e r i c a l Example Number 3 N u m e r i c a l Example Number 4 N u m e r i c a l Example Number 5 Summary and C o n c l u s i o n s 85 85 87 91 92 95 98 100 101 104 110 115 117 119 120 1 2 2 122 127 131 132 132 140 144 149 153 155 OPTIMAL INVESTMENT STRATEGIES FOR THE SELLERS OF EQUITY LINKED L I F E INSURANCE POLICIES WITH ASSET VALUE GUARANTEES. 160 7.1 7.2 7.3 7.4 7.5 Introduction Hedging P o l i c y F e a s i b i l i t y o f the P o l i c y Numerical Example Summary and C o n c l u s i o n s BIBLIOGRAPHY 160 160 164 167 186 187 viii APPENDIX A. B. C. D. E. F. Page Computer Program No. 1: Option Prices on Stocks with Continuous Dividend. Payments 191 Computer Program No. 2: Option Prices on Stocks with Discrete Dividend Payments 193 Solution Algorithm: Option Prices i n the Constant Continuous Premium Contract 197 Computer Program No.3: Option Prices i n the Constant Continuous Premium Contract 200 Computer Program No.4: Option Prices i n the Periodic Premium Contract 202 Computer Program No.5: Computation of the Equilibrium Value of the Guarantee with Known Date of Expiration.... 205 Computer Program No.6: Computation of the Equilibrium Value of the Contract and the Risk Premium 208 H. Select M o r t a l i t y Tables 210 I. Computer Program No.7: Risk Premium versus Term of Contract 211 G. J. Computer Program No.8: Risk Premium versus Age of the Purchaser at Entry 213 K. Computer Program No.9: Investment P o l i c y 214 L. Investment P o l i c y L . l F e a s i b i l i t y of the P o l i c y L.2 Cash P o s i t i o n 219 219 220 ix LIST OF TABLES TABLE ••' Page 3.1 Trans Canada Pipeline. Warrant Values 51 3.2 Trans Canada P i p e l i n e Warrant: Regressions 53 3.3 Trans Canada P i p e l i n e Warrant Values 54 4.1 C r i t i c a l Stock Prices as a Function of Dividends and Time to Expiration 70 C r i t i c a l Stock Prices as a Function of the Variance Rate and Time to Expiration 73 C r i t i c a l Stock Prices as a Function of the Riskless Interest Rate and Time to Maturity 75 ATT Theoretical Warrant Prices for S p e c i f i c Stock Prices as a Function of Time to Expiration 77 4.2 4.3 4.4 4.5 Data Used i n the Estimation of Theoretical Warrant Prices 80 4.6 Warrant Prices 81 4.7 Ratio of Warrant Prices to Market Prices 82 5.1 Section of an Hypothetical M o r t a l i t y Table 116 6.1 M o r t a l i t y Table: Canadian Assured L i f e s , 1958-64 * 6.2 Computation of the Equilibrium Value of the Guarantee 2 6 with Known Date of Expiration: Example No.l 135 6.3 Risk Premium versus Term of Contract 138 6.4 Risk Premium versus Age of Purchaser at Entry 139 6.5 Computation of the Equilibrium Value of the Guarantee with Known Date of Expiration: Example No.2 141 6.6 Risk Premium versus Term of Contract 142 6.7 Risk Premium versus Age of Purchaser at Entry 143 6.8 Computation of the Equilibrium Value of the Guarantee with Known Date of Expiration: Example No.3 146 Table 6.9 Page Risk Premium versus Term of Contract 147 6.10 Risk Premium versus Age of Purchaser at Entry 148 6.11 Computation of the Equilibrium Value of the Guarantee with Known Date of Expiration: Example No.4 151 6.12 Risk Premium versus Term of Contract and Age of Purchaser at Entry 152 6.13 Computation of the Equilibrium Value of the Guarantee with Known Date of Expiration: Example No.5 154 6.14 Risk Premium versus Age of Purchaser at Entry 156 6.15 Summary of Annual Premium f o r the Put 157 7.1 Investment P o l i c y 172 7.2 Investment P o l i c y 173 7.3 Investment P o l i c y 174 7.4 Investment P o l i c y 175 7.5 Investment P o l i c y 176 7.6 Investment P o l i c y 177 7.7 Investment P o l i c y 178 7.8 Investment P o l i c y 179 7.9 Investment P o l i c y 1^0 7.10 Investment i n the Riskless Asset 181 7.11 Proportion Invested i n the Reference P o r t f o l i o 183 7.12 C r i t i c a l Values of the Reference P o r t f o l i o 185 x i LIST OF FIGURES FIGURE Page 3.1 A Mesh of Points Laid Over a Rectangular Region 40 4.1 C r i t i c a l Stock Price 63 4.2 C r i t i c a l Stock Price with Premium P 66 4.3 C r i t i c a l Stock Prices as a Function of Dividends and Time to Expiration 71 C r i t i c a l Stock Prices as a Function of the Variance Rate and Time to Expiration 74 4.4 4.5 C r i t i c a l Stock Prices as a Function of the Riskless Rate and Time to Expiration 76 4.6 ATT Theoretical Warrant Prices f o r S p e c i f i c Stock Prices 78 5.1 P a r t i a l D i f f e r e n t i a l Equation and Boundary Conditions for the Periodic Premium Contract 7.1 Investment 7.2 7.3 Investment i n the Riskless Asset Proportion Invested i n the Reference P o r t f o l i o i n the R i s k l e s s Asset Ill 167 182 184 xii ACKNOWLEDGEMENTS The c r e d i t f o r undertaking and completing t h i s d i s s e r t a t i o n must be shared by the candidate with h i s mentors on the Faculty of Commerce and Business Administration at the University of B r i t i s h Columbia. Most important among the l a t t e r i s Professor Michael J . Brennan. I f any one can be credited with the genesis of the idea for the research that culminated i n this report, i t i s he. As chairman of the d i s s e r t a t i o n committee, Dr. Brennan provided generous and indispensable guidance to the candidate. His sustained i n t e r e s t i n the study was a source of support so e s s e n t i a l f o r an undertaking of t h i s kind. Professor Phelim P. Boyle also merits a s p e c i a l expression of gratitude f o r h i s invaluable guidance to the candidate with respect to the f i e l d of insurance. Without h i s contribution of concepts and f a c t u a l information r e l a t i v e to this d i s c i p l i n e , the project could not have been launched. Mr. A l v i n G. Fowler, Associate Director of the Computer Centre at UBC, provided e s s e n t i a l i n s t r u c t i o n i n the solution of p a r t i a l d i f f e r e n t i a l equations by numerical methods, for which the author i s much indebted to him. As members of the d i s s e r t a t i o n committee, Professors Robert W. White and William T. Ziemba read the d i s s e r t a t i o n and offered h e l p f u l suggestions toward i t s improvement. Professor Ernest Greenwood, University of C a l i f o r n i a (Berkeley), kindly consented to examine the manuscript from the viewpoint of s t y l e . His e d i t o r i a l comments resulted i n a more readable product. The laborious task of typing the manuscript, was expertly performed under heavy time pressures by Miss Joan Adair. To a l l the l a t t e r the candidate wishes to extend his g r a t e f u l appreciation. During the period while the candidate was conducting the research herein reported he was receiving f i n a n c i a l support from The Canada Council. The candidate expresses h i s gratitude to that body. Throughout the years that he was a graduate student at the University of B r i t i s h Columbia the candidate was the fortunate r e c i p i e n t of the friendship and encouragement of Professors Peter A. Lusztig and Maurice D. L e v i . I t was t h e i r moral support which helped the candidate to surmount the obstacles so c h a r a c t e r i s t i c of one's i n i t i a l phase of graduate education. candidate owes a l a s t i n g debt of gratitude. To both of them the xiv To G l o r i a CHAPTER 1: 1.1 INTRODUCTION Preamble This study i s concerned with options: an option i s a security which gives i t s owner the r i g h t to purchase or s e l l another security at a predetermined price under certain conditions. The objective of this study i s twofold: (1) The valuation of options on dividend paying stocks; (2) The p r i c i n g of a new l i f e insurance contract. and These two apparently unrelated problems are shown to have a common underlying structure and the same valuation techniques can be applied i n both cases. The t h e o r e t i c a l foundation of this study i s the stock option p r i c i n g model developed by Black and Scholes [3] and extended by Merton [32]. This option valuation model i s derived from the market equilibrium or arbitrage p r i n c i p l e that, i n equilibrium, there are no r i s k l e s s p r o f i t s to be made with a zero net investment. A zero net investment p o r t f o l i o i s obtained by taking long and short positions on the stock, the option and the r i s k l e s s asset. A p a r t i a l d i f f e r e n t i a l equation governing the value of the option i s the r e s u l t of this analysis, which together with the boundary condition can be used under certain conditions to derive an a n a l y t i c a l expression f o r the value of the option. Most of the options discussed i n this study give r i s e to p a r t i a l d i f f e r e n t i a l equations f o r which no closed form a n a l y t i c a l solutions exist (or for which closed form solutions have not been found). to solve these equations by numerical methods. I t i s shown how The equations i n question are l i n e a r parabolic p a r t i a l d i f f e r e n t i a l equations and although solution procedure^ are f u l l y discussed i n s c i e n t i f i c and mathematical texts they have i 2 not been treated extensively i n the finance l i t e r a t u r e . The numerical edure used here consists i n approximating p a r t i a l derivatives by proc- finite differences to obtain expressions which can be handled by the computer. 1.2 The Valuation of Options on Dividend Paying Stocks Assuming " i d e a l market conditions" Black and Scholes [13] developed an equilibrium option p r i c i n g model for the case where the underlying stock pays no dividends. They showed that i t i s possible to form a hedged p o r t f o l i o by taking a long p o s i t i o n on the stock and a short p o s i t i o n on the option (or v i c e versa) where the return of the hedged p o r t f o l i o i s independent of the p r i c e of the stock, that i s , r i s k l e s s . In equilibrium, to avoid arbitrage p r o f i t s , the return on the hedged p o r t f o l i o must then be equal to the r i s k free rate of i n t e r e s t . By maintaining continuously the hedged p o r t f o l i o they obtain a p a r t i a l d i f f e r e n t i a l equation the value of the option. governing Given the value of the option at expiration (boundary condition) they solve the d i f f e r e n t i a l equation and obtain a formula for the value of the option as a function of the current p r i c e of the stock, time to maturity and known parameters (the r i s k l e s s i n t e r e s t rate and the variance of the rate of return on the stock both assumed constant, and the exercise p r i c e of the option). Merton [32] shows that i n the case where the stock i s assumed to pay continuous dividends, the hedging process described by Black and Scholes can also be applied and a d i f f e r e n t p a r t i a l d i f f e r e n t i a l equation obtained. He points out that, i n general, this p a r t i a l d i f f e r e n t i a l equation can not be solved by a n a l y t i c a l methods ( i . e . , i t does not have a closed form s o l u t i o n ) . A numerical procedure f o r p r i c i n g options on stocks with dividend payments i s developed i n this study. continuous In p a r t i c u l a r , the p a r t i a l d i f f e r e n t i a l equation mentioned i n the preceding paragraph i s solved numerically i n the case of constant continuous dividends. The present author f e e l s that the numerical approach developed here w i l l have a number of f r u i t f u l applications i n the theory of options i n finance. As mentioned, this thesis deals with two such problems: the valuation of options on dividend paying stocks and the p r i c i n g of a new l i f e insurance contract. Some options are exercisable only at maturity (European type) , but most options are exercisable at any time p r i o r to expiration (American type). Merton [32] has shown that i f a stock pays no dividends or the option i s "dividend payout p r o t e c t e d , " i t w i l l never pay to exercise an American option before maturity and,hence, the value of an American option i s equal to the value of i t s European counterpart. But i f the stock pays dividends and the option i s not dividend p r o t e c t e d , i t may pay to exercise the American option before maturity because the option holder foregoes the dividend paid to the stockholder and,hence,its value may counterpart. be greater than i t s European Merton has also shown that i f a stock pays discrete dividends, perhaps i t w i l l pay to exercise an American option j u s t before the stock goes ex-dividend, but never i n between dividend payments dates. Black[5] has proved that the value of an option on a stock that pays discrete dividends i s also governed by the same p a r t i a l d i f f e r e n t i a l equation derived by Black and Scholes f o r the no-dividend case. The boundary conditions, however, change at each dividend payment date to r e f l e c t the fact that i t may to exercise the American option at those points i n time. recognized by Merton [32]. pay This was also No closed form solution has been found i n t h i s case. A general numerical solution to the p a r t i a l d i f f e r e n t i a l equation governing the value of an option on a stock which pays discrete dividends 4 i s developed i n this d i s s e r t a t i o n . In addition, the optimal strategy for exercising American options i s derived. For a s u f f i c i e n t l y large value of the stock, c l e a r l y l a r g e r than the exercise p r i c e , i t may pay to exercise the American option at dividend payments dates. This study shows how to determine the " c r i t i c a l stock p r i c e " above which i t w i l l pay to exercise the option. 1.3 The P r i c i n g of a New L i f e Insurance Contract T r a d i t i o n a l l y l i f e insurance contracts have provided for benefits fixed i n terms of the l o c a l unit of account. The prolonged period of sustained i n f l a t i o n during the l a s t three decades has served to diminish the usefulness of such usually long-term currency denominated contracts. As a r e s u l t , l i f e insurance companies have increasingly moved toward the study and issuance of equity based products, the benefits of which depend upon the performance of some reference p o r t f o l i o of common stocks. The idea behind this move i s that, i n the long run, equities would provide a better hedge against i n f l a t i o n than a fixed monetary amount. If a l l the investment r i s k associated with owing common stock i s l e f t to the policyholder, he has no guarantee as to the asset value of h i s contract. Turner [44] argues that i t i s both reasonable and appropriate for a l i f e insurance company to o f f e r an a d d i t i o n a l assurance under ^equity based products whereby part of the investment r i s k i s assumed by the company. This has been increasingly done i n the United Kingdom and Canada. This d i s s e r t a t i o n deals with the p r i c i n g of equity-linked l i f e insurance contracts with asset value guarantee.. In these contracts the benefits depend upon the performance of a p o r t f o l i o of common stocks ; the insurance company, however, guarantees a minimum l e v e l of benefits (payable 5 on death or maturity) regardless of the investment performance of the reference p o r t f o l i o . In an equity based l i f e insurance policy with this type of provision the insurance company assumes part of the investment r i s k as w e l l as the mortality r i s k . When the contract becomes a claim (either because of death or maturity), the policyholder i s e n t i t l e d to receive an amount equal either to the value of the reference p o r t f o l i o at that time or to the guaranteed sum, whichever i s the greater. The benefits of the contract can then be viewed .as the sum of a sure amount, the guarantee, plus the amount by which the value of the reference portf o l i o exceeds the guarantee. This l a t t e r amount may be regarded as the value, at the time when the benefits are payable, of an immediately exercisable c a l l option which permits the holder to purchase the reference p o r t f o l i o f o r the pre-determined guaranteed amount. I f the value of the reference p o r t f o l i o i s smaller than the guaranteed amount, the c a l l option w i l l not be exercised and w i l l have zero value, while i f i t i s larger,the c a l l options w i l l be exercised and i t s exercise value w i l l be the difference between the value of the reference p o r t f o l i o and the guaranteed amount. The benefits of the contract at expiration can also be viewed as the sum of the value of the reference p o r t f o l i o plus the amount by which the guarantee exceeds the value of the reference p o r t f o l i o . This l a t t e r amount may be regarded as the value of an immediately exercisable put option which permits the holder to s e l l the reference p o r t f o l i o f o r the pre-determined guaranteed amount. I f the value of the reference p o r t f o l i o i s greater than the guaranteed amount, the put option w i l l not be exercised and w i l l have zero value, while i f i t i s smaller, the put w i l l be exercised and i t s exercise value w i l l be the difference between the guaranteed amount and the value of the reference p o r t f o l i o . Three separate types of equity-linked l i f e insurance p o l i c i e s with asset value guarantee must be considered: (1) The single premium contract i n which the whole investment i n the reference p o r t f o l i o i s made at the date of issue of the p o l i c y . It i s shown that i n this case the value of the c a l l option corresponds precisely to the value of a c a l l option on a non-dividend paying stock f o r which a closed form solution has been derived by Black and Scholes. (2) The constant continuous premium contract i n which a constant rate of investment i n the reference p o r t f o l i o i s also assumed. continuous I t w i l l be shown that i n this case i t i s possible to obtain a p a r t i a l d i f f e r e n t i a l equation governing the value of the c a l l option very s i m i l a r to the one derived by Merton [32] f o r an option on a continuous dividend paying stock, but where the premium has to be considered as a negative dividend. numerical solution to the d i f f e r e n t i a l equation i s also (3) A developed. The p e r i o d i c premium contract i n which the investments on the reference p o r t f o l i o are also made p e r i o d i c a l l y . From the viewpoint of p r a c t i c a l applications this type of contract i s the most important one. I t w i l l be shown that the valuation of the c a l l option f o r the periodic premium contract i s analogous to the valuation of a c a l l option on a stock that pays discrete dividends, where the premium represents a negative dividend. The p a r t i a l d i f f e r e n t i a l equation and boundary conditions are derived and the solution procedure i s developed. From the point of view of the insurance company the important variable for p r i c i n g equity-linked l i f e insurance contracts with asset value is the value of the put option. guarantee The value of the put option represents the "cost" to the insurance company of giving the guarantee. I t w i l l be 7 shown how to compute the value of the put option from the value of the c a l l option for the types of contracts mentioned above. Another important problem in relation to these equity-based l i f e insurance policies is the determination by insurance companies of the appropriate investment policies to enable them to hedge against the major risks associated with the provision of the guarantee. It w i l l be shown how the equilibrium option pricing model used in this study determines, by the hedging process described in the preceding section, the optimal investment strategy to be followed by insurance companies i f they want to hedge against the risk assumed by giving the guarantee. The need for an appropriate model for pricing equity-linked l i f e insurance policies with asset value guarantee has long been apparent in the actuarial literature.'' The model developed in this study gives normative rules for pricing such contracts. rules are equilibrium prices; Moreover, the prices determined by these that i s , they would prevail in a perfectly competitive market and would have the property that the insurance company charging them and following the investment policy determined by the model w i l l bear no risk and w i l l make neither profit nor loss. It is the hope of the author of this study that the model w i l l find practical applications in the l i f e insurance industry. 1.4 Outline of the Thesis A survey of some of the relevant option literature is given in Chapter 2. Black and Scholes' [3 ] equilibrium option pricing model is See for example Leckie [22], Turner [44], Di Paolo [10] and Kahn [18]. 8 presented along l i n e s given by Merton [32]. Merton's extension of the model to the case where the stock pays continuous dividends i s given and the p a r t i a l d i f f e r e n t i a l equation governing the value of the option i s derived. This chapter concludes by discussing recent extensions of the equilibrium option p r i c i n g model. The numerical solution of parabolic p a r t i a l d i f f e r e n t i a l using the f i n i t e differences method i s presented i n Chapter 3. equations The algorithm f o r the value of an option on a stock which pays constant solution continuous dividends i s developed and the numerical method i s applied to p r i c e a p a r t i c ular option. A solution algorithm f o r the value of options on stocks with d i s c r e t e dividends i s developed i n Chapter 4. these options are also considered. The optimal strategies f o r e x e r c i s i n g Using the American Telephone and Telegraph 1975 Warrant as an example, the s e n s i t i v i t y of the c r i t i c a l stock p r i c e to changes i n dividends, variance rates and i n t e r e s t rates i s examined. Chapter 5 i s devoted to the development of the model f o r equilibrium p r i c i n g of asset value guarantees on equity-linked l i f e insurance contracts. In t h i s chapter the basic concepts of t r a d i t i o n a l and equity based l i f e insurance p o l i c i e s are presented and the a e t u a l i a l l i t e r a t u r e covering the l a t t e r i s surveyed. Following t h i s , the concept of the amount as an option i s developed guaranteed and the models for valuing the option component of three types of contracts (single premium, premium and p e r i o d i c premium) are presented. equilibrium p r i c i n g of the investment continuous The put option as the r i s k i s then considered. mortality r i s k i s introduced to f i n d the Finally, average equilibrium value of the contract and the " r i s k premium" that insurance companies must charge f o r g i v i n g the guarantee. The solution algorithm for the model presented i n Chapter 5 i s derived i n Chapter 6. Some numerical examples are then given and the s e n s i t i v i t y of the value of the guarantee to various parameters i s examined. parameters are: (1) the variance of the rate of return on the reference portfolio; (2) the r i s k l e s s i n t e r e s t rate; guarantee; (4) the age of the purchaser of the p o l i c y ; of These (3) the nature of the minimum and (5) the term the contract. Chapter 7 i s concerned with the d e r i v a t i o n of appropriate investment p o l i c i e s for l i f e insurance companies to enable them to hedge against the r i s k s assumed by giving the guarantee i n equity based l i f e insurance p o l i c i e s . I t i s shown how the answer to this problem i s given by the hedging process described by Black and Scholes [3 ]. This chapter also studies the question whether the premiums received by an insurance company for a given p o l i c y are always s u f f i c i e n t to finance the long p o s i t i o n s i n the hedged p o r t f o l i o or sometimes i n s u f f i c i e n t so that a d d i t i o n a l borrowing i s required. r i s k i s also introduced to a r r i v e at the actual optimal investment Mortality strategy. 10 CHAPTER 2: 2.1 THE PRICING OF OPTIONS . Introduction An option i s a security giving i t s owner the right to buy or s e l l an asset at a predetermined p r i c e , subject to certain conditions, within a s p e c i f i e d period of time. I f the option can be exercised any time up to the date of i t s expiration or maturity, i t i s c a l l e d an "American option." I f i t can be exercised only at expiration, i t i s c a l l e d a "European option." The previously agreed price that i s paid f o r the asset when the" option i s exercised i s c a l l e d the "exercise p r i c e . " can be surrendered The l a s t date on which the option or exercised i s c a l l e d the "maturity date" or "expiration date." A standard " c a l l option" gives i t s owner the r i g h t to buy and a standard "put option" gives him the right to s e l l one share of common stock at a given exercise p r i c e . There exists other option contracts which are d i f f e r e n t combinations of put and c a l l options. A warrant i s a type of c a l l option which (1) i s issued by a corporation rather than by an i n d i v i d u a l , (2) has a longer l i f e than the normal c a l l option, and (3) i s not protected against dividends, that i s , i t s owner does not receive the dividends paid on the associated asset (whereas some c a l l options are dividend protected). Throughout most of t h i s d i s s e r t a t i o n unprotected c a l l options and warrants w i l l be discussed with the generic work "option" being used to denote this type of s e c u r i t i e s . Put options w i l l also be considered when discussing l i f e insurance p o l i c i e s i n Chapters 5 and 6, but then the d i s t i n c tion between the c a l l option and the put option w i l l be c l e a r l y indicated. The problem of evaluating stock options has been studied by economists and s t a t i s t i c i a n s since the beginning introduced of the century when Bachelier [2] the theory of Brownian motion i n stock price fluctuations. 11 These studies may be c l a s s i f i e d into three broad categories: (1) Relevant v a r i a b l e or s t a t i s t i c a l approaches; (2) Expected value theories; and (3) Market equilibrium theory. In the s t a t i s t i c a l approach to option valuation a l l variables that might possibly influence option p r i c e s are considered and the s t a t i s t i c a l l y s i g n i f i c a n t ones are i d e n t i f i e d . In the expected value approach to option valuation the mathematical r e l a t i o n s h i p between option prices and the underlying stock p r i c e s i s analyzed, and equations for estimating expected future option values are developed. The t h e o r e t i c a l value of the option i s then assumed to be the discounted present value of this future value. The market equilibrium approach to option valuation also considers the mathematical r e l a t i o n s h i p between the option p r i c e and the associated common stock p r i c e , but the r e l a t i o n s h i p i s obtained by imposing market equilibrium conditions. I t i s based on the arbitrage p r i n c i p l e that i n equilibrium there are no r i s k l e s s p r o f i t s to be made. This chapter summarizes some of the important published work on the d i f f e r e n t approaches to option valuation with p a r t i c u l a r emphasis on the market equilibrium approach which has been the motivation of this d i s s e r t a t i o n . Then, the Black-Scholes [3] option valuation model i s presented extensions by Merton [32] are considered. and relevant This option p r i c i n g model provides the t h e o r e t i c a l basis as w e l l as the s t a r t i n g point for this t h e s i s . The chapter concludes with a consideration of further developments i n the formulation of equilibrium option p r i c i n g models. 12 2.2 S t a t i s t i c a l Approach to Option Valuation Shelton's [38] model f o r the valuation of warrants i s a typical example of the s t a t i s t i c a l approach -to option valuation. He postulates that the warrant p r i c e i s bounded from below by i t s conversion value and bounded from above by a value equal to three-fourths of the current stock price. The upper boundary rests upon the argument that when a stock i s s e l l i n g at four times the exercise p r i c e , the leverage advantage i s so small so that the premium over conversion value disappears. His s t a t i s t i c a l analysis tests to what extent the relevant factors influence the p r i c e of the warrant within these boundaries. The factors tested by Shelton i n a stepwise multiple regression were: (1) dividend y i e l d on the associated stock; (2) whether the warrant i s l i s t e d or not (dummy v a r i a b l e ) ; (3) the length of time to expiration of the warrant; (4) whether the warrant sold for more or less than $5.00 (dummy v a r i a b l e ) ; (5) the h i s t o r i c v o l a t i l i t y of the common stock (measured by the average of the r a t i o s of yearly highs to lows of stock p r i c e s f o r three years) and; (6) ' the recent trend of the stock p r i c e (measured by the percentage change of the stock p r i c e over the previous year). Shelton found that the f i r s t three of the above v a r i a b l e s are highly s i g n i f i c a n t i n explaining warrant p r i c e s , while the l a s t three v a r i a b l e s are not s i g n i f i c a n t . I t i s i n t e r e s t i n g to note that neither v o l a t i l i t y nor recent trend of the related stock the way i proved to be s i g n i f i c a n t , though i n which they were measured should be taken into account. 13 In h i s 1965 study on the valuation of warrants Kassouf [19] described the relationship between observed warrant prices and i t s underlying observed warrant prices and i t s underlying common stock by the formula y = (x + l ) y = warrant price/exercise p r i c e x = stock price/exercise p r i c e z = estimated parameter (has a value between 1 and «°) Z V z - 1 where In the s t a t i s t i c a l analysis to estimate the value of the c r i t i c a l v a r i a b l e z, Kassouf considered four variables which might influence warrant p r i c e s : (1) the length of time to expiration; (2) the dividend y i e l d on the underlying stock; (3) the p o t e n t i a l d i l u t i o n of the common stock r e s u l t i n g from the exercise of a l l outstanding warrants; (4) and the recent price h i s t o r y of the common stock. In a more recent paper Kassouf [20]presented the l a t e s t r e s u l t s of his s t a t i s t i c a l approach to warrant valuation. In a multiple regression, based on 222 observations f o r the period 1945 to 1964, he obtains the following equation f o r estimating the value of z: z = 1.2221 + 5.3131/T + 14.8135R + 0.2765D + 0.4401 X/A + 0.4131 K,n(X/X..) 14 where T = number of months to maturity R = dividend y i e l d on the stock (annual) D = number of outstanding warrants/number of outstanding common shares X = p r i c e of common share A = exercise price of warrant Xj = p r i c e of common share 11 months previous. He found a l l variables to be s i g n i f i c a n t (at a 0.05 level) and a multiple c o r r e l a t i o n c o e f f i c i e n t of 0.63. Kassouf did not include any measure of r i s k i n h i s regression. His warrant p r i c i n g formula also implies that warrants w i l l never be exercised p r i o r to maturity because the warrant p r i c e would drop to the exercise value (X-A) only i f z goes to i n f i n i t y . In a very i n t e r e s t i n g book: "Beat the Market, A S c i e n t i f i c Stock Market System," Thorp and Kassouf [43] describe how i t should be possible to make substantial p r o f i t s by s e l l i n g warrants short and hedging by buying long the underlying stock. They point out that a hedged investment turns out to be p r o f i t a b l e and yet less r i s k y since the price of the warrant and the price of the stock tend to move i n sympathy with each other. When the stock and warrant prices move i n one d i r e c t i o n the loss i n one p o s i t i o n (short or long) i s l a r g e l y covered by the gain i n the other p o s i t i o n (long or short). The s e l e c t i o n of the appropriate candidates for this hedging operation i s based on K a s s o u f s s t a t i s t i c a l approach to warrant valuation as described above. 15 The concept developed by Thorp and Kassouf which states that i t i s possible to reduce r i s k by forming a hedged p o s i t i o n between the warrant and i t s underlying stock, i s one of the c r u c i a l ideas used by BlackScholes [3] to developed the fundamental model of the market equilibrium approach to valuation. sections. This model w i l l be discussed i n d e t a i l i n subsequent What Thorp and Kassouf f a i l to pursue i s the f a c t that i n equilibrium the expected return on a p e r f e c t l y hedged p o s i t i o n , created by going short i n one s e c u r i t y and long i n the other s e c u r i t y , must be equal to the return on a r i s k l e s s asset. This l a s t important concept was f i r s t recognized by Black-Scholes [3]. 2.3 Expected Value Approach to Option Valuation This approach i s a straight forward adaptation of the t r a d i t i o n a l bond and stock valuation models which proceed by discounting an expected stream of dividends or coupon payments and terminal p r i c e at a predetermined discount rate. Sprenkle [39] and Ayres [1] attempt to f i n d the expected value of a warrant and to measure r i s k preferences of investors i n the markets for warrants. To compute the expected value of a warrant using a n a l y t i c a l methods i t i s necessary to specify the p r o b a b i l i t y d i s t r i b u t i o n of future stock p r i c e s . Sprenkle assumed that successive changes i n stock prices follow the random walk model and that the p r o b a b i l i t y d i s t r i b u t i o n of stock p r i c e changes i s l o g normally d i s t r i b u t e d . The expected value of the warrant when exercised i s then given by: (2.1) where E[W] S W £(S) = = = = C = Jc (S-C) US)dS random v a r i a b l e representing stock p r i c e ; random variable representing warrant p r i c e ; p r o b a b i l i t y density function of future stock prices (lognormal), and exercise p r i c e . 16 Sprenkle assumes that investors perceive r i s k i n the warrant market i n the variable "leverage," defined as the r a t i o of the standard percentage changes i n warrant prices to the standard deviation of deviation of percentage changes i n stock p r i c e s , minus one. The primary purpose of Sprenkle's work i s to provide estimation of investors' r i s k preferences empirical and he does this by comparing the actual p r i c e s of warrants with the computed values obtained by the expected exercise value of the warrant at expiration. discounting But he i s unable to estimate empirically the required discount f a c t o r . Ayres' objective i s to investigate the existence of r i s k aversion i n the warrant markets. He also formulates h i s warrant valuation model based on the assumption of lognormally d i s t r i b u t e d stock p r i c e changes and computes the value of the warrant by discounting the expected exercise value of the warrant at maturity at the expected rate of return on the warrant. But when analyzing the r e l a t i o n s h i p between r i s k and return to determine the existence of r i s k aversion, he assumes that the warrant investor perceives a (the standard deviation of the lognormal d i s t r i b u t i o n of the r a t i o s of stock prices) as the measure of r i s k i n the successive stock. In t h e i r warrant valuation models neither Sprenkle nor Ayres take into account e x p l i c i t l y the fact that a warrant (American type) i s exercisable at any time p r i o r to maturity. Samuelson [36] and McKean [27] discuss, from the economic and mathematical standpoints respectively, a warrant valuation model which also takes into account the non-negative extra value to the warrant-holder of the r i g h t to exercise a warrant at any (American type) . time i n the i n t e r v a l p r i o r to i t s maturity 17 Under the assumptions of constant expected rate of return on the stock (a) and of constant expected rate of return on the warrant (g) Samuelson and McKean derive a general formulation f o r evaluating warrants. They assume that the p r o b a b i l i t y d i s t r i b u t i o n of possible values of the stock when the warrant i s exercised i s lognormal and take the expected value of this d i s t r i b u t i o n cutting i t o f f at the exercise p r i c e as i n equation (2.1). They then discount this expected value to the present using as a discount rate the expected rate of return on the warrant (B). But the problem of determining the values of a and 3 has not been solved. 2.4 Market Equilibrium Approach to Option Valuation This approach was taken f o r the f i r s t time i n Black-Scholes' seminal paper "The P r i c i n g of Options and Corporate L i a b i l i t i e s " [3]. Being e s s e n t i a l to the understanding of this d i s s e r t a t i o n , the Black-Scholes' (B-S) model w i l l be presented here i n greater d e t a i l following c l o s e l y t h e i r o r i g i n a l paper. In deriving their formula f o r the value of an option i n terms of the p r i c e of the underlying stock, B-S make the following assumptions ("ideal conditions"): a) The r i s k l e s s i n t e r e s t rate i s known and i s constant through (Merton time. ' 1 [32] extends B-S model to include stochastic i n t e r e s t rates.) B-S a c t u a l l y use the short-term i n t e r e s t rate, but being constant and known the long-term i s the same as the short-term. Merton [32] shows that the relevant rate i s that of a pure discount bond with the same maturity date as the option. 18 b) The stock price follows a random walk i n continuous time. The p r o b a b i l i t y d i s t r i b u t i o n of possible stock prices at the end of any f i n i t e i n t e r v a l i s log-normal. stock i s constant. The variance rate of the return on the The stochastic s p e c i f i c a t i o n of stock prices w i l l be discussed i n greater d e t a i l i n subsequent sections. (In Merton's extension of the B-S model the variance rate of the return on the stock i s allowed to be a known deterministic function of time.) c) The stock pays no dividends or other d i s t r i b u t i o n s . Merton [32] obtains the p a r t i a l d i f f e r e n t i a l equation governing the value of the option i n the case of continuous indications on how dividends and Merton [32] and Black [ 5] give to solve the problem i n the case of d i s c r e t e dividends. These topics w i l l be treated i n subsequent sections. d) The option i s "European," that i s , i t can only be exercised at maturity. Merton [32] proves that i f an American option i s "dividend protected," i t should never be exercised before maturity and he studies the conditions under which an American option w i l l not be exercised before maturity even i f i t i s not "dividend protected." e) There are no transaction costs i n buying or s e l l i n g the stock or the option. f) The borrowing and the lending rates are equal and there are no r e s t r i c t i o n s on borrowing. g) S h o r t s e l l i n g i s allowed without r e s t r i c t i o n s and without p e n a l t i e s . Thorp [42] shows that r e s t r i c t i o n s against the use of proceeds of shortsales do not invalidate B-S a n a l y s i s . Black-Scholes argue that, "under these assumptions, the value of the option w i l l depend only on the price of the stock and time and variables ! I I i 19 that are taken to be known constants. Thus, i t i s possible to create a hedged position, consisting of a long p o s i t i o n i n the stock and a short p o s i t i o n i n the option, the value of which w i l l not depend on the p r i c e of the stock, but w i l l depend only on time and the values of known constants." I f the value of the option as a function of the stock p r i c e S and time t i s expressed as W(S,t), the number of options that must be sold short against one share of stock long may be denoted by l/-r^r or equivalently by (2.2) l/W (S,t) s where the subscript refers to the p a r t i a l derivative of W(S,t) with respect to i t s f i r s t argument. To see that the value of such a hedged p o s i t i o n does not depend on the p r i c e of the stock, B-S point out that the r a t i o of the change i n the option value to the change i n the stock p r i c e , AW/AS, i s approximated AS i s small. by Wg(S,t) when Then, i f the stock price changes by an amount AS, the option p r i c e w i l l change, approximately, by an amount Wg(S,t)AS, and the number of options given by expression (2.2) w i l l change i n value by an amount AS. Thus, the change i n the value of a long p o s i t i o n i n one share of stock w i l l be approximately o f f s e t by the change i n value of a short p o s i t i o n i n 1/Wg options. Black-Scholes conclude that, "As the variables S and t change, the number of options to be sold short to create a hedged p o s i t i o n with one share of stock changes. I f the hedge i s maintained continuously, then the approximation mentioned above becomes exact, and the return on the hedged p o s i t i o n i s completely independent of the change i n the value of the stock. return on the hedged p o s i t i o n becomes c e r t a i n . " 2 See Black-Scholes [3] p. 641. In fact, the 20 The value of the equity i n the hedged p o s i t i o n i s : (2.3) S - W/W g and the change i n the value of the equity i n a short i n t e r v a l At i s : (2.4) AS - AW/Wg Since the return on the equity i n the hedged p o s i t i o n i s c e r t a i n , the return must be equal to rAt (where r i s the r i s k l e s s i n t e r e s t r a t e ) . Then the change i n the equity (2.4) must be equal to the value of the equity (2.3) times rAt. (2.5) AS - AW/W S = (S-W/W )rAt s Assuming that the short p o s i t i o n i s changed continuously, i t i s possible to use stochastic calculus to expand AW, introduce i t s value i n (2.5) and s i m p l i f y to obtain the p a r t i a l d i f f e r e n t i a l equation the value of the option. governing The d e t a i l s of the derivation are not given here , because the B-S formula i s derived i n the next section along the l i n e s given by Merton [32], t h i s being more convenient for our subsequent analysis. In t h i s approach to option valuation no assumptions are made with respect to the expected return on the stock and the expected return on the option. The mathematical r e l a t i o n s h i p between the option value and the value of the associated stock and the time to maturity of the option i s obtained by the arbitrage p r i n c i p l e that i n market equilibrium conditions there are no r i s k l e s s p r o f i t s to be made. This approach to option valuation, based on sound economic theory, has been one of the important seventies; advances i n the finance l i t e r a t u r e of the i t s p o t e n t i a l applications into the p r i c i n g of corporate l i a b i l i t i e s and other kinds of options have not yet been f u l l y explored. The market equilibrium approach to option valuation w i l l be followed i n the rest of this study. 21 2.5 M e r t o n ' s D e r i v a t i o n o f t h e B-S F o r m u l a The paying expected r a t e o f r e t u r n p e r u n i t o f time on a s t o c k a t each p o i n t i n time t i s d e f i n e d by (2.6) E 01 and non-dividend t [ ^ ( t S ( t ] ) o f t h e r a t e o f r e t u r n p e r u n i t o f time i s d e f i n e d by [( \ 2 a - h = the variance (2.7) S ( t + S(t+ ^ (t = - S ( t -ah)*] ) h where S ( t ) i s t h e p r i c e o f t h e s t o c k a t time t , h i s a s m a l l i n c r e m e n t a n d "E " i s t h e c o n d i t i o n a l e x p e c t a t i o n the limit as h tends t o zero, a i s c a l l e d expected r a t e o f r e t u r n and a the operator. time In instantaneous the instantaneous variance of return. Under m i l d assumptions Merton [31] shows t h a t i t i s p o s s i b l e t o w r i t e down t h e s t o c k r e t u r n d y n a m i c s i n s t o c h a s t i c d i f f e r e n c e form as ( 2 8 s(t h) - ) + t s ( t ) __ a h + where y ( t ) i s a random p r o c e s s E [y] t (2.9) a y ( t ) / with = 0 and EJy ] 2 = 1 Defining a stochastic process, (2.10) z(t+h) - z ( t ) the r limit = ? ( t ) , by y(t)«fr~ o f (2.10) as h tends t o zero d e s c r i b e s a Wiener process of 22 Brownian motion defined by the stochastic d i f f e r e n t i a l equation: (2.11) dz = y(t)/dt with E [dz] = (2.12) 0 and E [dz] 2 fc = dt The instantaneous rate of return on the stock can be obtained by taking the l i m i t of (2.8) as h tends to dt. The stochastic d i f f e r e n t i a l equation f o r the instantaneous rate of return i s then: adt + adz "Processes such as (2.13) are c a l l e d It6 processes and while they are continuous, they are not d i f f e r e n t i a b l e . In (2.13) a may be a stochastic v a r i a b l e but a i s assumed constant. In Merton's [32] more general formulation of the model a may be a known, deterministic function of time. Given a constant rate of i n t e r e s t i t i s reasonable to assume^ that the value of the option, W, i s a function of the stock p r i c e S and the length of time to expiration, T: (2.14) W = W(S,T) Given the d i s t r i b u t i o n a l assumptions on the stock p r i c e , i t i s necessary to apply Ito's Lemma to obtain an expression f o r the change i n the option p r i c e over time. Itd's Lemma i s the stochastic-analog of See Merton [31] p. 873. For a discussion of stochastic d i f f e r e n t i a l equations of the Ito type see McKean [28] . A function i s not d i f f e r e n t i a b l e i f the derivatives of the function with respect of i t s argument, i n this case time, ;do not exist, at one or more points i n the domain. 4 See Merton [32 ] . 23 Taylor's expansion i n calculus; i t states how to d i f f e r e n t i a t e functions of Wiener processes."* Applying Ito's Lemma to (2.14), the stochastic d i f f e r e n t i a l equation for option p r i c e changes i s given by: (2.15) dW = W dS + W dT + h W ( d S ) s T 2 ss where subscripts denote p a r t i a l derivatives and (2.16) (dS) = (2.17) dT dW 2 2 = -dt Substituting (2.18) a S dt 2 (2.13) (2.16) and (2.17) into (2.15): = W S(adt+adz) - W dt + h a S 2 s T 2 W dt ss Rearranging terms (2.18) can be written as (2.19) dW ~ = Bdt + ydz where (2.20) (2.21) 3 = ^[aSW + % a S 2 s = gsw 2 W ss - W] T s W Notice that the stochastic d i f f e r e n t i a l equation (2.19) f o r the instantaneous return on the option has the same stochastic process dz (standard Wiener process) as the stochastic d i f f e r e n t i a l equation (2.13) for the instantaneous return on the stock. This fact allows the formation of a hedged p o r t f o l i o between the stock and the option where the stochastic component of the return i s eliminated, See McKean [28] and Merton [30]. as indicated by B-S. 24 Consider forming a p o r t f o l i o by investing i n the stock, the option and the r i s k l e s s asset, such that the t o t a l investment i n the p o r t f o l i o i s zero. This can be achieved by financing long positions i n the s e c u r i t i e s with borrowing and using the proceeds of short sales. Let a^ = instantaneous amount invested i n the stock a2 = instantaneous amount invested i n the option 33 = instantaneous amount invested i n the r i s k free asset. Then the condition for zero t o t a l investment i n the p o r t f o l i o can be written as (2.22) a L + a 2 + a = 3 0 The instantaneous (2.23) dy = a i jr- d o l l a r return on the p o r t f o l i o , dy, can be written as + a 2 Substituting (2.13), dy ^ + a rdt 3 (2.19) and a^(adt + adz) + a ( 8 d t + ydz) = 2 (2.22) into (2.23) - (a^ + a ) 2 rdt and s i m p l i f y i n g : (2.24) dy = [a (a-r) + a ( B - r ) ] d t + [aja + a y]dz x 2 A strategy a^ = a^ 2 can be chosen such that the c o e f f i c i e n t of dz i n (2.24) i s always zero (normalization): (2.25) *a a] + a * 2 Y = 0 Then, the t o t a l return on the p o r t f o l i o , dy*, would be that i s , the hedged p o s i t i o n would have been formed. non-stochastic; Since the aggregate investment i n the p o r t f o l i o i s zero, to avoid arbitrage p r o f i t s the return on the p o r t f o l i o must be zero. 25 Therefore: (2.26) a i * ( a - r ) + a *(B-r) = 0 2 Eliminating a^ and a from (2.25) and (2.26) the market equilibrium 2 condition i s obtained: ( 2 . 8=r a-r 2 7 ) 1 a = Substituting the values of B and y from (2.20) and (2.21) respectively into (2.27) and simplifying gives the p a r t i a l d i f f e r e n t i a l equation governing the value of the option on a non-dividend paying stock: ho S (2.28) 2 2 W + rS W ss s - W T - Wr = 0 subject to the boundary conditions: (2.29) W(S,0) = Max[0, S-E] W(0,T) = 0 and (2.30) Notice that the expected instantaneous rate of return on the stock, a, has dropped out from equation (2.28). Equation (2.28) i s a second order l i n e a r p a r t i a l d i f f e r e n t i a l equation of the parabolic type. To be able to solve (2.28) the boundary conditions must be s p e c i f i e d . Boundary condition (2.29) expresses the fact that at the maturity of the option i t s value w i l l be equal to the stock price minus the exercise p r i c e , E, or zero of the former i s negative. This value i s c a l l e d the exercise value of the option. Boundary condition (2.30) expresses the fact that the value of the option corresponding to a stock price equal to zero i s zero i r r e s p e c t i v e of the 6 p a r t i a l d i f f e r e n t i a l equations w i l l be treated i n somewhat more d e t a i l in Chapter 3 when numerical solutions are discussed. 1 : 26 outstanding l i f e of the option.^ I f Che stock price i s zero i t can never recover B-S solve the d i f f e r e n t i a l equation (2.28) with boundary conditions (2.29) and (2.30) by transforming them by s u b s t i t u t i o n into the heattransfer equation of physics, which has a known closed form s o l u t i o n . The solution i s given by the formula: (2.31) W(S,T) = SN(d ) - E e ~ r T L N(d ) 2 where 9 s d l = g + (r + ho ) T 1 off d 2 = N(d) d l = - off -x I d - — / 2 e 2 dx, i s the cumulative normal density function. /2TT Black-Scholes point out, "Note that the expected return on the stock does not appear i n equation (2.31). stock price i s independent The option value as a function of the of the expected return on the stock. The expected return on the option, however, w i l l depend on the expected return on the stock. The f a s t e r the stock price r i s e s , the f a s t e r the option p r i c e w i l l r i s e through the functional relationship (2.31)." 8 Also note that the option p r i c e does not depend on the r i s k preferences of investors. 2.6 Extension of the Model To analyze the e f f e c t of dividends on unprotected options, Merton [32] T h i s boundary condition i s r e a l l y not necessary to solve the differenti a l equation. I t can be obtained mathematically from (2.28) and (2.29) as w i l l be seen l a t e r on. 7 See Black-Scholes [ 3] p. 644. 27 assumes a continuous dividend per share per unit time, D(S,T), when the stock p r i c e i s S and the time to maturity of the option i s T. Differ- e n t i a l equation (2.13) gives the t o t a l instantaneous rate of return on the stock. Thus, i f the stock pays dividends the d i f f e r e n t i a l equation should be written as: rn (2.32) dS + Ddt = A,. A adt +. adz or (2.33) -|§ = ( - | ) d t + adz a The instantaneous expected rate of return from p r i c e appreciation i s (a - D/g) a n d the instantaneous rate of return from p r i c e appreciation i s given by (2.33) The s t o c h a s t i c d i f f e r e n t i a l equation f o r option p r i c e changes i s s t i l l given by (2.15) with (2.16) and (2.17), because the value of the option i s s t i l l a function of the p r i c e of the stock S and time to maturity T ( i . e . W = W(S,T)). Substituting (2.33), (2.16) and (2.17) into (2.15) and rearranging terms, the d i f f e r e n t i a l equation f o r option p r i c e changes can be written as: (2.34) Again dw = [ho S 2 2 W ss + (aS-D)W - W ]dt + aSWgdz g T (2.19) can be written, but the new d e f i n i t i o n s of 0 and y are: (2.35) 3 (2.36) y = ^iha S 2 2 W ss + (aS-D)W - W~] g T =^S. The instantaneous d o l l a r return on the zero equity p o r t f o l i o i s now given by /o o7\ A (2.37) dy = dS-4-Ddt ai — + a 2 dW . + a 3 . rdt 28 where the t o t a l rate of return on the stock given by (2.32) has to be considered. Substituting (2.32), terms, the instantaneous (2.38) dy = (2.19) and (2.22) into (2.37) and rearranging d o l l a r return on the p o r t f o l i o can be written as: [aj_(a-r) + a ( 0 - r ) ] dt + [a a+a2Y]dz 2 1 Using the same investment strategy as i n the l a s t section, the condition for no arbitrage p r o f i t s i s (2.39) « a-r X a Substituting the values of g and y from (2.35) and (2.36) r e s p e c t i v e l y into (2.39) and s i m p l i f y i n g the r e s u l t i n g expression, the d i f f e r e n t i a l equation governing the value of the option i s obtained: (2.40) ho S 2 2 W + (rS-D)W ss s - W T - rW = 0 subject to the boundary conditions (2.41) W(S,0) = Max[0,S-E] and (2.42) W(0,T) = 0 Merton [ 32] has shown that when the stock pays no dividends or the option i s dividend protected,it w i l l never pay to exercise an American option p r i o r to maturity option. and,hence, i t has the same value as a European The option i s always worth more " a l i v e " than "dead." Therefore, formula (2.31) i s v a l i d j u s t as w e l l f o r an American option as f o r a European option under the conditions s p e c i f i e d i n the preceeding section. However when the stock pays dividends or there i s a lack of protection against payouts to the common stock,there may be a difference between the American and European option values which implies a p o s i t i v e p r o b a b i l i t y of premature exercise. 29 T h e r e f o r e , d i f f e r e n t i a l equation (2.40) with boundary condition (2.41) and (2.42) govern the value of a European option. For an American option an a d d i t i o n a l arbitrage boundary condition i s required: (2.43) W(S,T) > Max[0,S-E] Condition (2.43) implies that the value of the option could never be lower than i t s exercise value because otherwise A general closed form s o l u t i o n to equation i t would be exercised. (2.40) subject to the boundary conditions (2.41) and (2.42) f o r the value of an option on a dividend paying stock has not been found. Only i n two p a r t i c u l a r cases have e x p l i c i t formulae f o r the value of the option been obtained. One of the cases i n which a closed form s o l u t i o n has been obtained f o r the European option i s that i n which the underlying stock pays a constant dividend y i e l d , that is,where the functional r e l a t i o n s h i p of D(S,T) i s given by (2.44) D(S,T) Samuelson = pS where p > 0 constant. [36] and Samuelson and Merton [37] have shown that i n this case, for the American option, there i s always a p o s i t i v e p r o b a b i l i t y of premature exercising and, hence, the arbitrage boundary condition (2.43) w i l l be binding f o r s u f f i c i e n t l y large stock prices and no closed form s o l u t i o n w i l l e x i s t . Therefore, the American option can be worth more "dead" than " a l i v e . " The s o l u t i o n of (2.40) f o r the European option can be obtained by substituting D = pS into (2.40) and proceeding solution. (2.45) The s o l u t i o n formula, given by Merton [32], i s : SN(d,) - E e W(S,T) -rT N(d ) where: £n - (2.46) + (r-p + ha )T 2 off d i n the same way as the B-S 2 = dj - off and N(d) i s as defined i n (2.31). 2 30 The second case i n which a p a r t i c u l a r closed form s o l u t i o n has been found i s that i n which (2.47) the stock pay constant continuous D(S,T) = dividends, that is,where D, where D i s constant. Merton [ 32] has shown that i n this case premature exercising may may not occur, depending upon the values for D, r , E and T; or and that a s u f f i c i e n t condition for no premature exercising i s given by: (2.48) D < Er If conditions (2.48) obtains, the s o l u t i o n for the European option value w i l l be the s o l u t i o n for the American option. Merton [32] points out that a closed form s o l u t i o n to (2.44) i n the case (2.47) has not yet been found for f i n i t e time to expiration and gives the s o l u t i o n to the perpetual option when (2.48) obtains. One of the objectives of t h i s d i s s e r t a t i o n i s to develop a numerical solution procedure for equation continuous (2.40) subject to (2.41) and (2.42) i n the constant dividend case when condition (2.48) obtains. the main topic of Chapter 3. This i s The numerical method developed i s a p p l i c a b l e , i n f a c t , to any known functional r e l a t i o n s h i p of D(S,T) and not only to the constant dividend case. Also, the numerical procedure can be applied, as i s done i n Chapter 4, to solve for the more r e a l i s t i c case of discrete dividend payments and to f i n d the optimal exercising strategy when premature exercising i s p r o f i t a b l e for the American option. 2.7 More Recent Developments One of the most c r i t i c a l assumptions of the Black-Scholes analysis i s the stochastic s p e c i f i c a t i o n of stock p r i c e changes, namely that the instantaneous rate of return on the stock follows an Ito process of the form 31 (2.13) ^ = adt + adz where dz i s a Wiener process or Brownian motion. This s p e c i f i c a t i o n can be taken to imply that the r e l a t i v e change i n the stock p r i c e over a small i n t e r v a l w i l l be given by a d r i f t component, adt, plus a random increment with mean zero and variance a dt. A fundamental r e s t r i c t i o n imposed by the use of stochastic d i f f e r e n t i a l equation (2.13) to describe stock p r i c e changes,is that within a small i n t e r v a l of time, ( t , t+dt), S(t) w i l l move i n a random fashion, but with high probabi l i t y , approaching to one as dt tends to zero, that S(t+dt) w i l l be i n an q a r b i t r a r i l y small neighborhood of S ( t ) . In essence, process (2.13) and, hence, of the B-S equation stock p r i c e dynamics follows a continuous path. a short i n t e r v a l of time the v a l i d i t y of (2.31) depends on whether That i s , i n the stock p r i c e can only change by a small amount.^ Cox and Ross argue that, "New information, however, tends to a r r i v e at a market i n d i s c r e t e lumps rather than i n a smooth flow, and assets i n such markets are l i k e l y to have discontinuous jumps i n value, thus v i o l a t i n g the basic assumption of a process such as (2.13). Such behaviour i s c h a r a c t e r i s t i c of many economic situations and i s p a r t i c u l a r l y relevant when p o l i t i c a l events or s t o c h a s t i c a l l y applied government or i n s t i t u t i o n a l constraints are of primary importance."^ In two recent papers, Cox and Ross [ 8] and Merton [34] have studied the valuation of options when stock price movements can be 9 See Cox and Ross [8 ] p. 2, 10 S e e Merton [34] p. 3. ^See Cox and Ross [ 8 ] p. 3. independently 32 better described by a jump process. Both studies use a Poisson process to describe the random a r r i v a l of an important piece of new information about the stock. Cox and Ross concentrate t h e i r analysis on the assumption that stock p r i c e changes follow a jump process rather than a d i f f u s i o n It6 process. They specify the jump process as: (2.49) — = odt + ( k - l ) d n where II i s a Poisson process and d n takes the value zero with p r o b a b i l i t y 1-Xdt and the value one with p r o b a b i l i t y to Xdt. X measures the "rate of p r o b a b i l i t y flow" f o r the jump and k-1 i s the "jump amplitude." When no jump occurs S changes the exponential rate a, but i f a jump occurs, S changes by (k-l)S to kS. They study the option valuation problem when stock p r i c e changes follow a jump process as (2.49) by attempting to form hedged p o r t f o l i o s with the stock, the option and the r i s k l e s s asset. "dimension" Since they consider only one of r i s k (the Poisson process) the hedged p o s i t i o n can be obtained and maintained a f t e r a jump by a discontinuous adjustment to a new p o s i t i o n . They obtain a complex formula f o r the value of the option that i s independent of the i n t e n s i t y of the process X which determines jumps. the expected number of Unfortunately, however, t h e i r valuation formula i s dependent on the expected rate of return on the stock, a. Merton [34] considers the more general case i n which the t o t a l change i n the stock price i s the sum of two types of changes: (1) the "normal" v i b r a t i o n s i n price assumed to follow a standard Brownian motion or Wiener process. geometric (2) The "abnormal" vibrations i n price due to the a r r i v a l of important new information about the stock that has more 33 than a marginal e f f e c t on p r i c e , assumed to follow a Poisson process. He then writes the t o t a l change i n stock p r i c e as: (2.50) dS S = (a-Ak)dt + adz + dq where a i s the instantaneous expected return on the stock; a i s the instantaneous variance of return, conditional on no occurance of the Poisson event; dz i s a standard Wiener process; process. dq and dz are assumed to be of a r r i v a l s per unit of time. dq i s an independent independent. Poisson X i s the mean number k = E ( y - l ) where (y-1) i s the random v a r i a b l e percentage change i n the stock p r i c e i f the Poisson event occurs and E i s the expectation operator. Merton also proceeds i n h i s analysis by forming p o r t f o l i o s which consist of the stock, the option,and the r i s k l e s s asset, but,given the complexity of the stochastic s p e c i f i c a t i o n of stock p r i c e changes assumed,he proves that i t i s impossible to f i n d weights which w i l l give r i s k l e s s return on the p o r t f o l i o , and hence, the B-S "no-arbitrage" technique cannot be employed. He then obtains a formula f o r the value of an option by imposing the a d d i t i o n a l assumptions that s e c u r i t i e s are priced so as to s a t i s f y the C a p i t a l Asset P r i c i n g Model and that the jump component of a security's return i s uncorrelated with the market. The second assumption implies that the jump component of a stock's r i s k can be d i v e r s i f i e d away and, as Merton points out, this i s a testable proposition. The models employed i n this d i s s e r t a t i o n assume that stock prices dynamics can be described by an Ito process of the form 2.8 (2.13). Conclusions An equilibrium option valuation model developed by Black-Scholes and 34 extended by Merton has been p r e s e n t e d i n t h i s c h a p t e r . The model r e p r e s e n t s a breakthrough i n the f i n a n c e l i t e r a t u r e not o n l y because i t g i v e s a r a t i o n a l b a s i s f o r p r i c i n g o p t i o n s , but a l s o because of i t s p o t e n t i a l use i n the v a l u a t i o n of o t h e r k i n d s of c o n t i n g e n t c l a i m s such as c o r p o r a t e liabilitiesJ There have been some r e l a t i v e l y s u c c e s s f u l e m p i r i c a l t e s t s of the model by B l a c k - S c h o l e s on the Chicago The [4 ] on c a l l - o p t i o n d a t a , Leonard [23] on w a r r a n t s , G a l a i Board Options Exchange d a t a , and I n g e r s o l l f a c t t h a t most companies do pay p o l i c y i s never a p r o p o r t i o n a l one d i v i d e n d s and [16] on Dual Funds. that t h e i r dividend ( i . e . with constant dividend y i e l d ) , c a l l s f o r a more g e n e r a l s o l u t i o n to the e q u i l i b r i u m o p t i o n v a l u a t i o n model. next two [ 14] c h a p t e r s of t h i s d i s s e r t a t i o n d e a l w i t h t h i s problem. The In Chapter 3 n u m e r i c a l methods f o r v a l u i n g o p t i o n s on s t o c k s w i t h continuous d i v i d e n d payments a r e developed when D < E r . In essence, the problem s o l v e d i s the one o f f i n d i n g a g e n e r a l n u m e r i c a l method of s o l u t i o n to p a r t i a l equation (2.40) g o v e r n i n g w i t h the more r e a l i s t i c the v a l u e of the o p t i o n . The problem of d e a l i n g case of p r i c i n g an o p t i o n on s t o c k s w i t h d i v i d e n d payments i s p r e s e n t e d and s o l v e d i n Chapter 4. chapter. e q u i l i b r i u m o p t i o n p r i c i n g model p r e s e n t e d i n t h i s c h a p t e r and numerical solution m o d i f i e d and adapted linked l i f e developed periodic A l s o the o p t i m a l s t r a t e g i e s f o r e x e r c i s i n g American o p t i o n s a r e d i s c u s s e d i n t h i s The differential procedures developed i n Chapters to s o l v e an a c t u a r i a l problem: 3 and 4 are The 5 and some a p p l i c a t i o n s a r e g i v e n i n Chapter See, f o r example, Merton's c o r p o r a t e debt. then the p r i c i n g of e q u i t y - i n s u r a n c e p o l i c i e s w i t h a s s e t v a l u e guarantees. i n Chapter the theory i s 6. [33] study on the p r i c i n g of " r i s k y " 35 CHAPTER 3: PROCEDURES FOR PRICING OPTIONS ON STOCKS WITH CONTINUOUS DIVIDEND PAYMENTS. 1 3.1 Introduction In the preceding chapter an equilibrium option valuation model was presented. By imposing the arbitrage p r i n c i p l e that i n equilibrium no r i s k l e s s p r o f i t s can be made with a zero net investment a partial differ- e n t i a l equation governing the value of an option was obtained. The value of the option - and hence also the value of i t s p a r t i a l derivatives with respect to stock value and time to maturity - i s a function of the value of the stock, time to maturity>and known constants. A closed form solution to the d i f f e r e n t i a l equation has been found when the underlying stock pays no dividends and also i n a few very r e s t r i c t ive cases when dividends are paid. The purpose of this chapter i s to give a general procedure for valuing options on stock with continuous dividend payments. The method i s based e s s e n t i a l l y on numerical solutions to p a r t i a l d i f f e r e n t i a l equations. After a b r i e f introduction to second-order partial equations, the f i n i t e differences method i s developed differential as i t applies to the numerical solution of parabolic p a r t i a l d i f f e r e n t i a l equations such as the one governing the value of an option. The solution algorithm i s then developed and a computer program for the constant continuous dividend case i s used to evaluate the Trans Canada P i p e l i n e warrant maturing A p r i l 3, 1976. Without implicating him, the author would l i k e to thank Mr. A l Fowler, Associate Director of the Computer Centre at the U n i v e r s i t y of B r i t i s h Columbia, for h e l p f u l assistance i n solving p a r t i a l d i f f e r e n t i a l equations by numerical methods. 36 The numerical s o l u t i o n i s compared with the results obtained by applying the B-S formula i n the case where the stock pays no dividends. I t i s made c l e a r that the numerical procedure can be made as accurate as required by reducing the size of the stock p r i c e step and the time step. 2 3.2 Introduction to P a r t i a l D i f f e r e n t i a l Equations A l i n e a r , second-order p a r t i a l d i f f e r e n t i a l equation i n two independent variables S and T can be written i n the form: (3.1) AW gs + BW ST + CW TT + DW + EW + FW = G g T where A, B, C, D, E, F and G are functions of S and T, W i s the dependent v a r i a b l e and subscripts denot p a r t i a l differentiation. 3 As i n the case of ordinary d i f f e r e n t i a l equations , a d d i t i o n a l inform- ation (boundary conditions) must be supplied along with (3.1) i n order to pick out a s p e c i f i c s o l u t i o n from the family of possible solutions to (3.1). In general, three types of second-order p a r t i a l d i f f e r e n t i a l are defined: i) The equation i s e l l i p t i c i f 2 B - 4AC < 0 The equation i s parabolic i f B 2 ii) - 4AC = 0 2 iii) The equation i s hyperbolic i f B - 4AC > 0 This section and the next draw on McCracken and Dorn [25]. 3 In ordinary d i f f e r e n t i a l equations the dependent v a r i a b l e i s a function of only one independent v a r i a b l e . equations 37 3.3 Difference Equations The d e f i n i t i o n of the derivative of a function of a single v a r i a b l e is: dy _ lim dx h-K) y(x+h) - y(x) h In a computer i t i s not possible to take the limit', however, h can be set at some small value and attempts can be made to prove that the approximation i s sufficiently continues close (accuracy) and that the error does not grow as the process (stability). The method can be described as replacing a derivative with a difference. Let us begin by considering only differences i n the S d i r e c t i o n . The Taylor series expansion for W ( S , T ) Q about the point (S ,T ) o Q can be written as W(S,T ) = W(S ,T ) + ( S - S ) W ( S , T ) + - ^ f ^ o o o S Q o Wg (n,T ) o where n l i e s between S and S . Hereafter S o we w i l l use n as a dummy to denote the Q constant that w i l l make the Taylor expansion exact. Let S = S + h, then o W (S ,T ) S 0 W ( S 0 n In other words, i f W (3.2) + h » n> T W ( S h G "> o) , - | T w (n,T ss o) i s approximated by (the equality sign i s used for s i m p l i c i t y ) W ( S , T ) = W(S +h,T )^- W ( S , T ) S 0 Q 0 0 0 0 the truncation error i s (3.3) E t = - | W ( ,T ) s s n 0 S 0 < n < S +h 0 Equation ( 3 . 2 ) was obtained by s u b s t i t u t i n g S = S +h i n the c series expansion, the r e s u l t i s c a l l e d a forward d i f f e r e n c e . difference can be obtained by l e t t i n g S= S Q - h. Taylor A backward 38 The result i s A^ (3.4) u r«j T ^ - w ( s W (S ,T ) S 0 - Q 0 > T ) - w(s -h,T ) — 0 It i salso possible 0 0 t o average (3.2) and (3.4) and g e t a t h i r d a p p r o x i m a t i o n f o r Wg ^ (3.5) n ^ m \ _ W(S +h,T ) - W ( S - h T ) 0 W (S ,T ) S To 0 0 0 - 0 > 0 ^ determine t h e t r u n c a t i o n e r r o r o f t h i s a p p r o x i m a t i o n , r e c a l l W(S,T ) = W ( S , T ) + 0 0 ( S - S 0 W D W (S ,T ) + ) S 0 ( S 0 ~ S ( ? ) 2 W S S (S ,T ) 0 0 that + SSS^' o) T Now l e t S = S + h and then S = S - h and s u b t r a c t D D t h e two r e s u l t i n g equations. The t r u n c a t i o n e r r o r i s seen to be: h (3.6) E T The (3.7) W S S 2 W ss(n,T ) = - | S S -h < n < S 0 0 = (S ,T ) 0 0 W (S S 0 + h,T ) 0 h i n the forward d i r e c t i o n . d i f f e r e n c e s a r e used. by +h d i f f e r e n c e e q u a t i o n f o r Wgg, u s i n g a forward d i f f e r e n c e , i s W^(SO TQ) ? I f forward d i f f e r e n c e s a r e now s u b s t i t u t e d biased 0 for W , s the r e s u l t would be I n order to avoid t h i s effect,backward The backward d i f f e r e n c e f o r W s ( S , T ) i s g i v e n 0 0 (3.4) and (3.8) W (S +h,T ) = s 0 0 Substituting W(S +h T )^- W ( S T ) n ? n n ? n (3.4) and (3.8) i n t o (3.7) the d i f f e r e n c e f o r W obtained: (3.9) W (So,To) ss = W(S +h,T ) -2W(S ,T ) + W ( S - h T ) 0 0 n h 2 n 0 ? n ss is 39 To determine the truncation error, r e c a l l that ,2 W(S,T ) = W(S ,T ) + (S-S ) W (S ,T ) + - ^ - ^ 1 W ( S , T ) + Q 0 ^ T6 ^ 3 0 D S 0 w ( s , T ) + - ^24"SSS o»-o/ ' sss vlJ 0 'SS 0 SS w 4 0 s s s s 0 0 ( ,T ) n 0 Now l e t S = S +h and then S = S -h and add the two r e s u l t i n g equations. Q The (3.10) E 0 truncation error i s seen to be: T —W s s s s (n,T ). S -h < n < S +h 0 Q Q An e n t i r e l y analogous development leads to difference equations f o r the derivatives i n the T - d i r e c t i o n . For example, take the step s i z e i n the T - d i r e c t i o n to be k, the backward difference for WVp i s (3.11) W ( S T ) = < n » n > - W(S ,T -k) k w T Q) s T 0 n With these expressions n the p a r t i a l d i f f e r e n t i a l equation (3.1) can be completely rewritten i n terms of differences. The r e s u l t i n g difference equation may now be approached by several methods depending on the type of equation ( e l l i p t i c , parabolic or hyperbolic), on the boundary conditions, on the size of the steps, etc. In subsequent sections a detailed presentation of the method best suited to solve the second order parabolic p a r t i a l d i f f e r e n t i a l equation derived i n Chapter 2 w i l l be made. For a comprehensive description of d i f f e r e n t approaches see Forsythe and Wasow [11] and McCracken and Dorn [25]. In general a rectangle of width S n and height T i s considered (the m c r i t e r i a to decide the maximum S, S , and the maximum T, T , depend n m problem and on the boundary conditions). equal i n t e r v a l s of width h = S / . n n m equal i n t e r v a l s of height k = T / . ° mm n The width S n oh the i s divided into n S i m i l a r l y the height T m i s divided into 40 Figure 3.1: A Mesh of Points Laid Over a Rectangular Region. m W i,j (i,j) 2 1 0 1 n s. The mesh i n t e r s e c t i o n s are numbered as shown i n Figure 3.1 and following (3.12) S the notation i s introduced: i = ih Tj = j k where i = 0, ,n where j = 0,, ,m W(S ,T ) = W(ih,jk) = W i j ±>j A difference equation for each mesh point w i l l be written and then the r e s u l t i n g system of simultaneous equations w i l l be solved. conditions w i l l give the values of W at the edge The boundary of the rectangle or a relationship between extreme values. 3.4 The P a r t i a l D i f f e r e n t i a l Equation Governing the Value of an Option Recall from Chapter 2 that the equilibrium value of an option, W, of which underlying stock pays a constant continuous dividend rate D, i s given by i i solution to the second order parabolic p a r t i a l d i f f e r e n t i a l equation the 41 (3.13) ha S W 2 2 ss + (rS-D) W - rW - W = 0 s T with boundary conditions (3.14) W(0,T) = 0 and (3.15) W(S,0) = max[0,S-E] where a i s the variance rate f o r the return on the stock assumed 2 r i s the r i s k l e s s constant constant, i n t e r e s t rate, E i s the exercise p r i c e , S i s the stock price,and T i s the time to maturity of the option. If the difference approximations (3.5), (3.7) and (3.11) are used f o r Wg, Wgg ,and r e s p e c t i v e l y , and the notation defined i n (3.12) i s used, the p a r t i a l d i f f e r e n t i a l equation (3.13) can be approximated i n difference form as 9 9 Wi+i -i - 2Wi i + Wi-l i h a S, L_LL1 h W , - A _ W,- rW k (3.16) 2 2 + ( rSl -D) 2 ^i+1 i ~ ^ l - l A — ? J o 2 u h A Simplifying (3.16) : (3.17) a Wi_ ± l f j +b W ± +c ±>j ± W 1 + l f j = W^^! i = 0, . . . n j = 0, . .. m where /o 1 R v (3.18) . (rSj-D)k a a - ± (3.19) a Sj k 2 ^ 2 2h2- b = (1+rk) + ° 2 s ± h i 2 k a n d z c. . ( r S i - D ) k _ q 2 k 2h 2h2 (3.20) 2 = S i 1 Recalling from (3.12) that S ± = i h , (3.18), (3.19) and (3.20) can be written as (3.18) a, = hrkx - ^ - %a ki 2 2 2n 1 i j (3.19) b = (1+rk) + a k i 2 ± 2 and 42 (3.20) c, = -Jsrki + ^ 2n ko ki 2 1 2 For any point ( i , j ) r e l a t i o n s h i p (3.17) can be represented schematically by drawing the four points that are related by the difference equation and showing f o r each point i t s c o e f f i c i e n t i n (3.17). The r e s u l t i s c a l l e d the s t e n c i l of the method. a i c i -1 i-1 3.5 i i+1 Truncation Error' The truncation error, E^, due to the approximation of (3.13) by (3.17) can be calculated with (3.3), (3.6) and (3.10): (3.21) E T = - ^ aS h 2 2 2 ± W ssss (n,T ) - |(rS -D) h j ± 2 W sss (n,T ) j + |w (s.,c) TT with S and ± - h< n < S ± + h Tj - k < C < Tj By appropriate s e l e c t i o n of the step sizes h and k, the truncation error can be made as small as desirable. 3.6 S t a b i l i t y and Convergence It has been indicated e a r l i e r that as h and k tend to zero the difference equation approaches the d i f f e r e n t i a l equation. However we 43 have not yet taken up the question whether the solution of the difference equation approaches (converges) the solution of the equation as h + 0 and k -> 0. differential In other words, whether the solution of (3.17) converges to the solution of (3.13). This i s the problem of convergence. Another important question i s the s t a b i l i t y of the method, that i s , the manner i n which an error i s propagated ("grows"). The order of magnitude of the cumulative departure of the s o l u t i o n caused by errors committed at d i f f e r e n t p o i n t s , i s used as a measure of the degree of s t a b i l i t y . The convergence of the solution and the s t a b i l i t y of the procedure assure the uniqueness of the s o l u t i o n . The method selected to solve (3.13) i s such that (3.17) i s probably stable 2 and convergent f o r a l l values of A = k/h . We postulate the l a t t e r to be true; no one has proved i t i n the general case of variable c o e f f i c i e n t s . For a detailed treatment of convergence and s t a b i l i t y see Forsythe and Wasow [11]. 3.7 Boundary Conditions The boundary conditions (3.14) and (3.15) expressed i n the notation defined i n (3.12) can be written as (3.22) W j 0} W ± (3.23) Q =0 = 0 for a l l j f or o < i < f 4 W. „ = ih-E i,o Subsequently f o r f- < i < n h i t w i l l be seen that the boundary conditions (3.22) and (3.23), which give the values of W f o r S=0 and f o r T=0 sufficient to solve the difference equation (3.17). respectively, are not An a d d i t i o n a l constraint i s needed at the upper l i m i t of S (for i=n). The a d d i t i o n a l boundary condition i s occasioned by considering the fact that 44 for any given time to expiration the slope of the curve r e l a t i n g the price of the option to the price of the underlying stock tends to one as the price of 4 the stock tends to i n f i n i t y , that is,the p a r t i a l d e r i v a t i v e Wg tends to one: lim Wg = 1 S -> oo Then for a s u f f i c i e n t l y large S the following approximation (3.24) W can be made: = 1 s The upper l i m i t of S can always be chosen large enough for (3.24) to hold w i t h i the p r e c i s i o n required at the boundary. As a "rule of thumb" this upper l i m i t should be three or four times the exercise price E. This i s a conservative limit,because Wg i s p r a c t i c a l l y equal to one for values of S greater than twice the exercise p r i c e . Then,using the backward difference(3.4),(3.24) can be re-written i n the new notation; (3.25) W (S , Tj) = s W n > j n ' W t l h " 1 > j = 1 f or a l l j or (3.26) -W _i j + W j = h n } for a l l j n> With boundary conditions (3.22), (3.23) and (3.26) and the difference equation (3.17) a system of l i n e a r equations can be formulated which w i l l give the values of W f o r time j , s t a r t i n g from the values of W for time In this way i t i s possible to proceed i n stepwise fashion to solve f o r the values of W for the whole g r i d . 3-8 System of Linear Equations The difference equation (3.17) generates equations as i varies from 1 to n-1. a system of l i n e a r In the f i r s t equation ( f o r i = 1) 4 I f the s u f f i c i e n t condition for no premature exercising of the option (2.48) holds the W-S curve w i l l never cut the l i n e W = S-E. Then W > S-E, but W < S and W(S) i s convex (see Merton [32]). Therefore: lim W = 1. g S ->- 00 45 the f i r s t term (a^ W j ) i n the equation i s zero because W ,j Qj (3.22)). 0 = ^ (from The l a s t equation (for i = n) i s obtained from the boundary condition (3.26). Here i t can be seen that this additional constraint was needed to be able to solve the system of n equations with n unknowns. b l a w- + W +b W lj;j 2 C lfj l W =W j _ i 2jj 2 L 2>j +c 2 W = 3 J U ,i-1 2 (3.27) a .W „ . + b . W , .+ c . W . =W . . n-1 n-2,3 n-1 n - l , j n-1 n,j n-l,j- W ..+W .= h n-l,J n,j Let f. = W. . . 1 y 3~ •!• for i = l,...,n-l 1 f n = h (3.28) a b n n = -1 = 1 and omitting the subscripts j f o r s i m p l i c i t y , the system (3.27) can be written as b l W l a 2 W l+ + c l 2 " W b W 2 2 + c 2 3 W = f f l 2 (3.29) a ,+W „ +b .W . +c ,W =f . n-1 n-2 n-1 n-1 n-1 n n-1 a n W .+b W = f n-1 n n n The matrix of c o e f f i c i e n t s of the system (3.29) has zeros everywhere except on the main diagonal and on the two diagonals p a r a l l e l to i t on either side. Such a matrix i s c a l l e d t r i d i a g o n a l . Fortunately the system represented by (3.29) i s of such a simple nature that a very elementary 46 procedure, Gauss' elimination method, leads to a simple solution. By successive subtraction of each equation from a suitable multiple of the succeeding one, the system (3.29) can be transformed into a simpler A tfc & one of the same type characterized by c o e f f i c i e n t s a^ , b^ , c^ , f ^ of instead a^, b i , c-£, f i , and where i n p a r t i c u l a r a^ = 0 i = 2,...n. The i t e r a t i v e process i s as follows: i) The f i r s t equation i s l e f t unchanged. ii) The second equation i s m u l t i p l i e d by ^ i . and the f i r s t equation i s sub5 tracted from i t , to obtain (the f i r s t term drops out) 3 2 [h. b a 9 2 - c ] W t 1 2 2 a + Co 1 W = *2 3 2 f - 2 f, 1 Now l e t , b bi * 2 = , ~ ~a 2 L h l c 2 „ * _ bi c - _ i c 2 2 = II h *2* a ~ *1 2 Then the transformed second equation would be b iii) 2* 2 W + c 2* 3 W = f 2* Then the t h i r d equation i s m u l t i p l i e d by b^/a-j and the transformed equation i s subtracted from i t . Again the f i r s t term of the t h i r d equation drops out. iv) This procedure i s continued u n t i l the l a s t equation. "*This procedure i s only followed i f a i s d i f f e r e n t from zero. = 0 the second equation i s also l e f t unchanged. 2 a 2 i 6• S|ee footnote 5. second If 47 The transformed system of equations i s now l * b l+ w c l* 2 w = b * W 2 2 + c * W 2 f l* = f * 3 2 (3.30) b n - l * W_! n + c _!* W n C n W n = f _i* n = f * In the f i r s t equation (3.31) b j * = bi, ci* = ci and f j * =f i and i n general (3.32) b *— i* - b i - c _i* i b * = ± for ± i=2,...n a (3.33) * c,* = i (3.34) f *= b i-l* c. ajL i b• i * f ± a f o r 1=2,...n-1 ± - f _!* ± for i= 2,...n i The solution of the transformed system i s now immediate: starting with the l a s t equation (3.35) W n = f */b * n n the Wj_ are given successively by (3.36) W ± = [f *_ c * W ] ± b ± 1+1 i=n-l,...,l i The procedure described solves f o r the values of Wj^j (i= l,...n) starting grid, from the values of Wj^j_i (i= l , . . . n ) . Q , i s computed by using the boundary condition (3.23) which gives the values of W± 3.9 The f i r s t l i n e of the iQ (i= o,...n). Solution Algorithm Using the developments of the preceding sections, a computer program n 48 was written to solve by numerical procedures p a r t i a l d i f f e r e n t i a l equation (3.13) with boundary conditions (3.14) and (3.15). The input required are : the variance rate f o r the return on the stock, the r i s k l e s s i n t e r e s t rate, the dividend rate, the exercise p r i c e of the option, the stock p r i c e increment (h), the time increment (k), the number of d i f f e r e n t stock prices, and the number of d i f f e r e n t times to maturity of the option. The output of the program i s the option p r i c e matrix. This matrix gives the numerical s o l u t i o n to the p a r t i a l d i f f e r e n t i a l equation f o r d i f f e r e n t stock p r i c e s and times to expiration. The program i s shown i n Appendix A. For condition (3.24) to hold within the p r e c i s i o n required at this boundary, the upper l i m i t of stock p r i c e chosen should be s u f f i c i e n t l y large i n r e l a t i o n to the exercise p r i c e of the option. At this point the slope of the curve describing the r e l a t i o n s h i p between the option p r i c e and the stock p r i c e f o r a given time to maturity i s set at one. Before proceeding i t i s necessary to use the algorithm to value an American option, to check the s u f f i c i e n t condition f o r no premature exercising discussed i n Chapter 2: (2.48) 3.10 D < Er Numerical Example To test the numerical procedure developed i n the preceding section, 49 available data' on the Trans Canada P i p e l i n e Warrant were used. To determine the variance rate of return on the Trans Canada P i p e l i n e (TCP) common stock, stock prices and dividends for 24 months were'considered (January 1969 to December 1970). The continuously compounded rate of return per month, R(t), on the stock i s : (3.37) R(t) E £n [ ^ ^ p - l where: S(t) : p r i c e of the stock at the end of month t D(t) : dividends paid during month t . Using the rates of return given by (3.37) f o r the period mentioned, the variance rate per month was computed (0.005712). on three to f i v e years The average government bonds at the end of January yield 1971 (5.42%) 8 was used as a proxy for the r i s k l e s s constant i n t e r e s t rate. The TCP Warrant has an exercise price of $41.00 and w i l l expire on A p r i l 3, 1976. During the whole period of available data (July '64 to January '71) the common stock paid a quarterly dividend of $0.25 per share. Thus the approximation of a continuous constant dividend of $1.00 per year seems quite reasonable. To test the accuracy of the numerical s o l u t i o n developed i n t h i s chapter, the algorithm was used to evaluate the TCP Warrant s e t t i n g the dividend rate equal to zero (D=0). Recall from Chapter 2 that i n the case where the stock pays no dividend Black-Scholes obtained a closed form (exact) solution to p a r t i a l d i f f e r e n t i a l equation (3.13), given by formula (2.31). For 100 d i f f e r e n t stock prices (from $1.00 to $100.00) and 100 d i f f e r e n t times to See Foster [12]. g The continuously compounded rate should be used. Chapter 4. 7 See footnote 4 i n 50 maturity (from 1 month to 100 months) the value of the warrant was evaluated using the B-S formula and the numerical procedure, using i n both cases the i above mentioned values of the parameters a , r and E. 2 Two warrant price matrices of 100 by 100 (that i s , 10,000 option prices) were obtained, one for each method of solution f o r the no dividend case. A t h i r d warrant price matrix of the same dimensions and c h a r a c t e r i s t i c s was computed by the numerical procedure f o r the value of the option, taking into consideration the dividends paid by the TCP common stock. Table 3.1 shows the warrant values obtained for the three cases f o r a stock p r i c e of $41.00 and times to expiration from !g year (6 months) to 8 years (96 months). The stock price chosen f o r the table was the exercise p r i c e of the warrant, because at this value the premium on the warrant i s at i t s maximum. The B-S solution and the numerical s o l u t i o n f o r the no dividend case do not d i f f e r from each 9 other by more than seven cents. The considerable impact of even a small dividend payment on the value of the warrant can be seen i n the l a s t column of Table 3.1. To give a more complete comparison of the methods, the warrant prices obtained by the numerical procedure f o r the no dividend case were regressed, for each time period, on the warrant prices obtained by the B-S formula. 99 regressions were obtained, one f o r each d i f f e r e n t time (months) to maturity, and each regression was based on 99 warrant values, one for each d i f f e r e n t stock p r i c e . It should be emphasized that the parameters that give p r e c i s i o n to the numerical solution were i n t e n t i o n a l l y chosen quite crude to show the robustness of the procedure. The stock p r i c e step was $1.00, the time step was one month,and the maximum stock price was chosen at a value only 2.5 times the exercise p r i c e . In subsequent applications of numerical procedures these parameters have been chosen so as to increase the p r e c i s i o n of the solutions obtained. TABLE 3.1 Trans Canada P i p e l i n e Warrant Values E = r = a2= D = Time to $41.00 5.42% per year .005712 per month $1 per year Warrant Value ( i n $) f o r S = $41.00 Maturity ( i n months) B-S No Div. 6 3.56 3.49 3.22 12 5.33 5.28 4.73 18 6.78 6. 74 5.92 24 8.06 8.02 6.91 30 9.23 9. 19 7.79 36 10.31 10.28 8.59 42 11.32 11.29 9.32 48 12.28 12.25 10.00 54 13.18 13.16 10.63 60 14.04 14.02 11.22 66 14.87 14.85 11.77 72 15.66 15.65 12.30 78 16.41 16.41 12.80 84 17.14 17.14 13.27 90 17.84 17.85 13.73 96 18.51 18.54 14.16 B-S No Div. NS No Div. NS With Div. NS No Div. B-S solution with no dividends numerical s o l u t i o n with no dividends numerical s o l u t i o n with a constant continuous dividend rate of $1 per year. NS With Div. 52 For the purpose of this study the numerical solution w i l l be considered 2 s a t i s f a c t o r y i f f o r each regression the R obtained i s very high, the intercept i s not s i g n i f i c a n t l y d i f f e r e n t from zero, and the slope i s not s i g n i f i c a n t l y d i f f e r e n t from one. I t can also be expected that, given the nature of the numerical s o l u t i o n , the results should become less accurate as time to e x p i r a t i o n i s increased. 2 The R obtained f o r a l l the regressions were greater than 0.9999. The intercepts f o r the f i r s t twenty-two periods (months) were not s i g n i f i c a n t l y d i f f e r e n t from zero and were smaller, i n absolute value, than one cent; period twenty-three but by they became greater, i n absolute value, than one cent and s i g n i f i c a n t l y d i f f e r e n t from zero. In the l a s t period, with more than eight years to maturity, the intercept was - 0.0734 and s i g n i f i c a n t l y d i f f e r e n t from zero. The slopes started with 1.0001 i n the f i r s t period and gradually increased to 1.0087 for the l a s t period. Table 3.2 shows the R s, the intercepts and slopes of the regressions f o r times to e x p i r a t i o n from % year to 8 years, every s i x periods (months). It should be noted that the stock price step chosen f o r the numerical solution was one d o l l a r and the time step was one month. of By decreasing the size these steps, i t i s possible to reduce the truncation error (3.21) and to make the numerical solution as accurate as required. Table 3.3 i l l u s t r a t e s this by reproducing the computations of Table 3.1, but using a stock price step of $0.25 and a time step of h, of one month. B-S a n a l y t i c a l solution decreases In this case the maximuir. iifference with the to four cents. TABLE 3.2 Trans Canada P i p e l i n e Warrant Regressions: Warrant V a l u e s B-S v e r s u s NS Time t o RSQ A B Maturity 6 .999997 -.0043 1.0001 12 .999998 -.0064 •1.0002 18 .999998 -.0083 1.0002 24 1.000000 -.0106 1.0004 30 1.000000 -.0138 1.0007 36 .999998 -.0179 1.0012 42 1.000000 -.0229 1.0018 48 .999998 -.0285 1.0025 54 .999999 -.0344 1.0032 60 .999993 -.0405 1.0040 66 .999995 -.0466 1.0048 72 .999998 -.0524 1.0056 78 1.000000 -.0579 1.0063 84 .999993 -.0630 1.0071 90 .999998 -.0675 1.0078 96 .999992 -.0716 1.0084 RSQ : R squared A : intercept B : slope o f the r e g r e s s i o n TABLE 3.3 Trans Canada P i p e l i n e Warrant E = r = a= D = Stock p r i c e step = Time s t e p = 2 $41.00 5.42% p e r y e a r 0.005712 p e r month $0.25 $0.25 0.25 month Warrant V a l u e Time t o Maturity ( i n months) Values B-S No D i v . ( i n $) f o r S = $41.00 NS No D i v . NS With D i v . 6 3.56 3.55 3.28 12 5.33 5.31 4.76 18 6.78 6.76 5.93 24 8.06 8.04 6.93 30 9.23 9.21 7.80 36 10.31 10.28 8.60 42 11.32 11.29 9.32 48 12.28 12.25 10.00 54 13.18 13.15 10.62 60 14.04 14.01 11.21 66 14.87 14.83 11.76 15.66 15.62 12.28 78 16.41 16.38 12.78 84 17.14 17.11 13.25 90 17.84 17.82 13.69 96 18.51 18.50 14.12 72 B-S No D i v . = NS No D i v . = NS With D i v . = . B-S s o l u t i o n w i t h no d i v i d e n d s n u m e r i c a l s o l u t i o n w i t h no d i v i d e n d s numerical s o l u t i o n with a constant c o n t i n u o u s d i v i d e n d r a t e o f $1.00 p e r y e a r . 55 3.11 Conclusions A method for solving a specific parabolic partial differential equation has been developed in this chapter: the one that governs the equilibrium value of an option- As an example of the method, numerical results have been obtained in the special case where the underlying stock is assumed to pay a constant continuous dividend. However, the method can be applied to deal with the case of any type of dividend policy. The method is thus extremely powerful. When solving partial differential equations by analytical methods, small changes in the coefficients or boundary conditions at times produce substantial changes in the analytic methods required and in the solutions obtained, and at other times make i t impossible or very difficult to obtain closed form solutions. This is the case, for example, in the Black-Scholes formulation where closed form solutions have been found in the no dividend situation, in the constant dividend yield situation, and in the constant dividend - perpetual warrant. ^ 1 However, in more significant cases, such as constant dividend rate and discrete dividends, no closed form solution has been found. The f l e x i b i l i t y of the numerical method, on the other hand, is striking. Most changes can be handled rapidly and efficiently. In Chapter 4 the problem of pricing an option, when its underlying stock pays discrete dividends and when there may be a finite probability of exercising before maturity, w i l l be studied. In Chapters 5 and 6, the related problem of pricing an equity linked l i f e insurance policy with asset value guarantee w i l l be discussed. It w i l l be argued that the same equilibrium option pricing model can be applied with some modifications and that premium payments can be interpreted as negative dividends. ^See Chapter 2. i ! 56 The method developed i n t h i s chapter can also be used i n the case i n which the dividend rate i s a known function of time and of the value of the stock, that i s , D = D(S,T), instead of constant. The only difference i s that the values of a^, b^, C i should r e f l e c t the changes i n D as S and T change along the g r i d . Promising applications of this method to the valuation of " c a l l and convertible s e c u r i t i e s are also features"^ envisioned. The " c a l l feature" gives a corporation the option to repurchase i t s debt at a stated p r i c e before maturity. 57 CHAPTER 4: OPTIMAL STRATEGIES FOR EXERCISING OPTIONS ON STOCKS WITH DISCRETE DIVIDENDS 4.1 Introduction In Chapter 2 i t was shown that a closed form s o l u t i o n f o r the value of an option on a dividend paying stock has been found only i n the case of constant dividend y i e l d , that i s , a dividend proportional to the value 1 of the stock', but i n this case there i s always a p o s i t i v e p r o b a b i l i t y of premature exercising (for large values of the stock price) and,hence, the value of the American option may be greater than the corresponding of the European option given by the closed form s o l u t i o n . value The American option can be worth more "dead" than " a l i v e . " In Chapter 3 a numerical method was developed to f i n d the value of the option i n the case i n which the underlying stock pays a constant was not a very r e s t r i c t i v e assumption), continuous (this dividend and where the s u f f i c i e n t conditions f o r no premature exercising (2.48), given by Merton [32], are met. In p r a c t i c e , however, the dividend p o l i c y of corporations i s not proportional (constant dividend y i e l d ) nor continuous (constant dividend rate) although the second approximation i s c e r t a i n l y better than the f i r s t . Corporations normally pay dividends quarterly or semi-annually, i . e . at discrete i n t e r v a l s , and are very reluctant to increase or reduce dividends even i n the case of s u b s t a n t i a l changes i n earnings or stock ^In Chapter 2 i t was also mentioned that Merton [32] gives a closed form s o l u t i o n i n the r e s t r i c t i v e case of a perpetual warrant when i t s underlying stock pays a constant continuous dividend and where the s u f f i c i e n t condition f o r no premature exercising obtains. 58 prices. The s t a b i l i t y of dividends has been widely dealt with i n the l i t e r a t u r e of finance [24,46,47], The objective of this chapter i s twofold: (1) to f i n d , by modifying the procedure developed i n Chapter 3, the value of an American option of which the underlying stock pays discrete dividends, and (2) to determine the c r i t i c a l stock p r i c e above which the option w i l l be worth more "dead" than "alive." Taking the American Telephone and Telegraph (ATT) 1975 warrant as an i l l u s t r a t i v e case, the s e n s i t i v i t y of the c r i t i c a l stock p r i c e to changes i n dividends variance rates, and i n t e r e s t rates are studied. f F i n a l l y , a comparison i s made between the actual ATT warrant p r i c e s at d i f f e r e n t points i n time and the t h e o r e t i c a l value obtained by the numerical procedure and by the Black-Scholes solutions i n both the no dividend case and i n the constant dividend y i e l d case. k .2 Theory Under certain conditions to be detailed subsequently, i t pays to exercise an American option before maturity i f the underlying stock pays dividends. Merton [32] has shown that exercise may be optimal only immediately before the stock's ex-dividend date. on a f i n i t e number of dates: Consequently, i t might pay to exercise the option only j u s t before the stock goes ex-dividend. Thus, instead of having an i n f i n i t e number of possible exercise dates, only the f i n i t e number of dividend dates must be considered. To simplify the presentation and with no loss of generality, the following notation w i l l be be used: 59 present time; moment when the o p t i o n i s to be v a l u e d , time o f the f i r s t l c time o f the i t t n dividend, dividend, time o f t h e l a s t d i v i d e n d b e f o r e m a t u r i t y , n-1 time o f m a t u r i t y o f t h e o p t i o n . "n As the n u m e r i c a l s o l u t i o n t o the p a r t i a l d i f f e r e n t i a l e q u a t i o n starts from e x p i r a t i o n date where the boundary c o n d i t i o n s a r e known ( i n the s p i r i t o f dynamic programming), t h e d i s c r e t e divid- ends and time i n t e r v a l s between d i v i d e n d s w i l l be d e f i n e d from the end: Pn-1 l fc n L o c P _ n D Dx 2 n-1 l z Consequently, D _^ 2 ^-2 2 tn-l i s the f i r s t dividend and n maturity ( s t a r t i n g from the present time t ) and Q 1 = t n„ - nt.- l _ c t _i n and - n ~ tn-i+l " n - i t T n-1 = t2 " T n " ti - t the t o t a l t -2 time l t c to m a t u r i t y i s tn the l a s t dividend before 60 As i n Chapter 2 , a l l the " i d e a l conditions" i n the market for the stock and for the option w i l l be assumed, i n p a r t i c u l a r that the variance rate of the return on the stock, a , i s constant and that the r i s k l e s s i n t e r e s t rate, r , i s also constant. Within the f i n a l time i n t e r v a l from t _ ^ to t ,where no dividends n n are paid the value of the option, W]_, i s the solution to the p a r t i a l d i f f e r e n t i a l equation (where T i s time to expiration): (4.1) ko S 2 2 W ss + rSW - rW - W s x =0 ' 2 with boundary conditions (4.2) W (0,T) = 0 W (S,0) = max[0, S-E] L and (4.3) 1 These r e s u l t s were derived i n Chapter 2. The solution to t h i s equation can be obtained by the Black-Scholes formula f o r the nodividend case or by the numerical method developed i n Chapter 3 imposing the condition that the constant continuous dividend rate be equal to zero. Subsequently i t w i l l be seen that the second approach i s the better one,because i t sets the stage f o r the numerical solution f o r the rest of the time i n t e r v a l s where a closed form solution has not been found. The value of the option at time t _ ^ ex-dividend can then be n written as W^(S,xi). 2 The subscripts i = 1, ...n have been used i n the boundary condition to indicate which time i n t e r v a l they refer to. Warning: these subscripts should not be confused with p a r t i a l derivatives. The d i f f e r e n t i a l equation (4.1) has been written without subscripts,because i t refers to a l l time i n t e r v a l s . This also holds true for boundary condition (4.2). 61 Within any period between dividend payments the value of the option can be expressed as a function of:the stock p r i c e , time to maturity,and known parameters. payments Also within any period between dividend a l l the assumptions made f o r the derivation of the B-S option p r i c i n g model hold. Thus, i t i s possible to form a hedged p o s i t i o n between the stock and the option by going long i n one and short i n the other, independent of the value of the stock. In other words, the ' whole derivation of the B-S model i s also v a l i d i n this case and the value of the option i s governed by p a r t i a l d i f f e r e n t i a l equation ( 4 . 1 ) . The boundary conditions, however, are d i f f e r e n t and, therefore, so i s the solutic for the value of the warrant. At each dividend date a new boundary condition i s imposed to take into account the fact that the p r i c e of the stock w i l l sudden] drop (when the stock goes ex-dividend) by the amount of the dividend and the fact that the option value f o r the reduced ex-dividend, stock p r i c e cannot be lower than the exercise value given by the corresponding cum-dividend stock p r i c e minus the exercise p r i c e , because otherwise the option w i l l be exercised. Within any period between dividend payments the option should never be exercised,because i t s value i s governed by ( 4 . 1 ) and i s always worth more " a l i v e " than "dead." option immediately But i t may be p r o f i t a b l e to exercise the before i t goes ex-dividend i f the value of the option, when the stock actually goes ex-dividend, i s less than was i t s exercise value (S-E) p r i o r to i t going ex-dividend. The option can then be assumed to expire at each dividend date, where the option-holder has the "option" to r e t a i n the option or to get the exercise value. These results were obtained by Merton [32] and Black [5 ]. 62 Consequently, the value of the option in the time interval between t _ n 2 and t _j (or any other subsequent time interval) is also n given by the solution to the partial differential equation (4.1), but where the maturity date is assumed to be t _ j . n At this point in time a decision w i l l have to be made whether to keep the option or whether to exercise i t . changed. Also, the boundary conditions at time t _^ have n If we assume that the change in the stock price at t _^, when n the stock goes from cum-dividend to ex-dividend, is equal to the amount of the dividend, D ^ , the new boundary condition is (4.4) W (S,0) 2 = Max[0, S - E , W J ^ C S - D J . T I ) ] that is,the boundary condition at t _ i is the greater of: zero, the exercise n value (cum-dividend) or the value of the option when the stock goes ex-dividend. If the f i r s t is greater than the second, the option w i l l be exercised at t _^; otherwise the option w i l l be retained. n The c r i t i c a l stock price, S , has been defined as the value of the stock C cum-dividend for which the exercise value is equal to the option value when the stock goes ex-dividend. S C For values of the stock greater than i t w i l l pay to exercise the' option. The other boundary condition (4.2) does not change in this time interval or in the subsequent ones. The solution to equation (4.1) with boundary conditions (4.2) and (4.4) can be easily obtained applying the numerical technique developed in Chapter 3. the option at time t _ n 2 Again i t is possible to find the value of ex-dividend W ( S , 2 T) dynamic programming approach until time tQ. 2 and to continue with this 63 In general, boundary condition (4.4) for time interval i can be written as (4.5) W (S,0) = Max[0, S-E, w ^ C S - D ^ ! , ± T ^ ) ] This boundary condition can be represented in the classical W-S diagram as shown in Figure 4.1. Figure 4.1: C r i t i c a l Stock Price In Figure 4.1 curve: OEAB is Max[0, S-E], OAC and OAB is W-j^S-Di.!) , is MaxfOEAB, OAC]. OAB is naturally the new boundary condition and A is the c r i t i c a l stock price. Following Samuelson's [36] and McKean's [27] studies on warrant pricing, Chen [ 6 , 7] derives a functional equation for the value of a warrant by applying a dynamic programming technique, but using the expected value approach. recognizes in [ 7],is i l The problem with this approach, as Chen that "In applying the warrant valuation model to compute the theoretical value of a warrant, the required rate of return 64 on w a r r a n t s ( i . e . the d i s c o u n t r a t e 0) must be known." The s o l u t i o n to the p a r t i a l d i f f e r e n t i a l e q u a t i o n w i t h changing c o n d i t i o n s developed i n t h i s c h a p t e r , i s based seminal idea that i n e q u i l i b r i u m position the expected (by g o i n g l o n g on the s t o c k and e q u a l to the r e t u r n on the r i s k l e s s expected on B l a c k and numerical boundary Scholes' r e t u r n on a hedged s h o r t on the o p t i o n ) must be asset and i s independent r e t u r n s on the s t o c k and on the o p t i o n as w e l l as of of'the the homogeneity o f e x p e c t a t i o n s . 4.3 Solution Algorithm The s o l u t i o n procedure w i t h boundary c o n d i t i o n s (4.2) f o r p a r t i a l d i f f e r e n t i a l equation and 4.5) f o r the v a l u e of an o p t i o n where the u n d e r l y i n g s t o c k pays d i s c r e t e d i v i d e n d s , i s analogous i n d i c a t e d i n Chapter this to t h a t 3 f o r the case o f c o n s t a n t c o n t i n u o u s case the continuous d i v i d e n d r a t e , D, p r o c e s s to make the n e c e s s a r y adjustments d i v i d e n d s . In i s e q u a l to z e r o , but a t each d i s c r e t e d i v i d e n d p a y i n g date i t i s n e c e s s a r y t o stop i n the As has a l r e a d y been mentioned, a t each d i v i d e n d p a y i n g date above which i t w i l l pay (3.24) w i l l be the same a l l the boundary c o n d i t i o n (3.23) w i l l be each time a d i s c r e t e d i v i d e n d i s p a i d . boundary c o n d i t i o n s , there a r e no equations presented the to e x e r c i s e the o p t i o n . a l o n g the i t e r a t i v e p r o c e s s , but (4.5) interval. s t o c k p r i c e as the v a l u e of the s t o c k , The boundary c o n d i t i o n s (3.22) and r e p l a c e d by iterative i n the boundary c o n d i t i o n s to be a b l e t o c o n t i n u e the i t e r a t i v e p r o c e s s i n the next time a l g o r i t h m w i l l g i v e the c r i t i c a l (4.1) i n Chapter 3. changes i n the system of Given linear the 65 The computer program presented i n Appendix A was adapted to the problem discussed i n this chapter, i n p a r t i c u l a r boundary condition (4.5) was computed at each discrete dividend date. At each dividend date the program gives f o r every stock price two option p r i c e s : the ex-dividend and the cum-dividend option p r i c e s . These two prices d i f f e r , because even when i t does not pay to exercise cum-dividend, the option p r i c e cum-dividend w i l l be the same f o r stock p r i c e S as the option p r i c e ex-dividend w i l l be f o r stock p r i c e S-D^. Thus, f o r a given stock p r i c e S the option p r i c e w i l l always be higher ex-dividend. At each dividend date the program also computes the c r i t i c a l stock p r i c e cumdividend. I f the curve OAC i n Figure 4.1 does not cut the l i n e EB within the range of stock prices selected f o r the computations,the program p r i n t s an a r b i t r a r i l y large number as the c r i t i c a l p r i c e . To f a c i l i t a t e the comparison between the two parts of boundary condition (4.5) (the exercise value and the value of the option exdividend f o r S-D-£),the stock p r i c e increment has to be chosen as an i n t e g r a l D. f r a c t i o n of the dividend (that i s h = _ i , where n i s a p o s i t i v e i n t e g e r ) . In the program shown i n Appendix B i t has been assumed that the discrete dividend i s the same i n every dividend date, although different known dividends could be incorporated with minor changes. Apart from the information required f o r the program explained i n Chapter 3, i t i s now necessary to know the amounts of the discrete dividends and t h e i r dates of payments. The output of the program gives the values of the option at d i f f e r e n t points i n time and for d i f f e r e n t stock prices and gives the c r i t i c a l stock prices at each dividend date. The program also plots the 66 c r i t i c a l stock p r i c e versus time to e x p i r a t i o n . A s l i g h t l y modified version of the program plots option prices for a given stock price as time to maturity changes. F i n a l l y , the program allows for a d i f f e r e n t d e f i n i t i o n of c r i t i c a l stock p r i c e . The exercising c r i t e r i a could be modified to specify that the option should be exercised cum-dividend i f the value of the option ex-dividend (for S-D^) i s no greater than P d o l l a r s above the exercise value, i . e . i f the premium on the option i s less than P. The c r i t i c a l stock p r i c e with premium P, S Figure 4.2: 0 , i s shown i n Figure 4.2. C r i t i c a l Stock P r i c e with Premium P. E-P E S S„ Except f o r the l i n e W = S-(E-P) which determines S S at D, cp Figure 4.2 i s the same as Figure 4.1 (where P = 0). This modified d e f i n i t i o n of c r i t i c a l stock price was made to study the s e n s i t i v i t y of the c r i t i c a l price to changes i n P and,thus,the "closeness" of the two parts of the boundary condition (OAC and OEB i n 67 F i g u r e 4.2) i n t h e "neighborhood" o f the c r i t i c a l 4.4 A p p l i c a t i o n to t h e ATT Warrant stock price. : Sensitivity Analysis The American Telephone and T e l e g r a p h warrant m a t u r i n g was s e l e c t e d as an example to study p r i c e s t o changes i n d i v i d e n d s , stock i n v a r i a n c e r a t e s , and i n i n t e r e s t r a t e s . 1970 e n t i t l i n g the h o l d e r s s u b s c r i b e t o 31,375,540 s h a r e s o f common s t o c k a t $52.00 a share b e g i n n i n g November 15, 1970 t o May 15, 1975. protected against d i l u t i o n a g a i n s t cash d i v i d e n d s . The e x e r c i s e p r i c e was ( s t o c k s p l i t s and s t o c k d i v i d e n d s ) ,but n o t The t o t a l amount o f shares o u t s t a n d i n g a t the time o f t h e i s s u e was 549,266,114, so t h e p o t e n t i a l in May 15, 1975 the s e n s i t i v i t y o f c r i t i c a l The ATT warrant was i s s u e d i n A p r i l , to J the number o f shares i s 5.71%. dilution I n the year preceding the i s s u e the s t o c k p r i c e range had been 46^ t o 58 /8 and t h e i n d i c a t e d d i v i d e n d 1 at t h e time was $2.60 payable q u a r t e r l y (January 1, A p r i l 1, J u l y 1 and October 1 ) . In view o f these f a c t s t h e p e r i o d c o n s i d e r e d f o r t h e study was November 15, 1970 to May 15, 1975. The v a r i a n c e o f t h e monthly r a t e of-, r e t u r n was computed from the CRSP tape u s i n g 60 o b s e r v ations for from J u l y 1965 t o June 1970. t h i s p e r i o d was 0.0017. The v a r i a n c e r a t e p e r month F o r the r i s k l e s s i n t e r e s t r a t e ^ the 3 In the remainder o f t h e t e x t whenever r e f e r e n c e i s made to the c r i t i c a l s t o c k p r i c e , the s t a n d a r d d e f i n i t i o n w i t h P e q u a l to zero i s implied, unless otherwise c l e a r l y s p e c i f i e d . 4 To be p r e c i s e , the r i s k l e s s i n t e r e s t r a t e s h o u l d have been a d j u s t e d to a c o n t i n u o u s compounding b a s i s . T h i s adjustment was not made i n what f o l l o w s , because i t would not have changed t h e n a t u r e of the r e s u l t s . 68 average y i e l d on 3 to 5 years United States government s e c u r i t i e s [41] for the corresponding month was chosen. For November 1970 this value was 6.37 per annum. 4.4.1 Dividends and C r i t i c a l Stock Prices Using the ATT data, within the stock p r i c e range considered ( i n this case from $0 to $260) a c r i t i c a l stock p r i c e with a value of $54.75 f o r the l a s t d i v i d end date was obtained; at none of the other dividend dates d i d a c r i t i c a l stock . p r i c e e x i s t (the curve OAC i n Figure 4.1 becomes asymptotic to EAB) or was greater than $260. This r e s u l t i s not s u r p r i s i n g , because with a dividend of $2.60 per year (5% of the exercise price) the s u f f i c i e n t condition f o r no premature exercising '(2.48) i n the continuous equivalent to this problem i s s a t i s f i e d . Only i n the l a s t dividend date, one and a h a l f months before maturity, does the discreteness of the problem become significant. Other things being equal, the value of an option i s an increasing function of the variance rate of return of the underlying stock and a decreasing function of dividend payments on the stock. The p r i c e of ATT common stock i s considered, r e l a t i v e to other NYSE stocks, to be reasonably stable. I t also has a high dividend y i e l d r e l a t i v e to other stocks. Thus, i n r e l a t i o n to other stocks, the variance rate of the ATT r e l a t i v e l y low and the dividend y i e l d r e l a t i v e l y high. stock i s Both of these factors should make the premium of the ATT warrant over i t s exercise value r e l a t i v e l y small. I f i t does not pay to exercise the option before maturity i n this case, where the premium i s r e l a t i v e l y small, the case w i l l be even 69 stronger f o r options on stocks with higher variance rate and lower dividend yield. To study the e f f e c t of dividends on c r i t i c a l stock prices as time to maturity changes,the same ATT data were used, but dividends of $0.90, $1.00 and $1.10 per quarter were assumed. The r e s u l t s are tabulated i n Table 4.1 and a graph i s shown i n Figure 4.3. expected,the As c r i t i c a l stock p r i c e i s a decreasing function of dividends and an increasing function of time to maturity, withvariance rate and r i s k l e s s i n t e r e s t rate constant. It i s i n t e r e s t i n g to note that the c r i t i c a l stock p r i c e with i a premium of only ten cents can produce s i g n i f i c a n t l y d i f f e r e n t r e s u l t s from the ones shown i n Table 4.1 and Figure 4.3. This i s due to the fact that the curve OAC i n Figure 4.1 i s very close to the l i n e EAB and almost p a r a l l e l to i t . " * 4.4.2 Variance Rates and C r i t i c a l Stock P r i c e s To test the s e n s i t i v i t y of the c r i t i c a l stock p r i c e to changes i n the variance rate of return on the associated stock f o r d i f f e r e n t times to expiration, the ATT warrant data were again used. However, this time a quarterly dividend of one d o l l a r was assumed, because, as was mentioned e a r l i e r , with the actual dividend of $0.65 no c r i t i c a l p r i c e existed within the To make sure that the accuracy of the numerical method was enough to give r e l i a b l e results the computer program was run on double p r e c i s i o n and a f t e r 216 time steps (because the time increment was chosen to be h of a month and the number of months considered was 54) the option prices never d i f f e r e d by more than one cent i n r e l a t i o n to the single precision run. Also, i n the case of the c r i t i c a l stock prices, this difference was not greater than one cent. TABLE 4.1 C r i t i c a l Stock Prices as a Function of Dividends and Time to Expiration (variance rate and r i s k l e s s i n t e r e s t rate constant). Time to Expiration (in months) C r i t i c a l Stock Price with dividends paid quarterly ( i n $) 0.90 1.00 1.10 1.5 53.87 53.63 53.46 4.5 59.35 57.74 56.84 7.5 62.08 59.76 58.47 10.5 64.16 61.24 59.63 13.5 65.88 • 62.42 60.53 16.5 67.36 '63.41 61.27 19.5 68.67 64.27 61.90 22.5 69.84 65.01 62.44 25.50 70.91 65.68 62.91 28.50 71.89 66.28 63.31 31.50 72.79 66.81 63.67 34.50 73.63 67.30 64.02 37.50 74.41 67.75 64.28 40.50 75.14 68.16 64.58 43.50 75.83 68.54 64.77 46.50 76.47 68.88 65.04 49.50 77.08 69.21 65.18 52.50 77.65 69.51 65.38 Variance Rate = 0.0017 per month Risk Free Rate = 6.37% per annum FIGURE 4.3: CRITICAL STOCK PRICES AS A FUNCTION OF 84.00 DIVIDENDS AND TIME TO EXPIRATION 80.00 D=0.9 76.00 # # 72.00 D= 1. 0 # CRITICAL STOCK PRICE 68.00- # # # # # # # 6 4 . 00 # # # # # D=1. 1 # 60.00 # # # # 56.00 52.00-... .00 # 8.00 16.00 I 24-00 32.00 40.00 TIHE TO EXPIRATION 48.00 72 range of stock prices considered. Variance rates of 0.001, 0.003 and 0.005 per month were used to compute c r i t i c a l stock prices. The results are tabulated in Table 4.2 and in the graph in Figure 4.4. As expected,the c r i t i c a l stock price is an increasing function of the variance rate for a given time to expiration, because the warrant price , which enters into the boundary condition (4.5), is an increasing function of the variance. 4.4.3 Interest Rates and C r i t i c a l Stock Prices Finally, to study the sensitivity of c r i t i c a l stock price to changes in the riskless interest rate 'for different times to expiration and with constant variance rate and dividends, the ATT warrant data were used assuming interests of 2%, 3% and 4% per annum; these low interest rates had to be used to get c r i t i c a l prices within the range of stock prices selected . Other things being equal, the value of an option and, hence,also the c r i t i c a l stock price are higher, the higher is the riskless interest rate. Thus, c r i t i c a l stock prices should be an increasing function of interest rates. This is the case as is shown in Table 4.3 and Figure 4.5. 4.4.4 ATT Theoretical Warrant Prices In Table 4.4 and Figure 4.6 the ATT theoretical warrant prices (cum-dividends) for some specific stock prices are shown for the dividend dates. During the l i f e of the warrant the price of ATT stock was always within the $39 to $65 range and $52 was the exercise price. TABLE 4.2 C r i t i c a l Stock P r i c e s as a F u n c t i o n o f the V a r i a n c e Rate and Time to M a t u r i t y (for constant i n t e r e s t r a t e and d i v i d e n d s ) . C r i t i c a l Stock P r i c e s ( i n $) V a r i a n c e Rate per Month 0.005 0.003 0.001 Time to Expiration ( i n months) 1.5 53.00 54.65 56.04 4.5 55.86 60.74 64.83 7.5 57.22 63.90 69.65 10.5 58.17 66.30 73.40 13.5 58.92 68.27 76.54 16.5 59.54 69.96 79.28 19.5 60.06 71.46 81.71 22.5 60.51 72.80 83.91 25.5 60.91 74.01 85.93 28.5 61.27 75.12 88.79 31.5 61.58 76.14 89.52 34.5 61.84 77.07 91.14 37.5 62.12 77.93 92.65 40.5 62.32 78.74 94.06 43.5 62.57 79.48 95.39 46.5 62.73 80.18 96.66 49.5 62.88 80.83 97.86 52.5 63.11 81.44 98.99 R i s k Free Rate = 6.37% p e r annum Dividends = $1.00 p e r q u a r t e r FIGURE 4.4: CRITICAL STOCK PRICES A S A FUNCTION 112.00 THE VARIANCE RATE AND OF TIME TO EXPIRATION 104.00 96. 00 VAS=0.005 # # 88.00 # CRITICAL STOCK PRICE 80.00# # # # # VAR=0.003 # # 72.00 64.00 VAR=0.001 # # # # 56.00 # # # # # # # # 48. 00-, .00 8. 00 16.00 -•I ... J . . . . . . . . . |... 24.00 32.00 40.00 TIME TO EXPIRATION 48.00 TABLE 4.3 Critical Function Interest Maturity variance Time to Expiration (in months) Stock Prices as a of the Riskless Rate and Time to ( f o r constant rate and dividends). C r i t i c a l Stock Prices (in $) Riskless Rate (per year) 2% 3% 4% 1.5 53.80 53.94 54.15 4.5 56.39 57.19 58.56 7.5 57.73 58.90 60.86 10.5 58.69 60.14 62.59 13.5 59.44 61.11 64.01 16.5 60.05 61.94 65.22 19.5 60.56 62.66 66.28 22.5 60.99 63.28 67.25 25.5 61.40 63.83 68.11 28.5 61.72 64.32 68.89 31.5 62.05 64.78 69.63 34.5 62.27 65.19 70.31 37.5 62.58 65.53 70.94 40.5 62.73 65.92 71.52 43.5 62.95 66.19 72.07 46.5 63.21 66.54 72.60 49.5 63.29 66.74 73.09 52.5 63.39 67.07 73.55 Variance Rate = 0.0017 per month Dividends = $0.65 per quarter FIGURE 4.5: CRITICAL STOCK PRICES AS A FUNCTION OF 84.00 THE RISKLESS RATE AND T I H E TO EXPIRATION 80.00- 76. 00- 72. 00- R=4% CRITICAL STOCK PRICE 68.00- H=3X # # # # # # # # # # 64.00 # # # t 60. 00 # # R=2% # # # # 56.00 52.00-... .00 8.00 16.00 .. |. . . . . . . . . | f ... 24.00 32.00 40.00 TIME TO EXPIRATION 48.00 TABLE 4.4 ATT Theoretical Warrant Prices f o r S p e c i f i c Stock Prices ($39, $45.5, $52, $58.5, $65) as a function of Time to Expiration. Time to Expiration (in months) $39 Warrant Values (in $) Stock Prices $52 $58.5 $45.5 $65 1.50 0.00 0.01 0.90 6.50 13.00 4.50 0.00 0.12 1.77 6.79 13.17 7.50 0.01 0.32 2.38 7.18 13.37 10.50 0.04 0.55 2.88 7.57 13.60 13.50 0.08 0.79 3.31 7.94 13.84 16.50 0.13 1.03 3.' 6 9 8.28 14.08 19.50 0.20 1.26 4.04 8.61 14.33 22.50 0.28 1.48 4.36 8.92 14.58 25.50 0.36 1.69 4.66 9.21 14.82 28.50 0.44 1.89 4.94 9.49 15.06 31.50 0.53 2.08 5.20 9.76 15.29 34.50 0.62 2.27 5.45 10.01 15.52 37.50 0.71 2.45 5.69 10.25 15.74 40.50 0.80 2.63 5.91 10.49 15.95 43.50 0.89 2.79 6.13 10.71 16.16 46.50 0.98 2.96 6.34 10.93 16.36 49.50 1.06 3.11 6.54 11.14 16.56 52.50 1.15 3.26 6.73 11.34 16.75 Variance Rate = 0.0017 per month Risk Free Rate = 6.37% per year Dividends = $0.65 per quarter 78 FIGURE 4 . 6 : 32.00- ATT THEORETICAL WARRANT FOR S P E C I F I C STOCK PRICES PRICES 28.00 24. 00 20.00 S=65.0 WARRANT PRICE 16.00- # # # # # # # # # # # S=58.5 # 12.00 # # # 8.00 # # # # S=52.0 # # 4.00 # # # # # # # f # # # # # # # # S=39.0 -# # # # # # S=45.5 * # # . 0 0 - . #. .00 # # # # # # # # # $• • $j • • $< 8.00 16.00 24.00 32.00 40.00 TIME TO EXPIRATION 4 8. 00 79 4.4.5 Comparing Warrant Values The f i n a l test on the ATT warrant consisted of a comparison between the actual warrant price i n the market and the t h e o r e t i c a l value obtained by three d i f f e r e n t i) procedures: The numerical solution with discrete dividends presented i n t h i s chapter. ii) The Black-Scholes formula,disregarding any dividend payment. i i i ) The Black-Scholes formula,assuming a constant dividend yield. To avoid the problem of establishing goes ex-dividend, the warrant was exactly when the stock estimated with the three above mentioned methods and compared with the market price j u s t midway between two dividend payments, s t a r t i n g on November 15, when exercising was permitted and ending i n November 15, 1970 1974. For the d i f f e r e n t estimations the actual r i s k l e s s interest rate , the indicated quarterly dividend, and the closing stock price for the p a r t i c u l a r date were used; but the same variance rate for the period '65 to '70 was used. This data are tabulated i n Table 4.5. In Table 4.6 the warrant market price i s compared with the three theoretical methods for d i f f e r e n t time to maturity. In Table 4.7 this comparison i s made between the ratios of the t h e o r e t i c a l values and the market p r i c e . 6 I I See [41]. TABLE 4.5 Data Used i n the Estimation of T h e o r e t i c a l Warrant P r i c e s . Date Time t o Expiration ( i n months) Riskless Rate (% p e r y e a r ) Quarterly Dividend ( i n $) Stock Price ( i n $) Nov 16-70 54 6.37 0.65 45 Feb 16-71 51 5.31 0.65 52.375 May 14-71 48 6.02 0.65 47.125 Aug 16-71 45 6.39 0.65 44.5 Nov 15-71 42 5.50 0.65 42.25 Feb 15-72 39 5.51 0.65 44.0 May 15-72 36 5.69 0.65 42.5 Aug 15-72 33 5.92 0.65 41.875 Nov 15-72 30 6.03 0.70 50.25 Feb 15-73 27 6.61 0.70 51.125 May 15-73 24 6.78 0.70 53.125 Aug 15-73 21 7.75 0.70 47.5 Nov 15-73 18 6.96 0.70 47.5 Feb 15-74 15 6.77 0.77 51.75 May 15-74 12 8.24 0.77 47.0 Aug 15-74 9 8.64 0.77 42.35 Nov 15-74 6 7.65 0.85 47.5 81 TABLE 4.6 Warrant Prices Warrant Prices Time to Expiration Market Price (in $) Numerical Method B-S No Dividends B-S With Dividends 54.000 8.000 3.270 8.580 2,600 51.000 11.500 6.090 12.550 5.380 48.000 9.675 3.720 8.730 3.090 45.000 9.000 2.630 6.710 2.120 42.000 7.125 1.430 4.210 1.090 39.000 7,675 1.810 4. 750 1.460 36.000 6.875 1.310 3-610 1.030 33.000 5.675 1.090 3,000 0.850 30.000 7.500 3.890 7.590 3. 500 27.000 7.250 4.400 8.000 4.040 24.000 6.675 5.420 8.960 5.070 21.000 4.875 2.440 4. 640 2.200 18.000 4.750 1.930 3. 630 1.750 15.000 4.375 3. 410 5.540 3. 180 12.000 2.750 1. 240 2. 280 1. 120 9.000 1.500 0.160 0.340 0.130 6.000 1.250 0.570 0.950 0.510 TABLE 4.7 R a t i o o f Warrant P r i c e s to Market Prices Warrant P r i c e / M a r k e t P r i c e Numerical Method B-S No Dividends 54.000 0.409 1.072 0.325 51.000 0.530 1.091 0.468 48.000 0.384 0.902 0.319 45.000 0.292 0.746 0.236 42.000 0.201 0.591 0.153 39.000 0.236 0.619 0. 190 36.000 0.191 0.525 0. 150 33.000 0.192 0.529 0.150 30.000 0.519 1.012 0.467 27.000 0.607 1.103 0.557 24.000 0.812 1.342 0.760 21.000 0.501 0.952 0.451 18.000 0.406 0.764 0.368 15.000 0.779 1.266 0.727 12.000 0.451 0.829 0.407 9.000 0.107 0.227 0.087 6.000 0.456 0.760 0.408 Time t o Expiration B-S With Dividends 83 A c c o r d i n g to these r e s u l t s account the two f o r m u l a t i o n s t h a t take into d i v i d e n d payments g i v e c o n s i s t e n t l y lower v a l u e s than those i n the market. I t i s i n t e r e s t i n g to n o t e the s i m i l a r i t y between the n u m e r i c a l s o l u t i o n and the B l a c k - S c h o l e s s o l u t i o n f o r c o n s t a n t d i v i d e n d y i e l d , a l t h o u g h i n a l l cases t h e former price. g i v e s a v a l u e c l o s e r t o the market The B l a c k - S c h o l e s s o l u t i o n w i t h o u t c l o s e r to the market p r i c e . the w a r r a n t w i t h o u t c o n s i d e r i n g d i v i d e n d s i s much P o s s i b l y t h i s i s because t h e market v a l u e s t a k i n g p r o p e r account o f the d i v i d e n d payments o f the underlying stock. There may be t h r e e b a s i c reasons by the model p r i c e s may d i f f e r from market p r i c e s : i) The model i s n o t an a c c u r a t e d e s c r i p t i o n o f the market. The assumptions made a r e too r e s t r i c t i v e . ii) The model g i v e s the " r i g h t " v a l u e s , b u t the market e i t h e r "under" o r " o v e r v a l u e s " the o p t i o n . be some p r o f i t o p p o r t u n i t i e s . I f t h i s i s the case, t h e r e s h o u l d Black-Scholes [4] t e s t t h i s hypothesis for c a l l options. iii) The model i s an a c c u r a t e d e s c r i p t i o n o f the market, b u t t h e h i s t o r i c a l v a r i a n c e used i n t h e model i s n o t t h e same v a r i a n c e used by the market t o p r i c e t h e o p t i o n . opportunities. Here a l s o t h e r e s h o u l d be p r o f i t T h i s a r e a o f r e s e a r c h i s c u r r e n t l y b e i n g expanded i n a s e p a r a t e study by t h e p r e s e n t author and o t h e r s . 4.5 Conclusions I n t h i s c h a p t e r the n u m e r i c a l s o l u t i o n t o p a r t i a l equations developed i n Chapter differential 3 has been m o d i f i e d s u c c e s s f u l l y to f i n d the values of an option i n which the underlying stock pays discrete dividends.^ In Chapters 5 and 6 the method w i l l be applied with some modifications to the valuation of equity-linked l i f e insurance contracts. The analysis of the e f f e c t of a changing exercise p r i c e on the value of an option i s very s i m i l a r to that of the e f f e c t of discrete dividends. For further discussion of this issue see Merton [32]. 85 CHAPTER 5: THE EQUILIBRIUM PRICING OF ASSET VALUE GUARANTEES ON EQUITY LINKED LIFE INSURANCE POLICIES: THEORY. Introduction to L i f e Insurance.^ To provide some background f o r the discussion of equity-linked l i f e insurance p o l i c i e s , i t i s convenient to begin t h i s chapter with a b r i e f discussion of some elementary l i f e insurance underlying any kind of insurance concepts. The basic concept i s the pooling of independent r i s k s . Hence,this concept can also be applied to the hazard of death. large group of l i v e s are insured,the If a sufficiently expected number of deaths i n a given year can be quite accurately predicted,using the law of large numbers. There.are four basic t r a d i t i o n a l types of l i f e insurance term, whole l i f e , endowment,and annuity. contracts: The basic function of the f i r s t three types of contracts i s to protect against premature death. Annuity contracts protect the policyholder against the r i s k of longevity and are not discussed here. Each of the three basic types w i l l be discussed i n turn. Term insurance provides l i f e insurance protection f o r a l i m i t e d of time. The face value (or sum assured) of the p o l i c y becomes payable i f death should occur within the s p e c i f i e d period the event of s u r v i v a l . period and nothing i s paid i n The period of protection may be one year or shorter, or i t may run to age s i x t y - f i v e . Term insurance may be regarded as temporary insurance more nearly comparable to property and casualty insurance any of the other types of l i f e insurance insurance contracts. and as such i s contracts than are The premium f o r term i s r e l a t i v e l y low,because these contracts do not cover the period This section draws on M c G i l l [26], 86 of old age when death i s most l i k e l y to occur. In contrast to term insurance which pays benefits only i f the insured terminates within a s p e c i f i e d number of years, whole l i f e life insurance provides f o r the payment of the face amount upon the death of the insured regardless of when i t may occur. I t i s usual for l e v e l premiums to be payable throughout the duration of the contract. monthly, quarterly, semi-annually or yearly. Premiums can be paid In some contracts premiums are payable for a l i m i t e d period (e.g. to age 65) or even as a single lump sum. The t h i r d type of contract, endowment insurance, provides f o r the payment of the face value of the p o l i c y on death within a c e r t a i n period or s u r v i v a l to the end of the s p e c i f i e d period. Term and whole l i f e insurance provide f o r payment of sum assured only i n the event of death. Under endowment insurance the sum assured i s payable i f the insured i s s t i l l l i v i n g at the end of the s p e c i f i e d period. The endowment insurance contract emphasizes the investment element. It i s e s s e n t i a l l y a savings plan with insurance to protect the program against premature death of the insured. When computing the rate of return on the investment of an endowment insurance contract, i t must be remembered that only part of the premium goes towards investment, the rest being used for protection. Because the r i s k of death increases with age,it i s necessary for an insurance company to b u i l d up reserves under l e v e l premium, whole l i f e , a n d endowment p o l i c i e s . I f a policyholder does not wish to continue payment under these contracts,the insurance company w i l l o f f e r him various options. One of these calls for the payment of a surrender value corresponding to 87 the reserve of the p o l i c y . In North America these surrender values are guaranteed by law f o r the duration of the contract. This has profound implications f o r the investment p o l i c i e s of the insurance companies. In p a r t i c u l a r i t means that common stocks are not suitable as a major component of the p o r t f o l i o . For example i n Canada there i s a 25% l i m i t a t i o n on the amount of common stock i n a conventional l i f e insurance p o r t f o l i o . In the United States the l i m i t a t i o n i s 5 to 10% of the p o r t f o l i o depending on the state. This i s one of the reasons why the return on conventional l i f e 2 insurance contracts has been r e l a t i v e l y 5.2 Equity Based L i f e low. Insurance The t r a d i t i o n a l l i f e insurance contracts considered i n the l a s t section provide f o r benefits fixed i n terms of the l o c a l unit of account. However,the prolonged period of sustained but i r r e g u l a r i n f l a t i o n since 1945 has served to diminish the usefulness of such long-term denominated contracts. currency As a r e s u l t , insurance companies have increasingly moved towards the issuance of equity based contracts the benefits of which depend upon the performance of some reference p o r t f o l i o of common stocks (mutual fund), i t being thought that e q u i t i e s , at l e a s t i n the long run, would provide a hedge against i n f l a t i o n . Equity based l i f e insurance was around 1953 first introduced i n the Netherlands and the f i r s t equity linked l i f e insurance product was offered in the United Kingdom i n the Autumn of 1957. The 1970 e d i t i o n of "Equity- Linked L i f e Assurance Tables," published by Stone & Cox Publications Ltd., See for example Kensicki [21] where the i n t e r n a l rate of return on a t y p i c a l whole l i f e p a r t i c i p a t i n g p o l i c y i s estimated to be 3.57% per annum. 88 presents summaries of some 141 equity-based l i f e insurance plans (109 annual premium plans and 32 single premium plans) offered by 93 l i f e companies in 3 the United Kingdom. In Canada the assets corresponding to these equity based products are held separately from the assets corresponding to the traditional l i f e insurance products. 4 These segregated funds,as they are called,are,under certain conditions, exempt from the 25% limitation mentioned earlier. Canadian companies s e l l equity based endowment contracts. Many Each year a proportion of the premium is invested directly in the equity or reference portfolio. The balance of the premium is used to cover expenses, any guarantees attached to the policy, and a margin for profit. In the United States l i f e insurance contracts, which vary according to the investment performance of a separate account, are known as variable l i f e insurance contracts. Since 1970 the question of whether variable l i f e insurance is or is not a security subject to the regulation of the Securities and Exchange Commission (SEC), has been the subject of discussion between the insurance industry and the SEC. At the time of writing, the issue is not resolved and at the moment no l i f e company in the United States sells this product. The insurance companies would naturally prefer to be exempted from the SEC regulations. In the United Kingdom this type of product is known as unit-linked insurance and has proven very popular. Many of the products now available in Canada have developed from contracts f i r s t issued in the United Kingdom. The examples discussed in this thesis w i l l be drawn mainly from Canada and to a lesser extent from the United Kingdom. 3 See Turner [45]. ^See remarks by S.H. Cooper on paper by E . T . Squires [40] to the Institute of Actuaries for a more complete discussion of this point. 89 In Canada the usual equity-based endowment policy contains a minimum guarantee of 75% (or 100%) of the gross premiums paid to the date of death or maturity. If the value of the accumulated units is greater than the guaranteed benefit ,the greater amount is payable. If, however, the value of the units is less than the minimum guaranteed amount,the insurance company has to make up the shortfall from its general l i f e fund. Part of the impetus for the inclusion of these guarantees sprang from a desire to avoid these contracts being classified as securities. For example,in 1970 the Ontario Securities Commission indicated that they would claim jurisdiction over certain types of equity contracts which did not include minimum benefits of at least 75% of gross premiums paid at the date of death or maturity. The Federal Department of Insurance, however, is anxious to ensure that insurance companies remain solvent and so seeks to avoid the inclusion of guarantees that would put the solvency of the l i f e insurance company in jeopardy. Guarantees on death and maturity are not permitted for policies with terms less than 10 years. are not to exceed 100% of gross premiums paid. Furthermore,maturity guarantees This thesis attempts to show that there are other ways of minimizing the risk associated with the provision of such guarantees. In an equity based contract with this type of provision the insurance company undertakes to bear part of the investment risk as well as mortality risk. This is a radical departure from the types of risks to which insurance companies have traditionally been exposed and which have been studied by actuaries. The essence of mortality risk is that individual lives are independent so that risks may be almost entirely eliminated by pooling or diversification: the actuary is then justified in replacing 90 probability distributions of benefits by their expected values and discounting these at an appropriate interest rate. Investment risk on the other hand affects a l l contracts simultaneously and in the same direction: effect. the pooling of risks w i l l have no elimination or diversification In other words,investment risks are not insurable risks; a single event, such as a dramatic decline in stock prices, w i l l simultaneously render the insurance company liable under the asset guarantee of a l l its maturing contracts. It is clearly no longer sufficient simply to compute the expected value of the asset guarantee and to discount that back to the present. This chapter is concerned exclusively with the pricing of the asset value guarantee on equity linked l i f e insurance policies and in Chapter 6 some numerical examples are presented. Chapter 7 w i l l be concerned with the derivation of appropriate investment policies for insurance companies to enable them to hedge against the investment risk associated with the provision of such guarantees. As a preliminary, the approach followed in this chapter w i l l be motivated by a brief comparison of the actuary's and the economist's approaches to the pricing of financial contracts. Then a brief survey of some of the actuarial literature on asset guaranteed equity linked l i f e insurance contracts w i l l be presented. Following t h i s , i t w i l l be shown that the benefit payable under an asset guaranteed equity linked policy can be represented as the combination of the payment of a predetermined cash benefit and the payment of an immediately exercisable c a l l option. It w i l l then be shown how the market equilibrium model of option valuation developed by Black-Scholes [ 3] and extended by Merton [32], detailed in 91 Chapter 2, can be used to determine the equilibrium value of the option: the value of the insurance contract is then the sum of the values of the fixed payment and of the c a l l option. Solution procedures for the single premium contract, the continuous premium contract, and the periodic premium contract are developed and discussed. 5.3 Alternative Approaches to Pricing A stylized characterization of the traditional actuarial approach to pricing an insurance contract is that the expected costs of the contract are f i r s t calculated and then discounted at an assumed interest rate to arrive at a present value of the costs. Risk considerations usually do not enter directly, but may be introduced in a judgmental fashion by relating the premium loading to the higher moments of the cost distribution and by adjusting the appropriate level of reserves. Such a procedure may work well as long as the risks considered are largely independent,so that the law of large numbers and the principle of diversification may be expected to apply. It is a wholly inappropriate method, however,for allowing for investment risk,both because the investment risks of a l l contracts are quite obviously not independent and because the securities markets themselves determine a relationship between risk and return which should be reflected in the insurance company's pricing of investment risk. The economist's instinct,on the other hand,is to attempt to determine equilibrium prices for financial contracts of which insurance contracts are but a special case. Equilibrium prices are those which may be expected to obtain in a perfectly competitive and frictionless securities markets. As such,they ignore important matters such as transaction costs and sales costs. 92 Nevertheless, the computation of such equilibrium p r i c e s , i t i s contended, provides a normative basis f o r p r i c i n g contracts. In h i s search f o r equilibrium prices the economist i s guided by the arbitragi p r i n c i p l e that i n a competitive market i n equilibrium there are no r i s k l e s s p r o f i t : to be made. This p r i n c i p l e has two important implications which w i l l be employed l a t e r : f i r s t , a composite contract must s e l l at a price equal to the sum of the prices of i t s constituent p r i m i t i v e contracts; and secondly, i t i s not possible to earn a p o s i t i v e r i s k l e s s return with zero net investment. The f i r s t implication enables us to assert that the p r i c e of the asset guaranteed contract i s equal to the sum of the prices of i t s constituent contracts, i . e . a conventional insurance contract and a contract the benefit of which i s an immediately exercisable c a l l option. The second implication enables us to follow Black-Scholes and Merton i n deriving a d i f f e r e n t i a l equation governing the value of the c a l l option. 5.4 B r i e f Survey of the A c t u a r i a l L i t e r a t u r e In recent years there has been a number of papers i n the a c t u a r i a l l i t e r a t u r e concerning the problem of equity linked l i f e insurance p o l i c i e s . As Turner [45] points out, many actuaries i n the United States and Canada are now deeply involved i n the development of equity-based l i f e insurance products. The problems with equity linked products which provide guarantees on death and maturity, as indicated by Leckie [22l» are the c a l c u l a t i o n of premiums and the determination of suitable reserves. "Traditionally actuaries have always considered mean values i n t h e i r assumption, but with these types of guarantee, i t i s necessary to consider the whole range of possible outcomes of investment performance and attach probabilities to each value." He continues "It is quite safe to say that not nearly enough theoretical work has been done on the nature of these guarantees, which provide some highly intriguing problems." In an important actuarial paper Turner [44] argues that under most equity-based products the contract-holder assumes the f u l l investment risk having no guarantee as to the asset value of his contract and that "it is both reasonable and appropriate for a l i f e insurance company to offer an additional assurance under such products whereby the investment risk is assumed by the company." Turner's paper considers only the case in which a minimum asset value is guaranteed at the end of a specified investment period, for example, maturity of the contract. Using the historical distribution of rates of returns, he develops a general simulation model which is used to estimate the probability distribution of the insurance company's l i a b i l i t y under the guarantee. Di Paolo [ 1 0 ] also employs a Monte Carlo technique to simulate future stock behavior which is then used to evaluate the adequacy of the investment risk premium"' for a certain type of equity-based endowment that guarantees a minimum death and maturity value. Di Paolo considers the case where the sum I n the actuarial literature the expression "risk premium" denotes the additional amount (percentage) that insurance companies charge to the holders of equity-linked l i f e insurance policies when a minimum guarantee is given. In the terminology of finance i t could be called "put premium." Both expressions - risk premium and put premium - w i l l be used interchangeably in what follows. The concept w i l l be defined more rigorously later on. 3 . 94 of the risk premium and investment component is constant and looks at the risk of insolvency under various assumptions. As mentioned above, Turner and Di Paolo use the probability distribution function of the rate of return on common shares, consistent with the random walk hypothesis, in a simulation exercise to provide a probability d i s t r i b ution of future investment results and of the amounts needed under the maturity guarantee. Then a risk premium can be calculated from the amounts required and their probabilities. Finally, Kahn [18] shows how simulation can be used to develop financial projections for variable l i f e insurance with a guaranteed minimum death benefit, but he complements this approach by showing how recourse to mathematical analyses based on risk theory can be used to see the impact of stock price changes more clearly. He assumed that the rate of return on common stock is log-normally distributed and then derives an expression for the expected value of the excess of the minimum guaranteed benefit under the option over the value of this benefit in the absence of a guarantee. Kahn's approach is very similar to the one used by Sprenkle [39] to estimate the expected value of a warrant at maturity and as such has the same short-comings, i . e . , the expected value of the option is a function of the expected rate of return on the common stock which is not known. Further, the problem considered is restricted to determin- ing the cost of a guaranteed minimum death benefit under a single premium variable l i f e insurance policy. Kahn's paper i s , nevertheless, the f i r s t attempt in the actuarial literature to find an analytic solution to the problem of evaluating asset value guarantees on equity linked l i f e insurance policies. It would appear, however, that no completely satisfactory solution to the 95 problem has yet been found. 5.5. The Guarantee as an Option Consider a policy the benefit of which paid at time t consists of the greater of the market value of some reference portfolio x(t) or a guaranteed sum g(t). Note that the guarantee is implicitly allowed to be a function of time. Then the benefit at time t, b(t), can be written as (5.1) b(t) = max {x(t), g(t)} \ or (5.2) b(t) = g(t) + max {(x(t) - g(t)), 0} Equation (5.2) re-expresses the benefit as the sum of the sure amount g(t) plus the amount by which the value of the reference portfolio exceeds the guarantee. This latter amount may be regarded as the value at time t of an immediately exercisable c a l l option which permits the holder to purchase the reference portfolio for the pre-determined amount g(t). If x(t) < g(t),the option w i l l not be exercised and w i l l have zero value, while i f x(t) > g(t),the option w i l l be exercised and its exercise value is x(t) - g(t). Thus,the present value of the benefits is the present value of the guarantee plus the value of a c a l l option on the reference portfolio. The present value of the guarantee can be easily obtained by discounting its value at maturity and the value of the c a l l option can be obtained by employing the procedures developed in preceding chapters, as w i l l be shown in subsequent sections. 96 Denote the value at time T of a c a l l option exercisable at time t at the exercise price g(t) by W(X(T), t-x, g(t)). Note that the value of the option has been written as a function of the current value of the reference portfolio, of the remaining l i f e or time to expiration of the option,(t-x),and . of the guarantee or exercise price g(t). At expiration the value of the option can be written as (5.3) w(x(t), 0, g(t)) = max {(x(t) - g(t)), 0} Equation (5.3) expresses the fact that at expiration the c a l l option is worth the greater of its value i f exercised, x(t) - g(t), or zero. Looking at the problem from a different angle the benefit as defined by (5.1) can also be written as (5.4) b(t) = x(t) + max{(g(t) - x(t)), 0} Equation (5.4) re-expresses the benefit as the sum of the value of the reference portfolio at time t, x(t), plus the amount by which the guarantee exceeds the value of the reference portfolio. This latter amount may be regarded as the value at time t of an immediately exercisable put option which permits the holder to s e l l the reference portfolio for the pre-determined amount g(t). If x(t) 2. g(t),the put option w i l l not be exercised and w i l l have zero value, while i f x(t) < g(t),the put w i l l be exercised and its exercise value is g(t) - x(t). Thus,the present value of the benefits can also be viewed as the present value of the reference portfolio plus the value of a put option on the reference portfolio. The present value of the future market value of the reference port- folio must be equal to the present value of the investment contributions, since this is the price that must be paid for that future value. The value of the put option can then be computed by equating the present value of the benefits obtained as the sum of the present value of the guarantees and the c a l l option, mentioned above, and the present value of the benefits also viewed as the sum of the present value of the reference portfolio and the put option. 97 Denote the value at time x of a put option exercisable at time t with exercise p r i c e g(t) by p(x(x), t-x, g(t)). Again the value of the option has been written as a function of the current value of the reference portf o l i o , of the time to expiration and of the guarantee or exercise p r i c e . At expiration the value of the put option can be written as (5.5) p ( x ( t ) , 0, g(t)) Equation = max (5.5) expresses [(g(t) - x ( t ) ) , 0] the,fact that at expiration the put option i s worth the greater of i t s value i f exercised, g(t) - x ( t ) , or zero. From the above discussion i t can be seen that the present value of the benefits at the time of issue of the p o l i c y , P V ( b ( t ) ) , can be expressed D as the sum of the present value of the guarantee and the value of a c a l l option: (5.6) PV (b(t)) Q = g(t)e" r t + w(x(0), t , g(t)) and also as the sum of the present value of the reference p o r t f o l i o , P V ( x ( t ) ) , Q and the value of a put option: (5.7) PV (b(t)) 0 = P V ( x ( t ) ) + p(x(0), t, g(t)) Q Notice that i n equation (5.6) continuous discounting has been used to be consistent with the option p r i c i n g model employed i n subsequent sections. From the viewpoint of l i f e insurance companies the value of the put option i s the most relevant f i g u r e ,because i t represents the t o t a l present value of the amounts that the insurance company must charge for giving the guarantee. The value of the put option can be obtained from (5.6) and (5.8) p(x(0), t, g(t)) =' w(x(0), t, g(t))+ g ( t ) e " r t (5.7): - PV (x(t)) Q If the value of the c a l l option i s known,the value of the put option can be obtained from equation (5.8). The problem i s then to find the value . 98 of the c a l l option. The following section deals with this problem. It should be emphasized that the analyses developed in this section apply when the exercise date (death or maturity) is known. It has been assumed that the expiration date (because of death or maturity), t, is a known parameter. The discussion has been directed to price the "investment risk" assumed by the insurance company for giving the guarantee, without any consideration yet of the "mortality risk" involved in the l i f e insurance policy. The mortality risk is introduced at a later stage. Accordingly, the problem of pricing an equity-linked l i f e insurance policy with asset value guarantee (1) w i l l be solved in three stages: The valuation of the c a l l option component of an insurance contract with known exercise date (death or maturity). This is what has been called the investment risk assumed by the l i f e insurance company. (2) The estimation of the probabilities of death during each year of the term of the contract. This is the actuarial problem related to the mortality risk assumed by the l i f e insurance company. (3) The calculation of the premiums to be paid by the policy-holder when both investment risk and mortality risk are considered. These problems w i l l be treated in turn in subsequent sections. 5.6 Valuing the Option Component of an Insurance Contract The equilibrium option valuation model presented in Chapter 2 w i l l be used to price the c a l l option component of the l i f e insurance contract. Therefore, the same basic assumptions made to derive that model are also implicit here. Even though l i f e insurance policies are not traded in the secondary market as some options are, the existence of a large number of l i f e insurance companies competing 99 f o r these p r o d u c t s ensures produce the same e f f e c t t h a t a r b i t r a g e p r o f i t s are competed away and, as a c o m p e t i t i v e s e c u r i t i e s market does. the a r b i t r a g e arguments p r e s e n t e d i n Chapter o p t i o n a r e assumed to a p p l y The guarantee, first Thus, a l l 2 to d e r i v e the v a l u e of an here. Three s e p a r a t e types of c o n t r a c t must be (1) hence, considered: type i s of the s i n g l e premium c o n t r a c t w i t h an a s s e t v a l u e i n which the whole investment at a s i n g l e p o i n t i n time. In t h i s case i n the r e f e r e n c e p o r t f o l i o i s made the v a l u e of the o p t i o n p r e c i s e l y to the v a l u e o f a c a l l o p t i o n on a n o n - d i v i d e n d corresponds paying s e c u r i t y f o r which a c l o s e d form s o l u t i o n has been d e r i v e d by B l a c k - S c h o l e s and Merton, as shown by e q u a t i o n (2) The second (2.31) i n Chapter 2. type o f c o n t r a c t i s t h a t of t h e c o n s t a n t c o n t i n u o u s where e q u a l p e r i o d i c premium and e q u a l p e r i o d i c investment p o r t f o l i o areapproximated by assuming a c o n s t a n t continuous i n the r e f e r e n c e p o r t f o l i o . behaviour The i n the r e f e r e n c e r a t e of d i f f e r e n t i a l equation governing o f t h i s o p t i o n corresponds premium to t h a t d e r i v e d by Merton investment the p r i c e [32] f o r an o p t i o n on a d i v i d e n d p a y i n g s t o c k when the d i v i d e n d i s n e g a t i v e , but s l i g h t l y d i f f e r e n t boundary c o n d i t i o n s to take i n t o account with the f a c t t h a t even i f the v a l u e of the r e f e r e n c e p o r t f o l i o goes to zero a t a c e r t a i n p o i n t i n time, i t s t a r t s growing immediately premium payments. i t may While no be s o l v e d s i m p l y and method d i s c u s s e d i n Chapter (3) The third a f t e r w a r d s w i t h the a d d i t i o n of c l o s e d form s o l u t i o n e x i s t s f o r t h i s to a h i g h degree of a c c u r a c y by the new equation, numerical 3. type o f c o n t r a c t i s t h a t of the e q u a l p e r i o d i c premium and equal p e r i o d i c investment of c o n s t a n t c o n t i n u o u s i n the r e f e r e n c e p o r t f o l i O j b u t where the r a t e of investment assumption i n the r e f e r e n c e p o r t f o l i o can be made,because the time between premium payments i s l a r g e (one y e a r f o r not 100 example). The d i f f e r e n t i a l equation governing o p t i o n between premium payments c o r r e s p o n d s for a non-dividend in security. the reference portfolio) to r e f l e c t is very similar dividend. very A t each premium payment (and investment i n the reference portfolio. The situation t o t h a t o f v a l u i n g an o p t i o n o f w h i c h u n d e r l y i n g s t o c k d i v i d e n d s where t h e premium r e p r e s e n t s a n e g a t i v e to the procedure The S i n g l e Premium For this t o t h a t d e r i v e d by B l a c k - S c h o l e s T h i s e q u a t i o n may a l s o b e s o l v e d b y n u m e r i c a l similar of d a t e new b o u n d a r y c o n d i t i o n s h a v e t o b e i m p o s e d the increased position pays d i s c r e t e 5.7 paying the price behaviour this developed i n Chapter methods 4. Contract c o n t r a c t the whole investment made w h e n t h e c o n t r a c t i s p u r c h a s e d ; to t h e v a l u e o f a c a l l i n the reference p o r t f o l i o i s and t h e v a l u e o f t h e o p t i o n o p t i o n on a non-dividend paying security corresponds given by e q u a t i o n (2.31). In the notation of this can be e x p r e s s e d (5.9) w(x(x), where d, 1 d 2 = = N(d) = as t-x, g(t)) = [ £n ^ Q - + l ~ ^ ~ a r^r- • 2ir t r -<» x(r) N(d ) x g(t)e" (i) i ty i e l d s actual price 2 x e ^ A d\ A s was shown i n C h a p t e r i s that N(d ) 2 N(d) i s the cumulative normal d i s t r i b u t i o n (5.9) r ( t _ T ) (r+^a )(t-x)]/oVt-x g(t) d c h a p t e r , t h e v a l u e o f t h e o p t i o n a t any time x 2, the s i g n i f i c a n c e function. of the valuation the market e q u i l i b r i u m v a l u e o f the o p t i o n such i s n o t g i v e n by (5.9), then riskless equation that i f the a r b i t r a g e p r o f i t s may b e 101 made (the basic assumptions of the model may particular: (ii) be r e c a l l e d , i n no transaction or s e l l i n g costs and no taxes), i t expresses the value of the option i n terms of the observable current p r i c e of the reference p o r t f o l i o , x, the rate of i n t e r e s t which i s known, the time to expiration of the option and the variance rate of return on the reference p o r t f o l i o . too may be estimated by examining the past v a r i a b i l i t y of the return on reference 5.8 Only the variance rate i s unobservable and i t the portfolio. The Constant Continuous Premium Contract The e s s e n t i a l difference between the multiple premium contract and the single premium contract discussed above,is that with the former the change i n value of the reference p o r t f o l i o depends not only on the rate of return on that p o r t f o l i o , b u t also on the rate of addition to the p o r t f o l i o through investment. The analysis here d i f f e r s from that given above i n the stochastic d i f f e r e n t i a l equation f o r x. The following a d d i t i o n a l notation i s introduced: SCO the value of one unit of the p o r t f o l i o at time T . N(x) the number of units owned by the p o l i c y holder at time T . D the constant instantaneous rate of investment i n the reference p o r t f o l i o . reference Then the value of the reference p o r t f o l i o x ( x ) (5.10) X(T) = i s given by: N(T) • S(x) and by Ito's Lemma**: \ Given this functional r e l a t i o n s h i p , the same expression would be obtained using ordinary calculus. 102 dx = NdS + SdN or ( 5 ^' u dx ) l L ) . x dS S dN N The value of a single unit of the reference p o r t f o l i o i s assumed to follow a stochastic d i f f e r e n t i a l equation analogous to equation (2.13), which,in the notation used i n this chapter,becomes : 7 (5.12) 4f = adx + adz The d i f f e r e n t i a l increase i n the number of units held on behalf of the policyholder i s given by: d N " f = d T ^ d T since D i s the constant instantaneous rate of investment. Then: (5.13) f - 2* Substituting (5.13) and (5.12) into (5.11), the change i n value of the reference p o r t f o l i o i s given by (5.14) — X = (o4 -) dr + adz X Comparing equation (5.14) with equation (2.33) i t can e a s i l y be seen that the stochastic d i f f e r e n t i a l equation f o r the instantaneous rate of return on the reference p o r t f o l i o corresponds to that derived by Merton for the instantaneous rate of return on a stock paying a continuous dividend, when the dividend i s negative . I n this chapter the variable "time" has been denoted by x and the "maturity date," which i s also a variable here, has been denoted by t. Thus, time to maturity has been denoted by T = t-r . For a fixed maturity date: dT = -dx . 7 103 Consequently the equilibrium value of the option, which i s a function of the value of the reference p o r t f o l i o , time to expiration and the exercise price, can be solved i n a manner similar continuous to the valuation of an option on a dividend paying stock developed i n Chapter 2. The option value must then s a t i s f y the p a r t i a l d i f f e r e n t i a l equation (5.15) 2 2 ho x w + (rx+D)w X X - wr - w = 0 JL X subject to the boundary conditions: (5.16) w(x,0,g) = max[(x-g), 0] (5.17) Dw (0,T,g) - w(0,T,g)r - w (0,T,g) (5.18) w (°°,T,g) x T = x = 0 1 The subscript x denotes p a r t i a l derivatives with respect to the value of the reference p o r t f o l i o and the subscript T denotes p a r t i a l d e r i v a t i v e s with respect to time to maturity between equation (T = t - t ) . Notice that the only d i f f e r e n c e (5.15) and (2.40) i s the sign of D (here the continuous premium payment invested i n the reference p o r t f o l i o , there the continuous dividend payment). Thus, by arguments i d e n t i c a l to those presented i n Chapter 2 the value of the c a l l option must follow (5.15) i f arbitrage p o s s i b i l i t i e s are to be eliminated. (5.15) combined with the boundary conditions governing the value of the option at expiration and at values of the reference p o r t f o l i o equal to zero and i n f i n i t y , determine the equilibrium value of the option at any time p r i o r to maturity. Boundary condition (5.16) i s the same as (5.3) and determines the value of the option at expiration. The boundary condition (5.17) f o r zero values of the reference p o r t f o l i o can be obtained d i r e c t l y by setting x=0 i n p a r t i a l d i f f e r e n t i a l 104 equation (5.15). Note that in this case the value of the option for a value of the reference portfolio equal to zero,is not zero. This is because even i f the value of the reference portfolio goes to zero at a certain point in time, i t starts growing immediately with the additional new premium payments (this was not the case for the option on a continuous dividend paying stock). Boundary condition (5.18) for infinite values of the reference is identical to boundary condition (3.24) for the value of an option on a continuous dividend paying stock. This boundary condition is s t i l l v a l i d , because the value of the c a l l option: (1) is a convex function of the value of the reference portfolio; (2) is bounded from above by the present value of the premiums invested in the reference portfolio (current value of the reference portfolio plus the present value of future premiums to be paid) since i t w i l l never pay to buy the option at a higher price than the present value of the reference portfolio; and (3) is bounded from below by its exercise value since i t would never pay to exercise the option, even i f i t were possible. While no explicit closed form solution for (5.15) subject to boundary conditions (5.16), (5.17) and (5.18) exists, i t can be solved numerically using the procedure developed in Chapter 3 to find the equilibrium value of an option on a continuous dividend paying stock. The only change that must be introduced to the procedure described in Chapter 3 is the new boundary 9 condition (5.17) . 5.9 The Periodic Premium Contract From the viewpoint of practical applications the periodic premium 8 See Merton P2 ] . 9 The solution algorithm with the new boundary condition is shown in Appendix C and the computer program is shown in Appendix D. 105 contract i s the most important one. Premiums are commonly paid monthly or yearly for this type of contract. Accordingly, i n the numerical examples presented i n Chapter 6 we w i l l concentrate on this case. As i n the case of the continuous premium contract, i n the periodic premium case the change i n value of the reference p o r t f o l i o depends not only on the rate of return on that p o r t f o l i o , but also on the rate of addition to the p o r t f o l i o through periodic investment. The value of the option, W(X(T) , t-x, g(t)) or w(x, t-.-r, g) f o r s i m p l i c i t y , w i l l also obey the basic Black-Scholes p a r t i a l d i f f e r e n t i a l equation. (5.19) 2 2 ho x w + rxw xx x - wr - w„ T = 0 But the boundary condition w i l l be d i f f e r e n t . payment date new In p a r t i c u l a r at each premium boundary conditions w i l l have to be established to r e f l e c t the fact that a d d i t i o n a l investments have been made i n the reference portfolio. The problem i s very similar to that of valuing an option where the underlying stock pays d i s c r e t e dividends, as presented by Merton [32] and Black [ 5] and treated i n d e t a i l i n Chapter 4, but i n this case the' premium can be thought of as a negative dividend. The fact that the premium payment i s an addition to the reference p o r t f o l i o determines some important changes i n the boundary conditions with respect to the ones developed i n Chapter 4 for the discrete dividend case. before maturity Also here i t w i l l not pay to exercise the option even i f i t were possible, because the reference p o r t f o l i o i s growing faster due to the additions of premium payments. The distinction between an American and a European option i s not relevant,because both w i l l be exercised at maturity only. This w i l l be proved l a t e r on. But both options, the one i n the discrete dividend case and the one i n the periodic premium case, obey the same p a r t i a l d i f f e r e n t i a l equation (5.19). 106 The value of the option w i l l always be governed by equation (5.19). The solution to this equation, however, w i l l depend on the boundary conditions imposed and these boundary conditions change for different periods between premium payment dates. The numerical solution of the problem requires a dynamic programming type of procedure , analogous to the one used in Chapter 4, which starts from the expiration date of the option where the boundary condition is known and then works the solution back to the desired date. After the last premium payment the boundary condition is given by (5.20) w(x, 0, g) = max[x-g, 0] At expiration the value of the option is the greater of (x-g) or zero. In any other period between premium payments one of the boundary conditions w i l l be given by the value of the option just before the next premium payment date. Let ^w be the value of the option just before the payment of the premium and w be the value of the option just after the payment of the premium, and let T^ be the time to expiration at the premium payment date and D the amount of premium invested in the reference portfolio. As discussed in Chapter 4, i t w i l l never pay to exercise the option in between premium payments dates (dividend dates in Chapter 4) where equation (5.19) governs the value of the option. It w i l l be shown that in this case i t also never pays to exercise the option at premium payment dates even i f this were possible. Suppose that i t could be exercised, then at any premium payment date the value of the option just before the payment of the premium is given by: (5.21) b w(x, T , g) = max[ w(x+D, T , g), (x-g), 0] a But in this case the value of the option after the payment of the premium, I w(x+D), Tp, g), w i l l always be greater than the exercise value before the a 107 payment of the premium, (x-g), because (5.22) w (x+D, T , g) > a p w (x, T , g) a p and (5.23) w (x, T , g) > max[(x-g), 0] a p Therefore, (5.24) w (x+D, T , g) > max[(x-g), 0] a p Substituting (5.24) into (5.21) the boundary condition at any premium payment date i s obtained: (5.25) w (x, T , g) = v (x+D, T , g) b From (5.24) i t can also be seen that i t w i l l never pay to exercise the option at a premium payment date and hence at any other point i n time before maturity. Thus, the value of an American option i s equal to i t s European counterpart. The boundary condition f o r zero values of the reference p o r t f o l i o w i l l also d i f f e r from the boundary condition f o r zero values of the stock obtained for the discrete dividend s i t u a t i o n discussed i n Chapter 4. In this case, as i n the continuous premium contract discussed i n the preceding section, the value of the option i s not necessarily equal to zero when the value of the reference p o r t f o l i o i s equal to zero. I t i s necessary to consider the effect on the value of the option of future premium payments. This boundary condition i s obtained by l e t t i n g x = 0 i n (5.19) : (5.26) wr + w T = 0 or (5.27) w (0, T, g) T = -rw(0, T, g) 108 The value of the option f o r the value of the reference p o r t f o l i o equal to zero decreases i n value with time to maturity at the r i s k l e s s interest r a t e . ^ 1 The,economic j u s t i f i c a t i o n f o r this "perverse" behaviour of the value of the option at the boundary i s the following: the value of the option f o r a value of the reference p o r t f o l i o equals to zero j u s t before any premium payment i s given by boundary condition (5.25): (5.28) w b (0, T , g) p = w a (D, T , g) > 0 I f the value of the reference p o r t f o l i o goes to zero at any point i n time between premium payments, i t s value w i l l remain zero u n t i l the next premium payment(assuming that the premium i s then invested i n another reference p o r t f o l i o with the same variance r a t e ) . But at the next premium payment the value of the option i s given by (5.28). Therefore, the value of the option at the boundary (value of reference p o r t f o l i o equal to zero) can be obtained by discounting (5.28) at the r i s k l e s s rate to the point i n time being considered. This i s exactly what i s implied by (5.26) or (5.27). Boundary condition (5.26) or (5.27) contradict usual condition W-j- > 0. This apparent contradiction arises from the nature of p a r t i a l d i f f e r e n t i a l equation (5.19) or (2.40). One of the basic assumptions underlying these equations i s that the return on the stock or reference p o r t f o l i o i s log-normally d i s t r i b u t e d . This d i s t r i b u t i o n implies that i f at the current point i n time the value of the stock (or reference p o r t f o l i o ) i s d i f f e r e n t from zero, the p r o b a b i l i t y of i t becoming zero i n the future i s zero. Therefore, the d i f f e r e n t i a l equation i s r e a l l y not defined at the boundary (for x = 0 or S = 0). To obtain t h e i r closed form solution Black-Scholes do not require this boundary condition and their formula, which gives the value of the option as a function of the current price of the stock, s a t i s f i e s the condition that W-K) when S-K). The same boundary condition (W(0,T) = 0) has economic j u s t i f i c a t i o n (see Merton [32]) and so they are used without further comment i n Chapters 3 and 4 when evaluating options on dividend paying stocks. When numerical methods are employed, however, the solution procedure s t a r t s from the maturity date (where the value of the option i s known for d i f f e r e n t stock prices) and to proceed with the stepwise procedure, values of the option for zero stock 109 For the period after the last premium payment, boundary condition (5.27) is equivalent to (5.29) w(0, T, g) = 0 w(0, 0, g) =0 because (5.30) Notice that (5.29) was the only boundary condition for the stock price equal to zero in the discrete dividend case. It seems appropriate to summarize the results obtained: the value of the c a l l option in the periodic premium contract is governed by the partial differential equation: (5.19) %ax z w + rxw - wr - w T xx x = 0 J- subject to the boundary conditions: (1) For the last period (after the last premium payment): (5.20) w(x, 0, g) (5.29) w(0, T, g) (5.18) w (oo, x = max [x-g, 0] =0 T, g) = 1 (Continued) 'prices must be assumed, even though the equation is not defined at this boundary. In the options discussed in this chapter, where the value of the option may be positive even for zero values of the reference portfolio, the boundary can pose a theoretical problem due to the "perverse behaviour," i . e . Wf < 0, implied by (5.26). Fortunately, from the viewpoint of practical applications the problem does not exist,because the numerical solution for the value of the option is very insensitive to small changes in this boundary condition. To test this, three different boundary conditions were used in a specific numerical example: (1) w^ = -rw, (2) = 0 and (3) w^ = rw which in the notation of Chapter 3 can be written as (1) W j = W j_^(l-rk), W ,j = W j _ i and (3) W i - W J _ J (1+rk). The results'obtained for the value of the option after 15 years (with a time step of one month) was exactly the same up to the fifth decimal place in the three cases. The analysis given in this footnote also applies to the continuous premium contract discussed in the preceding section. Q) 0j Q 0 J Qj 0 110 (2) For a l l the other periods between premium payments: (5.25) w (x, T , g) b p (5.27) w x (0, T, g) (5.18) w x (~, T, g) = w (x+D, T , g) a = p -rw(0, T, g) = 1 These r e s u l t s are also shown graphically on Figure 5.1. The value of the c a l l option for a value of the reference p o r t f o l i o equal to zero j u s t before the f i r s t premium payment, ^w (0, t , g), i s the solution to our problem. The value of the reference p o r t f o l i o i s equal to zero,because no monies have been invested i n i t yet. No closed form s o l u t i o n exists f o r this equation subject to i t s boundary conditions, but i t may be solved by modifying the numerical procedure developed i n Chapter 4 to take into account the new boundary conditions. This procedure involves a dynamic programming type of approach s t a r t i n g from the maturity of the option and going back u n t i l the moment j u s t before the f i r s t premium payment (the moment of the signature of the contract). As indicated above, the aim of t h i s discussion i s to f i n d the value of the option at that point i n time f o r a value of the reference portf o l i o equal to zero. The solution algorithm f o r this problem i s developed i n Chapter 6 and the computer- program i s shown i n Appendix E. 5.10 The C a l l Option, The Put Option and the P r i c i n g of Investment Risk The periodic premium contract i s the most general and important case of equity linked l i f e insurance policy with asset value guarantee, so that hereafter the discussion w i l l concentrate on t h i s . I The r e s u l t s obtained Ill Figure 5.1: P a r t i a l D i f f e r e n t i a l Equation and Boundary Conditions f o r the Periodic Premium Contract %a x 2 Time to Expiration w 2 xx + rxw x - wr - w T = 0 first premium payment date w(x,T,g ) w(x,T ,g) = any . premium D payment date last premium payment date p D_ o II w(x+D,T ,g) b w(x,T p 1>g 00 H ) w(x+D,T ) 1>g oo H A O Maturity date a 0 w(x,0,g) = max[x-g,0] Value of the Reference P o r t f o l i o . 112 can be extended very easily to the other two types of contracts: the single premium contract and the continuous premium contract, which are simpler in nature and have fewer practical applications. It is of interest to find the equilibrium value of the option at the moment the l i f e insurance policy is issued,because that w i l l be the time when the premium for the contract is established. Thus, one particular value of the option price matrix obtained by the numerical solution is of \ special interest: the value of the option just before the f i r s t premium payment for a value of the reference portfolio equal to zero, that Is b w(0, t-x, g(t)). Assuming that the policy is issued at x= 0 and simplifying the notation without any risk of ambiguity, this value can be written as w(t). w(t) i s , therefore, the present value of a c a l l option that w i l l pay at maturity the greater of x(t) - g(t) or zero. Assuming death is certain i n t years the present value of the guaranteed amount g(t), is given in continuous discounting, in order to be consistent with the option pricing model, by: (5.31) PV (g(t)) = Q g(t)e~ rt Thus, the total equilibrium present value of the benefits or present value of the contract can be written as (5.32) PV (b(t)) Q = g(t)e" rt + w(t) Equation (5.32) is identical with equation (5.6) obtained in Section 5.5. The present value of the contract is equal to the present value of the guarantee plus the value of the c a l l option. In Section 5.5 i t was also shown that the total equilibrium present value of the contract can be expressed as the sum of the present value of the reference portfolio and the value of a put option. The present value 113 of the future market value of the reference p o r t f o l i o , P V ( x ( t ) ) , must be equal Q to the present value of the contributions to the reference p o r t f o l i o since this i s the price that must be paid f o r that future value: (5.33) PV (x(t)) t-1 I D e~ = Q r k k=0 I f the value of the put option corresponding to ^ ( 0 , t, g) , i . e . , ^p(0, t, g), i s written as p ( t ) , the present value of the contract can also be written as: (5.34) PV (b(t)) = Q t-1 £De~ k=0 r K +p(t) Combining (5.32) and (5.34) the value of the put option can be obtained: (5.35) p(t) = g(t)e r t + w(t) - t-1 . £D e" ; r k=0 As mentioned e a r l i e r , the value of the put option represents the cost to the insurance company of giving the guarantee. p r i c i n g of the investment I t i s the equilibrium r i s k assumed by the insurance company. I t i s of i n t e r e s t to c a l c u l a t e the p e r i o d i c premium f o r the put, y ( t ) , that i s ,the periodic amount that should be added to the premium payments to compensate the insurance company f o r assuming the investment benefits are s t i l l assumed payable with certainty risk. at time t . The The present value of these periodic premiums f o r the put i s equal to the present value of the put option, that i s : t-1 _ . (5.36) p(t) = y ( t ) I e k=0 r k or, solving f o r the value of the periodic premium f o r the put: (5.37) y(t) = _P r -rk ( t ) t L k=0 e 114 A n a l o g o u s l y , the p e r i o d i c premium f o r the c a l l , z ( t ) , can be defined as (5.38) w(t) = z(t) t-1 _ I e r k=0 The t o t a l p e r i o d i c premium, v ( t ) , can be w r i t t e n a s : (5.39) v(t) = P Vt-1 ( b ( t ),) V -rk n I e k=0 or, equally as: (5.40) v(t) = D + y(t) The t o t a l p e r i o d i c premium has two components: invested i n the r e f e r e n c e (1) the amount p o r t f o l i o , D, and (2) the amount charged f o r assuming the investment r i s k , y ( t ) . Finally, s u b s t i t u t i n g (5.37) and (5.38) i n t o (5.35) and s i m p l i f y i n g gives: (5.41) y(t) + D = z ( t ) + g(t) — -rt f — Y -rk k=0 The c o n t i n u o u s time v e r s i o n of (5.41) can be w r i t t e n a s (5.42) y(t) + D = z(t) + g(t) r 1 1 ; -rt _ 1-e 6 r t il The computer program shown i n Appendix F c a l c u l a t e s the v a l u e c a l l option, the p r e s e n t value of the guarantee, the t o t a l p r e s e n t o f the value of The c o n t i n u o u s time v e r s i o n has the advantage o f b e i n g a d i f f e r e n t i a b l e f u n c t i o n of t . A study on the r e l a t i o n s h i p between the p e r i o d i c premium on the p u t , time to m a t u r i t y and the amount o f the guarantee (as a p r o p o r t i o n o f the investment component of the premia p a i d ) c o u l d be undertaken. 115 the contract ( i . e . , the c a l l option plus the PV of the guarantee), the present value of the reference p o r t f o l i o , the value of the put option, the t o t a l periodic premium, the p e r i o d i c premium f o r the put and the periodic premium f o r the c a l l . In the preceding sections the method f o r p r i c i n g the option component of an insurance contract with known exercise date (death or maturity) has been presented. In addition,to assume part of the investment r i s k derived from the option element of equity-linked l i f e insurance p o l i c i e s with asset value guarantee, insurance companies also bear the m o r t a l i t y r i s k derived from the hazard of death. The exercise date i s r e a l l y not known; i t i s also a stochastic v a r i a b l e with a c e r t a i n p r o b a b i l i t y d i s t r i b u t i o n . Subsequent sections deal with the problem of m o r t a l i t y r i s k . 12 5.11 The Measurement of Mortality A l i f e insurance p o l i c y c a l l s f o r the payment of c e r t a i n benefits, b(t), with s p e c i f i e d p r o b a b i l i t i e s . The determination of these p r o b a b i l i t i e s by the usual a c t u a r i a l methods w i l l now be considered. The p r o b a b i l i t y that a new l i f e , aged 0, w i l l survive to a t t a i n age x i s referred to as the s u r v i v a l function, S(x). S(x) i s a continuous function of x, defined on the i n t e r v a l 0 < x £ w, which decreases from the value S(0) = 1 to the value S(w) = 0. The designation of a terminal age smallest value of x f o r which S(x) vanishes, i s merely a convenient ing w, simplify- device. The s u r v i v a l function can be used to determine the expected number of survivors at any age x s t a r t i n g from any assumed group of newly-born l i v e s . 12 For a more thorough exposition of the topic see Jordan [17]. 116 Let Ix represent the number of survivors at each age x and dx represent deaths i n the year of age x to x+1. (5.43) Then £x = k • S(x) , where k i s a p o s i t i v e constant representing the s i z e of the i n i t i a l group. (5.44) dx = Ix - lx+l The value of k i s c a l l e d the radix of the mortality table. to the value of Z Q I t corresponds and i s generally taken to be some large i n t e g r a l number. The i n t e r p r e t a t i o n of Ix as a "number l i v i n g " or "number surviving" and of dx as a "number dying",is a convenient a i d i n v i s u a l i z i n g some of the r e l a t i o n s h i p s that follow. But i t should be borne i n mind that neither Ix nor dx has any absolute meaning, the values of both being dependent on the value of the radix chosen i n the construction of the table. The tabulated values of ix that appear i n mortality tables are usually f o r i n t e g r a l values of x only, although from d e f i n i t i o n (5.43) Hx i s a continuous function of x. A section of an hypothetical example of a mortality table i s shown i n Table 5.1 (the values i n this table are not r e a l i s t i c ones). TABLE 5.1 13 Section of an Hypothetical M o r t a l i t y Table Age x Ix dx 0 1 2 3 4 5 6 7 8 9 10 100,000 99,499 98,995 98,484 97,980 97,468 96,954 96,437 95,917 95,394 94,868 501 504 506 509 512 514 517 520 523 526 528 See Jordan [17] p. 8. 117 In practice the mortality tables are usually constructed on a purely empirical basis from statistical studies of mortality data. Probabilities of death and survival may be obtained from the I x and dx columns of the mortality table. The probability that a l i f e aged x w i l l survive to age x+n is denoted by px, and n ft x+n px = — — n IK (5.45) / r \ r r The probability that a l i f e aged x w i l l die within n years is denoted by qx, and n 5.46) qx n = 1- px n n = Jlx r The probability that a l i f e aged x w i l l survive for n years and die in the (n+l) , ,,,. tb n r ( 5 - n|« 4 7 ) x year is denoted by | q x , and dx+n ftx+n - ftx+n+1 = IT = ^ The probabilities of death and survival defined above w i l l be used in the pricing of asset value guarantees on equity linked l i f e insurance policies. 5.12 Mortality Risk Since the main objective of this chapter is to deal with the problem of investment risk, we have abstracted from mortality risk by assuming that a sufficient number of contracts are sold so that the amount of benefits paid per contract at time t is given by: (5.48) a(t) • b(t) where a(t) is the probability of the benefits under any one contract being payable at time t and reflects both mortality experience and the maturity 118 of the contract. Assume an i n d i v i d u a l aged x purchasing a p o l i c y which matures i n n years. Assume also, f o r s i m p l i c i t y , that i f the i n d i v i d u a l dies during year t,the benefits are payable at the end of year t; and that i f he dies during the l a s t year of the p o l i c y or does not die during the p o l i c y period,the benefits are payable at maturity. Using the p r o b a b i l i t i e s of death and s u r v i v a l defined i n the preceding section,the p r o b a b i l i t y of an i n d i v i d u a l aged x of dying during year t i s given by (5.49) a(t) = ^jqx = * X + t ~\~ ^ fort=l, ...,n-l and the p r o b a b i l i t y that an i n d i v i d u a l aged x w i l l survive u n t i l the l a s t year of the p o l i c y , that i s the p r o b a b i l i t y of the benefits being payable at maturity, i s given by / rr / > \ / \ (5.50) a(n) = The f i r s t &x+n-l - £x+n , £x+n + — term i n the r i g h t hand side of equation (5.50) i n d i c a t e s the p r o b a b i l i t y that he w i l l d i e during the l a s t year of the contract and the second term indicates the p r o b a b i l i t y of s u r v i v a l . Expression (5.50) can also be written as: (5.51) a(n) = ^px - ^ ± The values of a ( t ) w i l l depend on the mortality table considered appropriate f o r the c a l c u l a t i o n , on the age of the i n d i v i d u a l purchasing the l i f e insurance contract and on the term of the contract (and n a t u r a l l y also on other variables that determine the choice of the mortality table such as sex and race) and can be e a s i l y computed using equations (5.49) and (5.51) given 119 the age of the i n d i v i d u a l , the l i f e of the contract and the mortality table. Bearing appropriate this i n mind i t i s possible to combine the analysis presented i n this section with that developed e a r l i e r i n the chapter to a r r i v e at a value f o r the contract which takes both mortality and investment r i s k into account. 5.13 The Equilibrium Value of the Contract: and M o r t a l i t y Risk Investment Risk With the developments of the preceding sections i t i s now possible to combine investment r i s k with mortality r i s k to obtain the equilibrium value of an equity-linked l i f e insurance contract with minimum guarantee. The average contract c a l l s f o r the c e r t a i n to be paid (but uncertain i n amount ) stream of b e n e f i t s : (5.52) o(t) b(t) = a(t) g(t) + a(t) w(x, 0, g) (t=l, ..., n) Hence,the problem of valuing an asset guarantee contract reduces to the problems of valuing the sure stream of payments a(t) g(t) and of valuing the uncertain stream a(t) w(x, 0, g ) . The t o t a l present value of this average contract, V, can then be written as: (5.53) V = n I a(t) g(t) e " t=l n r t + £ a(t) w(t) t=l or equivalently as: (5.54) V = n I a(t) PV (b(t)) Q t=l The t o t a l present value of the average contract given by (5.53) and (5.54) i s a c t u a l l y the equilibrium present value of the contract where both investment r i s k and mortality r i s k have been taken into consideration. 120 The (5.55) average Y n £ t=l = periodic premium f o r the put, Y, can be expressed a(t) y(t) as: where y(t) i s the periodic premium f o r the put with known date of expiration t. The expected periodic premium f o r the put given by (5.55) as a percentage of the p e r i o d i c premium invested i n the reference p o r t f o l i o i s c a l l e d the " r i s k premium" i n the a c t u a r i a l l i t e r a t u r e . be c a l l e d also "put premium." (Y/D) Here i t s h a l l The r i s k premium or put premium i s the figure needed by l i f e insurance companies to p r i c e equity based l i f e p o l i c i e s with asset value guarantee. insurance I t represents the amount (as a percentage of the premium invested i n the reference p o r t f o l i o ) that the insurance company has to charge f o r giving the guarantee when both investment and m o r t a l i t y r i s k are taken into consideration. For a given pattern of guarantees g(t) the values of w(t) and y(t) are independent of mortality r i s k and can e a s i l y be tabulated. The values of a ( t ) , which represent mortality r i s k , are also r e a d i l y obtainable from the appropriate m o r t a l i t y table given the age of the purchaser of the p o l i c y and the duration of the contract. The computer program shown i n Appendix G computes the equilibrium present value of the contract and the r i s k premium. 5.14 Conclusions The need f o r an appropriate model for the determination of p r i c e s f o r equity linked l i f e insurance p o l i c i e s with asset value guarantee has long been apparent. See f o r example Leckie [22] and Turner This chapter has shown how [44]. the models and methods of Chapters 2, 3 and 4 can be applied to this problem to y i e l d normative rules for p r i c i n g such contracts. prices; Moreoever, the p r i c e s determined by these rules are equilibrium that i s , they are the prices which would p r e v a i l i n a p e r f e c t l y 121 competitive market and have the property that i f the insurance company were to charge them and were also to follow an investment policy determined by the model, i t w i l l bear no r i s k and make no p r o f i t or loss. Further consideration w i l l be given to this problem i s Chapter 7. The model w i l l be applied i n Chapter 6 to a number of d i f f e r e n t situations i n order to analyze the s e n s i t i v i t y of the r i s k premium to changes i n : (1) the variance of the rate of return on the reference p o r t f o l i o ; (2) the r i s k l e s s i n t e r e s t rate; (3) the nature of the guarantee; (4) the age of the purchaser at entry; (5) the term of the contract. and 122 . CHAPTER 6: THE EQUILIBRIUM PRICING OF ASSET VALUE GUARANTEES ON PERIODIC PREMIUM EQUITY-LINKED LIFE INSURANCE POLICIES: 6.1 APPLICATIONS Introduction A general model for the valuation of equity-linked l i f e insurance contracts with minimum death benefit and maturity developed i n the preceding chapter. benefit guarantees was The model i s based on the e a r l i e r work of Black and Scholes and further developed by Merton on the p r i c i n g of options, and as such has some of the same l i m i t a t i o n s . I t assumes perfect markets with no transaction costs, no s e l l i n g costs or other market imperfect- ions. option In other words, i f a l l the basic assumptions underlying the p r i c i n g model discussed i n d e t a i l i n Chapter 2 are v a l i d , the model described in Chapter 5 gives the equilibrium prices of equity-linked l i f e insurance contracts with asset value guarantees. They are equilibrium prices i n the sense that at these prices there are no r i s k l e s s p r o f i t s to be made, whereas at any other prices r i s k l e s s p r o f i t s can be made under these i d e a l market conditions. It i s the contention of t h i s study, however, that these equilibrium prices provide a normative basis f o r p r i c i n g equity based l i f e insurance contracts with minimum guarantee. In fact they represent the f i r s t approach to the valuation of such contracts with a sound t h e o r e t i c a l basis. However, without the numerical methods of s o l u t i o n developed i n Chapters 3 and k the t h e o r e t i c a l model i n i t s e l f would not be of much p r a c t i c a l importance. In essence, the use of a sound t h e o r e t i c a l model for the p r i c i n g of options and the development of numerical methods of solutions have permitted us to solve the problem of evaluating investment r i s k assumed by the 123 insurance company i n equity-linked l i f e insurance contracts with asset value guarantee. In this chapter, a f t e r developing the basic s o l u t i o n algorithm f o r the p e r i o d i c period contract, we find the equilibrium value of some of the common types of guarantees. of these values with regard In addition, we examine the s e n s i t i v i t y to some of the important underlying variables. The variables that influence the equilibrium value of the contract are: (i) the variance of the rate of return on the reference (ii) the r i s k l e s s i n t e r e s t rate; (iii) the nature of the minimum guarantee; (iv) the choice of mortality t a b l e ; (v) the age of the purchaser of the policy; and (vi) the term of the contract. portfolio; 1 Some v a r i a t i o n has been allowed i n a l l these variables i n the numerical examples presented i n t h i s chapter. The value of the c a l l option and,therefore the value of the l i f e insurance p o l i c y w i l l depend on the c h a r a c t e r i s t i c s of the reference p o r t f o l i o or unit t r u s t i n which the investment component of the premiums are invested. The variance rate of return of the reference p o r t f o l i o i s a key v a r i a b l e i n the valuation of the c a l l option. Thus, the choice by a l i f e insurance company of unit t r u s t or reference p o r t f o l i o ( v i z . , the variance r a t e ) , w i l l determine the l e v e l of equilibrium r i s k premiums. ^In p r a c t i c e the actual cost of the guarantee w i l l depend on the mortality experience, which may or may not be r e f l e c t e d i n the mortality table chosen. The difference between "ex-ante" and "ex-post" values applies to some of the other variables as w e l l . This point w i l l not, however, be pursued here. 124 The company should use an estimate of the variance of the rate of return of the reference p o r t f o l i o chosen i n a l l i t s c a l c u l a t i o n s and should also bear i n mind that the higher the variance the higher w i l l be the " r i s k premiums" that i t w i l l have to charge. with Canadian companies can expect that a w e l l d i v e r s i f i e d equity p o r t f o l i o the variance rate would approximate the Toronto Stock Exchange I n d u s t r i a l Index. examples, except one, Therefore, iy* a l l the numerical the variance used i s that of the Toronto Stock Exchange (TSE) I n d u s t r i a l Index with dividends reinvested for the period September to August 1973. 1968 In the exception a larger variance i s used to show the influence of this v a r i a b l e . I t should be remembered that the option p r i c i n g model used assumes a constant r i s k l e s s i n t e r e s t rate. To demonstrate i t s e f f e c t of this v a r i a b l e , 2 two values considered r e l a t i v e l y extreme i n the a c t u a r i a l f i e l d were selected namely, 4% and 8%. The s e l e c t i o n of the appropriate r i s k l e s s rate i s a d i f f i c u l t task f o r l i f e insurance companies issuing long term p o l i c i e s . Life insurance companies, however, have to face this problem whether they are evaluating conventional p o l i c i e s or equity-linked ones with minimum guarantee. Three d i f f e r e n t kinds of guarantees at death or maturity were considered i n accordance with the more usual contracts issued i n Canada These i n t e r e s t rates are continuous compounding rates. 3 The e f f e c t of i n f l a t i o n on the r e a l value of the guarantee or exercise price and on the s e l e c t i o n of r i s k l e s s rate (nominal or r e a l ) , i s a topic which has not been studied well and requires further research. In what follows the problems created by i n f l a t i o n are not considered. 125 and the United Kingdom. set In the f i r s t .case the amount of the guarantee i s to be 100% of the sums invested i n the reference p o r t f o l i o , that i s , the minimum guarantee "grows" with the premiums invested i n the unit t r u s t . The second case, considered very common i n Canada, i s s i m i l a r to the f i r s t case ,but the amount of the guarantee i s set to be only 75% of the premiums invested i n the reference p o r t f o l i o . These two cases have been chosen because they are representative of the more common p o l i c i e s issued. Any other guarantee as a percentage of the premium invested i n the reference p o r t f o l i o can be treated analogously. The t h i r d case considered i s that of a fixed minimum guarantee at death or maturity independent of the premiums invested i n the reference p o r t f o l i o when the policy expires because of death or maturity. Summarizing, the three key variables considered were: (1) Variance Rate: corresponding values of 0.01846 and 0.04 were used, the f i r s t to estimated variance i n the TSE I n d u s t r i a l Index. (2) Interest Rate: values of 4% and 8% were employed i n the examples. (3) Types of Guarantee: 100% of the sums invested i n the reference p o r t f o l i o , 75% of the sums, and a f i x e d amount were considered. The mortality Table used for the examples was the Canadian Assured L i f e s 1958-64 Select f o r males published by the Canadian Insurance Table 6.1 Companies. shows the mortality data (ultimate) used. A l l the examples presented i n this chapter correspond premium contract discussed i n d e t a i l i n Chapter 5. to the p e r i o d i c To simplify Only the "ultimate" form between the ages of 20 and 80 were used. For an explanation of Select Mortality Tables see Appendix H. 126 Table 6.1; AGE 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Mortality Table: Canadian Assured Lifes, 1958-64 (Males) Ultimate £ _x 985731.44 984650.06 983540.38 982429.94 981344.38 980301.19 979309.13 978369.00 977471.81 976602.88 975743.44 974874.06 973975.25 973027.56 972013.69 970917.25 969724.00 968425.50 967013.56 965477.94 963801.88 961958.13 959916.81 957641.81 955092.56 952225.38 948994.50 945357.94 941281.56 936744.56 931733.94 AGE 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 £ _x 926241.38 920253.25 913740.56 906658.19 898948.88 890543.69 881356.88 871284.69 860206.31 847993.13 834523.56 819698.25 803450.19 785752,63 766612.50 746058.06 724125.44 700851.31 676273.88 650442.25 623420.25 595289.69 566144.88 536093.38 505250.81 473738.31 441687.56 409255.31 376635.63 344066.06 127 the presentation without loss of generality i t w i l l be assumed that the investment component of the premium invested i n the reference p o r t f o l i o i s equal to one monetary unit ( d o l l a r or pound) and that the premium i s paid once a year at the beginning of the year. e a s i l y evaluated. Monthly payments could be Also, to simplify the discussion, i f death occurs during a c e r t a i n year ,the l i q u i d a t i o n of the p o l i c y (sale of reference p o r t f o l i o and payment of the guarantee) i s assumed to take place at the end of that year. For the purposes of t h i s study we w i l l ignore withdrawals, i . e . , p o l i c i e s whose holders v o l u n t a r i l y discontinue premium payments and receive the value of the investment account less a deduction. F i n a l l y , r e c a l l that i n the equilibrium model developed i n Chapter 5 and in the applications presented i n t h i s chapter transaction costs and commissions have not been considered. 6.2 Solution Algorithm f o r the P e r i o d i c Premium Contract The s o l u t i o n algorithm f o r the p e r i o d i c premium contract with known expiration date (death or maturity) w i l l be developed i n this section. The procedure i s based on the numerical s o l u t i o n to parabolic p a r t i a l d i f f e r e n t i a l equations presented i n Chapters 3 and 4. In the preceding chapter i t was shown that the value of the c a l l option i n the periodic premium contract i s governed by the p a r t i a l d i f f e r e n t i a l equation: (6.1) 2 2 ho x w + rxw xx - wr - w m x =0 T subject to the boundary conditions: (1) (6.2) For the period a f t e r the l a s t premium payment: w(x, 0, g) = max[x-g, 0] 128 (6.3) w(0, T, g) (6.4) w (oo, x T, = 0 g) = 1 For this period the problem does not present any d i f f i c u l t i e s because equation (6.1) subject to boundary conditions (6.2), (6.3) and (6.4) corresponds exactly to the basic Black-Scholes model with no dividends, which can be solved a n a l y t i c a l l y with formula (2.31) or by the numerical procedures shown i n Chapters 3 and 4. (2) For the other periods between premium payments: (6.5) b w(x, T , g) = w(x+D, T , g) a (6.6) w (0, T, g) = - r w(0, T, g) (6.7) w (~, T, g) = 1 T x . Boundary conditions (6.5) and (6.6) are d i f f e r e n t from the corresponding ones i n Chapters 3 and 4 and,therefore, the procedures employed i n those chapters must be modified to s a t i s f y these new conditions. Using the f i n i t e differences method f o r solving p a r t i a l d i f f e r e n t i a l equations i n the same way as i n Chapter 3 and 4, equation (6.1) can be approximated (6.8) i n difference form as: a. w. , . + b. w. . + c. l - l , j l l i , j l w.,, . l + l , j where (6.9) a ± (6.10) b ± (6.11) c ± = %rki- ha\i = (1+rk) + a k i 2 = -hrk± - Jia ki 2 2 2 (6.12) w(x,, T.) 2 = w(ih, j k ) = w, , = w. . , i , J - l 129 with h being the reference p o r t f o l i o p r i c e increment and k the time to maturity increment. At any premium payment date, denoted by j , boundary condition can be written i n the notation defined by (6.13) ,w. . b 1,3 where D/h = w.,_ . a i+D/h,3 (6.5) (6.12) as: p i s an integer (the reference p o r t f o l i o p r i c e increment, h, has to be chosen as an i n t e g r a l factor of the p e r i o d i c premium payment, D). Boundary condition (6.6) and (6.7) can be written i n the same notation respectively, as: (6.14) w . = (1-rk) w (6.15) . . for a l l j 0,3-1 0,3 -w . . + w n-1,3 . = h for a l l j n,3 In (6.15) n i s the l a s t point of the g r i d and has to be large enough for (6.15) to hold within the p r e c i s i o n required. As a " r u l e of thumb" x has to be more than three times the exercise p r i c e , n The difference equation (6.8) l i n e a r equations by making i = l , i s obtained can be used to generate a system of n-1. The l a s t equation (for i=n) from boundary condition (6.15) a. w . + b. w, . + c. w„ . 1 0,3 1 1,3 1 2,j a 2 0 w. . + b w» . + c w„ . 1,3 2 2,3 2 3,3 0 ' 1,3-1 = = 0 w 9 , 2,3-1 (6.16) . a . w „.+b n-1 n-2,3 . w ,.+c n-1 n-1,3 - w ,w . n-1 n,3 , . + w . n-1,3 n,3 =w = n-1,3-1 h 130 (6.16) i s a system of n equations with (n+1) unknowns (w. 1 i=0, .... n). But the value of w . for >3 . c a n be obtained from boundary condition (6.14) The system of equations (6.16) can then be written as ^1 a 2 l W °1 2 + w x W + b w 2 ^1 = + c 2 2 w = f , w „ + b , w ,+c ,w = n-1 n-2 n-1 n-1 n-1 n f a f 3 2 (6.17) a n w , + b w = n-1 n n , n-1 n where f1 1 = w, . , - a.(l-rk)w . . l.J-l 1 o,j-l f. = w. f = h = -1 = 1 a b n n n . , f o r i=2, .... and where the subscripts j (from w. n-1 .) have been omitted f o r s i m p l i c i t y . The matrix of c o e f f i c i e n t s of the system (6.17) i s t r i d i a g o n a l and, thus, Gauss' elimination method can be conveniently applied to solve f o r the values of w^(i=l, ... n). With this procedure i t i s possible to f i n d the values of w, . i,J (i=0, ... n) s t a r t i n g from the values w. . .(1=0, ... n). The f i r s t l i n e of the grid f o r each period between premium payments i s obtained from the l a s t l i n e of the preceding period using boundary condition (6.13). A new boundary condition (6.13) i s obtained at each premium payment date and 131 the i t e r a t i v e process progresses i n between premium payments as described above. I t should be emphasized that the procedure s t a r t s from the maturity date of the option and advances i n backward d i r e c t i o n i n time to the date of issue of the contract. The computer program shown i n Appendix E i s based i n the s o l u t i o n algorithm presented above. The s o l u t i o n algorithm f o r option prices i n the constant continuous premium contract i s given i n Appendix C and the respective computer program i s shown i n Appendix D. 6.3 The Variance Rate of the TSE I n d u s t r i a l Index In accordance with our procedure of employing Canadian data f o r the applications of the model, the TSE I n d u s t r i a l Index was used as a proxy of the reference p o r t f o l i o . The data to calculate the variance of the rate of return f o r this index were taken from the "Canadian Monthly Security Data F i l e (U.B.C.)" which records values from June, 1961 to August 1973. The continuously compounded rate of return on the index with dividends reinvested, R, i s : I.U+Y.) (6.18) J R E Zn 3 where: I. = Value of the index at the end of time i . = Average dividend y i e l d during time j . J Y_. Using the p r i c e r e l a t i v e s given by (6.18) the variance rate per year ? for the rate of return was calculated for the whole period included i n the F i l e (June '61 to August '73) and for the l a s t f i v e years (September to August '73). '68 The r e s u l t i n g values were 0.01516 and 0.01846 respectively. 132 The value f o r the l a t e r period was was the one used i n the example,because i t considered to be a better estimate of future variance rates. In one example a higher variance rate of 0.04 was chosen a r b i t r a r i l y to show the e f f e c t of this v a r i a b l e on the r i s k premium. 6.4 Put Premium or "Risk Premium" The v a r i a b l e chosen f o r analyzing and comparing the p o l i c i e s the expected annual premium on the put or r i s k premium. point of the insurance company this v a r i a b l e has attracted was From the viewthe most attention, because i t represents the p r i c e that the insurance companies would have to charge f o r bearing the investment risk. Other variables such as the value of the c a l l option, the present value of the guaranteed amount, the t o t a l present value of the contract, the present value of the reference p o r t f o l i o or the value of the put option which may be required for studying other aspects of the problem (transaction and s e l l i n g costs f o r example) are readily a v a i l a b l e from the program. 6.5 Numerical Example No. 1 The parameters used i n example no. 1 were: (1) Variance Rate: (2) Risk Free Rate: (3) Guarantee: the variance rate of the TSE I n d u s t r i a l Index; 4% per annum (continuous); 100% of the investment component of the premiums paid at death or maturity. Table 6.2 shows the computations of the equilibrium value of the guarantee with known date of expiration (death or maturity) from one to t h i r t y years. The entries i n Table 6.2 correspond to the values of 133 the variables defined in Chapter 5 for the parameters of example no. 1; and can be summarized as follows: Column (1) : known expiration date of the contract, t, from 1 to 30 years. Column (2) : value of the c a l l option, w(t). Column (3) : present value of the guarantee, g(t) e Column (4) : total present value of the contract, PV (b(t) -rt = Q —rt w(t) + g(t) e . Corresponds to Column (2) plus Column (3). Column (5) : present value of the reference portfolio, PV (x(t)) = Q k=0 Column (6) : value of the put option, p(t) = PV (b(t)) - PV (x(t)). Q Q Corresponds to Column (4) minus Column (5). Column (7) : total annual premium for the contract, v(t) = D + y(t). Corresponds to the annuity with present value equal to the total present value of the contract (Column (4)). Column (8) : annual premium for the put, y(t). Corresponds to the annuity with present value equal to the value of the put option (Column (6)). Column (9) : annual premium for the c a l l , z(t). Corresponds to the annuity with present value equal to the value of the c a l l option (Column (2)). Tables such as 6.2 were computed for each of the examples given in this chapter using the computer program shown in Appendix F, which is the same as the one shown in Appendix E for solving for the periodic premium 134 contract, but where the process, i s repeated f o r d i f f e r e n t expiration dates with changing guarantees''*'. It should be emphasized that Table 6.2 deals with the "investment r i s k " aspect of the p o l i c y without any consideration of the "mortality r i s k " aspect. As can be seen from Table 6.2, the value of the c a l l option increases from $0.07 f o r a one-year contract with a guarantee of to $8.98 f o r a thirty-year contract with a guarantee of $30.00. this period the annual premium on the put decreases from 3.5% one year contract with $1 guarantee to 1.1% f o r a 30-year with $30 $1.00 In for a contract guarantee. "To obtain accurate results care should be taken i n the s e l e c t i o n of the reference p o r t f o l i o price step that i t not be large r e l a t i v e to the exercise p r i c e . As a " r u l e of thumb" the highest reference p o r t f o l i o value should be approximately 3 to 4 times the exercise p r i c e , so i f 400 increments are used, 100 of them should be below the exercise p r i c e . I t must be remembered also that the reference p o r t f o l i o p r i c e increment has to be an i n t e g r a l factor of the annual premium invested i n the reference p o r t f o l i o to permit the p r a c t i c a l a p p l i c ation of boundary condition (5.29). To solve f o r these two problems simultaneously the reference p o r t f o l i o p r i c e increment i s determined within the program and changes f o r d i f f e r e n t maturities. ^To be consistent with the option p r i c i n g model, a l l compounding and discounting of values are done continuously. Thus the r i s k l e s s i n t e r e s t rate considered i s the continuous rate. However, the numerical s o l u t i o n j by using f i n i t e differences instead of p a r t i a l derivatives.implies discrete (although for short i n t e r v a l s ) compounding. For example,if the time increment i s one month,a monthly compounding i s implied. By decreasing the size of the time increment the discrete compounding can be made as close to continuous compounding as desired. I I i i 135 Table 6.2: Computation of the Equilibrium Value of the Guarantee with Known Date of Expiration: Example No. 1. Variance Rate Riskless Rate Guarantee = = = No. years to Expiration Value Call P.V. Guarantee (1) (2) (3) 0. 07439 0.18082 0. 31923 0.48583 0. 67762 0. 89208 1. 12696 1. 38029 1.65021 1.93505 2. 23326 2.54340 2.86413 3. 19399 3.53224 3. 87723 4. 22873 4. 58483 4. 94503 5. 30849 5.67545 6. 04340 6. 41246 6. 78205 7. 15165 7.52218 7.89018 8. 25659 8.62118 8.98341 0. 96079 1.84623 2.66076 3.40858 4.09365 4. 71977 5. 29049 5. 80919 6. 27909 6. 70320 7.08440 7.42540 7. 72877 7.99693 8.23218 8.43668 8. 61249 8. 76155 8.88567 8. 98658 9. 06593 9. 12523 9. 16594 9. 18944 9. 19699 9. 18983 9. 16909 9. 13584 9. 09111 9. 03584 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 I i 1 1 0.01846 0.04000 100% of investment component on death or maturity Total P.V. Contract (4) 1.03518 2.02706 2.97999 3.89441 4.77127 5.61184 6.41745 7.18948 7.92929 8.63825 9.31767 9.96880 10.59290 11.19092 11.76441 12.31392 12.84122 13.34638 13.83069 14.29507 14.74137 15.16863 15.57840 15.97149 . 16.34863 16.71201 17.05927 17.39243 17.71228 18.01924 P.V. Reference Portfolio Value Put Total Annual Premium Annual Premium Put Annual Premium Call (5) (6) (7) (8) (9) 0.03518 0.03380 0.03332 0.03277 0.03208 0.03127 0.03036 0.02940 0.02841 0.02739 0.02637 0.02535 0.02435 0.02335 0.02239 1.02143 0.02053 0.01962 0.01874 0.01788 0.01712 0.01632 0.01556 0.01482 0.01411 0.01352 0.01287 0.01224 0.01165 0.01108 0.07439 0.09222 0.11069 0.12884 0.14658 0.16393 0.18094 0.19763 0.21403 0.23015 0.24600 0.26160 0.27697 0.29207 0.30697 0.32161 0.33607 0.35027 0.36424 0.37799 0.39159 0.40492 0.41803 0.43093 0.44362 0.45619 0.46847 0.48053 0.49240 0.50407 1.00000 1.96079 2. 88391 3. 77083 4. 62297 5.44170 6. 22833 6. 98411 7. 71026 8.40793 9. 07825 9. 72229 10. 34107 10. 93559 11. 50680 12. 05561 12. 58291 13. 08952 13. 57627 14. 04394 14, 49327 14. 92498 15. 33976 15. 73828 16. 12117 16. 48904 16. 84250 17. 18210 17. 50838 17. 82185 0.03518 0.06627 0.09609 0.12358 0.14830 0.17015 0.18912 0.20537 0.21904 0.23032 0.23941 0.24651 0.25183 0.25533 0.25761 0.25830 0.25832 0.25686 0.25442 0.25113 0.24811 0.24365 0.23864 0.23321 0.22746 0.22296 0.21677 0.21033 0.20390 0.19739 1.03518 1.03380 1.03332 1.03277 1.03208 1.03127 1.03036 1.02940 1.02841 1.02739 1.02637 1.02535 1.02435 1.02335 1.02239 0. 25830 1.02053 1.01962 1.01874 1.01788 1.01712 1.01632 1.01556 1.01482 1.01411 1.01352 1.01287 1.01224 1.01165 1.01108 136 All this means that i f a person ,assumed immortal f o r the present, 2 invests $1 per year for 20 years i n a mutual fund with variance rate a and i f at the end of that time wants to receive $20 or the value of h i s units, whichever i s the larger, he would have to pay, i n equilibrium, to the insurance company a r i s k premium of 1.8% annually. In other words ,he would have to pay annual premiums of $ 1.018 instead of $1, of which $0,018 would be the equilibrium price charged by the insurance company for taking the investment r i s k . In Chapter 5 i t was shown that once the equilibrium value of the guarantee with known date of expiration has been obtained (such as those in Table 6.2), mortality factors can e a s i l y be incorporated by computing expected values using the appropriate p r o b a b i l i t i e s of death and s u r v i v a l during the term of the contract. Given the age of the purchaser of the p o l i c y at entry and the term of the p o l i c y , the p r o b a b i l i t i e s of death during the l i f e of the contract can be computed from mortality figures, such as those i n Table 6.1, using the formulas derived i n Chapter 5. When both investment and mortality r i s k s are considered, the r e s u l t i n g put premiums or r i s k premiums are the value required by l i f e insurance companies to price equity-linked l i f e insurance contracts with asset value guarantees. For p r a c t i c a l purposes i t i s of i n t e r e s t to f i n d the dependence of these r i s k premiums on the age of the purchaser and the term of the contract. A computer program was prepared to tabulate and graph the r i s k premium versus the term of the contract f o r a given age of the purchaser of the policy. In This program i s shown i n Appendix I. addition, another computer program tabulates and graphs the r i s k 137 premium versus the age of the purchaser of the policy f o r a given term of the contract. This program i s shown i n Appendix J . Table 6.3 shows the relationship between the r i s k premium and the term of the contract for purchasers of ages 20, 30, 40 and 50,respectively. p r a c t i c a l purposes this i s the most i n t e r e s t i n g age range. For These r e s u l t s can be b r i e f l y summarized i n the following table: Term of Contract Level of Annual Risk Premiums (%) Age of Purchaser at Entry (years) (in years) 20 30 40 50 10 2.74 2.74 2.75 2.77 20 1.81 1.82 1.86 30 1. 15 1.19 1.31 : 1.98 1.60 The r i s k premium on a 10-year p o l i c y i s almost completely i n s e n s i t i v e to the age of the purchaser ( f o r ages between 20 and 50). Even on a 20-year p o l i c y the r i s k premium i s quite i n s e n s i t i v e to the age of the purchaser. Only on a 30-year p o l i c y does the r i s k premium increase a 20-year o l d to 1.60% f o r a 50-year s i g n i f i c a n t l y from 1.15% f o r o l d purchaser. These conclusions are a consequence of the patterns of mortality, because i n a l l cases there i s a high p r o b a b i l i t y that no benefits w i l l be paid u n t i l the maturity of the contract. They can be seen even better i n Table 6.4 which shows the r e l a t i o n - ship between the r i s k premium and the age of the purchaser f o r contracts of terms 10, 20 and 30 years respectively. For a l l ages, however, the r i s k premium i s s e n s i t i v e to changes i n the term of the contract. This i s a d i r e c t consequence of the influence of time to maturity on the value of the option for the nature of guarantee offered i n example no. 1. 138 Table 6.3: Risk Premium versus Term of Contract Guarantee = M o r t a l i t y Table Risk Free Rate Variance Rate = = = Term of Contract (in years) 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3.518 3.380 3.332 3.277 3.209 3.128 3.038 2.943 2.844 2.743 2.642 2.541 2.443 2.344 2.249 2.155 2.066 1.977 1.890 1.806 1.732 1.654 1.580 1.508 1.439 1.382 1.320 1.259 1.203 1.149 100% of investment death or maturity CA 1958-64 0.04 0.01846 (TSE) component on Level of Annual Risk Premium (%) Age of Purchaser at Entry (years) 30 3.518 3.380 3.332 3.277 3.209 3.128 3.038 : 2.942 2.844 2.743 2.643 2.542 2.444 2.345 2.251 2.158 2.070 1.982 1.897 1.815 1.742 1.666 1.594 1.525 1.459 1.405 1.345 1.289 1.236 1.186 40 50 3.518 3.380 3.332 3.278 3.209 3.129 3.040 2.946 2.849 2.750 2.651 2.553 2.458 2.363 2.273 2.183 2.100 2.017 1.937 1.860 1.793 1.724 1.660 1.598 1.540 1.493 1.443 1.395 1.353 1.313 3.518 3.381 3.333 3.279 3.212 3.134 3.047 2.956 2.864 2.770 2.677 2.585 2.497 2.411 2.330 2.251 2.179 2.109 2.042 1.980 1.927 1.873 1.825 1.780 1.739 1.707 1.674 1.644 1.618 1.595 Table 6.4: Risk Premium versus Age of Purchaser at Entry = Guarantee Mortality Table Risk Free Rate Variance Rate Age of Purchaser at Entry (years) 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 10 100% of investment death or maturity = CA 1958-64 = 0.04 = 0.01846 (TSE) component on Level of Annual Risk Premium (%) Term of Contract (years) 2.743 2.743 2.743 2.743 2.743 2.743 2.743 2.743 2.743 2.743 2.743 2.744 2.744 2.744 2.745 2.745 2.746 2.747 2.748 2.749 2.750 2.751 2.753 2.754 2.756 2.758 2.769 2.762 2.764 2.767 2.770 20 1.806 1.806 1.806 1.806 1.807 1.807 1.808 1.809 1.811 1.812 1.815 1.817 1.820 1.823 1.827 1.831 1.836 1.841 1.847 1.853 1.860 1.868 1.877 1.886 1.896 1.908 1.920 1.933 1.947 1.963 1.980 30 1.149 1.150 1.152 1.154 1.156 1.159 1.163 1.168 1.173 1.179 1.186 1.193 1.202 1.211 1.222 1.234 1.247 1.261 1.277 1.294 1.313 1.333 1.355 1.379 1.404 1.432 1.461 1.491 1.524 1.559 1.595 140 6.6 Numerical Example No. 2 The parameters used i n example no. 2 are the same as those used i n example no. 1 with the exception from 4% to that the i n t e r e s t rate was increased 8%: (1) Variance Rate : (2) Risk Free Rate : 8% per annum; (3) Guarantee 100% : paid at death or the variance rate of the TSE I n d u s t r i a l Index; of the investment component of the premiums maturity. With the other parameters l e f t unchanged, the i n t e r e s t rate was increased to 8% i n order to investigate the e f f e c t of the r i s k l e s s i n t e r e s t rate on the r i s k premium. 4% can be considered as a "low" rate and 8% as a "high" rate; so both rates can be regarded as "extremes" of the possible range of i n t e r e s t rates. The computation of the equilibrium value of the guarantee with known date of expiration i s shown i n Table 6.5. examples that follow t h i s table was twenty years. In t h i s and i n the numerical computed for maturities from one to Mainly to save computer time the calculations were r e s t r i c t e d to contracts with maturities of twenty years and l e s s . In t h i s case the premium becomes r e l a t i v e l y small a f t e r twenty years. When the mortality r i s k i s introduced, Table 6.6 shows the r e l a t i o n s h i p between the r i s k premium and the term of the contract for purchasers of ages 20, 30, 40 and 50. From another angle Table 6.7 between the r i s k premium and terms 10 and 20 years. follows: shows the r e l a t i o n s h i p the age of the purchaser for contracts of These r e s u l t s can be summarized i n the table that 141 T a b l e 6.5: Computation o f t h e E q u i l i b r i u m V a l u e o f the Guarantee w i t h Known Date o f E x p i r a t i o n : Example No. 2 V a r i a n c e Rate = R i s k l e s s Rate = Guarantee = No. y e a r s to Expiration Value Call P.V. Guarantee (1) (2) (3) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0. 09871 0. 25161 0. 45440 0.69903 0. 97838 1.28614 1. 61666 1.96491 2. 32641 2. 69724 3.07401 3. 45368 3. 83364 4. 21169 4. 58586 4. 95473 5. 31679 5. 67108 6. 01675 6. 35307 0. 92312 1. 70429 2. 35988 2. 90460 3. 35160 3. 71270 3. 99846 4. 21834 4. 38077 4. 49329 4. 56261 4. 59472 4. 59491 4. 56792 4. 51792 4. 44860 4. 36324 4. 26471 4. 15553 4. 03794 0.01846 0.08000 100% o f investment component on death o r maturity, P.V. Reference Portfolio Value Put Total Annual Premium Annual Premium Put Annual Premium Call (4) (5) (6) (7) (8) (9) 1.02183 1.95590 2. 81428 3. 60363 4. 32998 4. 99884 5. 61513 6. 18326 6. 70718 7. 19053 7. 63662 8. 04839 8. 42856 8.77961 9. 10378 9. 40333 9.68003 9.93579 10. 17228 10. 39101 1. 00000 1. 92312 2. 77526 3. 56189 4. 28804 4. 95836 5. 57714 6. 14835 6. 67564 7. 16239 7. 61172 8.02650 8.40940 8. 76285 9. 08913 9. 39032 9. 66836 9. 92502 10. 16195 10.38066 0.02183 0.03279 0.03902 0.04174 0.04194 0.04048 0.03799 0.03491 0.03154 0.02813 0.02490 0.02189 0.01916 0.01676 0.01465 0.01301 0.01167 0.01077 0.01034 0.01035 1. 02183 1. 01705 1. 01406 1. 01172 1.00978 1. 00816 1. 00681 1. 00568 1.00472 1. 00393 1. 00327 1.00273 1. 00228 1.00191 1.00161 1.00138 1. 00121 1. 00108 1.00102 1. 00100 0.02183 0.01705 0.01406 0.01172 0.00978 0.00816 0.00681 0.00568 0.00472 0.00393 0.00327 0.00273 0.00228 0.00191 0.00161 0.00138 0.00121 0.00108 0.00102 0.00100 0.09871 0.13084 0.16373 0.19625 0.22816 0.25939 0.28987 0.31958 0.34849 0.37658 0.40385 0.43028 0.45588 0.48063 0.50454 0.52764 0.54992 0.57139 0.59209 0.61201 Total P.V. Contract Table 6.6: Risk Premium versus Term of Contract = Guarantee M o r t a l i t y Table Risk Free Rate Variance Rate Term of Contract (in years) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 20 = = = 100% of investment death or maturity CA 1958-64 0.08 0.01846 (TSE) component on Level of Annual Risk Premium (%) Age of Purchaser at Entry (years) 2.183 1.706 1.407 1.174 0.981 0.820 0.686 0.573 0.478 0.400 0.335 0.281 0.237 0.200 0.171 0.148 0.131 0.119 0.113 0.111 30 2.183 1.705 1.407 1.174 0.980 0.819 0.685 , 0.573 0.478 0.400 0.334 0.281 0.237 0.201 0.171 0.149 0.132 0.120 0.114 0.112 40 2.183 1.706 1.408 1.176 0.983 0.823 0.690 0.580 0.486 0.409 0.345 0.293 0.250 0.215 0.187 0.166 0.150 0.138 0.133 0.131 50 2.183 1.708 1.413 1.183 0.994 0.838 0.709 0.602 0.512 0.439 0.379 0.331 0.291 0.259 0.234 0.215 0.202 0.191 0.187 0.185 143 Table 6.7: Risk Premium versus Age of Purchaser at Entry Guarantee = Mortality Table Risk Free Rate Variance Rate Age of Purchaser at Entry (years) 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 1 1 100% of investment component on death or maturity = CA 1958-64 = 0.08 = 0.01846 (TSE) Level of Annual Risk Premium (%) Term of Contract (years) 10 20 0.400 0.400 0.400 0.400 0.399 0.399 0.399 0.399 0.399 0.399 0.400 0.400 0.400 0.401 0.402 0.403 0.404 0.405 0.406 0.407 0.409 0.411 0.413 0.416 0.418 0.421 0.424 0.428 . 0.431 0.435 0.439 0.111 0. I l l 0.110 0.110 0.110 0.110 0.110 0.110 0.110 0.110 0.112 0.113 0. 114 0.115 0.117 0.118 0.120 0.122 0.125 0.128 0.131 0.134 0.138 0.143 0.147 0.153 0.158 0.164 0.171 0.178 0.185 144 Term of Contract Level of Annual Risk Premium Age (in years) of Purchaser at Entry (%) (years) 20 30 40 50 10 .40 .40 .41 .44 20 . 11 . 11 . 13 .19 Again the r i s k premium i s quite i n s e n s i t i v e to the age of the purchaser (between 20 and 50 years old) and more s e n s i t i v e to the term of the contract. The most important conclusion from this example i s the s u b s t a n t i a l e f f e c t of the r i s k l e s s i n t e r e s t rate on the equilibrium value of the r i s k premium. I f the i n t e r e s t rate i s increased from 4% to 8%,the r i s k premium decreases f o r a 20-year old purchaser on a 10-year contract from 2.74% 0.40% and for a 50-year old purchaser on a 20-year contract from to 1.98% to 0.19%. As discussed i n Chapter 2, the appropriate r i s k l e s s i n t e r e s t rate for Black-Scholes and Merton's option p r i c i n g model i s that of a r i s k l e s s bond with the same maturity as the option. Thus, i t may perhaps be suitable to vary the i n t e r e s t with the term of the contract. These problems, which require further research, are not pursued i n t h i s study. The s e l e c t i o n of the appropriate i n t e r e s t rate i s then a very important factor i n the p r i c i n g of r i s k premiums and l i f e insurance companies should give c a r e f u l consideration to this f a c t o r . 6.7 Numerical Example No. 3 The parameters used i n example no. 3 are the same as those i n example no. 2 with the exception that the variance rate was increased 145 from 0.01846 (TSE) to 0.04 to highlight the influence of the variance rate on the valuation of r i s k premiums: (1) Variance Rate (2) Risk Free Rate : (3) Guarantee : : paid at death or 0.04 per year; 8% per year; 100% of the investment component of the premiums maturity. The computation of the equilibrium value of the guarantee with known date of expiration i s shown i n Table 6.8 c a l l option and p a r t i c u l a r l y i n example no. 2. and,as expected,the value of the the annual premium on the put are higher than The annual premium on the put i s an increasing function of the variance of the rate of return on the reference portfolio. When mortality r i s k i s also taken into consideration,Table 6.9 shows the r e l a t i o n s h i p between the r i s k premium and the term of the contract for purchasers of ages 20, 40 and 60. Table 6.10 shows the r e l a t i o n s h i p between the r i s k premium and the age of the purchaser f o r contracts of terms 10, These r e s u l t s can be summarized i n the 15 and 20 years. following table: Term of Contract Level of Annual Risk Premium Age of Purchaser at Entry (%) (years) (in years) 20 30 40 50 60 10 2.05 2.05 2.06 2.11 2.24 20 0.90 0.90 0.96 1.13 1.49 This example i l l u s t r a t e s the substantial influence on the valuation of r i s k premiums of the variance portfolio: when the variance rate of the return on the rate increases reference from 0.01846 (TSE) to 0.04, the r i s k premium increases for a 20 year old on a 10-year term from 0.40% 146 T a b l e 6.8: Computation o f the E q u i l i b r i u m V a l u e o f the Guarantee w i t h Known Date o f E x p i r a t i o n : Example No. 3 V a r i a n c e Rate R i s k l e s s Rate Guarantee No. y e a r s to Expiration Value Call P.V. Guarantee (1) (2) (3) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.12061 0.29378 0.51539 0.77636 1.06909 1.38734 1.72548 2.07887 2.44319 2.81492 3.19102 3.56876 3.94581 4.32020 4.69006 5.05441 5.41111 5.76044 6.10122 6.43273 0. 92312 1. 70429 2. 35988 2. 90460 3. 35160 3. 71270 3. 99846 4. 21834 4. 38077 4. 49329 4. 56261 4. 59472 4. 59491 4. 56792 4. 51792 4. 44860 4. 36324 4. 26471 4. 15553 4. 03794 => = = 0.04000 0.08000 100% o f investment component on death o r m a t u r i t y Total P.V. Contract (4) 1.04373 1, 99807 2. 87527 3. 68095 4. 42070 5. 10004 5. 72394 6. 29721 6. 82396 7. 30821 7. 75364 8. 16347 8.54072 8. 88812 9. 20798 9. 50301 9. 77435 10. 02515 10. 25676 10.47067 P.V. Reference Portfolio Value Put Total Annual Premium Annual Premium Put Annual Premium Call (5) (6) (7) (8) (9) 0.04373 0.07495 0.10001 0.11907 0.13266 0.13168 0.14680 0.14886 0.14832 0.14582 0.14192 0.13697 0.13133 0.12527 0.11885 0.11269 0.10599 0.10013 0.09481 0.09001 1. 04373 1. 03897 1. 03604 1. 03343 1. 03094 1. 02857 1. 02632 1. 02421 1.02222 1.02036 1.01864 1.01706 1.01562 1. 01430 1.01308 1. 01200 1.01096 1. 01009 1. 00933 1. 00867 0.04373 0.03897 0.03604 0.03343 0.03094 0.02857 0.02632 0.02421 0.02222 0.02036 0.01864 0.01706 0.01562 0.01430 0.01308 0.01200 0.01096 0.01009 0.00933 0.00867 0.12601 0.15276 0.18571 0.21796 0.24932 0.27980 0.30938 0.33812 0.36599 0.39301 0.41922 0.44462 0.46921 0.49301 0.51601 0.53826 0.55967 0.58040 0.60040 0.61968 1.00000 1.92312 2.77526 3.56189 4.28804 4.95836 5.57714 6.14835 6.67564 7.16239 7.61172 8.02650 8.40940 8.76285 9.08913 9.39032 9.66836 9.92502 10.16195 10.38066 Table 6.9: Risk Premium versus Term of Contract = Guarantee Mortality Table Risk Free Rate Variance Rate. 100% of investment death or maturity = CA 1958-64 = 0.08 =0.04 component on Level of Annual Risk Premium (%) Age of Purchaser at Entry (years) Term of Contract (in years) 20 40 60 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 4.373 3.898 3.605 3.345 3.097 2.861 2.638 2.429 2.231 2.047 1.877 1.720 1.578 1.448 1.327 1.221 1.119 1.033 0.959 0.894 4.373 3.898 3.606 3.347 3.100 2.866 2,644 2.437 2.243 2.062 1.896 1.744 1.607 1.481 1.367 1.266 1.170 1.090 1.022 0.963 4.373 3.905 3.621 3.374 3.143 2.929 2.731 2.551 1.286 2.238 2.106 1.990 1.889 1.801 1.724 1.659 1.601 1.556 1.519 1.490 Table 6.10: Risk Premium versus Age of Purchaser at Entry Guarantee = Mortality Table Risk Free Rate Variance Rate = = = Age of Purchaser at Entry (years) 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 i i 10 100% of investment death or maturity CA 1958-64 .08 0.04 component on Level of Annual Risk Premium (%) Term of Contract (years) 2.047 2.047 2.046 2.046 2.046 2.046 2.045 2.045 2.046 2.046 2.047 2.047 2.048 2.049 2.050 2.052 2.053 2.055 2.057 2.059 2.062 2.065 2.069 2.073 2.077 2.082 2.087 2.093 2.098 2.104 2.111 15 20 1.327 1.327 1.327 1.327 1.326 1.326 1.326 1.327 1.328 1.329 1.330 1.332 1.334 1.337 1.339 1.343 1.346 1.351 1.355 1.361 1.367 1.373 1.381 1.389 1.398 1.408 1.418 1.429 1.442 1.445 1.470 0.894 0.894 0.894 0.894 0.894 0.894 0.895 0.896 0.898 0.900 0.903 0.908 0.910 0.914 0.919 0.924 0.930 0.937 0.945 0.953 0.963 0.973 0.985 0.098 1.012 1.027 1.043 1.061 1.081 1.102 1.125 149 to 2.05% and for a 50 year old on a 20-year terra from 0.19% to 1.13%. The fact that the equilibrium r i s k premium i s an increasing function of the v o l a t i l i t y (variance rate) of the reference p o r t f o l i o , s h o u l d be c a r e f u l l y considered by l i f e insurance companies when selecting the unit trust. Sometimes the insurance company picks i t s own p o r t f o l i o as the reference p o r t f o l i o . I t i s conceivable that l i f e insurance companies could select two or more unit trusts with d i f f e r e n t h i s t o r i c a l variance rates so as to be able to o f f e r to t h e i r c l i e n t s d i f f e r e n t p o l i c i e s , which would n a t u r a l l y demand d i f f e r e n t r i s k premiums. 6.8 Numerical Example No. 4 In the f i r s t three examples the nature of the guarantee remained unchanged, that i s at a guarantee of 100% of the investment premiums paid at death or maturity. of only 75% of investment maturity. no. 7 component of the' Example no. 4 deals with a guarantee component of the premiums paid at death or The other parameters used are the same as those i n example 1: (1) Variance Rate the variance rate of the TSE I n d u s t r i a l Index; (2) Risk Free Rate 4% per year; (3) Guarantee 75% of the investment component of the premiums paid at death or maturity. In Canada a common guarantee i s 75% of the t o t a l premiums paid. This i s a more generous guarantee than 75% of the investment component. 150 For example no. 4 the computation of the equilibrium value of the guarantee with known date of expiration, i s shown i n Table 6.11. Comparing this table with Table 6.2 of example no. 1 i t can be observed that even though the value of the c a l l option i s always higher i n the former, which i s natural because the exercise p r i c e i s lower other things being equal, the present value of the guarantee i s lower, thereby making the t o t a l present value of the contract lower. As the present value of the reference p o r t f o l i o remains unchanged, the value of the put and,therefore,the annual premium on the put are lower i n the case of the 75% guarantee than i n the case of the 1 This r e s u l t can be seen d i r e c t l y by considering that the guarantee i n this example i s less generous and so i t w i l l cost l e s s . I t i s i n t e r e s t i n g to note that i n this example the annual premium on the put increases with time to maturity . When m o r t a l i t y r i s k i s also taken into consideration,Table 6.12 shows the relationship between the r i s k premium and the term of the contract for purchasers of ages 20, 30, 40, 50 and 60. These r e s u l t s are summarized i n the following table: Term of Level of Annual Risk Premium (%) Contract Age of Purchaser at Entry (years) (in years) 20 30 40 50 60 10 0.22 0.22 0.22 0.22 0.21 20 0.23 0.24 0.23 0.23 0.22 These r e s u l t s contradict the "common b e l i e f " i n the a c t u a r i a l l i t e r a t u r e that the r i s k premium should decrease with the term of the contract. The relationship between the r i s k premium, the nature of the guarantee, and the time to maturity requires further research. 151 Table 6.11: Computation of the Equilibrium Value of the Guarantee with Known Date of Expiration: Example No. 4 Variance Rate Riskless Rate Guarantee No. years to Expiration Value Call P.V. Guarantee (1) (2) (3) 0.27976 0.57709 0. 89051 1.21820 1.55866 1.91017 2.27135 2.64075 3.01723 3. 39943 3. 78635 4. 17696 4.57047 4.96598 5. 36207 5. 75939 6. 15547 6. 55199 6. 94574 7.33712 0.72059 1.38467 1.99557 2.55643 3.07024 3.53983 3.96786 4. 35689 4. 70932 5.02740 5. 31330 5.56905 5. 79658 5.99769 6. 17413 6. 32751 6. 45937 6. 57116 6. 66425 6. 73994 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 = = = 0.01846 0.04000 75% of investment component on death or maturity Total P.V. Contract (4) 1.00035 1.96177 2.88608 3. 77463 4. 62890 5.44999 6. 23921 6. 99764 7. 72655 8.42683 9.09965 9. 74601 10.36705 10.96368 11. 53620 12. 08690 12. 61484 13. 12315 13. 60999 14. 07706 P.V. Reference Portfolio Value Put Total Annual Premium Annual Premium Put Annual Premium Call (5) (6) (7) (8) (9) 0.00035 0.00098 0.00218 0.00381 0.00593 0.00829 0.01088 0.01353 0.01629 0.01890 0.02139 0.02372 0.02597 0.02808 0.02940 0.03129 0.03194 0.03363 0.03372 0.03312 1.00035 1.00050 1.00076 1.00101 1.00128 1.00152 1.00175 1.00194 1.00211 1.00225 1.00236 1.00244 1.00251 1.00257 1.00255 1.00259 1.00254 1.00257 1.00248 1.00236 0.00035 0.00050 0.00076 0.00101 0.00128 0.00152 0.00175 0.00194 0.00211 0.00225 0.00236 0.00244 0.00251 0.00257 0.00255 0.00259 0.00254 0.00257 0.00248 0.00236 0.27976 0.29432 0.30879 0.32306 0.33716 0.35102 0.36468 0.37811 0.39133 0.40431 0.41708 0.42963 0.44197 0.45411 0.46599 0.47774 0.48919 0.50055 0.51161 0.52244 1.00000 1.96079 2.88391 3.77083 4.62297 5.44170 6.22833 6.98411 7.71026 8.40793 9.07825 9.72229 10.34107 10.93559 11.50680 12.05561 12.58291 13.08952 13.57627 14.04394 Table 6.12: Risk Premium versus Term of Contract and Age of Purchaser at Entry Guarantee M o r t a l i t y Table Risk Free Rate Variance Rate Term of Contract (in years) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 20 0.035 0.050 0.076 0.101 0.128 0.152 0.174 0.193 0.210 0.224 0.235 0.243 0.250 0.256 0.254 0.256 0.253 0.256 0.247 0.235 75% of investment component on death or maturity = CA 1958-64 = 0.04 = 0.01846 (TSE) = Level of Annual Risk Premium (%) Age of Purchaser at Entry (years) 30 0.035 0.050 0.076 0.101 0.128 0.152 0. 175 0.193 0.210 0.224 0.235 0.243 0.250 0.256 0.254 0.258 0.253 0.256 0.247 0.239 40 0.035 0.050 0.076 0. 101 0.127 0.151 0. 174 0.192 0.209 0.223 0.233 0.241 0.248 0.253 0.251 0.255 0.251 0.253 0.245 0.234 50 0.035 0.050 0.076 0.100 0.126 0. 150 0. 172 0. 189 0.205 0.218 0.228 0.235 0.242 0.247 0.245 0.248 0.244 0.247 0.240 0.231 60 0.035 0.050 0.075 0.099 0.124 0. 145 0.166 0.182 0. 196 0.207 0.215 0.221 0.226 0.230 0.229 0.231 0.229 0.230 0.226 0.220 153 In. this case the age of the purchaser and the term of the contract make very l i t t l e difference for the r i s k premium and a uniform r i s k premium for a l l ages and terms of contracts can be considered. The magnitude of the r i s k premium i s r e l a t i v e l y small f o r the parameters selected. 6".9 Numerical Example No. 5 In this l a s t numerical example the nature of the guarantee has been changed again to show the f l e x i b i l i t y of the method f o r solving for many different situations. Instead of increasing the amount of the guarantee as new premiums are invested i n the reference p o r t f o l i o , this example 9 considers a fixed amount of minimum guarantee on death or maturity . A 20 year contract i s assumed where $1 i s invested i n the unit trust each year. A $20 minimum guarantee i s considered at death or maturity ently of the date when the b e n e f i t s are payable. independ- The other two parameters used are those of example no. 2: (1) Variance Rate : the variance rate of the TSE I n d u s t r i a l Index; (2) Risk Free Rate : 8% per year; (3) Guarantee : $20.00, fixed f o r a l l dates of expiration. The computation of the equilibrium value of the guarantee with known date of expiration for example no. 5 i s shown i n Table 6.13. Naturally the value of the c a l l option i s p r a c t i c a l l y equal to zero and the value of the put option i s very high f o r short maturities,because there i s p r a c t i c a l l y zero p r o b a b i l i t y that the reference p o r t f o l i o w i l l reach a value of $20 for those short maturities. The annual premium on the put i s , t h e r e f o r e , Contracts of this type are popular i n the United Kingdom. 154 T a b l e 6.13: Computation o f the E q u i l i b r i u m V a l u e o f the Guarantee w i t h Known Date o f E x p i r a t i o n : Example No. 5 V a r i a n c e Rate R i s k l e s s Rate Guarantee No. y e a r s to Expiration (1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 P.V. Value C a l l ' Guarantee (2) 0.00000 0.00000 0.00000 0.00000 0.00000 0.00009 0.00231 0.02142 0.10161 0.30506 0.67014 1.18823 1.81715 2.50637 3.21369 3.91016 4.57865 5.21048 •5.80211 6.35307 (3) 18. 46233 17. 04288 15. 73256 14. 52298 13. 40640 12. 37567 11. 42418 10. 54585 9. 73505 8.98658 8. 29566 7. 65786 7. 06910 6. 52560 6. 02389 5. 56075 5. 13323 4. 73857 4. 37425 4. 03794 = 0.01846 = 0.08000 =20.00 Total P.V. Contract (4) 18. 46233 17. 04288 15. 73256 14. 52298 13. 40640 12. 37576 11. 42650 10. 56727 9. 83666 9. 29165 8. 96580 8. 84609 8. 88625 9. 03197 9. 23758 9. 47092 9. 71188 9. 94904 10. 17636 10. 39101 P.V. Reference Portfolio Value Put Total Annual Premium Annual Premium Put Annual Premium Call (5) (6) (7) (8) (9) 1.00000 1. 92312 2. 77526 3. 56189 4. 28804 4. 95836 5. 57714 6. 14835 6. 67564 7. 16239 7. 61172 8. 02650 8. 40940 8. 76285 9. 08913 9. 39032 9. 66836 9. 92502 10.16195 10.38066 17. 46233 18. 46233 17.46233 7.86212 15. 11976 8. 86212 12. 95730 5. 66886 4.66886 10. 96109 4.07733 3.07733 9. 11837 3. 12647 2.12647 1.49594 7. 41741 2. 49594 04881 1.04881 5. 84936 2. 4. 41893 1. 71872 0.71872 3. 16102 1. 47352 0.47353 2. 12925 1. 29728 0.29728 1. 35408 1. 17789 0.17789 0. 81959 1. 10211 0.10211 0.47685 1. 05670 0.05670 0. 26912 1. 03071 0.03071 0. 14845 1. 01633 0.01633 0. 08059 1. 00858 0.00858 0. 04352 1. 00450 0.00450 0.00242 0.02402 1. 00242 0. 01441 1. 00142 0.00142 0. 01035 1. 00100 0.00100 0.00000 0.00000 0.00000 0.00000 0.00000 0.00002 0.00041 0.00348 0.01522 0.04259 0.08804 0.14804 0.21609 0.28602 0.35357 0.41640 0.47357 0.52498 0.57096 0.61201 155 also very high f o r lower maturities ; U p to the seventh year i t i s more than 100% of the annual investment component of the premium. Table 6.14 shows the r e l a t i o n s h i p between the r i s k premium and the age of the purchaser f o r this p a r t i c u l a r contract. In this case the age of the purchaser i s an important v a r i a b l e i n determining the r i s k premium. The r i s k premium i s highly s e n s i t i v e to the p r o b a b i l i t y of death i n e a r l i e r periods, because of the high values of annual premium on the put f o r these early maturities. For a 20 year old purchaser the r i s k premium i s 4.41% and decreases to 4.08% f o r a 30 year old due to the higher p r o b a b i l i t i e s of death i n the early twenties than i n the early t h i r t i e s . premium goes to 9.48% Then the r i s k f o r a 40 year old and to the extremely high value of 27.54% f o r a 50 year o l d . The nature of the fixed minimum guarantee presented i n this example i s quite d i f f e r e n t from a guarantee proportional to the investment component of the premiums paid as shown i n the other examples. The order of magnitude of the equilibrium r i s k premiums r e f l e c t s this difference. There i s probably a market f o r a great v a r i e t y of contracts,depending on the purchaser's attitude towards r i s k and h i s expectations about the future. 6.10 Summary and Conclusions The equilibrium p r i c i n g model of equity-linked l i f e insurance p o l i c i e s with asset value guarantees developed i n Chapter 5,has been applied i n this chapter to solve f i v e p a r t i c u l a r numerical examples. The examples have been chosen to represent a variety of possible guarantees and to show the e f f e c t of important parameters such as the r i s k l e s s i n t e r e s t rate and the variance rate of return on the reference p o r t f o l i o . Naturally, there exist many other possible type of guarantees and values for the parameters; Table 6.14: Risk Premium versus Age of Purchaser at Entry Guarantee = M o r t a l i t y Table Risk Free Rate Variance Rate Term of Contract = = = = Age of Purchaser at Entry (years) 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 20.00 (Fixed amount f o r a l l maturities) CA 1958-64 0.08 0.01846 (TSE) 20 Level of Annual Risk Premium (%) 4.410 4.417 4.349 4.228 4.082 3.940 3.829 3.772 3.782 3.868 4.028 4.261 4.566 4.936 5.370 5.863 6.413 7.028 7.728 8.535 9.477 10.565 11.815 13.236 14.832 16.599 18.523 20.587 22.772 25.084 27.544 157 but most of these cases can be s o l v e d by a d i r e c t a p p l i c a t i o n o f the same p r i c i n g model. The p r o p o r t i o n of the investment p a i d s e t as guarantee, the f o u r t h was which i n the f i r s t 100% and i n which i n the the r e s u l t i n g r i s k premiums may be s u b s t a n t - d i f f e r e n t . I t has a l s o been seen t h a t changing the r i s k l e s s i n t e r e s t r a t e the v a r i a n c e r a t e does not c o m p l i c a t e the computations, the r e s u l t i n g r i s k premiums To h i g h l i g h t old o f the premiums $20, make v e r y l i t t l e d i f f e r e n c e i n the a p p l i c a t i o n o f the model, a l t h o u g h , n a t u r a l l y , or t h r e e examples was 75%, o r the amount o f the f i x e d guarantee, f i f t h example was ially component although again are q u i t e d i f f e r e n t . the d i f f e r e n c e s between the examples ,assume a 40 y e a r p e r s o n p u r c h a s i n g a 20 y e a r the e q u i l i b r i u m r i s k premiums contract. The f o l l o w i n g t a b l e summarizes t h a t he would have to pay a c c o r d i n g to the s p e c i f i c a t i o n s o f each n u m e r i c a l example: L e v e l o f Annual R i s k Premium (%) 40 y e a r s o l d p u r c h a s e r - 20 y e a r s term Example No. 1 r = 4% o 2 = 0.01846 E = 100% 1.86% Example No. 2 r = 8% a 2 = 0.01846 E = 100% 0.13% Example No. 3 r = 8% a 2 = 0.04 E = 100% E = 75% 0.96% Example No. 4 r = 4% a 2 = 0.01846 0.23% Example No. 5 r = 8% a 2 = 0.01846 T a b l e 6.15 E = $20 9.48% g i v e s a summary o f the v a l u e s o b t a i n e d f o r the annual premiums f o r the put i n the f i v e examples c o n s i d e r e d ( f o r m a t u r i t i e s o f one 158 Table 6.15: Summary of Annual Premium f o r the Put Annual Premium f o r the Put (%) Example No.1 Example No.2 Example No.3 Example No.4 Example No.5 No. years to Expiration 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 i I 2 a = r = E = TSE 4% 100% TSE 8% 100% 4% 8% 100% TSE 4% 75% 3.52 3.38 3.33 3.28 3.21 3. 13 3.04 2.94 2.84 2.74 2.64 2.54 2.44 2.34 2.24 2.14 2.05 1.96 1.87 1.79 2.18 1.71 1.41 1.17 0.98 0.82 0.68 0.57 0.47 0.39 0.33 0.27 0.23 0. 19 0.16 0.14 0. 12 0.11 0.10 0.10 4.37 3.90 3.60 3.34 3.09 2.86 2.63 2.42 2.22 2.04 1.86 1.71 1.56 1.43 1.31 1.20 1.10 1.01 0.93 0.87 0.04 0.05 0.08 0. 10 0.13 0. 15 0. 18 0.19 0.21 0.23 0.24 0.24 0.25 0.26 0.26 0.26 0.25 0.25 0.25 0.24 TSE 8% $20.00 1746.23 . 786.21 466.89 307.73 212.64 149.59 104.88 71.87 47.35 29.73 17.79 10.21 5.67 3.07 1.63 0.86 0.45 0.24 0.14 0.10 159 to twenty years). This table permits a comparison of the equilibrium value of the guarantee, with known date of expiration, for the d i f f e r e n t examples. F i n a l l y , i t should be emphasized that the p r i c i n g model presented i n t h i s study i s an equilibrium model under the assumption of perfect markets. In i t s p r a c t i c a l applications market imperfections transaction costs, s e l l i n g costs, p r o f i t s and taxes should be such as considered. However,the basic d i f f i c u l t y facing insurance companies has been the c a l c u l a t i o n of the value of this guarantee. The transaction incurred by them can be found r e a d i l y by conventional costs methods. In p r a c t i c e insurance companies tend to charge a uniform r i s k premium for a given contract independent of age. Current p r a c t i c e i s to charge an a d d i t i o n a l 1% for a guarantee of 75 to 100% of premiums paid. I t has been shown that while this i s within the range of results produced i n this study, the r i s k premium depends on the following: age of the purchaser at entry, the term of the contract, the nature of the guarantee, the variance rate of the return on the reference p o r t f o l i o , and the r i s k l e s s i n t e r e s t rate. There i s no t h e o r e t i c a l j u s t i f i c a t i o n for charging a uniform r i s k premium. The model developed i n t h i s thesis gives a sound t h e o r e t i c a l basis and an expedient numerical procedure f o r p r i c i n g these contracts. 160 OPTIMAL INVESTMENT STRATEGIES FOR THE CHAPTER 7: SELLERS OF EQUITY LINKED LIFE INSURANCE POLICIES WITH ASSET VALUE GUARANTEES. 7.1 Introduction In Chapter 5 ah equilibrium model f o r p r i c i n g equity based l i f e insurance contracts with minimum guarantee was this model was developed. In Chapter 6 applied to some t y p i c a l contracts to determine the r i s k premium that insurance companies should charge, i n equilibrium, for bearing a d d i t i o n a l l y the investment r i s k associated with these contracts. This chapter i s concerned with the d e r i v a t i o n of appropriate investment p o l i c i e s for insurance companies to enable them to hedge against the major r i s k s associated with the provision of asset value guarantees i n equity l i n k e d l i f e insurance p o l i c i e s . In addition, i t w i l l examine the question whether the hedging p o l i c y can be followed by the l i f e insurance company without the necessity of borrowing a d d i t i o n a l funds. a numerical example w i l l be 7.2 To conclude the chapter presented. Hedging P o l i c y . When presenting the Black-Scholes' option p r i c i n g model i n Chapter 2, i t was shown how to create a hedged p o s i t i o n consisting of one share of common stock long and 1/Wg the p r i c e of the stock. options short, the value of which would not depend on This hedging process plus the fact that the rate of return on the equity p o s i t i o n , being r i s k l e s s , must be equal to the r i s k free i n t e r e s t rate were used as the basis f o r deriving p a r t i a l d i f f e r e n t i a l equations governing The analysis was paying stock. the value of an option on a non-dividend paying stock. extended to cover the case of an option on a dividend Furthermore i t was shown that this model could also be used to f i the value of an option on the reference p o r t f o l i o i n the case of equity 161 based l i f e insurance contracts. If the insurance company follows the investment policy determined by the model, through the hedging process described above, i t w i l l bear no risk and w i l l make no profit or loss. As discussed in earlier chapters,an equity linked l i f e insurance policy with asset value guarantee can be viewed as a regular l i f e insurance contract (endownment) plus a c a l l option on a reference portfolio. The insurance company is selling short a c a l l option on the reference portfolio. To eliminate a l l the risk assumed by selling short the c a l l option, the insurance company must form a hedged position by taking a long position in the reference portfolio. Assuming for the moment a known date of expiration of the contract, the long position in the reference portfolio must be (from (2.2) and (2.3) in Chapter 2): (7.1) xw x where x is the value of the reference portfolio and w is the partial x derivative of the value of the option with respect to the value of the reference portfolio. If the insurance company maintains this hedged position,it w i l l be immunized against loss (and p r o f i t ) . It is essential to note that the hedged position must be maintained continuously throughout the term of the contract for this immunization to be totally effective. This is done by adjusting the amounts invested in the reference portfolio and in the risk free asset. The value of the call option is an increasing function of the value of the reference portfolio, that is: 162 (7.2) 0 < w - and < 1 x - therefore (7.3) x w < x x - Hence, the optimal strategy dictates that not a l l the investment component be a c t u a l l y invested i n the reference p o r t f o l i o . This i s possible i n the case of a number of United Kingdom equity based contracts where the actual wording says "/invested or deemed to be invested." Notice that w x gives the proportion of the value of the reference p o r t f o l i o (x) which should be a c t u a l l y invested i n the reference p o r t f o l i o to maintain the hedged p o s i t i o n . discuss s p e c i f i c numerical This concept w i l l become clearer when we examples. The continuous adjustment of the p o s i t i o n i n the reference p o r t f o l i o gives r i s e to transaction costs. These are ignored i n the present study. For the single premium contract the amount invested i n the reference p(7.4) o r t f o l ixo w.has= anxa nN(d a l y t)i c a l expression given b y : x 1 x where d^ and N( ) are defined i n (5.9). For the continuous premium contract and the p e r i o d i c premium contract there i s no a n a l y t i c a l expression f o r w , but w X 2 can e a s i l y be obtained by X the numerical method described i n Chapter 5. It has been shown that that amount that must be invested i n the reference p o r t f o l i o at time T to form a hedged p o s i t i o n with one c a l l 1 See Black-Scholes [3,4]. 2 In fact w can be computed d i r e c t l y from w. defined i n Chapter 6: 1+1,3 " i - 1 - J w.x 2h x W Using the notation 163 option sold short, assuming known date of expiration t, (7.1) x w (x, is g) t-T, Let a(x,t) be the probability of the benefits under any one contract being payable at time t given that the policyholder is alive at time T . T varies from 0 to n-1 and t varies from T+1 to n (n being the term of the contract). v 3 Then, the average proportion invested in the reference portfolio at time x, w x (x)> is n (7.7) W (T) = I a(x,t) w (x, t-x, g) t=x+l and the average amount invested in the reference portfolio at time x, L(x), (7.8) is L(x) = x W (x) x 3 The probabilities a(x,t) can be computed using (5.49) and (5.51), taking into consideration that the l i f e aged x at the time the contract is issued is aged x+x at time x and that the time to maturity t is now t-x. Thus (7.5) a(x,t) = *x+t-l ~ £x+t x+x £ x+n-l = ~~ x+x f o r t = T f l > n _i for x = 0, n-1 for x = 0, n-1 £ (7.6) a(x,n) & Notice that x is here the usual actuarial symbol for age (there should be no confusion with the value of the reference portfolio). 164 7.3 F e a s i b i l i t y of the P o l i c y While i n p r i n c i p l e the hedging p o l i c y can always be followed to eliminate a l l r i s k s , a p r a c t i c a l question i s whether the p o l i c y can be followed without the necessity of borrowing a d d i t i o n a l funds, since some insurance companies may have constraints on such borrowing. To examine this question i t i s necessary to consider: (1) The assets and l i a b i l i t i e s of the insurance company f o r the contract at each point i n time. Provided that the hedging p o l i c y has been followed,these w i l l always be equal since the p o l i c y ensures that neither p r o f i t s nor losses are made. (2) The a l l o c a t i o n of the insurance company's assets between the reference p o r t f o l i o , the r i s k l e s s asset,and the present value of the future premiums receivable. The present value of the l i a b i l i t i e s at time x ( f o r known date of expiration t) can be viewed as the sum of the present value of the guarantee and the value of the c a l l option at time x: (7.9) PV of l i a b i l i t i e s = ge" r ( t ~ x ) + w(x, t-x, g) But the present value of the l i a b i l i t i e s at time x can also be viewed as the sum of the present value of the reference p o r t f o l i o , PV^(x(t)), and the value of the put option at time (7.10) PV of l i a b i l i t i e s = x^: P V ( x ( t ) ) + p(x, t-x, g) T The present value of the assets at time x i s given by the available funds (cash p o s i t i o n : invested i n the reference p o r t f o l i o and the r i s k l e s s 4 This analysis i s equivalent to the one presented i n Chapter 5 f o r the t o t a l present value of the contract. Notice that known date of expiration,t, has been assumed.Mortality considerations w i l l be added l a t e r on. 165 asset) and the p r e s e n t value o f the f u t u r e premiums r e c e i v a b l e a t time T , PVFP: (7.11) PV If of a s s e t s = funds a v a i l a b l e + PVFP the hedging s t r a t e g y has been f o l l o w e d , t h e l i a b i l i t i e s must be e q u a l to the p r e s e n t t h i s case the v a l u e of the funds a c t u a l l y i n v e s t e d p o r t f o l i o has Let value i(x,T) of the followed. (7.12) been shown to be x be value present value of the a s s e t s . i n the Also, the in reference wx the investment i n the r i s k l e s s a s s e t a t time reference of p o r t f o l i o of x T for a such t h a t the hedged s t r a t e g y i s Then: PV of assets = xw + i ( x , x ) + PVFP and (7.13) i(x,x) If = PV of l i a b i l i t i e s - x w - PVFP i ( x , x ) < 0,the b o r r o w i n g o f a d d i t i o n a l funds i s r e q u i r e d to follow the hedged p o l i c y . For the s i n g l e premium c o n t r a c t , i n which the whole premium i s p a i d when the p o l i c y i s i s s u e d , i t i s p o s s i b l e to o b t a i n for the net (7.14) w(x, investment i n the r i s k l e s s t-x, g) = x N(d ) 1 - ge" an a n a l y t i c a l e x p r e s s i o n asset,because: r ( t _ T ) N(d ) 2 and (7.15) PVFP = 0 Substituting = (7.4), ge" (7.9), (7.14) and (7.15) i n t o (7.13) we obtain: (7.16) i(x,t) Thus,it i s never n e c e s s a r y to borrow a d d i t i o n a l funds to f o l l o w the hedged r ( t _ T ) (1 - N ( d ) ) > 0 2 p o l i c y f o r the s i n g l e premium c o n t r a c t . 166 For the-multiple (continuous are no a n a l y t i c a l expressions and periodic) premium contracts there for i ( x , x ) . ' There are, nevertheless, some general properties, which are proven i n Appendix L, that can be summarized as follows: Theorem 1: The investment i n the r i s k l e s s asset i s a decreasing function of the value of the reference p o r t f o l i o (at any given time). This theorem i s also true f o r the single premium contract. Theorem 2: At any given date the maximum borrowing required to follow the hedged p o l i c y i s bounded by the present value of the future put premiums ( r i s k premiums) receivable. A d i r e c t consequence of theorem 2 i s : Corollary: I f the t o t a l value of the put option i s paid at the date of issue of the policy(as a kind of "loading" fee) and i f only the investment component of the premium i s paid continuously or p e r i o d i c a l l y through time, no borrowing w i l l be required to follow the hedged strategy. The p r a c t i c a l implications of this c o r o l l a r y w i l l be seen when we present a numerical example i n Section 7.5. The r e l a t i o n s h i p between i ( x , t ) and x, for a given T , i s shown i n Figure 7.1 The numerical procedure to obtain i(x,x) f o r the p e r i o d i c premium contract i s discussed i n Section 7.4. 167 Figure 7.1: Investment i n the Riskless Asset j OQ j : present value of the future put premiums receivable, x : c r i t i c a l value of the reference p o r t f o l i o . Q The value at which the i-curve cuts the x-axis ( x ) i n Figure 7.1 c can be defined as "the c r i t i c a l value of the reference p o r t f o l i o . " values of the reference p o r t f o l i o greater than x required. £ For some borrowing w i l l be The curve s h i f t s up as x increases, that i s , the maximum possible borrowing required decreases as the option approaches maturity. F i n a l l y , to determine the actual f e a s i b i l i t y of the policy when the hedging p o l i c y i s followed, i t i s necessary to introduce mortality r i s k . The cash p o s i t i o n f o r the contract when mortality i s considered i s derived i n Appendix L. 7.4 Numerical Example: As a numerical example i n this section we w i l l present a case of i the|periodic premium contract. This example gives the actual amounts 168 invested in the reference portfolio and risk free asset to maintain the hedged position. The parameters selected for the example have been those of example number 1 in Chapter 6 assuming, in addition, a 40-year old person purchasing a 10-year policy. As before i t w i l l be assumed that the investment component of the premium is equal to one dollar which is paid once a year at the beginning of the year. The parameters for the numerical example are then: (1) Variance Rate : the variance rate of the TSE Industrial Index (0.01846); (2) Risk Free Rate (3) Guarantee : : 4% per annum (continuous) 100% of the investment component of the premiums paid at death or maturity; (4) Age of purchaser at entry (5) Term of the policy : : 40 years old; and 10 years. To highlight the numerical results obtained, nine different possible performances of the reference portfolio were considered. We have considered the case of a constant return on the reference portfolio of: - 20%, - 15%, - 10%, - 5%, 0%, 5%, 10%, 15% and 20%. policy has been determined once a year The investment just after each premium payment. Tables 7.1 to 7.9 summarize the results obtained. 7 The entries in these tables correspond to: F u l l r e s u l t s are a v a i l a b l e upon request. I t must be kept i n mind that to eliminate r i s k , t h e hedged p o r t f o l i o should be maintained continuously through time. Here we are analyzing the p o r t f o l i o only once a year,but assuming that the hedge i s maintained continuously. 7 169 Column (1) : p o i n t i n time where the investment s t r a t e g y i s e v a l u a t e d , s t a r t i n g j u s t a f t e r the f i r s t premium payment a f t e r the l a s t premium payment Column (2) : (year 0) and ending just (year 9 ) . v a l u e of the r e f e r e n c e p o r t f o l i o a t the c o r r e s p o n d i n g point i n time, g i v e n the p e r c e n t a g e annual i n c r e a s e i n the v a l u e of the r e f e r e n c e portfolio. Column (3) : v a l u e o f the c a l l o p t i o n m o r t a l i t y data) o b t a i n e d Column (4) : by n u m e r i c a l (weighted a c c o r d i n g t o the procedures. p r o p o r t i o n o f the funds deemed t o be i n v e s t e d i n the r e f e r e n c e p o r t f o l i o a c t u a l l y i n v e s t e d i n i t , o b t a i n e d by u s i n g the v a l u e s i n Column ( 3 ) . ^ Column (5) : l o n g p o s i t i o n i n the r e f e r e n c e p o r t f o l i o a hedged p o s i t i o n ; ^ 1 Column (6) : : by m u l t i p l y i n g Column (2) by Column ( 4 ) . cash p o s i t i o n o f the i n s u r a n c e the c o r r e s p o n d i n g Column (7) obtained r e q u i r e d to m a i n t a i n company f o r t h i s c o n t r a c t a t p o i n t i n time. investment i n the r i s k l e s s a s s e t r e q u i r e d to f o l l o w the hedged s t r a t e g y ; t h i s i s the d i f f e r e n c e between the l o n g p o s i t i o n i n the reference p o r t f o l i o (Column (5)) and the cash p o s i t i o n (Column ( 6 ) ) . If the investment i s n e g a t i v e , i t i m p l i e s t h a t b o r r o w i n g i s r e q u i r e d . Tables 7.1 to 7.9 were o b t a i n e d No. 9 w h i c h g i v e s the v a l u e s from the output shown i n these o f Computer•Program t a b l e s , b u t f o r a whole range g F o l l o w i n g our c o n v e n t i o n a l n o t a t i o n , the performance o f the r e f e r e n c e p o r t f o l i o i s compounded y e a r l y . 9 See f o o t n o t e 2 and e q u a t i o n See equation (7.8). (7.7). 170 of d i f f e r e n t values of the reference p o r t f o l i o (400 values every 0.05 12 13 dollars). ' Computer Program No. 9 i s shown i n Appendix K. As can be seen from the tables, whenever the return on the reference p o r t f o l i o i s negative,the proportion of the premiums paid invested i n the reference p o r t f o l i o i s quite small and decreases with time to maturity. With a return of - 20% and - 15% the long p o s i t i o n i n the reference p o r t f o l i o one year before maturity i s equal to zero and with a - 10% return i t i s equal to only three cents. As a consequence of t h i s most of the funds received by the insurance company (cash position) are invested i n the r i s k l e s s asset. When the return on the reference p o r t f o l i o i s p o s i t i v e , however, most of the funds received by the insurance comapny are invested i n the reference p o r t f o l i o and i n a few cases, with returns of 15% and 20%, a d d i t i o n a l 14 borrowing i s required to maintain the hedged p o s i t i o n . As expected,the net borrowing required i s very small i n a l l cases. For the d i f f e r e n t performances of the reference p o r t f o l i o summarize the investment considered, Table 7.10 and Figure 7.2 i n the r i s k l e s s asset and Table 7.11 and Figure 7.3 summarize the proportion of the amount deemed to be invested i n the reference p o r t f o l i o , actually invested. The output of Computer Program No. 9 can also be used to f i n d interesting variable: another the value of the reference p o r t f o l i o above which " T o be able to use the output of Computer Program No. 9 without i n t e r p o l a t i o n , the value of the reference p o r t f o l i o corresponding to a certain percentage annual increase was approximated to the nearest factor of f i v e cents. 13 The apparent errors of one cent are due to rounding e r r o r s . See Tables 7.8 and 7.9. 171 net borrowing i s required to maintain the hedged p o r t f o l i o . This variable has been defined i n this study as the " c r i t i c a l value of the reference p o r t f o l i o " at a given point i n time and can be obtained by finding the f i r s t value of the reference p o r t f o l i o f o r which a p o s i t i v e net borrowing i s r e q u i r e d . ' 1 Table 7.12 shows the c r i t i c a l values of the reference p o r t f o l i o j u s t a f t e r every premium payment date. A f t e r the f i r s t premium payment (year 0) an i n f i n i t e return i s required to get to the c r i t i c a l value. A f t e r the second premium payment (year 1) a return of 265% i s required. As maturity i s approached the constant annual return on the reference p o r t f o l i o required to reach the c r i t i c a l value decreases; premium payment this return i s approximately 10%.^ a f t e r the l a s t I t should be kept i n mind, however, that the maximum amount to be borrowed i s bounded by the present value of the " r i s k premiums" unpaid, which decreases with time to maturity. ''Again the values have been approximated of f i v e cents. to the nearest factor '^This i s why some borrowing i s required with returns of 15% and 20%. 172 Investment P o l i c y Table 7.1 0.01846 0.04000 Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium f o r the Put Annual Increase i n Value of the Reference P o r t f o l i o Year Value of Reference Portfolio Weighted Proportion Value of Invested i n Option Reference Portfolio (1) (2) (3) (4) 0 1.00 1.91 1 1.80 2 100% of investment component on death or maturity 40 years 10 years 0.02742 - 20% Long P o s i t i o n i n Reference Portfolio Cash Position Investment i n Riskless Asset (5) (6) (7) 0.85 0.85 1.03 +0.17 1.78 0.83 1.49 1.89 +0.41 2.45 1.52 0.77 1.90 2.65 +0.76 3 2.95 1.14 0.68 2.02 3.34 + 1.32 4 3.35 0.73 0.55 1.84 4.05 +2.21 5 3.70 0.38 0.37 1.38 4.25 +3.47 6 3.95 0.13 0.18 0.70 5.81 +5.11 7 4.15 0.02 0.04 0.17 6.95 +6.78 8 4.35 0.00 0.00 0.01 8.24 +8.23 9 4.45 0.00 0.00 0.00 9.61 +9.61 173 Investment P o l i c y Table 7.2 0.01846 0.04000 Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase i n Value of the Reference P o r t f o l i o Year Value of Reference Portfolio Weighted Value of Option Proportion Invested i n Reference Portfolio (1) (2) (3) (4) 0 1.00 1.91 1 1.85 2 100% of investment component on death or maturity 40 years 10 years 0.02742 = - 15% Long P o s i t i o n i n Reference Portfolio Cash Position Investment i n Riskless Asset (5) (6) (7) 0.85 0.85 1.03 +0.17 1.83 0.83 1.54 1.93 +0.39 2.55 1.60 0,79 2.01 2.73 +0.72 3 3.20 1.31 0.72 2.32 3.52 +1.20 4 3.70 0.93 0.62 2.28 4.25 +1.97 5 4.15 0.57 0.47 1.96 5.04 +3.08 6 4.55 0.27 0.29 1.34 5.95 +4.61 7 4.85 0.07 0.11 0.55 7.00 +6.46 8 5.10 0.01 0.01 0.07 8.25 +8.17 9 5.35 0.00 0.00 0.00 9.61 +9.61 174 Table 7.3 Investment P o l i c y 0.01846 0.04000 Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase i n Value of the Reference P o r t f o l i o Year Value of Reference Portfolio Weighted Proportion Value of Invested i n Option Reference Portfolio 100% of investment component on death or maturity 40 years 10 years 0.02742 = - 10% Long P o s i t i o n i n Reference Portfolio Cash Position Investment i n Riskless Asset (1) (2) (3) 0 1.00 1.91 0.85 0.85 1.03 +0.17 1 1.90 1.87 0.84 1.99 1.97 +0.38 2 2.70 1.72 0.81 2.17 2.85 +0.68 3 3.45 1.50 0.76 2.62 3.70 +1.08 4 4.10 1.19 0.69 2.82 4.51 +1.69 5 4.70 0.86 0.59 2.75 5.33 +2.58 6 5.20 0.50 0.44 2.26 6.18 +3.92 7 5.70 0.23 0.26 1.49 7.16 +5.67 8 6.15 0.05 0.09 0.55 8.29 +7.74 9 6.50 0.00 0.00 0.03 9.61 +9.58 (4) (5) (6) (7) 175 Table 7.4 Investment P o l i c y Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase i n Value of the Reference P o r t f o l i o Year Value of Reference Portfolio (1) (2) (3) 0 1.00 1.91 0.85 1 1.95 1.91 2 2.85 3 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 - 5% Weighted Proportion Long P o s i t i o n Value of Invested i n i n Reference Option Reference Portfolio Portfolio Cash Position Investment i n Riskless Asse (6) (7) 0.85 1.03 +0.17 0.84 1.64 2.02 +0.38 1.84 0.82 2.34 2.97 +0.63 3.70 1.69 0.79 2.93 3.90 +0.97 4 4.50 1.48 0.75 3.36 4.80 +1.44 5 5.30 1.24 0.69 3.66 5.72 +2.05 6 6.05 0.95 0.61 3.69 6.63 +2.94 7 6.75 0.62 0.49 3.33 7.55 +4.23 8 7.40 0.29 0.32 2.38 8.54 +6.15 9 8.05 0.06 0.11 0.87 9.67 +8.80 (4) (5) 176 Table 7.5 : Investment P o l i c y Variance Rate Risk Free Rate Guarantee = = = Age of Purchaser at Entry Term of Contract Annual Premium f o r the Put Annual Increase i n Value of the Reference P o r t f o l i o = = = 0.02742 = 0% Year Value of Reference Portfolio Weighted Proportion Value of Invested i n Option Reference Portfolio (1) (2) (3) (4) 0 1.00 1.91 1 2.00 2 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years Cash Position Investment i n Riskless Asset (5) (6) (7) 0.85 0.85 1.03 +0.17 1.95 0.85 1.69 2.06 +0.37 3.00 1.96 0.84 2.51 3.10 +0.58 3 4.00 1.94 0.83 3.30 4.14 +0.84 4 5.00 1.87 0.81 4.05 5.19 +1.14 5 6.00 1.76 0.80 4.74 6.24 +1.50 6 7.00 1.61 0.76 5.35 7.29 +1.94 7 8.00 1.39 0.73 5.87 8.33 +2.46 8 9.00 1.12 0.70 6.25 9.36 +3.11 9 10.00 0.74 0.64 6.40 10.35 +3.95 \ Long P o s i t i o n i n Reference Portfolio 177 Investment Table 7.6 Policy 0.01846 0.04000 Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium f o r the Put Annual Increase i n Value of the Reference P o r t f o l i o Year Value of Reference Portfolio Weighted Proportion Value of Invested i n Option Reference Portfolio (1) (2) (3) (4) 0 1.00 1.91 1 2.05 2 100% of investment component on death or maturity 40 years 10 years 0.02742 = 5% Long P o s i t i o n i n Reference Portfolio Cash Position Investment i n Riskless Asset (5) (6) (7) 0.85 0.85 1.03 +0.17 1.99 0.85 1.75 2.10 +0.35 3.15 2.09 0.85 2.69 3.22 +0.54 3 4.30 2.19 0.85 3.67 4.39 +0.72 4 5.55 2.33 0.86 4.78 5.65 +0.86 5 6.80 2.43 0.87 5.90 6.90 +1.00 6 8.15 2.56 0.88 7.20 8.24 +1.04 7 9.55 2.68 0.90 8.63 9.62 +0.98 8 11.05 2.84 0.94 10.37 11.08 +0.71 9 12.60 3.01 0.98 12.36 12.61 +0.25 178 Investment P o l i c y Table 7.7 0.01846 0.04000 Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase i n Value of the Reference P o r t f o l i o Year Value of Reference Portfolio Weighted Value of Option Proportion Invested i n Reference Portfolio (1) (2) (3) (4) 0 1.00 1.91 1 2.10 2 100% of investment component on death or maturity 40 years 10 years 0.02742 = 10% Long P o s i t i o n i n Reference Portfolio Cash Position Investment i n Riskless Asset (5) (6) (7) 0.85 0.85 1.03 +0.17 2.04 0.86 1.80 2.14 +0.34 3.30 2.22 0.87 2.86 3.35 +0.50 3 4.65 2.49 0.88 4.10 4.70 +0.60 4 6.10 2.82 0.90 5.50 6.13 +0.64 5 7.70 3.24 0.93 7.12 7.71 +0.59 6 9.50 3.80 0.95 9.05 9.48 +0.43 7 11.45 4.49 0.98 11.20 11.42 +0.22 8 13.60 5.33 1.00 13.53 13.57 +0.04 9 15.95 6.34 1.00 15.95 15.95 0.00 179 Investment P o l i c y Table 7.8 Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium f o r the Put Annual Increase i n Value of the Reference P o r t f o l i o Year Value of Reference Portfolio Weighted Value of Option Proportion Invested i n Reference Portfolio (1) (2) (3) (4) 0 1.00 1.91 1 2.15 2 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 = 15% Long P o s i t i o n i n Reference Portfolio Cash Position Investment i n Riskless Asset (5) (6) (7) 0.85 0.85 1.03 +0.17 2.08 0.86 1.85 2.19 +0.34 3.45 2.35 0.88 3.03 3.48 +0.46 3 5.00 2.81 0.91 4.52 5.01 +0.49 4 6.75 3.41 0.93 6.30 6.73 +0.43 5 8.75 4.23 0.96 8.42 8.70 +0.29 6 11.05 5.31 0.99 10.88 10.99 +0.11 7 13.75 6.76 1.00 13.71 13.70 -0.01 8 16.80 8.53 1.00 16.80 16.77 -0.04 9 20.30 10.69* 1.00 20.30 20.30* (*) extrapolated values 0.00 180 Table 7.9 Investment P o l i c y Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium f o r the Put Annual Increase i n Value of the Reference P o r t f o l i o Year = 20% Value of Reference Portfolio Weighted Value of Option (1) (2) (3) (4) (5) (6) (7) 0 1.00 1.91 0.85 0.85 1.03 +0.17 1 2.20 2.12 0.87 1.91 2.23 +0.32 2 3.65 2.52 0.89 3.26 3.66 +0.41 3 5.35 3.13 0.92 4.94 5.33 +0.39 4 7.45 4.08 0.97 7.12 7.39 +0.27 5 9.95 5.40 0.98 9.79 9.87 +0.09 6 12.90 7.15 1.00 12.86 12.83 -0.03 7 16.50 9.51 1.00 16.50 16.45 -0.04 8 20.80 12.53* 1.00 20.80 20.77* -0.03 9 25.95 16.34* 1.00 25.95 25.95* 0.00 (*) extrapolated values Proportion Invested i n Reference Portfolio 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 Long P o s i t i o n i n Reference Portfolio Cash Position Investment i n Riskless Asset s 181 Table 7.10 : Investment i n the Riskless Asset Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium f o r the Put Year = 0.01846 = 0.0400 = 100% of investment component on death or maturity = 40 years = 10 years = 0.02742 Investment i n the Riskless Asset Annual Increase i n Value of Reference P o r t f o l i o (%) -20 -15 -10 -5 0 5 10 15 0 +0.17 +0.17 +0.17 +0.17 +0.17 +0.17 +0.17 +0. 17 +0.17 1 +0.41 +0.39 +0.38 +0.38 +0.37 +0.35 +0.34 +0.34 +0.32 2 +0.76 +0.72 +0.68 +0.63 +0.58 +0.54 +0.50 +0.46 +0.41 3 +1.32 +1.20 +1.08 +0.97 +0.84 +0.72 +0.60 +0.49 +0.39 4 +2.21 +1.97 +1.69 +1.44 +1.14 +0.86 +0.64 +0.43 +0.27 5 +3.47 +3.08 +2.58 +2.05 +1.50 +1.00 +0.59 +0.29 +0.09 6 .+5.11 +4.61 +3.92 +2.94 +1.94 +1.04 +0.43 +0.11 -0.03 7 +6.78 +6.46 +5.67 +4.23 +2.46 +0.98 +0.22 -0.01 -0.04 8 +8.23 +8.17 +7.74 +6.15 +3.11 +0.71 +0.04 -0.04 -0.03 9 +9.61 +9.61 +9.58 +8.80 +3.95 +0.25 0.00 0.00 20 0.00 182 FIGURE 7 . 2 : INVESTMENT IN THE RISKLESS ASSET 14.00- 12.00 10.00 8. 00 INVESTMENT 6.00- IN RISKLESS ASSET 0% 4.00 2.00 # # # .00- -2.00-.........|..... 00 2.00 # t 20% t # . . . . | . . . . . . . . . J 4.00 6.00 TIME I 8.00 10. 00 183 Table 7.11 : Proportion Invested In the Reference P o r t f o l i o = = = 0.01846 0.0400 100% of investment component on death or maturity = 40 years = 10 years = 0.02742 Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium. f o r the Put Proportion Invested i n the Reference P o r t f o l i o Annual Increase i n Value of Reference P o r t f o l i o (%) Year 5 10 15 20 0.85 0.85 0.85 0.85 .85 .85 .86 .86 0.87 .82 .84 .85 .87 .88 .89 .76 .79 .83 .85 .88 .91 .92 .62 .69 .75 .81 .86 .90 .93 .97 .37 .42 .59 .69 .80 .87 .93 .96 .98 6 .18 .29 .44 .61 .76 .88 .95 .99 1.00 7 .04 .11 .26 .49 .73 .90 .98 1.00 1.00 8 .00 .01 .09 .32 .70 .94 1.00 1.00 1.00 9 .00 .00 .00 . 11 .64 .98 1.00 1.00 1.00 -20 -15 -10 -5 0 0 0.85 0.85 0.85 0.85 0.85 : 1 .83 .83 .84 .84 2 .77 .79 .81 3 .68 .72 4 .55 5 184 FIGURE 7 . 3 : PROPORTION INVESTED IN THE REFERENCE PORTFOLIO 1.60- 1.40 1.20 20% 1.00 # # # # # # # # PROPORTION INVESTED IN REFERENCE PORTFOLIO .80# # # # # # 0% 60 40 20 -20% 00-i.. .00 I 2.00 i 4.00 . # . . . . # . . . . j . . . 6- 00 TIME 8.00 10.00 185 Table 7.12 : C r i t i c a l Values of the Reference Portfolio Variance Rate Risk Free Rate Guarantee = 0.01846 = 0.04 = 100% of investment component on death or maturity Age of Purchaser-at Entry = 40 years Term of Contract = 10 years Annual Premium for the Put = 0.02742 Year C r i t i c a l Value of the Reference Portfolio ($) 0 2.80 1 4.65 2 6.40 3 8.00 4 9.50 5 10.95 6 12.25 7 13.45 8 14.35 9 15.40 186 7.5 Summary and Conclusions The model developed i n Chapter 5 to f i n d the equilibrium p r i c e of an equity linked l i f e insurance contract with asset value guarantee can also be used to f i n d the optimal investment strategy f o r the s e l l e r s of these contracts to hedge against the investment r i s k assumed by giving the guarantee. In this chapter i t was shown how the Black-Scholes hedging process can be adapted to this purpose. I t was also shown that the amounts that have to be invested i n the reference p o r t f o l i o are smaller than the value of the reference p o r t f o l i o assigned to the policyholder. To make the prescribed investment strategy possible, l i f e insurance companies must have l e g a l f l e x i b i l i t y as to where to invest the premiums received from' the policyholders, because i f they were required by law to a c t u a l l y invest the investment component of the premium paid i n the reference p o r t f o l i o they would not be able to follow the indicated strategy. The model implies a continuous adjustment of the amounts invested. In p r a c t i c e this i s impossible. Further studies could investigate the optimal r e v i s i o n time, considering both the transaction costs and the e f f e c t s of these discontinuous adjustments on the elimination of r i s k . 187 Bibliography [1] Ayres, H . F . , "Risk Aversion in the Warrant Markets," Industrial Management Review, Vol. 5, No. 1, F a l l , 1963. [2] Bachelier, L . , "Theory of Speculation" (translation) in the Random Character of Stock Market Prices, ed. Cootner, M.I.T. Press, Cambridge, Mass., 1964. [3] Black, F. and Scholes, M . J . , "The Pricing of Options and Corporate L i a b i l i t i e s , " Journal of P o l i t i c a l Economy, Vol. 81, No. 3, May-June 1973. [A] Black, F. and Scholes, M . J . , "The Valuation of Option Contracts and a Test of Market Efficiency. Journal of Finance, Vol. 27, May 1972. 11 [5] Black, F. "The Pricing of Complex Options and Corporate L i a b i l i t i e s . " Unpublished draft, Graduate School of Business Administration, University of Chicago, June 1974. [6] Chen, H . Y . , "A Dynamic Programming Approach to the Valuation of Warrants." Unpublished Ph.D. Dissertation, Graduate School of Business Administration, University of California, Berkeley, 1969. [7] Chen, H . Y . , "A Model of Warrant Pricing in a Dynamic Market," Journal of Finance, Vol. 25, No. 5 (December 1970). [8] Cox, T.C. and Ross, S.A., "The Pricing of Options for Jump Processes," Working Paper No. 2-75, The Warton School of Business, University of Pennsylvania, 1975. [9] Dahlquist, G. and Bjorck, A . , Numerical Methods, Prentice-Hall Inc., 1974. [10] Di Paolo, F . , "An Application of Simulated Stock Market Trends to Investigate a Ruin Problem," Transactions Society of Actuaries, Vol. XXI, 1969. [11] Forsythe, G.E. and Wasow, W.R., Finite Difference Methods for Partial Differential Equations, John Wiley & Sons Inc. 1960. 188 [12]. Foster, M., "An Evaluation of the Black-Scholes Warrant Valuation Model and its Use as an Investment Decision Model." MBA graduating essay, University of British Columbia, June 1971. [13] Frankel, T . , "Variable Annuities, Variable Insurance and Separate Accounts," Boston University Law Review, Vol. 51, 1971. [14] Galai, D. "Pricing of Options and the Efficiency of the Chicago Board Options Exchange." Unpublished Ph.D. Dissertation, University of Chicago, March 1975. [15] Grant, A.T. and Kingsworth, G.A. "Unit Trusts and Equity Linked Endowment Assurances," Journal of the Institute of Actuaries, Vol. 93, Part III, No. 396, 1967. [16] Ingersoll, J . E . J r . , "A Theoretical and Empirical Investigation of the Dual Purpose Funds: An Application of Contingent Claims Analysis," Working Paper No. 782-75, Massachusetts Institute of Technology, A p r i l , 1975. [17] Jordan, C.W. J r . , "Society of Actuaries' Textbook on Life Contingencies," The Society of Actuaries, Chicago 1967. [18] Kahn, P.M., "Projections of Variable Life Insurance Operations," Transactions Society of Actuaries, Vol. XXIII, 1971. [19] Kassouf, S.T., "A Theory and a Econometric Model for Common Stock Purchase Warrants," Analytical Publishers Co., New York, 1965. [20] Kassouf, S.T., "Warrant Price Behavior: 1945 to 1964," The Financial Analysts Journal, JanuaryFebruary, 1968. [21] Kensicki, P.R., "Consumer Valuation of Life Insurance A Capital Budgeting Approach," The Journal of Risk and Insurance, December 1974. [22] Leckie, S.H., "Variable Annuities and Variable Life Insurance in the United States of America," Journal of the Institute of Actuaries Student's Society, Vol. 20, Part 2, October 1972. 189 [23] Leonard, R.J., [24] Lusztig, P.A. [25] McCracken, D.D. [26] McGill, D.M., L i f e Insurance, Richard D. Irwin, Inc., 1967. [27] McKean, H.P., "A Free Boundary Problem f o r the Heat Equation A r i s i n g From a Problem of Mathematical Economics," I n d u s t r i a l Management Review, V o l . 6., No. 2, Spring, 1965. [28] McKean, H.P., Stochasic Integrals New York: Press, 1969. [29] Merton, R.C., "Lifetime P o r t f o l i o Selection under Uncertainty: The Continuous Time Case," Review of Economics and S t a t i s t i c s , Vol. 51, No. 3 (August 1969). [30] Merton, R.C., "Optimum Consumption and P o r t f o l i o Rules i n a Continuous-Time Model," Journal of Economic Theory, V o l . 3, No. 4, December 1971. [31] Merton, R.C., "An Intertemporal C a p i t a l Asset P r i c i n g Model," Econometrica, V o l . 41, No. 5, (September 1973). [32] Merton, R.C., "The Theory of Rational Option P r i c i n g , " B e l l Journal of Economics and Management Science, V o l . 4, No. 1, Spring 1973. [33] Merton, R.C., "On the P r i c i n g of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, V o l . 29, May 1974. [34] Merton, R.C., "Option P r i c i n g when Underlying Stock Returns are Discontinuous," Working Paper No. 787-75, Massachusetts I n s t i t u t e of Technology, A p r i l 1975. [35] Praetz, P.D., "Random Walk and Investment Theory," Journal of the I n s t i t u t e of Actuaries Students' Society, Vol. 20, Part 3, June 197: "An Empirical Examination of a New General Equilibrium Model f o r Warrant Pricing." Unpublished M.S. Thesis, M.I.T., September 1971. and Schwab, B., and Dorn, W.S. "Managerial Finance i n a Canadian Setting." Holt, Rinehart and Winston of Canada, Limited, 1973. Numerical Methods and Fortram Programming. John Wiley & Sons, Inc., 1964. Academic 190 [36] Samuelson, P . A . , "Rational Theory of Warrant Pricing," Industrial Management Review, Vol. 6, No. 2, Spring 1965. [37] Samuelson, P.A. and Merton, R.C. "A Complete Model of Warrant Pricing that Maximizes U t i l i t y , " Industrial Management Review, Vol. 10, Winter, 1969. [38] Shelton, T . P . , "The Relation of the Price of a Warrant to the Price of Its Associated Stock," Financial Analysts Journal, May-June and July-August, 1967. [39] Sprenkle, CM. , "Warrant Prices as Indication of Expectation and Preferences," Yale Economic Essays, I ( F a l l , 1961). [40] Squires, R . T . , "Unit-Linked Assurance: Observations and Propositions," Journal of the Institute of Actuaries, Vol. 101, Part 1, No. 416, 1974. [41] Survey of Current Business: United States Department of Commerce. [42] Thorp, E . O . , "Extensions of the Black-Scholes Option Model," 39th Session of the International Statistical Institute, Vienna, Austria (August 1973). [43] Thorp, E.O. and Kassouf, S . T . , Beat the Market: A Scientific Stock Market System, Random House, Inc. New York, 1967. [44] Turner, S . H . , "Asset Value Guarantees under EquityBased Products," Transactions Society of Actuaries, Vol. XXI, 1969. [45] Turner, S . H . , "Equity-Based Life Insurance in the United Kingdom," Transactions Society of Actuaries, Vol. XXIII, 1971. [46] Van Home, T . C . , Financial Management and Policy. Prentice-Hall Inc., Third Edition, [47] Weston, T . F . and Brigham, E . F . , 1974. Managerial Finance. Holt, Rinehart and Winston, Inc. Fourth Edition, 1972. C C C C C C C C C C C C C C C C C C C C C C C C C 191 APPENDIX A COMPUTER PROGRAM NO. DESCRIPTION OF 1: OPTION PRICES ON STOCKS WITH DIVIDENDS PAYMENTS CONTINUOUS PARAMETERS H--WARRANT PRICE VAR—VARIANCE RATE FOR THE RETURN ON THE STOCK R F — R I S K L E S S INTEREST RATE D — D I V I D E N D RATE E--EXCERCICE PRICE XH—STOCK PRICE- INCREMENT X K — T I M E INCREMENT NS—NO. OF STOCK PRICES NT--NO. OF TIMES A,B,C-- COEFFFICENTS IN THE ORIGINAL EQUATIONS AA,BB,CC, COEFICIENIS OF MODIFIED EQUATIONS F--RHS ORIGINAL EQUATIONS FF--RHS MODIFIED EQUATIONS T K — V E C T O R OF TIMES DIMENSION 1 (100, 10 0),A (100),B (100) ,C (100) , AA (1 00) , BB(IOO) , 1CC (100) ,F (100) ,FF (100) ,TK (100) READ(5,100) VAR,RF,D,E,XH,XK,NS,NT 100 FORMAT(4F10.5/2F5.2,214) WRITE(6,100) VAR,fiF,D,E,XH,XK„NS,NT C C C I N I T I A L I Z E BOUNDARY VALUES OF H 1 2 C C C C C NS=NS+1 NT=NT+1 DO 1 J=1,NT W (1, J) =0.0 CONTINUE DO 2 1=1,NS S=XH*FLOAT (1-1) I F ( S - L E . E ) H(I,1)=0.0 I F ( S . G T . E ) W(I,1)=S-E T=0.0 CONTINUE DO 9 J=2,NT SOLVE SIM EQNS FOR J J I S BASE PERIOD Y1=0.5*RF*XK Y2=0.5*D*XK/XH Y3=0.5*VAR*XK NS1=NS-1 NS2=NS-2 DO 6 1=1,NS2 A (I)=Y1*I-Y2-Y3*I*I B (I)=1.0+2.0*Y1 + 2 . 0 * Y 3 * I * I C (I)=-Y1*I+Y2-Y3*I*I 6 C C C 1 9 2 TRANSFORM EQUATIONS 7 C C C CONTINUE DO 7 1=2,NS1 BB(I) =BB (1-1) *B (I) / A (I) - C C (1-1) CC (I) = B B ( I - 1 ) *C ( I ) / A (I) FF(I)=BB(I-1)*F(I)/A(I)-FF(I-1) BB(1)=B(1) CC(1)=C(1) FF(1)=F(1) CONTINUE SOLVE FOR W 8 101 9 C C C C W (NS, J) =FF (NS1)/BB (NS1) DO 8 IX=1,NS2 I=NS-IX W ( I , J ) = { F F ( I - 1 ) - C C (1-1) *W (1+1, J) ) / B B (1-1) T = F L O A T ( J - 1 ) *XK FORMAT(5X,* TIME TO E X P I R A T I O N = • , F 1 0 . 5 / / / 5 ( 5 X , 2 0 F 6 - 2 / ) / / / / ) CONTINUE OUTPUT OUTPUT STARTS WITB T=0.0 AND S=0.0 200 204 205 201 202 203 207 SSIG F (I) =W (1+1, J - 1 ) A(1)=0.0 A (NS1)=-1.0 B (NS1)=1.0 F (NS 1) =XH WRITE (6,200) V A R , R F , D , E , X H , X K , N S , N T FORMAT(1H1,SOX,•WARRANT PRICE H A T R I X ' / / / , 2 0 X , ' V A R I A N C E R A T E = • , 1 F 1 0 . S / 2 0 X , • R I S K L E S S R A T E = • , F 1 0 . 5 / 2 0 X , * DIVIDEND R A T E = • , F 1 0 . 5 / 420X,'EXERCISE P R I C E = » , F 1 0 . 5 / 2 2 0 X , ' S T O C K PRICE I N C R E M E N T ^ , F 1 0 . 5 / 2 0 X , • T I M E INCREMENT=*,F10.5/ 320X,'NUMBER OF P R I C E S = ' , I 4 / 2 0 X , ' N U M B E R OF T I M E S = ' , I 4 / ) NFIRST=1 NLAST=20 I F (NLAST-GT.NT) NLAST=NT DO 205 J=NFIKST,NLAST TK (J) =XK*FLOAT (J-1) CONTINUE WRITE (6, 201) (TK(J),J=NFIRST,NLAST) F O R M A T ( / / / 1 1 X,20 F6 . 2 / / / ) DO 202 I=1,NS SI=XH*FLOAT (1-1) WRITE (6,203) S I , (8(1,J) ,J=NFIRST,NLAST) FORHAT(1X,F6.2,4X,20F6.2) I F (NLAST. E Q . NT) GO TO 207 NFIRST=NFIRST+20 NLAST=NLAST+20 GO TO 204 STOP END 193 c C C C APPENDIX B c C C C C C C C C C C C C . C C C C C C C C C C C C C C C 100 110- 111 120 203 201 202 200 COMPUTER PROGRAM NO. 2: DESCRIPTION OPTION PRICES ON STOCKS HITH DISCRETE DIVIDEND PAYMENTS OF PARAMETERS W— WARRANT PRICE VAR—VARIANCE RATE FOR THE RETURN ON THE STOCK B F — R I S K L E S S INTEREST RATE D—DIVIDEND R A T E , MOST BE SET EQUAL TO 0 . 0 E — E X C E R C I C E PRICE XH —STOCK PRICE INCREMENT X K — T I M E INCREMENT NS—NO. OF STOCK PRICES A , B , C — COEFFFICENTS IN THE ORIGINAL EQUATIONS A A , B B , C C , COEFICIENTS OF MODIFIED EQUATIONS F—RHS ORIGINAL EQUATIONS F F — R H S MODIFIED EQUATIONS D I — D E S C R E T E DIVIDEND N T ( K ) - - V E C T O R OF THE NO. OF TIME PERIODS CONSIDERED IN EACH I N TERVAL BETWEEN DIVIDENDS NK—NO. OF INTERVALS TO BE CONSIDERED S C — C R I T I C A L STOCK P R I C E , FOR VALUES OF STOCK PRICE GREATER OF EQUAL TO SC THE OPTION SHOULD BE EXERCICED DIVIDEND-ON P—MINHIMON PREMIUM TO E X C E R C I S E , USED TO DETERMINE SC DIMENSION W (400, 20) , A (400) , B (400) , C (400) , A A ( 4 0 0 ) , B B ( 4 0 0 ) , 1CC(400) , F ( 4 Q 0 ) , F F ( 4 0 0 ) , NT (20) , SD (400) , S C (20) , S T (400) , 2TD(20) DIMENSION SCA(20) ,TDA{20) •, READ(5,100) V A R , R F , D , E , X H , X K , N S WRITE(6,100)VAR,RF,D,E,XH,XK,NS FORMAT ( 4 F 1 0 . 5 / 2 F 5 . 2, 214) READ (5,110) NK, (NT (K) ,K=1,NK) FORMAT(I4/10I4/10I4) WRITE (6, 110) NK, (NT (K) ,K=1,NK) NK1=NK-1 READ (5,111) DI WRITE(6,111)DI FORMAT(F10.5) READ ( 5 , 120) P HRITE ( 6 , 1 2 0 ) P FORMAT(F8.3) WRITE(6,200)VAR,RF,D,E,XH,XK,NS,NK H R I T E ( 6 , 2 0 1 ) (NT (K) ,K=1,NK) WRITE(6,2Q2) DI WRITE (6,203) P FORMAT(20X,* MIN PREMIUM TO E X E R C I S E * , F 1 0 . 5 ) FORMAT(20X,* TIME PERIODS PER I N T E R V A L = » , 2 0 1 4 ) F O R M A T ( 2 0 X , ' D E S C R E T E DIVIDEND= * , F 1 0 . 5 ) FORMAT(1H1,40X, OPTION P R I C E S : DESCRETE DIVIDENDS C A S E » / / / / 2 0 X , 1' VARIANCE RATE= * ,.P10. 5 / 2 O X , * RISKLESS RATE= • , F 1 0 . 5 / 2 0 . X , 2'DIVIDEND RATE= * , F 1 0 . 5 / 2 0 X , ' E X E R C I S E P R I C E = « , F 1 0 . 5 / 2 0 X , f 194 3»ST0CK PRICE I N C R E M E N T ^ , F 1 0 . 5 / 2 0 X , T I M E INCREMENT =•,Fl0.5/20X, 4'NUMBER OF P R I C £ S = • , I 4 / 2 0 X , » N U M B E R OF TIME INTERVALS=•,14) f C C C C FULL OUTPUT — IPRINT=1 SUMMARIZED OUTPUT — IPRINT=0 ! IPRINT=0 C C C I N I T I A L I Z E BOUNDARY VALUES OF 8 1 2 300 C C C C C NS=NS+1 NT 1=NT (1) +NT (2) DO 1 J=1,NT1 H (1,J) =0.0 CONTINUE DO 2 1=1,NS ST (I) =XB*FLOAT (1-1) I F (ST (I) .LE.E) 9 ( 1 , 1 ) =0.0 I F (ST (I) .GT.E) « ( I , 1 ) = S T ( I ) - E T=0-0 I F ( I P R I N T . E Q . 1) WRITE (6,300) T, (H ( I , 1) ,1=1, NS) FORMAT(//20X,'TIME TO EXPIRATION=«,F10.5//20(5X,20F6.2/)) CONTINUE DO 12 K=1,NK NTKK=NT (K) +1 DO 9 J=2,NTKK SOLVE SIM EQNS FOR J J I S BASE PERIOD 6 C C . C 7 20 ¥1=0.5*RF*XK Y2=0.5*D*XK/XH ¥3=0.5*VAR*XK NS1=NS-1 NS2=NS-2 DO 6 1=1,NS2 A (I) = Y 1 * I - Y 2 - Y 3 * I * I B(I)=1.0+2.0*Y1+2.0*Y3*I*I C{I)=-Y1*I+¥2-Y3*I*I F <I)=W (1+1,J-1) A (1) =0.0 A (NS1)=-1.0 B(NS1) = 1.0 F(NS1)=XH TRANSFORM EQUATIONS CONTINUE DO 21 I=2,NS1 I F ( A B S (A (I) ) .LT. 0.0001) GO TO 20 BB (I) =BB (1-1) *B ( I ) / A ( I ) - C C (1-1) CC (I) =BB ( I - 1) *C (I) /A (I) FF(I)=BB(I-1)*F(I)/A(I)-FF(I-1) I F ( A B S (BB (I) ) .LT.10000.) GO TO 21 BB(I) =0.0001*BB (I) CC(I)=0.0001*CC(I) FF (I) =0.0001*FF (I) GO TO 21 I F ( I P B I N T . E Q . 1) WRITE(6,60) I , A ( I ) , B ( I ) , C ( I ) , F ( I ) 195 60 FORMAT(/ 1X,'1=•,14,1X,*A=•,F15.6,IX,'B=',F15.6,1X,•C=•, 1F15.6,1X,•F=« F15.6) BB(I)=B(I) CC(I)=C(I) F F ( I ) =F(I) CONTINUE BB(1)=B(1) CC(1)=C(1) FF(1)=F(1) CONTINUE / 21 C C C SOLVE FOR W 8 31 30 700 702 703 9 C C C W (NS, J) =FF (NS1)/BB (NS1) DO 8 IX=1,NS2 I=NS-IX W(I,J) = (FF(I-1)-CC(I-1)*W(I+1,J))/BB(I-1) I F ( K . E Q . 1 ) G O TO 31 T=FLOAT ( J - 1 + NTK) *XK GO TO 30 T=FLOAT ( J - 1 ) *XK NTK=0 CONTINUE I F ( I P R I N T . E Q . 1) WRITE (6,300) T , (W ( I , J ) ,1=1, NS) I F ( J . L T . N T K K ) GO TO . 9 WRITE (6,700) T F O R M A T ( / / 5 X , • T I M E TO E X P I R A T I O N = ' , F 1 0 . 5 / ) TD(K) =T WRITE(6,702) (ST (I) , I = 1 , N S , 2 0 ) FORMAT(/IX,*SP E X - D I V . » , 2 0 F 6 . 2 ) WRITE (6,703) (W ( I , J ) , I = 1 , N S , 2 0 ) FORMAT(/1X,•OP E X - D I V 2 0 F 6 . 2 / ) CONTINUE NTK=NTK+NT(K) I F ( K . EQ.NK) GO TO 16 NEW BOUNDARY VALUES OF W 10 C C C NXH=DI/XH DO 10 1=1,NS II=I-NXH IF(II.LT.1)11=1 WD (I) =W ( I I , NTKK) - W (I , 1) CRITICAL STOCK PRICE 17 19 18 WITH PREMIUM P IE=E/XH I F (WD (IE) . L E . P) GO TO 19 DO 17 I = I E , N S I F (WD (I) . G T . P) GO TO 17 I F (WD (I) . L E . P ) SC (K)=ST (I) - (P-WD (I) ) *XH/(WD ( 1 - 1 ) -WD (I) ) GO TO 18 CONTINUE SC(K)=200. GO TO 18 SC(K)=E CONTINUE DO 99 1=1,NS II=I-NXH I F ( I I . L T . 1)11=1 99 705 704 116 12 16 900 901. C C C PRINT A PLOT 70 71 72 SSIG I F (WD (I) . G E . O . O ) W ( I , 1 ) = W ( I I , N T K K ) IF(WD(I) . L T . O . O ) W ( I , 1)=W ( 1 , 1 ) I F ( I P R I N T . E Q . 1) WRITE (6,300) T , ( i ( I , 1) ,1=1, SS) WRITE (6, 705) (ST (I) , I = 1 , N S , 2 0 ) FORMAT ( / 1 X , * S P C U M - D I V » , 2 0 F 6 . 2 ) WRITE(6,704) (W ( I , 1) , 1 = 1 , N S , 20) FORMAT ( / I X , * OP C U M - D I V , 2 0 F 6 . 2) WRITE(6, 1 16) SC(K) F O R M A T ( / / 5 X , • C R I T I C A L STOCK PRICE C U H - D I V = • , F 8 - 3 / / ) CONTINUE CONTINUE WRITE(6,900) (TD (K) ,K=1,NK1) F O R M A T ( / / 1 X , ' T TO E X ' , 2 0 F 6 . 2 ) WRITE (6,901) (SC(K) ,K=1,NK1) FORMAT(/1X, ' C R I T S P • , 2 0 F 6 . 2 / / ) DO 70 J = 2 , 1 9 TDA (J) =TD (J-1). TDA(1)=0. DO 71 J = 2 , 1 9 SCA(J)=SC(J-1) SCA(1)=50. CALL S C A L E ( T D A , 1 9 , 10. , XMIN,DX, 1) CALL S C A L E ( S C A , 1 9 , 1 0 . , Y M I N , D Y , 1 ) CALL AXIS (0. , 0. , • TIME TO EXPIRATION' , - 18 , 1 0. , 0 . , XMIN, DX) CALL A X I S ( 0 . , 0 . , ' C R I T I C A L STOCK P R I C E * , 2 0 , 1 Q . , 9 0 . , Y H I N , D Y ) DO 72 J = 2 , 1 9 CALL SYMBOL (TDA (J) , S C A (J) , 0 . 1 4 , 4 , 0 . , - 1 ) CALL PLOTND STOP END 196 197 APPENDIX C: Solution Algorithm: Option Prices in the Constant Continuous Premium Contract. The partial differential equation governing the value of the c a l l option in the constant continuous premium contract was derived in Chapter 5. The equation obtained was: (C.l) ho x 2 w 2 + (rx+D) w - wr - w XX X = 0 -L subject to the boundary conditions (C.2) w(x,0,g) = Max[x-g, 0] (C.3) Dw (0,T,g) - w(0,T,g)r - w (0,T,g) (C.4) w O,T,g) x T = 0 = 1 x In finite difference form, partial differential equation (C.l) can be written as: (C.5) a. w. , .'+ D, w. . + c. w.,, x l-l,j i i,j l l+l,j = w. . . i,J-l where (C.6) a. = %rki + JgDk/h - %a ki (C.7) b = (1+rk) + a k i (C.8) c = -hrki - %Dk/h - %a ki (C.9) w(x , T ) 2 I ± ± 2 2 2 2 = w(ih, jk) = 2 w. , Boundary conditions (C.2), (C.3) and (C.4) can be expressed in finite difference form, respectively, /w. (C.10)j 1 n , = 0 for = ih-g as: 0 < i < g/h u 0 for g/h < i < n 198 , i n Dk = vu.ii; w ^ Q D k + h k h r + w h j x + D k + h k r + h w o , j - i for a l l j (C.12) -w . . + w . = n-1,3 n,j h for a l l j Difference equation (C.5) can be used to generate a system of l i n e a r equations by making i = 1, ... n-1. The l a s t equation (for i = n) i s obtained from boundary condition (C.12) a,w . + b,w. . + c,w . 1 0,3 1,3 1 2,j A 0 1 a„w. . + b w 0 2 1,3 = w . + c„w, . o 2 2,j 2 3,3 1 » J L u ,j-l = 2 (C.13) a ,w „ .+ b .w . . + c .w . = n-1 n-2,3 n-1 n-1,3 n-1 n,3 - w . . + n-1,3 w , . , n-1,3-1 w . = h n,3 The system of n l i n e a r equations (C.13) has (n+1) unknowns (w, . f o r »3 But the value of W Q ^ can be obtained from boundary i = 0, ..., n). condition (C.11). The system of equation (C.13) can then be written as: Dka. (bi 1 + ™ — _ • _ ui—";—c) Dk + hkr + h &2™i + D 2 2 W + °2 3 W w i 1+ c i W = 12 0 = f, 1 ^2 (C.14) a , w - + b , w . + c , w n-1 n-2 n-1 n-1 n-1 n a w ,+ b w n n-1 n n = f , n-1 = f n r 199 where h a^ l f = W l , j - 1 " Dk + hkr + h f. = w. f = h = -1 a b n n n = . , w 0,j-1 f o r i = 2, .... n-1 1 and where the subscripts j have been omitted f o r s i m p l i c i t y . The matrix of c o e f f i c i e n t s of the system (C.14) i s t r i d i a g o n a l , so Gauss' elimination can be employed to solve f o r the values of w^(i = 1, . .., n ) . The values of w. . can be obtained s t a r t i n g from the values of w. . ,. The f i r s t l i n e of the grid, w. , i s computed by using boundary condition (C.10). n The computer program shown i n Appendix D i s based on t h i s algorithm. solution 200 c C C C C C C C C C C C C C. C C C C 100 C C 2 C C 6 C APPENDIX D COMPUTER PROGRAM N O . 3 : DESCRIPTION O P T I O N P R I C E S I N T H E CONSTANT PREMIUM CONTRACT CONTINUOUS OF PARAMETERS H — O P T I O N VALUE V A R — V A R I A N C E R A T E FOR T H E RETURN OF T H E R E F E R E N C E P O R T F O L I O R F — R I S K L E S S INTEREST RATE D — C O N T I N U O U S PREMIUM R A T E ( W I T H - SIGN) X H — R P P R I C E INCREMENT X K — T I M E INCREMENT N S — N O . OF RP P R I C E S N T — N O . OF T I M E S DIMENSION H ( 4 0 0 , 200) , A (400) , B (400) , C (4 00) , A A (400) , BB (400) , 1CC (400) , F ( 4 0 0 ) , F F ( 4 0 0 ) , T K ( 2 0 0 ) READ(5,100) VAR,RF,D,£,XH,XK,NS,NT FORMAT(4F10.5/2F10.5,214) I N I T I A L I Z E BOUNDARY V A L U E S O F H NS=NS+1 NT=NT+1 DO 2 1 = 1 , N S S=XH*FLOAT (1-1) I F ( S . L E . E ) B (1,1) =0.0 I F ( S . G T . E ) B (1,1) = S - E T=0-0 W (1,1) =0.0 DIV= (-D*XK+XH*XK*RF+XH) DO 9 J = 2 , N T S O L V E S I M EQNS FOR J Y1=0.5*RF*XK Y2=0.5*D*XK/XH Y3=0.5*VAR*XK NS2=NS-2 NS1-NS-1 DO 6 1 = 1 , NS2 A (I)=Y1*I-Y2-Y3*I*I B (I)=1.0 + 2.0*Y1+ 2 . 0 * Y 3 * I * I C (I)=-Y1*I+Y2-Y3*I*I F (I) =B ( I + 1 , J - 1 ) B (1) =B (1) - D * X K * A ( 1 ) / D I V F ( 1 ) = F ( 1 ) - H (1 , J - 1 ) * X H * A ( 1 ) / D I V A(NS1)=-1.0 B (NS1) = 1 . 0 F(NS1)=XH TRANSFORM E Q U A T I O N S BB(1)=B(1) C C (1) = C ( 1 ) FF(1) =F(1) DO 7 1 = 2 , N S 1 BB (I) =BB ( 1 - 1 ) * B ( I ) / A (I) - C C ( 1 - 1 ) CC (I) =BB ( 1 - 1 ) * C (I) / A (I) 7 C 8 9 C C 200 101 $SI<3 F F ( I ) = B B ( I - 1 ) *F ( I ) / A ( I ) - F F ( 1 - 1 ) S O L V E FOR W W (NS, J ) = F F (NS1) / B B (NS1) DO 8 I X = 1 , N S 2 I=NS-IX W ( I , J ) = ( F F (1-1) -CC (1-1) *W (1+1, J ) ) /BB ( 1 - 1 ) W ( 1 , J)=-W ( 2 , J ) * D * X K / D I V + X H * W ( 1 , J - 1 ) / D I V T=FLOAT(J-1)*XK CONTINDE OUTPUT- , S T A R T S WITH T=0.0 AND R P = 0 . 0 201 WRITE(6,200)VAR,RF,D,E,XH,XK,NS,NT F O R M A T ( 1 H 1 , 3 0 X , * V A L U E OF AN E Q U I T Y L I N K E D O P T I O N WITH CONTINUOUS 1PREHIUH P A Y M E N T S V / / 2 0 X , • V A R I A N C E RATE=',F10.5/20X,»RISKLESS RATE 2 = ' , F 1 0 . 5 / 2 0 X , ' C O N T I N U O U S PREMIUM B A T E = ' , F 1 0 . 5 / 2 0 X , * G U A R A N T I E D AMT 2 = ' / F 1 0 . 5 / 2 0 X , ' R E F E R E N C E P O R T F O L I O P R I C E INCREMENT= • , F1 0 . 5 / 2 0 X , . 3 ' T I M E I N C R E M E N T = ' , F 1 0 . 5 / 2 0 X , ' N O . OF RP P R I C E S = ' , I 4 / 2 0 X , ' N O . OF T I M 4ES=',IU///) WRITE ( 6 , 101) T, (W ( I , N T ) ,1=1, NS) F O R M A T ( 5 X , • T I M E TO E X P I R A T I O N = • , F 1 0 . 5 / / 2 0 ( 5 X , 2 0 F 6 . 2 / ) / / ) STOP END C 202 c c C C APPENDIX E c C C C C C C C C C C C C C C C C C C C COMPUTER PROGRAM NO. 4 : DESCRIPTION OPTION PRICES CONTRACT IN THE PERIODIC PREMIUM OF PARAMETERS H —OPTION VALUE VAR—VARIANCE RATE FOR THE RETURN ON THE REFERENCE PORTFOLIO R F — R I S K L E S S RATE X H — REFERENCE PORTFOLIO PRICE INCREMENT X K — T I M E INCREMENT NS—NO. OF REFERENCE PORTFOLIO PRICES D—PREMIUM INVESTED PERIODICALY ON THE REFERENCE PORTFOLIO E—GUARANTIED AMOUNT ND—NO. OF PREMIUM PAYMENTS T D — T I M E BETWEEN PREMIUM PAYMENTS NT—NO. OF TIME PERIODS BETWEEN PREMIUM PAYMENTS 100 101 DIMENSION W (400,50) , A ( 4 0 0 ) , B (400) , C (400) , AA (400) , B B (400) , ICC (400) , F (400) , F F (400) , S T (400) READ(5,100)VAR,RF,E,D,XH,NS,ND,TD,NT FORMAT(3F10.5/2F10.5,I5/I5,F10.5,I5) XK=TD/FLOAT(NT) WRITE (6,101)VAR,RF,E,D,XH,XK,NS,ND,TD,NT F O R M A T ( 1 H 1 , 3 0 X , • V A L U E OF AN EQUITY LINKED OPTION WITH P E R I O D I C , • 1PREMIUH P A Y M E N T S « / / / 2 0 X , « V A R I A N C E R A T E = ' , F 1 0 . 5 / 2 0 X , * R I S K L E S S ' ,• 2 R A T E = * , F 1 0 . 5 / 2 0 X , * GUARANTIED A M O U N T ' , F 1 0 . 5 / 2 0 X , * PREMIUM(INVESTED 3IN EQUITY)= ' , F 1 0 . 5 / 2 0 X , ' R E F E R E N C E PORTFOLIO PRICE INCREMENTS * , 4F10.5/2QX,•TIME INCREMENT^,F10.5/20X,•NUMBER OF REFERENCE' , • 5PORTFOLIO PRICES= * , I 5 / 2 0 X , ' N U M B E R OF PREMIUM P A Y M E N T S • , 1 5 / 6 2 O X , ' T I M E BETWEEN PREMIUM P A Y M E N T S = ' , F 1 0 . 5 / 2 0 X , • N O . OF T I M E ' ,» 7PERIODS BETWEEN PREMIUM P A Y M E N T S « , 1 5 / / / ) 3 3 3 C C C C C FULL OUTPUT — IPRINT=1 INTERMEDIATE OUTPUT—IPRINT=2 SUMMARIZED OUTPUT — IPRINT=0 IPRINT=0 C C , C 2 300 INITIALIZE BOUNDARY VALUES OF W NS=NS+1 DO 2 1=1,NS ST(I)=XH*FLOAT(I-1) I F (ST (I) . L E . E ) W ( I , 1) =0.0 I F (ST (I) . G T . E ) W ( I , 1) = S T ( I ) - E T=0.0 I F ( I P R I N T . E Q . 1) W R I T E ( 6 , 3 0 0 ) T , (W ( I , 1) , 1=1 , NS) F O R M A T ( / / 2 0 X , ' T I M E TO E X P I R A T I O N ' , F 1 0 . 5 / / 2 0 ( 5 X , 2 0 F 6 . 2 / ) ) NXH=D/XH NT1=NT+1 DO 12 K=1,ND 3 DO 9 J=2,NT1 C C C C C SOLVE SIM EQNS FOR J J IS 6 C C C BASE PERIOD W (1,J)=W (1, J-1) * E X P ( - R F * X K ) Y1=0.5*RF*XK Y3=0.5*VAR*XK WS1=NS-1 NS2=NS-2 DO 6 1=1,NS2 A (I) = ¥ 1 * I - Y 3 * I * I B(I)=1.0+2.Q*Y1+2.0*Y3*I*I C (I)=-Y1*I-Y3*I*I F (I) =H (1+1, J - 1 ) F ( 1 ) = W ( 2 , J - 1 ) - A ( 1 ) *W (1,J) A (NS1)=-1.0 B (NS1) =1.0 F (NS1) =XH TRANSFORM 7 20 60 21 EQUATIONS DO 21 1=2,NS1 I F (ABS (A (I) ) . L T . 0.0001) GO TO 20 BB (I) =BB (1-1) *B ( I ) / A ( I ) - C C (1-1) CC (I) =BB (1-1) * C ( I ) / A (I) FF(I)=BB(I-1)*F(I)/A(I)-FF(I-1) I F ( A B S ( B B ( I ) ) . L T . 1 0 0 0 0 . ) G O TO 21 BB(I) = 0 . 0 0 0 1 * B B ( I ) CC (I) =0. 000 1*CC (I) FF(I)=0.0001*FF(I) GO TO 21 I F ( I P R I N T . E Q . 1) WRITE (6,60) I , A (I) , B (I) , C (I) , F (I) FORMAT (/ 1 X , M = « ,14, I X , • A= • , F 1 5 . 6, 1X, • B=« , F 1 5 . 6, 1X, « C= • , 1F15.6,IX,«F=',F15.6) BB(I)=B(I) CC(I)=C(I) FF(I) =F(I) CONTINUE BB(1)=B(1) CC(1).=C(1) FF(1) =F(1) CONTINUE C SOLVE FOR W c c 8 31 30 W (NS, J ) = F F ( N S 1 ) / B B (NS1) DO 8 IX=1,NS2 I=NS-IX W ( I , J ) = ( F F ( I - 1 ) - C C (1-1)*W (1+1, J) ) / B B (1-1) I F (K. EQ. 1) GO TO 31 T=FLOAT(J-1+NTK)*XK GO TO 30 T=FLOAT(J-1)*XK NTK=0 CONTINUE I F ( I P R I N T . E Q . 1) WRITE (6,300) T , (W ( I , J ) ,1=1, NS) I F ( J . L T . NT1) GO TO 9 I F ( I P R I N T . EQ. 2) WRITE (6 ,700) T 203 7 0 0 ; FORMAT ( / / 5 X , • TIME TO EXPIRATION=' , F1 0. 5/) I F (IPRINT. E Q . 2) WRITE (6,702) (ST (I) , 1=1 , NS, 20) 702 FORMAT(/1X , *RP A - P R E M . ' , 2 0 F 6 - 2 ) I F ( I P R I N T . E Q . 2) WRITE (6,703) (8 ( I , J ) ,1 = 1 , N S , 2 0 ) 703 FORMAT(/1X,'OP A-PREM.•,20F6.2/) 9 CONTINUE NTK=NTK+ NT C C C 204 NEW BOUNDARY VALUES OF W NSD=NS-NXH DO 10 1=1,USD 11=1+NXH 10 W (I,1)=3(II,NT1) NSD1=NSD+1 DO 11 I=NSD1,NS 11=1-1 11 W ( 1 , 1 ) =W (11,1)+XH I F ( I P R I N T . E Q . 1) WRITE (6,300) T , (W ( I , 1).,1=1,US) I F ( I P R I N T . E Q . 2) WRITE(6,705) (ST (I) , I = 1 , N S , 2 0 ) 705 F O R M A T ( / I X , • RP B - P R E M . * , 2 0 F 6 . 2 ) I F ( I P R I N T . E Q . 2) WHITE (6,704) (W (1,-1). , 1 = 1 , HS, 20) 704 F O R M A T ( / 1 X , * OP B - P R E M ' , 2 0 F 6 . 2 ) 12 CONTINUE I F ( I P R I N T . E Q . 2 ) W R I T E ( 6 , 3 0 0 ) T , (W ( 1 , 1) , 1=1, NS) GA=E*EXP (-RF*T) PV=W(1,1)+GA ND1=ND-1 AN=0. I F (ND. EQ. 1) GO TO 34 DO 33 1=1,ND1 BN=EXP(-RF*I) 33 AN=AN+BN 34 CONTINUE PVRP=1.* (1. + AN) PUT-PV-PVBP P=PV/(1.+AN) APP=P-1. APC=W (1, 1 ) / ( 1 . + A N ) WRITE ( 6 , 3 0 1 ) W ( 1 , 1 ) , G A , P V , P V R P , P U T , P , A P P , A P C 301 F O R M A T ( / / 4 0 X , * R E S U L T S ' / / / 2 0 X , ' V A L U E OF THE CALL O P T I O N = * , F 1 0 . 5 / 1 2 0 X , ' P V OF GUARANTIED A M O U N T = « , F T Q . 5 / 2 0 X , » T O T A L PV OF THE CONTRACT 2 = ' , F 1 0 . S / 2 0 X , ' P V OF THE REFERENCE P O R T F O L I O = » , F 1 0 . 5 / 2 0 X , 3 ' V A L U E OF THE PUT O P T I O N = ' , F 1 0 . 5 / 2 Q X , ' T O T A L ANNUAL PREMIUM—', 4 F 1 0 . 5 / 2 0 X , ' A N N U A L PREMIUM FOR P U T = • , F 1 0 . 5 / 2 0 X , • A N N U A L PREMIUM FOR 5CALL=',F10.5////) STOP END SSIG c C C C C C C C C C C C C C C C C C C C C C C C - 205 APPENDIX F COMPUTER PROGRAM DESCRIPTION OF NO. 5: COMPUTATION OF T H E E Q U I L I B R I U M V A L U E OF T H E GUARANTEE WITH KNOWN D A T E OF EXPIRATION PARAMETERS W — O P T I O N VALUE V A R — V A R I A N C E R A T E FOR T H E RETURN ON THE R E F E R E N C E P O R T F O L I O R F — R I S K L E S S RATE XH— REFERENCE PORTFOLIO P R I C E INCREMENT X K — T I M E INCREMENT N S — N O . OF R E F E R E N C E P O R T F O L I O P R I C E S D — P R E M I U M I N V E S T E D P E R I O D I C A L Y ON T H E R E F E R E N C E P O R T F O L I O E—GUARANTIED AMOUNT ND--NO. OF PREMIUM PAYMENTS T D — T I M E BETWEEN PREMIUM PAYMENTS N T — N O . OF T I M E P E R I O D S BETWEEN PREMIUM PAYMENTS DIMENSION W ( 4 0 0 , 5 0 ) ,A (400) , B (400) ,C (400) , AA (400) , BB (400) , 1CC (400) , F (400) , F F (400) ,ST ( 4 0 0 ) DIMENSION GA (30) , PV (30) , PVRP (30) ,PUT (30) , P (30) , APP (30) , 1APC(30) ,CA(30) READ ( 5 , 100) VAR,RF,D,NS,TD, NT 100 F O R M A T ( 3 F 1 0 . 5 , I 5 , F 1 0 . 5 , I 5 ) XK=TD/FLOAT(NT) WRITE ( 6 , 1 0 1 ) V A R , R F , D , X K , N S , T D , N T 101 F O R M A T ( 1 H 1 , 3 0 X , * V A L U E OF AN E Q U I T Y L I N K E D O P T I O N WITH P E R I O D I C • , * 1PREMIUM PAYMENTS *///20X,* V A R I A N C E R A T E = * , F 1 0 . 5 / 2 0 X , * R I S K L E S S * , « 2RATE=',F10.5/20X, * 'PREMIUM ( I N V E S T E D 31N E Q U I T Y = » , . « r F 1 0 . 5 / 2 0 X , ' T I H E INCREMENT=', F 1 0 . 5 / 2 0 X , » NUMBER OF R E F E R E N C E ' , ' 5PORTFOLIO P R I C E S = ' , I 5 / 6 2 0 X , ' T I M E BETWEEN PREMIUM P A Y M E N T S = * , F 1 0 . 5 / 2 0 X , • N O . OF-TIME' ,» 7 P E R I O D S BETWEEN PREMIUM PAYMENTS ',15///) NS=NS+1 WRITE(6,301) 301 FORMAT(///3X,* NO. , Y E A R S * , 4 X , ' V A L U E C A L L * 3 X , ' P V GUARANT.',2X, 1 ' T O T A L P V ' , 5 X , ' P V P O R T F O L I O * , I X , 'VALUE PUT ' ,.4X , ' T O T A L PREM',3X, 2* A PREH P U T * , 3 X , ' A PREM C A L L ' / / / ) DO 1000 ND=1,20 E=0.75*ND*D NXH= 1 2 5 . / F L O A T (ND) XH=D/FLOAT(NXH) 3 r C C C INITIALIZE 2 BOUNDARY VALUES OF W DO 2 1=1,NS ST(I)=XH*FLOAT(I-1) I F ( S T ( I ) . L E . E ) W ( I , 1) =0.0 I F (ST ( I ) . GT. E) W ( I , 1) =ST (I) - E T=0.0 NT1=NT+1 DO 12 K=1,ND DO 9 J=2,NT1 SOLVE SIM EQNS FOB J J I S BASE PERIOD W ( 1 , J ) = 8 ( 1 , J - 1 ) *EXP(RF*XK) Y1=0.5*RF*XK Y3=Q.5*VAR*XK NS1=NS-1 NS2=NS-2 DO 6 1=1,NS2 A (I)=Y1*I-Y3*I*I B ( I ) = 1 . 0 + 2.0*Y1 + 2 . 0 * Y 3 * I * I C (I)=-Y1*I-Y3*I*I F(I)=8(I+1,J-1) F (1) =8 (2, J - 1 ) - A ( 1 ) * 8 ( 1 , J ) A (NS1) = - 1 . 0 B (NS1)=1.0 / F(NS1)=XH TRANSFORM EQUATIONS DO 21 1=2,NS1 I F ( A B S ( A (I) ) . L T . 0. 0001) GO TO 20 BB (I) =BB (1-1) *B (I) / A (I) - C C (1-1) CC (I) =BB (1-1) *C (I) / A (I) F F (I) =BB (1-1) * F (I) / A (I) - F F (I-1) I F (ABS (BB (I) ) . L T . 10000.) GO TO 21 BB (I) =0. 0001*BB (I) CC (I) =0. 000 1*CC (I) FF(I)=0.0001*FF(I) GO TO 21 CONTINUE BB(I)=B(I) CC(I)=C(I) FF(I)=F(I) CONTINUE BB(1)=B(1) CC(1)=C(1) FF(1)=F(1) CONTINUE SOLVE FOR 8 8 (NS, J) =FF (NS 1 ) / B B (NS 1) DO 8 IX=1,NS2 I=NS-IX 8 ( I , J ) = (FF (1-1) - C C ( I - 1 ) * W ( I + 1 , J ) ) / B B (1-1) I F (K. EQ. 1) GO TO 31 T = F L O A T ( J - 1 + NTK) *XK GO TO 30 T=FLOAT (J-1) *XK NTK=0 CONTINUE I F ( J . L T - N T 1 ) G O TO 9 CONTINUE NTK=NTK+NT / NE8 BOUNDARY VALUES OF 8 NSD=NS-NXH DO 10 1=1,NSD '207 II=I+NXH 10 I ( I , 1) =.W ( I I , NT1) NSD1=NSD+1 DO 11 I=NSD1,NS 11=1-1 11 W ( I , 1) =W ( I I , 1)+XH 12 CONTINUE CA(ND)=W (1,1) GA(ND)=E*EXP(-RF*T) PV (ND) =CA (ND) +GA (ND) ND1=ND-1 AN=0. I F ( N D . E Q . 1 ) G O TO 34 DO 33 I=1,ND1 BN=EXP(-RF*I) 33 AN=AN+BN 34 CONTINUE PVRP (ND) =1.+AN PUT (ND) =PV (ND)-PVRP (ND) P(ND)=PV (ND)/(1.+AN) APP (ND) =P (ND) - 1 . APC(ND) =CA (ND)/(1.+AN) 35 WRITE(6,302) ND, CA (ND) ,GA (ND) , P V (ND) ,PVRP(ND) ,PUT(ND) ,P(ND) , 1APP(ND) , A P C (ND) 302 FORMAT(5X,I5,2X,8(3X,F10.5)) 303 FORMAT (F10.5) WRITE (7,303) CA (ND) WRITE (8,303) PV (ND) WRITE (9,303)PUT(ND) WRITE (2, 303) APP (ND) 1000 CONTINUE STOP END $SIG 208 APPENDIX G COMPUTER PROGRAM NO. ,6: COMPUTATION OF THE EQUILIBRIUM VALUE OF THE CONTRACT AND THE RISK PREMIUM DIMENSION PV(30) , APP (30) , XLX (81) , A ( 3 0 ) READ ANNUAL PREMIUM ON THE PUT AND TOTAL PRESENT VALUE DO 20 1=1,30 READ (2,1) APP (I) FORMAT(F10.5) READ (3, 1) PV (I) READ MORTALITY TABLE(ULTIMATE)STARTING AT AGE 20 AND ENDING AT AGE 80 (61 OBSERVATIONS) READ(5,2) (XLX(J) , J = 1 , 6 1 ) FORMAT(10 ( 6 F 1 2 . 2 / ) , F 1 2 . 2 ) READ AGE OF PURCHASER AND L I F E OF CONTRACT READ (5, 3) NAGE,NL FORMAT (213) NX=NAGE-19 NL1=NL-1 DO U K=1,NL1 NXT=NX+K NXT1=NXT-1 A (K)= (XLX(NXTI) -XLX(NXT) ) / X L X ( N X ) NXTL=NX+NL-1 A (NL) =XLX ( N X T L ) / X L X (NX) CHEK THAT PROBABILITIES SUM=0.0 DO 10 1=1,NL SUM=SUM + A ( I ) WRITE (6, 11) SUM FORMAT ( I X , F 1 0 . 5 ) SUM TO 1.0 CALCULATES EXPECTED VALUES P=0.0 V=0.0 DO 5 J = 1 , N L P=P+A (J) *APP (J) V=V + A(J) *PV (J) OUTPUT WRITE (6,6) N AGE, NL FORMAT ( 1 H 1 / / , 1 0 X , 'EQUITY LINKED L I F E INSURANCE POLICY WITH 1 VALUE G U A R A N T E E V / / 2 0 X , 'AGE OF PURCHASER » , 2 X , I 3 / / 2 0 X , 3 ASSET 7 8 9 $SIG 2'TERM OF THE CONTRACT= • , 2X, 1 3 / / / 5 X , ' TIME ' , 5 X , ' TOTAL PV » , 5 X , 3'VALUE P U T ' , 5 X , ' A L P H A ' / ) DO 7 1=1,NL WRITE(6,8) I , PV (I) , APP (I) , A (I) FORMAT ( 5 X , I 3 , 5 X , F 1 0 . 5 , - i X , F 1 0 . 5 , 4 X , F 8 . 5 ) WRITE (6,9) P , V F O R M A T ( / / 2 0 X , * EXPECTED VALUE OF THE ANNUAL PREMIUM ON THE P U T = ' , 1 F 1 0 . 5 / / 2 0 X , ' E X P E C T E D TOTAL PRESENT VALUE OF THE P O L I C Y = • , F 1 0 . 5 / / ) STOP END 210 APPENDIX H: Select Mortality Tables* "The mortality of a group of insured lives exhibits certain distinctive characteristics which derive from the special nature of such a group. Before an insurance policy is issued, the insured must be satisfied that the applicant meets certain underwriting standards. Some applicants, because of health conditions or other factors, w i l l not be offered insurance on a standard basis, and some others may be considered uninsurable. As a result of this selection process, a group of lives insured on a standard basis does not constitute a random group, but rather a select group, a l l the members of which have i n i t i a l l y satisfied certain criteria of insurability. It follows that the mortality of such a group w i l l vary not only by age, but also by the duration of the insurance. The period of time during which the effects of selection are s t i l l significant is called the select period.'' This feature of mortality among insured lives is recognized in the construction of select mortality tables, which show the mortality variation both by age and duration. Usually a uniform select period for a l l ages at issue is assumed and the table is condensed into a select and ultimate i form." Table 5-1 shows the ultimate column (L ^) between the ages of 20 and x+ 80 of the CA 1958-64 (M) select. The approximation of using the ultimate form has been done throughout the study. *See Jordan [17] p. 24-25. 211 APPENDIX I COMPUTER PROGRAM NO. 7: RISK PREMIUM VERSUS TERM OF CONTRACT DIMENSION APP(30) , X L X (61) , A(30) DIMENSION RP (30) (30)~ DO 20 1=1,20 READ ( 2 , 1) APP (I) FORMAT(F10.5) READ (3,2) (XLX(J) , J = 1,61) FORMAT (10 (6F12. 2/) , F 1 2 . 2 ) READ (5,3) NAGE FORMAT (13) NX=NAGE-19 DO 100 NL=1,20 NL1=NL-1 I F (NL1.EQ..0) GO TO 15 DO -4 K=1,NL1 NXT=NX+K NXT1=NXT-1 A (K) = (XLX (NXT1) - X L X (NXT) ) / X L X (NX) NXTL=NX+NL-1 A (NL) =XLX (NXTL) / X L X (NX) P=0.0 DO 5 J = 1 , N L P=P+A (J) *APP (J) EP(NL)=P T (NL) =FLOAT(NL) SUM=0.0 DO 10 1=1,NL SUM=SUM+A (I) WRITE (6, 11) SUM FORMAT(IX,F10.5) GO TO 100 RP(1) =APP (1) T (1) =FLOAT (NL) CONTINUE WRITE (6,25) FORMAT(1H1,7X,'GUARANTEE: 75% OF INVESTMENT COMPONENT ON DEATH 1 MATURITY * / 8 X , * MORTALITY T A B L E : CA 1 9 5 8 - 6 4 ' / 8 X , ' R I S K FREE RATE: 2 0 4 ' / 8 X , • VARIANCE R A T E : 0 . 0 1 8 4 6 (TSE) ' / / / ) WRITE (6,6)NAGE FORMAT( 5 X , • R I S K PREMIUM VS TERM. OF CONTRACT'//8 X , 1'AGE OF PURCHASER= * , 2 X , I 3 / / / 5 X , ' T E R M OF C O N T R A C T ' , S X , 2'RISK PREMIUM'//) DO 7 NL=1,20 WRITE (6,8) NL,RP(NL) FORMAT (1 OX, 1 3 , 1 5 X , F 1 0.5) CALL SCALE(T,20,10.,XMIN,DX,1) CALL S C A L E ( B P , 2 0 , 1 0 . YMIN,DY,1) CALL AXIS ( 0 . , 0 . , ' T E R M OF C O N T R A C T ' , - 1 6 , 1 0 . , 0 . , X M I N , D X ) CALL AXIS (0. ,-0. , 'RISK PREMIUM',12,10.,90YMIN,DY) DO 31 1=1,20 CALL SYMBOL (T (I) , R P ( I ) , 0 . 1 4 , 4 , 0 . , - 1 ) CALL PLOTND f STOP END *SIG 212 c C C C C C C C APPENDIX J COMPUTER PROGRAM NO. 8: 213 RISK PREMIUM VERSUS AGE OF PURCHASER AT ENTRY DIMENSION APP (30) , XLX (61) , A (30) , RP (31) , AGE (31) READ (2,1) (APP (I) , 1 = 1,20) 1 FORMAT ( F 1 0 . 5 ) READ (3, 2) (XLX(J) , J = 1 , 6 1 ) 2 FORMAT ( 1 0 ( 6 F 1 2 . 2 / ) , F 1 2 . 2 ) READ (5,3) NL 3 FORMAT (13) NL1=NL-1 DO 200 NAGE=20,50 NX=NAGE-19 DO 4 K=1,NL1 NXT=NX+K NXT1=NXT-1 4 A(K)= ( X L X ( N X T 1 ) - X L X ( N X T ) ) / X L X ( N X ) NXTL=NX+NL-1 A (NL) =XLX (NXTL) / X L X (NX) P=0.0 DO 5 J = 1 , N L 5 P=P+A (J) * A P P ( J ) RP(NX) =P AGE (NX) =FL0AT (NAGE) 200 CONTINUE WRITE (6,25) 25 FORMAT ( 1 H 1 , 7 X , • G U A R A N T E E : 2 0 . 0 0 (FIXED AMOUNT FOR ALL MATURITIES) ' 1 / 8 X , • M O R T A L I T Y T A B L E : / C A 1 9 5 8 - 6 4 ' / 8 X , • R I S K FREE R A T E : 0 . 0 2 8 * / 8 X , * VARIANCE RATE: 0.01846 (TSE)*///) WRITE (6,6) NL 6 FORMAT(5X,•RISK PREMIUM VS AGE OF P U R C H A S E R ' / / 8 X , • T E R M OF CONTRACT 1 = « , 2 X , I 3 / / / 5 X , • A G E OF P U R C H A S E R • , 5 X , * RISK PREMIUM*//) DO 7 J=1,31 7 WRITE (6, 8) AGE (J) ,RP (J) 8 FORMAT(10X,F3.0,12X,F10.5) CALL SCALE(AGE,31,10.,XMIN,DX,1) CALL SCALE(RP,31,10.,YMIN,DY,1) CALL AXIS (0. , 0 . , * AGE OF PURCHASER • , - 1 6 , 1 0 . , 0 . , XMIN/DX) CALL A X I S ( 0 . , 0 . , * R I S K P R E M I U M * , 1 2 , 1 0 . , 9 0 . , Y M I N , D Y ) DO 31 1=1,31 31 CALL SYMBOL (AGE (I) , R P ( I ) , 0 . 14, 4, 0. , -1) CALL PLOTND STOP END $SIG c 214 C C APPENJDIX K COMPOTES PROGRAM c c C NO.: 9: INVESTMENT POLICY C C C C DESCRIPTION OF PARAMETERS C C W—OPTION C VAR—VARIANCE C R F — R I S K L E S S C X H — C X K — T I M E VALUE RATE REFERENCE NS—NO. C D—PREMIUM OF C E—GUARANTIED ND--NO. C TD--TIME C NT—NO. THE RETURN ON THE REFERENCE PORTFOLIO PORTFOLIO PRICE INCREMENT INCREMENT C C FOR RATE REFERENCE INVESTED OF PRICES ON THE REFERENCE PORTFOLIO AMOUNT PREMIUM BETWEEN OF PORTFOLIO PERIODICALY TIME PAYMENTS PREMIUM PERIODS PAYMENTS BETWEEN PREMIUM PAYMENTS C DIMENSION 1CC (400) W ( 4 0 0 , 5 0 ) , A (400) , F (400) , F F (400) DIMENSION 1 A P C (30) G A (30) , C A , B (400) , ST , C (400) , AA (400) , B B (400) , (400) , P V (30) , P V R P (30) , P U T (30) , P (30) , A P P (30) , (30) DIMENSION WY ( 1 0 , 4 0 0 , 1 0 ) 1CASH ( 4 0 0 , 1 0 ) , B A R ( 4 0 0 , DIMENSION AL(10,10) DIMENSION XLX(61) , E W Y ( 4 0 0 , 1 0 ) , W X ( 4 0 0 , 1 0 ) , X W X (4 0 0 , 1 0 ) , 10) READ(5,100)VAR,RF,D,NS,TD,NT TOO FORMAT(3F10-5,15,F10.5,15) READ ( 5 , 4 0 0 ) N A G E , NC 400 FORMAT (213) READ(2,402) 402 FORMAT (10 (XLX (J) ,J=1,61) ( 6 F 1 2 . 2/) ,F12-2) XK=TD/FLOAT(NT) WRITE 101 (6,101)VAR,RF,D,XK,NS,TD,NT FORMAT(1H1,30X,»VALUE 1PREMI0H OF AN EQUITY P A Y B E N T S ' / / / 2 0 X , * VARIANCE LINKED R A T E 3 OPTION 2 R A T E = ' , F 1 0 . 5 / 2 0 X , 3IN E Q U I T Y 3 PERIODIC',• ,» •PREMIUM(INVESTED ' , 4 F 1 0 . 5 / 2 0 X , ' T I M E I N C R E M E N T ' , F 1 0 . 5 / 2 0 X , * NUMBER 3 5PORTFOLIO P R I C E S 620X,'TIME BETWEEN 7PERI0DS WITH ' , F 1 0 . S / 2 0 X , ' R I S K L E S S » BETWEEN 3 OF REFERENCE' , » ' , 1 5 / PREMIUM PREMIUM P A Y M E N T S P A Y M E N T S 3 3 ' , F 1 0 . 5 / 2 O X , * NO. OF TIME' ,« ' , 1 5 / / / ) WRITE ( 6 , 4 0 1 ) N A G E , NC 401 FORMAT ( 2 0 X , ' A G E OF PURCHASER AT E N T R Y 3 ' , I 3 / 2 0 X , ' T E E S OF CONTRACT T , I 3 / / ) NS=NS+1 WRITE(6,301) 301 F O R M A T ( / / / 3 X , ' N O . 1'TOTAL 2'A PREM DO 1000 E=ND*D X H 3 0 . 0 5 NXH=20 P V , 5 X , » P V P U T ' , 3 X , ' A ND=1,NC Y E A R S ' , 4 X , ' V A L U E C A L L ' , 3 X , ' P V P O R T F O L I O ' , 1 X , • V A L U E PREM C A L L * / / / ) GUARANT.•,2X, P U T • , 4 X , • T O T A L P R E M ' , 3 X , 3 INITIALIZE BOUNDARY VALUES OF H DO 2 1=1,NS ST(I)=XH*FLOAT(I-1) I F (ST (I) - L E . E ) W ( I , 1) =0.0 I F (ST (I) . G T . E ) W ( I , 1) =ST(I) - E T=0.0 NT1=NT+l" DO 12 K=1,ND DO 9 J=2,NT1 SOLVE SIM EQNS FOR J J IS BASE PERIOD 8 ( 1 , J ) =B ( 1 , J - 1 ) *EXP(RF*XK) Y1=0.5*RF«XK Y3=0.5*VAR*XK NS1=NS-1 NS2=NS-2 DO 6 1=1,NS2 A (I)=¥1*I-Y3*I*I B ( I ) = 1 . 0 + 2.Q*Y1 + 2 . 0 * Y 3 * I * I C (I)=-Y1*I-Y3*I*I F (I)=H ( I + 1 , J - 1 ) F ( 1 ) = » ( 2 , J - 1 ) - A ( 1 ) *H(1,J) A(NS1)=-1.0 B (NS1) =1.0 F (NS1) =XH TRANSFORM EQUATIONS DO 21 1=2,NS1 I F (ABS (A (I) ) . L T . 0.0001) GO TO 20 BB (I) =BB ( I - 1) *B (I) / A (I) - G C (1-1) CC(I)=BB ( 1 - 1 ) * C ( I ) / A ( I ) F F ( I ) = B B ( I - 1 ) *F ( I ) / A (I),-FF ( 1 - 1 ) I F (ABS (BB (I) ) . L T - 1 0 0 0 0 . ) GO TO 21 BB (I) =0. 0001*BB (I) CC(I) = 0 . 0 0 0 1 * C C ( I ) F F (I) = 0 . 0 0 0 1 * F F ( I ) GO TO 21 CONTINUE BB(I)=B(I) CC(I)=C(I) FF(I)=F(I) CONTINUE BB(1)=B(1), CC(1)=C(1) FF(1)=F(1) CONTINUE. SOLVE FOR W W (NS,J) =FF ( N S 1 ) / B B (NS1) DO 8 IX=1,NS2 I=NS-IX W ( I , J ) = ( F F ( I - 1 ) - C C ( 1 - 1 ) *W ( 1 + 1 , J) ) / B B ( 1 - 1 ) I F ( K . E Q . 1 ) G O TO 31 31 30 9 66 C C C T=FLOAT (J-1+NTK)*XK GO TO 30 T=FLOAT(J-1) *XK NIK=0 CONTINUE I F ( J . L T . N T 1 ) G O TO 9 CONTINUE NTK=NTK+NT DO 66 1=1,400 BY (ND,I,K)=W ( I , N T 1 ) 216 NEW BOUNDARY VALUES OF W 10 11 12 33 34 35 302 1000 92 C 71 70 74 72 73 NSD=NS-NXH DO 10 1=1,NSD 11=1+NXH W ( I , 1)=8 ( I I , N T 1 ) NSD1=NSD+1 DO 11 I=NSD1,NS 11=1-1 W(I,1)=W(II,1)+XB Q CONTINUE CA (ND) =W (1,1) GA (ND) =E*EXP (-RF*T) PV (ND) =CA (ND)+GA (ND) ND1=ND-1 AN=0. I F (ND. EQ. 1) GO TO 34 DO 33 1=1,ND1 BN=EXP(-RF*I) AN=AN+BN CONTINUE PVRP (ND) =1. + AN PUT (ND) =PV (ND) -PVRP (ND) P (ND) =PV (ND) / (1.+AN) APP (ND) =P (ND) - 1 . APC (ND) =CA (ND) / ( 1.+AN) WRITE (6,302) ND,CA (ND) ,GA (ND) ,PV(ND) ,PVRP(ND) ,PUT(ND) ,P(ND) , 1 APP (ND) , A P C (ND) FORMAT(5X,I5,2X,8(3X,F10.5) ) CONTINUE FORMAT(5X,F7.3,5X,10(F7.3,2X)) COMPUTATION OF THE ALPHA MATRIX NX=NAGE-19 NC1-NC-1 DO 70 1=1,NC1 11=1-1 DO 71 J=I,NC1 AL ( I , J) = (XLX (NX+J-1) - X L X (NX+J) ) / X L X (NX+II) AL(I,NC)=XLX(NX + NC-1)/XLX(NX+II) AL(NC,NC)=1. WRITE (6,74) FORMAT(////3X,'YEAR*,20X,'ALPHA MATRIX'///) DO 72 1=1,NC WRITE (6,73) I , (AL ( I , J ) , J = I , N C ) FORMAT(5X,I2,5X,10(F7.5,2X)) DO 88 J=1,NC DO 88 1=1,400 = SUfl=0.0 DO 80 K=J,NC 80 88 81 82 SUM=SUM+AL(J,K) *WY ( K , I , K - J + 1 ) EWY(I,J)=SUM CONTINUE WRITE (6,81). F O R M A T ( / / / 3 0 X , ' O V E R A L L W MATRIX (WEIGHTED) DO 82 1=1,400 WRITE (6,92) ST (I) , (EWY ( I , J) , J=1,NC) 1 1 '///) COMPUTATION OF MATRIX OF PARTIAL DERIVATIVES WX 83 84 85 86 87 200 201 C 203 205 204 215 206 202 207 208 C C C DO 83 J=1,NC DO 83 1=2,399 WX(I,J) = ( E W Y ( I + 1 , J ) - E W Y ( I - 1 , J ) ),/(2.*XH) XWX ( I , J) =ST(I) * W X ( I , J ) WRITE (6,84) F O R M A T ( / / / 3 0 X , ' P A R T I A L DERIVATIVE WX M A T R I X ( W E I G H T E D ) • / / / ) DO 85 1=2,399 WRITE (6,92) ST (I) , (WX ( I , J ) , J = 1 , N C ) WRITE(6,86) F O R M A T ( / / / 3 0 X , * LONG POSITION I N THE REFERENCE P O R T F O L I O ' / / / ) DO 87 1=2,399 WRITE (6,92) ST ( I ) , (XWX ( I , J) , J=1,NC) COMPUTATION OF RISK PREMIUM YP=0.0 DO 200 ND=1,NC YP=YP+AL (1,ND) *APP(ND) WRITE ( 6 , 2 0 1 ) Y P FORMAT(///20X,'RISK PREMIUM *,F10» 5 / / / ) 3 COMPUTAION OF CASH POSITION DO 202 J=1,NC SUM1-0.0 S0M3=0.0 SUM=0-0 DO 203 K=J,NC S0M1=SUM1 + AL ( J , K ) *D*FLOAT (K) *EXP (- ( K - J + 1).*RF) DO 204 K=J,NC S0M2=0-0 K1=K-1 IF(J.EQ.NC) GO TO 215 DO 205 N=J,K1 SUM2=SUM2*EXP (- (N-J+1) *RF) SUM3=SUM3+AL(J,K)*SUM2 CONTINUE S0M=SUM1-(D+YP)*SUM3 DO 206 1=2,399 CASH ( I , J) =EW¥ ( I , J)+SUM CONTINUE WRITE (6,207) FORMAT(///40X,'CASH POSITION*///) DO 208 1=2,399 WRITE (6, 92) ST (I) , (CASH ( I , J) , J = 1 , N C ) NET BORROWING 209 DO 209 1=2,399 DO 209 J=1,NC BAR ( I , J ) =XWX ( I , J ) - C A S H ( I , J) WRITE (6,210) 210 211 $SIG FORMAT(///** OX,* NET BORROWING'///) DO 211 1=2,399 WRITE{6,92) ST (I) , (BAR(I,J) ,J=1,NC) STOP END 218 219 APPENDIX L: Investment P o l i c y F e a s i b i l i t y of the P o l i c y ; In this section we w i l l prove Theorems 1 and 2 of Chapter Form (7.9) and (L.l) i(x,-r) = (7.13) we ge~ ^ r t _ T 7. can write: ^ + w - x w - PVFP X Then (L.2) (w xx | i = 9x -x w < 0 xx > 0 because the w-x curve, has increasing slope) (L.2) proves Theorem 1. From (7.10) and (7.13) we can write: (L.3) i(x,x) = PV (x(t)) + p - x w T - PVFP X For s i m p l i c i t y we w i l l analyze the continuous premium contract; for i t we have: (X(T)) (L.4) PV (L.5) PVFP = = x + /^" Di dX T /J" (D+y) Substituting (L.4) (L.6) i(x,x) = rX e~ dX = rX and L.5) x + p - x W y ''==' x + f ^[l-e- [l-e" r ( t - T r ( t _ T ) ] ] ) into (L.3): - ^-[l-e" r ( t " T ) ] The value of i f o r x=0 i s : (L.7) i(0,x) = p(0,t-x,g) - - ^ - [ l - e " r ( t _ T ) ] For low values of the reference p o r t f o l i o i n r e l a t i o n to the exercise p r i c e (x<< g) the value of the put i s a decreasing function of time to maturity. At maturity p(0,0,g) = g and at time T , p(0,t-t,g) < -r(t-x) ge . Also from w X = l+p„ and w X value of p at time x i s p(0,t-x,g). x < 1 : p — X < 0. — Thus, the maximum At any time p r i o r to x the value of 220 the put can not be greater than p(0,t-x,g) because this would imply a known negative return on the put (which i s impossible by the arbitrage p r i n c i p l e ) . Therefore: (L.8) p(0,t-T,g) > p(0,t,g) Also (L.9) = f p(0,t,g) [l-e" ]> * r t l-e- r ( t - T ) From (L.8) and (L.9): (L.10) p(0,t,g) > ^ [ l - e " r ( t _ T ) ] From (L.10) and (L.7) (L.ll) i(0,x) > 0 For "low" values of the reference p o r t f o l i o the long p o s i t i o n on the reference can be financed from the premiums paid: there i s a p o s i t i v e investment i n the r i s k l e s s asset. The value of i f o r x=°° can be estimated by considering that (L.12) w (°°,t-T,g) (L.13) 1(»,T) = x = 1 - ^[l-e" and r ( t " p(~,t-x,g) T ) ] = 0 < 0 For "very high" values of the reference p o r t f o l i o some borrowing i s required to follow the hedged p o l i c y : the amount to be borrowed i s the present value of the future put premiums to be received. From ( L . l l ) , L.2 (L.13) and Theorem 1, Theorem 2 obtains. Cash P o s i t i o n The cash p o s i t i o n of the contract or funds a v a i l a b l e at time x f o r a contract with known date of expiration t , c ( x , x , t ) , can be written(from (7.9) and (7.11)) a s 221 (L.14) c(x,T,t) = w(x,t-T,g) + ge r ( t _ T ) - PVFP For the periodic premium contract the present value of the future premiums receivable can be written as (L.15) PVFP t-f-l = I (D+y) e ~ k-1 I r k When the expiration of the contract i s not known with certainty, the actual cash p o s i t i o n , C(x,T), can be obtained from (5.55), (7.5), (7.6), (L.14) and (L.15): (L.16) C ( X , T ) n = £ a(x,t) c(x,x,t) t=x+l C ( X , T ) = or n (L.16) I t=x+l ri (t,t) w(x,t-T,g) + I a(x,t) g(t) e - r ( t - T ) t=T+l n t-x-1 - (D+Y) I a(x,t)[ I e" ] t=x+l k=l rk Notice that the annual put premium on the put with known date of expiration, y ( t ) , has been replaced i n (L.16) by the actual put premium charged, Y. Formula (L.16) has been used i n Computer Program No. 9 to compute the cash p o s i t i o n .
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Generalized option pricing models : numerical solutions...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Generalized option pricing models : numerical solutions and the pricing of a new life insurance contract Schwartz, Eduardo Saul 1975
pdf
Page Metadata
Item Metadata
Title | Generalized option pricing models : numerical solutions and the pricing of a new life insurance contract |
Creator |
Schwartz, Eduardo Saul |
Date Issued | 1975 |
Description | The option pricing model developed by Black and Scholes and extended by Merton gives rise to partial differential equations governing the value of an option. When the underlying stock pays no dividends - and in some very restrictive cases when it does -, a closed form solution to the partial differential equation subject to the appropriate boundary condition, has been obtained. But, in more relevant cases such as the one in which the stock pays discrete dividends, no closed form solution has been found. This study shows how to solve those equations by numerical methods. The equations in question are linear parabolic partial differential equations. Although solution procedures are well known in Science and Mathematics, they have not been treated extensively in the finance literature. The numerical procedure used consists in approximating partial derivatives by finite differences to obtain expressions which can be handled by the computer. A general numerical solution to the partial differential equation governing the value of an option on a stock which pays discrete dividends, is developed in this dissertation. In addition, the optimal strategies for exercising American options are derived. For a sufficiently large value of the stock clearly larger than the exercise price, it may pay to exercise the American option at dividend payment dates. This study shows how to determine the "critical stock price" above which it will pay to exercise the option. Traditionally life insurance contracts have provided for benefits fixed in terms of the local unit of account. The prolonged period of inflation during the last three decades has served to diminish the usefulness of such usually long-term currency denominated contracts. As a result, life insurance companies have increasingly moved toward the study and issuance of equity based products the benefits of which depend upon the performance of some reference portfolio of common stock. This dissertation deals with the pricing of equity-linked life insurance contracts with asset value guarantee. In these contracts the benefits depend upon the performance of a portfolio of common stock. The insurance company, however, guarantees a minimum level of benefits (payable on death or maturity) regardless of the investment performance of the reference portfolio. In an equity based life insurance policy with this type of provision the insurance company assumes part of the investment risk as well as the mortality risk. When the contract becomes a claim (either because of death of maturity), the policyholder is entitled to receive an amount equal either to the value of the reference portfolio at that time or to the guaranteed sum, whichever is the greater. This study shows how the benefits of the contract can be viewed in terms of options and it obtains the partial differential equations governing the value of these options, which can also be solved by the above mentioned numerical methods. It is shown that the premium payments can be interpreted as negative dividend payments on the reference portfolio. Another important problem in relation to these policies is the determination of the appropriate investment policies for insurance companies to enable them to hedge against the major risks associated with the provision of the guarantee. It is shown how the equilibrium option pricing model determines the optimal investment strategy to be followed by insurance companies. The need for an appropriate model for pricing equity-linked life insurance policies with asset value guarantee, has long been apparent in the actuarial literature. The model developed in this study gives normative rules for pricing such contracts. Moreover, the prices determined by these rules are equilibrium prices, that is, they would prevail in a perfectly competitive market and would have the property that if the insurance company were to charge them and were to follow the investment policy determined by the model, it will bear no risk and will make neither profit nor loss. It is the hope of the author of this study that the model will find practical applications in the life insurance industry. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0100067 |
URI | http://hdl.handle.net/2429/19712 |
Degree |
Doctor of Philosophy - PhD |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1975_A1 S38_3.pdf [ 10.98MB ]
- Metadata
- JSON: 831-1.0100067.json
- JSON-LD: 831-1.0100067-ld.json
- RDF/XML (Pretty): 831-1.0100067-rdf.xml
- RDF/JSON: 831-1.0100067-rdf.json
- Turtle: 831-1.0100067-turtle.txt
- N-Triples: 831-1.0100067-rdf-ntriples.txt
- Original Record: 831-1.0100067-source.json
- Full Text
- 831-1.0100067-fulltext.txt
- Citation
- 831-1.0100067.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0100067/manifest