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Generalized option pricing models : numerical solutions and the pricing of a new life insurance contract Schwartz, Eduardo Saul 1975

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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., University of British Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in 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 th i s thesis in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L i b r a r y sha l l make it f ree ly ava i lab le for reference and study. I further agree that permission for extensive copying of th i s thes is for s cho lar ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or p u b l i c a t i o n of th is thes is for f i n a n c i a l gain sha l l not be allowed without my writ ten permiss ion. Department of C c y T^vc^v^ O ^ - -The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ABSTRACT The option pricing model developed by Black and Scholes and extended by Merton gives rise to partial d i f f e r e n t i a l equations governing the value of an option. When the underlying stock pays no dividends - and in some very restrictive cases when i t 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 d i f f e r e n t i a l equations. Although solution procedures are well known i n Science and Mathematics, they have not been treated extensively in the finance literature. The numerical procedure used consists in approximating partial derivatives by f i n i t e differences to obtain expressions which can be handled by the computer. A general numerical solution to the pa 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, i s 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, 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 price" above which i t w i l l pay to exercise the option. Traditionally l i f e insurance contracts have provided for benefits fixed in terms of the local unit of account. The prolonged period of infl a t i o n during the last three decades has served to diminish the usefulness of such usually long-term currency denominated contracts. As a result, 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 portfolio of common stock. This dissertation deals with the pricing of equity-linked l i f e 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 l i f e 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 i s entitled to receive an amount equal either to the value of the reference portfolio 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 in terms of options and i t obtains the partial 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 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 i s 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 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. Moreover, the prices determined by these rules are equilibrium prices, that i s , they would prevail in a perfectly 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 risk and w i l l make neither profit nor loss. It i s 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. v i TABLE OF CONTENTS Page CHAPTER 1. INTRODUCTION 1 1.1 Preamble 1 1-2 The Valuation of Options on Dividend Paying Stocks.... 2 1.3 The Pricing of a New Life Insurance Contract 4 1.4 Outline of the Thesis 7 2. THE PRICING OF OPTIONS 10 2.1 Introduction 10 2.2 S t a t i s t i c a l Approach to Option Valuation 12 2.3 Expected Value Approach to Option Valuation 15 2.4 Market Equilibrium Approach to Option Valuation 17 2.5 Merton ?s Derivation of the B-S Formula 21 2.6 Extension of the Model 26 2.7 More Recent Developments 30 2.8 Conclusions 33 3. PROCEDURES FOR PRICING OPTIONS ON STOCKS WITH CONTINUOUS DIVIDEND PAYMENTS 35 3.1 Introduction 35 3.2 Introduction to Partial Differential Equations 36 3.3 Difference Equations 37 3.4 The Par t i a l Differential Equation Governing the Value of an Option 40 3.5 Truncation Error 42 3.6 Stability and Convergence 42 3.7 Boundary Conditions 43 3.8 System of Linear Equations 44 3.9 Solution Algorithm 47 3.10 Numerical Example 48 3.11 Conclusions 55 4. OPTION STRATEGIES FOR EXERCISING OPTIONS ON STOCKS WITH DISCRETE DIVIDENDS 57 4.1 Introduction 57 4.2 Theory 58 4.3 Solution Algorithm 64 4.4 Application to the ATT Warrant: Sensitivity Analysis. 67 4.4.1 Dividends and C r i t i c a l Stock Prices 68 4.4.2 Variance Rates and C r i t i c a l Stock Prices 69 4.4.3 Interest Rates and C r i t i c a l Stock Prices 72 4.4.4 ATT Theoretical Warrant Prices 72 4.4.5 Comparing Warrant Values 79 v i i Chapter page 4.5 Conclusions 83 5. THE EQUILIBRIUM PRICING OF ASSET VALUE GUARANTEES ON EQUITY LINKED LIFE INSURANCE POLICIES: THEORY 85 5.1 Introduction to L i f e Insurance 85 5.2 Equity Based L i f e Insurance 87 5.3 A l t e r n a t i v e Approaches to P r i c i n g 91 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 92 5.5 The Guarantee as an Option 95 5.6 Valuing the Option Component of an Insurance Contract. 98 5.7 The Single Premium Contract 100 5.8 The Constant Continuous Premium Contract 101 5.9 The P e r i o d i c Premium Contract 104 5.10 The C a l l Option, The Put Option-and the P r i c i n g of Investment Risk 110 5.11 The Measurement of M o r t a l i t y 115 5.12 M o r t a l i t y Risk 117 5.13 The Equilibrium Value of the Contract: Investment Risk and M o r t a l i t y Risk 119 5.14 Conclusions 120 6. THE EQUILIBRIUM PRICING OF ASSET VALUE GUARANTEES ON PERIODIC PREMIUM EQUITY LINKED LIFE INSURANCE POLICIES: APPLICATIONS 1 2 2 6.1 Introduction 122 6.2 Solution Algorithm f o r the P e r i o d i c Premium Contract. 127 6.3 The Variance Rate of the TSE I n d u s t r i a l Index 131 6.4 Put Premium or "Risk Premium" 132 6.5 Numerical Example Number 1 132 6.6 Numerical Example Number 2 140 6.7 Numerical Example Number 3 144 6.8 Numerical Example Number 4 149 6.9 Numerical Example Number 5 153 6.10 Summary and Conclusions 155 7. OPTIMAL INVESTMENT STRATEGIES FOR THE SELLERS OF EQUITY LINKED LIFE INSURANCE POLICIES WITH ASSET VALUE GUARANTEES. 160 7.1 Introduction 160 7.2 Hedging P o l i c y 160 7.3 F e a s i b i l i t y of the P o l i c y 164 7.4 Numerical Example 167 7.5 Summary and Conclusions 186 BIBLIOGRAPHY 187 v i i i APPENDIX Page A. Computer Program No. 1: Option Prices on Stocks with Continuous Dividend. Payments 191 B. Computer Program No. 2: Option Prices on Stocks with Discrete Dividend Payments 193 C. Solution Algorithm: Option Prices in the Constant Continuous Premium Contract 197 D. Computer Program No.3: Option Prices in the Constant Continuous Premium Contract 200 E. Computer Program No.4: Option Prices in the Periodic Premium Contract 202 F. Computer Program No.5: Computation of the Equilibrium Value of the Guarantee with Known Date of Expiration.... 205 G. Computer Program No.6: Computation of the Equilibrium Value of the Contract and the Risk Premium 208 H. Select Mortality Tables 210 I. Computer Program No.7: Risk Premium versus Term of Contract 211 J. Computer Program No.8: Risk Premium versus Age of the Purchaser at Entry 213 K. Computer Program No.9: Investment Policy 214 L. Investment Policy 219 L . l F e a s i b i l i t y of the Policy 219 L.2 Cash Position 220 ix LIST OF TABLES TABLE ••' Page 3.1 Trans Canada Pipeline. Warrant Values 51 3.2 Trans Canada Pipeline Warrant: Regressions 53 3.3 Trans Canada Pipeline 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 4.2 C r i t i c a l Stock Prices as a Function of the Variance Rate and Time to Expiration 73 4.3 C r i t i c a l Stock Prices as a Function of the Riskless Interest Rate and Time to Maturity 75 4.4 ATT Theoretical Warrant Prices for Specific Stock Prices as a Function of Time to Expiration 77 4.5 Data Used in 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 Mortality Table 116 6.1 Mortality Table: Canadian Assured Lifes, 1958-64 * 2 6 6.2 Computation of the Equilibrium Value of the Guarantee 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 Page 6.9 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 for the Put 157 7.1 Investment Policy 172 7.2 Investment Policy 173 7.3 Investment Policy 174 7.4 Investment Policy 175 7.5 Investment Policy 176 7.6 Investment Policy 177 7.7 Investment Policy 178 7.8 Investment Policy 179 7.9 Investment Policy 1^ 0 7.10 Investment i n the Riskless Asset 181 7.11 Proportion Invested in the Reference Portfolio 183 7.12 C r i t i c a l Values of the Reference Portfolio 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 4.4 C r i t i c a l Stock Prices as a Function of the Variance Rate and Time to Expiration 74 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 for Specific Stock Prices 78 5.1 Partial Differential Equation and Boundary Conditions for the Periodic Premium Contract I l l 7.1 Investment i n the Riskless Asset 167 7.2 Investment i n the Riskless Asset 182 7.3 Proportion Invested i n the Reference Portfolio 184 x i i ACKNOWLEDGEMENTS The credit for undertaking and completing this dissertation must be shared by the candidate with his mentors on the Faculty of Commerce and Business Administration at the University of Bri t i s h Columbia. Most important among the latter i s Professor Michael J . Brennan. If 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 dissertation committee, Dr. Brennan provided generous and indispensable guidance to the candidate. His sustained interest in the study was a source of support so essential for an undertaking of this kind. Professor Phelim P. Boyle also merits a special expression of gratitude for his invaluable guidance to the candidate with respect to the f i e l d of insurance. Without his contribution of concepts and factual information relative to this discipline, the project could not have been launched. Mr. Alvin G. Fowler, Associate Director of the Computer Centre at UBC, provided essential instruction in the solution of partial 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 dissertation committee, Professors Robert W. White and William T. Ziemba read the dissertation and offered helpful suggestions toward i t s improvement. Professor Ernest Greenwood, University of California (Berkeley), kindly consented to examine the manuscript from the viewpoint of style. His editorial comments resulted in 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 latter the candidate wishes to extend his grateful appreciation. During the period while the candidate was conducting the research herein reported he was receiving financial support from The Canada Council. The candidate expresses his gratitude to that body. Throughout the years that he was a graduate student at the University of British Columbia the candidate was the fortunate recipient of the friendship and encouragement of Professors Peter A. Lusztig and Maurice D. Levi. It was their moral support which helped the candidate to surmount the obstacles so characteristic of one's i n i t i a l phase of graduate education. To both of them the candidate owes a lasting debt of gratitude. xiv To Gloria CHAPTER 1: INTRODUCTION 1.1 Preamble This study is concerned with options: an option is a security which gives i t s owner the right to purchase or s e l l another security at a predeter-mined price under certain conditions. The objective of this study i s twofold: (1) The valuation of options on dividend paying stocks; and (2) The pricing of a new l i f e insurance contract. 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 theoretical foundation of this study i s the stock option pricing 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 principle that, in equilibrium, there are no riskless profits to be made with a zero net investment. A zero net investment portfolio i s obtained by taking long and short positions on the stock, the option and the riskless asset. A partial differential equation governing the value of the option is the result of this analysis, which together with the boundary condition can be used under certain conditions to derive an analytical expression for the value of the option. Most of the options discussed in this study give rise to part i a l d i f f e r e n t i a l equations for which no closed form analytical solutions exist (or for which closed form solutions have not been found). It is shown how to solve these equations by numerical methods. The equations in question are linear parabolic partial d i f f e r e n t i a l equations and although solution proc-edure^ are ful l y discussed in s c i e n t i f i c and mathematical texts they have i 2 not been treated extensively in the finance literature. The numerical proc-edure used here consists in approximating partial derivatives by f i n i t e differences to obtain expressions which can be handled by the computer. 1.2 The Valuation of Options on Dividend Paying Stocks Assuming "ideal market conditions" Black and Scholes [13] developed an equilibrium option pricing model for the case where the underlying stock pays no dividends. They showed that i t is possible to form a hedged portfolio by taking a long position on the stock and a short position on the option (or vice versa) where the return of the hedged portfolio is independent of the price of the stock, that i s , riskless. In equilibrium, to avoid arbitrage profits, the return on the hedged portfolio must then be equal to the risk free rate of interest. By maintaining continuously the hedged portfolio they obtain a partial d i f f e r e n t i a l equation governing the value of the option. Given the value of the option at expiration (boundary condition) they solve the differential equation and obtain a formula for the value of the option as a function of the current price of the stock, time to maturity and known parameters (the riskless interest rate and the variance of the rate of return on the stock both assumed constant, and the exercise price of the option). Merton [32] shows that in the case where the stock is assumed to pay continuous dividends, the hedging process described by Black and Scholes can also be applied and a different p a r t i a l differential equation obtained. He points out that, in general, this p a r t i a l differential equation can not be solved by analytical methods (i.e., i t does not have a closed form solution). A numerical procedure for pricing options on stocks with continuous dividend payments is developed in this study. In particular, the partial d i f f e r e n t i a l equation mentioned in the preceding paragraph is solved numerically in the case of constant continuous dividends. The present author feels that the numerical approach developed here w i l l have a number of f r u i t f u l applications in the theory of options in finance. As mentioned, this thesis deals with two such problems: the valuation of options on dividend paying stocks and the pricing 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 prior to expiration (American type). Merton [32] has shown that i f a stock pays no dividends or the option is "dividend payout protected,"it w i l l never pay to exercise an American option before maturity and,hence, the value of an American option is equal to the value of i t s European counterpart. But i f the stock pays dividends and the option is not dividend protected,it 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 be greater than i t s European counterpart. 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 just before the stock goes ex-dividend, but never in between dividend payments dates. Black[5] has proved that the value of an option on a stock that pays discrete dividends is also governed by the same parti a l differential equation derived by Black and Scholes for the no-dividend case. The boundary conditions, however, change at each dividend payment date to reflect the fact that i t may pay to exercise the American option at those points in time. This was also recognized by Merton [32]. No closed form solution has been found in this case. A general numerical solution to the partial differential equation governing the value of an option on a stock which pays discrete dividends 4 is developed in this dissertation. In addition, the optimal strategy for exercising American options i s derived. For a sufficiently large value of the stock, clearly larger than the exercise price, 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 price" above which i t w i l l pay to exercise the option. 1.3 The Pricing of a New Lif e Insurance Contract Traditionally l i f e insurance contracts have provided for benefits fixed in terms of the local unit of account. The prolonged period of sustained inflation during the last three decades has served to diminish the usefulness of such usually long-term currency denominated contracts. As a result, 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 portfolio of common stocks. The idea behind this move i s that, in the long run, equities would provide a better hedge against in f l a t i o n than a fixed monetary amount. If a l l the investment risk associated with owing common stock is l e f t to the policyholder, he has no guarantee as to the asset value of his contract. Turner [44] argues that i t is both reasonable and appropriate for a l i f e insurance company to offer an additional assurance under ^equity based products whereby part of the investment risk i s assumed by the company. This has been increasingly done in the United Kingdom and Canada. This dissertation deals with the pricing of equity-linked l i f e insurance contracts with asset value guarantee.. In these contracts the benefits depend upon the performance of a portfolio of common stocks ; the insurance company, however, guarantees a minimum level of benefits (payable 5 on death or maturity) regardless of the investment performance of the reference portfolio. In an equity based l i f e 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 or maturity), the policyholder i s entitled to receive an amount equal either to the value of the reference portfolio 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 port-folio exceeds the guarantee. This latter 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 portfolio for the pre-determined guaranteed amount. If the value of the reference portfolio is 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 is 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 portfolio and the guaranteed amount. The benefits of the contract at expiration can also be viewed as the sum of the value of the reference portfolio plus the amount by which the guarantee exceeds the value of the reference portfolio. This latter amount may be regarded as the value of an immediately exercisable put option which permits the holder to s e l l the reference portfolio for the pre-determined guaranteed amount. If the value of the reference portfolio 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 is 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 portfolio. Three separate types of equity-linked l i f e insurance policies with asset value guarantee must be considered: (1) The single premium contract in which the whole investment in the reference portfolio i s made at the date of issue of the policy. It i s shown that in 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 for which a closed form solution has been derived by Black and Scholes. (2) The constant continuous premium contract in which a constant continuous rate of investment in the reference portfolio i s also assumed. It w i l l be shown that in this case i t i s possible to obtain a parti 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 similar to the one derived by Merton [32] for an option on a continuous dividend paying stock, but where the premium has to be considered as a negative dividend. A numerical solution to the differential equation i s also developed. (3) The periodic premium contract in which the investments on the reference portfolio are also made periodically. From the viewpoint of practical applications this type of contract i s the most important one. It w i l l be shown that the valuation of the c a l l option for 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 parti a l differential 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 pricing equity-linked l i f e insurance contracts with asset value guarantee is the value of the put option. The value of the put option represents the "cost" to the insurance company of giving the guarantee. It w i l l be 7 shown how to compute the value of the put option from the value of the cal l option for the types of contracts mentioned above. Another important problem in relation to these equity-based l i fe 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 wi 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 invest-ment 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 fe 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 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 wi l l bear no risk and wil l make neither profit nor loss. It is the hope of the author of this study that the model wi l l find practical applic-ations in the l i fe 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 lines given by Merton [32]. Merton's extension of the model to the case where the stock pays continuous dividends is given and the partial differential equation governing the value of the option is derived. This chapter concludes by discussing recent extensions of the equilibrium option pricing model. The numerical solution of parabolic partial differential equations using the f i n i t e differences method is presented in Chapter 3. The solution algorithm for the value of an option on a stock which pays constant continuous dividends is developed and the numerical method is applied to price a partic-ular option. A solution algorithm for the value of options on stocks with discrete dividends i s developed i n Chapter 4. The optimal strategies for exercising these options are also considered. Using the American Telephone and Telegraph 1975 Warrant as an example, the sensitivity of the c r i t i c a l stock price to changes in dividends, variance rates and interest rates i s examined. Chapter 5 i s devoted to the development of the model for equilibrium pricing of asset value guarantees on equity-linked l i f e insurance contracts. In this chapter the basic concepts of traditional and equity based l i f e insurance policies are presented and the aetualial literature covering the latter i s surveyed. Following this, the concept of the guaranteed amount as an option is developed and the models for valuing the option component of three types of contracts (single premium, continuous premium and periodic premium) are presented. The put option as the equilibrium pricing of the investment risk is then considered. Finally, mortality risk is introduced to find the average equilibrium value of the contract and the "risk premium" that insurance companies must charge for giving the guarantee. The solution algorithm for the model presented in Chapter 5 is derived in Chapter 6. Some numerical examples are then given and the sensitivity of the value of the guarantee to various parameters i s examined. These parameters are: (1) the variance of the rate of return on the reference portfolio; (2) the riskless interest rate; (3) the nature of the minimum guarantee; (4) the age of the purchaser of the policy; and (5) the term of the contract. Chapter 7 is concerned with the derivation of appropriate investment policies for l i f e insurance companies to enable them to hedge against the risks assumed by giving the guarantee in equity based l i f e insurance policies. It is 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 policy are always sufficient to finance the long positions i n the hedged portfolio or sometimes insufficient so that additional borrowing is required. Mortality risk i s also introduced to arrive at the actual optimal investment strategy. 10 CHAPTER 2: THE PRICING OF OPTIONS . 2.1 Introduction An option is a security giving i t s owner the right to buy or s e l l an asset at a predetermined price, subject to certain conditions, within a specified period of time. If the option can be exercised any time up to the date of i t s expiration or maturity, i t i s called an "American option." If i t can be exercised only at expiration, i t i s called a "European option." The previously agreed price that i s paid for the asset when the" option i s exercised i s called the "exercise price." The last date on which the option can be surrendered or exercised i s called the "maturity date" or "expiration date." A standard " c a l l option" gives i t s owner the right 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 price. There exists other option contracts which are different 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 individual, (2) has a longer l i f e than the normal c a l l option, and (3) is 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 this dissertation 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 securities. Put options w i l l also be considered when discussing l i f e insurance policies in Chapters 5 and 6, but then the distinc-tion between the c a l l option and the put option w i l l be clearly indicated. The problem of evaluating stock options has been studied by economists and statisticians since the beginning of the century when Bachelier [2] introduced the theory of Brownian motion in stock price fluctuations. 11 These studies may be classified into three broad categories: (1) Relevant variable 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 prices are considered and the s t a t i s t i c a l l y significant ones are identified. In the expected value approach to option valuation the mathematical relationship between option prices and the underlying stock prices is analyzed, and equations for estimating expected future option values are developed. The theoretical 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 mathe-matical relationship between the option price and the associated common stock price, but the relationship i s obtained by imposing market equilibrium conditions. It is based on the arbitrage principle that in equilibrium there are no riskless profits to be made. This chapter summarizes some of the important published work on the different approaches to option valuation with particular emphasis on the market equilibrium approach which has been the motivation of this dissertation. Then, the Black-Scholes [3] option valuation model i s presented and relevant extensions by Merton [32] are considered. This option pricing model provides the theoretical basis as well as the starting point for this thesis. The chapter concludes with a consideration of further developments in the formulation of equilibrium option pricing models. 12 2.2 S t a t i s t i c a l Approach to Option Valuation Shelton's [38] model for 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 price 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 is selling at four times the exercise price, the leverage advantage is 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 price of the warrant within these boundaries. The factors tested by Shelton in a stepwise multiple regression were: (1) dividend yield on the associated stock; (2) whether the warrant is l i s t e d or not (dummy variable); (3) the length of time to expiration of the warrant; (4) whether the warrant sold for more or less than $5.00 (dummy variable); (5) the historic v o l a t i l i t y of the common stock (measured by the average of the ratios of yearly highs to lows of stock prices for three years) and; ' (6) the recent trend of the stock price (measured by the percentage change of the stock price over the previous year). Shelton found that the f i r s t three of the above variables are highly significant in explaining warrant prices, while the last three variables are not significant. It i s interesting to note that neither v o l a t i l i t y nor recent trend of the related stock proved to be significant, though the way in which they were measured should be taken into account. i 13 In his 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 Z + l ) V z - 1 where y = warrant price/exercise price x = stock price/exercise price z = estimated parameter (has a value between 1 and «°) 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 variable z, Kassouf considered four variables which might influence warrant prices: (1) the length of time to expiration; (2) the dividend yield on the underlying stock; (3) the potential dilution of the common stock resulting from the exercise of a l l outstanding warrants; and (4) the recent price history of the common stock. In a more recent paper Kassouf [20]presented the latest results 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 for the period 1945 to 1964, he obtains the following equation for 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 yield on the stock (annual) D = number of outstanding warrants/number of outstanding common shares X = price of common share A = exercise price of warrant Xj = price of common share 11 months previous. He found a l l variables to be significant (at a 0.05 level) and a multiple correlation coefficient of 0.63. Kassouf did not include any measure of risk in his regression. His warrant pricing formula also implies that warrants w i l l never be exercised prior to maturity because the warrant price 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 interesting book: "Beat the Market, A Scientific Stock  Market System," Thorp and Kassouf [43] describe how i t should be possible to make substantial profits by selling warrants short and hedging by buying long the underlying stock. They point out that a hedged investment turns out to be profitable and yet less risky since the price of the warrant and the price of the stock tend to move in sympathy with each other. When the stock and warrant prices move in one direction the loss in one position (short or long) i s largely covered by the gain in the other position (long or short). The selection of the appropriate candidates for this hedging operation i s based on Kassoufs 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 risk by forming a hedged position between the warrant and i t s underlying stock, is one of the crucial ideas used by Black-Scholes [3] to developed the fundamental model of the market equilibrium approach to valuation. This model w i l l be discussed in detail in subsequent sections. What Thorp and Kassouf f a i l to pursue i s the fact that i n equilibrium the expected return on a perfectly hedged position, created by going short in one security and long in the other security, must be equal to the return on a riskless asset. This last 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 traditional bond and stock valuation models which proceed by discounting an expected stream of dividends or coupon payments and terminal price at a predetermined discount rate. Sprenkle [39] and Ayres [1] attempt to find the expected value of a warrant and to measure risk preferences of investors in the markets for warrants. To compute the expected value of a warrant using analytical methods i t is necessary to specify the probability distribution of future stock prices. Sprenkle assumed that successive changes in stock prices follow the random walk model and that the probability distribution of stock price changes i s log-normally distributed. The expected value of the warrant when exercised is then given by: (2.1) E[W] = Jc (S-C) US)dS where S = random variable representing stock price; W = random variable representing warrant price; £(S) = probability density function of future stock prices (lognormal), and C = exercise price. 16 Sprenkle assumes that investors perceive risk in the warrant market in the variable "leverage," defined as the ratio of the standard deviation of percentage changes in warrant prices to the standard deviation of percentage changes in stock prices, minus one. The primary purpose of Sprenkle's work is to provide empirical estimation of investors' risk preferences and he does this by comparing the actual prices of warrants with the computed values obtained by discounting the expected exercise value of the warrant at expiration. But he i s unable to estimate empirically the required discount factor. Ayres' objective is to investigate the existence of risk aversion in the warrant markets. He also formulates his warrant valuation model based on the assumption of lognormally distributed stock price 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 relationship between risk and return to determine the existence of risk aversion, he assumes that the warrant investor perceives a (the standard deviation of the lognormal distribution of the ratios of successive stock prices) as the measure of risk in the stock. In their 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) is exercisable at any time prior 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 right to exercise a warrant at any time in the interval prior to i t s maturity (American type) . 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 for evaluating warrants. They assume that the probability distribution of possible values of the stock when the warrant i s exercised i s lognormal and take the expected value of this distribution cutting i t off at the exercise price as in 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 for the f i r s t time in Black-Scholes' seminal paper "The Pricing of Options and Corporate L i a b i l i t i e s " [ 3 ] . Being essential to the understanding of this dissertation, the Black-Scholes' (B-S) model w i l l be presented here in greater detail following closely their original paper. In deriving their formula for the value of an option in terms of the price of the underlying stock, B-S make the following assumptions ("ideal conditions"): a) The riskless interest rate i s known and is constant through time.1' (Merton [32] extends B-S model to include stochastic interest rates.) B-S actually use the short-term interest rate, but being constant and known the long-term is 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. 1 8 b) The stock price follows a random walk in continuous time. The probability distribution of possible stock prices at the end of any f i n i t e interval is log-normal. The variance rate of the return on the stock is constant. The stochastic specification of stock prices w i l l be discussed in greater detail in subsequent sections. (In Merton's extension of the B-S model the variance rate of the return on the stock is allowed to be a known deterministic function of time.) c) The stock pays no dividends or other distributions. Merton [32] obtains the partial differential equation governing the value of the option in the case of continuous dividends and Merton [32] and Black [ 5] give indications on how to solve the problem in the case of discrete dividends. These topics w i l l be treated in subsequent sections. d) The option is "European," that i s , i t can only be exercised at maturity. Merton [32] proves that i f an American option is "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 is not "dividend protected." e) There are no transaction costs i n buying or se l l i n g the stock or the option. f) The borrowing and the lending rates are equal and there are no restrictions on borrowing. g) Shortselling i s allowed without restrictions and without penalties. Thorp [42] shows that restrictions against the use of proceeds of short-sales do not invalidate B-S analysis. 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 is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, the value of which w i l l not depend on the price of the stock, but w i l l depend only on time and the values of known constants." If the value of the option as a function of the stock price S and time t is 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/Ws(S,t) where the subscript refers to the partial 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 position does not depend on the price of the stock, B-S point out that the ratio of the change in the option value to the change in the stock price, AW/AS, is approximated by Wg(S,t) when AS i s small. Then, i f the stock price changes by an amount AS, the option price 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 in value by an amount AS. Thus, the change in the value of a long position in one share of stock w i l l be approximately offset by the change in value of a short position in 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 position with one share of stock changes. If the hedge is maintained continuously, then the approximation mentioned above becomes exact, and the return on the hedged position i s completely independent of the change in the value of the stock. In fact, the return on the hedged position becomes certain." 2 See Black-Scholes [3] p. 641. 20 The value of the equity in the hedged position i s : (2.3) S - W/Wg and the change in the value of the equity in a short interval At i s : (2.4) AS - AW/Wg Since the return on the equity in the hedged position i s certain, the return must be equal to rAt (where r i s the riskless interest rate). 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/WS = (S-W/Ws)rAt Assuming that the short position i s changed continuously, i t i s possible to use stochastic calculus to expand AW, introduce i t s value in (2.5) and simplify to obtain the partial d i f f e r e n t i a l equation governing the value of the option. The details of the derivation are not given here , because the B-S formula is derived in the next section along the lines given by Merton [32], this being more convenient for our subsequent analysis. In this 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 relationship 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 principle that in market equilibrium conditions there are no riskless profits to be made. This approach to option valuation, based on sound economic theory, has been one of the important advances in the finance literature of the seventies; i t s potential applications into the pricing of corporate l i a b i l i t i e s and other kinds of options have not yet been fu l l y explored. The market equilibrium approach to option valuation w i l l be followed in the rest of this study. 21 2.5 Mer 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 e x p e c t e d r a t e o f r e t u r n p e r u n i t o f t i m e on a n o n - d i v i d e n d p a y i n g s t o c k a t each p o i n t i n t i m e t i s d e f i n e d by (2.6) E t [ S ( t + ^ ( t - S ( t ) ] 0 1 = h and t h e v a r i a n c e 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 t i m e i s d e f i n e d by (2.7) 2 \ [( S ( t +^ ( t- S ( t ) - a h ) * ] a = 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 t i m e t , h i s a s m a l l t i m e i n c r e m e n t and "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 o p e r a t o r . I n the l i m i t as h tend s t o z e r o , a i s c a l l e d t h e i n s t a n t a n e o u s e x p e c t e d r a t e o f r e t u r n and a t h e i n s t a n t a n e o u s v a r i a n c e o f r e t u r n . Under m i l d a s s u m p t i o n s M e r t o n [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 dynamics i n s t o c h a s t i c d i f f e r e n c e f o r m as ( 2 8 ) s ( t + h ) t - s ( t ) __ a h + a y ( t ) / r where y ( t ) i s a random p r o c e s s w i t h E t [ y ] = 0 (2.9) and E J y 2 ] = 1 D e f i n i n g a s t o c h a s t i c p r o c e s s , ? ( t ) , by (2.10) z ( t + h ) - z ( t ) = y(t)«fr~ the l i m i t o f (2.10) as h t e n d s t o z e r o d e s c r i b e s a Wiener p r o c e s s o f 22 Brownian motion defined by the stochastic differential equation: (2.11) dz = y(t)/dt with E [dz] = 0 (2.12) and E f c[dz] 2 = dt The instantaneous rate of return on the stock can be obtained by taking the limit of (2.8) as h tends to dt. The stochastic d i f f e r e n t i a l "Processes such as (2.13) are called It6 processes and while they are continuous, they are not differentiable. In (2.13) a may be a stochastic variable but a is 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 interest i t i s reasonable to assume^ that the value of the option, W, is a function of the stock price S and the length of time to expiration, T: Given the distributional assumptions on the stock price, i t i s necessary to apply Ito's Lemma to obtain an expression for the change in the option price over time. Itd's Lemma is 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 is not differentiable i f the derivatives of the function with respect of i t s argument, in this case time, ;do not exist, at one or more points in the domain. equation for the instantaneous rate of return i s then: adt + adz (2.14) W = W(S,T) 4 See Merton [32 ] . 23 Taylor's expansion in calculus; i t states how to differentiate 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 price changes i s given by: (2.15) dW = WsdS + WTdT + h W s s(dS) 2 where subscripts denote partial derivatives and (2.16) (dS) 2 = a 2S 2dt (2.17) dT = -dt Substituting (2.13) (2.16) and (2.17) into (2.15): (2.18) dW = WsS(adt+adz) - WTdt + h a 2S 2 W s sdt Rearranging terms (2.18) can be written as dW ~ = Bdt + ydz 3 = ^[aSWs + % a 2S 2 Wss - WT] = gsw s W Notice that the stochastic d i f f e r e n t i a l equation (2.19) for 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 portfolio between the stock and the option where the stochastic component of the return is eliminated, as indicated by B-S. (2.19) where (2.20) (2.21) See McKean [28] and Merton [30]. 24 Consider forming a portfolio by investing in the stock, the option and the riskless asset, such that the total investment in the portfolio i s zero. This can be achieved by financing long positions in the securities with borrowing and using the proceeds of short sales. Let a^ = instantaneous amount invested in the stock a2 = instantaneous amount invested in the option 33 = instantaneous amount invested in the risk free asset. Then the condition for zero total investment in the portfolio can be written as (2.22) a L + a 2 + a 3 = 0 The instantaneous dollar return on the portfolio, dy, can be written as (2.23) dy = ai jr- + a 2 ^ + a 3rdt Substituting (2.13), (2.19) and (2.22) into (2.23) dy = a^(adt + adz) + a 2(8dt + ydz) - (a^ + a 2) rdt and simplifying: (2.24) dy = [a x(a-r) + a 2(B-r)]dt + [aja + a 2y]dz A strategy a^ = a^ can be chosen such that the coefficient of dz in (2.24) is always zero (normalization): (2.25) a]*a + a 2 * Y = 0 Then, the total return on the portfolio, dy*, would be non-stochastic; that i s , the hedged position would have been formed. Since the aggregate investment in the portfolio is zero, to avoid arbitrage profits the return on the portfolio must be zero. 25 Therefore: (2.26) a i * ( a - r ) + a 2 * (B-r) = 0 Eliminating a^ and a2 from (2.25) and (2.26) the market equilibrium condition i s obtained: ( 2 . 2 7 ) 8=r = 1 a-r a Substituting the values of B and y from (2.20) and (2.21) respectively into (2.27) and simplifying gives the pa 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: (2.28) ho2S2 Wss + rS Ws - WT - Wr = 0 subject to the boundary conditions: (2.29) W(S,0) = Max[0, S-E] and (2.30) W(0,T) = 0 Notice that the expected instantaneous rate of return on the stock, a, has dropped out from equation (2.28). Equation (2.28) is a second order linear 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 specified. 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 price, E, or zero of the former is negative. This value i s called 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 is zero irrespective of the 61 : 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 in somewhat more detail in Chapter 3 when numerical solutions are discussed. 26 outstanding l i f e of the option.^ If Che stock price is zero i t can never recover B-S solve the differential equation (2.28) with boundary conditions (2.29) and (2.30) by transforming them by substitution into the heat-transfer equation of physics, which has a known closed form solution. The solution is given by the formula: (2.31) W(S,T) = SN(dL) - Ee~ r T N(d 2) where s 9 d l = g + (r + ho1) T off d 2 = dl - off -x 2 I d N(d) = - — / e 2 dx, is the cumulative normal density function. /2TT Black-Scholes point out, "Note that the expected return on the stock does not appear in equation (2.31). The option value as a function of the stock price i s independent 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 faster the stock price rises, the faster the option price w i l l rise through the functional relationship (2.31)." 8 Also note that the option price does not depend on the risk preferences of investors. 2.6 Extension of the Model To analyze the effect of dividends on unprotected options, Merton [32] 7This boundary condition is really not necessary to solve the different-i a l equation. It can be obtained mathematically from (2.28) and (2.29) as w i l l be seen later on. See Black-Scholes [ 3] p. 644. 27 assumes a continuous dividend per share per unit time, D(S,T), when the stock price is S and the time to maturity of the option i s T. Differ-ential equation (2.13) gives the total instantaneous rate of return on the stock. Thus, i f the stock pays dividends the differential equation should be written as: rn dS + Ddt A,. . A (2.32) = adt + adz or (2.33) -|§ = ( a - |)dt + adz The instantaneous expected rate of return from price appreciation i s (a - D/g) a n d the instantaneous rate of return from price appreciation i s given by (2.33) The stochastic di f f e r e n t i a l equation for option price 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 price 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 dif f e r e n t i a l equation for option price changes can be written as: (2.34) dw = [ho2S2 Wss + (aS-D)Wg - WT]dt + aSWgdz Again (2.19) can be written, but the new definitions of 0 and y are: (2.35) 3 = ^iha2S2 Wss + (aS-D)Wg - W~T] (2.36) y = ^ S . The instantaneous dollar return on the zero equity portfolio is now given by /o o7\ A dS-4-Ddt dW . . (2.37) dy = ai — + a 2 + a 3 rdt 28 where the total rate of return on the stock given by (2.32) has to be considered. Substituting (2.32), (2.19) and (2.22) into (2.37) and rearranging terms, the instantaneous dollar return on the portfolio can be written as: (2.38) dy = [aj_(a-r) + a 2(0-r)] dt + [a1a+a2Y]dz Using the same investment strategy as in the last section, the condition for no arbitrage profits is (2.39) « X a-r a Substituting the values of g and y from (2.35) and (2.36) respectively into (2.39) and simplifying the resulting expression, the di f f e r e n t i a l equation governing the value of the option i s obtained: (2.40) ho2S2 Wss + (rS-D)Ws - WT - 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 prior to maturity and,hence, i t has the same value as a European option. The option i s always worth more "alive" than "dead." Therefore, formula (2.31) i s valid just as well for an American option as for a European option under the conditions specified in 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 positive probability of premature exercise. 29 Therefore,differential equation (2.40) with boundary condition (2.41) and (2.42) govern the value of a European option. For an American option an additional arbitrage boundary condition is 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 i t would be exercised. A general closed form solution to equation (2.40) subject to the boundary conditions (2.41) and (2.42) for the value of an option on a dividend paying stock has not been found. Only in two particular cases have explicit formulae for the value of the option been obtained. One of the cases in which a closed form solution has been obtained for the European option i s that in which the underlying stock pays a constant dividend yield, that is,where the functional relationship of D(S,T) is given by (2.44) D(S,T) = pS where p > 0 constant. Samuelson [36] and Samuelson and Merton [37] have shown that in this case, for the American option, there i s always a positive probability of premature exercising and, hence, the arbitrage boundary condition (2.43) w i l l be binding for sufficiently large stock prices and no closed form solution w i l l exist. Therefore, the American option can be worth more "dead" than "alive." The solution of (2.40) for the European option can be obtained by substituting D = pS into (2.40) and proceeding in the same way as the B-S solution. The solution formula, given by Merton [32], i s : (2.45) W(S,T) SN(d,) - Ee -rT N(d 2) where: (2.46) £n - + (r-p + ha2)T off d 2 = dj - off and N(d) is as defined in (2.31). 30 The second case in which a particular closed form solution has been found is that in which the stock pay constant continuous dividends, that is,where (2.47) D(S,T) = D, where D is constant. Merton [ 32] has shown that in this case premature exercising may or may not occur, depending upon the values for D, r, E and T; and that a sufficient condition for no premature exercising is given by: (2.48) D < Er If conditions (2.48) obtains, the solution for the European option value w i l l be the solution for the American option. Merton [32] points out that a closed form solution to (2.44) in the case (2.47) has not yet been found for f i n i t e time to expiration and gives the solution to the perpetual option when (2.48) obtains. One of the objectives of this dissertation i s to develop a numerical solution procedure for equation (2.40) subject to (2.41) and (2.42) in the continuous constant dividend case when condition (2.48) obtains. This i s the main topic of Chapter 3. The numerical method developed is applicable, in fact, to any known functional relationship of D(S,T) and not only to the constant dividend case. Also, the numerical procedure can be applied, as is done in Chapter 4, to solve for the more r e a l i s t i c case of discrete dividend payments and to find the optimal exercising strategy when premature exercising i s profitable 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 is the stochastic specification of stock price 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 is a Wiener process or Brownian motion. This specification can be taken to imply that the relative change in the stock price over a small interval 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 restriction imposed by the use of stochastic d i f f e r e n t i a l equation (2.13) to describe stock price changes,is that within a small interval of time, (t, t+dt), S(t) w i l l move in a random fashion, but with high probab-i l i t y , approaching to one as dt tends to zero, that S(t+dt) w i l l be in an q ar b i t r a r i l y small neighborhood of S(t). In essence, the va l i d i t y of process (2.13) and, hence, of the B-S equation (2.31) depends on whether stock price dynamics follows a continuous path. That i s , in a short interval of time the stock price can only change by a small amount.^ Cox and Ross argue that, "New information, however, tends to arrive at a market in discrete lumps rather than in a smooth flow, and assets in such markets are l i k e l y to have discontinuous jumps in value, thus violating the basic assumption of a process such as (2.13). Such behaviour is characteristic of many economic situations and is particularly relevant when p o l i t i c a l events or stochastically applied government or institutional constraints are of primary importance."^ In two recent papers, Cox and Ross [ 8] and Merton [34] have independently studied the valuation of options when stock price movements can be 9 See Cox and Ross [8 ] p. 2, 1 0See Merton [34] p. 3. ^See Cox and Ross [8] p. 3. 32 better described by a jump process. Both studies use a Poisson process to describe the random arrival of an important piece of new information about the stock. Cox and Ross concentrate their analysis on the assumption that stock price changes follow a jump process rather than a diffusion It6 process. They specify the jump process as: (2.49) — = odt + (k-l ) dn where II is a Poisson process and dn takes the value zero with probability 1-Xdt and the value one with probability to Xdt. X measures the "rate of probability flow" for the jump and k-1 is 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 price changes follow a jump process as (2.49) by attempting to form hedged portfolios with the stock, the option and the riskless asset. Since they consider only one "dimension" of risk (the Poisson process) the hedged position can be obtained and maintained after a jump by a discontinuous adjustment to a new position. They obtain a complex formula for the value of the option that i s independent of the intensity of the process X which determines the expected number of jumps. Unfortunately, however, their valuation formula is dependent on the expected rate of return on the stock, a. Merton [34] considers the more general case in which the total change in the stock price is the sum of two types of changes: (1) the "normal" vibrations in price assumed to follow a standard geometric Brownian motion or Wiener process. (2) The "abnormal" vibrations in price due to the arrival of important new information about the stock that has more 33 than a marginal effect on price, assumed to follow a Poisson process. He then writes the total change in stock price as: (2.50) dS S = (a-Ak)dt + adz + dq where a is the instantaneous expected return on the stock; a is the instantaneous variance of return, conditional on no occurance of the Poisson event; dz is a standard Wiener process; dq is an independent Poisson process. dq and dz are assumed to be independent. X is the mean number of arrivals per unit of time. k = E(y-l) where (y-1) is the random variable percentage change in the stock price i f the Poisson event occurs and E is the expectation operator. Merton also proceeds in his analysis by forming portfolios which consist of the stock, the option,and the riskless asset, but,given the complexity of the stochastic specification of stock price changes assumed,he proves that i t i s impossible to find weights which w i l l give riskless return on the portfolio, and hence, the B-S "no-arbitrage" technique cannot be employed. He then obtains a formula for the value of an option by imposing the additional assumptions that securities are priced so as to satisfy the Capital Asset Pricing 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 risk can be diversified away and, as Merton points out, this i s a testable proposition. The models employed in this dissertation assume that stock prices dynamics can be described by an Ito process of the form (2.13). 2.8 Conclusions An equilibrium option valuation model developed by Black-Scholes and 34 extended by Merton has been presented i n t h i s chapter. The model represents a breakthrough i n the finance l i t e r a t u r e not only because i t gives a r a t i o n a l basis f o r p r i c i n g options, but also because of i t s p o t e n t i a l use i n the valuation of other kinds of contingent claims such as corporate l i a b i l i t i e s J There have been some r e l a t i v e l y s uccessful empirical t e s t s of the model by Black-Scholes [4 ] on c a l l - o p t i o n data, Leonard [23] on warrants, G a l a i [ 14] on the Chicago Board Options Exchange data, and I n g e r s o l l [16] on Dual Funds. The f a c t that most companies do pay dividends and that t h e i r dividend p o l i c y i s never a proportional one ( i . e . with constant dividend y i e l d ) , c a l l s f o r a more general s o l u t i o n to the e q u i l i b r i u m option v a l u a t i o n model. The next two chapters of t h i s d i s s e r t a t i o n deal with t h i s problem. In Chapter 3 numerical methods f o r valuing options on stocks with continuous dividend payments are developed when D < Er. In essence, the problem solved i s the one of f i n d i n g a general numerical method of s o l u t i o n to p a r t i a l d i f f e r e n t i a l equation (2.40) governing the value of the option. The problem of dealing with the more r e a l i s t i c case of p r i c i n g an option on stocks with p e r i o d i c dividend payments i s presented and solved i n Chapter 4. Also the optimal s t r a t e g i e s f o r e x e r c i s i n g American options are discussed i n t h i s chapter. The e q u i l i b r i u m option p r i c i n g model presented i n t h i s chapter and the numerical s o l u t i o n procedures developed i n Chapters 3 and 4 are then modified and adapted to solve an a c t u a r i a l problem: the p r i c i n g of equity-l i n k e d l i f e insurance p o l i c i e s with asset value guarantees. The theory i s developed i n Chapter 5 and some a p p l i c a t i o n s are given i n Chapter 6. See, f o r example, Merton's [33] study on the p r i c i n g of " r i s k y " corporate debt. 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 principle that in equilibrium no riskless profits can be made with a zero net investment a partial d i f f e r -ential 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 - is a function of the value of the stock, time to maturity>and known constants. A closed form solution to the dif f e r e n t i a l equation has been found when the underlying stock pays no dividends and also in 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 is based essentially on numerical solutions to parti a l d i f f e r e n t i a l equations. After a brief introduction to second-order par t i a l d i f f e r e n t i a l equations, the f i n i t e differences method is developed as i t applies to the numerical solution of parabolic par 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 is used to evaluate the Trans Canada Pipeline warrant maturing April 3, 1976. Without implicating him, the author would like to thank Mr. Al Fowler, Associate Director of the Computer Centre at the University of British Columbia, for helpful assistance in solving partial d i f f e r e n t i a l equations by numerical methods. 36 The numerical solution is compared with the results obtained by applying the B-S formula in the case where the stock pays no dividends. It is made clear that the numerical procedure can be made as accurate as required by reducing the size of the stock price step and the time step. 2 3.2 Introduction to Partial Differential Equations A linear, 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 in the form: (3.1) AWgs + BWST + CWTT + DWg + EWT + FW = G where A, B, C, D, E, F and G are functions of S and T, W i s the dependent variable and subscripts denot partial differentiation. 3 As in the case of ordinary d i f f e r e n t i a l equations , additional inform-ation (boundary conditions) must be supplied along with (3.1) i n order to pick out a specific solution from the family of possible solutions to (3.1). In general, three types of second-order partial d i f f e r e n t i a l equations are defined: 2 i) The equation is e l l i p t i c i f B - 4AC < 0 2 i i ) The equation is parabolic i f B - 4AC = 0 2 i i i ) The equation is hyperbolic i f B - 4AC > 0 This section and the next draw on McCracken and Dorn [25]. 3 In ordinary di f f e r e n t i a l equations the dependent variable is a function of only one independent variable. 3 7 3 . 3 Difference Equations The definition of the derivative of a function of a single variable i s : dy _ lim y(x+h) - y(x) dx h-K) h In a computer i t is 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 is sufficiently close (accuracy) and that the error does not grow as the process continues ( s t a b i l i t y ) . The method can be described as replacing a derivative with a difference. Let us begin by considering only differences in the S direction. The Taylor series expansion for W ( S , T Q ) about the point ( S o , T Q ) can be written as W(S,T o) = W (S o,T o) + ( S - S Q ) W S ( S o,T o) + - ^ f ^ WgS(n,To) where n li e s between S and S Q. Hereafter we w i l l use n as a dummy to denote the constant that w i l l make the Taylor expansion exact. Let S = S + h, then o W S(S 0,T 0) - W ( S n + h » T n > h - W ( S"> To) , - | w s s(n,T o ) In other words, i f W G i s approximated by (the equality sign is used for simplicity) ( 3 . 2 ) W S ( S 0,T Q) = W(S 0+h,T 0)^- W (S 0,T 0) the truncation error i s ( 3 . 3 ) E t = - | W s s( n,T 0) S 0 < n < S 0+h Equation ( 3 . 2 ) was obtained by substituting S = Sc+h in the Taylor series expansion, the result is called a forward difference. A backward  difference can be obtained by letting S = S Q - h. 38 The r e s u l t i s A^  u r«j T ^ - w ( s 0 >T 0) - w(s 0-h,T 0) (3.4) W S ( S 0 , T Q ) - — I t i s also p o s s i b l e to average (3.2) and (3.4) and get a t h i r d approximation f o r Wg ^ n ^ m \ _ W(S 0+h,T 0) - W(S 0-h >T 0) (3.5) W S ( S 0 , T 0 ) - ^ To determine the truncation e r r o r of t h i s approximation, r e c a l l that W(S,T 0) = W ( S 0,T 0) + ( S - S D ) W S ( S 0 , T 0 ) + ( S ~ S ( ? ) 2 W S S ( S 0 , T 0 ) + W S S S ^ ' T o ) Now l e t S = S D+h and then S = S D - h and subtract the two r e s u l t i n g equations. The truncation e r r o r i s seen to be: h 2 (3.6) E T = - | W S ss(n,T 0) S 0-h < n < S 0 +h The d i f f e r e n c e equation f o r Wgg, using a forward d i f f e r e n c e , i s (3.7) W S S ( S 0 , T 0 ) = W S ( S 0 + h , T 0 ) h - W ^ ( S O ? T Q ) I f forward di f f e r e n c e s are now substituted for Ws, the r e s u l t would be biased i n the forward d i r e c t i o n . In order to avoid t h i s effect,backward differences are used. The backward d i f f e r e n c e for Ws(S 0,T 0) i s given by (3.4) and (3.8) W s(S 0+h,T 0) = W(S n+h ?T n)^- W(S n ?T n) Sub s t i t u t i n g (3.4) and (3.8) into (3.7) the di f f e r e n c e for Wss i s obtained: (3.9) W s s(So,To) = W(S 0+h,T 0) -2W(Sn,Tn) + W(S 0-h ?T n) h 2 39 To determine the truncation error, r e c a l l that ,2 'SS W(S,TQ) = W(S0,T0) + (S-SD) W S(S 0,T 0) + - ^ - ^ 1 W S S(S 0,T 0) + ^ T ^ 3 w s s s ( s 0 , T 0 ) + - ^ - 4 w s s s s ( n , T 0 ) 6 "SSSvlJo»-o/ ' 24 Now let S = SQ+h and then S = S0-h and add the two resulting equations. The truncation error i s seen to be: (3.10) E T — W s s s s ( n , T 0 ) . S Q-h < n < SQ+h An entirely analogous development leads to difference equations for the derivatives in the T - direction. For example, take the step size in the T - direction to be k, the backward difference for WVp is (3.11) W T(S Q )T 0) = w< sn» Tn> - W(Sn,Tn-k) k With these expressions the pa r t i a l differential equation (3.1) can be completely rewritten in terms of differences. The resulting 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 partial d i f f e r -ential equation derived in Chapter 2 w i l l be made. For a comprehensive description of different approaches see Forsythe and Wasow [11] and McCracken and Dorn [25]. In general a rectangle of width S n and height T m is considered (the cr i t e r i a to decide the maximum S, S n, and the maximum T, T m, depend oh the problem and on the boundary conditions). The width S n is divided into n equal intervals of width h = S n/ n. Similarly the height T m is divided into m equal intervals of height k = T / . n ° mm 40 Figure 3.1: A Mesh of Points Laid Over a Rectangular Region. m 2 1 0 W i , j ( i , j ) 1 s. n The mesh intersections are numbered as shown in Figure 3.1 and the following notation is introduced: (3.12) S i = ih where i = 0, ,n Tj = jk where j = 0,, ,m W(Si,Tj) = W(ih,jk) = W±>j A difference equation for each mesh point w i l l be written and then the resulting system of simultaneous equations w i l l be solved. The boundary conditions w i l l give the values of W at the edge of the rectangle or a relationship between extreme values. 3.4 The Partial Differential 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, is given by the i i solution to the second order parabolic partial d i f f e r e n t i a l equation 41 (3.13) h a2 S 2 Wss + (rS-D) Ws - rW - WT = 0 with boundary conditions (3.14) W(0,T) = 0 and (3.15) W(S,0) = max[0,S-E] where a2 is the variance rate for the return on the stock assumed constant, r i s the riskless constant interest rate, E i s the exercise price, S i s the stock price,and T is the time to maturity of the option. If the difference approximations (3.5), (3.7) and (3.11) are used for Wg, Wgg ,and respectively, and the notation defined in (3.12) i s used, the partial differential equation (3.13) can be approximated in difference form as 9 9 Wi+i -i - 2Wi i + Wi-l i ^i+1 i ~ ^ l - l A (3.16) h a2 S,2 L _ L L 1 + ( r S l-D) ? J o u — h 2 2 h W,- A _ W,- A - rW k Simplifying (3.16) : (3.17) a ± Wi_ l f j + b ± W±>j + c ± W 1 + l f j = W^^! i = 0, ...n j = 0, . .. m where /o 1 R v a . (rSj -D)k a 2Sj 2k (3.18) a ± - ^ 2h2-(3.19) b ± = (1+rk) + ° 2 s i 2 k a n d h z (3.20) c. = . ( r S i - D ) k _ q2 S i 2k 1 2h 2h2 Recalling from (3.12) that S ± = ih, (3.18), (3.19) and (3.20) can be written as (3.18) a, = hrkx - ^  - % a 2 k i 2 1 2 n i j (3.19) b ± = (1+rk) + a 2 k i 2 and 42 (3.20) c, = -Jsrki + ^  - ko2ki2  1 2n For any point ( i , j ) relationship (3.17) can be represented schematically by drawing the four points that are related by the difference equation and showing for each point i t s coefficient in (3.17). The result i s called the stencil of the method. a i c i -1 i-1 i i+1 3.5 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 = - ^  a2S±2h2 W s s s s(n,T j) - | ( r S ± - D ) h 2 W s s s(n,T j) + |wTT(s.,c) with S± - h < n < S± + h and Tj - k < C < Tj By appropriate selection of the step sizes h and k, the truncation error can be made as small as desirable. 3.6 Stability and Convergence It has been indicated earlier that as h and k tend to zero the difference equation approaches the differential equation. However we 43 have not yet taken up the question whether the solution of the difference equation approaches (converges) the solution of the d i f f e r e n t i a l equation as h + 0 and k -> 0. In other words, whether the solution of (3.17) converges to the solution of (3.13). This is 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 in which an error i s propagated ("grows"). The order of magnitude of the cumulative departure of the solution caused by errors committed at different points,is 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 solution. The method selected to solve (3.13) is such that (3.17) is probably stable 2 and convergent for a l l values of A = k/h . We postulate the latter to be true; no one has proved i t in the general case of variable coefficients. 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 in the notation defined in (3.12) can be written as (3.22) W 0 }j =0 for a l l j W± Q = 0 f or o < i < f (3.23) 4 W. „ = ih-E for f- < i < n i,o h -Subsequently i t w i l l be seen that the boundary conditions (3.22) and (3.23), which give the values of W for S=0 and for T=0 respectively, are not sufficient to solve the difference equation (3.17). An additional constraint is needed at the upper limit of S (for i=n). The additional boundary condition is occasioned by considering the fact that 44 for any given time to expiration the slope of the curve relating 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 partial derivative Wg tends to one: lim Wg = 1 S -> oo Then for a sufficiently large S the following approximation can be made: (3.24) Ws = 1 The upper limit of S can always be chosen large enough for (3.24) to hold withi the precision required at the boundary. As a "rule of thumb" this upper limit should be three or four times the exercise price E. This is a conservative limit,because Wg is practically equal to one for values of S greater than twice the exercise price. Then,using the backward difference(3.4),(3.24) can be re-written in the new notation; (3.25) W s(S n, Tj) = W n > j 'h W t l " 1 > j = 1 f o r a l l j o r (3.26) -W n_i }j + W n >j = h for a l l j With boundary conditions (3.22), (3.23) and (3.26) and the difference equation (3.17) a system of linear equations can be formulated which w i l l give the values of W for time j , starting from the values of W for time In this way i t i s possible to proceed in stepwise fashion to solve for the values of W for the whole grid. 3-8 System of Linear Equations The difference equation (3.17) generates a system of linear equations as i varies from 1 to n-1. In the f i r s t equation (for i = 1) 4 If the sufficient condition for no premature exercising of the option (2.48) holds the W-S curve w i l l never cut the line W = S-E. Then W > S-E, but W < S and W(S) is convex (see Merton [32]). Therefore: lim Wg = 1. S ->- 0 0 45 the f i r s t term (a^ W Q jj) in the equation is zero because W0,j = ^ (from (3.22)). The last equation (for i = n) is 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. bl w-lj;j + C l W2jj = WL j _ i a 2 Wlfj + b 2 W2>j + c 2 W 3 J = U2,i-1 (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. . . for i = l , . . . , n - l 1 1 y 3~ •!• (3.28) f = h n a = -1 n b = 1 n and omitting the subscripts j for simplicity, the system (3.27) can be written as (3.29) b l W l + c l W2 " f l a 2 W l + b 2 W2 + c 2 W3 = f 2 a , + W „ + b . W . + c , W = f . n-1 n-2 n-1 n-1 n-1 n n-1 a W . + b W = f n n-1 n n n The matrix of coefficients of the system (3.29) has zeros everywhere except on the main diagonal and on the two diagonals parallel to i t on either side. Such a matrix i s called tridiagonal. Fortunately the system represented by (3.29) is 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 coefficients a^ , b^ , c^ , f^ instead of a^, b i , c-£, f i , and where in particular a^ = 0 i = 2,...n. The iterative process is as follows: i) The f i r s t equation i s l e f t unchanged. i i ) The second equation is multiplied by ^ i . and the f i r s t equation is sub-5 3 2 tracted from i t , to obtain (the f i r s t term drops out) [h. b 9 - c t] W2 + Co W3 = f 2 - f , a 2 2 1 a 2 1 *2 1 Now l e t , * b i , b2 = ~ L h 2 ~ cl a 2 „ * _ bi c 2 - _ i c 2 *2* = II h ~ * 1 a2 Then the transformed second equation would be b2* W2 + c2* W 3 = f2* i i i ) Then the third equation is multiplied by b^/a-j and the transformed second equation is subtracted from i t . Again the f i r s t term of the third equation drops out. iv) This procedure is continued u n t i l the last equation. "*This procedure is only followed i f a 2 is different from zero. If a 2 = 0 the second equation is also l e f t unchanged. i 6 • S|ee footnote 5. 47 The transformed system of equations i s now b l * w l + c l * w2 = f l * b 2* W2 + c 2* W3 = f 2 * (3.30) b n - l * Wn_! + c n_!* Wn = f n _ i * C Wn = f n * In the f i r s t equation (3.31) b j * = bi, ci* = ci and f j * = f i and in general b * — i * (3.32) b ±* = - b i - c±_i* for i=2,...n a i * b i - l * (3.33) c,* = c. for 1=2,...n-1 i ajL i b • i * (3.34) f ± * = f ± - f ± _ ! * for i= 2,...n a i The solution of the transformed system is now immediate: starting with the last equation (3.35) Wn = f n * / b n * the Wj_ are given successively by (3.36) W± = [ f ± * _ c ±* W 1 + 1] i = n - l , . . . , l b i The procedure described solves for the values of Wj^j (i= l,...n) starting from the values of Wj^j_i (i= l,...n). The f i r s t line of the grid, Q , is computed by using the boundary condition (3.23) which gives the values of W±iQ (i= o,...n). 3.9 Solution Algorithm Using the developments of the preceding sections, a computer program 48 was written to solve by numerical procedures partial 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 for the return on the stock, the riskless interest rate, the dividend rate, the exercise price of the option, the stock price increment (h), the time increment (k), the number of different stock prices, and the number of different times to maturity of the option. The output of the program i s the option price matrix. This matrix gives the numerical solution to the part i a l d i f f e r e n t i a l equation for different stock prices and times to expiration. The program i s shown in Appendix A. For condition (3.24) to hold within the precision required at this boundary, the upper limit of stock price chosen should be sufficiently large in relation to the exercise price of the option. At this point the slope of the curve describing the relationship between the option price and the stock price for a given time to maturity i s set at one. Before proceeding to use the algorithm to value an American option, i t is necessary to check the sufficient condition for no premature exercising discussed in Chapter 2: (2.48) D < Er 3.10 Numerical Example To test the numerical procedure developed in the preceding section, 49 available data' on the Trans Canada Pipeline Warrant were used. To determine the variance rate of return on the Trans Canada Pipeline (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) : price 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) for the period mentioned, the variance rate per month was computed (0.005712). The average yield on three to five years government bonds at the end of January 1971 (5.42%) 8 was used as a proxy for the riskless constant interest rate. The TCP Warrant has an exercise price of $41.00 and w i l l expire on April 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 solution developed in this chapter, the algorithm was used to evaluate the TCP Warrant setting the dividend rate equal to zero (D=0). Recall from Chapter 2 that in the case where the stock pays no dividend Black-Scholes obtained a closed form (exact) solution to partial d i f f e r e n t i a l equation (3.13), given by formula (2.31). For 100 different stock prices (from $1.00 to $100.00) and 100 different times to 7See Foster [12]. g The continuously compounded rate should be used. See footnote 4 in Chapter 4. 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 in both cases the i above mentioned values of the parameters a 2, r and E. Two warrant price matrices of 100 by 100 (that i s , 10,000 option prices) were obtained, one for each method of solution for the no dividend case. A third warrant price matrix of the same dimensions and characteristics was computed by the numerical procedure for 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 for a stock price of $41.00 and times to expiration from !g year (6 months) to 8 years (96 months). The stock price chosen for the table was the exercise price 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 solution for the no dividend case do not diffe 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 in the last column of Table 3.1. To give a more complete comparison of the methods, the warrant prices obtained by the numerical procedure for 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 for each different time (months) to maturity, and each regression was based on 99 warrant values, one for each different stock price. It should be emphasized that the parameters that give precision to the numerical solution were intentionally chosen quite crude to show the robustness of the procedure. The stock price 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 price. In subsequent applications of numerical procedures these parameters have been chosen so as to increase the precision of the solutions obtained. TABLE 3.1 Trans Canada Pipeline Warrant Values E = $41.00 r = 5.42% per year a2= .005712 per month D = $1 per year Time to Maturity (in months) Warrant Value (in $) for S = $41.00 B-S No Div. NS No Div. NS With 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. B-S solution with no dividends numerical solution with no dividends numerical solution with a constant continuous dividend rate of $1 per year. 52 For the purpose of this study the numerical solution w i l l be considered 2 satisfactory i f for each regression the R obtained is very high, the intercept is not significantly different from zero, and the slope i s not significantly different from one. I t can also be expected that, given the nature of the numerical solution, the results should become less accurate as time to expiration is increased. 2 The R obtained for a l l the regressions were greater than 0.9999. The intercepts for the f i r s t twenty-two periods (months) were not significantly different from zero and were smaller, in absolute value, than one cent; but by period twenty-three they became greater, in absolute value, than one cent and significantly different from zero. In the last period, with more than eight years to maturity, the intercept was - 0.0734 and significantly different from zero. The slopes started with 1.0001 in the f i r s t period and gradually increased to 1.0087 for the last period. Table 3.2 shows the R s, the intercepts and slopes of the regressions for times to expiration from % year to 8 years, every six periods (months). It should be noted that the stock price step chosen for the numerical solution was one dollar and the time step was one month. By decreasing the size of 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 illustrates 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. In this case the maximuir. iifference with the B-S analytical solution decreases to four cents. TABLE 3.2 Trans Canada P i p e l i n e Warrant Regressions: Warrant Values B-S versus NS Time to Maturity RSQ A B 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 of the regression A : inte r c e p t B : slope TABLE 3.3 Trans Canada P i p e l i n e Warrant Values E = $41.00 r = 5.42% per year a2= 0.005712 per month D = $0.25 Stock p r i c e step = $0.25 Time step = 0.25 month Time to Maturity ( i n months) Warrant Value ( i n $) f o r S = $41.00 B-S No Div. NS No Div. NS With Div. 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 72 . 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 B-S No Div. = B-S s o l u t i o n with no dividends NS No Div. = numerical s o l u t i o n with no dividends NS With Div. = numerical s o l u t i o n with a constant continuous dividend rate of $1.00 per year. 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 lexibil ity 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, wi l l be studied. In Chapters 5 and 6, the related problem of pricing an equity linked l i fe insurance policy with asset value guarantee wi l l be discussed. It wi 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 in this chapter can also be used in the case in which the dividend rate is 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 is that the values of a^, b^, C i should reflect the changes in D as S and T change along the grid. Promising applications of this method to the valuation of " c a l l features"^ and convertible securities are also envisioned. The " c a l l feature" gives a corporation the option to repurchase i t s debt at a stated price 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 solution for the value of an option on a dividend paying stock has been found only in the case of constant dividend y i e l d , 1 that i s , a dividend proportional to the value of the stock', but in this case there is always a positive probability of premature exercising (for large values of the stock price) and,hence, the value of the American option may be greater than the corresponding value of the European option given by the closed form solution. The American option can be worth more "dead" than "alive." In Chapter 3 a numerical method was developed to find the value of the option i n the case in which the underlying stock pays a constant (this was not a very restrictive assumption), continuous dividend and where the sufficient conditions for no premature exercising (2.48), given by Merton [32], are met. In practice, however, the dividend policy of corporations i s not proportional (constant dividend yield) nor continuous (constant dividend rate) although the second approximation i s certainly better than the f i r s t . Corporations normally pay dividends quarterly or semi-annually, i.e. at discrete intervals, and are very reluctant to increase or reduce dividends even in the case of substantial changes in earnings or stock ^In Chapter 2 i t was also mentioned that Merton [32] gives a closed form solution in the restrictive case of a perpetual warrant when i t s underlying stock pays a constant continuous dividend and where the sufficient condition for no premature exercising obtains. 58 prices. The s t a b i l i t y of dividends has been widely dealt with in the literature of finance [24,46,47], The objective of this chapter is twofold: (1) to find, by modifying the procedure developed in 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 price 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 sensitivity of the c r i t i c a l stock price to changes in dividends fvariance rates, and interest rates are studied. Finally, a comparison is made between the actual ATT warrant prices at different points in time and the theoretical value obtained by the numerical procedure and by the Black-Scholes solutions in both the no dividend case and in the constant dividend yield 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. Consequently, i t might pay to exercise the option only on a f i n i t e number of dates: just 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 c l t n-1 "n present time; moment when the option i s to be valued, time of the f i r s t dividend, time of the i t n dividend, time of the l a s t dividend before maturity, time of maturity of the option. As 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 s t a r t s from e x p i r a t i o n date where the boundary conditions are known (in the s p i r i t of dynamic programming), the d i s c r e t e d i v i d -ends and time i n t e r v a l s between dividends w i l l be defined from the end: Pn-1 P n_ 2 D 2 Dx ln Ln-1 fco c l z2 ^ - 2 tn-l tn Consequently, Dn_^ is the f i r s t dividend and the last dividend before maturity (starting from the present time t Q ) and 1 = t „ - t. n _ c n - l t n _ i - t n -2 ~ tn-i+l " t n - i Tn-1 = t2 " tl T n " ti - t c and the t o t a l time to maturity i s 60 As in Chapter 2,all the "ideal conditions" in the market for the stock and for the option w i l l be assumed, in particular that the variance rate of the return on the stock, a , is constant and that the riskless interest rate, r, i s also constant. Within the f i n a l time interval from t n _ ^ to tn,where no dividends are paid the value of the option, W]_, is the solution to the parti a l d i f f e r e n t i a l equation (where T i s time to expiration): (4.1) ko2S2 Wss + rSWs - rW - Wx =0 ' 2 with boundary conditions (4.2) WL(0,T) = 0 and (4.3) W1(S,0) = max[0, S-E] These results were derived in Chapter 2. The solution to this equation can be obtained by the Black-Scholes formula for the no-dividend case or by the numerical method developed in 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 for the numerical solution for the rest of the time intervals where a closed form solution has not been found. The value of the option at time t n _ ^ ex-dividend can then be written as W^(S,xi). 2 The subscripts i = 1, ...n have been used in the boundary condition to indicate which time interval they refer to. Warning: these subscripts should not be confused with partial derivatives. The differential equation (4.1) has been written without subscripts,because i t refers to a l l time intervals. This also holds true for boundary condition (4.2). 6 1 Within any period between dividend payments the value of the option can be expressed as a function of:the stock price, time to maturity,and known parameters. Also within any period between dividend payments a l l the assumptions made for the derivation of the B-S option pricing model hold. Thus, i t i s possible to form a hedged position between the stock and the option by going long in one and short in the other, independent of the value of the stock. In other words, the ' whole derivation of the B-S model i s also valid in this case and the value of the option i s governed by parti a l d i f f e r e n t i a l equation ( 4 . 1 ) . The boundary conditions, however, are different and, therefore, so is the solutic for the value of the warrant. At each dividend date a new boundary condition is imposed to take into account the fact that the price 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 for the reduced ex-dividend, stock price cannot be lower than the exercise value given by the corresponding cum-dividend stock price minus the exercise price, 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 is always worth more "alive" than "dead." But i t may be profitable to exercise the option immediately 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) prior 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 retain 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 n_j (or any other subsequent time interval) is also given by the solution to the partial differential equation (4.1), but where the maturity date is assumed to be t n _ j . At this point in time a decision wi l l have to be made whether to keep the option or whether to exercise i t . Also, the boundary conditions at time tn_^ have changed. If we assume that the change in the stock price at tn_^, when 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 2 ( S , 0 ) = Max[0, S - E , W J ^ C S - D J . T I ) ] that is,the boundary condition at t n _ i is the greater of: zero, the exercise value (cum-dividend) or the value of the option when the stock goes ex-dividend. If the first is greater than the second, the option wi l l be exercised at tn_^; otherwise the option wi l l be retained. The cr i t ica l stock price, S C , has been defined as the value of the stock cum-dividend for which the exercise value is equal to the option value when the stock goes ex-dividend. For values of the stock greater than S C i t wi 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. Again i t is possible to find the value of the option at time t n _ 2 ex-dividend W 2 ( S , T2) and to continue with this dynamic programming approach until time tQ. 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: Crit ical Stock Price In Figure 4.1 curve: OEAB is Max[0, S-E], OAC is W-j^S-Di.!) , and OAB is MaxfOEAB, OAC]. OAB is naturally the new boundary condition and A is the cr i t ica 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. The problem with this approach, as Chen recognizes in [ 7],is that "In applying the warrant valuation model to i l compute the theoretical value of a warrant, the required rate of return 64 on warrants ( i . e . the discount rate 0) must be known." 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 with changing boundary conditions developed i n t h i s chapter,is based on Black and Scholes' seminal idea that i n e q u i l i b r i u m the expected return on a hedged p o s i t i o n (by going long on the stock and short on the option) must be equal to the return on the r i s k l e s s asset and i s independent of'the expected returns on the stock and on the option as w e l l as of the homogeneity of expectations. 4.3 Solution Algorithm The s o l u t i o n procedure f o r p a r t i a l d i f f e r e n t i a l equation (4.1) with boundary conditions (4.2) and 4.5) f o r the value of an option where the underlying stock pays d i s c r e t e dividends,is analogous to that indi c a t e d i n Chapter 3 f o r the case of constant continuous dividends. In t h i s case the continuous dividend rate, D, i s equal to zero, but at each d i s c r e t e dividend paying date i t i s necessary to stop i n the i t e r a t i v e process to make the necessary adjustments i n the boundary conditions to be able to continue the i t e r a t i v e process i n the next time i n t e r v a l . As has already been mentioned, at each dividend paying date the algorithm w i l l give the c r i t i c a l stock p r i c e as the value of the stock, above which i t w i l l pay to exercise the option. The boundary conditions (3.22) and (3.24) w i l l be the same a l l along the i t e r a t i v e process, but the boundary condition (3.23) w i l l be replaced by (4.5) each time a d i s c r e t e dividend i s paid. Given the boundary conditions, there are no changes i n the system of l i n e a r equations presented i n Chapter 3. 65 The computer program presented in Appendix A was adapted to the problem discussed in this chapter, in particular boundary condition (4.5) was computed at each discrete dividend date. At each dividend date the program gives for every stock price two option prices: the ex-dividend and the cum-dividend option prices. These two prices differ, because even when i t does not pay to exercise cum-dividend, the option price cum-dividend w i l l be the same for stock price S as the option price ex-dividend w i l l be for stock price S-D^ . Thus, for a given stock price S the option price 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 price cum-dividend. If the curve OAC in Figure 4.1 does not cut the line EB within the range of stock prices selected for the computations,the program prints an arb i t r a r i l y large number as the c r i t i c a l price. 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 ex-dividend for S-D-£),the stock price increment has to be chosen as an integral D. fraction of the dividend (that i s h = _ i , where n is a positive integer). In the program shown in Appendix B i t has been assumed that the discrete dividend is the same in every dividend date, although different known dividends could be incorporated with minor changes. Apart from the information required for the program explained in Chapter 3, i t is now necessary to know the amounts of the discrete dividends and their dates of payments. The output of the program gives the values of the option at different points in time and for different 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 price versus time to expiration. A slightly modified version of the program plots option prices for a given stock price as time to maturity changes. Finally, the program allows for a different definition of c r i t i c a l stock price. 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^ ) is no greater than P dollars above the exercise value, i.e. i f the premium on the option is less than P. The c r i t i c a l stock price with premium P, S , is shown in Figure 4.2. Figure 4.2: C r i t i c a l Stock Price with Premium P. 0 E-P E S S„ S Except for the line W = S-(E-P) which determines S at D, cp Figure 4.2 is the same as Figure 4.1 (where P = 0). This modified definition of c r i t i c a l stock price was made to study the sensitivity of the c r i t i c a l price to changes in P and,thus,the "closeness" of the two parts of the boundary condition (OAC and OEB in 67 Figure 4.2) i n the "neighborhood" of the c r i t i c a l stock p r i c e . J 4.4 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 Analysis The American Telephone and Telegraph warrant maturing May 15, 1975 was selected as an example to study the s e n s i t i v i t y of c r i t i c a l stock p r i c e s to changes i n dividends, i n variance rates, and i n i n t e r e s t rates. The ATT warrant was issued i n A p r i l , 1970 e n t i t l i n g the holders to subscribe to 31,375,540 shares of common stock at $52.00 a share beginning November 15, 1970 to May 15, 1975. The exercise p r i c e was protected against d i l u t i o n (stock s p l i t s and stock dividends) ,but not against cash dividends. The t o t a l amount of shares outstanding at the time of the issue was 549,266,114, so the p o t e n t i a l d i l u t i o n i n the number of shares i s 5.71%. In the year preceding the issue the stock p r i c e range had been 46^ to 58 1/8 and the in d i c a t e d dividend at the time was $2.60 payable quarterly (January 1, A p r i l 1, Ju l y 1 and October 1). In view of these f a c t s the period considered f o r the study was November 15, 1970 to May 15, 1975. The variance of the monthly rate of-, return was computed from the CRSP tape using 60 observ-ations from J u l y 1965 to June 1970. The variance rate per month for t h i s period was 0.0017. For 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 of the text whenever reference i s made to the c r i t i c a l stock p r i c e , the standard d e f i n i t i o n with P equal 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 precise, the r i s k l e s s i n t e r e s t rate should have been adjusted to a continuous compounding bas i s . This adjustment was not made i n what follows, because i t would not have changed the nature of the r e s u l t s . 68 average yi e l d on 3 to 5 years United States government securities [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 price range considered (in this case from $0 to $260) a c r i t i c a l stock price with a value of $54.75 for the last divid end date was obtained; at none of the other dividend dates did a c r i t i c a l stock . price exist (the curve OAC in Figure 4.1 becomes asymptotic to EAB) or was greater than $260. This result i s not surprising, because with a dividend of $2.60 per year (5% of the exercise price) the sufficient condition for no premature exercising '(2.48) in the continuous equivalent to this problem i s satisfied. Only in the last dividend date, one and a half 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 price of ATT common stock is considered, relative to other NYSE stocks, to be reasonably stable. It also has a high dividend yield relative to other stocks. Thus, i n relation to other stocks, the variance rate of the ATT stock i s relatively low and the dividend yield relatively high. Both of these factors should make the premium of the ATT warrant over i t s exercise value relatively small. If i t does not pay to exercise the option before maturity in this case, where the premium is relatively small, the case w i l l be even 69 stronger for options on stocks with higher variance rate and lower dividend yield. To study the effect 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 results are tabulated in Table 4.1 and a graph i s shown in Figure 4.3. As expected,the c r i t i c a l stock price i s a decreasing function of dividends and an increasing function of time to maturity, withvariance rate and riskless interest rate constant. It is interesting to note that the c r i t i c a l stock price with i a premium of only ten cents can produce significantly different results from the ones shown in Table 4.1 and Figure 4.3. This i s due to the fact that the curve OAC in Figure 4.1 is very close to the line EAB and almost par a l l e l to it."* 4.4.2 Variance Rates and C r i t i c a l Stock Prices To test the sensitivity of the c r i t i c a l stock price to changes in the variance rate of return on the associated stock for different times to expiration, the ATT warrant data were again used. However, this time a quarterly dividend of one dollar was assumed, because, as was mentioned earlier, with the actual dividend of $0.65 no c r i t i c a l price existed within the To make sure that the accuracy of the numerical method was enough to give reliable results the computer program was run on double precision and after 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 differed by more than one cent in relation to the single precision run. Also, in 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 riskless interest rate constant). Time to Expiration (in months) C r i t i c a l Stock Price with dividends paid quarterly (in $) 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 84.00 FIGURE 4.3: CRITICAL STOCK PRICES AS A FUNCTION OF DIVIDENDS AND TIME TO EXPIRATION 80.00 76.00 72.00 D=0.9 # # D= 1. 0 CRITICAL STOCK PRICE 68.00-# # # # 6 4 . 00 60.00 56.00 # # # # # # # # D=1. 1 # # # # # # # 52.00-... .00 I 8.00 16.00 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 cr i t ica l stock prices. The results are tabulated in Table 4.2 and in the graph in Figure 4.4. As expected,the cr i t ical 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 Crit ical Stock Prices Finally, to study the sensitivity of cr i t ical 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 cr i t ica l prices within the range of stock prices selected . Other things being equal, the value of an option and, hence,also the cr i t ica l stock price are higher, the higher is the riskless interest rate. Thus, cr i t ical 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 fe 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 Function of the Variance Rate and Time to Maturity (for constant i n t e r e s t rate and dividends). Time to Ex p i r a t i o n ( i n months) C r i t i c a l Stock P r i c e s ( i n $) Variance Rate per Month 0.001 0.003 0.005 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 Risk Free Rate = Dividends = 6.37% per annum $1.00 per quarter 112.00 FIGURE 4.4: CRITICAL STOCK PRICES A S A FUNCTION OF THE VARIANCE RATE AND TIME TO EXPIRATION 104.00 96. 00 88.00 VAS=0.005 # # # CRITICAL STOCK PRICE 80.00-72.00 64.00 56.00 # VAR=0.003 # # VAR=0.001 # # # # # # # # # # # 48. 00-, # # # # # .00 8. 00 16.00 -•I . . . J . . . . . . . . . | . . . 24.00 32.00 40.00 TIME TO EXPIRATION 48.00 TABLE 4.3 C r i t i c a l Stock Prices as a Function of the Riskless Interest Rate and Time to Maturity (for constant variance rate and dividends). Time to Cr i t i c a l Stock Prices (in $) Expiration Riskless Rate (per y ear) (in months) 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 84.00 FIGURE 4.5: CRITICAL STOCK PRICES AS A FUNCTION OF THE RISKLESS RATE AND T I H E TO EXPIRATION 80.00-76. 00-72. 00- R=4% CRITICAL STOCK PRICE 68.00-64.00 60. 00 56.00 t # # # # # 52.00-... .00 H=3X # # # # # # R=2% # # # # # # # # 8.00 16.00 . . | . . . . . . . . . | f . . . 24.00 32.00 40.00 TIME TO EXPIRATION 48.00 TABLE 4.4 ATT Theoretical Warrant Prices for Specific Stock Prices ($39, $45.5, $52, $58.5, $65) as a function of Time to Expiration. Time to Expiration (in months) Warrant Values (in $) Stock Prices $39 $45.5 $52 $58.5 $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 Risk Free Rate Dividends = 0.0017 per month = 6.37% per year = $0.65 per quarter 78 FIGURE 4 . 6 : ATT THEORETICAL WARRANT PRICES 32 .00 -28.00 24. 00 20.00 WARRANT PRICE 16.00-12.00 8.00 4.00 FOR SPECIFIC STOCK PRICES # # # # # # # # # # # * # # # # # # # # # # # # # # # # # # # # # # # # S=65.0 # # # # S=58.5 # # # S=52.0 # # S=45.5 # f # # # # S=39.0 -# # # # . 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 in the market and the theoretical value obtained by three different procedures: i) The numerical solution with discrete dividends presented in this chapter. i i ) 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 exactly when the stock goes ex-dividend, the warrant was estimated with the three above mentioned methods and compared with the market price just midway between two dividend payments, starting on November 15, 1970 when exercising was permitted and ending in November 15, 1974. For the different estimations the actual riskless interest rate , the indicated quarterly dividend, and the closing stock price for the particular date were used; but the same variance rate for the period '65 to '70 was used. This data are tabulated in Table 4.5. In Table 4.6 the warrant market price i s compared with the three theoretical methods for different time to maturity. In Table 4.7 this comparison is made between the ratios of the theoretical values and the market price. 6See [41]. I I TABLE 4.5 Data Used in the Estimation of Theoretical Warrant Prices. Date Time to Riskless Quarterly Stock E x p i r a t i o n Rate Dividend P r i c e ( i n months) (% per year) (i n $) ( 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 (in $) Time to Market Numerical B-S No B-S With Expiration Price Method Dividends 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 Ratio of Warrant Pr i c e s to Market Pr i c e s Warrant Price/Market P r i c e Time to Numerical B-S No B-S With Ex p i r a t i o n Method Dividends 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 83 According to these r e s u l t s the two formulations that take i n t o account dividend payments give c o n s i s t e n t l y lower values than those i n the market. I t i s i n t e r e s t i n g to note the s i m i l a r i t y between the numerical s o l u t i o n and the Black-Scholes s o l u t i o n f o r constant dividend y i e l d , although i n a l l cases the former gives a value c l o s e r to the market p r i c e . The Black-Scholes s o l u t i o n without considering dividends i s much cl o s e r to the market p r i c e . P o s s i b l y t h i s i s because the market values the warrant without taking proper account of the dividend payments of the underlying stock. There may be three 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 not an accurate d e s c r i p t i o n of the market. The assumptions made are too r e s t r i c t i v e . i i ) The model gives the " r i g h t " values, but the market e i t h e r "under" or "overvalues" the option. I f t h i s i s the case, there should be some p r o f i t opportunities. Black-Scholes [4] tes t t h i s hypothesis f o r c a l l options. i i i ) The model i s an accurate d e s c r i p t i o n of the market, but the h i s t o r i c a l variance used i n the model i s not the same variance used by the market to p r i c e the option. Here a l s o there should be p r o f i t opportunities. This area of research i s cu r r e n t l y being expanded i n a separate study by the present author and others. 4.5 Conclusions In t h i s chapter the numerical s o l u t i o n to p a r t i a l d i f f e r e n t i a l equations developed i n Chapter 3 has been modified s u c c e s s f u l l y to f i n d the values of an option in 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 effect of a changing exercise price on the value of an option is very similar to that of the effect 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 Li f e Insurance.^ To provide some background for the discussion of equity-linked l i f e insurance policies , i t i s convenient to begin this chapter with a brief discussion of some elementary l i f e insurance concepts. The basic concept underlying any kind of insurance i s the pooling of independent risks. Hence,this concept can also be applied to the hazard of death. If a sufficiently large group of lives are insured,the expected number of deaths in a given year can be quite accurately predicted,using the law of large numbers. There.are four basic traditional types of l i f e insurance contracts: term, whole l i f e , endowment,and annuity. 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 risk 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 for a limited period of time. The face value (or sum assured) of the policy becomes payable i f death should occur within the specified period and nothing i s paid i n the event of survival. The period of protection may be one year or shorter, or i t may run to age sixty-five. Term insurance may be regarded as temporary insurance and as such i s more nearly comparable to property and casualty insurance contracts than are any of the other types of l i f e insurance contracts. The premium for term insurance is relatively low,because these contracts do not cover the period This section draws on McGill [26], 86 of old age when death is most l i k e l y to occur. In contrast to term insurance which pays benefits only i f the l i f e insured terminates within a specified number of years, whole l i f e insurance provides for the payment of the face amount upon the death of the insured regardless of when i t may occur. It is usual for level premiums to be payable throughout the duration of the contract. Premiums can be paid monthly, quarterly, semi-annually or yearly. In some contracts premiums are payable for a limited period (e.g. to age 65) or even as a single lump sum. The third type of contract, endowment insurance, provides for the payment of the face value of the policy on death within a certain period or survival to the end of the specified period. Term and whole l i f e insurance provide for payment of sum assured only in the event of death. Under endowment insurance the sum assured is payable i f the insured i s s t i l l l i v i n g at the end of the specified period. The endowment insurance contract emphasizes the investment element. It i s essentially 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 risk of death increases with age,it is necessary for an insurance company to build up reserves under level premium, whole life,and endowment policies. If a policyholder does not wish to continue payment under these contracts,the insurance company w i l l offer him various options. One of these calls for the payment of a surrender value corresponding to 87 the reserve of the policy. In North America these surrender values are guaranteed by law for the duration of the contract. This has profound implications for the investment policies of the insurance companies. In particular i t means that common stocks are not suitable as a major component of the portfolio. For example in Canada there is a 25% limitation on the amount of common stock in a conventional l i f e insurance portfolio. In the United States the limitation is 5 to 10% of the portfolio depending on the state. This is one of the reasons why the return on conventional l i f e 2 insurance contracts has been relatively low. 5.2 Equity Based Li f e Insurance The traditional l i f e insurance contracts considered in the last section provide for benefits fixed in terms of the local unit of account. However,the prolonged period of sustained but irregular i n f l a t i o n since 1945 has served to diminish the usefulness of such long-term currency denominated contracts. As a result, insurance companies have increasingly moved towards the issuance of equity based contracts the benefits of which depend upon the performance of some reference portfolio of common stocks (mutual fund), i t being thought that equities, at least in the long run, would provide a hedge against inflat i o n . Equity based l i f e insurance was f i r s t introduced in the Netherlands around 1953 and the f i r s t equity linked l i f e insurance product was offered in the United Kingdom in the Autumn of 1957. The 1970 edition of "Equity- Linked Life Assurance Tables," published by Stone & Cox Publications Ltd., See for example Kensicki [21] where the internal rate of return on a typical whole l i f e participating policy i s estimated to be 3.57% per annum. 8 8 presents summaries of some 141 equity-based l i fe insurance plans (109 annual premium plans and 32 single premium plans) offered by 93 l i fe 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 fe insurance products. These segregated funds,as they are called,are,under 4 certain conditions, exempt from the 25% limitation mentioned earlier. Many Canadian companies sel l equity based endowment contracts. 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 fe insurance contracts, which vary according to the investment performance of a separate account, are known as variable l i fe insurance  contracts. Since 1970 the question of whether variable l i fe 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 fe 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 first issued in the United Kingdom. The examples discussed in this thesis wi 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 fe 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 fe insurance company in jeopardy. Guarantees on death and maturity are not permitted for policies with terms less than 10 years. Furthermore,maturity guarantees are not to exceed 100% of gross premiums paid. 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: the pooling of risks wil l have no elimination or diversification effect. In other words,investment risks are not insurable risks; a single event, such as a dramatic decline in stock prices, wi 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 fe insurance policies and in Chapter 6 some numerical examples are presented. Chapter 7 wi 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 wi 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 fe insurance contracts wi l l be presented. Following this , i t wi 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 cal l option. It wil 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 call 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 first 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 prices, i t is contended, provides a normative basis for pricing contracts. In his search for equilibrium prices the economist i s guided by the arbitragi principle that in a competitive market in equilibrium there are no riskless profit: to be made. This principle has two important implications which w i l l be employed later: 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 primitive contracts; and secondly, i t i s not possible to earn a positive riskless return with zero net investment. The f i r s t implication enables us to assert that the price 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 in deriving a diffe r e n t i a l equation governing the value of the c a l l option. 5.4 Brief Survey of the Actuarial Literature In recent years there has been a number of papers in the actuarial literature concerning the problem of equity linked l i f e insurance policies. As Turner [45] points out, many actuaries in the United States and Canada are now deeply involved in 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 calculation of premiums and the determination of suitable reserves. "Traditionally actuaries have always considered mean values in their 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 ful 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 fe 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 iabi l i ty 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 3 In the actuarial literature the expression "risk premium" denotes the additional amount (percentage) that insurance companies charge to the holders of equity-linked l i fe 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 - wil l be used interchangeably in what follows. The concept wi l l be defined more rigorously later on. . 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 distrib-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 fe 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 fe insurance policy. Kahn's paper is , nevertheless, the f irst attempt in the actuarial literature to find an analytic solution to the problem of evaluating asset value guarantees on equity linked l i fe 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 cal 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 wi l l not be exercised and wi l l have zero value, while i f x(t) > g(t),the option wi 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 cal 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 cal l option can be obtained by employing the procedures developed in preceding chapters, as wi l l be shown in subsequent sections. 96 Denote the value at time T of a cal 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 fe 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 call 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 sel l the reference portfolio for the pre-determined amount g(t). If x(t) 2. g(t),the put option wi l l not be exercised and wi l l have zero value, while i f x(t) < g(t),the put wi 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 call 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 price 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 port-f o l i o , of the time to expiration and of the guarantee or exercise price. At expiration the value of the put option can be written as (5.5) p(x(t), 0, g(t)) = max [(g(t) - x(t)), 0] Equation (5.5) expresses the,fact that at expiration the put option is 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 policy, PV D(b(t)), can be expressed as the sum of the present value of the guarantee and the value of a c a l l option: (5.6) PV Q(b(t)) = g ( t ) e " r t + w(x(0), t , g(t)) and also as the sum of the present value of the reference portfolio, PV Q(x(t)), and the value of a put option: (5.7) PV 0(b(t)) = PV Q(x(t)) + p(x(0), t, g(t)) Notice that in equation (5.6) continuous discounting has been used to be consistent with the option pricing model employed in subsequent sections. From the viewpoint of l i f e insurance companies the value of the put option i s the most relevant figure ,because i t represents the total 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 (5.7): (5.8) p(x(0), t, g(t)) =' w(x(0), t, g(t))+ g ( t ) e " r t - PV Q(x(t)) If the value of the c a l l option is known,the value of the put option can be obtained from equation (5.8). The problem is then to find the value . 98 of the cal 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 fe insurance policy. The mortality risk is introduced at a later stage. Accordingly, the problem of pricing an equity-linked l i fe insurance policy with asset value guarantee wi l l be solved in three stages: (1) The valuation of the call 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 fe 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 fe 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 wi 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 wi l l be used to price the cal l option component of the l i fe insurance contract. Therefore, the same basic assumptions made to derive that model are also implicit here. Even though l i fe insurance policies are not traded in the secondary market as some options are, the existence of a large number of l i fe insurance companies competing 99 for these products ensures that arbitrage p r o f i t s are competed away and, hence, produce the same e f f e c t as a competitive s e c u r i t i e s market does. Thus, a l l the arbitrage arguments presented i n Chapter 2 to derive the value of an option are assumed to apply here. Three separate types of contract must be considered: (1) The f i r s t type i s of the s i n g l e premium contract with an asset value guarantee, i n which the whole investment i n the reference p o r t f o l i o i s made at a s i n g l e point i n time. In t h i s case the value of the option corresponds p r e c i s e l y to the value of a c a l l option on a non-dividend paying s e c u r i t y f o r which a closed form s o l u t i o n has been derived by Black-Scholes and Merton, as shown by equation (2.31) i n Chapter 2. (2) The second type of contract i s that of the constant continuous premium where equal p e r i o d i c premium and equal p e r i o d i c investment i n the reference p o r t f o l i o areapproximated by assuming a constant continuous rate of investment i n the reference p o r t f o l i o . The d i f f e r e n t i a l equation governing the p r i c e behaviour of t h i s option corresponds to that derived by Merton [32] f o r an option on a dividend paying stock when the dividend i s negative, but with s l i g h t l y d i f f e r e n t boundary conditions to take into account the f a c t that even i f the value of the reference p o r t f o l i o goes to zero at a c e r t a i n point i n time, i t s t a r t s growing immediately afterwards with the a d d i t i o n of new premium payments. While no closed form s o l u t i o n e x i s t s f o r t h i s equation, i t may be solved simply and to a high degree of accuracy by the numerical method discussed i n Chapter 3. (3) The t h i r d type of contract i s that of the equal p e r i o d i c premium and equal p e r i o d i c investment i n the reference p o r t f o l i O j b u t where the assumption of constant continuous rate of investment i n the reference p o r t f o l i o can not be made,because the time between premium payments i s large (one year f o r 100 e x a m p l e ) . The d i f f e r e n t i a l e q u a t i o n g o v e r n i n g t h e p r i c e b e h a v i o u r o f t h i s o p t i o n between premium payments c o r r e s p o n d s 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 f o r a n o n - d i v i d e n d p a y i n g s e c u r i t y . A t each premium payment (and i n v e s t m e n t i n t h e r e f e r e n c e p o r t f o l i o ) d a t e new boundary c o n d i t i o n s have t o be imposed t o r e f l e c t t h e i n c r e a s e d p o s i t i o n i n t h e r e f e r e n c e p o r t f o l i o . The s i t u a t i o n i s v e r y s i m i l a r 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 pays d i s c r e t e 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 d i v i d e n d . T h i s e q u a t i o n may a l s o be s o l v e d by n u m e r i c a l methods v e r y s i m i l a r t o t h e p r o c e d u r e d e v e l o p e d i n C h a p t e r 4. 5.7 The S i n g l e Premium C o n t r a c t F o r t h i s c o n t r a c t t h e whole i n v e s t m e n t i n t h e r e f e r e n c e p o r t f o l i o i s made when t h e c o n t r a c t i s p u r c h a s e d ; and t h e v a l u e o f t h e o p t i o n c o r r e s p o n d s t o t h e 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 p a y i n g s e c u r i t y g i v e n by e q u a t i o n (2.31). I n t h e n o t a t i o n o f t h i s 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 t i m e x can be e x p r e s s e d as (5.9) w ( x ( x ) , t - x , g ( t ) ) = x ( r ) N ( d x ) - g ( t ) e " r ( t _ T ) N ( d 2 ) where d, = [ £n ^ Q - + ( r + ^ a 2 ) ( t - x ) ] / o V t - x 1 g(t) d2 = d l ~ a ^ t ~ x N(d) = r^r- r e ^ A d\ • 2ir -<» N(d) i s t h e c u m u l a t i v e n o r m a l d i s t r i b u t i o n f u n c t i o n . As was shown i n C h a p t e r 2, t h e s i g n i f i c a n c e o f t h e v a l u a t i o n e q u a t i o n (5.9) i s t h a t ( i ) i t y i e l d s t he market e q u i l i b r i u m v a l u e o f t h e o p t i o n s u c h t h a t i f t h e a c t u a l p r i c e i s n o t g i v e n by (5.9), t h e n r i s k l e s s a r b i t r a g e p r o f i t s may be 101 made (the basic assumptions of the model may be recalled, in particular: no transaction or selling costs and no taxes), ( i i ) i t expresses the value of the option in terms of the observable current price of the reference portfolio, x, the rate of interest which is known, the time to expiration of the option and the variance rate of return on the reference portfolio. Only the variance rate is unobservable and i t too may be estimated by examining the past v a r i a b i l i t y of the return on the reference portfolio. 5.8 The Constant Continuous Premium Contract The essential difference between the multiple premium contract and the single premium contract discussed above,is that with the former the change in value of the reference portfolio depends not only on the rate of return on that portfolio,but also on the rate of addition to the portfolio through investment. The analysis here differs from that given above in the stochastic differential equation for x. The following additional notation is introduced: SCO the value of one unit of the reference portfolio at time T . N ( x ) the number of units owned by the policy-holder at time T . D the constant instantaneous rate of investment in the reference portfolio. Then the value of the reference portfolio x ( x ) is given by: (5.10) X ( T ) = N ( T ) • S ( x ) and by Ito's Lemma**: \ Given this functional relationship, the same expression would be obtained using ordinary calculus. 102 dx = NdS + SdN or ( 5 u ) dx . dS dN ^ ' l L ) x S N The value of a single unit of the reference portfolio 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 in this chapter,becomes7: (5.12) 4f = adx + adz The d i f f e r e n t i a l increase in the number of units held on behalf of the policyholder is given by: d N " f d T = ^ d T since D is the constant instantaneous rate of investment. Then: (5.13) f - 2 * Substituting (5.13) and (5.12) into (5.11), the change in value of the reference portfolio i s given by (5.14) — = (o4 -) dr + adz X X Comparing equation (5.14) with equation (2.33) i t can easily be seen that the stochastic d i f f e r e n t i a l equation for the instantaneous rate of return on the reference portfolio corresponds to that derived by Merton for the instantaneous rate of return on a stock paying a continuous dividend, when the dividend is negative . 7In this chapter the variable "time" has been denoted by x and the "maturity date," which is 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 . 103 Consequently the equilibrium value of the option, which is a function of the value of the reference portfolio, time to expiration and the exercise price, can be solved in a manner similar to the valuation of an option on a continuous dividend paying stock developed in Chapter 2. The option value must then satisfy the partial d i f f e r e n t i a l equation 2 2 (5.15) ho x w + (rx+D)w - wr - w = 0 X X X JL subject to the boundary conditions: (5.16) w(x,0,g) = max[(x-g), 0] (5.17) Dwx(0,T,g) - w(0,T,g)r - wT(0,T,g) = 0 (5.18) wx(°°,T,g) = 1 The subscript x denotes part i a l derivatives with respect to the value of the reference portfolio and the subscript T denotes part i a l derivatives with respect to time to maturity (T = t - t ) . Notice that the only difference between equation (5.15) and (2.40) is the sign of D (here the continuous premium payment invested in the reference portfolio, there the continuous dividend payment). Thus, by arguments identical to those presented in 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 portfolio equal to zero and i n f i n i t y , determine the equilibrium value of the option at any time prior to maturity. Boundary condition (5.16) is the same as (5.3) and determines the value of the option at expiration. The boundary condition (5.17) for zero values of the reference portfolio can be obtained directly by setting x=0 in partial 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 val id, because the value of the cal 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 wi 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 8See 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 is the most important one. Premiums are commonly paid monthly or yearly for this type of contract. Accordingly, in the numerical examples presented in Chapter 6 we w i l l concentrate on this case. As in the case of the continuous premium contract, in the periodic premium case the change in value of the reference portfolio depends not only on the rate of return on that portfolio, but also on the rate of addition to the portfolio through periodic investment. The value of the option, W ( X ( T ) , t-x, g(t)) or w(x, t-.-r, g) for simplicity, 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. 2 2 (5.19) ho x w + rxw - wr - w„ = 0 xx x T But the boundary condition w i l l be different. In particular at each premium payment date new boundary conditions w i l l have to be established to reflect the fact that additional investments have been made in the reference portfolio. The problem is very similar to that of valuing an option where the underlying stock pays discrete dividends, as presented by Merton [32] and Black [ 5] and treated in detail in Chapter 4, but in this case the' premium can be thought of as a negative dividend. The fact that the premium payment is an addition to the reference portfolio determines some important changes in the boundary conditions with respect to the ones developed in Chapter 4 for the discrete dividend case. Also here i t w i l l not pay to exercise the option before maturity even i f i t were possible, because the reference portfolio i s growing faster due to the additions of premium payments. The distinction between an American and a European option is not relevant,because both w i l l be exercised at maturity only. This w i l l be proved later on. But both options, the one in the discrete dividend case and the one in the periodic premium case, obey the same parti a l d i f f e r e n t i a l equation (5.19). 1 0 6 The value of the option wil l always be governed by equation (5.19). The solution to this equation, however, wil 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 wi 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 wi 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 wil 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 it 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) bw(x, T , g) = max[aw(x+D, T , g), (x-g), 0] But in this case the value of the option after the payment of the premium, I aw(x+D), Tp, g), wi l l always be greater than the exercise value before the 107 payment of the premium, (x-g), because (5.22) w (x+D, T , g) > w (x, T , g) a p 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) bw (x, T , g) = v (x+D, T , g) 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 in time before maturity. Thus, the value of an American option is equal to i t s European counterpart. The boundary condition for zero values of the reference portfolio w i l l also dif f e r from the boundary condition for zero values of the stock obtained for the discrete dividend situation discussed in Chapter 4. In this case, as in the continuous premium contract discussed in the preceding section, the value of the option i s not necessarily equal to zero when the value of the reference portfolio i s equal to zero. It i s necessary to consider the effect on the value of the option of future premium payments. This boundary condition i s obtained by letting x = 0 in (5.19) : (5.26) wr + wT = 0 or (5.27) wT(0, T, g) = -rw(0, T, g) 108 The value of the option for the value of the reference portfolio equal to zero decreases in value with time to maturity at the riskless interest rate. 1^ The,economic j u s t i f i c a t i o n for this "perverse" behaviour of the value of the option at the boundary i s the following: the value of the option for a value of the reference portfolio equals to zero just before any premium payment i s given by boundary condition (5.25): (5.28) bw (0, T p, g) = aw (D, T , g) > 0 If the value of the reference portfolio goes to zero at any point in 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 in another reference portfolio with the same variance rate). 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 portfolio equal to zero) can be obtained by discounting (5.28) at the riskless rate to the point in 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 parti a l differential equation (5.19) or (2.40). One of the basic assumptions underlying these equations is that the return on the stock or reference portfolio is log-normally distributed. This distribution implies that i f at the current point in time the value of the stock (or reference portfolio) i s different from zero, the probability of i t becoming zero in the future is zero. Therefore, the di f f e r e n t i a l equation i s really not defined at the boundary (for x = 0 or S = 0). To obtain their 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, satisfies the condition that W-K) when S-K). The same boundary condition (W(0,T) = 0) has economic ju s t i f i c a t i o n (see Merton [32]) and so they are used without further comment in Chapters 3 and 4 when evaluating options on dividend paying stocks. When numerical methods are employed, however, the solution procedure starts from the maturity date (where the value of the option is known for different 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 because (5.30) w(0, 0, g) = 0 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 cal l option in the periodic premium contract is governed by the partial differential equation: (5.19) %azx w + rxw - wr - wT = 0 xx x J-subject to the boundary conditions: (1) For the last period (after the last premium payment): (5.20) w(x, 0, g) = max [x-g, 0] (5.29) w(0, T, g) = 0 (5.18) wx(oo, 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) WQ )j = W Q j j_^(l-rk), W 0,j = W 0 j j _ i and (3) WQ i - W 0 J 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. 110 (2) For a l l the other periods between premium payments: (5.25) bw (x, T p, g) = aw (x+D, T p, g) (5.27) wx (0, T, g) = -rw(0, T, g) (5.18) wx (~, T, g) = 1 These results are also shown graphically on Figure 5.1. The value of the c a l l option for a value of the reference portfolio equal to zero just before the f i r s t premium payment, ^ w (0, t, g), is the solution to our problem. The value of the reference portfolio is equal to zero,because no monies have been invested in i t yet. No closed form solution exists for this equation subject to i t s boundary conditions, but i t may be solved by modifying the numerical procedure developed in Chapter 4 to take into account the new boundary conditions. This procedure involves a dynamic programming type of approach starting from the maturity of the option and going back u n t i l the moment just before the f i r s t premium payment (the moment of the signature of the contract). As indicated above, the aim of this discussion i s to find the value of the option at that point in time for a value of the reference port-folio equal to zero. The solution algorithm for this problem i s developed in Chapter 6 and the computer- program i s shown in Appendix E. 5.10 The Call Option, The Put Option and the Pricing of Investment Risk The periodic premium contract is 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 this. The results obtained I I l l Figure 5.1: Partial Differential Equation and Boundary Conditions for the Periodic Premium Contract Time to  Expiration f i r s t premium payment date any . premium D payment date last premium D_ payment date %a 2x 2 w x x + rxwx - wr - wT = 0 Maturity date 0 o II oo H A O w(x,T,g ) w(x,Tp,g) = aw(x+D,Tp,g) bw(x,T 1 > g) w(x+D,T1>g) w(x,0,g) = max[x-g,0] 00 H Value of the Reference Portfolio. 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 fe insurance policy is issued,because that wi 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 first premium payment for a value of the reference portfolio equal to zero, that Is bw(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) is , therefore, the present value of a cal l option that wi l l pay at maturity the greater of x(t) - g(t) or zero. Assuming death is certain in 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) PVQ(g(t)) = g(t)e~ r t Thus, the total equilibrium present value of the benefits or present value of the contract can be written as (5.32) PVQ(b(t)) = g(t)e" r t + 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 cal 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 portfolio, PV Q(x(t)), must be equal to the present value of the contributions to the reference portfolio since this is the price that must be paid for that future value: t-1 (5.33) PV Q(x(t)) = I D e ~ r k k=0 If the value of the put option corresponding to ^ ( 0 , t, g) , i.e., ^p(0, t, g), is written as p(t), the present value of the contract can also be written as: t-1 (5.34) PV Q(b(t)) = £ D e ~ r K + p ( t ) k=0 Combining (5.32) and (5.34) the value of the put option can be obtained: t-1 . (5.35) p(t) = g(t)e r t + w(t) - £ D ; e " r k=0 As mentioned earlier, the value of the put option represents the cost to the insurance company of giving the guarantee. It is the equilibrium pricing of the investment risk assumed by the insurance company. It i s of interest to calculate the periodic premium for the put, y ( t ) , that i s ,the periodic amount that should be added to the premium payments to compensate the insurance company for assuming the investment risk. The benefits are s t i l l assumed payable with certainty at time t. The present value of these periodic premiums for the put is equal to the present value of the put option, that i s : t-1 _ . (5.36) p(t) = y(t) I e r k k=0 or, solving for the value of the periodic premium for the put: (5.37) y(t) = t _ P ( t ) r -rk L e k=0 114 Analogously, the p e r i o d i c premium f o r the c a l l , z ( t ) , can be defined as t-1 _ (5.38) w(t) = z(t) 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 as: PV n ( b ( t ) ) (5.39) v ( t ) = t-1 , V -rk 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: (1) the amount invested i n the reference 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 ) . F i n a l l y , s u b s t i t u t i n g (5.37) and (5.38) into (5.35) and s i m p l i f y i n g gives: - r t (5.41) y ( t ) + D = z(t) + g(t) — f — Y -rk k=0 The continuous time v e r s i o n of (5.41) can be w r i t t e n a s 1 1 ; - r t (5.42) y(t) + D = z(t) + g(t) r 6 _ r t 1-e i l-The computer program shown i n Appendix F c a l c u l a t e s the value of the c a l l option, the present value of the guarantee, the t o t a l present value of The continuous time v e r s i o n has the advantage of being a d i f f e r e n t i a b l e function 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 put, time to maturity and the amount of the guarantee (as a proportion of the investment component of the premia paid) could 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 portfolio, the value of the put option, the total periodic premium, the periodic premium for the put and the periodic premium for the c a l l . In the preceding sections the method for pricing 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 risk derived from the option element of equity-linked l i f e insurance policies with asset value guarantee, insurance companies also bear the mortality risk derived from the hazard of death. The exercise date is really not known; i t i s also a stochastic variable with a certain probability distribution. Subsequent sections deal with the problem of mortality risk. 12 5.11 The Measurement of Mortality A l i f e insurance policy calls for the payment of certain benefits, b(t), with specified probabilities. The determination of these probabilities by the usual actuarial methods w i l l now be considered. The probability that a new l i f e , aged 0, w i l l survive to attain age x is referred to as the survival function, S(x). S(x) is a continuous function of x, defined on the interval 0 < x £ w, which decreases from the value S(0) = 1 to the value S(w) = 0. The designation of a terminal age w, smallest value of x for which S(x) vanishes, is merely a convenient simplify-ing device. The survival function can be used to determine the expected number of survivors at any age x starting from any assumed group of newly-born lives. 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 in the year of age x to x+1. Then (5.43) £x = k • S(x) , where k is a positive constant representing the size of the i n i t i a l group. (5.44) dx = Ix - lx+l The value of k is called the radix of the mortality table. It corresponds to the value of Z Q and is generally taken to be some large integral number. The interpretation of Ix as a "number li v i n g " or "number surviving" and of dx as a "number dying",is a convenient aid in visualizing some of the relationships that follow. But i t should be borne in mind that neither Ix nor dx has any absolute meaning, the values of both being dependent on the value of the radix chosen in the construction of the table. The tabulated values of ix that appear in mortality tables are usually for integral values of x only, although from definition (5.43) Hx i s a continuous function of x. A section of an hypothetical example of a mortality table i s shown in 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 Mortality Table Age x Ix dx 0 100,000 501 1 99,499 504 2 98,995 506 3 98,484 509 4 97,980 512 5 97,468 514 6 96,954 517 7 96,437 520 8 95,917 523 9 95,394 526 10 94,868 528 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 fe aged x wi l l survive to age x+n is denoted by npx, and r / r \ ft x+n (5.45) px = — — n r IK The probability that a l i fe aged x wi l l die within n years is denoted by nqx, and 5.46) qx = 1 - px = n n n r Jlx The probability that a l i fe aged x wi l l survive for n years and die in the (n+l) t b year is denoted by n |qx, and , r , , , . dx+n ftx+n - ftx+n+1 ( 5 - 4 7 ) n|« x = IT = ^ The probabilities of death and survival defined above wi l l be used in the pricing of asset value guarantees on equity linked l i fe 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 individual aged x purchasing a policy which matures in n years. Assume also, for simplicity, that i f the individual dies during year t,the benefits are payable at the end of year t; and that i f he dies during the last year of the policy or does not die during the policy period,the benefits are payable at maturity. Using the probabilities of death and survival defined in the preceding section,the probability of an individual aged x of dying during year t i s given by (5.49) a(t) = ^ j q x = * X + t ~ \ ~ ^ f o r t = l , . . . , n - l and the probability that an individual aged x w i l l survive u n t i l the last year of the policy, that i s the probability of the benefits being payable at maturity, i s given by / r r / > \ / \ &x+n-l - £x+n , £x+n (5.50) a(n) = + — The f i r s t term in the right hand side of equation (5.50) indicates the probability that he w i l l die during the last year of the contract and the second term indicates the probability of survival. Expression (5.50) can also be written as: (5.51) a(n) = ^ p x - ^ ± The values of a(t) w i l l depend on the mortality table considered appropriate for the calculation, on the age of the individual purchasing the l i f e insurance contract and on the term of the contract (and naturally also on other variables that determine the choice of the mortality table such as sex and race) and can be easily computed using equations (5.49) and (5.51) given 119 the age of the individual, the l i f e of the contract and the appropriate mortality table. Bearing this in mind i t is possible to combine the analysis presented in this section with that developed earlier in the chapter to arrive at a value for the contract which takes both mortality and investment risk into account. 5.13 The Equilibrium Value of the Contract: Investment Risk and Mortality Risk  With the developments of the preceding sections i t i s now possible to combine investment risk with mortality risk to obtain the equilibrium value of an equity-linked l i f e insurance contract with minimum guarantee. The average contract calls for the certain to be paid (but uncertain in amount ) stream of benefits: (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 total present value of this average contract, V, can then be written as: n n (5.53) V = I a(t) g(t) e" r t + £ a(t) w(t) t=l t=l or equivalently as: n (5.54) V = I a(t) PV Q(b(t)) t=l The total present value of the average contract given by (5.53) and (5.54) is actually the equilibrium present value of the contract where both investment risk and mortality risk have been taken into consideration. 120 The average periodic premium for the put, Y, can be expressed as: n (5.55) Y = £ a(t) y(t) where y(t) is the periodic premium for the t=l put with known date of expiration t. The expected periodic premium for the put given by (5.55) as a percentage of the periodic premium invested in the reference portfolio (Y/D) is called the "risk premium" in the actuarial literature. Here i t shall be called also "put premium." The risk premium or put premium is the figure needed by l i f e insurance companies to price equity based l i f e insurance policies with asset value guarantee. It represents the amount (as a percentage of the premium invested in the reference portfolio) that the insurance company has to charge for giving the guarantee when both investment and mortality 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 risk and can easily be tabulated. The values of a(t), which represent mortality risk, are also readily obtainable from the appropriate mortality table given the age of the purchaser of the policy and the duration of the contract. The computer program shown in Appendix G computes the equilibrium present value of the contract and the risk premium. 5.14 Conclusions The need for an appropriate model for the determination of prices for equity linked l i f e insurance policies with asset value guarantee has long been apparent. See for example Leckie [22] and Turner [ 4 4 ] . This chapter has shown how the models and methods of Chapters 2, 3 and 4 can be applied to this problem to yield normative rules for pricing such contracts. Moreoever, the prices determined by these rules are equilibrium prices; that i s , they are the prices which would prevail in a perfectly 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 risk and make no profit 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 in Chapter 6 to a number of different situations in order to analyze the sensitivity of the risk premium to changes i n : (1) the variance of the rate of return on the reference portfolio; (2) the riskless interest rate; (3) the nature of the guarantee; (4) the age of the purchaser at entry; and (5) the term of the contract. 122 . CHAPTER 6: THE EQUILIBRIUM PRICING OF ASSET VALUE GUARANTEES ON PERIODIC PREMIUM EQUITY-LINKED LIFE INSURANCE POLICIES: APPLICATIONS 6.1 Introduction A general model for the valuation of equity-linked l i f e insurance contracts with minimum death benefit and maturity benefit guarantees was developed in the preceding chapter. The model is based on the earlier work of Black and Scholes and further developed by Merton on the pricing of options, and as such has some of the same limitations. It assumes perfect markets with no transaction costs, no selling costs or other market imperfect-ions. In other words, i f a l l the basic assumptions underlying the option pricing model discussed in detail in Chapter 2 are valid, 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 in the sense that at these prices there are no riskless profits to be made, whereas at any other prices riskless profits can be made under these ideal market conditions. It i s the contention of this study, however, that these equilibrium prices provide a normative basis for pricing 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 theoretical basis. However, without the numerical methods of solution developed in Chapters 3 and k the theoretical model in i t s e l f would not be of much practical importance. In essence, the use of a sound theoretical model for the pricing of options and the development of numerical methods of solutions have permitted us to solve the problem of evaluating investment risk assumed by the 123 insurance company in equity-linked l i f e insurance contracts with asset value guarantee. In this chapter, after developing the basic solution algorithm for the periodic period contract, we find the equilibrium value of some of the common types of guarantees. In addition, we examine the sensitivity of these values with regard 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 portfolio; ( i i ) the riskless interest rate; ( i i i ) the nature of the minimum guarantee; (iv) the choice of mortality table 1; (v) the age of the purchaser of the policy; and (vi) the term of the contract. Some variation has been allowed in a l l these variables in the numerical examples presented in this chapter. The value of the c a l l option and,therefore the value of the l i f e insurance policy w i l l depend on the characteristics of the reference portfolio or unit trust in which the investment component of the premiums are invested. The variance rate of return of the reference portfolio i s a key variable in the valuation of the c a l l option. Thus, the choice by a l i f e insurance company of unit trust or reference portfolio (viz., the variance rate), w i l l determine the level of equilibrium risk premiums. ^In practice the actual cost of the guarantee w i l l depend on the mortality experience, which may or may not be reflected in the mortality table chosen. The difference between "ex-ante" and "ex-post" values applies to some of the other variables as well. 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 portfolio chosen in a l l i t s calculations and should also bear in mind that the higher the variance the higher w i l l be the "risk premiums" that i t w i l l have to charge. Canadian companies can expect that with a well diversified equity portfolio the variance rate would approximate the Toronto Stock Exchange Industrial Index. Therefore, iy* a l l the numerical examples, except one, the variance used is that of the Toronto Stock Exchange (TSE) Industrial Index with dividends reinvested for the period September 1968 to August 1973. In the exception a larger variance is used to show the influence of this variable. It should be remembered that the option pricing model used assumes a constant riskless interest rate. To demonstrate i t s effect of this variable, 2 two values considered relatively extreme in the actuarial f i e l d were selected namely, 4% and 8%. The selection of the appropriate riskless rate is a d i f f i c u l t task for l i f e insurance companies issuing long term policies. L i f e insurance companies, however, have to face this problem whether they are evaluating conventional policies or equity-linked ones with minimum guarantee. Three different kinds of guarantees at death or maturity were considered in accordance with the more usual contracts issued in Canada These interest rates are continuous compounding rates. 3 The effect of inflation on the real value of the guarantee or exercise price and on the selection of riskless rate (nominal or real), is a topic which has not been studied well and requires further research. In what follows the problems created by inflation are not considered. 125 and the United Kingdom. In the f i r s t .case the amount of the guarantee is set to be 100% of the sums invested in the reference portfolio, that i s , the minimum guarantee "grows" with the premiums invested in the unit trust. The second case, considered very common in Canada, is similar to the f i r s t case ,but the amount of the guarantee is set to be only 75% of the premiums invested in the reference portfolio. These two cases have been chosen because they are representative of the more common policies issued. Any other guarantee as a percentage of the premium invested in the reference portfolio can be treated analogously. The third case considered is that of a fixed minimum guarantee at death or maturity independent of the premiums invested in the reference portfolio when the policy expires because of death or maturity. Summarizing, the three key variables considered were: (1) Variance Rate: values of 0.01846 and 0.04 were used, the f i r s t corresponding to estimated variance in the TSE Industrial Index. (2) Interest Rate: values of 4% and 8% were employed in the examples. (3) Types of Guarantee: 100% of the sums invested in the reference portfolio, 75% of the sums, and a fixed amount were considered. The mortality Table used for the examples was the Canadian Assured Lifes 1958-64 Select for males published by the Canadian Insurance Companies. Table 6.1 shows the mortality data (ultimate) used. A l l the examples presented in this chapter correspond to the periodic premium contract discussed in detail in Chapter 5. 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; Mortality Table: Canadian Assured Lifes, 1958-64 (Males) Ultimate AGE £ AGE £ _x _x 20 985731.44 51 926241.38 21 984650.06 52 920253.25 22 983540.38 53 913740.56 23 982429.94 54 906658.19 24 981344.38 55 898948.88 25 980301.19 56 890543.69 26 979309.13 57 881356.88 27 978369.00 58 871284.69 28 977471.81 59 860206.31 29 976602.88 60 847993.13 30 975743.44 61 834523.56 31 974874.06 62 819698.25 32 973975.25 63 803450.19 33 973027.56 64 785752,63 34 972013.69 65 766612.50 35 970917.25 66 746058.06 36 969724.00 67 724125.44 37 968425.50 68 700851.31 38 967013.56 69 676273.88 39 965477.94 70 650442.25 40 963801.88 71 623420.25 41 961958.13 72 595289.69 42 959916.81 73 566144.88 43 957641.81 74 536093.38 44 955092.56 75 505250.81 45 952225.38 76 473738.31 46 948994.50 77 441687.56 47 945357.94 78 409255.31 48 941281.56 79 376635.63 49 936744.56 80 344066.06 50 931733.94 127 the presentation without loss of generality i t w i l l be assumed that the investment component of the premium invested in the reference portfolio is equal to one monetary unit (dollar or pound) and that the premium is paid once a year at the beginning of the year. Monthly payments could be easily evaluated. Also, to simplify the discussion, i f death occurs during a certain year ,the liquidation of the policy (sale of reference portfolio and payment of the guarantee) i s assumed to take place at the end of that year. For the purposes of this study we w i l l ignore withdrawals, i.e., policies whose holders voluntarily discontinue premium payments and receive the value of the investment account less a deduction. Finally, recall that in the equilibrium model developed in Chapter 5 and in the applications presented in this chapter transaction costs and commissions have not been considered. 6.2 Solution Algorithm for the Periodic Premium Contract The solution algorithm for the periodic premium contract with known expiration date (death or maturity) w i l l be developed in this section. The procedure i s based on the numerical solution to parabolic partial d i f f e r -ential equations presented in Chapters 3 and 4. In the preceding chapter i t was shown that the value of the c a l l option in the periodic premium contract i s governed by the part i a l d i f f e r -ential equation: 2 2 (6.1) ho x w + rxw - wr - wm =0 xx x T subject to the boundary conditions: (1) For the period after the last premium payment: (6.2) w(x, 0, g) = max[x-g, 0] 128 (6.3) w(0, T, g) = 0 (6.4) w x ( o o , T, 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 analytically with formula (2.31) or by the numerical procedures shown in Chapters 3 and 4. (2) For the other periods between premium payments: (6.5) bw(x, T , g) = aw(x+D, T , g) (6.6) wT(0, T, g) = -r w(0, T, g) . (6.7) wx(~, T, g) = 1 Boundary conditions (6.5) and (6.6) are different from the corresponding ones in Chapters 3 and 4 and,therefore, the procedures employed in those chapters must be modified to satisfy these new conditions. Using the f i n i t e differences method for solving p a r t i a l d i f f e r e n t i a l equations in the same way as in Chapter 3 and 4, equation (6.1) can be approximated in difference form as: (6.8) a. w. , . + b. w. . + c. w . , , . = w. . , l l - l , j l i , j l l + l , j i , J - l where (6.9) a ± = % r k i - ha\i2 (6.10) b± = (1+rk) + a 2 k i 2 (6.11) c ± = -hrk± - J i a 2 k i 2 (6.12) w(x,, T.) = w(ih, jk) = w, , 129 with h being the reference portfolio price increment and k the time to maturity increment. At any premium payment date, denoted by j , boundary condition (6.5) can be written in the notation defined by (6.12) as: (6.13) ,w. . = w.,_ . b 1,3 a i+D/h,3p where D/h is an integer (the reference portfolio price increment, h, has to be chosen as an integral factor of the periodic premium payment, D). Boundary condition (6.6) and (6.7) can be written in the same notation respectively, as: (6.14) w . = (1-rk) w . . for a l l j 0,3 0 ,3-1 (6.15) -w . . + w . = h for a l l j n-1,3 n,3 In (6.15) n is the last point of the grid and has to be large enough for (6.15) to hold within the precision required. As a "rule of thumb" x has to be more than three times the exercise price, n The difference equation (6.8) can be used to generate a system of linear equations by making i=l, n-1. The last equation (for i=n) i s obtained from boundary condition (6.15) a. w . + b. w, . + c. w„ . ' 1 0,3 1 1,3 1 2,j = 1,3-1 a 0 w. . + b 0 w» . + c 0 w„ . = w9 , 2 1,3 2 2,3 2 3,3 2,3-1 (6.16) . a . w „ . + b . w , . + c ,w . = w n-1 n-2,3 n-1 n-1,3 n-1 n,3 n-1,3-1 - w , . + w . = h n-1,3 n,3 130 (6.16) is a system of n equations with (n+1) unknowns (w. . for 1 > 3 i=0, .... n). But the value of w .can be obtained from boundary condition (6.14) The system of equations (6.16) can then be written as ^1 W l + °1 W2 = ^1 a 2 wx + b 2 w2 + c 2 w3 = f 2 (6.17) a , w „ + b , w , + c ,w = f , n-1 n-2 n-1 n-1 n-1 n n-1 a w , + b w = f n n-1 n n n where f1 = w, . , - a.(l-rk)w . . 1 l . J - l 1 o , j - l f. = w. . , for i=2, .... n-1 f = h n a = -1 n b = 1 n and where the subscripts j (from w. .) have been omitted for simplicity. The matrix of coefficients of the system (6.17) is tridiagonal and, thus, Gauss' elimination method can be conveniently applied to solve for the values of w^(i=l, ... n). With this procedure i t is possible to find the values of w, . i , J (i=0, ... n) starting from the values w. . .(1=0, ... n). The f i r s t line of the grid for each period between premium payments is obtained from the last line of the preceding period using boundary condition (6.13). A new boundary condition (6.13) is obtained at each premium payment date and 131 the iterative process progresses in between premium payments as described above. It should be emphasized that the procedure starts from the maturity date of the option and advances in backward direction in time to the date of issue of the contract. The computer program shown in Appendix E i s based in the solution algorithm presented above. The solution algorithm for option prices in the constant continuous premium contract i s given in Appendix C and the respective computer program i s shown in Appendix D. 6.3 The Variance Rate of the TSE Industrial Index In accordance with our procedure of employing Canadian data for the applications of the model, the TSE Industrial Index was used as a proxy of the reference portfolio. The data to calculate the variance of the rate of return for 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 : (6.18) R E Zn where: I. = Value of the index at the end of time i . J Y_. = Average dividend yield during time j . Using the price relatives given by (6.18) ?the variance rate per year for the rate of return was calculated for the whole period included in the F i l e (June '61 to August '73) and for the last five years (September '68 to August '73). The resulting values were 0.01516 and 0.01846 respectively. I .U+Y.) J 3 132 The value for the later period was the one used in the example,because i t was 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 effect of this variable on the risk premium. 6.4 Put Premium or "Risk Premium" The variable chosen for analyzing and comparing the policies was the expected annual premium on the put or risk premium. From the view-point of the insurance company this variable has attracted the most attention, because i t represents the price that the insurance companies would have to charge for 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 total present value of the contract, the present value of the reference portfolio or the value of the put option which may be required for studying other aspects of the problem (transaction and selling costs for example) are readily available from the program. 6.5 Numerical Example No. 1 The parameters used in example no. 1 were: (1) Variance Rate: the variance rate of the TSE Industrial Index; (2) Risk Free Rate: 4% per annum (continuous); (3) Guarantee: 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 thirty years. The entries in 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 cal l option, w(t). - r t Column (3) : present value of the guarantee, g(t) e Column (4) : total present value of the contract, PVQ(b(t) = —rt w(t) + g(t) e . Corresponds to Column (2) plus Column (3). Column (5) : present value of the reference portfolio, PVQ(x(t)) = k=0 Column (6) : value of the put option, p(t) = PVQ(b(t)) - PVQ(x(t)). 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 ca l l , z(t). Corresponds to the annuity with present value equal to the value of the cal 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, is repeated for different expiration dates with changing guarantees''*'. It should be emphasized that Table 6.2 deals with the "investment risk" aspect of the policy without any consideration of the "mortality risk" aspect. As can be seen from Table 6.2, the value of the c a l l option increases from $0.07 for a one-year contract with a guarantee of $1.00 to $8.98 for a thirty-year contract with a guarantee of $30.00. In this period the annual premium on the put decreases from 3.5% for a one year contract with $1 guarantee to 1.1% for a 30-year contract with $30 guarantee. "To obtain accurate results care should be taken in the selection of the reference portfolio price step that i t not be large relative to the exercise price. As a "rule of thumb" the highest reference portfolio value should be approximately 3 to 4 times the exercise price, so i f 400 increments are used, 100 of them should be below the exercise price. It must be remembered also that the reference portfolio price increment has to be an integral factor of the annual premium invested in the reference portfolio to permit the practical applic-ation of boundary condition (5.29). To solve for these two problems simultaneously the reference portfolio price increment is determined within the program and changes for different maturities. ^To be consistent with the option pricing model, a l l compounding and discounting of values are done continuously. Thus the riskless interest rate considered is the continuous rate. However, the numerical solutionj by using f i n i t e differences instead of partial derivatives.implies discrete (although for short intervals) compounding. For example,if the time increment is one month,a monthly compounding is 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 = 0.01846 Riskless Rate = 0.04000 Guarantee = 100% of investment component on death or maturity No. years Value P.V. Total P.V. Value Total Annual Annual to C a l l Guarantee P.V. Reference Put Annual Premium Premium Expiration Contract Portfolio Premium Put Call (1) (2) (3) (4) (5) (6) (7) (8) (9) 1 0. 07439 0. 96079 1.03518 1. 00000 0.03518 1. 03518 0.03518 0.07439 2 0.18082 1. 84623 2.02706 1. 96079 0.06627 1. 03380 0.03380 0.09222 3 0. 31923 2. 66076 2.97999 2. 88391 0.09609 1. 03332 0.03332 0.11069 4 0.48583 3. 40858 3.89441 3. 77083 0.12358 1. 03277 0.03277 0.12884 5 0. 67762 4. 09365 4.77127 4. 62297 0.14830 1. 03208 0.03208 0.14658 6 0. 89208 4. 71977 5.61184 5. 44170 0.17015 1. 03127 0.03127 0.16393 7 1. 12696 5. 29049 6.41745 6. 22833 0.18912 1. 03036 0.03036 0.18094 8 1. 38029 5. 80919 7.18948 6. 98411 0.20537 1. 02940 0.02940 0.19763 9 1. 65021 6. 27909 7.92929 7. 71026 0.21904 1. 02841 0.02841 0.21403 10 1. 93505 6. 70320 8.63825 8.40793 0.23032 1. 02739 0.02739 0.23015 11 2. 23326 7. 08440 9.31767 9. 07825 0.23941 1. 02637 0.02637 0.24600 12 2. 54340 7. 42540 9.96880 9. 72229 0.24651 1. 02535 0.02535 0.26160 13 2. 86413 7. 72877 10.59290 10. 34107 0.25183 1. 02435 0.02435 0.27697 14 3. 19399 7. 99693 11.19092 10. 93559 0.25533 1. 02335 0.02335 0.29207 15 3. 53224 8.23218 11.76441 11. 50680 0.25761 1. 02239 0.02239 0.30697 16 3. 87723 8. 43668 12.31392 12. 05561 0.25830 0. 25830 1.02143 0.32161 17 4. 22873 8. 61249 12.84122 12. 58291 0.25832 1. 02053 0.02053 0.33607 18 4. 58483 8. 76155 13.34638 13. 08952 0.25686 1. 01962 0.01962 0.35027 19 4. 94503 8.88567 13.83069 13. 57627 0.25442 1. 01874 0.01874 0.36424 20 5. 30849 8. 98658 14.29507 14. 04394 0.25113 1. 01788 0.01788 0.37799 21 5. 67545 9. 06593 14.74137 14, 49327 0.24811 1. 01712 0.01712 0.39159 22 6. 04340 9. 12523 15.16863 14. 92498 0.24365 1. 01632 0.01632 0.40492 23 6. 41246 9. 16594 15.57840 15. 33976 0.23864 1. 01556 0.01556 0.41803 24 6. 78205 9. 18944 15.97149 15. 73828 0.23321 1. 01482 0.01482 0.43093 25 7. 15165 9. 19699 . 16.34863 16. 12117 0.22746 1. 01411 0.01411 0.44362 26 7. 52218 9. 18983 16.71201 16. 48904 0.22296 1. 01352 0.01352 0.45619 27 7. 89018 9. 16909 17.05927 16. 84250 0.21677 1. 01287 0.01287 0.46847 28 8. 25659 9. 13584 17.39243 17. 18210 0.21033 1. 01224 0.01224 0.48053 29 8.62118 9. 09111 17.71228 17. 50838 0.20390 1. 01165 0.01165 0.49240 30 8. 98341 9. 03584 18.01924 17. 82185 0.19739 1. 01108 0.01108 0.50407 I i 1 1 136 A l l this means that i f a person ,assumed immortal for the present, 2 invests $1 per year for 20 years in a mutual fund with variance rate a and i f at the end of that time wants to receive $20 or the value of his units, whichever is the larger, he would have to pay, in equilibrium, to the insurance company a risk 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 risk. 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 easily be incorporated by computing expected values using the appropriate probabilities of death and survival during the term of the contract. Given the age of the purchaser of the policy at entry and the term of the policy, the probabilities of death during the l i f e of the contract can be computed from mortality figures, such as those in Table 6.1, using the formulas derived in Chapter 5. When both investment and mortality risks are considered, the resulting put premiums or risk 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 practical purposes i t is of interest to find the dependence of these risk premiums on the age of the purchaser and the term of the contract. A computer program was prepared to tabulate and graph the risk premium versus the term of the contract for a given age of the purchaser of the policy. This program is shown in Appendix I. In addition, another computer program tabulates and graphs the risk 137 premium versus the age of the purchaser of the policy for a given term of the contract. This program is shown in Appendix J. Table 6.3 shows the relationship between the risk premium and the term of the contract for purchasers of ages 20, 30, 40 and 50,respectively. For practical purposes this i s the most interesting age range. These results can be bri e f l y summarized in the following table: Term of Level of Annual Risk Premiums (%) Contract 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 : 1.98 30 1. 15 1.19 1.31 1.60 The risk premium on a 10-year policy is almost completely insensitive to the age of the purchaser (for ages between 20 and 50). Even on a 20-year policy the risk premium i s quite insensitive to the age of the purchaser. Only on a 30-year policy does the risk premium increase significantly from 1.15% for a 20-year old to 1.60% for a 50-year old purchaser. These conclusions are a consequence of the patterns of mortality, because in a l l cases there is a high probability that no benefits w i l l be paid un t i l the maturity of the contract. They can be seen even better in Table 6.4 which shows the relation-ship between the risk premium and the age of the purchaser for contracts of terms 10, 20 and 30 years respectively. For a l l ages, however, the risk premium is sensitive to changes in the term of the contract. This i s a direct consequence of the influence of time to maturity on the value of the option for the nature of guarantee offered in example no. 1. 138 Table 6.3: Risk Premium versus Term of Contract Guarantee = 100% of investment component on death or maturity Mortality Table = CA 1958-64 Risk Free Rate = 0.04 Variance Rate = 0.01846 (TSE) Term of Level of Annual Risk Premium (%) Contract Age of Purchaser at Entry (years) (in years) 20 30 40 50 1 3.518 3.518 3.518 3.518 2 3.380 3.380 3.380 3.381 3 3.332 3.332 3.332 3.333 4 3.277 3.277 3.278 3.279 5 3.209 3.209 3.209 3.212 6 3.128 3.128 3.129 3.134 7 3.038 3.038 : 3.040 3.047 8 2.943 2.942 2.946 2.956 9 2.844 2.844 2.849 2.864 10 2.743 2.743 2.750 2.770 11 2.642 2.643 2.651 2.677 12 2.541 2.542 2.553 2.585 13 2.443 2.444 2.458 2.497 14 2.344 2.345 2.363 2.411 15 2.249 2.251 2.273 2.330 16 2.155 2.158 2.183 2.251 17 2.066 2.070 2.100 2.179 18 1.977 1.982 2.017 2.109 19 1.890 1.897 1.937 2.042 20 1.806 1.815 1.860 1.980 21 1.732 1.742 1.793 1.927 22 1.654 1.666 1.724 1.873 23 1.580 1.594 1.660 1.825 24 1.508 1.525 1.598 1.780 25 1.439 1.459 1.540 1.739 26 1.382 1.405 1.493 1.707 27 1.320 1.345 1.443 1.674 28 1.259 1.289 1.395 1.644 29 1.203 1.236 1.353 1.618 30 1.149 1.186 1.313 1.595 Table 6.4: Risk Premium versus Age of Purchaser at Entry Guarantee = 100% of investment component on death or maturity Mortality Table = CA 1958-64 Risk Free Rate = 0.04 Variance Rate = 0.01846 (TSE) Age of Purchaser Level of Annual Risk Premium (%) at Entry Term of Contract (years) (years) 10 20 30 20 2.743 1.806 1.149 21 2.743 1.806 1.150 22 2.743 1.806 1.152 23 2.743 1.806 1.154 24 2.743 1.807 1.156 25 2.743 1.807 1.159 26 2.743 1.808 1.163 27 2.743 1.809 1.168 28 2.743 1.811 1.173 29 2.743 1.812 1.179 30 2.743 1.815 1.186 31 2.744 1.817 1.193 32 2.744 1.820 1.202 33 2.744 1.823 1.211 34 2.745 1.827 1.222 35 2.745 1.831 1.234 36 2.746 1.836 1.247 37 2.747 1.841 1.261 38 2.748 1.847 1.277 39 2.749 1.853 1.294 40 2.750 1.860 1.313 41 2.751 1.868 1.333 42 2.753 1.877 1.355 43 2.754 1.886 1.379 44 2.756 1.896 1.404 45 2.758 1.908 1.432 46 2.769 1.920 1.461 47 2.762 1.933 1.491 48 2.764 1.947 1.524 49 2.767 1.963 1.559 50 2.770 1.980 1.595 140 6.6 Numerical Example No. 2 The parameters used in example no. 2 are the same as those used in example no. 1 with the exception that the interest rate was increased from 4% to 8%: (1) Variance Rate : the variance rate of the TSE Industrial Index; (2) Risk Free Rate : 8% per annum; (3) Guarantee : 100% of the investment component of the premiums paid at death or maturity. With the other parameters l e f t unchanged, the interest rate was increased to 8% in order to investigate the effect of the riskless interest rate on the risk 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 interest rates. The computation of the equilibrium value of the guarantee with known date of expiration is shown in Table 6.5. In this and in the numerical examples that follow this table was computed for maturities from one to twenty years. Mainly to save computer time the calculations were restricted to contracts with maturities of twenty years and less. In this case the premium becomes relatively small after twenty years. When the mortality risk i s introduced, Table 6.6 shows the relationship between the risk premium and the term of the contract for purchasers of ages 20, 30, 40 and 50. From another angle Table 6.7 shows the relationship between the risk premium and the age of the purchaser for contracts of terms 10 and 20 years. These results can be summarized in the table that follows: 141 Table 6.5: Computation of the Equ i l i b r i u m Value of the Guarantee with Known Date of E x p i r a t i o n : Example No. 2  Variance Rate = 0.01846 Riskless Rate = 0.08000 Guarantee = 100% of investment component on death or maturity, No. years Value P.V. T o t a l P.V. Value T o t a l Annual Annual to C a l l Guarantee P.V. Reference Put Annual Premium Premium Ex p i r a t i o n Contract P o r t f o l i o Premium Put C a l l (1) (2) (3) (4) (5) (6) (7) (8) (9) 1 0. 09871 0. 92312 1. 02183 1. 00000 0.02183 1. 02183 0.02183 0.09871 2 0. 25161 1. 70429 1. 95590 1. 92312 0.03279 1. 01705 0.01705 0.13084 3 0. 45440 2. 35988 2. 81428 2. 77526 0.03902 1. 01406 0.01406 0.16373 4 0.69903 2. 90460 3. 60363 3. 56189 0.04174 1. 01172 0.01172 0.19625 5 0. 97838 3. 35160 4. 32998 4. 28804 0.04194 1. 00978 0.00978 0.22816 6 1.28614 3. 71270 4. 99884 4. 95836 0.04048 1. 00816 0.00816 0.25939 7 1. 61666 3. 99846 5. 61513 5. 57714 0.03799 1. 00681 0.00681 0.28987 8 1. 96491 4. 21834 6. 18326 6. 14835 0.03491 1. 00568 0.00568 0.31958 9 2. 32641 4. 38077 6. 70718 6. 67564 0.03154 1. 00472 0.00472 0.34849 10 2. 69724 4. 49329 7. 19053 7. 16239 0.02813 1. 00393 0.00393 0.37658 11 3. 07401 4. 56261 7. 63662 7. 61172 0.02490 1. 00327 0.00327 0.40385 12 3. 45368 4. 59472 8. 04839 8.02650 0.02189 1. 00273 0.00273 0.43028 13 3. 83364 4. 59491 8. 42856 8.40940 0.01916 1. 00228 0.00228 0.45588 14 4. 21169 4. 56792 8.77961 8. 76285 0.01676 1. 00191 0.00191 0.48063 15 4. 58586 4. 51792 9. 10378 9. 08913 0.01465 1. 00161 0.00161 0.50454 16 4. 95473 4. 44860 9. 40333 9. 39032 0.01301 1. 00138 0.00138 0.52764 17 5. 31679 4. 36324 9.68003 9. 66836 0.01167 1. 00121 0.00121 0.54992 18 5. 67108 4. 26471 9.93579 9. 92502 0.01077 1. 00108 0.00108 0.57139 19 6. 01675 4. 15553 10. 17228 10. 16195 0.01034 1. 00102 0.00102 0.59209 20 6. 35307 4. 03794 10. 39101 10.38066 0.01035 1. 00100 0.00100 0.61201 Table 6.6: Risk Premium versus Term of Contract Guarantee = 100% of investment component on death or maturity Mortality Table = CA 1958-64 Risk Free Rate = 0.08 Variance Rate = 0.01846 (TSE) Term of Level of Annual Risk Premium (%) Contract Age of Purchaser at Entry (years) (in years) 20 30 40 50 1 2.183 2.183 2.183 2.183 2 1.706 1.705 1.706 1.708 3 1.407 1.407 1.408 1.413 4 1.174 1.174 1.176 1.183 5 0.981 0.980 0.983 0.994 6 0.820 0.819 0.823 0.838 7 0.686 0.685 , 0.690 0.709 8 0.573 0.573 0.580 0.602 9 0.478 0.478 0.486 0.512 10 0.400 0.400 0.409 0.439 11 0.335 0.334 0.345 0.379 12 0.281 0.281 0.293 0.331 13 0.237 0.237 0.250 0.291 14 0.200 0.201 0.215 0.259 15 0.171 0.171 0.187 0.234 16 0.148 0.149 0.166 0.215 17 0.131 0.132 0.150 0.202 18 0.119 0.120 0.138 0.191 19 0.113 0.114 0.133 0.187 20 0.111 0.112 0.131 0.185 143 Table 6.7: Risk Premium versus Age of Purchaser at Entry Guarantee = 100% of investment component on death or maturity Mortality Table = CA 1958-64 Risk Free Rate = 0.08 Variance Rate = 0.01846 (TSE) Age of Purchaser at Entry (years) Level of Annual Risk Premium (%) Term of Contract 10 (years) 20 20 0.400 0.111 21 0.400 0. I l l 22 0.400 0.110 23 0.400 0.110 24 0.399 0.110 25 0.399 0.110 26 0.399 0.110 27 0.399 0.110 28 0.399 0.110 29 0.399 0.110 30 0.400 0.112 31 0.400 0.113 32 0.400 0. 114 33 0.401 0.115 34 0.402 0.117 35 0.403 0.118 36 0.404 0.120 37 0.405 0.122 38 0.406 0.125 39 0.407 0.128 40 0.409 0.131 41 0.411 0.134 42 0.413 0.138 43 0.416 0.143 44 0.418 0.147 45 0.421 0.153 46 0.424 0.158 47 0.428 0.164 48 . 0.431 0.171 49 0.435 0.178 50 0.439 0.185 1 1 144 Term of Level of Annual Risk Premium (%) Contract Age of Purchaser at Entry (years) (in years) 20 30 40 50 10 .40 .40 .41 .44 20 . 11 . 11 . 13 .19 Again the risk premium is quite insensitive to the age of the purchaser (between 20 and 50 years old) and more sensitive to the term of the contract. The most important conclusion from this example is the substantial effect of the riskless interest rate on the equilibrium value of the risk premium. If the interest rate is increased from 4% to 8%,the risk premium decreases for a 20-year old purchaser on a 10-year contract from 2.74% to 0.40% and for a 50-year old purchaser on a 20-year contract from 1.98% to 0.19%. As discussed in Chapter 2, the appropriate riskless interest rate for Black-Scholes and Merton's option pricing model i s that of a riskless bond with the same maturity as the option. Thus, i t may perhaps be suitable to vary the interest with the term of the contract. These problems, which require further research, are not pursued in this study. The selection of the appropriate interest rate is then a very important factor in the pricing of risk premiums and l i f e insurance companies should give careful consideration to this factor. 6.7 Numerical Example No. 3 The parameters used in example no. 3 are the same as those in 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 risk premiums: (1) Variance Rate : 0.04 per year; (2) Risk Free Rate : 8% per year; (3) Guarantee : 100% of the investment component of the premiums paid at death or maturity. The computation of the equilibrium value of the guarantee with known date of expiration i s shown in Table 6.8 and,as expected,the value of the c a l l option and particularly the annual premium on the put are higher than in example no. 2. The annual premium on the put is an increasing function of the variance of the rate of return on the reference portfolio. When mortality risk i s also taken into consideration,Table 6.9 shows the relationship between the risk premium and the term of the contract for purchasers of ages 20, 40 and 60. Table 6.10 shows the relationship between the risk premium and the age of the purchaser for contracts of terms 10, 15 and 20 years. These results can be summarized in 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 2.05 2.05 2.06 2.11 2.24 20 0.90 0.90 0.96 1.13 1.49 This example illustrates the substantial influence on the valuation of risk premiums of the variance rate of the return on the reference portfolio: when the variance rate increases from 0.01846 (TSE) to 0.04, the risk premium increases for a 20 year old on a 10-year term from 0.40% 146 Table 6.8: Computation of the Eq u i l i b r i u m Value of the Guarantee with Known Date of E x p i r a t i o n : Example No. 3  Variance Rate => 0.04000 Riskless Rate = 0.08000 Guarantee = 100% of investment component on death or maturity No. years Value P.V. T o t a l P.V. Value T o t a l Annual Annual to C a l l Guarantee P.V. Reference Put Annual Premium Premium Ex p i r a t i o n Contract P o r t f o l i o Premium Put C a l l (1) (2) (3) (4) (5) (6) (7) (8) (9) 1 0.12061 0. 92312 1. 04373 1.00000 0.04373 1. 04373 0.04373 0.12601 2 0.29378 1. 70429 1, 99807 1.92312 0.07495 1. 03897 0.03897 0.15276 3 0.51539 2. 35988 2. 87527 2.77526 0.10001 1. 03604 0.03604 0.18571 4 0.77636 2. 90460 3. 68095 3.56189 0.11907 1. 03343 0.03343 0.21796 5 1.06909 3. 35160 4. 42070 4.28804 0.13266 1. 03094 0.03094 0.24932 6 1.38734 3. 71270 5. 10004 4.95836 0.13168 1. 02857 0.02857 0.27980 7 1.72548 3. 99846 5. 72394 5.57714 0.14680 1. 02632 0.02632 0.30938 8 2.07887 4. 21834 6. 29721 6.14835 0.14886 1. 02421 0.02421 0.33812 9 2.44319 4. 38077 6. 82396 6.67564 0.14832 1. 02222 0.02222 0.36599 10 2.81492 4. 49329 7. 30821 7.16239 0.14582 1. 02036 0.02036 0.39301 11 3.19102 4. 56261 7. 75364 7.61172 0.14192 1. 01864 0.01864 0.41922 12 3.56876 4. 59472 8. 16347 8.02650 0.13697 1. 01706 0.01706 0.44462 13 3.94581 4. 59491 8.54072 8.40940 0.13133 1. 01562 0.01562 0.46921 14 4.32020 4. 56792 8. 88812 8.76285 0.12527 1. 01430 0.01430 0.49301 15 4.69006 4. 51792 9. 20798 9.08913 0.11885 1. 01308 0.01308 0.51601 16 5.05441 4. 44860 9. 50301 9.39032 0.11269 1. 01200 0.01200 0.53826 17 5.41111 4. 36324 9. 77435 9.66836 0.10599 1. 01096 0.01096 0.55967 18 5.76044 4. 26471 10. 02515 9.92502 0.10013 1. 01009 0.01009 0.58040 19 6.10122 4. 15553 10. 25676 10.16195 0.09481 1. 00933 0.00933 0.60040 20 6.43273 4. 03794 10.47067 10.38066 0.09001 1. 00867 0.00867 0.61968 Table 6.9: Risk Premium versus Term of Contract Guarantee = 100% of investment component on death or maturity Mortality Table = CA 1958-64 Risk Free Rate = 0.08 Variance Rate. =0 .04 Term of Level of Annual Risk Premium (%) Contract Age of Purchaser at Entry (years) (in years) 20 40 60 1 4.373 4.373 4.373 2 3.898 3.898 3.905 3 3.605 3.606 3.621 4 3.345 3.347 3.374 5 3.097 3.100 3.143 6 2.861 2.866 2.929 7 2.638 2,644 2.731 8 2.429 2.437 2.551 9 2.231 2.243 1.286 10 2.047 2.062 2.238 11 1.877 1.896 2.106 12 1.720 1.744 1.990 13 1.578 1.607 1.889 14 1.448 1.481 1.801 15 1.327 1.367 1.724 16 1.221 1.266 1.659 17 1.119 1.170 1.601 18 1.033 1.090 1.556 19 0.959 1.022 1.519 20 0.894 0.963 1.490 Table 6.10: Risk Premium versus Age of Purchaser at Entry Guarantee = 100% of investment component on death or maturity Mortality Table = CA 1958-64 Risk Free Rate = .08 Variance Rate = 0.04 Age of Purchaser Level of Annual Risk Premium (%) at Entry Term of Contract (years) (years) 10 15 20 20 2.047 1.327 0.894 21 2.047 1.327 0.894 22 2.046 1.327 0.894 23 2.046 1.327 0.894 24 2.046 1.326 0.894 25 2.046 1.326 0.894 26 2.045 1.326 0.895 27 2.045 1.327 0.896 28 2.046 1.328 0.898 29 2.046 1.329 0.900 30 2.047 1.330 0.903 31 2.047 1.332 0.908 32 2.048 1.334 0.910 33 2.049 1.337 0.914 34 2.050 1.339 0.919 35 2.052 1.343 0.924 36 2.053 1.346 0.930 37 2.055 1.351 0.937 38 2.057 1.355 0.945 39 2.059 1.361 0.953 40 2.062 1.367 0.963 41 2.065 1.373 0.973 42 2.069 1.381 0.985 43 2.073 1.389 0.098 44 2.077 1.398 1.012 45 2.082 1.408 1.027 46 2.087 1.418 1.043 47 2.093 1.429 1.061 48 2.098 1.442 1.081 49 2.104 1.445 1.102 50 2.111 1.470 1.125 i i 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 risk premium is an increasing function of the v o l a t i l i t y (variance rate) of the reference portfolio,should be carefully considered by l i f e insurance companies when selecting the unit trust. Sometimes the insurance company picks i t s own portfolio as the reference portfolio. It is conceivable that l i f e insurance companies could select two or more unit trusts with different h i s t o r i c a l variance rates so as to be able to offer to their clients different policies, which would naturally demand different risk premiums. 6.8 Numerical Example No. 4 In the f i r s t three examples the nature of the guarantee remained unchanged, that is at a guarantee of 100% of the investment component of the' premiums paid at death or maturity. Example no. 4 deals with a guarantee of only 75% of investment component of the premiums paid at death or maturity. 7 The other parameters used are the same as those in example no. 1: (1) Variance Rate the variance rate of the TSE Industrial 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 is 75% of the total premiums paid. This is 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, is shown in 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 is always higher in the former, which is natural because the exercise price is lower other things being equal, the present value of the guarantee i s lower, thereby making the total present value of the contract lower. As the present value of the reference portfolio remains unchanged, the value of the put and,therefore,the annual premium on the put are lower in the case of the 75% guarantee than in the case of the 10i This result can be seen directly by considering that the guarantee in this example is less generous and so i t w i l l cost less. It i s interesting to note that in this example the annual premium on the put increases with time to maturity . When mortality risk i s also taken into consideration,Table 6.12 shows the relationship between the risk premium and the term of the contract for purchasers of ages 20, 30, 40, 50 and 60. These results are summarized in 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 results contradict the "common belief" in the actuarial literature that the risk premium should decrease with the term of the contract. The relationship between the risk 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 = 0.01846 Riskless Rate = 0.04000 Guarantee = 75% of investment component on death or maturity No. years Value P.V. Total P.V. Value Total Annual Annual to Call Guarantee P.V. Reference Put Annual Premium Premium Expiration Contract Portfolio Premium Put Call (1) (2) (3) (4) (5) (6) (7) (8) (9) 1 0. 27976 0.72059 1. 00035 1.00000 0.00035 1. 00035 0.00035 0.27976 2 0. 57709 1. 38467 1. 96177 1.96079 0.00098 1. 00050 0.00050 0.29432 3 0. 89051 1. 99557 2. 88608 2.88391 0.00218 1. 00076 0.00076 0.30879 4 1. 21820 2. 55643 3. 77463 3.77083 0.00381 1. 00101 0.00101 0.32306 5 1. 55866 3. 07024 4. 62890 4.62297 0.00593 1. 00128 0.00128 0.33716 6 1. 91017 3. 53983 5. 44999 5.44170 0.00829 1. 00152 0.00152 0.35102 7 2. 27135 3. 96786 6. 23921 6.22833 0.01088 1. 00175 0.00175 0.36468 8 2. 64075 4. 35689 6. 99764 6.98411 0.01353 1. 00194 0.00194 0.37811 9 3. 01723 4. 70932 7. 72655 7.71026 0.01629 1. 00211 0.00211 0.39133 10 3. 39943 5. 02740 8. 42683 8.40793 0.01890 1. 00225 0.00225 0.40431 11 3. 78635 5. 31330 9. 09965 9.07825 0.02139 1. 00236 0.00236 0.41708 12 4. 17696 5. 56905 9. 74601 9.72229 0.02372 1. 00244 0.00244 0.42963 13 4. 57047 5. 79658 10.36705 10.34107 0.02597 1. 00251 0.00251 0.44197 14 4. 96598 5. 99769 10.96368 10.93559 0.02808 1. 00257 0.00257 0.45411 15 5. 36207 6. 17413 11. 53620 11.50680 0.02940 1. 00255 0.00255 0.46599 16 5. 75939 6. 32751 12. 08690 12.05561 0.03129 1. 00259 0.00259 0.47774 17 6. 15547 6. 45937 12. 61484 12.58291 0.03194 1. 00254 0.00254 0.48919 18 6. 55199 6. 57116 13. 12315 13.08952 0.03363 1. 00257 0.00257 0.50055 19 6. 94574 6. 66425 13. 60999 13.57627 0.03372 1. 00248 0.00248 0.51161 20 7. 33712 6. 73994 14. 07706 14.04394 0.03312 1. 00236 0.00236 0.52244 Table 6.12: Risk Premium versus Term of Contract and Age of Purchaser at Entry  Guarantee = 75% of investment component on death or maturity Mortality Table = CA 1958-64 Risk Free Rate = 0.04 Variance Rate = 0.01846 (TSE) Term of Level of Annual Risk Premium (%) Contract Age of Purchaser at Entry (years) (in years) 20 30 40 50 60 1 0.035 0.035 0.035 0.035 0.035 2 0.050 0.050 0.050 0.050 0.050 3 0.076 0.076 0.076 0.076 0.075 4 0.101 0.101 0. 101 0.100 0.099 5 0.128 0.128 0.127 0.126 0.124 6 0.152 0.152 0.151 0. 150 0. 145 7 0.174 0. 175 0. 174 0. 172 0.166 8 0.193 0.193 0.192 0. 189 0.182 9 0.210 0.210 0.209 0.205 0. 196 10 0.224 0.224 0.223 0.218 0.207 11 0.235 0.235 0.233 0.228 0.215 12 0.243 0.243 0.241 0.235 0.221 13 0.250 0.250 0.248 0.242 0.226 14 0.256 0.256 0.253 0.247 0.230 15 0.254 0.254 0.251 0.245 0.229 16 0.256 0.258 0.255 0.248 0.231 17 0.253 0.253 0.251 0.244 0.229 18 0.256 0.256 0.253 0.247 0.230 19 0.247 0.247 0.245 0.240 0.226 20 0.235 0.239 0.234 0.231 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 risk premium and a uniform risk premium for a l l ages and terms of contracts can be considered. The magnitude of the risk premium is relatively small for the parameters selected. 6".9 Numerical Example No. 5 In this last 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 for solving for many different situations. Instead of increasing the amount of the guarantee as new premiums are invested in the reference portfolio, this example 9 considers a fixed amount of minimum guarantee on death or maturity . A 20 year contract is assumed where $1 is invested in the unit trust each year. A $20 minimum guarantee i s considered at death or maturity independ-ently of the date when the benefits are payable. The other two parameters used are those of example no. 2: (1) Variance Rate : the variance rate of the TSE Industrial Index; (2) Risk Free Rate : 8% per year; (3) Guarantee : $20.00, fixed for a l l dates of expiration. The computation of the equilibrium value of the guarantee with known date of expiration for example no. 5 is shown in Table 6.13. Naturally the value of the c a l l option is practically equal to zero and the value of the put option is very high for short maturities,because there is practically zero probability that the reference portfolio w i l l reach a value of $20 for those short maturities. The annual premium on the put is,therefore, Contracts of this type are popular in the United Kingdom. 154 Table 6.13: Computation of the Eq u i l i b r i u m Value of the Guarantee with Known Date of E x p i r a t i o n : Example No. 5  Variance Rate = 0.01846 Riskless Rate = 0.08000 Guarantee = 2 0 . 0 0 No. years Value P.V. T o t a l P.V. Value T o t a l Annual Annual to C a l l ' Guarantee P.V. Reference Put Annual Premium Premium Ex p i r a t i o n Contract P o r t f o l i o Premium Put C a l l (1) (2) (3) (4) (5) (6) (7) (8) (9) 1 0.00000 18. 46233 18. 46233 1. 00000 17. 46233 18. 46233 17.46233 0.00000 2 0.00000 17. 04288 17. 04288 1. 92312 15. 11976 8. 86212 7.86212 0.00000 3 0.00000 15. 73256 15. 73256 2. 77526 12. 95730 5. 66886 4.66886 0.00000 4 0.00000 14. 52298 14. 52298 3. 56189 10. 96109 4.07733 3.07733 0.00000 5 0.00000 13. 40640 13. 40640 4. 28804 9. 11837 3. 12647 2.12647 0.00000 6 0.00009 12. 37567 12. 37576 4. 95836 7. 41741 2. 49594 1.49594 0.00002 7 0.00231 11. 42418 11. 42650 5. 57714 5. 84936 2. 04881 1.04881 0.00041 8 0.02142 10. 54585 10. 56727 6. 14835 4. 41893 1. 71872 0.71872 0.00348 9 0.10161 9. 73505 9. 83666 6. 67564 3. 16102 1. 47352 0.47353 0.01522 10 0.30506 8.98658 9. 29165 7. 16239 2. 12925 1. 29728 0.29728 0.04259 11 0.67014 8. 29566 8. 96580 7. 61172 1. 35408 1. 17789 0.17789 0.08804 12 1.18823 7. 65786 8. 84609 8. 02650 0. 81959 1. 10211 0.10211 0.14804 13 1.81715 7. 06910 8. 88625 8. 40940 0.47685 1. 05670 0.05670 0.21609 14 2.50637 6. 52560 9. 03197 8. 76285 0. 26912 1. 03071 0.03071 0.28602 15 3.21369 6. 02389 9. 23758 9. 08913 0. 14845 1. 01633 0.01633 0.35357 16 3.91016 5. 56075 9. 47092 9. 39032 0. 08059 1. 00858 0.00858 0.41640 17 4.57865 5. 13323 9. 71188 9. 66836 0. 04352 1. 00450 0.00450 0.47357 18 5.21048 4. 73857 9. 94904 9. 92502 0.02402 1. 00242 0.00242 0.52498 19 •5.80211 4. 37425 10. 17636 10.16195 0. 01441 1. 00142 0.00142 0.57096 20 6.35307 4. 03794 10. 39101 10.38066 0. 01035 1. 00100 0.00100 0.61201 155 also very high for lower maturities ; Up to the seventh year i t is more than 100% of the annual investment component of the premium. Table 6.14 shows the relationship between the risk premium and the age of the purchaser for this particular contract. In this case the age of the purchaser is an important variable in determining the risk premium. The risk premium is highly sensitive to the probability of death in earlier periods, because of the high values of annual premium on the put for these early maturities. For a 20 year old purchaser the risk premium is 4.41% and decreases to 4.08% for a 30 year old due to the higher probabilities of death in the early twenties than in the early t h i r t i e s . Then the risk premium goes to 9.48% for a 40 year old and to the extremely high value of 27.54% for a 50 year old. The nature of the fixed minimum guarantee presented in this example is quite different from a guarantee proportional to the investment component of the premiums paid as shown in the other examples. The order of magnitude of the equilibrium risk premiums reflects this difference. There is probably a market for a great variety of contracts,depending on the purchaser's attitude towards risk and his expectations about the future. 6.10 Summary and Conclusions The equilibrium pricing model of equity-linked l i f e insurance policies with asset value guarantees developed in Chapter 5,has been applied in this chapter to solve five particular numerical examples. The examples have been chosen to represent a variety of possible guarantees and to show the effect of important parameters such as the riskless interest rate and the variance rate of return on the reference portfolio. 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 = 20.00 (Fixed amount for a l l maturities) Mortality Table = CA 1958-64 Risk Free Rate = 0.08 Variance Rate = 0.01846 (TSE) Term of Contract = 20 Age of Purchaser at Entry (years) Level of Annual Risk Premium (%) 20 4.410 21 4.417 22 4.349 23 4.228 24 4.082 25 3.940 26 3.829 27 3.772 28 3.782 29 3.868 30 4.028 31 4.261 32 4.566 33 4.936 34 5.370 35 5.863 36 6.413 37 7.028 38 7.728 39 8.535 40 9.477 41 10.565 42 11.815 43 13.236 44 14.832 45 16.599 46 18.523 47 20.587 48 22.772 49 25.084 50 27.544 157 but most of these cases can be solved by a d i r e c t a p p l i c a t i o n of the same p r i c i n g model. The proportion of the investment component of the premiums paid set as guarantee, which i n the f i r s t three examples was 100% and i n the fourth was 75%, or the amount of the f i x e d guarantee, which i n the f i f t h example was $20, make very 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 of the model, although, n a t u r a l l y , the r e s u l t i n g r i s k premiums may be substant-i a l l y d i f f e r e n t . I t has also been seen that changing the r i s k l e s s i n t e r e s t rate or the variance rate does not complicate the computations, although again the r e s u l t i n g r i s k premiums are quite d i f f e r e n t . To h i g h l i g h t the differences between the examples ,assume a 40 year ol d person purchasing a 20 year contract. The following table summarizes the e q u i l i b r i u m r i s k premiums that he would have to pay according to the s p e c i f i c a t i o n s of each numerical example: Level of Annual Risk Premium (%) 40 years o l d purchaser - 20 years term Example No. 1 r = 4% o2 = 0.01846 E = 100% 1.86% Example No. 2 r = 8% a2 = 0.01846 E = 100% 0.13% Example No. 3 r = 8% a2 = 0.04 E = 100% 0.96% Example No. 4 r = 4% a2 = 0.01846 E = 75% 0.23% Example No. 5 r = 8% a2 = 0.01846 E = $20 9.48% Table 6.15 gives a summary of the values obtained f or the annual premiums for the put i n the f i v e examples considered ( for maturities of one 158 Table 6.15: Summary of Annual Premium for the Put Annual Premium for the Put (%) Example No.1 Example No.2 Example No.3 Example No.4 Example No.5 No. years 2 a = TSE TSE 4% TSE TSE to r = 4% 8% 8% 4% 8% Expiration E = 100% 100% 100% 75% $20.00 1 3.52 2.18 4.37 0.04 1746.23 . 2 3.38 1.71 3.90 0.05 786.21 3 3.33 1.41 3.60 0.08 466.89 4 3.28 1.17 3.34 0. 10 307.73 5 3.21 0.98 3.09 0.13 212.64 6 3. 13 0.82 2.86 0. 15 149.59 7 3.04 0.68 2.63 0. 18 104.88 8 2.94 0.57 2.42 0.19 71.87 9 2.84 0.47 2.22 0.21 47.35 10 2.74 0.39 2.04 0.23 29.73 11 2.64 0.33 1.86 0.24 17.79 12 2.54 0.27 1.71 0.24 10.21 13 2.44 0.23 1.56 0.25 5.67 14 2.34 0. 19 1.43 0.26 3.07 15 2.24 0.16 1.31 0.26 1.63 16 2.14 0.14 1.20 0.26 0.86 17 2.05 0. 12 1.10 0.25 0.45 18 1.96 0.11 1.01 0.25 0.24 19 1.87 0.10 0.93 0.25 0.14 20 1.79 0.10 0.87 0.24 0.10 i I 159 to twenty years). This table permits a comparison of the equilibrium value of the guarantee, with known date of expiration, for the different examples. Finally, i t should be emphasized that the pricing model presented in this study is an equilibrium model under the assumption of perfect markets. In i t s practical applications market imperfections such as transaction costs, selling costs, profits and taxes should be considered. However,the basic d i f f i c u l t y facing insurance companies has been the calculation of the value of this guarantee. The transaction costs incurred by them can be found readily by conventional methods. In practice insurance companies tend to charge a uniform risk premium for a given contract independent of age. Current practice is to charge an additional 1% for a guarantee of 75 to 100% of premiums paid. It has been shown that while this i s within the range of results produced in this study, the risk 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 portfolio, and the riskless interest rate. There is no theoretical j u s t i f i c a t i o n for charging a uniform risk premium. The model developed in this thesis gives a sound theoretical basis and an expedient numerical procedure for pricing these contracts. CHAPTER 7: 160 OPTIMAL INVESTMENT STRATEGIES FOR THE SELLERS OF EQUITY LINKED LIFE INSURANCE POLICIES WITH ASSET VALUE GUARANTEES. 7.1 Introduction In Chapter 5 ah equilibrium model for pricing equity based l i f e insurance contracts with minimum guarantee was developed. In Chapter 6 this model was applied to some typical contracts to determine the risk premium that insurance companies should charge, in equilibrium, for bearing additionally the investment risk associated with these contracts. This chapter is concerned with the derivation of appropriate investment policies for insurance companies to enable them to hedge against the major risks associated with the provision of asset value guarantees in equity linked l i f e insurance policies. In addition, i t w i l l examine the question whether the hedging policy can be followed by the l i f e insurance company without the necessity of borrowing additional funds. To conclude the chapter a numerical example w i l l be presented. 7.2 Hedging Policy . When presenting the Black-Scholes' option pricing model in Chapter 2, i t was shown how to create a hedged position consisting of one share of common stock long and 1/Wg options short, the value of which would not depend on the price of the stock. This hedging process plus the fact that the rate of return on the equity position, being riskless, must be equal to the risk free interest rate were used as the basis for deriving par t i a l d i f f e r e n t i a l equations governing the value of an option on a non-dividend paying stock. The analysis was extended to cover the case of an option on a dividend paying stock. Furthermore i t was shown that this model could also be used to f i the value of an option on the reference portfolio in the case of equity 161 based l i fe insurance contracts. If the insurance company follows the investment policy determined by the model, through the hedging process described above, i t wi l l bear no risk and wil l make no profit or loss. As discussed in earlier chapters,an equity linked l i fe insurance policy with asset value guarantee can be viewed as a regular l i fe insurance contract (endownment) plus a call option on a reference portfolio. The insurance company is selling short a cal l option on the reference portfolio. To eliminate a l l the risk assumed by selling short the call 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 wx is the partial 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 wi l l be immunized against loss (and profit). 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 < 1 - x -and therefore (7.3) x w < x x -Hence, the optimal strategy dictates that not a l l the investment component be actually invested in the reference portfolio. This i s possible in the case of a number of United Kingdom equity based contracts where the actual wording says "/invested or deemed to be invested." Notice that wx gives the proportion of the value of the reference portfolio (x) which should be actually invested in the reference portfolio to maintain the hedged position. This concept w i l l become clearer when we discuss specific numerical examples. The continuous adjustment of the position in the reference portfolio gives rise to transaction costs. These are ignored i n the present study. For the single premium contract the amount invested in the reference portfolio has an analytical expression given by 1: For the continuous premium contract and the periodic premium contract there is no analytical expression for w , but w can easily be obtained by X X 2 the numerical method described in Chapter 5. It has been shown that that amount that must be invested in the reference portfolio at time T to form a hedged position with one c a l l 1See Black-Scholes [3,4]. 2 In fact wx can be computed directly from w. Using the notation defined in Chapter 6: (7.4) x w. x = x N(d x) where d^ and N( ) are defined in (5.9). w. x 1+1,3 " W i - 1 - J 2h 163 option sold short, assuming known date of expiration t, is (7.1) x w (x, t - T , g) 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 v 3 contract). 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), is (7.8) L(x) = x Wx(x) 3 The probabilities a(x,t) can be computed using (5.49) and (5.51), taking into consideration that the l i fe 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 f o r t = T f l > n _ i £x+x for x = 0, n-1 £x+n-l (7.6) a(x,n) = ~~ for x = 0, n-1 &x+x 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 Feasibility of the Policy While in principle the hedging policy can always be followed to eliminate a l l risks, a practical question is whether the policy can be followed without the necessity of borrowing additional 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 for the contract at each point in time. Provided that the hedging policy has been followed,these w i l l always be equal since the policy ensures that neither profits nor losses are made. (2) The allocation of the insurance company's assets between the reference portfolio, the riskless 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 (for 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 = g e " 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 portfolio, PV^(x(t)), and the value of the put option at time x^: (7.10) PV of l i a b i l i t i e s = PV T(x(t)) + p(x, t-x, g) The present value of the assets at time x is given by the available funds (cash position: invested in the reference portfolio and the riskless 4 This analysis is equivalent to the one presented in Chapter 5 for the total present value of the contract. Notice that known date of expiration,t, has been assumed.Mortality considerations w i l l be added later on. 165 asset) and the present value of the future premiums receivable at time T , PVFP: (7.11) PV of assets = funds a v a i l a b l e + PVFP I f the hedging strategy has been followed,the present value of the l i a b i l i t i e s must be equal to the present value of the assets. A l s o , i n t h i s case the value of the funds a c t u a l l y invested i n the reference p o r t f o l i o has been shown to be x wx-Let i ( x , T ) be the investment i n the r i s k l e s s asset at time T f o r a value of the reference p o r t f o l i o of x such that the hedged strategy i s followed. Then: (7.12) PV of assets = x w + i(x,x) + PVFP and (7.13) i(x,x) = PV of l i a b i l i t i e s - x w - PVFP I f i(x,x) < 0,the borrowing of a d d i t i o n a l funds i s required to follow the hedged p o l i c y . For the s i n g l e premium contract, i n which the whole premium i s paid when the p o l i c y i s issued, i t i s p o s s i b l e to obtain an a n a l y t i c a l expression fo r the net investment i n the r i s k l e s s asset,because: (7.14) w(x, t-x, g) = x N(d 1) - g e " r ( t _ T ) N(d 2) and (7.15) PVFP = 0 S u b s t i t u t i n g (7.4), (7.9), (7.14) and (7.15) into (7.13) we obtain: (7.16) i ( x , t ) = g e " r ( t _ T ) (1 - N(d 2)) > 0 Thus,it i s never necessary to borrow a d d i t i o n a l funds to follow the hedged p o l i c y f o r the s i n g l e premium contract. 166 For the-multiple (continuous and periodic) premium contracts there are no analytical expressions for i(x,x).' There are, nevertheless, some general properties, which are proven in Appendix L, that can be summarized as follows: Theorem 1: The investment in the riskless asset i s a decreasing function of the value of the reference portfolio (at any given time). This theorem is also true for the single premium contract. Theorem 2: At any given date the maximum borrowing required to follow the hedged policy is bounded by the present value of the future put premiums (risk premiums) receivable. A direct consequence of theorem 2 i s : Corollary: If the total 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 periodically through time, no borrowing w i l l be required to follow the hedged strategy. The practical implications of this corollary w i l l be seen when we present a numerical example in Section 7.5. The relationship between i(x,t) and x, for a given T , i s shown in Figure 7.1 The numerical procedure to obtain i(x,x) for the periodic premium contract i s discussed in Section 7.4. 167 Figure 7.1: Investment in 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 portfolio. Q The value at which the i-curve cuts the x-axis (x c) in Figure 7.1 can be defined as "the c r i t i c a l value of the reference portfolio." For values of the reference portfolio greater than x £ some borrowing w i l l be required. The curve shifts up as x increases, that i s , the maximum possible borrowing required decreases as the option approaches maturity. Finally, to determine the actual f e a s i b i l i t y of the policy when the hedging policy is followed, i t i s necessary to introduce mortality risk. The cash position for the contract when mortality is considered i s derived in Appendix L. 7.4 Numerical Example: As a numerical example in 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 wi 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 : 4% per annum (continuous) (3) Guarantee : 100% of the investment component of the premiums paid at death or maturity; (4) Age of purchaser at entry : 40 years old; and (5) Term of the policy : 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%. The investment policy has been determined once a year just after each premium payment.7 Tables 7.1 to 7.9 summarize the results obtained. The entries in these tables correspond to: F u l l results are available upon request. 7 I t must be kept in mind that to eliminate risk,the hedged portfolio should be maintained continuously through time. Here we are analyzing the portfolio only once a year,but assuming that the hedge is maintained continuously. 169 Column (1) : point i n time where the investment strategy i s evaluated, 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 (year 0) and ending j u s t a f t e r the l a s t premium payment (year 9). Column (2) : value of the reference p o r t f o l i o at the corresponding point i n time, given the percentage annual increase i n the value of the reference p o r t f o l i o . Column (3) : value of the c a l l option (weighted according to the m o r t a l i t y data) obtained by numerical procedures. Column (4) : proportion of the funds deemed to be invested i n the reference p o r t f o l i o a c t u a l l y invested i n i t , obtained by using the values i n Column (3).^ Column (5) : long p o s i t i o n i n the reference p o r t f o l i o required to maintain a hedged p o s i t i o n ; 1 ^ obtained by m u l t i p l y i n g Column (2) by Column (4). Column (6) : cash p o s i t i o n of the insurance company f o r t h i s contract at the corresponding point i n time. Column (7) : investment i n the r i s k l e s s asset required to follow the hedged strategy; t h i s i s the d i f f e r e n c e between the long 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 ) ) . I f the investment i s n e g a t i v e , i t implies that borrowing i s required. Tables 7.1 to 7.9 were obtained from the output of Computer•Program No. 9 which gives the values shown i n these tables,but f o r a whole range g Following our conventional notation, the performance of the reference p o r t f o l i o i s compounded yearly. 9 See footnote 2 and equation (7.7). See equation (7.8). 170 of different values of the reference portfolio (400 values every 0.05 12 13 dollars). ' Computer Program No. 9 is shown in Appendix K. As can be seen from the tables, whenever the return on the reference portfolio is negative,the proportion of the premiums paid invested in the reference portfolio i s quite small and decreases with time to maturity. With a return of - 20% and - 15% the long position in the reference portfolio one year before maturity is equal to zero and with a - 10% return i t is equal to only three cents. As a consequence of this most of the funds received by the insurance company (cash position) are invested in the riskless asset. When the return on the reference portfolio i s positive, however, most of the funds received by the insurance comapny are invested in the reference portfolio and in a few cases, with returns of 15% and 20%, additional 14 borrowing is required to maintain the hedged position. As expected,the net borrowing required is very small i n a l l cases. For the different performances of the reference portfolio considered, Table 7.10 and Figure 7.2 summarize the investment in the riskless asset and Table 7.11 and Figure 7.3 summarize the proportion of the amount deemed to be invested in the reference portfolio, actually invested. The output of Computer Program No. 9 can also be used to find another interesting variable: the value of the reference portfolio above which "To be able to use the output of Computer Program No. 9 without interpolation, the value of the reference portfolio corresponding to a certain percentage annual increase was approximated to the nearest factor of five cents. 13 The apparent errors of one cent are due to rounding errors. See Tables 7.8 and 7.9. 171 net borrowing is required to maintain the hedged portfolio. This variable has been defined in this study as the " c r i t i c a l value of the reference portfolio" at a given point in time and can be obtained by finding the f i r s t value of the reference portfolio for which a positive net borrowing i s required. 1' Table 7.12 shows the c r i t i c a l values of the reference portfolio just after every premium payment date. After 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. After the second premium payment (year 1) a return of 265% is required. As maturity i s approached the constant annual return on the reference portfolio required to reach the c r i t i c a l value decreases; after the last premium payment this return i s approximately 10%.^ It should be kept i n mind, however, that the maximum amount to be borrowed i s bounded by the present value of the "risk premiums" unpaid, which decreases with time to maturity. ''Again the values have been approximated to the nearest factor of five cents. '^This i s why some borrowing i s required with returns of 15% and 20%. 172 Table 7.1 Investment Policy Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase in Value of the Reference Portfolio 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 - 20% Year Value of Weighted Proportion Long Position Reference Value of Invested in in Reference Portfolio Option (1) 0 1 2 3 4 5 6 7 8 9 (2) 1.00 1.80 2.45 2.95 3.35 3.70 3.95 4.15 4.35 4.45 (3) 1.91 1.78 1.52 1.14 0.73 0.38 0.13 0.02 0.00 0.00 Reference Portfolio (4) 0.85 0.83 0.77 0.68 0.55 0.37 0.18 0.04 0.00 0.00 Portfolio (5) 0.85 1.49 1.90 2.02 1.84 1.38 0.70 0.17 0.01 0.00 Cash Investment Position in Riskless Asset (6) 1.03 1.89 2.65 3.34 4.05 4.25 5.81 6.95 8.24 9.61 (7) +0.17 +0.41 +0.76 + 1.32 +2.21 +3.47 +5.11 +6.78 +8.23 +9.61 173 Table 7.2 Investment Policy Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase in Value of the Reference Portfolio 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 = - 15% Year Value of Weighted Proportion Long Position Reference Value of Invested in in Reference Portfolio Option (1) 0 1 2 3 4 5 6 7 8 9 (2) 1.00 1.85 2.55 3.20 3.70 4.15 4.55 4.85 5.10 5.35 (3) 1.91 1.83 1.60 1.31 0.93 0.57 0.27 0.07 0.01 0.00 Reference Portfolio (4) 0.85 0.83 0,79 0.72 0.62 0.47 0.29 0.11 0.01 0.00 Portfolio (5) 0.85 1.54 2.01 2.32 2.28 1.96 1.34 0.55 0.07 0.00 Cash Investment Position in Riskless Asset (6) 1.03 1.93 2.73 3.52 4.25 5.04 5.95 7.00 8.25 9.61 (7) +0.17 +0.39 +0.72 +1.20 +1.97 +3.08 +4.61 +6.46 +8.17 +9.61 174 Table 7.3 Investment Policy Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase in Value of the Reference Portfolio 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 = - 10% Year Value of Weighted Proportion Long Position Reference Value of Invested in in Reference Portfolio Option (1) 0 1 2 3 4 5 6 7 8 9 (2) 1.00 1.90 2.70 3.45 4.10 4.70 5.20 5.70 6.15 6.50 (3) 1.91 1.87 1.72 1.50 1.19 0.86 0.50 0.23 0.05 0.00 Reference Portfolio (4) 0.85 0.84 0.81 0.76 0.69 0.59 0.44 0.26 0.09 0.00 Portfolio (5) 0.85 1.99 2.17 2.62 2.82 2.75 2.26 1.49 0.55 0.03 Cash Investment Position in Riskless Asset (6) 1.03 1.97 2.85 3.70 4.51 5.33 6.18 7.16 8.29 9.61 (7) +0.17 +0.38 +0.68 +1.08 +1.69 +2.58 +3.92 +5.67 +7.74 +9.58 175 Table 7.4 Investment Policy Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase in Value of the Reference Portfolio 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 - 5% Year Value of Weighted Proportion Long Position Reference Value of Invested in in Reference Cash Investment Position in Riskless Portfolio Option Reference Portfolio Portfolio Asse (1) (2) (3) (4) (5) (6) (7) 0 1.00 1.91 0.85 0.85 1.03 +0.17 1 1.95 1.91 0.84 1.64 2.02 +0.38 2 2.85 1.84 0.82 2.34 2.97 +0.63 3 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 +8.80 176 Table 7.5 : Investment Policy Variance Rate = Risk Free Rate = Guarantee = Age of Purchaser at Entry = Term of Contract = Annual Premium for the Put = Annual Increase in Value of the Reference Portfolio = 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 0% Year Value of Weighted Proportion Long Position Cash Investment Reference Value of Invested in in Reference Position in Riskless Portfolio Option Reference Portfolio Asset Portfolio (1) (2) (3) (4) (5) (6) (7) 0 1.00 1.91 0.85 0.85 1.03 +0.17 1 2.00 1.95 0.85 1.69 2.06 +0.37 2 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 \ 177 Table 7.6 Investment Policy Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase in Value of the Reference Portfolio 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 = 5% Year Value of Weighted Proportion Long Position Reference Value of Invested in in Reference Portfolio Option Reference Portfolio Cash Investment Position in Riskless Asset (1) 0 1 2 3 4 5 6 7 8 9 (2) 1.00 2.05 3.15 4.30 5.55 6.80 8.15 9.55 11.05 12.60 (3) 1.91 1.99 2.09 2.19 2.33 2.43 2.56 2.68 2.84 3.01 Portfolio (4) 0.85 0.85 0.85 0.85 0.86 0.87 0.88 0.90 0.94 0.98 (5) 0.85 1.75 2.69 3.67 4.78 5.90 7.20 8.63 10.37 12.36 (6) 1.03 2.10 3.22 4.39 5.65 6.90 8.24 9.62 11.08 12.61 (7) +0.17 +0.35 +0.54 +0.72 +0.86 +1.00 +1.04 +0.98 +0.71 +0.25 178 Table 7.7 Investment Policy Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase in Value of the Reference Portfolio 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 = 10% Year Value of Weighted Proportion Long Position Cash Investment Reference Portfolio Value of Option Invested in Reference Portfolio in Reference Portfolio Position in Riskless Asset (1) (2) (3) (4) (5) (6) (7) 0 1.00 1.91 0.85 0.85 1.03 +0.17 1 2.10 2.04 0.86 1.80 2.14 +0.34 2 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 Table 7.8 Investment Policy Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase in Value of the Reference Portfolio 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 = 15% Year Value of Weighted Proportion Long Position Reference Value of Invested in in Reference Portfolio Option (1) 0 1 2 3 4 5 6 7 8 9 (2) 1.00 2.15 3.45 5.00 6.75 8.75 11.05 13.75 16.80 20.30 (3) 1.91 2.08 2.35 2.81 3.41 4.23 5.31 6.76 8.53 10.69* Reference Portfolio (4) 0.85 0.86 0.88 0.91 0.93 0.96 0.99 1.00 1.00 1.00 Portfolio (5) 0.85 1.85 3.03 4.52 6.30 8.42 10.88 13.71 16.80 20.30 Cash Investment Position in Riskless Asset (6) 1.03 2.19 3.48 5.01 6.73 8.70 10.99 13.70 16.77 20.30* (7) +0.17 +0.34 +0.46 +0.49 +0.43 +0.29 +0.11 -0.01 -0.04 0.00 (*) extrapolated values 180 Table 7.9 Investment Policy Variance Rate Risk Free Rate Guarantee Age of Purchaser at Entry Term of Contract Annual Premium for the Put Annual Increase in Value of the Reference Portfolio 0.01846 0.04000 100% of investment component on death or maturity 40 years 10 years 0.02742 = 20% Year Value of Weighted Proportion Long Position Reference Value of Invested in in Reference Portfolio Option (1) 0 1 2 3 4 5 6 7 8 9 (2) 1.00 2.20 3.65 5.35 7.45 9.95 12.90 16.50 20.80 25.95 (3) 1.91 2.12 2.52 3.13 4.08 5.40 7.15 9.51 12.53* 16.34* Reference Portfolio (4) 0.85 0.87 0.89 0.92 0.97 0.98 1.00 1.00 1.00 1.00 Portfolio (5) 0.85 1.91 3.26 4.94 7.12 9.79 12.86 16.50 20.80 25.95 Cash Investment Position in Riskless Asset s (6) 1.03 2.23 3.66 5.33 7.39 9.87 12.83 16.45 20.77* 25.95* (7) +0.17 +0.32 +0.41 +0.39 +0.27 +0.09 -0.03 -0.04 -0.03 0.00 (*) extrapolated values 181 Table 7.10 : Investment in the Riskless Asset Variance Rate = 0.01846 Risk Free Rate = 0.0400 Guarantee = 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 Investment in the Riskless Asset Annual Increase in Value of Reference Portfolio (%) -20 -15 -10 -5 0 5 10 15 20 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 0.00 182 FIGURE 7.2 : INVESTMENT IN THE RISKLESS ASSET 14 .00-12.00 10.00 8. 00 INVESTMENT IN RISKLESS ASSET 6 .00-4.00 2.00 # . 0 0 -# # # t t - 2 . 0 0 - . . . . . . . . . | . . . . . . . . . | . . . . . . . . . J 0% 20% # 00 2.00 4.00 I 6.00 TIME 8.00 10. 00 183 Table 7.11 : Proportion Invested In the Reference Portfolio Variance Rate = 0.01846 Risk Free Rate = 0.0400 Guarantee = 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 Proportion Invested in the Annual Increase in Value of Reference Reference Portfolio Portfolio (%) -20 -15 -10 -5 0 5 10 15 20 0 0.85 0.85 0.85 0.85 0.85 : 0.85 0.85 0.85 0.85 1 .83 .83 .84 .84 .85 .85 .86 .86 0.87 2 .77 .79 .81 .82 .84 .85 .87 .88 .89 3 .68 .72 .76 .79 .83 .85 .88 .91 .92 4 .55 .62 .69 .75 .81 .86 .90 .93 .97 5 .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 1.00 184 FIGURE 7 . 3 : PROPORTION INVESTED IN THE REFERENCE PORTFOLIO 1.60-1.40 1.20 1.00 # # # # # # PROPORTION INVESTED IN REFERENCE PORTFOLIO . 8 0 - # # # # # 60 40 20 # 0 0 - i . . .00 I 2.00 i 20% # # 0% -20% 4.00 6- 00 TIME . # . . . . # . . . . j . . . 8.00 10.00 185 Table 7.12 : Crit ical Values of the Reference Portfolio Variance Rate = 0.01846 Risk Free Rate = 0.04 Guarantee = 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 Crit ical 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 in Chapter 5 to find the equilibrium price of an equity linked l i f e insurance contract with asset value guarantee can also be used to find the optimal investment strategy for the sellers of these contracts to hedge against the investment risk assumed by giving the guarantee. In this chapter i t was shown how the Black-Scholes hedging process can be adapted to this purpose. It was also shown that the amounts that have to be invested in the reference portfolio are smaller than the value of the reference portfolio assigned to the policyholder. To make the prescribed investment strategy possible, l i f e insurance companies must have legal 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 actually invest the investment component of the premium paid in the reference portfolio they would not be able to follow the indicated strategy. The model implies a continuous adjustment of the amounts invested. In practice this is impossible. Further studies could investigate the optimal revision time, considering both the transaction costs and the effects of these discontinuous adjustments on the elimination of risk. 187 Bibliography [1] [2] [3] [A] [5] [6] [7] [8] Ayres, H.F. , Bachelier, L . , Black, F. and Scholes, M.J . , Black, F. and Scholes, M.J . , Black, F. Chen, H.Y., Chen, H.Y., Cox, T.C. and Ross, S.A., "Risk Aversion in the Warrant Markets," Industrial Management Review, Vol. 5, No. 1, Fa l l , 1963. "Theory of Speculation" (translation) in the Random Character of Stock Market Prices, ed. Cootner, M.I.T. Press, Cambridge, Mass., 1964. "The Pricing of Options and Corporate Liabil i t ies ," Journal of Polit ical Economy, Vol. 81, No. 3, May-June 1973. "The Valuation of Option Contracts and a Test of Market Efficiency. 1 1 Journal of Finance, Vol. 27, May 1972. "The Pricing of Complex Options and Corporate Liabil i t ies ." Unpublished draft, Graduate School of Business Administration, University of Chicago, June 1974. "A Dynamic Programming Approach to the Valuation of Warrants." Unpublished Ph.D. Dissertation, Graduate School of Business Administration, University of California, Berkeley, 1969. "A Model of Warrant Pricing in a Dynamic Market," Journal of Finance, Vol. 25, No. 5 (December 1970). "The Pricing of Options for Jump Processes," Working Paper No. 2-75, The Warton School of Business, University of Pennsylvania, 1975. [9] [10] [11] Dahlquist, G. and Bjorck, A . , Di Paolo, F . , Forsythe, G.E. and Wasow, W.R., Numerical Methods, Prentice-Hall Inc., 1974. "An Application of Simulated Stock Market Trends to Investigate a Ruin Problem," Transactions Society of Actuaries, Vol. XXI, 1969. Finite Difference Methods for Partial Differential Equations, John Wiley & Sons Inc. 1960. 188 [12]. Foster, M., [13] [14] [15] Frankel, T . , Galai, D. Grant, A.T. and Kingsworth, G.A. "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. "Variable Annuities, Variable Insurance and Separate Accounts," Boston Univer-sity Law Review, Vol. 51, 1971. "Pricing of Options and the Efficiency of the Chicago Board Options Exchange." Unpublished Ph.D. Dissertation, Univer-sity of Chicago, March 1975. "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 . , [17] [18] [19] [20] [21] Jordan, C.W. J r . , Kahn, P.M., Kassouf, S.T., Kassouf, S.T., Kensicki, P.R., "A Theoretical and Empirical Investig-ation of the Dual Purpose Funds: An Application of Contingent Claims Analysis," Working Paper No. 782-75, Massachusetts Institute of Technology, Apri l , 1975. "Society of Actuaries' Textbook on Life Contingencies," The Society of Actuaries, Chicago 1967. "Projections of Variable Life Insurance Operations," Transactions Society of Actuaries, Vol. XXIII, 1971. "A Theory and a Econometric Model for Common Stock Purchase Warrants," Analytical Publishers Co., New York, 1965. "Warrant Price Behavior: 1945 to 1964," The Financial Analysts Journal, January-February, 1968. "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. and Schwab, B., [25] McCracken, D.D. and Dorn, W.S. [26] McGill, D.M., [27] McKean, H.P., [28] [29] [30] [31] [32] [33] [34] McKean, H.P., Merton, R.C., Merton, R.C., Merton, R.C., Merton, R.C., Merton, R.C., Merton, R.C., [35] Praetz, P.D., "An Empirical Examination of a New General Equilibrium Model for Warrant Pricing." Unpublished M.S. Thesis, M.I.T., September 1971. "Managerial Finance in a Canadian Setting." Holt, Rinehart and Winston of Canada, Limited, 1973. Numerical Methods and Fortram Programming. John Wiley & Sons, Inc., 1964. Lif e Insurance, Richard D. Irwin, Inc., 1967. "A Free Boundary Problem for the Heat Equation Arising From a Problem of Mathematical Economics," Industrial Management Review, Vol. 6., No. 2, Spring, 1965. Stochasic Integrals New York: Academic Press, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous Time Case," Review of Economics and Statistics, Vol. 51, No. 3 (August 1969). "Optimum Consumption and Portfolio Rules in a Continuous-Time Model," Journal of Economic Theory, Vol. 3, No. 4, December 1971. "An Intertemporal Capital Asset Pricing Model," Econometrica, Vol. 41, No. 5, (September 1973). "The Theory of Rational Option Pricing," Bell Journal of Economics and Management Science, Vol. 4, No. 1, Spring 1973. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, Vol. 29, May 1974. "Option Pricing when Underlying Stock Returns are Discontinuous," Working Paper No. 787-75, Massachusetts Institute of Technology, April 1975. "Random Walk and Investment Theory," Journal of the Institute of Actuaries Students' Society, Vol. 20, Part 3, June 197: 190 [36] Samuelson, P.A. , [37] Samuelson, P.A. and Merton, R.C. [38] Shelton, T .P . , [39] Sprenkle, CM. , [40] Squires, R.T. , [41] [42] Thorp, E .O. , [43] Thorp, E.O. and Kassouf, S.T. , [44] Turner, S.H., [45] Turner, S.H., [46] Van Home, T . C . , [47] Weston, T.F. and Brigham, E . F . , "Rational Theory of Warrant Pricing," Industrial Management Review, Vol. 6, No. 2, Spring 1965. "A Complete Model of Warrant Pricing that Maximizes Uti l i ty," Industrial Management Review, Vol. 10, Winter, 1969. "The Relation of the Price of a Warrant to the Price of Its Associated Stock," Financial Analysts Journal, May-June and July-August, 1967. "Warrant Prices as Indication of Expectation and Preferences," Yale Economic Essays, I (Fal l , 1961). "Unit-Linked Assurance: Observations and Propositions," Journal of the Institute of Actuaries, Vol. 101, Part 1, No. 416, 1974. Survey of Current Business: United States Department of Commerce. "Extensions of the Black-Scholes Option Model," 39th Session of the International Statistical Institute, Vienna, Austria (August 1973). Beat the Market: A Scientific Stock Market System, Random House, Inc. New York, 1967. "Asset Value Guarantees under Equity-Based Products," Transactions Society of Actuaries, Vol. XXI, 1969. "Equity-Based Life Insurance in the United Kingdom," Transactions Society of Actuaries, Vol. XXIII, 1971. Financial Management and Policy. Prentice-Hall Inc., Third Edition, 1974. Managerial Finance. Holt, Rinehart and Winston, Inc. Fourth Edition, 1972. 191 C C C APPENDIX A C C C COMPUTER PROGRAM NO. 1: OPTION PRICES ON STOCKS WITH CONTINUOUS C DIVIDENDS PAYMENTS C C DESCRIPTION OF PARAMETERS C C H--WARRANT PRICE C VAR—VARIANCE RATE FOR THE RETURN ON THE STOCK C RF—RISKLESS INTEREST RATE C D—DIVIDEND RATE C E--EXCERCICE PRICE C XH—STOCK PRICE- INCREMENT C XK—TIME INCREMENT C NS—NO. OF STOCK PRICES C NT--NO. OF TIMES C A,B,C-- COEFFFICENTS IN THE ORIGINAL EQUATIONS C AA,BB,CC, COEFICIENIS OF MODIFIED EQUATIONS C F--RHS ORIGINAL EQUATIONS C FF--RHS MODIFIED EQUATIONS C TK—VECTOR OF TIMES C 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 INITIALIZE BOUNDARY VALUES OF H C NS=NS+1 NT=NT+1 DO 1 J=1,NT 1 W (1, J) =0.0 CONTINUE DO 2 1=1,NS S=XH*FLOAT (1-1) IF(S-LE.E) H(I,1)=0.0 2 IF(S.GT.E) W(I,1)=S-E T=0.0 CONTINUE DO 9 J=2,NT C C SOLVE SIM EQNS FOR J C C J IS BASE PERIOD C 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*Y3*I*I C (I)=-Y1*I+Y2-Y3*I*I 6 F (I) =W (1+1, J-1) A(1)=0.0 1 9 2 A (NS1)=-1.0 B (NS1)=1.0 F (NS 1) =XH C C TRANSFORM EQUATIONS C CONTINUE DO 7 1=2,NS1 BB(I) =BB (1-1) *B (I) / A (I) - C C (1-1) CC (I) =BB(I-1) *C ( I ) / A (I) 7 F F ( I ) = B B ( I - 1 ) * F ( I ) / A ( I ) - F F ( I - 1 ) BB(1)=B(1) CC(1)=C(1) FF(1)=F(1) CONTINUE C C SOLVE FOR W C W (NS, J) =FF (NS1)/BB (NS1) DO 8 IX=1,NS2 I=NS-IX 8 W ( I ,J ) = {FF(I-1) - C C (1-1) *W (1+1, J) ) / B B (1-1) T=FLOAT(J-1) *XK 101 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 / ) / / / / ) 9 CONTINUE C C OUTPUT C OUTPUT STARTS WITB T=0.0 AND S=0.0 C WRITE (6,200) V A R , R F , D , E , X H , X K , N S , N T 200 FORMAT(1H1,SOX,•WARRANT PRICE H A T R I X ' / / / , 2 0 X , ' V A R I A N C E RATE=• , 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 / 4 2 0 X , ' E X E R C I S E P R I C E = » , F 1 0 . 5 / 220X, 'STOCK 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 PRICES=' , I4 /20X, 'NUMBER OF T I M E S = ' , I 4 / ) NFIRST=1 NLAST=20 204 I F (NLAST-GT.NT) NLAST=NT DO 205 J=NFIKST,NLAST TK (J) =XK*FLOAT (J-1) 205 CONTINUE WRITE (6, 201) (TK(J) ,J=NFIRST,NLAST) 201 F O R M A T ( / / / 1 1 X,20 F6 . 2 / / / ) DO 202 I=1,NS SI=XH*FLOAT (1-1) 202 WRITE (6,203) S I , (8(1,J) ,J=NFIRST,NLAST) 203 F O R H A T ( 1 X , F 6 . 2 , 4 X , 2 0 F 6 . 2 ) IF (NLAST. EQ. NT) GO TO 207 NFIRST=NFIRST+20 NLAST=NLAST+20 GO TO 204 207 STOP END SSIG c 193 C C APPENDIX B C c C COMPUTER PROGRAM NO. 2: OPTION PRICES ON STOCKS HITH DISCRETE C DIVIDEND PAYMENTS C C C C DESCRIPTION OF PARAMETERS C C W— WARRANT PRICE C VAR—VARIANCE RATE FOR THE RETURN ON THE STOCK C BF—RISKLESS INTEREST RATE C D—DIVIDEND RATE, MOST BE SET EQUAL TO 0.0 C . E—EXCERCICE PRICE C XH —STOCK PRICE INCREMENT C XK—TIME INCREMENT C NS—NO. OF STOCK PRICES C A , B , C — COEFFFICENTS IN THE ORIGINAL EQUATIONS C A A , B B , C C , COEFICIENTS OF MODIFIED EQUATIONS C F—RHS ORIGINAL EQUATIONS C FF—RHS MODIFIED EQUATIONS C DI—DESCRETE DIVIDEND C NT(K)- -VECTOR OF THE NO. OF TIME PERIODS CONSIDERED IN EACH I N -C TERVAL BETWEEN DIVIDENDS C NK—NO. OF INTERVALS TO BE CONSIDERED C SC—CRITICAL STOCK PRICE, FOR VALUES OF STOCK PRICE GREATER OF C EQUAL TO SC THE OPTION SHOULD BE EXERCICED DIVIDEND-ON C P—MINHIMON PREMIUM TO EXCERCISE, USED TO DETERMINE SC C DIMENSION W (400, 20) , A (400) ,B (400) , C (400) ,AA(400) ,BB(400) , 1CC(400) ,F(4Q0) ,FF(400) , 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 W R I T E ( 6 , 1 0 0 ) V A R , R F , D , E , X H , X K , N S 100 FORMAT ( 4 F 1 0 . 5 / 2 F 5 . 2, 214) READ (5,110) NK, (NT (K) ,K=1,NK) 110- FORMAT(I4/10I4/10I4) WRITE (6, 110) NK, (NT (K) ,K=1,NK) NK1=NK-1 READ (5,111) DI WRITE(6,111)DI 111 FORMAT(F10.5) READ (5, 120) P HRITE (6,120)P 120 FORMAT(F8.3) W R I T E ( 6 , 2 0 0 ) V A R , R F , D , E , X H , X K , N S , N K HRITE(6,201) (NT (K) ,K=1,NK) WRITE(6,2Q2) DI WRITE (6,203) P 203 FORMAT(20X,* MIN PREMIUM TO EXERCISE* ,F10 .5 ) 201 FORMAT(20X,* TIME PERIODS PER I N T E R V A L = » , 2 0 1 4 ) 202 FORMAT(20X,'DESCRETE DIVIDEND= * ,F10 .5 ) 200 FORMAT(1H1,40X, f OPTION PRICES: DESCRETE DIVIDENDS C A S E » / / / / 2 0 X , 1' VARIANCE RATE= * ,.P10. 5 /2OX, * RISKLESS RATE= • ,F10.5 /20 .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 , 1 9 4 3»ST0CK PRICE INCREMENT^,F10.5/20X, fTIME INCREMENT =•,Fl0.5/20X, 4'NUMBER OF PRIC£S=•,I4/20X,»NUMBER OF TIME INTERVALS=•,14) C C FULL OUTPUT — IPRINT=1 C SUMMARIZED OUTPUT — IPRINT=0 ! C IPRINT=0 C C INITIALIZE BOUNDARY VALUES OF 8 C NS=NS+1 NT 1=NT (1) +NT (2) DO 1 J=1,NT1 1 H (1,J) =0.0 CONTINUE DO 2 1=1,NS ST (I) =XB*FLOAT (1-1) IF (ST (I) .LE.E) 9(1,1) =0.0 2 I F (ST (I) .GT.E) «(I, 1)=ST(I)-E T=0-0 IF(IPRINT.EQ. 1) WRITE (6,300) T, (H ( I , 1) ,1=1, NS) 300 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 C C SOLVE SIM EQNS FOR J C C J IS BASE PERIOD C ¥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) =Y1*I-Y2-Y3*I*I B(I)=1.0+2.0*Y1+2.0*Y3*I*I C{I)=-Y1*I+¥2-Y3*I*I 6 F <I)=W (1+1,J-1) A (1) =0.0 A (NS1)=-1.0 B(NS1) = 1.0 F(NS1)=XH C C . TRANSFORM EQUATIONS C CONTINUE DO 21 I=2,NS1 IF(ABS (A (I) ) .LT. 0.0001) GO TO 20 BB (I) =BB (1-1) *B (I)/A (I)-CC (1-1) CC (I) =BB ( I - 1) *C (I) /A (I) 7 F F ( I ) = B B ( I - 1 ) * F ( I ) / A ( I ) - F F ( I - 1 ) IF(ABS (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 20 IF(IPBINT.EQ. 1) WRITE(6,60) I,A(I) ,B(I) ,C(I) ,F(I) 195 60 FORMAT(/ 1 X , ' 1 = • , 1 4 , 1 X , * A = • , F 1 5 . 6 , I X , ' B = ' , F 1 5 . 6 , 1 X , • C = • , 1 F 1 5 . 6 , 1 X , • F = « / F 1 5 . 6 ) BB(I)=B(I) CC(I)=C(I) FF(I ) =F(I) 21 CONTINUE BB(1)=B(1) CC(1)=C(1) FF(1)=F(1) CONTINUE C C SOLVE FOR W C W (NS, J) =FF (NS1)/BB (NS1) DO 8 IX=1,NS2 I=NS-IX 8 W(I,J) = ( F F ( I - 1 ) - C C ( I - 1 ) * W ( I + 1 , J ) ) / B B ( I - 1 ) I F ( K . E Q . 1 ) G O TO 31 T=FLOAT ( J - 1 + NTK) *XK GO TO 30 31 T=FLOAT (J-1) *XK NTK=0 30 CONTINUE I F ( IPRINT.EQ. 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 700 F O R M A T ( / / 5 X , • T I M E TO EXPIRATION=' ,F10 .5 / ) TD(K) =T WRITE(6,702) (ST (I) , I=1,NS,20) 702 F O R M A T ( / I X , * S P E X - D I V . » , 2 0 F 6 . 2 ) WRITE (6,703) (W ( I , J ) , I=1,NS,20) 703 FORMAT(/1X, •OP E X - D I V 2 0 F 6 . 2 / ) 9 CONTINUE NTK=NTK+NT(K) I F ( K . EQ.NK) GO TO 16 C C NEW BOUNDARY VALUES OF W C NXH=DI/XH DO 10 1=1,NS II=I-NXH I F ( I I . L T . 1 ) 1 1 = 1 10 WD (I) =W ( I I , NTKK) - W (I , 1) C C CRITICAL STOCK PRICE WITH PREMIUM P C IE=E/XH I F (WD (IE) . L E . P) GO TO 19 DO 17 I=IE,NS I F (WD (I) . GT. 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 17 CONTINUE SC(K)=200. GO TO 18 19 SC(K)=E 18 CONTINUE DO 99 1=1,NS II=I-NXH I F ( I I . L T . 1)11=1 IF (WD (I) . GE.O.O) W (I ,1)=W(II,NTKK) 99 IF(WD(I) . L T . O . O ) W (I , 1)=W (1 , 1 ) 196 I F ( IPRINT.EQ. 1) WRITE (6,300) T , ( i ( I , 1) ,1=1, SS) WRITE (6, 705) (ST (I) , I=1,NS,20) 705 FORMAT ( /1X ,*SP C U M - D I V » , 2 0 F 6 . 2 ) WRITE(6,704) (W ( I , 1) ,1=1,NS, 20) 704 FORMAT ( / I X , * OP C U M - D I V , 20F6. 2) WRITE(6, 1 16) SC(K) 116 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 / / ) 12 CONTINUE 16 CONTINUE WRITE(6,900) (TD (K) ,K=1,NK1) 900 F O R M A T ( / / 1 X , ' T TO EX' ,20F6.2) WRITE (6,901) (SC(K) ,K=1,NK1) 901 . FORMAT(/1X, 'CRIT S P • , 2 0 F 6 . 2 / / ) C C PRINT A PLOT C DO 70 J=2,19 70 TDA (J) =TD (J-1). TDA(1)=0. DO 71 J=2,19 71 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=2,19 72 CALL SYMBOL (TDA (J) ,SCA (J) , 0 . 1 4 , 4 , 0 . , - 1 ) CALL PLOTND STOP END SSIG 197 APPENDIX C: Solution Algorithm: Option Prices in the Constant Continuous Premium Contract.  The partial differential equation governing the value of the call option in the constant continuous premium contract was derived in Chapter 5. The equation obtained was: (C.l) ho2x2 w + (rx+D) w - wr - w = 0 X X X -L subject to the boundary conditions (C.2) w(x,0,g) = Max[x-g, 0] (C.3) Dwx(0,T,g) - w(0,T,g)r - wT(0,T,g) = 0 (C.4) w x O ,T ,g ) = 1 In finite difference form, partial differential equation (C.l) can be written as: (C.5) a. w. , .'+ D, w. . + c. w.,, = w. . . x l - l , j i i , j l l+l,j i , J - l where (C.6) a. = %rki + JgDk/h - %a 2ki 2 I (C.7) b ± = (1+rk) + a 2 k i 2 (C.8) c± = -hrki - %Dk/h - %a 2ki 2 (C.9) w(x , T ) = w(ih, jk) = w . , Boundary conditions (C.2), (C.3) and (C.4) can be expressed in finite difference form, respectively, as: /w. n = 0 for 0 < i < g/h (C.10)j 1 , u 0 = ih-g for g/h < i < n 198 , i n = Dk h  v u . i i ; w Q^ D k + h k r + h w x j + D k + h k r + h w o , j - i for a l l j (C.12) -w . . + w . = h for a l l j n-1,3 n,j Difference equation (C.5) can be used to generate a system of linear equations by making i = 1, ... n-1. The last equation (for i = n) is obtained from boundary condition (C.12) a,wA . + b,w. . + c,w0 . 1 0,3 1 1,3 1 2,j = w 1 » J L a„w. . + b0wo . + c„w, . 2 1,3 2 2,j 2 3,3 = u 2 , j - l (C.13) a , w „ . + b . w . . + c . w . = w , . , n-1 n-2,3 n-1 n-1,3 n-1 n,3 n-1,3-1 - w . . + w . = h n-1,3 n,3 The system of n linear equations (C.13) has (n+1) unknowns (w, . for »3 i = 0, ..., n). But the value of W Q ^  can be obtained from boundary condition (C.11). The system of equation (C.13) can then be written as: Dka. ( b i + ™ — _ • _ u i—";—c) w i + c i W 0 = f, 1 Dk + hkr + h 1 1 2 1 &2™i + D 2 W 2 + °2 W3 = 2^ (C.14) a , w - + b , w . + c , w = f , n-1 n-2 n-1 n-1 n-1 n n-1 a w , + b w r n n-1 n n = f n 199 where h a^ f l = W l , j - 1 " Dk + hkr + h w0,j-1 f. = w. . , for i = 2, .... n-1 f = h n a = -1 n b n = 1 and where the subscripts j have been omitted for simplicity. The matrix of coefficients of the system (C.14) i s tridiagonal, so Gauss' elimination can be employed to solve for the values of w^(i = 1, . .., n). The values of w. . can be obtained starting from the values of w. . ,. The f i r s t line of the grid, w. n , is computed by using boundary condition (C.10). The computer program shown in Appendix D is based on this solution algorithm. 200 c C APPENDIX D C C C COMPUTER PROGRAM N O . 3 : OPTION P R I C E S IN THE CONSTANT CONTINUOUS C PREMIUM CONTRACT C C C D E S C R I P T I O N OF PARAMETERS C C H — O P T I O N VALUE C V A R — V A R I A N C E R A T E FOR THE RETURN OF THE R E F E R E N C E P O R T F O L I O C R F — R I S K L E S S I N T E R E S T R A T E C . D—CONTINUOUS PREMIUM R A T E ( W I T H - SIGN) C X H — R P P R I C E INCREMENT C X K — T I M E INCREMENT C N S — N O . OF RP P R I C E S C N T — N O . OF TIMES DIMENSION H (400 , 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 ) R E A D ( 5 , 1 0 0 ) V A R , R F , D , £ , X H , X K , N S , N T 100 F O R M A T ( 4 F 1 0 . 5 / 2 F 1 0 . 5 , 2 1 4 ) C I N I T I A L I Z E BOUNDARY VALUES OF H C NS=NS+1 NT=NT+1 DO 2 1=1,NS S=XH*FLOAT (1 -1) I F ( S . L E . E ) B (1 , 1 ) = 0 . 0 2 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 C SOLVE SIM EQNS FOR J C Y 1 = 0 . 5 * R F * X K Y 2 = 0 . 5 * D * X K / X H Y 3 = 0 . 5 * V A R * X K NS2=NS-2 N S 1 - N S - 1 DO 6 1=1 , NS2 A ( I ) = Y 1 * I - Y 2 - Y 3 * I * I B ( I ) = 1 . 0 + 2 . 0 * Y 1 + 2 . 0 * Y 3 * I * I C ( I ) = - Y 1 * I + Y 2 - Y 3 * I * I 6 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 ( N S 1 ) = - 1 . 0 B (NS1) = 1.0 F ( N S 1 ) = X H C TRANSFORM EQUATIONS B B ( 1 ) = B ( 1 ) CC (1) =C(1) F F ( 1 ) =F(1) DO 7 1=2,NS1 BB (I) =BB (1 -1 ) * B (I) / A (I) - C C (1 -1 ) CC (I) =BB (1 -1) * C (I) / A (I) 7 F F ( I ) = B B ( I - 1 ) *F ( I ) / A ( I ) - F F (1-1) C SOLVE FOR W W (NS, J) =FF (NS1) /BB (NS1) 201 DO 8 IX=1,NS2 I=NS-IX 8 W ( I , J) = (FF (1-1) -CC (1-1) *W (1+1, J) ) /BB (1-1) W ( 1 , J)=-W ( 2 , J ) *D*XK/DIV+XH*W(1,J-1)/DIV T=FLOAT(J-1)*XK 9 CONTINDE C OUTPUT- , STARTS WITH T=0.0 AND RP=0.0 C WRITE(6,200)VAR,RF,D,E,XH,XK,NS,NT 200 FORMAT(1H1,30X,*VALUE OF AN EQUITY LINKED OPTION WITH CONTINUOUS 1PREHIUH PAYMENTSV//20X,•VARIANCE RATE=',F10.5/20X,»RISKLESS RATE 2=',F10.5/20X,'CONTINUOUS PREMIUM BATE=',F10.5/20X,* GUARANTIED AMT 2='/F10.5/20X,'REFERENCE PORTFOLIO PRICE INCREMENT= • , F1 0.5/20X,. 3'TIME INCREMENT=',F10.5/20X,'NO. OF RP PRICES=',I4/20X,'NO. OF TIM 4ES=',IU///) WRITE ( 6 , 101) T, (W (I,NT) ,1=1, NS) 101 FORMAT(5X,•TIME 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 $SI<3 202 C c c C APPENDIX E C c C COMPUTER PROGRAM NO. 4 : OPTION PRICES IN THE PERIODIC PREMIUM C CONTRACT C C C C DESCRIPTION OF PARAMETERS C C H —OPTION VALUE C VAR—VARIANCE RATE FOR THE RETURN ON THE REFERENCE PORTFOLIO C RF—RISKLESS RATE C X H — REFERENCE PORTFOLIO PRICE INCREMENT C XK—TIME INCREMENT C NS—NO. OF REFERENCE PORTFOLIO PRICES C D—PREMIUM INVESTED PERIODICALY ON THE REFERENCE PORTFOLIO C E—GUARANTIED AMOUNT C ND—NO. OF PREMIUM PAYMENTS C TD—TIME BETWEEN PREMIUM PAYMENTS C NT—NO. OF TIME PERIODS BETWEEN PREMIUM PAYMENTS C DIMENSION W (400,50) ,A(400) ,B (400) , C (400) , AA (400) ,BB (400) , ICC (400) , F (400) , FF (400) , S T (400) R E A D ( 5 , 1 0 0 ) V A R , R F , E , D , X H , N S , N D , T D , N T 100 F O R M A T ( 3 F 1 0 . 5 / 2 F 1 0 . 5 , I 5 / I 5 , F 1 0 . 5 , I 5 ) XK=TD/FLOAT(NT) WRITE ( 6 , 1 0 1 ) V A R , R F , E , D , X H , X K , N S , N D , T D , N T 101 FORMAT(1H1,30X,•VALUE OF AN EQUITY LINKED OPTION WITH P E R I O D I C , • 1PREMIUH PAYMENTS«/ / / 20X,«VARIANCE R A T E = ' , F 1 0 . 5 / 2 0 X , * RISKLESS' , • 2RATE=* ,F10 .5 /20X,* GUARANTIED A M O U N T 3 ' , 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 * , 4 F 1 0 . 5 / 2 Q X , • T I M E I N C R E M E N T ^ , F 1 0 . 5 / 2 0 X , • N U M B E R OF REFERENCE' , • 5PORTFOLIO PRICES= * , I5 /20X, 'NUMBER OF PREMIUM P A Y M E N T S 3 • , 1 5 / 62OX, 'TIME BETWEEN PREMIUM P A Y M E N T S = ' , F 1 0 . 5 / 2 0 X , • N O . OF TIME' , » 7PERIODS BETWEEN PREMIUM P A Y M E N T S 3 « , 1 5 / / / ) C C FULL OUTPUT — IPRINT=1 C INTERMEDIATE OUTPUT—IPRINT=2 C SUMMARIZED OUTPUT — IPRINT=0 C IPRINT=0 C C , I N I T I A L I Z E BOUNDARY VALUES OF W C NS=NS+1 DO 2 1=1,NS ST(I)=XH*FLOAT(I-1) IF (ST (I) . L E . E ) W ( I , 1) =0.0 2 I F (ST (I) . G T . E ) W ( I , 1) = S T ( I ) - E T=0.0 I F ( IPRINT.EQ. 1) WRITE(6 ,300)T, (W ( I , 1) , 1=1 , NS) 300 F O R M A T ( / / 2 0 X , ' T I M E TO E X P I R A T I O N 3 ' , F 1 0 . 5 / / 2 0 ( 5 X , 2 0 F 6 . 2 / ) ) NXH=D/XH NT1=NT+1 DO 12 K=1,ND DO 9 J=2,NT1 C C C C C J IS BASE PERIOD SOLVE SIM EQNS FOR J 203 W (1,J)=W (1, J-1) *EXP(-RF*XK) 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 6 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 C 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) 7 F F ( I ) = B B ( I - 1 ) * F ( I ) / A ( I ) - F F ( I - 1 ) I F ( A B S ( B B ( I ) ) . L T . 1 0 0 0 0 . ) G O TO 21 BB(I) =0.0001*BB(I) CC (I) =0. 000 1*CC (I) FF(I )=0 .0001*FF(I ) GO TO 21 20 I F ( IPRINT.EQ. 1) WRITE (6,60) I , A (I) , B (I) , C (I) , F (I) 60 FORMAT (/ 1 X , M = « ,14, IX, • A= • , F15. 6, 1X, • B=« , F15 . 6, 1X, « C= • , 1 F 1 5 . 6 , I X , « F = ' , F 1 5 . 6 ) BB(I)=B(I) CC(I)=C(I) FF(I ) =F(I) 21 CONTINUE BB(1)=B(1) CC(1).=C(1) FF(1) =F(1) CONTINUE W (NS, J )=FF(NS1) /BB (NS1) DO 8 IX=1,NS2 I=NS-IX 8 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 31 T=FLOAT(J-1)*XK NTK=0 30 CONTINUE I F ( IPRINT.EQ. 1) WRITE (6,300) T , (W ( I ,J ) ,1=1, NS) I F ( J . L T . NT1) GO TO 9 I F (IPRINT. EQ. 2) WRITE (6 ,700) T C C TRANSFORM EQUATIONS C c c SOLVE FOR W 700; FORMAT ( / / 5 X , • TIME TO EXPIRATION=' , F1 0. 5/) I F (IPRINT. EQ. 2) WRITE (6,702) (ST (I) , 1=1 , NS, 20) 204 702 FORMAT(/1X , *RP A - P R E M . ' , 2 0 F 6 - 2 ) IF ( IPRINT.EQ. 2) WRITE (6,703) (8 ( I , J ) ,1 = 1,NS,20) 703 F O R M A T ( / 1 X , ' O P A - P R E M . • , 2 0 F 6 . 2 / ) 9 CONTINUE NTK=NTK+ NT C C NEW BOUNDARY VALUES OF W C 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 ( IPRINT.EQ. 1) WRITE (6,300) T , (W ( I , 1).,1=1,US) I F ( IPRINT.EQ. 2) WRITE(6,705) (ST (I) , I=1,NS,20) 705 F O R M A T ( / I X , • RP B-PREM.* ,20F6 .2 ) I F ( IPRINT.EQ. 2) WHITE (6,704) (W (1,-1). ,1=1, HS, 20) 704 FORMAT(/1X,* OP B-PREM' ,20F6 .2 ) 12 CONTINUE I F ( I P R I N T . E Q . 2 ) WRITE(6 ,300)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.+AN) 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 FORMAT( / /40X,* R E S U L T S ' / / / 2 0 X , ' V A L U E OF THE CALL OPTION=*,F10 .5 / 120X, 'PV OF GUARANTIED AMOUNT=«,FTQ.5 /20X,»TOTAL 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 'VALUE 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 5 C A L L = ' , F 1 0 . 5 / / / / ) STOP END SSIG c C - 205 C APPENDIX F C C C COMPUTER PROGRAM NO. 5: COMPUTATION OF THE EQUILIBRIUM VALUE C OF THE GUARANTEE WITH KNOWN DATE OF C EXPIRATION C C C DESCRIPTION OF PARAMETERS C C W—OPTION VALUE C VAR—VARIANCE RATE FOR THE RETURN ON THE REFERENCE PORTFOLIO C R F — R I S K L E S S RATE C X H — REFERENCE PORTFOLIO PRICE INCREMENT C X K — T I M E INCREMENT C NS—NO. OF REFERENCE PORTFOLIO PRICES C D—PREMIUM INVESTED PERIODICALY ON THE REFERENCE PORTFOLIO C E—GUARANTIED AMOUNT C ND--NO. OF PREMIUM PAYMENTS C T D — T I M E BETWEEN PREMIUM PAYMENTS C NT—NO. OF TIME PERIODS BETWEEN PREMIUM PAYMENTS C DIMENSION W (400,50) ,A (400) , B (400) ,C (400) , AA (400) , BB (400) , 1CC (400) , F (400) , FF (400) ,ST (400) 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 FORMAT(3F10.5,I5,F10.5,I5) XK=TD/FLOAT(NT) WRITE (6,101)VAR,RF,D,XK,NS,TD,NT 101 FORMAT(1H1,30X,* VALUE OF AN EQUITY LINKED OPTION WITH PERIODIC•,* 1PREMIUM PAYMENTS *///20X,* VARIANCE RATE=*,F10.5/20X,* RISKLESS * , « 2RATE=',F10.5/20X, * 'PREMIUM (INVESTED 31N E Q U I T Y = » , . «rF10.5/20X,'TIHE INCREMENT=', F10. 5/20X, » NUMBER OF REFERENCE' , ' 5PORTFOLIO PRICES=',I5/ 620X,'TIME BETWEEN PREMIUM PAYMENTS=*,F10.5/20X,•NO. OF-TIME' ,» 7PERIODS BETWEEN PREMIUM PAYMENTS 3',15///) NS=NS+1 WRITE(6,301) 301 FORMAT(///3X,* NO. ,YEARS*,4X,'VALUE CALL* r3X,'PV GUARANT.',2X, 1 'TOTAL PV',5X,'PV PORTFOLIO* , IX, 'VALUE PUT ' ,.4X , ' TOTAL PREM',3X, 2* A PREH PUT*,3X,'A PREM CALL'///) DO 1000 ND=1,20 E=0.75*ND*D NXH= 125./FLOAT (ND) XH=D/FLOAT(NXH) C C I N I T I A L I Z E BOUNDARY VALUES OF W C DO 2 1=1,NS ST(I)=XH*FLOAT(I-1) I F (ST (I) .LE.E) W ( I , 1) =0.0 2 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 IS 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*Y3*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 IF(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 (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) F F ( 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) /BB (NS 1) DO 8 IX=1,NS2 I=NS-IX / 8 ( I , J ) = (FF (1-1) - C C (I-1)*W(I+1,J) ) /BB (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 ( 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. IF(ND.EQ.1)GO 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) ,PV (ND) ,PVRP(ND) ,PUT(ND) ,P(ND) , 1APP(ND) ,APC (ND) 302 F O R M A T ( 5 X , I 5 , 2 X , 8 ( 3 X , F 1 0 . 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(30) 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=1,61) FORMAT(10 (6F12.2/) ,F12 .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) ) /XLX(NX) NXTL=NX+NL-1 A (NL) =XLX (NXTL)/XLX (NX) CHEK THAT PROBABILITIES SUM TO 1.0 SUM=0.0 DO 10 1=1,NL SUM=SUM + A(I) WRITE (6, 11) SUM FORMAT ( IX,F10 .5) CALCULATES EXPECTED VALUES P=0.0 V=0.0 DO 5 J=1,NL 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 ASSET 1 VALUE G U A R A N T E E V / / 2 0 X , 'AGE OF PURCHASER 3» , 2 X , I 3 / / 2 0 X , 2'TERM OF THE CONTRACT= • , 2X, 1 3 / / / 5 X , ' TIME ' , 5 X , ' TOTAL PV » , 5X, 3'VALUE P U T ' , 5 X , ' A L P H A ' / ) DO 7 1=1,NL 7 WRITE(6,8) I , PV (I) , APP (I) , A (I) 8 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 9 FORMAT( / /20X,* EXPECTED VALUE OF THE ANNUAL PREMIUM ON THE PUT=' , 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 $SIG 2 1 0 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, wi 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 in i t ia l ly satisfied certain criteria of insurability. It follows that the mortality of such a group wi 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 x +^) between the ages of 20 and 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) ,XLX (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/) ,F12.2) 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) -XLX (NXT) ) / X L X (NX) NXTL=NX+NL-1 A (NL) =XLX (NXTL) / X L X (NX) P=0.0 DO 5 J=1,NL 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 TABLE: CA 1 9 5 8 - 6 4 ' / 8 X , ' R I S K FREE RATE: 2 0 4 ' / 8 X , • VARIANCE RATE: 0.01846 (TSE) ' / / / ) WRITE (6,6)NAGE FORMAT( 5X, •RISK PREMIUM VS TERM. OF CONTRACT'//8 X, 1'AGE OF PURCHASER= * , 2 X , I 3 / / / 5 X , ' T E R M OF CONTRACT' ,SX, 2'RISK PREMIUM'/ / ) DO 7 NL=1,20 WRITE (6,8) NL,RP(NL) FORMAT (1 OX, 13 ,15X,F1 0.5) CALL S C A L E ( T , 2 0 , 1 0 . , X M I N , D X , 1 ) CALL S C A L E ( B P , 2 0 , 1 0 . f 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 P R E M I U M ' , 1 2 , 1 0 . , 9 0 Y M I N , D Y ) DO 31 1=1,20 CALL SYMBOL (T (I) ,RP(I) , 0 . 1 4 , 4 , 0 . , -1) CALL PLOTND STOP END 212 *SIG c 213 C APPENDIX J C C C COMPUTER PROGRAM NO. 8: RISK PREMIUM VERSUS AGE OF PURCHASER C AT ENTRY C C DIMENSION APP (30) , XLX (61) , A (30) , RP (31) , AGE (31) READ (2,1) (APP (I) ,1 = 1,20) 1 FORMAT (F10.5) READ (3, 2) (XLX(J) ,J=1,61) 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)= (XLX(NXT1)-XLX(NXT)) /XLX(NX) NXTL=NX+NL-1 A (NL) =XLX (NXTL) / X L X (NX) P=0.0 DO 5 J=1,NL 5 P=P+A (J) *APP(J) RP(NX) =P AGE (NX) =FL0AT (NAGE) 200 CONTINUE WRITE (6,25) 25 FORMAT (1H1,7X,•GUARANTEE: 20.00 (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 28* /8X, * VARIANCE RATE: 0.01846 ( T S E ) * / / / ) 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 PURCHASER•,5X,* RISK PREMIUM*//) DO 7 J=1,31 7 WRITE (6, 8) AGE (J) ,RP (J) 8 FORMAT(10X,F3 .0 ,12X,F10 .5 ) CALL S C A L E ( A G E , 3 1 , 1 0 . , X M I N , D X , 1 ) CALL S C A L E ( R P , 3 1 , 1 0 . , Y M I N , D Y , 1 ) CALL AXIS (0. , 0 . , * AGE OF PURCHASER • , -16 , 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) ,RP(I) , 0. 14, 4, 0. , -1) CALL PLOTND STOP END $SIG c C 214 C A P P E N J D I X K c c C C O M P O T E S P R O G R A M N O . : 9 : I N V E S T M E N T P O L I C Y C C C C D E S C R I P T I O N O F P A R A M E T E R S C C W — O P T I O N V A L U E C V A R — V A R I A N C E R A T E F O R T H E R E T U R N O N T H E R E F E R E N C E P O R T F O L I O C R F — R I S K L E S S R A T E C X H — R E F E R E N C E P O R T F O L I O P R I C E I N C R E M E N T C X K — T I M E I N C R E M E N T C N S — N O . O F R E F E R E N C E P O R T F O L I O P R I C E S C 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 O N T H E R E F E R E N C E P O R T F O L I O C E — G U A R A N T I E D A M O U N T C N D - - N O . O F P R E M I U M P A Y M E N T S C T D - - T I M E B E T W E E N P R E M I U M P A Y M E N T S C N T — N O . O F T I M E P E R I O D S B E T W E E N P R E M I U M P A Y M E N T S C D I M E N S I O N W ( 4 0 0 , 5 0 ) , A ( 4 0 0 ) , B ( 4 0 0 ) , C ( 4 0 0 ) , A A ( 4 0 0 ) , B B ( 4 0 0 ) , 1 C C ( 4 0 0 ) , F ( 4 0 0 ) , F F ( 4 0 0 ) , S T ( 4 0 0 ) D I M E N S I O N G A ( 3 0 ) , P V ( 3 0 ) , P V R P ( 3 0 ) , P U T ( 3 0 ) , P ( 3 0 ) , A P P ( 3 0 ) , 1 A P C ( 3 0 ) , C A ( 3 0 ) D I M E N S I O N WY ( 1 0 , 4 0 0 , 1 0 ) , E W Y ( 4 0 0 , 1 0 ) , W X ( 4 0 0 , 1 0 ) , X W X ( 4 0 0 , 1 0 ) , 1 C A S H ( 4 0 0 , 1 0 ) , B A R ( 4 0 0 , 1 0 ) D I M E N S I O N A L ( 1 0 , 1 0 ) D I M E N S I O N X L X ( 6 1 ) R E A D ( 5 , 1 0 0 ) V A R , R F , D , N S , T D , N T T O O F O R M A T ( 3 F 1 0 - 5 , 1 5 , F 1 0 . 5 , 1 5 ) R E A D ( 5 , 4 0 0 ) N A G E , N C 4 0 0 F O R M A T ( 2 1 3 ) R E A D ( 2 , 4 0 2 ) ( X L X ( J ) , J = 1 , 6 1 ) 4 0 2 F O R M A T ( 1 0 ( 6 F 1 2 . 2 / ) , F 1 2 - 2 ) X K = T D / F L O A T ( N T ) W R I T E ( 6 , 1 0 1 ) V A R , R F , D , X K , N S , T D , N T 1 0 1 F O R M A T ( 1 H 1 , 3 0 X , » V A L U E O F A N E Q U I T Y L I N K E D O P T I O N W I T H P E R I O D I C ' , • 1 P R E M I 0 H P A Y B E N T S ' / / / 2 0 X , * V A R I A N C E R A T E 3 ' , F 1 0 . S / 2 0 X , ' R I S K L E S S » , » 2 R A T E = ' , F 1 0 . 5 / 2 0 X , • P R E M I U M ( I N V E S T E D 3 I N E Q U I T Y 3 ' , 4 F 1 0 . 5 / 2 0 X , ' T I M E I N C R E M E N T 3 ' , F 1 0 . 5 / 2 0 X , * N U M B E R O F R E F E R E N C E ' , » 5 P O R T F O L I O P R I C E S 3 ' , 1 5 / 6 2 0 X , ' T I M E B E T W E E N P R E M I U M P A Y M E N T S 3 ' , F 1 0 . 5 / 2 O X , * N O . O F T I M E ' , « 7 P E R I 0 D S B E T W E E N P R E M I U M P A Y M E N T S 3 ' , 1 5 / / / ) W R I T E ( 6 , 4 0 1 ) N A G E , N C 4 0 1 F O R M A T ( 2 0 X , ' A G E O F P U R C H A S E R A T E N T R Y 3 ' , I 3 / 2 0 X , ' T E E S O F C O N T R A C T 3 T , I 3 / / ) N S = N S + 1 W R I T E ( 6 , 3 0 1 ) 3 0 1 F O R M A T ( / / / 3 X , ' N O . Y E A R S ' , 4 X , ' V A L U E C A L L ' , 3 X , ' P V G U A R A N T . • , 2 X , 1 ' T O T A L P V , 5 X , » P V P O R T F O L I O ' , 1 X , • V A L U E P U T • , 4 X , • T O T A L P R E M ' , 3 X , 2 ' A P R E M P U T ' , 3 X , ' A P R E M C A L L * / / / ) D O 1 0 0 0 N D = 1 , N C E = N D * D X H 3 0 . 0 5 N X H = 2 0 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) Y 1 = 0 . 5 * R F « X K Y3=0.5*VAR*XK NS1=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 (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 ) FF(I )=BB(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) =0.0001*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 W 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 ) I F ( K . E Q . 1 ) G O TO 31 T=FLOAT (J-1+NTK)*XK GO TO 30 31 T=FLOAT(J-1) *XK 216 NIK=0 30 CONTINUE I F ( J . L T . N T 1 ) G O TO 9 9 CONTINUE NTK=NTK+NT DO 66 1=1,400 66 BY (ND,I,K)=W (I,NT1) C C NEW BOUNDARY VALUES OF W C NSD=NS-NXH DO 10 1=1,NSD 11=1+NXH 10 W ( I , 1)=8 (II ,NT1) NSD1=NSD+1 DO 11 I=NSD1,NS 11=1-1 11 W(I,1)=W(II,1)+XB Q 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 (ND. EQ. 1) GO TO 34 DO 33 1=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) ,PV(ND) ,PVRP(ND) ,PUT(ND) ,P(ND) , 1 APP (ND) ,APC (ND) 302 F O R M A T ( 5 X , I 5 , 2 X , 8 ( 3 X , F 1 0 . 5 ) ) 1000 CONTINUE 92 F O R M A T ( 5 X , F 7 . 3 , 5 X , 1 0 ( F 7 . 3 , 2 X ) ) C COMPUTATION OF THE ALPHA MATRIX NX=NAGE-19 NC1-NC-1 DO 70 1=1,NC1 11=1-1 DO 71 J=I,NC1 71 AL ( I , J) = (XLX (NX+J-1) -XLX (NX+J) ) / X L X (NX+II) 70 AL(I ,NC)=XLX(NX + NC-1)/XLX(NX+II) AL(NC,NC)=1. WRITE (6,74) 74 F O R M A T ( / / / / 3 X , ' Y E A R * , 2 0 X , ' A L P H A M A T R I X ' / / / ) DO 72 1=1,NC 72 WRITE (6,73) I , (AL ( I , J ) ,J=I ,NC) 73 F O R M A T ( 5 X , I 2 , 5 X , 1 0 ( F 7 . 5 , 2 X ) ) DO 88 J=1,NC DO 88 1=1,400 = SUfl=0.0 DO 80 K=J,NC 80 SUM=SUM+AL(J,K) *WY ( K , I , K - J + 1 ) EWY(I,J)=SUM 88 CONTINUE 1 1 WRITE (6,81). 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 82 WRITE (6,92) ST (I) , (EWY ( I , J) , J=1,NC) COMPUTATION OF MATRIX OF PARTIAL DERIVATIVES WX DO 83 J=1,NC DO 83 1=2,399 WX(I,J) = (EWY(I+1 ,J ) -EWY(I -1 ,J ) ) , / (2 .*XH) 83 XWX ( I , J) =ST(I) *WX(I ,J) WRITE (6,84) 84 F O R M A T ( / / / 3 0 X , ' P A R T I A L DERIVATIVE WX MATRIX(WEIGHTED)• / / / ) DO 85 1=2,399 85 WRITE (6,92) ST (I) , (WX ( I , J ) ,J=1,NC) WRITE(6,86) 86 F O R M A T ( / / / 3 0 X , * LONG POSITION IN THE REFERENCE P O R T F O L I O ' / / / ) DO 87 1=2,399 87 WRITE (6,92) ST ( I ) , (XWX ( I , J) , J=1,NC) COMPUTATION OF RISK PREMIUM YP=0.0 DO 200 ND=1,NC 200 YP=YP+AL (1,ND) *APP(ND) WRITE (6,201)YP 201 F O R M A T ( / / / 2 0 X , ' R I S K P R E M I U M 3 * , F 1 0 » 5 / / / ) C COMPUTAION OF CASH POSITION DO 202 J=1,NC SUM1-0.0 S0M3=0.0 SUM=0-0 DO 203 K=J,NC 203 S0M1=SUM1 + AL (J ,K) *D*FLOAT (K) *EXP (- ( K - J + 1).*RF) DO 204 K=J,NC S0M2=0-0 K1=K-1 I F ( J . E Q . N C ) GO TO 215 DO 205 N=J,K1 205 SUM2=SUM2*EXP (- (N-J+1) *RF) 204 SUM3=SUM3+AL(J,K)*SUM2 215 CONTINUE S0M=SUM1-(D+YP)*SUM3 DO 206 1=2,399 206 CASH ( I , J) =EW¥ ( I , J)+SUM 202 CONTINUE WRITE (6,207) 207 F O R M A T ( / / / 4 0 X , ' C A S H P O S I T I O N * / / / ) DO 208 1=2,399 208 WRITE (6, 92) ST (I) , (CASH ( I , J) ,J=1,NC) C C NET BORROWING C DO 209 1=2,399 DO 209 J=1,NC 209 BAR ( I ,J ) =XWX (I , J ) - C A S H ( I , J) WRITE (6,210) 210 FORMAT(///** OX,* NET BORROWING'///) DO 211 1=2,399 211 WRITE{6,92) ST (I) , (BAR(I,J) ,J=1,NC) 218 STOP END $SIG 219 APPENDIX L: Investment Policy Feasibility of the Policy; In this section we w i l l prove Theorems 1 and 2 of Chapter 7. Form (7.9) and (7.13) we can write: (L.l) i(x,-r) = ge~ r^ t _ T^ + w - x w - PVFP X Then (L.2) | i = -x w < 0 9x xx (w x x > 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 - PVFP T X For simplicity we w i l l analyze the continuous premium contract; for i t we have: (L.4) PV (X (T ) ) = x + /^" TDi r XdX ''==' x + f [ l - e " r ( t _ T ) ] (L.5) PVFP = /J" (D+y) e~ r XdX = ^ [ l - e - r ( t - T ) ] Substituting (L.4) and L.5) into (L.3): (L.6) i(x,x) = x + p - x W y - ^ - [ l - e " r ( t " T ) ] The value of i for 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 portfolio in relation to the exercise price (x<< g) the value of the put is 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 = l+p„ and w < 1 : p < 0. Thus, the maximum X X x — X — value of p at time x is p(0,t-x,g). At any time prior 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 principle). Therefore: (L.8) p(0,t-T,g) > p(0,t,g) Also (L.9) p(0,t,g) = f [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 portfolio the long position on the reference can be financed from the premiums paid: there is a positive investment in the riskless asset. The value of i for x=°° can be estimated by considering that (L.12) wx(°°,t-T,g) = 1 and p(~,t-x,g) = 0 (L.13) 1(»,T) = - ^ [ l - e " r ( t " T ) ] < 0 For "very high" values of the reference portfolio some borrowing i s required to follow the hedged policy: the amount to be borrowed i s the present value of the future put premiums to be received. From ( L . l l ) , (L.13) and Theorem 1, Theorem 2 obtains. L.2 Cash Position The cash position of the contract or funds available at time x for 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.15) PVFP = I (D+y) e ~ r k (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 t - f - l I k-1 When the expiration of the contract i s not known with certainty, the actual cash position, C ( x , T ) , can be obtained from (5.55), (7.5), (7.6), (L.14) and (L.15): n (L.16) C ( X , T ) = £ a(x,t) c(x,x,t) t=x+l or n ri (L.16) C ( X , T ) = I (t,t) w(x,t-T,g) + I a(x,t) g(t) t=x+l t = T + l e - r ( t - T ) n t-x-1 - (D+Y) I a(x,t)[ I e" r k] t=x+l k=l Notice that the annual put premium on the put with known date of expiration, y(t) , has been replaced in (L.16) by the actual put premium charged, Y. Formula (L.16) has been used in Computer Program No. 9 to compute the cash position. 

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