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UBC Theses and Dissertations

Discrete hedging in insurance risk management Sator, Imre Emil 1976

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DISCRETE HEDGING IN INSURANCE R ISK MANAGEMENT by IMRE EMIL SATOR B. Comm., U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF . MASTER OF SCIENCE IN BUSINESS ADMINISTRATION i n THE FACULTY OF COMMERCE AND BUSINESS ADMINISTRATION We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BR IT ISH COLUMBIA J u l y , 1976 (T) Imre B m i l S a t o r , 1976 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thou t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 1 ABSTRACT B a s e d u p o n t h e B l a c k - S c h o l e s o p t i o n p r i c i n g m o d e l , S c h w a r t z d e v e l o p e d a n e q u i l i b r i u m p r i c i n g d e f i n i t i o n o f t h e e q u i t y - l i n k e d l i f e i n s u r a n c e c o n t r a c t w i t h a n a s s e t v a l u e g u a r a n t e e . U n d e r t h e c o n d i t i o n s o f t h i s c o n t r a c t , t h e b e n e f i c i a r y may e l e c t t o r e c e i v e t h e v a l u e o f a r e f e r e n c e p o r t f o l i o o f s e c u r i t i e s o r a minimum g u a r a n t e e d a m o u n t , w h i c h e v e r i s g r e a t e r . I n t h i s s e n s e , t h e c o n t r a c t i s synonymous t o i n v e s t m e n t i n a m u t u a l f u n d a n d a t e r m i n s u r a n c e p o l i c y , The g u a r a n t e e p r o v i s i o n , h o w e v e r , g i v e s r i s e t o n o n d i v e r s i f i a b l e r i s k . I n t h e e v e n t o f a g e n e r a l m a r k e t c o l l a p s e , t h e company becomes s i m u l t a n e o u s l y l i a b l e f o r t h e g u a r a n t e e o n a l l m a t u r e c o n t r a c t s . S u c h a n e v e n t u a l i t y c o u l d p r o v e t o b e d i s a s t e r o u s . The e q u i l i b r i u m m o d e l p r o p o s e s a h e d g i n g s t r a t e g y w h i c h e l i m i n a t e s t h e p r o b a b i l i t y o f t h i s t y p e o f a l o s s . A t a n y p o i n t i n t i m e , t h e b e n e f i t s o f t h e c o n t r a c t may b e v i e w e d a s t h e p r e s e n t v a l u e o f t h e g u a r a n t e e p l u s t h e v a l u e o f a c a l l o p t i o 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 o n v e r s e l y , i t c a n a l s o b e s t a t e d a s t h e p r e s e n t v a l u e o f t h e r e f e r e n c e p o r t f o l i o p l u s t h e v a l u e o f t h e p u t o p t i o n . S i n c e t h e c a l l o p t i o n i s s o l d s h o r t , a f u l l y h e d g e d p o s i t i o n may b e e s t a b l i s h e d b y t h e a p p r o p r i a t e 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 . M a i n t e n a n c e o f t h i s p o s i t i o n i m p l i e s t h a t n o g a i n s i i or losses w i l l occur. In p r a c t i s e , however, continuous hedging i s impossible because of transaction costs. Adoption of a p o l i c y of p e r i o d i c r e v i s i o n to r e - e s t a b l i s h the hedged p o s i t i o n w i l l r e s u l t i n gains and losses. Exposure to r i s k , therefore, i s not eliminated. This d i s s e r t a t i o n deals with the problem of r i s k exposure r e s u l t i n g from a d i s c r e t e r e v i s i o n strategy. Through the employment of simulation techniques, the impact of various r e v i s i o n s t r a t e g i e s and transaction cost l e v e l s on the losses i s examined. As a basis of comparison, a naive strategy was also developed. That i s , under the naive strategy, the company buys the market p o r t f o l i o with the premium and holds i t u n t i l maturity. The case under consideration i s that of the s i n g l e premium contract with a known maturity date, The r e s u l t s of the analysis e s t a b l i s h the dominance of the p e r i o d i c r e v i s i o n strategy over the naive. Furthermore, f o r lower transaction cost l e v e l s , the d i s p e r s i o n of losses i s reduced as the number of revisions i s increased. Although the simulation model does not provide an optimal s o l u t i o n , i t does provide the framework f o r the establishment of a management strategy which i s consistent with the firm's perception of acceptable r i s k . I t i s hoped that the proposed strategy w i l l f i n d acceptance not only by the insurance industry but also i n the areas of mutual and pension fund management. V i i i TABLE OF CONTENTS CHAPTER . PAGE INTRODUCTION ^ 1 1. OVERVIEW 5 1.1 B a c k g r o u n d 5 1.2 R i s k — 7 1.3 L i t e r a t u r e O v e r v i e w * r * 9 2. THE EQUILIBRIUM MODEL 14-2.1 The P r i c i n g o f an O p t i o n 1 4 2.2 The B l a c k - S c h o l e s V a l u a t i o n F o r m u l a 1 6 2.3 The S c h w a r t z D i s s e r t a t i o n 24 2.4 The S i n g l e Premium Case 25 2.5 Summary 31 3. DEVELOPMENT OF THE SIMULATION MODEL — 32 3.1 B a s i c C o n c e p t s o f S i m u l a t i o n «* 32 3.2 The R e l e v a n c e o f P o r t f o l i o C o m p o s i t i o n 33 3.3 The R e t u r n on t h e P o r t f o l i o 35 3.4 The S i m u l a t i o n P r og r am 39 4. ANALYSIS OF RESULTS ~ * 52 4.1 I n t e r m e d i a t e R e s u l t s 52 4 . 2 The N a i v e S t r a t e g y 55 4 . 3 The R e v i s i o n S t r a t e g y : O v e r a l l L o s s e s 57 4 .4 The R e v i s i o n S t r a t e g y : D i s a s t e r L o s s e s 60 4 . 5 The R e v i s i o n S t r a t e g y : L a r g e s t L o s s e s >- 65 4 .6 Summary 67 i v CHAPTER PAGE 5 . CONCLUSIONS 7 0 BIBLIOGRAPHY 77 V APPENDIX PAGE A . S i m u l a t i o n P r o g r am 81 B. Summary S t a t i s t i c s : I n t e r m e d i a t e C a l c u l a t i o n s 92 T a b l e 1 94 T a b l e 2 95 T a b l e 3 96 C. Summary S t a t i s t i c s : N a i v e S t r a t e g y 97 T a b l e 1 98 T a b l e 2 „ 99 D. Summary S t a t i s t i c s : O v e r a l l L o s s e s 100 T a b l e 1. T r a n s a c t i o n C o s t \% 101 T a b l e 2. T r a n s a c t i o n C o s t 1%! •* 102 T a b l e 3. T r a n s a c t i o n C o s t 2% 103 T a b l e 4 . T r a n s a c t i o n C o s t 2hi 1 0 4 T a b l e 5. T r a n s a c t i o n C o s t 0% — 105 E. Summary S t a t i s t i c s : D i s a s t e r L o s s e s 106 T a b l e 1. T r a n s a c t i o n C o s t 1% 107 T a b l e 2. T r a n s a c t i o n C o s t lh% — 1 ° 8 T a b l e 3, T r a n s a c t i o n C o s t 2% 109 T a b l e 4 . T r a n s a c t i o n C o s t 2h% 110 T a b l e 5. T r a n s a c t i o n C o s t 0% 111 F. Summary S t a t i s t i c s : L a r g e s t L o s s e s - ~ — 112 T a b l e 1. T r a n s a c t i o n C o s t 1% ^ H 3 T a b l e 2. T r a n s a c t i o n C o s t lh% * H 4 T a b l e 3. T r a n s a c t i o n C o s t 2% 115 T a b l e 4. T r a n s a c t i o n C o s t 2h% • 1 1 6 T a b l e 5. T r a n s a c t i o n C o s t 0% 2 1 7 v i ACKNOWLEDGEMENTS The most d i f f i c u l t task f a c i n g the student o f any d i s c i p l i n e i s t o adequately express h i s g r a t i t u d e and indebtedness to those i n d i v i d u a l s whose guidance and encouragement was a p r e r e q u i s i t e t o the completion o f h i s academic undertaking. A s p e c i a l expression o f g r a t i t u d e must be extended t o Profe s s o r s Michael J . Brennan and Eduardo S. Schwartz o f the F a c u l t y o f Commerce and Business A d m i n i s t r a t i o n o f the U n i v e r s i t y o f B r i t i s h Columbia f o r t h e i r i n v a l u a b l e counsel throughout the course o f study. Dr. Brennan, as chairman o f the d i s s e r t a t i o n committee, not o n l y d i r e c t e d the course o f the a n a l y s i s , but must a l s o be c r e d i t e d w i t h i t s i n c e p t i o n . Dr. Schwartz provided constant guidance throughout the development o f the a n a l y s i s , along w i t h a thorough e x p l a n a t i o n o f h i s own d i s s e r t a t i o n , which i s an i n t e g r a l p a r t o f t h i s t h e s i s . G r a t i t u d e must a l s o be extended to P r o f e s s o r s P. Boyle and P. Larkey, who consented to reading the manuscript and acted as members of the d i s s e r t a t i o n committee. I t i s a l s o a p r i v i l e g e to recognize the years o f encouragement and a s s i s t a n c e g i v e n by P r o f e s s o r s W. T. Stanbury, R. W. White and W. F. J . Wood, f o r which a l a s t i n g debt o f g r a t i t u d e i s owed. L a s t l y , to the t y p i s t , my w i f e , a pers o n a l note o f thanks f o r enduring the r e v i s i o n s and more. 1 I n t r o d u c t i o n With the emergence o f e q u i t y l i n k e d l i f e insurance c o n t r a c t s i n Canada and the United Kingdom as a v i a b l e a l t e r n a t i v e to s t r a i g h t insurance, a number of important i s s u e s a r i s e w i t h respect to the v a l u e and the r i s k s a s s o c i a t e d w i t h t h i s type o f a c o n t r a c t . I t can be argued w i t h some j u s t i f i c a t i o n t h a t t h i s type of a c o n t r a c t i s not r e a l l y a l i f e insurance instrument, but a s e c u r i t y or asset i n the case o f the s i n g l e premium c o n t r a c t , o r an investment schedule i n the case o f the p e r i o d i c premium c o n t r a c t . This controversy has only r e c e n t l y been r e s o l v e d i n the U n i t e d S t a t e s , w i t h the S e c u r i t i e s Exchange Commission e s t a b l i s h i n g the procedures f o r the r e g i s t r a t i o n o f the s a l e of these instruments. To January 1976, only the E q u i t a b l e L i f e Assurance S o c i e t y has r e g i s t e r e d w i t h the S e c u r i t i e s Exchange Commission and p u b l i s h e d i t s i n t e n t i o n to s e l l the product i n a l i m i t e d market. In i t s s i m p l e s t form, the e q u i t y l i n k e d l i f e insurance c o n t r a c t may be viewed as an instrument w i t h a f i x e d t e r m i n a l asset v a l u e , p l u s a premium, the magnitude o f which depends on the performance o f a reference p o r t f o l i o o f s e c u r i t i e s . The insurance component i s the f i x e d t e r m i n a l asset v a l u e , or a guarantee by the company t h a t i n the event o f a c o n s i d e r a b l e stock market d e c l i n e , the b e n e f i c i a r y w i l l s t i l l be able 2 to elect to receive a guaranteed amount, which is determined at the creation of the contract. Obviously, the lower bound of the premium is zero. Put another way, for the investor, the benefits of the contract is either the guaranteed amount, or the value of the reference portfolio of securities, whichever is greater. The reference portfolio may be any portfolio of securities or mutual fund, or in the extreme, a portfolio consisting of one stock. Under equilibrium conditions, only the variance rate of the reference portfolio is of importance, not the return, for reasons which become evident in the discussion in Chapter 2. There is no requirement that the company invest a l l the premiums in this portfolio, and in fact i t has been shown that i t would be suboptimal to do so. (11 ) Briefly, the investor purchases the present value of the guaranteed amount plus the right to exercise his option with respect to the reference portfolio at the termination of the contract, should its value exceed the guaranteed amount. The company, on the other hand, is faced with a different set of problems. The most obvious is the pricing of the equity linked l i f e insurance contract. The second is the management of the funds invested in the reference portfolio. Thirdly, since a portion of the premium is invested in securities, the company must develop a strategy to minimize the probability of bankruptcy or disaster in the event of a market collapse. Portfolio theory suggests that most of the unsystematic risk associated with a security may be diversified away by selecting a large enough portfolio, or that i t may be completely eliminated by buying the 3 market p o r t f o l i o . From the point of "view of the insurance company, however, neither of these alternatives seem satisfactory because of the guaranteed amount. In the event of a market collapse, the beneficiary of the contract w i l l obviously elect to receive the guaranteed amount, as he would i n a l l cases where th i s amount exceeded the value of the reference p o r t f o l i o . I t should be noted, however, that disaster occurs only i f the value of the reference p o r t f o l i o plus the amount invested i n the r i s k free asset i s less than the guaranteed amount. The objective function of the company, therefore, may be viewed as the development of a strategy to minimize the p r o b a b i l i t y of t h i s type of bankruptcy. The theoretical framework for such a strategy has been proposed by E. Schwartz i n his doctoral dissertation. In a very general sense, he shows that the valuation of the equity linked l i f e insurance contract i s c losely related to the option p r i c i n g problem. This interpretation gives r i s e to a hedging strategy. Theoretically, i f a f u l l y hedged po s i t i o n i s maintained by continuous adjustment and the appropriate price i s charged f o r the contract, then no gains or losses can occur. At a p r a c t i c a l l e v e l , t h i s strategy i s not feasible because of transaction costs. A discrete hedging p o l i c y may, however, prove to be a t t r a c t i v e , i n the sense that the p r o b a b i l i t y of the disaster losses discussed previously, may be reduced. The objective of t h i s thesis i s to examine the hypothesis that the p r o b a b i l i t y of disaster losses may be reduced by adopting a discrete hedging p o l i c y i n the management of equity linked l i f e insurance contracts. 4 A l t h o u g h numerous a l t e r n a t i v e s e x i s t , t h e a n a l y s i s w i l l f o c u s e n t i r e l y o n t h e s i n g l e p e r i o d , s i n g l e p r e m i u m c o n t r a c t . I n o r d e r t o t e s t t h e h y p o t h e s i s , c o m p u t e r s i m u l a t i o n t e c h n i q u e s w e r e a d o p t e d . The d e v e l o p m e n t o f t h e a n a l y s i s c o n f o r m s t o t h e f o l l o w i n g o u t l i n e . C h a p t e r 1 p r o v i d e s a b r i e f d i s c u s s i o n o f t h e e v o l u t i o n o f e q u i t y l i n k e d l i f e i n s u r a n c e c o n t r a c t s . A r e v i e w o f some o f t h e l i t e r a t u r e p e r t i n e n t t o t h i s t o p i c i s a l s o p r e s e n t e d i n t h i s s e c t i o n . C h a p t e r 2 f o c u s e s m a i n l y o n t h e B l a c k - S c h o l e s o p t i o n v a l u a t i o n f o r m u l a a n d t h e r e l e v a n t s e c t i o n s o f t h e S c h w a r t z d i s s e r t a t i o n , w h i c h i n e f f e c t f o r m s t h e f o u n d a t i o n s o f t h i s a n a l y s i s . C h a p t e r 3 c o n t a i n s a d i s c u s s i o n o f s i m u l a t i o n a n d t h e s i m u l a t i o n m o d e l e m p l o y e d . C h a p t e r 4 s u m m a r i z e s t h e f i n d i n g s o f t h e t h e s i s a n d p r o v i d e s r e c o m m e n d a t i o n s f o r f u r t h e r a n a l y s i s . C h a p t e r 5 i s a r e s t a t e m e n t o f t h e p r o b l e m a n d a b r i e f c o n c l u s i o n . A n a p p e n d i x h a s b e e n p r o v i d e d f o r t h e r e l e v a n t s t a t i s t i c s a n d t h e s i m u l a t i o n p r o g r a m . 5 C h a p t e r 1 1.1 Background I n t e r e s t i n g l y enough, t h e f i r s t l i f e i n s u r a n c e c o n t r a c t s m a r k e t e d by companies, were i n f a c t t e r m i n s u r a n c e p o l i c i e s , as opposed t o w h o le l i f e i n s t r u m e n t s . A l t h o u g h t h e r e a s o n f o r t h i s phenomenon i s somewhat u n c l e a r , i t has b e e n a r g u e d t h a t a g e n e r a l m i s u n d e r s t a n d i n g o f t h e t h e o r y o f i n s u r a n c e o r t h e i n f a n t s t a t e o f t h e t h e o r y c a u s e d companies t o t a k e t h i s p o s i t i o n . B r i e f l y , t e r m i n s u r a n c e , as t h e name i m p l i e s , p r o v i d e s c o v e r a g e f o r a s p e c i f i e d p e r i o d . I n t h i s s e n s e , t e r m i n s u r a n c e i s c o n c e r n e d p r i m a r i l y w i t h t h e p r o b a b i l i t y o f a c c i d e n t a l d e a t h and n a t u r a l d e a t h w i t h i n t h e s p e c i f i e d p e r i o d o f t i m e , n o t w i t h t h e e v e n t u a l c e r t a i n t y . As s u c h , i t may be v i e w e d as a b e t between t h e company and t h e i n s u r e d , t h e company t a k i n g t h e p o s i t i o n t h a t t h e i n d i v i d u a l w i l l s u r v i v e t h e c o n t r a c t p e r i o d , t h e l a t t e r t h a t he w i l l n o t . The w h o l e l i f e c o n t r a c t , on t h e o t h e r hand, r e c o g n i z e s t h e e v e n t u a l i t y o f d e a t h as a c e r t a i n t y . The company assumes t h e l i a b i l i t y f o r t h e f a c e v a l u e o f t h e c o n t r a c t a t i t s c r e a t i o n , and t h e r e f o r e must b u i l d up a r e s e r v e d u r i n g t h e l i f e o f t h e c o n t r a c t i n o r d e r t o meet i t s o b l i g a t i o n a t i t s t e r m i n a t i o n . The premiums a s s o c i a t e d w i t h t h e s e c o n t r a c t s v a r y c o n s i d e r a b l y . Term i n s u r a n c e premiums t e n d t o be l o w e r t h a n w h ole l i f e , r e f l e c t i n g t h e f a c t t h a t t h e company may n o t be l i a b l e f o r t h e f a c e v a l u e o f t h e c o n t r a c t , and t h e r e a l i t y t h a t aged i n d i v i d u a l s a r e e x c l u d e d f r o m t h i s 6 type of an instrument. With the evolution of nonforfeiture clauses in the contracts and the inclusion of loan provisions, the whole l i f e policy became a very versatile instrument from a marketing point of view. The value of the loan option is self evident, the nonforfeiture clause can be interpreted as the surrender value of the contract i f i t is terminated before maturity, and of course the insurance aspect needs no explanation. Although intuitively appealing as a package, serious doubts have been raised concerning the real value of these benefits, given the investment required in the policy. Furthermore, the long run average rate of return on whole l i f e contracts has been about 3% to 5%, suggesting that any saving or investment motive which may have been a part of the decision making criteria, was i l l founded. The third type of policy, the endowment contract, offers some interesting features to the insured. In the most general sense i t may be viewed as a savings plan with a guarantee attached to i t . This type of insurance attempts to incorporate certain features of both the term and whole l i f e contracts. The policy generally provides for the payment of the face value to the beneficiary in the event of premature death of the insured, ( ie. within the specified time of the contract ). If the insured is alive at the termination of the contract, the insured sum is s t i l l paid by the company. In essence then, i t is simply an insured savings plan. 7 The r a p i d increase i n i n f l a t i o n r a t e s s i n c e 1945 and the sharp f l u c t u a t i o n s have caused p o l i c y holders and f o r t h a t matter insurance companies t o review the t e r m i n a l b e n e f i t s t h a t c u r r e n t p o l i c i e s p urport to y i e l d . As mentioned p r e v i o u s l y , a 3 - 5% r e t u r n on a whole l i f e p o l i c y i s anything but s a t i s f a c t o r y , g iven t h a t i n f l a t i o n r a t e s have been as h i g h as 131 per year. Those on f i x e d incomes, o r i n d i v i d u a l s cashing i n endowment type p o l i c i e s have i n c u r r e d c o n s i d e r a b l e l o s s e s due to the s p i r a l i n g i n f l a t i o n r a t e s . I t can be argued t h a t insurance companies should have done very w e l l , repaying u n i n f l a t e d d o l l a r s w i t h i n f l a t e d ones, but the returns d e c l a r e d by companies does not seem t o r e f l e c t t h i s . I t remains to be seen what impact c u r r e n t e f f o r t s t o reduce the r a t e o f i n f l a t i o n w i l l have on the c o n t r a c t s signed i n the l a s t f i v e or s i x years. 1.2 R i s k Although the r i s k o f death i s o f t e n aluded t o by insurance salesmen as a j u s t i f i c a t i o n f o r the premiums charged, i n a t h e o r e t i c a l sense t h i s arguement i s i n c o n s i s t e n t . B a s i c a l l y , the theory o f insurance i s synonymous to p o r t f o l i o theory. The p o o l i n g of independent r i s k i m p l i e s t h a t the unsystematic p o r t i o n , o r the r i s k unique to a p a r t i c u l a r case may be d i v e r s i f i e d away. Consequently, only the r i s k inherent i n the i n d u s t r y remains. The key to the argument i s the independence f a c t o r . I f t h i s c r i t e r i o n i s observed, then the number of deaths per p e r i o d may be c a l c u l a t e d w i t h c o n s i d e r a b l e accuracy, assuming t h a t there i s a 8 s u f f i c i e n t y l a r g e enough sample. The o n l y t h i n g t h a t remains, t h e r e f o r e , i s the matching of the l i q u i d a t i o n o f i n t e r e s t earning assets and claims against the company, as they become due. I f the theory d i d not h o l d , then there c o u l d be no j u s t i f i c a t i o n f o r the l i a b i l i t y exemption clauses such as war, n a t u r a l d i s a s t e r s , a ct o f God, e t c . , i n c l u d e d i n a l l p o l i c i e s . In essence, these phenomena v i o l a t e the independence assumption. As the above argument i m p l i e s , under the c o n d i t i o n s o f the p o l i c i e s discussed thus f a r , the insurance company assumes a l l the r i s k . P r i m a r i l y because o f the r a p i d i n c r e a s e i n i n f l a t i o n and the general d i s s a t i s f a c t i o n o f p o l i c y holders w i t h the subsequent d e c l i n e i n the v a l u e of the p o l i c y , insurance companies began t o market an e q u i t y l i n k e d instrument f i r s t i n the Netherlands i n the e a r l y f i f t i e s , then l a t e r i n B r i t a i n and Canada, and now i n the United S t a t e s . The primary d i f f e r e n c e between t h i s type of p o l i c y and the conventional instrument i s t h a t a p o r t i o n o f the r i s k i s borne by the i n s u r e d . As d i s c u s s e d p r e v i o u s l y , the b e n e f i t s o f the c o n t r a c t may be viewed as the value of a reference p o r t f o l i o o f s e c u r i t i e s , or a guaranteed amount, whichever i s g r e a t e r . The theory behind t h i s approach i s t h a t the i n d i v i d u a l i s w i l l i n g to assume a p a r t o f the r i s k because of the b e l i e f t h a t s e c u r i t i e s p r o v i d e a hedge against i n f l a t i o n . Since the r i s k s t r u c t u r e i s d i f f e r e n t from the conventional approach, the v a l u a t i o n o f t h i s instrument and the determination of the premium s t r u c t u r e presents some unique problems. In a narrow sense, the p o l i c y i s s i m i l a r t o buying a mutual fund and term insurance as a complement, from the p o i n t o f view o f the i n s u r e d . I t 9 should be noted t h a t t h i s type o f instrument provides the departure p o i n t from g e n e r a l l y accepted insurance theory. The r i s k a s s o c i a t e d w i t h the asset value guarantee can not be d i v e r s i f i e d away. A general market d e c l i n e w i l l r e s u l t i n a catastrophe as the guarantee w i l l be e x e r c i s e d under a l l maturing c o n t r a c t s . Although a c o n s i d e r a b l e number o f papers have been w r i t t e n on the s u b j e c t o f e q u i t y l i n k e d l i f e insurance p o l i c i e s , most o f them pursue what may be d e f i n e d as a naive approach. That i s , most o f them focus on the problem o f e s t a b l i s h i n g adequate r e s e r v e s , without c o n s i d e r i n g a l l the parameters of the problem. In the t r a d i t i o n a l sense, t h i s meant average or mean reserve requirements. I f companies enjoyed the same degree of experience w i t h the e q u i t y l i n k e d products as they do w i t h whole l i f e c o n t r a c t s , then such a s t r a t e g y may w e l l be acceptable t o a c e r t a i n extent. T h i s , however, i s not the case. 1.3 L i t e r a t u r e Overview Sq u i r e s , ( 1 6 ) i n a 1974 paper c o r r e c t l y i d e n t i f i e s the inadequacies of e x i s t i n g models attempting to e x p l a i n stock market behavior. U n f o r t u n a t e l y , h i s assumptions about the market are a l s o open to c r i t i c i s m , which tend t o negate h i s c o n c l u s i o n s . In p a r t i c u l a r , i t w i l l be shown th a t i n the case of s i n g l e premium p o l i c i e s , i t becomes i r r e l e v a n t when the p o l i c y i s e f f e c t e d . His a s s e r t i o n i s t h a t i f the p o l i c y i s e f f e c t e d when the market i s a t i t s peak, then the company i s 10 subjected t o s u b s t a n t i a l r i s k s during the subsequent trough. This a s s e r t i o n c l e a r l y ignores the random walk or e f f i c i e n t market hypothesis. I f the random walk i s an accurate d e s c r i p t i o n o f p r i c e behavior, o f which there i s c o n s i d e r a b l e evidence, then a "market peak" can only be i d e n t i f i e d i n • r e t r o s p e c t . Put another way, i f the random walk hypothesis h o l d s , then i t i s impossible to determine i f today's stock p r i c e i s at i t s "peak" because i n order t o accomplish t h a t , tomorrow's p r i c e must be known. This i s not p o s s i b l e because of the c o n d i t i o n : P ( X t> 0 ) = P ( Xt< 0 ) = .5 t h a t i s , the p r o b a b i l i t y o f a p o s i t i v e p r i c e change equals the p r o b a b i l i t y o f a negative p r i c e change ( i e . 0.5 ). I f such i s the case, then the best estimate of tomorrow's p r i c e must be today's p r i c e . This can be d e r i v e d by a l g e b r a . In e f f e c t then, s i n c e tomorrow's p r i c e i s not known f o r c e r t a i n , troughs and peaks cannot be i d e n t i f i e d . The major p o i n t to note however, i s t h a t h i s approach completely ignores the i m p l i c a t i o n s of the hedging s t r a t e g y . The major c o n t r i b u t i o n s t o the understanding o f the nature of the e q u i t y l i n k e d l i f e insurance c o n t r a c t were s i m u l a t i o n models developed by Turner (18), DiPaolo (6 ) , and Kahn (8 ). T h e i r work may be viewed as the p o i n t of departure from the conventional i n t e r p r e t a t i o n o f the v a l u a t i o n problem. G e n e r a l i z i n g , t h e i r approach may be i n t e r p r e t e d as aggressive, whereas the t r a d i t i o n a l p o s i t i o n i s defensive. As Turner (17 ) s t a t e s , i n reference t o a paper by Sidney Benjamin, " the s t a t e d approach 11 to v a l u a t i o n , t h a t i s , to the establishment o f a d d i t i o n a l reserves f o r asset value guarantees, i s t o determine on each v a l u a t i o n date any reserves,;; which, i n the o p i n i o n o f the actuary, would be r e q u i r e d c o n s i d e r i n g the nature o f the guarantees provided and the f i n a n c i a l s i t u a t i o n a t t h a t time". But c l e a r l y , the most important element o f the d e c i s i o n making c r i t e r i a i s l a c k i n g i n t h i s d e f i n i t i o n , t h a t i s , the expected performance o f the reference p o r t f o l i o , o r i n a more general sense, the expected performance o f the market. Since the b a s i c theory of the e q u i t y l i n k e d product i s t h a t the i n v e s t o r i s w i l l i n g to assume a p o r t i o n o f the r i s k i n order to a t t a i n a higher y i e l d , t h i s must be the l e a s t d e s i r a b l e a l t e r n a t i v e from both p o i n t s o f view. The reserve requirement, t h e r e f o r e , must be a f u n c t i o n o f the expected r a t e o f r e t u r n on the market, and the p r o b a b i l i t y o f a t t a i n i n g an e q u i t y p o s i t i o n g r e a t e r than the amount of the guarantee. As p o i n t e d out i n : the v a r i o u s c r i t i q u e s o f the aforementioned papers, there i s a general r e l u c t a n c e on the p a r t o f a c t u a r i e s t o accept the cur r e n t theory o f c a p i t a l a s s e t p r i c i n g . This may account f o r the h e s i t a t i o n observed i n viewing the instrument as an o p t i o n p r i c i n g problem, as opposed t o a reserve problem. Turner (17 ) b a s i c a l l y views the problem i n three stages. F i r s t , he recognizes t h a t some conclusions must be made concerning the nature o f the p r o b a b i l i t y d e n s i t y f u n c t i o n of s e c u r i t y r e t u r n s . Second, he focuses on the e v a l u a t i o n o f the net r i s k premium o f an asset v a l u e guarantee a t the end of the c o n t r a c t p e r i o d , g i v e n the e q u i t y l i n k e d 12 instrument. T h i r d l y , he analyzes the s e n s i t i v i t y o f the net r i s k premium t o changes i n u n d e r l y i n g parameters, such as investment p e r i o d , charges against the r e t u r n on e q u i t y , taxes, and decrements i n m o r t a l i t y and withdrawals. The o v e r a l l i m p l i c a t i o n s of h i s a n a l y s i s i s the p r e s e n t a t i o n of a framework which f o r c e s a c t u a r i e s i n t o viewing the problem of e q u i t y l i n k e d c o n t r a c t s w i t h an a s s e t value guarantee i n a more q u a n t i t a t i v e or a n a l y t i c s e t t i n g than the t r a d i t i o n a l approach. DiPaolo r e l i e s on Monte Ca r l o techniques to generate or simulate s e c u r i t y t r e n d s , which are then u t i l i z e d to evaluate the adequacy o f the investment r i s k premium charged, f o r an e q u i t y based endowment p o l i c y . His b a s i c assumption i s t h a t a r i s k premium i s deemed to be adequate i f the p r o b a b i l i t y o f the r i s k fund being i n a s t a t e of r u i n i s s m a l l a f t e r the l a s t c o n t r a c t matures. Ruin, i n t h i s sense, occurs i f the r i s k fund i n c u r s l o s s e s a f t e r the t e r m i n a t i o n o f the l a s t c o n t r a c t . As i s the case w i t h most s i m u l a t i o n models, h i s does not provide an optimal s o l u t i o n r a t h e r a d i s t r i b u t i o n o f the v a r i o u s outcomes. Nevertheless the model does recognize the f a c t t h a t the r a t e of r e t u r n on the s e c u r i t i e s i s an i n s e p a r a b l e f a c t o r i n the management of the funds i n v o l v e d . Kahn's approach t o the problem i s s i m i l a r to t h a t o f Turner and DiPaolo. He a l s o u t i l i z e s s i m u l a t i o n and a n a l y t i c techniques t o generate a market tren d or r e t u r n and uses the r e s u l t s to p r o j e c t v a r i o u s insurance a l t e r n a t i v e s . In g e n e r a l , h i s f i n d i n g s show the 13 extreme s e n s i t i v i t y o f earnings o f a v a r i a b l e l i f e insurance company w i t h respect to investment performance. He a l s o shows th a t the c o s t of a minimum death b e n e f i t guarantee v a r i e s w i d e l y w i t h investment performance. In g e n e r a l , the s i g n i f i c a n c e of these analyses l i e s i n the i n t e r p r e t a t i o n or d e f i n i t i o n o f the problem. As s t a t e d p r e v i o u s l y , the departure from the t r a d i t i o n a l defensive p o s i t i o n t h a t the over-r i d i n g f a c t o r i s the establishment of r e s e r v e s , must be viewed as a breakthrough. A l l three authors e x p l i c i t l y recognize t h a t the c r i t i c a l i s s u e i s the performance o f the investment p o r t f o l i o . Under t h e i r assumption, t h e r e f o r e , the p r o b a b i l i t y o f r u i n i s a f u n c t i o n of the p r o b a b i l i t y of a s u s t a i n e d market d e c l i n e or a general c o l l a p s e . The f o l l o w i n g chapter w i l l show th a t i n a t h e o r e t i c a l sense i t i s p o s s i b l e to view the problem i n such a framework th a t the p r o b a b i l i t y o f r u i n may be completely e l i m i n a t e d through a process o f hedging. In order to accomplish t h i s , a comprehensive overview of the Black-Scholes o p t i o n p r i c i n g model w i l l be presented, f o l l o w e d by a d i s c u s s i o n of the major i s s u e s o f the Schwartz d i s s e r t a t i o n ; This w i l l p r o v i d e the necessary background f o r the s i m u l a t i o n model employed i n t h i s a n a l y s i s . 14 Chapter 2 The E q u i l i b r i u m Model 2.1 The P r i c i n g o f an Option In the most general sense, an o p t i o n may be d e f i n e d as the r i g h t t o buy or s e l l an as s e t , s u b j e c t to c e r t a i n c o n d i t i o n s , w i t h i n a s p e c i f i e d p e r i o d of time. The p r i c e p a i d f o r the o p t i o n i s g e n e r a l l y r e f e r r e d to as the s t r i k i n g or e x e r c i s e p r i c e . The l a s t day on which i t can be e x e r c i s e d i s the mat u r i t y or e x p i r a t i o n date. There are b a s i c a l l y two types o f op t i o n s ; the European, and the American. An American o p t i o n may be e x e r c i s e d any time up to and i n c l u d i n g the mat u r i t y date, whereas the European can only be e x e r c i s e d on the s p e c i f i e d f u t u r e date. The s i m p l e s t k i n d o f o p t i o n i s the r i g h t to buy a s i n g l e share of common stock, o r c a l l o p t i o n . I t s counterpart, the put o p t i o n , i s the r i g h t t o s e l l one share of common to another p a r t y . I t can be r e a d i l y seen t h a t a number o f combinations o f the two b a s i c o p t i o n types are p o s s i b l e , depending on the o b j e c t i v e s o f the i n d i v i d u a l . For the purposes of t h i s p o r t i o n o f the a n a l y s i s , however, the focus w i l l be on the c a l l o p t i o n . C l e a r l y , a r e l a t i o n s h i p must e x i s t between the value o f the op t i o n and the p r i c e o f the u n d e r l y i n g s e c u r i t y . I t can be expected t h a t the higher the p r i c e o f the stock, the gr e a t e r should be the value of the o p t i o n . I f the stock p r i c e i s c o n s i d e r a b l y g r e a t e r than the 15 e x e r c i s e p r i c e , then, the o p t i o n w i l l probably be e x e r c i s e d . More f o r m a l l y , a t t h i s p o i n t , the value o f the o p t i o n w i l l be approximately equal to the p r i c e o f the stock minus the p r i c e o f a pure discount bond t h a t matures on the same date as the o p t i o n , and has a face value equal t o the s t r i k i n g p r i c e o f the o p t i o n . That i s : ( 2-1 ) V 0 t = PE - B ( e" r t * ) Where VO^ i s the value o f the o p t i o n a t time t ; PE the p r i c e o f the - r t * s e c u r i t y a t time t ; and the expr e s s i o n B ( e ) the p r i c e o f the discount bond. Since the p r o b a b i l i t y o f e x e r c i s i n g the o p t i o n becomes very h i g h as the m a t u r i t y date approaches ( i e . as per the above assumption ) , the process may be viewed as a d e f e r r e d purchase p l a n . The value o f the discount bond at the c r e a t i o n o f the o p t i o n represents the amount which must be i n v e s t e d i n the r i s k f r e e a s s e t i n order to i n s u r e t h a t s u f f i c i e n t funds are a v a i l a b l e to e x e r c i s e the o p t i o n a t ma t u r i t y . In essence, t h i s i s the same as a d e f e r r e d p l a n . On the other hand, i f the stock p r i c e , PE i s c o n s i d e r a b l y l e s s than the s t r i k i n g p r i c e , the o p t i o n w i l l probably e x p i r e , so t h a t i t s value should be near zero. Furthermore, i f the e x p i r a t i o n date i s f a r o f f , then the p r i c e o f the discount bond w i l l be low, implying t h a t the o p t i o n value w i l l be approximately the same as the stock p r i c e . I f the e x p i r a t i o n date i s near, then the o p t i o n value should approximately equal the d i f f e r e n c e between the stock p r i c e 16 and the e x e r c i s e p r i c e , or zero i f the stock p r i c e i s l e s s than the s t r i k i n g p r i c e . Normally, i t can be expected t h a t the value o f the o p t i o n should d e c l i n e , i f there i s no change i n the stock p r i c e . W i t h i n t h i s framework i t can be expected t h a t the o p t i o n i s more v o l a t i l e than the stock. That i s , f o r a gi v e n percent change i n the p r i c e o f the stock, a l a r g e r percent change w i l l occur i n the value of the o p t i o n , given t h a t m a t u r i t y i s h e l d constant. I t should be noted, however, t h a t the r e l a t i v e v o l a t i l i t y o f the o p t i o n i s not constant as i t depends on stock p r i c e and m a t u r i t y . 2.2 The Black-Scholes V a l u a t i o n Formula The o r i g i n s o f the Black-Scholes v a l u a t i o n formula may be found i n the works o f Sprenkle (1961), Ayres (1963), Samuelson (1965), et c . These e a r l i e r works d e a l t p r i m a r i l y w i t h the v a l u a t i o n o f warrants, but f o r a l l i n t e n t s and purposes, the theory i s e q u a l l y a p p l i c a b l e to other o p t i o n s . The major problem w i t h these e a r l i e r f o rmulations i s the f a c t t h a t some o f the parameters were l e f t undefined. The key assumption t h a t they do u t i l i z e , however, i s a conclusiQn;..of:.the work o f Thorpe and Kassouf (1967). They note t h a t the v a l u a t i o n o f the warrant hinges upon the r a t i o o f stock options to shares needed t o create a hedged p o s i t i o n by going s h o r t i n one s e c u r i t y and long i n the other. As Black and Scholes p o i n t out, what they f a i l e d t o recognize 17 was the f a c t t h a t the expected r a t e o f r e t u r n on such a p o s i t i o n must be equal to the r i s k f r e e r a t e o f r e t u r n . Given t h i s e q u i l i b r i u m c o n d i t i o n , they proceed to develop t h e i r t h e o r e t i c a l v a l u a t i o n formula. Before proceeding to t h e i r model, however, a review o f t h e i r assumptions i s i n order. G e n e r a l l y , i d e a l market c o n d i t i o n s are assumed f o r the stock and the o p t i o n . More s p e c i f i c a l l y : a) The s h o r t term i n t e r e s t r a t e i s assumed t o be known and constant through time. This may be r e l a x e d under c e r t a i n c o n d i t i o n s . b) Stock p r i c e s f o l l o w a random walk i n continuous time w i t h a v a r i a n c e r a t e p r o p o r t i o n a l to the square o f the stock p r i c e . That i s , the d i s t r i b u t i o n o f stock p r i c e s i s l o g normal and the va r i a n c e r a t e o f the r e t u r n on the stock i s constant. c) There are no dividends or any other d i s t r i b u t i o n s . d) The o p t i o n i s European, ( i e . e x e r c i s a b l e a t m a t u r i t y o n l y ) . e) No t r a n s a c t i o n s c o s t s i n buying or s e l l i n g the stock or the o p t i o n . ( This w i l l be r e l a x e d i n the subsequent a n a l y s i s ) . 18 £) I t i s p o s s i b l e to borrow a t the sh o r t term r a t e and buy any p o r t i o n o f a s e c u r i t y . g) No p e n a l t i e s f o r short s e l l i n g . I t can be r e a d i l y seen t h a t under these assumptions, the value o f the o p t i o n depends only on the stock p r i c e and time, and on parameters which are taken as known.xonstants. Under such circumstances i t i s p o s s i b l e to form a hedged p o s i t i o n by s h o r t i n g the o p t i o n and t a k i n g a long p o s i t i o n i n the stock such t h a t the value o f the o p t i o n w i l l not depend on the stock p r i c e but on time and the constants. That i s , more f o r m a l l y the value o f the o p t i o n may be expressed as: ( 2-2 ) w ( x, t ) or as a f u n c t i o n o f the stock p r i c e x and time t . To form the hedged p o s i t i o n , the number of options t h a t must be s o l d a g a i n s t one stock may be w r i t t e n as: ( 2-3 ) 1 / W l ( x, t ) The s u b s c r i p t denotes the p a r t i a l w i t h respect to the f i r s t argument. To show t h a t the value o f the hedged p o s i t i o n does not depend on stock p r i c e , note t h a t w^ (' x , t ) i s the r a t i o o f the change i n the op t i o n value to the change i n the stock p r i c e . That i s , i f x changes by A x, the o p t i o n p r i c e w i l l change by w., ( x , t ) Ax, so t h a t the 19 value o f l/w^ ( x , t ) options w i l l change by Ax. Therefore a change i n the value o f the long p o s i t i o n i n x w i l l be approximately o f f s e t by the change i n the value o f the short p o s i t i o n i n 1/w^ o p t i o n s . I f c o n t i n u i t y i s assumed, then i t can be shown t h a t the approximations become exact and the r e t u r n on the hedged p o s i t i o n i s completely independent o f the changes i n the value o f the stock. That i s , the r e t u r n on the hedged p o s i t i o n becomes c e r t a i n . I t i s important to note t h a t the argument i s c o n s i s t e n t w i t h e x i s t i n g p o r t f o l i o and market theory. Under the random walk and constant v a r i a n c e r a t e assumptions the covariance between the r e t u r n s on the e q u i t y and the stock w i l l be zero. The same argument a p p l i e s to the market p o r t f o l i o concept. Consequently, under a continuous adjustment p o l i c y , the r i s k i n the hedged p o s i t i o n i s zero. Even i f continuous adjustment does not occur, i t i s expected t h a t the r i s k w i l l be s m a l l . The c r i t i c i a l f a c t o r , however, i s t h a t i t may be completely d i v e r s i f i e d away by h o l d i n g a p o r t f o l i o o f hedged p o s i t i o n s . These g e n e r a l i z a t i o n s have d e f i n i t e i m p l i c a t i o n s f o r the r e s t o f t h i s a n a l y s i s . The value o f the e q u i t y i n the p o s i t i o n , g i v e n one share long and 1/w^ options short i s d e f i n e d as: ( 2-4 ) Ve = x - w/w. 20 and the change i n Ve over a s h o r t i n t e r v a l t as: ( 2-5 ); AVe = Ax - Aw/*^ Under the assumption o f continuous adjustment, Aw can be expanded through s t o c h a s t i c c a l c u l u s and be shown th a t the value o f the e q u i t y becomes: 2 2 ( 2 - 6 ) -( h^Y\_ v x + w 2 -* A t/ Wi 2 where the s u b s c r i p t s r e f e r t o p a r t i a l d e r i v a t i v e s and v i s the varia n c e r a t e of r e t u r n on the stock. Furthermore, s i n c e the r e t u r n on the e q u i t y i s knownfor c e r t a i n to be r A t , the change i n the e q u i t y can be expressed as: ( 2-7 ) ( x - w/w.^  r A t Equating (2-6) and (2-7), dropping t and r e a r r a n g i n g g i v e s the f o l l o w i n g d i f f e r e n t i a l equation f o r the value o f the o p t i o n : 2 2 ( 2-8 ) w 2 = rw - rxw^ - h (v x w^ ) Assuming t * t o be the m a t u r i t y date and c the e x e r c i s e p r i c e and the f o l l o w i n g boundary c o n d i t i o n s , ( 2-9 ) w( x , t * ) = x-c f o r x ^ c and w( x , t * ) = 0 f o r x < c and s o l v i n g . (2-8) s u b j e c t t o (2-9)_ r e s u l t s i n the o p t i o n v a l u a t i o n formula. ( 2 ) . The formula may be s t a t e d as: ( 2-10 ) w(x,t) = x N (d 1) - ce 1"^"**) N (d 2) where ( 2-11 ) d = l n ( x / c ) + (r + % v 2 ) ( t * - t ) v ( t * - t ) % A = l n f x / c ) + (r - h v 2 ) ( t * - t ) 2 v ( t * - t ) % and N(d) i s the cumulative normal d e n s i t y f u n c t i o n g i v e n by: ( 2-12 ) N(d) = 1/.(2TT ) 2 e " " 2 W dx d , r -,2 Taking the p a r t i a l d e r i v a t i v e of equation (2-10 ) w i t h respect t o the f i r s t argument and s i m p l i f y i n g , r e s u l t s i n the f o l l o w i n g d e f i n i t i o n ( 2-13 ) w 1 (x,t) = N (d 1) which i s o f p a r t i c u l a r importance t o t h i s a n a l y s i s , as the expression defines the r a t i o of stock to options i n the hedged p o s i t i o n . 22 Black and Scholes continue by p o i n t i n g out t h a t as can be seen from expression ( 2-10 ) , the v a l u e o f the o p t i o n i s not a f u n c t i o n o f the expected r e t u r n on the s t o c k . T h i s , however, does not negate the p r o p o s i t i o n t h a t the expected r e t u r n on the o p t i o n i s a f u n c t i o n o f the expected r e t u r n on the stock. In e f f e c t , then,the f o r m u l a t i o n confirms t h a t the p r i c e o f the o p t i o n i s independent o f the i n v e s t o r s ' u t i l i t y f u n c t i o n s . As presented i n a previous argument, from equations ( 2-10 ) and ( 2-13 ) i t can be shown th a t the v o l a t i l i t y o f the o p t i o n i s always g r e a t e r than the v o l a t i l i t y o f the stock. That i s , s i n c e the r a t i o ( 2-14 ) xw 1 / w i s always g r e a t e r than one, i t i m p l i e s t h a t the r e l a t i v e v o l a t i l i t i e s w i l l m a i n t a i n a r e l a t i o n s h i p such t h a t the v o l a t i l i t y o f the o p t i o n w i l l always exceed the v o l a t i l i t y o f the s e c u r i t y . Black and Scholes a l s o show t h a t equation ( 2-8 ) may be d e r i v e d by u s i n g the c a p i t a l asset p r i c i n g model,but f o r the purposes of t h i s a n a l y s i s , the previous development w i l l s u f f i c e . I t i s worthwhile t o note the observed e m p i r i c a l t e s t r e s u l t s r e p o r t e d by Black and Scholes when they compared the t h e o r e t i c a l v a l u a t i o n p r e d i c t i o n to a c t u a l c a l l - o p t i o n data. They r e p o r t t h a t the observed values tend t o d e v i a t e from the p r e d i c t e d values i n a systematic manner. 23 The purchasers o f c a l l options tend to overpay, but the w r i t e r s o f the o p t i o n r e c e i v e approximately what the v a l u a t i o n formula p r e d i c t s . According to Black and Scholes, the d i f f e r e n c e must be the t r a n s a c t i o n s c o s t s . They a l s o note t h a t the observed d i f f e r e n c e s tend to be g r e a t e r f o r options on low r i s k stocks than hi g h r i s k s e c u r i t i e s . The t r a n s a c t i o n costs,however, remove the p o t e n t i a l p r o f i t o p p o r t u n i t i e s i m p l i e d . Since i t s d e r i v a t i o n , the o p t i o n v a l u a t i o n formula has r e c e i v e d a c o n s i d e r a b l e amount of a t t e n t i o n from academics. As s t a t e d i n the assumptions, Black and Scholes r e s t r i c t e d the a n a l y s i s to non-dividend paying s e c u r i t i e s . Merton (10 ) extends the model and shows t h a t t h i s assumption may be r e l a x e d to stocks paying continuous dividends. He a l s o shows th a t i f there are no d i v i d e n d s , then i t w i l l never pay to e x e r c i s e an American o p t i o n before the m a t u r i t y date, implying t h a t the v a l u a t i o n formula i s e q u a l l y v a l i d f o r the l a t t e r o p t i o n . With respect to s e n s i t i v i t y , he notes t h a t the value o f the o p t i o n w i l l i n c r e a s e c o n t i n u o u s l y t o the extent t h a t the m a t u r i t y t , the r i s k f r e e 2 r a t e r , or the v a r i a n c e r a t e v i n c r e a s e s . The upper bound, he concludes, must be the stock p r i c e . B r i e f l y , the v a l u a t i o n formula must be considered as one of the most important breakthroughs i n f i n a n c e . I t not o n l y o f f e r s a s i m p l i f i e d s o l u t i o n to a r a t h e r complex problem, but a l s o causes a re-examination o f the e x i s t i n g f i n a n c i a l theory. For example, i n the area o f corporate f i n a n c e i t can be shown r a t h e r e a s i l y t h a t the stock 24 h o l d e r s , i n e f f e c t , have an o p t i o n on the assets o f the f i r m assuming bonds to be outstanding. Furthermore, each bond i n t e r e s t payment may be viewed as an o p t i o n c o n t r a c t . In g e n e r a l , i t can be shown t h a t the t o t a l number of options i n t h i s context i s equal to n + 1, where n equals the number of i n t e r e s t payments to be made. C l e a r l y , the arguments may be extended to c o n v e r t i b l e "instruments, e t c . , but these become r a t h e r complicated. Having developed the o p t i o n v a l u a t i o n formula to t h i s p o i n t , i t becomes necessary to i n t e g r a t e i t i n t o the e q u i t y l i n k e d l i f e insurance framework. In order to achieve t h i s , the next s e c t i o n w i l l focus on the Schwartz d i s s e r t a t i o n r e f e r r e d t o i n the previous chapter. 2.3 The Schwartz D i s s e r t a t i o n This d i s s e r t a t i o n o f f e r s one o f the most important challenges to the l i f e insurance i n d u s t r y . In essence, i t c a l l s upon the i n d u s t r y to completely re-evaluate i t s p o s i t i o n w i t h respect t o the management of e q u i t y l i n k e d l i f e insurance c o n t r a c t s . In t h i s sense i t a l s o challenges r e g u l a t o r y bodies to review t h e i r i n t e r p r e t a t i o n o f the concept o f r i s k . The d i s s e r t a t i o n may be d i v i d e d i n t o two d i s t i n c t areas. One, the p r e v i o u s l y c i t e d problems concerning the v a l u a t i o n of c e r t a i n options and two, the i n t e r p r e t a t i o n o f the e q u i t y l i n k e d l i f e insurance 25 instrument i n the o p t i o n p r i c i n g framework. In the f i r s t p a r t , the author develops numerical methods to evaluate options on s e c u r i t i e s paying d i s c r e t e dividends. He proceeds to show t h a t under c e r t a i n c o n d i t i o n s , i t pays to e x e r c i s e American options p r i o r to the m a t u r i t y date ( i e . the s e c u r i t y pays d i s c r e t e dividends ). B r i e f l y , he shows tha t a " c r i t i c a l stock p r i c e " can be determined, above which i t pays to e x e r c i s e the o p t i o n . The second p a r t of the d i s s e r t a t i o n deals w i t h e q u i t y l i n k e d instruments w i t h an asset value guarantee, i n the o p t i o n p r i c i n g framework. W i t h i n t h i s framework, the optimal investment s t r a t e g y o f the f i r m i s developed. Along the same l i n e s , p a r t i a l d i f f e r e n t i a l equations are given f o r the o p t i o n components o f the constant, continuous premium c o n t r a c t , the p e r i o d i c premium c o n t r a c t and the s i n g l e premium c o n t r a c t . For the purposes o f t h i s a n a l y s i s , the s i n g l e premium c o n t r a c t i s the r e l e v a n t focus o f d i s c u s s i o n . The S i n g l e Premium Case R e s t a t i n g e a r l i e r d e f i n i t i o n s , an e q u i t y l i n k e d l i f e insurance c o n t r a c t w i t h an asset v a l u e guarantee, i s an instrument p r o v i d i n g the b e n e f i t o f e i t h e r the value of a reference p o r t f o l i o o f s e c u r i t i e s , or a guaranteed amount, whichever i s g r e a t e r . A c a l l o p t i o n permits the h o l d e r t o purchase an ass e t a t a given p r i c e , whereas the put o p t i o n 26 i s the r i g h t t o s e l l an asset f o r a c e r t a i n or predetermined amount. W i t h i n t h i s framework, the c o n t r a c t may be desc r i b e d as f o l l o w s . I f , upon m a t u r i t y , the value o f the reference p o r t f o l i o exceeds the guaranteed amount, the b e n e f i c i a r y may e x e r c i s e a c a l l o p t i o n on the reference p o r t f o l i o . In t h i s sense, the e x e r c i s e p r i c e i s the guaranteed amount. The value of the o p t i o n must t h e r e f o r e be the d i f f e r e n c e between the guarantee and the p o r t f o l i o value. The c a l l o p t i o n , o b v i o u s l y w i l l not be e x e r c i s e d i f the value of the guarantee exceeds the value of the p o r t f o l i o . Put another way, the value of the put o p t i o n p l u s the value of the refer e n c e p o r t f o l i o must equal the value o f the guarantee p l u s the value of the c a l l o p t i o n . E i t h e r d e f i n i t i o n may be viewed as the b e n e f i t s o f the c o n t r a c t . I t should be noted t h a t the p r e v i o u s l y s t a t e d assumptions o f the Black-Scholes model, i n p a r t i c u l a r , t h a t the investment i s made i n non-dividend paying s e c u r i t i e s , s t i l l apply to the above argument. A l s o , g i v e n t h a t the investment i n the reference p o r t f o l i o i s made at the time the c o n t r a c t i s purchased, Schwartz shows t h a t the value o f the o p t i o n a t any p o i n t i n time corresponds e x a c t l y t o the Black-Scholes c a l l o p t i o n f o r m u l a t i o n . That i s , the p a r t i a l d i f f e r e n t i a l equation may be expressed as: 2 2 ( 2-15 ) h v x w,,' + rxw n - rw ^ w ? = 0 27 w i t h the boundary c o n d i t i o n s : ( 2-16 ) w (x,t) = max ( 0, x-g ) Therefore, a t any time T , the value o f the o p t i o n may be s t a t e d as: ( 2-17 ) w ( x ( T ) , t - T- , g ( t ) ) = X(T ) N(d x) - g ( t ) e " r ( t " T } N ( d 2 ) ( 2-18 ) where d1 = ( l n ( x ( T )/g(t) + ( r + %v 2) ( t - x )/v ( t - x ) h ( 2-19 ) d 2 = d x - v ( t - x ) h 2 C 2-20 ) N(d) = 1/C2T. ) % / o o e " % ( x ) dx and N(d) i s the cumulative normal d e n s i t y f u n c t i o n , as before. Although the asset value guarantee, g, has been s u b s t i t u t e d f o r the e x e r c i s e p r i c e , c, these equations are i d e n t i c a l to equations ( 2-10 ) to ( 2-12 ) , r e s p e c t i v e l y . The b a s i c conclusions t h a t the value o f the o p t i o n a t any time t , can be expressed i n terms o f the c u r r e n t p r i c e o f the reference p o r t f o l i o , the v a r i a n c e r a t e o f r e t u r n on the p o r t f o l i o , the time to m a t u r i t y and the r a t e o f i n t e r e s t , s t i l l h o l d s . Only the v a r i a n c e r a t e i s unobservable,but t h i s can be estimated from h i s t o r i c a l data. 28 Schwartz goes on to p o i n t out t h a t i f equation ( 2-17 ) d i d not h o l d i n the sense t h a t the c a l c u l a t e d v a l u e was not the market e q u i l i b r i u m value o f the o p t i o n , then a r b i t r a g e p r o f i t s would e x i s t . W i t h i n t h i s framework, the t o t a l value o f the c o n t r a c t t o the i n s u r e d , at the time the instrument i s created, can be expressed as the present v a l u e o f the guaranteed amount p l u s the v a l u e o f the c a l l o p t i o n : ( 2-21 ) PV n Cb(tO) = g(t) e " r t 4 w ( x ( 0 ) , t , g(t)) where P V Q ( b ( t ) ) i s the present value of the b e n e f i t s . S i m i l a r l y , the present value o f the b e n e f i t s may be expressed i n terms o f the present value o f the refe r e n c e p o r t f o l i o and the valu e o f the put op t i o n : ( 2-22 ) PV 0 ( b ( t ) ) = PV Q ( x ( t ) ) * p ( x ( 0 ) , t , g ( t ) ) Since the value o f the c a l l o p t i o n i s given by equation ( 2-17 ) , the value o f the put may be c a l c u l a t e d from ( 2-21 ) and ( 2-22 ). That i s : ( 2-23 ) p ( x ( 0 ) , t , g ( t ) ) = w(x(0), t , g ( t ) ) +' g ( t ) e " r t - P V Q ( x ( t ) ) 29 Under e q u i l i b r i u m c o n d i t i o n s , and s t a y i n g w i t h i n the s t a t e d assumptions o f the previous s e c t i o n , the present v a l u e of the b e n e f i t s may be viewed as the premium charged f o r the c o n t r a c t . I t i s important to note t h a t w i t h i n t h i s framework, the value o f the c a l l o p t i o n must be determined before the premium can be e s t a b l i s h e d . Since a l l the parameters o f the v a l u a t i o n formula, ( 2-17 ) , are known, or can be estimated, the procedure becomes mechanical. I t should be recognized, however, t h a t from the p o i n t o f view of the company, the most r e l e v a n t c a l c u l a t i o n i s the determination of the value of the put o p t i o n . In a s t r i c t sense, t h i s amount represents the charge f o r the guarantee, or the amount which the company r e c e i v e s f o r assuming the investment r i s k . I t was i n d i c a t e d i n the d i s c u s s i o n of the Black-Scholes o p t i o n p r i c i n g model t h a t a hedged p o s i t i o n may be formed, such t h a t no gains or l o s s e s w i l l be encountered as long as a continuous r e v i s i o n p o l i c y i s observed. This hedged p o s i t i o n was formed by going long i n the s e c u r i t y and short i n the c a l l o p t i o n . The same l o g i c i s a p p l i c a b l e to the insurance case i n q u e s t i o n . Since the company has s o l d a c a l l o p t i o n on the reference p o r t f o l i o s h o r t , i n order t o e l i m i n a t e the a s s o c i a t e d r i s k , i t must take a long p o s i t i o n i n the p o r t f o l i o . In order t o m a i n t a i n the hedged p o s i t i o n , i t must r e v i s e the long p o s i t i o n c ontinuously. That i s from equation ( 2-14 ) i t can be shown t h a t the p o s i t i o n i n the p o r t f o l i o i s equal t o ( 2-24 ) xw., (x,t) 30 where x i s the v a l u e of the reference p o r t f o l i o and w^ ( x , t ) , as b e f o r e , the p a r t i a l d e r i v a t i v e o f the o p t i o n value w i t h respect t o the f i r s t arguement. This may a l s o be i n t e r p r e t e d as the p r o p o r t i o n which a c u t a l l y must be i n v e s t e d i n the reference p o r t f o l i o . From t h i s i t f o l l o w s t h a t not a l l the funds have to be a c t u a l l y i n v e s t e d i n the p o r t f o l i o . This can be shown by the f o l l o w i n g arguments. F i r s t l y , the v a l u e o f the o p t i o n must always be l e s s than or equal to the value of the p o r t f o l i o , ( i e . w-< x ). Secondly, i t must exceed or be equal to the d i f f e r e n c e between the p o r t f o l i o v a l u e and the e x e r c i s e p r i c e ( i e . wr> x-e). T h i r d l y , i t i s assumed t h a t w(x,t) i s concave upward. Given these boundary c o n d i t i o n s , i t f o l l o w s t h a t the p a r t i a l d e r i v a t i v e w i t h respect to x i s i n c r e a s i n g always w i t h i n the s p e c i f i e d range. In f a c t the range o f the d e r i v a t i v e must be: ( 2-25 ) 0< w± ( x , t ) - ^ l I t w i l l equal zero i f (x-e) i s l e s s than or equal to zero. On the other hand, w^ (x,t) can o n l y equal one i f x becomes i n f i n i t e ( i e . w^( «>,t)=l). From ( 2-25 ) i t f o l l o w s t h a t : ( 2-26 ) x wn ( X,t ) < X 31 2.5 Summary B r i e f l y , i n t h i s s e c t i o n , i t was necessary to f o l l o w the Schwartz d i s s e r t a t i o n very c l o s e l y , as i t provides the t h e o r e t i c a l framework f o r the subsequent s i m u l a t i o n problem. In essence, i t i s t h i s framework t h a t provided the i n i t i a t i v e to b u i l d a s i m u l a t i o n model and t e s t the hypothesis, t h a t given t r a n s a c t i o n s c o s t s , a hedging s t r a t e g y o f t h i s nature w i l l s t i l l reduce d i s a s t e r l o s s e s . Because of t r a n s a c t i o n s c o s t s , a continuous model i s not p r a c t i c a l . Because of t h i s l i a b i l i t y , c e r t a i n assumptions must be made concerning the d i s t r i b u t i o n o f returns on the market or p o r t f o l i o , about the s i z e o f the t r a n s a c t i o n s costs per r e v i s i o n and the r e v i s i o n schedule i t s e l f . The next chapter o f the a n a l y s i s deals w i t h these assumptions, the foundations o f s i m u l a t i o n and the subsequent model developed f o r the hypothesis t e s t . I t a l s o considers the r e s u l t s of the v a r i o u s a l t e r n a t i v e s t e s t e d . 32 Chapter 3 Development of the S i m u l a t i o n Model  B a s i c Concepts o f S i m u l a t i o n In the most general sense, s i m u l a t i o n may best be des c r i b e d as the process o f d e s i g n i n g , b u i l d i n g , v a l i d a t i o n , a n a l y s i s and o p e r a t i o n of a formal model designed to represent only those features o f the system under study which are b e l i e v e d t o be s i g n i f i c a n t i n view o f the o b j e c t i v e s behind the i n v e s t i g a t i o n . In other words, i t i s the sy n t h e s i s and a n a l y s i s o f a system w i t h the f u n c t i o n i n g of the r e a l system being represented. This does not, however, mean t h a t the system being modeled must e x i s t . For example, c e r t a i n p h y s i c a l phenomena simply take too long f o r the a n a l y s t to observe, whereas a s i m u l a t i o n program can reduce the time f a c t o r such t h a t i t becomes very simple to study the problem. S i m u l a t i o n i s not intended to provide optimal s o l u t i o n s to the problem, r a t h e r i t permits the a n a l y s t to employ an a l g o r i t h m , the parameters of which may be a l t e r e d by the a n a l y s t so t h a t a range of s o l u t i o n s may be generated. In t h i s sense, the model i s p r e d i c t i v e , given t h a t c e r t a i n assumptions about the r e l e v a n t parameters have been made. In the case o f the e q u i t y l i n k e d l i f e insurance c o n t r a c t , given the co m p l e x i t i e s o f the problem and the p o t e n t i a l c o s t s i n v o l v e d , c l e a r l y , a co n s i d e r a b l e amount o f research and a n a l y s i s must be performed before a company should undertake such a pro p o s a l . Since the system discussed i n the previous chapter does not e x i s t i n the r e a l w o r l d , 33 and t h e r e f o r e i s unobservable, the a n a l y s t must b u i l d a model of the t h e o r e t i c a l framework and s i m u l a t e r e a l w o r l d e x t e r n a l i t i e s which can a f f e c t the model i n order t o determine the s e n s i t i v i t y o f the model t o these e x t e r n a l i t i e s . This i m p l i e s t h a t i t i s not enough t o q u a n t i f y the model parameters, but a l s o the r e l e v a n t e x t e r n a l i t i e s , i f p o s s i b l e ; i f not, then c e r t a i n assumptions must be made about them. These e x t e r n a l i t i e s are the t o p i c s o f d i s c u s s i o n i n the next s e c t i o n . 3.2 The Relevance of P o r t f o l i o Composition In the previous chapter, under the d i s c u s s i o n o f the t h e o r e t i c a l model, constant reference was being made t o the r e f e r e n c e p o r t f o l i o o f s e c u r i t i e s ; . The a c t u a l composition of the p o r t f o l i o was not considered a t a l l however. The obvious question t h e r e f o r e , i s the relevance o f the composition o f the p o r t f o l i o to the model being developed. P o r t f o l i o theory suggests t h a t the i n v e s t o r should simply buy the market p o r t f o l i o and r e v i s e i t o n l y to m a i n t a i n the r a t i o s . That i s , g i v e n the amount o f wealth a v a i l a b l e f o r investment i t should be d i s t r i b u t e d among the s e c u r i t i e s i n such a way t h a t the r a t i o o f the amount i n v e s t e d i n each s e c u r i t y to the t o t a l wealth i s the same as the r a t i o o f the v a l u e of the s e c u r i t i e s J of each company to the t o t a l value of the market p o r t f o l i o . I f the r a t i o s i n the market p o r t f o l i o change r e s u l t i n g from the reinvestment o f d i v i d e n d s , new i s s u e s , e t c . , then the investment p o r t f o l i o should be a l t e r e d so t h a t the r a t i o s remain the same. 34 I f the p o l i c y of continuous r e v i s i o n c o u l d be pursued, ( i e . no t r a n s a c t i o n s c o s t s ) , then the composition o f the r e f e r e n c e p o r t f o l i o i s i r r e l e v a n t s i n c e the amount i n v e s t e d i n the p o r t f o l i o depends on the c u r r e n t value of the p o r t f o l i o , not on an i m p l i c i t r i s k r e t u r n r e l a t i o n s h i p which has been e s t a b l i s h e d . Looking a t i t from another angle, s i n c e the o b j e c t i v e under the continuous r e v i s i o n s t r a t e g y i s to be f u l l y hedged a t a l l times, i t becomes i r r e l e v a n t what the value o f the p o r t f o l i o i s , or what the r e t u r n on the p o r t f o l i o has been. This must be t r u e , because as i t was s t a t e d b e f o r e , i f a f u l l y hedged p o s i t i o n i s maintained, then the insurance company w i l l not experience lo s s e s or gains as i t has not assumed any r i s k . In t h i s sense, only the v a r i a n c e o f the reference p o r t f o l i o i s o f importance because the value o f the c a l l o r put o p t i o n depends on the v a r i a n c e r a t e . Since i n a p r a c t i c a l s i t u a t i o n , o n ly a d i s c r e t e or p e r i o d i c r e v i s i o n p o l i c y can be pursued, because o f t r a n s a c t i o n s c o s t s , the company w i l l be exposed t o some r i s k . I n t h i s sense, the composition of the p o r t f o l i o does become r e l e v a n t , as does the r e t u r n on the p o r t f o l i o . The extent to which t h i s i s t r u e depends on the number o f r e v i s i o n s t h a t the company can undertake during the l i f e o f the c o n t r a c t . That i s , assuming t h a t the e x p i r a t i o n date i s known f o r c e r t a i n , the company i s s u b j e c t to p o r t f o l i o r e l a t e d r i s k from the l a s t r e v i s i o n p o i n t to the e x p i r a t i o n date, g i v e n t h a t a market c o l l a p s e has not occurred i n the previous p e r i o d s . 35 For the purposes of this analysis, i t is assumed that the insurance company forms the reference portfolio by buying the market portfolio, as represented by the Toronto Stock Exchange ( ie. TSE ). Given this assumption, i t is only logical that the variance rate to be employed by the model be the observed historical rate of the TSE. It should be noted, however, that the amount to be invested in the reference portfolio is s t i l l governed by the differential equations discussed in the previous chapter. It is assumed that the cost of maintaining the market portfolio is exogenous to the model under consideration. 3 . 3 The Return on the Portfolio Within the simulation framework, i t becomes necessary to generate a rate of return on the reference portfolio. The simplest way to achieve this objective is to employ a random number generator, which is capable of generating from various distributions. Given the Black-Scholes assumption that security prices are lognormally distributed, with a constant variance rate, i t is assumed that the returns on the market portfolio are lognormally distributed, with the same variance restriction. This is the position taken with respect to the simulation model. 36 Since most random number generators do not generate d i r e c t l y from a lognormal d i s t r i b u t i o n , but from a standard normal, transformations become necessary. The s i m p l e s t r e c o n c i l i a t i o n can be presented by reviewing the two-parameter d i s t r i b u t i o n ' s d e f i n t i o n , as g i v e n by A i t c h i s o n and Brown ( 1 ). Assuming an e s s e n t i a l l y p o s i t i v e v a r i a t e X(0< x<°° ) such t h a t : ( 3-1 ) Y = lnX 2 i s normally d i s t r i b u t e d w i t h mean u and v a r i a n c e v , then i t can be 2 s a i d t h a t X i s lognormally d i s t r i b u t e d , and w r i t e X i s A (u,v ) and 2 correspondingly Y i s N(u, v ). The d i s t r i b u t i o n o f X i s completely 2 s p e c i f i e d by the two parameters u and v . Obviously, t h i s i s the s i m p l e s t n a t u r a l d e f i n i t i o n . I t i s e v i d e n t , however, t h a t X cannot assume zero values as the t r a n s f o r m a t i o n Y = In X i s not d e f i n e d f o r X = 0. Since X and Y have the r e l a t i o n s h i p Y=lnX, the d i s t r i b u t i o n s o f X and Y are r e l a t e d by: ( 3-2 ) A(x) = N(lnx) (x> 0) ( 3-3 ) A (x) = 0 (x <0) ( 3-4 ) and d A (x)=£L/xv(2 ^ ) J s ) e x p ( - ( l / 2 v 2 ) ( l n x - u ) 2 ) dx (x>0) 37 which describes the frequency curve with a single mode at: -2 r -7 r -\ U - V ( 3-5 ) x = e The mean may be defined as: ( 3-6 ) x = e u + ^ 2 and the variance: ? ^ 2 r , 7 A 2 2u + v (e v - 1) ( 3 - 7 ) v = e v ^ - 2 2 ( 3-8 ) = x z n where n is the coefficient of variation of the distribution. The median is simply e u. It should be noted that the two-parameter lognormal distribution does possess reproductive properties, which is the justi f i c a t i o n for the assumption that the returns on securities i s distributed lognormally. That i s , i f the return R on security j , at time t, is given by: ( 3-9 ) R. = (P- " P- )/ P-3t 3 t J t - 1 J t ^ l 38 where P i s the p r i c e of the s e c u r i t y , and has a A( u,^) , i t f o l l o w s t h a t R has a . A d i s t r i b u t i o n from the c o r o l l a r y t h a t : ( 3-10) l n X 1 - l n X 2 = l n ^ / X ^ implying t h a t the lognormal d i s t r i b u t i o n w i l l have d i v i s i b l e r e p r o d u c t i v e p r o p e r t i e s . In order t o generate values from a lognormal d i s t r i b u t i o n w i t h a known mean SM and a standard d e v i a t i o n SX, only the f o l l o w i n g t r a n s f o r m a t i o n i s necessary: 0 ( 3-11) v = (In(1.0 * XS 2/XM 2 ( 3-12 ) and x = Tri (XM) - h ( v 2 ) Since the generator s e l e c t s a value s from S which i s N(0,1), the f o l l o w i n g e v a l u a t i o n occurs: ( 3-13 ) x = exp (X + v;s ) In the s i m u l a t i o n program, the mean r e t u r n was s p e c i f i e d as 8% and the va r i a n c e r a t e on the TSE as .01846. 39 I t i s recognized t h a t i n r e a l i t y , a computer random number generator i s at best a pseudo-random number generator. The b i a s which t h i s may i n t r o d u c e , ' i s probably so minimal t h a t i t i s not worthwhile to pursue the e f f e c t by performing randomness t e s t s . 3.4 The S i m u l a t i o n Program Given t h a t t r a n s a c t i o n s costs are t o be i n c l u d e d i n the a n a l y s i s , i t i s no longer v a l i d from a p r a c t i c a l p o i n t of view t o assume that continuous adjustment o f the r a t i o o f the long p o s i t i o n i n the reference p o r t f o l i o t o the s h o r t i n the c a l l o p t i o n on t h i s p o r t f o l i o i s p o s s i b l e . Since, as s t a t e d p r e v i o u s l y , the o b j e c t i v e of t h i s a n a l y s i s i s to examine the p o t e n t i a l l o s s e s which may be i n c u r r e d by the company under a d i s c r e t e r e v i s i o n p o l i c y , i t i s necessary to determine or d e f i n e the types o f losses which may occur. Reviewing the previous arguments, i t becomes evident t h a t two types o f p o t e n t i a l l o s s e s may occur. The f i r s t type may be viewed as simply a d d i t i o n a l c o s t s o f conducting business, a r i s i n g from the f a c t t h a t t r a n s a c t i o n s costs are now r e l e v a n t . In t h i s sense the word l o s s i s a misnomer as the company w i l l simply charge the p o l i c y holder f o r these a d d i t i o n a l c o s t s ; but f o r the sake o f s i m p l i c i t y , the aforementioned terminology w i l l be adhered t o . The magnitude o f these lo s s e s over the l i f e o f the c o n t r a c t w i l l be r e l a t e d to the s i z e of the imbalances 40 i n t h e hedged p o s i t i o n a t t h e t i m e o f r e v i s i o n and t h e number o f r e v i s i o n s p l a n n e d f o r , d u r i n g t h e c o n t r a c t p e r i o d . T h i s p o i n t w i l l be p u r s u e d f u r t h e r i n a n o t h e r s e c t i o n . The s e c o n d t y p e o f l o s s may be d e f i n e d as a " d i s a s t e r l o s s " . A l t h o u g h i t i s e x p e c t e d t h a t t h e o c c u r e n c e o f t h i s t y p e s o f l o s s s h o u l d be i n f r e q u e n t , n e v e r t h e l e s s , f r o m t h e p o i n t o f v i e w o f t h e company, t h e s e a r e v e r y i m p o r t a n t . T h i s t y p e o f l o s s c a n o c c u r f r o m a g e n e r a l c o l l a p s e i n t h e m a r k e t . L o o k i n g a t i t f r o m a n o t h e r a n g l e , i f t h e v a l u e o f t h e r e f e r e n c e p o r t f o l i o , p l u s t h e amount i n v e s t e d i n t h e r i s k f r e e a s s e t i s l e s s t h a n t h e g u a r a n t e e d amount, t h e company must e i t h e r have t h e a b i l i t y t o b o r r o w , o r f a c e b a n k r u p t c y . I t s h o u l d be n o t e d t h a t t h e c r i t e r i a o f t h e r e f e r e n c e p o r t f o l i o b e i n g g r e a t e r t h a n t h e g u a r a n t e e d amount i s o v e r s t a t i n g t h e r e q u i r e m e n t by t h e amount i n v e s t e d i n t h e r i s k f r e e a s s e t . As s t a t e d p r e v i o u s l y , i t i s n o t n e c e s s a r y , and as a m a t t e r o f f a c t , s u b o p t i m a l t o i n v e s t a l l o f t h e premium i n t h e r e f e r e n c e p o r t f o l i o . C o n s e q u e n t l y , t h e e x c e s s c a n be i n v e s t e d i n a r i s k f r e e a s s e t , assuming no a d d i t i o n a l c o s t s b e s i d e s t h e t r a n s a c t i o n s r e q u i r e m e n t s . The i n i t i a l p a r a m e t e r s o f t h e s i m u l a t i o n program were s e t a r b i t r a r i l y . More s p e c i f i c a l l y : M a r k e t r e t u r n 8% V a r i a n c e 0.01846 41 R i s k f r e e r a t e 6% Contract p e r i o d 10 years Number o f r e v i s i o n s 10 Transactions costs 1% Guaranteed amount $100.00 I n i t i a l v a l u e , Ref. P o r t . $100.00 Number of s i m u l a t i o n s 500 The r i s k f r e e r a t e , c o n t r a c t p e r i o d , guaranteed amount and i n i t i a l investment i n the reference p o r t f o l i o were h e l d constant f o r a l l the s i m u l a t i o n s . The number o f s i m u l a t i o n s were v a r i e d from 500 to 2,000 i n order to e s t a b l i s h the s t a b i l i t y of the r e s u l t s . In g e n e r a l , i t was found t h a t about a one one-hundredth cent change occurred i n the mean l o s s e s i f the number o f s i m u l a t i o n s were expanded from 500 t o 2,000. About the same magnitude change occurred i n the standard d e v i a t i o n o f these l o s s e s . Consequently, 500 s i m u l a t i o n s were adopted f o r a l l the runs, as the changes described were deemed to be i n s i g n i f i c a n t . I n i t i a l l y , the t r a n s a c t i o n s costs were pe r m i t t e d to vary from 1% to 2.5% by increments o f 0.5%. T h i s , i n e f f e c t , r e s u l t s i n a c o s t o f 2 to 5% to get i n and out of the market, which i s f a i r l y r e p r e s e n t a t i v e of r e a l i t y . These costs may be d i v i d e d i n the f o l l o w i n g manner: a) The cost o f buying the i n i t i a l p o r t f o l i o . b) R e v i s i o n c o s t s . c) The cost o f s e l l i n g the p o r t f o l i o . 42 In t h i s sense, i t i s assumed t h a t at the end o f the c o n t r a c t , the reference p o r t f o l i o must be l i q u i d a t e d . I t should a l s o be noted t h a t t r a n s a c t i o n s costs only apply to the reference p o r t f o l i o , not to the investment i n the r i s k f r e e asset. The l a t t e r i s assumed to be the equivalent o f a savings account, which t y p i c a l l y does not i n c u r r t r a n s a c t i o n s c o s t s . As s t a t e d p r e v i o u s l y , the mean r e t u r n on the p o r t f o l i o was assumed to be 8% w i t h a va r i a n c e of 0.01846. This was a l s o h e l d constant f o r the runs. Various arguments may be made about the appropriateness o f the assumed mean r e t u r n , but i t should be noted t h a t the c i r t i c a l assumption i s the v a r i a n c e r a t e . The r e v i s i o n parameter was p e r m i t t e d to vary across the s i m u l a t i o n s . I n i t i a l l y , annual r e v i s i o n s were adopted as the p o l i c y o f the insurance company, but i n subsequent s i m u l a t i o n s i t was changed to s i x month, four month and three month i n t e r v a l s . In essence the s i m u l a t i o n paramenters may be def i n e d as the t r a n s a c t i o n c o s t range and the r e v i s i o n p o l i c y range, as per the p r e d e f i n e d increments. In order to f a c i l i t a t e an e a s i e r understanding o f the computer program p r o v i d e d i n Appendix ( A ) , some of the n o t a t i o n o f the previous chapters w i l l be a l t e r e d to conform to t h a t o f the program. Since some of the mechanics o f the program are not r e l e v a n t t o t h i s d i s c u s s i o n , or may be summarized i n one equation, a d i c t i o n a r y o f the 43 v a r i a b l e s and f u n c t i o n s has been pr o v i d e d at the end o f the program i n order to c l a r i f y the l o g i c . The main p o r t i o n of the program may be viewed i n f o u r stages: a) The i n i t i a l p e r i o d . This i s synonymous to the c r e a t i o n o f the c o n t r a c t and the subsequent v a l u a t i o n of the b e n e f i t s . b) vThe r e v i s i o n p o l i c y . This p o r t i o n deals w i t h the r e v i s i o n of the hedged p o s i t i o n , g i v e n the h y p o t h e t i c a l market r e t u r n f o r the p e r i o d , as d e s c r i b e d by the f i r m ' s p o l i c y . c) Termination o f the c o n t r a c t . This s e c t i o n determines the f i r m ' s performance w i t h respect to the c o n t r a c t under c o n s i d e r a t i o n . In e f f e c t , i t e s t a b l i s h e s the f i r m ' s l i a b i l i t y t o the p o l i c y ' s b e n e f i c i a r y . d) O v e r a l l performance e v a l u a t i o n . The f i n a l s e c t i o n may be viewed as an e v a l u a t i o n of the performance o f a l a r g e p o r t f o l i o o f c o n t r a c t s , managed under the same c r i t e r i a ( i e . w i t h respect t o r e v i s i o n and t r a n s a c t i o n s c o s t s ) . Given the assumption that a t the time of the i n i t i a t i o n o f the c o n t r a c t , the value of the reference p o r t f o l i o X, i s equal t o the guaranteed amount, from equations ( 2-17 ) to ( 2-20 ), the value o f the o p t i o n may be determined as f o l l o w s : ( 3-14 ) s i n c e from ( 3-15 ) d^ becomes ( 3-16 ) and so t h a t the valu e o f the o p t i o n at time zero becomes: ( 3-17 ) OPT(O) = X(0) N(d 1) - g ( t ) e " r t N(d 2) where the previous v a r i a b l e d e f i n i t i o n s apply. Having determined the value o f the o p t i o n , the f i r m ' s t o t a l l i a b i l i t y may be w r i t t e n as: ( 3-18 ) L{0) = g ( t ) e " r t + OPT(O) and the a c t u a l amount i n v e s t e d i n the reference p o r t f o l i o x, as: ( 3-19 ) x (0) = X(0) • NCcy l n ( X ( 0 ) / g ( t ) = 0 ln(100/100) = 0 d x = ((r + v 2 ) ( t ) ) / v ( t ) i 2 d2 = d 1 - v ( t ) % 45 Since the i n i t i a l wealth p o s i t i o n must equal the i n i t i a l l i a b i l i t y o f the f i r m , a f t e r the investment i n the reference p o r t f o l i o i s made, the wealth p o s i t i o n becomes: ( 3-20 ) W(0) = L(0) - (x(0) * TR) where TR i s the i n c u r r e d t r a n s a c t i o n s c o s t , as a percentage. The amount i n v e s t e d i n the r i s k f r e e asset becomes: ( 3-21 ) RF(0) = W(0) - x(0) These equations summarize the company's p o s i t i o n a t the c r e a t i o n of the c o n t r a c t . I t should be noted t h a t s i n c e i t i s assumed t h a t the i n i t i a l v a lue o f the reference p o r t f o l i o equals the guaranteed amount, the i n i t i a l investment i n the reference p o r t f o l i o w i l l be i d e n t i c a l f o r a l l the s i m u l a t i o n s . The i n i t i a l wealth w i l l depend on the s i z e o f the t r a n s a c t i o n s costs i n the c u r r e n t c a l c u l a t i o n s , s i n c e L(0) and x(0) are constant i n equation (3-20). S i m i l a r l y , the amount i n v e s t e d i n the r i s k f r e e asset depends on the value o f W(0) i n equation (3-21). The second stage o f the program evaluates the above r e l a t i o n s h i p s a t d i s c r e t e p o i n t s i n time, as d e f i n e d by the r e v i s i o n p o l i c y under c o n s i d e r a t i o n . A t each r e v i s i o n p o i n t a r a t e o f r e t u r n 4 6 i s generated, so t h a t the value of the reference p o r t f o l i o a t time t i s : ( 3-22 ) X ( t ) = X ( t - l ) * Z ( At) where Z i s the simulated r e t u r n on the p o r t f o l i o . That i s , i f Z i s g r e a t e r than one, the r e s u l t i s a p r o f i t o r i n c r e a s e i n the value of the p o r t f o l i o . I f i t i s l e s s than one, a l o s s or r e d u c t i o n i n the value i s i m p l i e d . Since the value of the reference p o r t f o l i o has changed, the hedged p o s i t i o n has a l s o changed, t h e r e f o r e the investment i n the p o r t f o l i o must be a l t e r e d . From equations (3-14) to (3-17) the value of the o p t i o n may be c a l c u l a t e d f o r time t , and from (3-19) the hedged p o s i t i o n may be r e - e s t a b l i s h e d . The new wealth p o s i t i o n at time t can be d e f i n e d as: ( 3-23 ) W(t) = (W(t-l) - x ( t - l ) ) »• e r A Z \ + x ( t - l ) ? Z (A t ) - ( ( x ( t ) - x ( t - l ) ) ) TR The f i r m ' s l i a b i l i t y a t time t i s determined from equation (3-18). The t h i r d s e c t i o n o f the program determines the f i n a l f i n a n c i a l p o s i t i o n o f the f i r m w i t h respect to the c o n t r a c t . The f i n a l w ealth p o s i t i o n i s determined by equation (3-23), but i t should be noted t h a t the t e r m i n a l t r a n s a c t i o n s costs are c o n s i d e r a b l y l a r g e r than f o r the r e v i s i o n s , as the whole p o r t f o l i o i s l i q u i d a t e d . To c l a r i f y the p o i n t , the f i n a l wealth p o s i t i o n may be s i m p l i f i e d t o : ( 3-24 ) W(t*) = (W(t*-1) - x ( t * - l ) e r A t ; + x ( t * - l ) •. Z) r ( ( x ( t * - l ) Z) • TR) where t * i s the m a t u r i t y date. The value o f the o p t i o n a t m a t u r i t y ( 3-25 ) OPT(t*) = X ( t * ) * g f o r X ( t * ) >g, or: ( 3-26 ) OPT(t*) = 0 f o r X ( t * ) <g. The company's l i a b i l i t y , t h e r e f o r e , equals: C 3-27 ) L ( t * ) = g + OPT(t*) The l i a b i l i t y may a l s o be expressed as: ( 3-28 ) L ( t * ) = X ( t * ) f o r X.(t*) >g, or ( 3^29 ) L ( t * ) = g f o r X ( t * ) <g. I f the value o f the reference p o r t f o l i o i s g r e a t e r than the guarantee, then the p r o f i t to the f i r m i s : ( 3-30 ) PR(t*) = W(t*) « X ( t * ) otherwise i t i s : ( 3-31 ) PR(t*) = WCt*) - g Since no p r o v i s i o n s were made f o r t r a n s a c t i o n s costs a t the c r e a t i o n of the c o n t r a c t , the p r o f i t represents the amount the company must charge the in s u r e d at the c r e a t i o n o f the c o n t r a c t , over and above the other c o s t s , i n order t o break even. I n t h i s sense, the p r o f i t i s negative. I t should be noted, however, t h a t i n very unique cases the p r o f i t may i n f a c t be p o s i t i v e . This s i t u a t i o n can a r i s e i f the value o f the reference p o r t f o l i o i s l e s s than the guarantee, but the wealth p o s i t i o n i s gre a t e r . This i s p o s s i b l e s i n c e a p o r t i o n o f the premium i s i n v e s t e d i n the r i s k f r e e a s s e t , so t h a t i n c e r t a i n i n s t a n c e s , the f o l l o w i n g c o n d i t i o n s may r e s u l t : ( 3-32 ) X ( t * ) <g but ( 3-33 ) X ( t * ) + RF(t*)> g 49 where RF(t*) represents the t e r m i n a l value o£ the investment i n the r i s k f r e e a s s e t , so t h a t W(t*) i s a c t u a l l y g r e a t e r than the guaranteed amount. Since the outcome des c r i b e d by equation (3-32) has occurred, the b e n e f i c i a r y would n a t u r a l l y e l e c t to r e c e i v e the guaranteed amount, l e a v i n g the p o s i t i v e p r o f i t given by (3-31) to the company. The s c a r c i t y o f t h i s phenomenon i s evidenced by the f a c t t h a t i t only occurred under the c o n d i t i o n of annual r e v i s i o n and one percent t r a n s a c t i o n c o s t c r i t e r i a . Appendix B provides examples o f the i n i t i a l , intermediate and t e r m i n a l values d e r i v e d from the previous equations, f o r v a r i o u s r e v i s i o n s t r a t e g i e s and t r a n s a c t i o n s c o s t s . The summary page of the appendix provides a d e s c r i p t i o n o f the v a r i a b l e s l i s t e d and a r e c o n c i l i a t i o n o f those v a r i a b l e s t o the equations o f t h i s s e c t i o n . To t h i s p o i n t , the arguments presented have d e a l t w i t h only the s i n g l e s i m u l a t i o n o r c o n t r a c t . The r e s t of the program focuses on the accumulation and p r o c e s s i n g o f the r e l e v a n t i n f o r m a t i o n over the whole s i m u l a t i o n . For each s i m u l a t i o n the program s t o r e s the t e r m i n a l value of the reference p o r t f o l i o , the l o s s ( p r o f i t ) a s s o c i a t e d w i t h i t , and the c l a s s i f i c a t i o n of the l o s s i n t o t r a n s a c t i o n cost d e r i v e d , or d i s a s t e r l o s s , as per the c r i t e r i o n d i s c ussed p r e v i o u s l y . That i s , i f the l o s s occurs because the value of the reference p o r t f o l i o i s l e s s than the guaranteed amount, r e s u l t i n g i n the b e n e f i c i a r y 50 e x e r c i s i n g the guarantee, the loss i s deemed to be d i s a s t e r l o s s . Otherwise, the loss i s assumed to be the r e s u l t of the transactions costs incurred because of the r e v i s i o n p o l i c y under consideration. Upon the completion of the simulation under the e x i s t i n g parameter values, the program c a l c u l a t e s the mean and standard deviation of the losses and of the value of the reference p o r t f o l i o and summarizes the s t a t i s t i c s i n the form of table s . (Refer to Appendix D). Tables are also produced f o r the mean and standard deviation of the d i s a s t e r losses. ( Appendix E). In order to analyse the magnitude of ordinary or tra n s a c t i o n cost losses, the program a l s o produces tables of the actual values, mean and standard deviation of the la r g e s t f i v e percent and ten percent losses incurred, r e s p e c t i v e l y . These tables may be found i n Appendix F. The f i n a l s e c t i o n of the program summarizes the above s t a t i s t i c s across a l l the simulations. The tables generated represent the mean and standard deviation of the above described loss categories under the various combinations of transaction cost l e v e l s and r e v i s i o n p o l i c i e s . These may be r e f e r r e d to i n Appendix E. With s l i g h t m o d i f i c a t i o n to the program, the naive strategy was also tested. This a l t e r n a t i v e may simply be viewed as the formation of the reference p o r t f o l i o at the cr e a t i o n of the contract, with the assumption that no r e v i s i o n s w i l l take place during i t s l i f e . Under t h i s p o l i c y the only a d d i t i o n a l costs i n c u r r e d are the i n i t i a l and t e r m i n a l t r a n s a c t i o n s c o s t s . The r e s u l t s of t h i s run are presented i n Appendix C. 52 Chapter 4 A n a l y s i s of Re s u l t s The a n a l y s i s o f the r e s u l t s proceeds from an i n depth examination o f the intermediate r e l a t i o n s h i p s to the e v a l u a t i o n o f the s i m u l a t i o n output. Because o f the volume o f output provided by the s i m u l a t i o n , only the most r e l e v a n t s t a t i s t i c s have been accumulated i n t a b u l a r form i n the appendices. As a r e s u l t , an examination o f each o f the appendices i s presented. 4.1 Intermediate R e s u l t s Appendix B provides the r e s u l t s o f intermediate c a l c u l a t i o n s f o r one percent t r a n s a c t i o n s costs and annual, s i x and f o u r month r e v i s i o n s , r e s p e c t i v e l y . I t should be noted t h a t the i n i t i a l investment i n the reference p o r t f o l i o , X-J, the value o f the reference p o r t f o l i o , RPORT, the i n i t i a l w ealth WLTH, the value of the o p t i o n , OPVAL,and the l i a b i l i t y to the f i r m , LIAB, i s the same f o r a l l s i m u l a t i o n s . RAND-Z i s the r e t u r n on the reference p o r t f o l i o i n any p e r i o d under c o n s i d e r a t i o n . The f i r s t two examples o f Table 1 c l e a r l y i n d i c a t e t h a t the b e n e f i c i a r y w i l l not e x e r c i s e the guarantee, but e l e c t to r e c e i v e the te r m i n a l value o f the reference p o r t f o l i o , s i n c e i n both cases t h i s 53 exceeds the value o f the guarantee. In the t h i r d example, however, the guarantee i s e x e r c i s e d , as the value o f the reference p o r t f o l i o i s l e s s than the guarantee. T h i s , i n e f f e c t , i s an example o f the d i s a s t e r losses discussed i n the previous chapters. This t a b l e a l s o shows the impact o f the d e c l i n e i n the value o f the reference p o r t f o l i o w i t h respect t o the a c t u a l investment i n the p o r t f o l i o . The f l u c t u a t i o n s i n the re t u r n s and subsequent l o s s e s on the reference p o r t f o l i o r e s u l t i n the value o f the o p t i o n becoming zero, and the investment i n the p o r t f o l i o being reduced c o n s i d e r a b l y . On the other hand, when p o s i t i v e r e t u r n s are experienced, as gi v e n i n t a b l e s two and t h r e e , the investment i n the reference p o r t f o l i o i s almost equal to the value o f the p o r t f o l i o . That i s , the q u a n t i t y X ( t ) •. N(d-^) i s almost equal to one, and i n f a c t becomes one i n some of the cases. Since i n these cases the investment i n the reference p o r t f o l i o i s great e r than the a c t u a l wealth p o s i t i o n of the company, s m a l l amounts of borrowing must occur i n order to a r r i v e a t the t e r m i n a l c o n d i t i o n s of the p a r t i c u l a r s i m u l a t i o n . I t i s assumed t h a t t h i s borrowing i s accomplished a t zero cost f o r s i m p l i c i t y . The increase i n the l o s s e s , ( i e . the negative p r o f i t values ), can be a t t r i b u t e d to the increase i n t r a n s a c t i o n costs r e s u l t i n g from the a d d i t i o n a l r e v i s i o n s undertaken. I t i s important to recognize t h a t doubling or t r i p l i n g the number o f r e v i s i o n s does not n e c e s s a r i l y r e s u l t i n doubling o r t r i p l i n g the l o s s e s , r e s p e c t i v e l y . I n f a c t , the los s e s i n c u r r e d , as gi v e n i n 54 Table 3, are l e s s than those i n d i c a t e d i n Table 2, which r e s u l t from ten l e s s r e v i s i o n s . While the comparison of the r e s u l t s of two s i m u l a t i o n s i s inadequate t o draw concrete conclusions from, n e v e r t h e l e s s , the i m p l i c a t i o n i s present t h a t the l a r g e r the number or r e v i s i o n s , the l e s s the company i s exposed to abnormal gains or l o s s e s . Furthermore, i t f o l l o w s that the more o f t e n the p o r t f o l i o i s r e v i s e d , the s m a l l e r the r e q u i r e d change i n the investment i n the reference p o r t f o l i o , i n order to r e - e s t a b l i s h the hedged p o s i t i o n . The o v e r a l l i m p l i c a t i o n i s t h a t i f these a s s e r t i o n s can be s u b s t a n t i a t e d , then the f i r m ' s exposure to abnormal l o s s e s may be reduced by f o l l o w i n g the hedging s t r a t e g y , which i n e f f e c t i s the hypothesis of the a n a l y s i s . The intermediate r e s u l t s a l s o s u b s t a n t i a t e the argument that the v o l a t i l i t y o f the o p t i o n i s always g r e a t e r than the v o l a t i l i t y o f the s e c u r i t y or reference p o r t f o l i o . For example, an 11% change i n the value of the reference p o r t f o l i o , i n Table 1, r e s u l t s i n a 15.6% change i n the v a l u e of the o p t i o n . S i m i l a r l y , a 141 d e c l i n e i n the p o r t f o l i o value causes a 40% d e c l i n e i n the value o f the o p t i o n . I t should be noted t h a t the r e l a t i v e v o l a t i l i t y o f the o p t i o n i s dependent not only on the value o f the reference p o r t f o l i o , but a l s o on the m a t u r i t y p e r i o d . 55 4.2 The Naive Strategy Under the naive s t r a t e g y i t i s assumed t h a t the company simply i n v e s t s the t o t a l premium i n the reference p o r t f o l i o and does not pursue a r e v i s i o n s t r a t e g y , but holds the p o r t f o l i o u n t i l the t e r m i n a t i o n of the c o n t r a c t ( Appendix C). The i n i t i a l investment i n the p o r t f o l i o i s assumed to be $100.00 which i s only 21 cents l e s s than the c a l c u l a t e d premium under the r e v i s i o n s t r a t e g y o p t i o n . Since only i n i t i a l and t e r m i n a l t r a n s a c t i o n costs can be i n c u r r e d under t h i s s t r a t e g y , they are omitted from the c a l c u l a t i o n s f o r the sake o f s i m p l i c i t y . The number of s i m u l a t i o n s were s e t a t 500, as f o r the r e v i s i o n s t r a t e g y o p t i o n . The number of periods i s synonymous t o the r e v i s i o n p e r i o d concept. That i s , f o r example, 10 peri o d s i m p l i e s the c a l c u l a t i o n of the r e t u r n on the p o r t f o l i o a n n u a l l y , 20 p e r i o d s , semi-annually, e t c . Table 2 summarizes the d i s a s t e r l o s s e s the company may incur-by f o l l o w i n g t h i s s t r a t e g y . The number of l o s s e s r e f e r s to the number o f times the guarantee i s e x e r c i s e d over 500 s i m u l a t i o n s . The percentage simply re-expresses the number of l o s s e s i n terms o f the number o f s i m u l a t i o n s . The average l o s s i s the mean of the d i s a s t e r l o s s e s under the p a r t i c u l a r s t r a t e g y , the standard d e v i a t i o n , the d e v i a t i o n from t h a t mean. The outcomes under the v a r i o u s options o f the naive s t r a t e g y suggest t h a t the process i s random. That i s , i t i s i r r e l e v a n t how many periods are considered over the number of s i m u l a t i o n s . The average 56 l o s s e s and the d e v i a t i o n s do not d i s p l a y any t r e n d . I t i s i n t e r e s t i n g to note, however, the magnitude of the d e v i a t i o n s i n r e l a t i o n t o the magnitude o f the mean l o s s e s . This can be a t t r i b u t e d p a r t l y to the l i m i t e d number o f observations under c o n s i d e r a t i o n . When the i n d i v i d u a l l o s s magnitudes are considered, they appear to be random. The mean and the standard d e v i a t i o n of the l o s s e s become c r i t i c a l , as the naive s t r a t e g y s t a t i s t i c s are used as the benchmark to compare the r e v i s i o n s t r a t e g y a g a i n s t . The . f i r s t : . Table summarizes the l o s s e s i n terms o f the t o t a l number of s i m u l a t i o n s per s t r a t e g y performed, and the corresponding mean values of the reference p o r t f o l i o . Since under the naive s t r a t e g y the company cannot make a p r o f i t , the upper bound i s s e t a t zero. That i s , s i n c e there i s no investment i n a r i s k f r e e . a s s e t , the p o s s i b i l i t y o f t o t a l wealth exceeding the guaranteed amount when the value of the reference p o r t f o l i o i s l e s s than the guarantee cannot occur. As s t a t e d p r e v i o u s l y , t h i s outcome can only occur i f a hedging p o l i c y i s adopted. As expected, the average l o s s per 500 s i m u l a t i o n s i s very c l o s e t o zero. I n f a c t the range over the periods considered i s from -0.33 to -0.77 d o l l a r s . The l a r g e standard d e v i a t i o n s and the f a c t that zero i s a boundary c o n d i t i o n ( i e . no p r o f i t s p o s s i b l e ) , imply t h a t the d i s t r i b u t i o n i s very h i g h l y skewed t o the l e f t . In e f f e c t , the standard d e v i a t i o n provides a good i n d i c a t i o n o f the magnitude o f the l o s s e s i n c u r r e d , e s p e c i a l l y when compared to the average. 57 The average values o f the reference p o r t f o l i o over the number of periods i s a l s o w i t h i n expectations. The range o f the averages i s from $236.67 t o $255.03 which can be t r a n s l a t e d i n t o compounding $100.00 investment a t a continuous r a t e o f from 8.5% t o about 9.5%. This i s reasonable as the mean r e t u r n parameter on the reference p o r t f o l i o was s e t at 8%. One standard d e v i a t i o n from the mean t r a n s l a t e s i n t o a range o f about 4% t o 13%, which i s acceptable i n terms of the var i a n c e r a t e assumed. In g e n e r a l , i t may be concluded t h a t although the number o f d i s a s t e r l o s s e s compared t o the number of si m u l a t i o n s i s not very h i g h , ( i e . from 2 to 4.2%), g i v e n the average s i z e and d e v i a t i o n o f the l o s s e s , the s t r a t e g y cannot be considered as very e f f e c t i v e . The resits, o f the a n a l y s i s i s concerned w i t h comparing the r e v i s i o n s t r a t e g y to the naive procedure, and showing t h a t the former i s dominant over the l a t t e r . 4.3 The R e v i s i o n Strategy: O v e r a l l Losses Appendix D summarizes the o v e r a l l l o s s e s i n c u r r e d by the company under the va r i o u s r e v i s i o n s t r a t e g i e s . The average l o s s e s represent d i s a s t e r l o s s e s as w e l l as the t r a n s a c t i o n c o s t s a s s o c i a t e d w i t h r e v i s i o n s . The t a b l e s present summaries o f t r a n s a c t i o n s costs ranging from 1% t o 2.5%, as w e l l as the case i n which no t r a n s a c t i o n s c o s t s are i n c u r r e d . 58 This case i s s i m i l a r t o the naive s t r a t e g y discussed i n the previous s e c t i o n , except t h a t the investment i n the reference p o r t f o l i o i s r e v i s e d a t the end of each p e r i o d . As expected,as:the t r a n s a c t i o n c o s t s are increased the average l o s s a l s o i n c r e a s e s , h o l d i n g the number of r e v i s i o n s constant. The standard d e v i a t i o n s a s s o c i a t e d w i t h these means do not, however, increase a t a p r o p o r t i o n a l r a t e . That i s , doubling the t r a n s a c t i o n cost does not r e s u l t i n doubling the standard d e v i a t i o n . For example, doubling the r a t e from 1% to 2% increases the average l o s s from -6.98 to -13.66 i n the case of annual r e v i s i o n s , but the standard d e v i a t i o n only changes from 3.34 to 5.24 r e s p e c t i v e l y . I f t r a n s a c t i o n costs are h e l d constant, a comparison of the average losses over the v a r i o u s r e v i s i o n s t r a t e g i e s r e v e a l s t h a t the averages tend t o i n c r e a s e at a decreasing r a t e , as the r e v i s i o n p e r i o d i s i n c reased t e n p e r i o d s a t a time. This tends to h o l d f o r a l l the cases except the s p e c i a l case o f zero t r a n s a c t i o n s c o s t . In t h i s case, the average l o s s i s very c l o s e to zero. Furthermore, comparing the l a t t e r case t o the naive s t r a t e g y w i t h zero t r a n s a c t i o n s c o s t , r e s u l t s i n the c o n c l u s i o n t h a t i n a l l cases the r e v i s i o n s t r a t e g y i s dominant over the n a i v e , i n terms o f average l o s s e s . I f p o s i t i v e t r a n s a c t i o n costs are considered, then t h i s comparison cannot be made. 59 The standard d e v i a t i o n s o f the average lo s s e s under the various s t r a t e g i e s a l s o p r o v i d e v a l u a b l e i n f o r m a t i o n . Holding t r a n s a c t i o n costs constant and'increasing the number of r e v i s i o n s , r e s u l t s i n an i n i t i a l d e c l i n e i n the standard d e v i a t i o n , but as the number o f r e v i s i o n s reaches f o r t y to f i f t y o p e r c o n t r a c t p e r i o d , i t begins to r i s e . For the lower t r a n s a c t i o n c o s t l e v e l s , a f t e r e i g h t y r e v i s i o n s the d e v i a t i o n i s about the same or lower than f o r t e n r e v i s i o n s , but f o r the higher l e v e l s , i t exceeds the d e v i a t i o n a f t e r t e n r e v i s i o n s . I f the t r a n s a c t i o n costs are e l i m i n a t e d ( i e . s e t at 0%) , then i t becomes c l e a r t h a t the more o f t e n the p o r t f o l i o i s r e v i s e d , the lower the standard d e v i a t i o n of the l o s s e s . This i s e n t i r e l y c o n s i s t e n t w i t h expectations. Comparison o f these d e v i a t i o n s t o those d e r i v e d under the naive s t r a t e g y leads t o the c o n c l u s i o n t h a t the r e v i s i o n s t r a t e g y i s dominant. The d e v i a t i o n s are not o n l y d e c l i n i n g , but i n magnitude they are con s i d e r a b l y s m a l l e r than the ones d e r i v e d under the naive p l a n . As discussed p r e v i o u s l y , under the naive p l a n the d e v i a t i o n s do not d i s p l a y any tr e n d , but suggest t h a t they are random. The average value of the reference p o r t f o l i o , as given by the v a r i o u s t a b l e s , are w i t h i n expectations. The d o l l a r values t r a n s l a t e i n t o a 7.5 to 9.5% annual r e t u r n over t e n years on an i n i t i a l investment o f about a hundred d o l l a r s . This i s reasonable, g i v e n the o r i g i n a l assumptions about the mean r e t u r n and va r i a n c e on the market 60 p o r t f o l i o . I t should be noted t h a t the average value and d e v i a t i o n s of the reference p o r t f o l i o are the same f o r the naive s t r a t e g y and the zero and one percent r e v i s i o n s t r a t e g i e s because the same sequence of random numbers were generated i n these e v a l u a t i o n s . This f a c t tends t o r e i n f o r c e the c o n c l u s i o n t h a t i n terms o f t o t a l l o s s e s the r e v i s i o n s t r a t e g y i s s u p e r i o r t o t h a t of the naive p l a n , s i n c e the same percent l o s s e s are generated. The d i f f e r e n c e i n the r e s u l t s must then be a t t r i b u t e d to the e f f e c t s of f o l l o w i n g the hedging s t r a t e g y . The average values f o r the reference p o r t f o l i o s o f the other s t r a t e g i e s are d i f f e r e n t because a d i f f e r e n t sequence of random numbers are generated due t o the changes i n the r e v i s i o n p e r i o d s . The r e s u l t s are s t i l l acceptable, however, i n terms o f the o r i g i n a l parameters. 4.4 The R e v i s i o n Strategy: D i s a s t e r Losses As d e f i n e d p r e v i o u s l y , d i s a s t e r l o s s e s occur i f the guarantee i s e x e r c i s e d by the b e n e f i c i a r y because o f a general market d e c l i n e and the subsequent poor performance o f the reference p o r t f o l i o i n r e l a t i o n to the guarantee. Appendix E summarizes these l o s s e s over the va r i o u s t r a n s a c t i o n costs and r e v i s i o n s t r a t e g i e s . As noted w i t h the o v e r a l l l o s s e s , the average d i s a s t e r l o s s i n c u r r e d under the va r i o u s r e v i s i o n s t r a t e g i e s , (holding t r a n s a c t i o n costs c o n s t a n t ) , tend to increase a t a decreasing r a t e . The f a c t t h a t 61 the average l o s s e s do increase as the t r a n s a c t i o n costs are inc r e a s e d i s w i t h i n e x p e c t a t i o n s , and does not c o n f l i c t w i t h the above c o n c l u s i o n . I t i s not p o s s i b l e however, to i s o l a t e the incremental l o s s e s a t t r i b u t a b l e to the increase i n t r a n s a c t i o n c o s t s , ( i e . h o l d i n g r e v i s i o n p e r i o d s c o n s t a n t ) , w i t h i n the e x i s t i n g s i m u l a t i o n program. In order to accomplish t h i s , a co n s i d e r a b l e amount of reprograming i s necessary. I f zero t r a n s a c t i o n c o s t s are considered, (Table 5 ) , then both gains and los s e s appear under the average l o s s category, but they tend to be c l o s e to zero i n magnitude. The gains can be a t t r i b u t e d to the investment i n the r i s k f r e e a s s e t . That i s , although the b e n e f i c i a r y e l e c t s to r e c e i v e the guaranteed amount because the value o f the reference p o r t f o l i o i s l e s s than the guarantee, the t o t a l wealth p o s i t i o n o f the company i s not. The d i f f e r e n c e between the value o f the reference p o r t f o l i o and the wealth p o s i t i o n l i s the amount i n v e s t e d i n the r i s k f r e e asset. Comparison o f these r e s u l t s to those d e r i v e d under the naive s t r a t e g y r e v e a l s t h a t f o r t r a n s a c t i o n costs o f 01 t o 1%! the r e v i s i o n s t r a t e g y i s dominant over the na i v e , i n terms of average l o s s e s . The r e s u l t s tend to be i n c o n c l u s i v e f o r t r a n s a c t i o n costs o f 2 and 2%%, i n terms o f average l o s s e s . I t may be concluded, however, th a t i f the number o f r e v i s i o n s are kept t o t e n per c o n t r a c t p e r i o d , then even a t these t r a n s a c t i o n cost l e v e l s the r e v i s i o n s t r a t e g y i s s u p e r i o r to the naive. 62 A n a l y s i s o f the standard d e v i a t i o n s o f the average l o s s e s r e v e a l s the a c t u a l impact o f the r e v i s i o n s t r a t e g y . A t a l l l e v e l s of t r a n s a c t i o n costs the d e v i a t i o n s tend to improve as the number o f r e v i s i o n s i s increased. The improvement may be viewed as a d e c l i n e i n the magnitude o f the d e v i a t i o n as the number of r e v i s i o n s are increased, or i n terms o f the r a t i o o f the mean l o s s to the standard d e v i a t i o n a s s o c i a t e d w i t h the l o s s . That i s , although the standard d e v i a t i o n increases i n some cases as the number o f r e v i s i o n s are increase d , the r e l a t i o n s h i p o f the average l o s s to the d e v i a t i o n should be considered i n s t e a d o f the absolute magnitude o f the d e v i a t i o n . With respect t o the naive s t r a t e g y , i n a l l cases the r e v i s i o n p o l i c y i s dominant, i n terms of the d e r i v e d d e v i a t i o n s . The d i f f e r e n c e s are e s p e c i a l l y s i g n i f i c a n t a t the lower t r a n s a c t i o n c o s t l e v e l s . For example, at \% t r a n s a c t i o n c o s t s and 10 r e v i s i o n s the d e v i a t i o n i s 5.55, at 80 r e v i s i o n s i t i s 2.00. Under the naive s t r a t e g y , however, the r e s p e c t i v e d e v i a t i o n s are 10.95 and 11.11. For the case o f no t r a n s a c t i o n c o s t , the d e c l i n e ' i s even more dramatic, ranging from 5.50 t o 1.47, r e s p e c t i v e l y . Even at the h i g h e s t cost l e v e l considered, ( i e . 2%%), the r e s p e c t i v e d e v i a t i o n s are 6.19 and 3.47 which are s t i l l c o n s i d e r a b l y below the naive r e s u l t s . G e n e r a l i z i n g , the i m p l i c a t i o n s o f the average l o s s e s and the r e s p e c t i v e standard d e v i a t i o n s are th a t by i n c r e a s i n g the number o f r e v i s i o n s , the company reduces the d i s p e r i s o n o f the l o s s e s , which can 63 be viewed as a r e d u c t i o n o r r i s k . The i n c r e a s e i n the average l o s s e s as the number o f r e v i s i o n s i s i n c r e a s e d , on the other hand, can be i n t e r p r e t e d as the cost of reducing the r i s k . The measure of r i s k may a l s o be i n t e r p r e t e d as the magnitude of l o s s e s i n c u r r e d by the company, but the major weakness of c o n s i d e r i n g absolutes i s the random component. That i s , i f the l a r g e s t l o s s under each s t r a t e g y i s considered as an i n d i c a t o r of the r i s k a s s o c i a t e d w i t h t h a t s t r a t e g y , then the random component must be i s o l a t e d i n order to determine the e f f e c t i v e n e s s o f the r e v i s i o n s t r a t e g y . The f o l l o w i n g histograms summarize the d i s a s t e r l o s s e s experienced under the naive s t r a t e g y and the r e v i s i o n s t r a t e g y f o r the 20 p e r i o d a l t e r n a t i v e . This i s synonymous to r e v i s i n g the p o r t f o l i o every s i x months, or i n the case of the naive s t r a t e g y , generating a r e t u r n on the p o r t f o l i o semi-annually. I t i s evident from the histograms t h a t not only are the mean and standard d e v i a t i o n s reduced, but a l s o the absolute magnitude o f the .losses. In absolute terms, the r e d u c t i o n i n the l a r g e s t l o s s i s $29.22 ( i e . from 42.33 t o 13.11). A n a l y s i s o f the behavior of the l a r g e s t l o s s i s not pursued, however, because of the changes i n the r e t u r n generating sequence r e s u l t i n g from i n c r e a s i n g the number of r e v i s i o n s . Doubling the number of r e v i s i o n s r e s u l t s i n doubling the number o f random returns generated. T h i s , i n t u r n can r e s u l t i n the generation of an extremely l a r g e l o s s . Because of t h i s p o s s i b i l i t y , the more conservative approach o f examining the means and standard d e v i a t i o n s o f the l o s s e s i s adopted. 64 NAIVE STRATEGY 2 0 TRANSo COST- 0% NOo PERIODS- 2 0 MEAN LOSS- - 1 3 * 3 6 STOo DEVo- 1 2 o 6 9 15 10 * * * 0 LOSS {$) FREQUENCY . / 42 o / 3 6 , / 30 * * / 24 * * * * / 18 * * * / 12 # * * / * * * REVISION STRATEGY DISASTER LOSSES TRANSo COST- 1 ? NOo PERIODS- 20 MEAN LOSS- - 5 o 8 0 20 _ STD« DEV.- 3 . 5 8 15 o o # # # # # 10 ' * * o ^ t 4 $ I 4= o * * * * * * 5 * * * „ * * * * * * 0 * * * * * * * 0 * * * * 0 _ o o o / o o o / o o • . / 0 0 0 / o o o / o o o / 0 0 0 / o o o / LOSS ( $ ) 42 3 6 30 24 18 12 6 0 FREQUENCY 0 0 0 0 0 2 8 12 65 4.5 The R e v i s i o n Strategy: Largest Losses The f i n a l s e t of t a b l e s , presented i n Appendix F, summarize the l a r g e s t 5 and 10% los s e s over the 500 s i m u l a t i o n s per r e v i s i o n s t r a t e g y and the various l e v e l s o f t r a n s a c t i o n c o s t s . The average losses represent the means of the 25 and 50 l a r g e s t l o s s e s i n c u r r e d under each r e v i s i o n s t r a t e g y , r e s p e c t i v e l y . I n essence i t i s an a n a l y s i s of the t a i l end of the t o t a l l o s s d i s t r i b u t i o n . As i n the previous t a b l e s , the average l o s s e s increase w i t h the number of r e v i s i o n s , but the d e v i a t i o n s from the mean do not increase l i n e a r l y . As the t r a n s a c t i o n costs increase from 1 to 2%% the d e v i a t i o n s do in c r e a s e more r a p i d l y as the number as the number of r e v i s i o n s i s increased. Even a t 2%%, the d e v i a t i o n s only double i f the number of r e v i s i o n s i s increased from 10 to 80. For the case of zero t r a n s a c t i o n c o s t , the average l o s s and corresponding standard d e v i a t i o n d e c l i n e s as the number o f r e v i s i o n s i s increased. In essence the i m p l i c a t i o n s o f these t a b l e s i s t h a t i f the company i s able to reduce the t r a n s a c t i o n c o s t l e v e l ( i e . the percent c o s t ) , considerable savings w i l l r e s u l t , along w i t h a r e d u c t i o n i n the r i s k a s s o c i a t e d w i t h the l o s s e s . These r e s u l t s may be examined from another p o i n t o f view. I f the company adopts a p o l i c y o f charging the i n v e s t o r f o r the average o f the 25 or 50 l a r g e s t l o s s e s , per c o n t r a c t , a cons i d e r a b l e decrease 66 o c c u r s i n t h e r i s k a s s o c i a t e d w i t h t h e p o r t f o l i o o f c o n t r a c t s . The f o l l o w i n g t a b l e s u m m a r i z e s t h e minimum a n d maximum number o f l o s s e s ( i n p e r c e n t a g e t e r m s ) w h i c h o c c u r o v e r t h e v a r i o u s t r a n s a c t i o n c o s t l e v e l s , g i v e n t h e above s t r a t e g y . P e r c e n t L o s s e s 5% 10% T r a n s . M i n . M a x . M i n . M a x . C o s t L o s s L o s s L o s s L o s s 1% 1.6% 2.6% 3.0% 4.0% lh% 1.2% 2.4% 3.0% 4.6% 2% 1.4% 2.2% 2.8% 4.0% 2Js% 1.4% 2.2% 3.2% 4.2% T h a t i s , f o r e x a m p l e , i f t h e company i n c u r s t r a n s a c t i o n c o s t s o f 1%, u n d e r t h e p o l i c y o f c h a r g i n g t h e a v e r a g e o f t h e 25 l a r g e s t l o s s e s ( i e . 5% ) , i r r e g a r d l e s s o f w h i c h r e v i s i o n s t r a t e g y i t p u r s u e s , t h e minimum number o f c o n t r a c t s o n w h i c h i t w i l l i n c u r a l o s s w i l l b e 1.6% o r 8, t h e maximum, 2.6% o r 13 . C l e a r l y , f o r a s p e c i f i c r e v i s i o n s t r a t e g y , t h e p e r c e n t a g e c a n v a r y , b u t i t w i l l n o t e x c e e d t h e s e l i m i t s f o r a g i v e n t r a n s a c t i o n c o s t l e v e l . I t s h o u l d a l s o b e n o t e d t h a t t h e l i m i t s a r e r e l a t i v e l y c o n s t a n t f o r t h e v a r i o u s t r a n s a c t i o n c o s t l e v e l s ( a l t h o u g h t h e a v e r a g e l o s s e s i n c r e a s e as t h e t r a n s a c t i o n c o s t i n c r e a s e s ) . 67 I f zero t r a n s a c t i o n costs are assumed, the trends i n the r e s u l t s are very s i m i l a r to those obtained i n the a n a l y s i s o f the d i s a s t e r l o s s e s . As the number of r e v i s i o n s i s i n c r e a s e d , the average l o s s d e c l i n e s and the standard d e v i a t i o n o f the l o s s e s becomes smaller. I n e f f e c t , the d i s t r i b u t i o n o f the l o s s e s a t the f i v e and ten percent l e v e l becomes t i g h t as the number of r e v i s i o n s are increased. This r e s u l t i s c o n s i s t e n t w i t h the hypothesis of the a n a l y s i s . 4.6 Summary In t h i s s e c t i o n an attempt has been made to analyse the losses which occur when a hedging s t r a t e g y i s adopted by the company. As a b a s i s o f comparison, the r e s u l t s of the naive or buy and h o l d s t r a t e g y have a l s o been presented. Although the l o s s e s have been considered from a number o f d i f f e r e n t p o i n t s o f view, from the r e s u l t s i t i s evident t h a t the hedging s t r a t e g y i s dominant over the naive . The average los s e s may be viewed as the cost o f reducing the standard d e v i a t i o n o r r i s k a s s o c i a t e d w i t h the r e v i s i o n s t r a t e g y under c o n s i d e r a t i o n . With the naive s t r a t e g y , however, there i s no attempt made to reduce or e l i m i n a t e r i s k . Consequently, the a d d i t i o n a l r i s k a s s o c i a t e d w i t h the guarantee p r o v i s i o n i s a l s o ignored. 68 Comparison o f the v a r i o u s r e v i s i o n s t r a t e g i e s over the d i f f e r e n t t r a n s a c t i o n cost l e v e l s i n d i c a t e s t h a t the increases i n the average lo s s e s and t h e i r r e s p e c t i v e standard d e v i a t i o n s may be a t t r i b u t e d to the change i n the t r a n s a c t i o n c o s t . Although a d i r e c t l i n e a r r e l a t i o n s h i p cannot be concluded from the evidence s u p p l i e d by the program, the changes are c o n s i s t e n t enough t o support the above co n c l u s i o n . From the r e s u l t s i t i s evident t h a t i t i s i n the i n t e r e s t of the company to minimize the t r a n s a c t i o n cost per share, to the extent t h a t t h i s i s p o s s i b l e . Although the p o i n t was not pursued i n the a n a l y s i s , savings r e s u l t i n g from a r e d u c t i o n i n the t r a n s a c t i o n cost per share p r o v i d e the opportunity to i n c r e a s e the number o f r e v i s i o n s and consequently reduce the r i s k of l o s s e s . This i s not a f a c t a r i s i n g from the r e s u l t s o f the a n a l y s i s , but a hypothesis based on the observed trends. This aspect c o u l d be considered i n subsequent analyses. I n the case of lower t r a n s a c t i o n costs per share, ( i e . 0 to 1%%), the evidence supports the hypothesis t h a t by i n c r e a s i n g the number o f r e v i s i o n s , the d e v i a t i o n or r i s k o f l o s s i s reduced. I f the t r a n s a c t i o n costs are i n the range o f 2 t o 2%% per share traded, the t r e n d i s towards a d e c l i n e i n the d e v i a t i o n s as the r e v i s i o n s are increased, but the f l u c t u a t i o n s do cause concern. The f l u c t u a t i o n s tend to occur when the number o f r e v i s i o n s i s i n c r e a s e d to about 60 to 70 per p e r i o d . Up t o these l e v e l s , and a f t e r , the d e v i a t i o n 69 declines with the increases in the revisions. Analysis of the actual magnitude of losses generated during these simulations reveals that extreme losses were created by the random number generating process. There is also the possibility that at higher transaction cost levels a significant underinvestment or overinvestment in the referenence portfolio may be occuring. This may be the result of the fact that in the theoretical model no consideration is given to the potential impact of transaction costs on the required investment in the reference portfolio. The special case of zero transaction costs clearly shows the impact of the revision strategy on the losses and the deviations when compared to the naive strategy. As discussed previously, losses can only occur under these conditions i f the guarantee is exercised. From the evidence presented i t is clear that i f the number of revisions becomes very large, the average loss and the standard deviation should approach zero, resulting in no gains or losses for the company. 70 Chapter 5  Conclusions The Schwartz dissertation proposed that an equity linked l i f e insurance contract with an asset value guarantee may be explained in terms of the Black-Scholes option valuation framework. Within this framework a model was developed which proved that under conditions of equilibrium no gains or losses can accrue to the insurance company. That is, by maintaining a fully hedged position between the investment in the reference portfolio and the call option on the reference portfolio continuously, the probability of a loss or gain becomes zero. Since under equilibrium conditions transaction costs are ignored, direct application of the model to a practical situation is not possible. Furthermore, i f transaction costs are included in the model, a continuous hedging strategy is not possible because such a strategy implies infinite transaction costs. Therefore i f a discrete revision strategy is adopted, the company will be subject to losses and gains. In light of this, i t becomes necessary to develop a procedure to analyse the nature of the losses and to determine i f the discrete strategy is superior to some benchmark, such as a naive buy and hold strategy. Superiority in this context results from obtaining a significant reduction in the risk of loss by adopting the proposed strategy. 71 More s p e c i f i c a l l y , the o b j e c t i v e of t h i s d i s s e r t a t i o n was to prove t h a t by adopting a d i s c r e t e r e v i s i o n s t r a t e g y , an insurance company can reduce the d i s p e r s i o n o f losses which can a r i s e as a r e s u l t o f a general market d e c l i n e . As a b a s i s of comparison, i t was assumed t h a t the market p o r t f o l i o would be bought and h e l d . Put another way, i f management r e j e c t s the o p t i o n p r i c i n g i n t e r p r e t a t i o n of the problem, then i t i s not unreasonable t o assume buying the market p o r t f o l i o as a v i a b l e a l t e r n a t i v e . In a d d i t i o n , an attempt has been made to examine the i m p l i c a t i o n s o f v a r i a b l e t r a n s a c t i o n c o s t s . B r i e f l y , i t was shown th a t the hedging concept i s v a l i d w i t h i n t h i s framework as the company takes a long p o s i t i o n i n the reference p o r t f o l i o and a short p o s i t i o n i n the c a l l o p t i o n on the p o r t f o l i o . According to the Black-Scholes f o r m u l a t i o n , i n order t o form the hedged p o s i t i o n 1/w-^  options must be s o l d short against each share hel d . With respect to the insurance c o n t r a c t , s i n c e one o p t i o n on the reference p o r t f o l i o i s s o l d s h o r t , to form the hedged p o s i t i o n , only the determination o f the necessary investment i n the p o r t f o l i o i s r e q u i r e d . This was g i v e n by the expression: ( 5-1 ) xCt) = X ( t ) • N(d x) The s i m u l a t i o n model generated a r a t e of r e t u r n on the p o r t f o l i o a t s p e c i f i e d times, which caused the hedged p o s i t i o n e s t a b l i s h e d i n the previous p e r i o d to be no longer v a l i d . I n order to r e - e s t a b l i s h the 72 hedged p o s i t i o n , t h e company h a d t o e i t h e r s e l l a p o r t i o n o f t h e p o r t f o l i o o r buy a d d i t i o n a l s h a r e s , d e p e n d i n g on w h e t h e r a p o s i t i v e o r n e g a t i v e r e t u r n was g e n e r a t e d . I n t h i s way, t h e company i n c u r r e d n o t o n l y i n i t i a l and t e r m i n a l t r a n s a c t i o n c o s t s , b u t a l s o t h e c o s t s o f r e v i s i n g t h e p o r t f o l i o . A n a l y s i s o f t h e i n t e r m e d i a t e c a l c u l a t i o n s i n d i c a t e d t h a t i f p o s i t i v e r e t u r n s were g e n e r a t e d on t h e r e f e r e n c e p o r t f o l i o , t h e p e r c e n t i n v e s t m e n t i n t h e p o r t f o l i o a p p r o a c h e d 100. On t h e o t h e r hand, i n t h e c a s e o f l o s s e s on t h e p o r t f o l i o , t h e p r o p o r t i o n i n v e s t e d d e c l i n e d c o n s i d e r a b l y . T h i s i s e n t i r e l y w i t h i n e x p e c t a t i o n s . P u t a n o t h e r way, i f t h e v a l u e o f t h e r e f e r e n c e p o r t f o l i o i s i n c r e a s i n g o v e r t i m e , t h e n t h e company s h o u l d be a l m o s t f u l l y i n v e s t e d i n t h e p o r t f o l i o . I f i t i s l o s i n g on t h e p o r t f o l i o , however, i t s t a n d s t o r e a s o n t h a t t h e i n v e s t m e n t s h o u l d be r e d u c e d . I t s h o u l d be n o t e d t h a t t h e d i f f e r e n c e between t h e amount i n v e s t e d i n t h e r e f e r e n c e p o r t f o l i o and t h e t o t a l w e a l t h i s always i n v e s t e d i n a r i s k f r e e a s s e t . T h i s p r i n c i p l e a c c o u n t s f o r t h e f a c t t h a t even t h o u g h t h e b e n e f i c i a r y e x e r c i s e s t h e g u a r a n t e e , t h e company may s t i l l show a p r o f i t . T h a t i s , t h e t e r m i n a l v a l u e o f t h e r e f e r e n c e p o r t f o l i o may be l e s s t h a n t h e g u a r a n t e e d amount, r e s u l t i n g i n t h e g u a r a n t e e b e i n g e x e r c i s e d , b u t t h e sum o f t h e p o r t f o l i o v a l u e p l u s t h e amount i n v e s t e d i n t h e r i s k f r e e a s s e t c a n e x c e e d t h e g u a r a n t e e d amount. Under s u c h c i r c u m s t a n c e s t h e company w i l l show a p r o f i t i n s p i t e o f t h e f a c t t h a t t h e v a l u e o f 73 the reference p o r t f o l i o d i d not exceed the guaranteed amount. During the course of the simulations, t h i s s i t u a t i o n only occurred f o r low transaction cost l e v e l s . In the course of the an a l y s i s , two types of losses were examined. The f i r s t may be defined as transaction cost derived losses. These are simply the r e s u l t of the i n i t i a l and terminal transaction costs, as w e l l as the costs of r e v i s i n g the p o r t f o l i o . In other words, these are the costs of adopting the hedging strategy. The terminal value of the loss per simulation i s the amount which the insured must be charged f o r , i n order to ensure that the company breaks even at the termination of the contract. The second type of loss was defined as a d i s a s t e r l o s s . This type of loss occurs when the b e n e f i c i a r y exercises the guarantee, because the value of the reference p o r t f o l i o i s l e s s than t h i s amount. As mentioned above, the company experiences a loss under these conditions only i f the value of the p o r t f o l i o plus the amount invested i n the r i s k f r ee asset does not exceed the value of the guarantee. This type of a loss w i l l occur only i f there i s a general decline i n the market. In essence, the minimization of the dis p e r s i o n of t h i s type of a loss i s the objective function of the hedging strategy. Within the framework of these d e f i n i t i o n s , the losses were examined from a number of d i f f e r e n t points of view. F i r s t l y , consideration was given to the average loss and i t s d i s p e r s i o n over the 500 simulations 74 per s t r a t e g y . Secondly, the d i s a s t e r l o s s e s were i s o l a t e d and analysed i n terms o f the mean and standard d e v i a t i o n , per case. T h i r d l y , the l a r g e s t 5 and 10% o f the l o s s e s per case were examined, i n terms o f the means and d e v i a t i o n s . L a s t l y , a s p e c i a l case o f zero t r a n s a c t i o n costs were simulated f o r each case. The c r i t e r i o n f o r the acceptance of the hedging s t r a t e g y over the naive was e s t a b l i s h e d i n terms o f the behavior o f the d e v i a t i o n o f the l o s s e s over the va r i o u s a l t e r n a t i v e s . That i s , the d e v i a t i o n s o f the l o s s e s had t o be not only l e s s than those g i v e n under the naive s t r a t e g y , but they should a l s o d e c l i n e as the number of r e v i s i o n s was increased. The r e s u l t s of the a n a l y s i s i n d i c a t e t h a t the hedging s t r a t e g y i s dominant over the naive. While the average l o s s e s increase as the number of r e v i s i o n s i s increa s e d , the d i s p e r s i o n o f the l o s s e s decreases. I t i s expected t h a t the average l o s s should increase as the number o f r e v i s i o n s i s increa s e d , because o f the a d d i t i o n a l t r a n s a c t i o n c o s t s . The s m a l l e r d e v i a t i o n s imply t h a t the d i s t r i b u t i o n o f the los s e s i s c o n s i d e r a b l y t i g h t e r as the r e v i s i o n s are increased. In t h i s sense, the r i s k a s s o c i a t e d w i t h the s t r a t e g y i s reduced. That i s , the average l o s s may be viewed as the co s t o f reducing the r i s k to the l e v e l i n d i c a t e d by the standard d e v i a t i o n . Put another way, the a d d i t i o n a l l o s s i n c u r r e d by i n c r e a s i n g the number o f r e v i s i o n s i s the c o s t o f reducing the r i s k by the amount i n d i c a t e d by the incremental change i n the standard d e v i a t i o n . 75 From the r e s u l t s o f the a n a l y s i s , a number o f management s t r a t e g i e s may be developed. For example, i n the l a s t chapter c o n s i d e r a t i o n was gi v e n to the s t r a t e g y of charging the in s u r e d the average o f the l a r g e s t 5% l o s s e s over and above the value of the o p t i o n and the present v a l u e o f the guarantee. I t should be noted t h a t t h i s proposal i s simply an a l t e r n a t i v e . No e f f o r t has been made i n the course of t h i s a n a l y s i s to examine the m a r k e t a b i l i t y o f the instrument, g i v e n these a d d i t i o n a l c o s t s . Furthermore, the degree of r i s k which a company may assume depends on the r i s k a v e r s i o n o f management, not on an optimal s o l u t i o n which one may expect. The c o n s t r a i n t s , i n t h i s sense are exogenous t o the o u t l i n e d procedure. U t i l i t y f u n c t i o n s have been ignored i n the course o f the a n a l y s i s because a t any p o i n t i n time, the r e q u i r e d investment i n the reference p o r t f o l i o depends not on i t s expected v a l u e , but the cu r r e n t v a l u e . The expected r e t u r n on the instrument, does however, depend on the expected performance of the reference p o r t f o l i o . Furthermore, the l e v e l of acceptable l o s s e s a l s o depend on u t i l i t y f u n c t i o n s . I n t h i s a n a l y s i s , an attempt has been made to r e i n f o r c e the c o n t e n t i o n t h a t the o p t i o n p r i c i n g i n t e r p r e t a t i o n o f the e q u i t y l i n k e d l i f e insurance c o n t r a c t w i t h an asset v a l u e guarantee i s the c o r r e c t i n t e r p r e t a t i o n of the problem. Furthermore, i t has been shown th a t the adoption of the hedging s t r a t e g y y i e l d s s u p e r i o r r e s u l t s to those g i v e n by the conventional buy and h o l d o p t i o n . The problem 76 of mortality has been excluded from the analysis, primarily to simplify the results. It is recommended that in subsequent analyses of the problem, this variable be included to determine i t s impact on the results. Because of the exclusion of the mortality problem, this analysis may be viewed also as an investment i n a mutual fund and a term insurance policy on that investment. This analysis has provided a viable alternative to the management of risk within the existing framework, i t remains to be seen whether the necessary legal conditions w i l l be brought about in order to provide the f l e x i b i l i t y required to adopt the proposed strategy. Before such changes can be expected, authorities w i l l have to re-examine the existing legislation governing the management of risk. 77 BIBLIOGRAPHY A i t c h i s o n , J . and Brown, J.A.C., The Lognormal D i s t r i b u t i o n Cambridge U n i v e r s i t y P r e s s , 1963. Black, F. and Scholes, M.J., The P r i c i n g o f Options and  Corporate L i a b i l i t i e s . J o u r n a l o f P o l i t i c a l Economy, V o l . 81, No. 3, May-June 1973. Black, F. and Scholes, M.J., The V a l u a t i o n o f Option Contracts  and a Test o f Market E f f i c i e n c y . J o u r n a l o f Finance, V o l . 27, May 1972. 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H . ^ A s s e t V a l u e G u a r a n t e e s Unde r E q u i t y B a s ed P r o d u c t s . T r a n s a c t i o n s S o c i e t y o f A c t u a r i e s , V o l . X X I , 1969. T u r n e r , S ,H. ^ " E q u i t y B a s ed L i f e I n s u r a n c e i n t h e U n i t e d K ingdom. T r a n s a c t i o n s S o c i e t y o f A c t u a r i e s , V o l . X X I I I , 1971 . V a n H o m e , J . C . , F i n a n c i a l Management a n d P o l i c y . P r e n t i c e -H a l l I n c . Second E d i t i o n , 1971 . V a n H o m e , J . C . , F o u n d a t i o n s f o r F i n a n c i a l Management: A Book  "o f R e a d i n g s . R.D. I r w i n , I n c . , Homewood, I l l i n o i s , 1967. 80 21) Weston, T.F. and Brigham, E. F., Managerial Finance. Holt, Rinehart and Winston, Inc. Fourth Edition, 1972, i 81 Appendix A  Si m u l a t i o n Program This appendix contains the s i m u l a t i o n program which generates the supportive s t a t i s t i c s f o r the a n a l y s i s . The t a b l e s o f the f o l l o w i n g appendices are e d i t e d from the output o f t h i s program. I t should be noted t h a t i f C0NT=2 then the r e s u l t s of intermediate c a l c u l a t i o n s are al s o produced as output ( i e . as per Appendix B ) , otherwise, i f C0NT=1, only the t e r m i n a l values are provided as output. S i m i l a r l y , i f NAIV=2, then the simple or naive buy and h o l d p o l i c y i s evaluated ( i e . no r e v i s i o n s ) , otherwise, i f NAiV=l, the general model w i t h appropriate r e v i s i o n s i s evaluated and w r i t t e n . When e v a l u a t i n g the naive s t r a t e g y , the f o l l o w i n g changes must be made: 1) I n i t i a l i z e : TR=0.0 DO 1500 JM=1,1 DO 1400 NREV= 10, 10, 10 2) Add before the comment card 'BEGIN': IF (NAIV. EQ. 2) GO TO 1 3) Add a t the bottom o f page one of the program: 1 X ( l ) = 100.00 OPVAL (1) =0.0 WLTH(l) = 100.00 RPORT(l) = 100.00 I n i t i a l and t e r m i n a l t r a n s a c t i o n s costs are c a l c u l a t e d e x t e r n a l l y because considerable changes must be made i n the program i n order to accomplish t h i s i n t e r n a l l y . 83 D I M E N S I O N R P O R T ( 9 9 ) , W L T H ( 9 9 ) , O P V A L ( 9 9 ) , X ( 9 9 ) , R A N { 9 9 ) , X L A B ( 9 9 ) , 1 C S U M 1 5 0 0 ) , A V E ( 5 0 0 ) , P R O F { 5 0 0 ) , R P 0 R T 1 { 5 0 0 ) t A V E P R T ( 5 0 0 ) , V A R P F T ( 5 0 0 ) , 2 V A R P R T ( 5 0 0 ) , D U M ( 5 0 0 ) , 0 U M 1 ( 5 0 0 ) , X L O S S ( 5 0 0 ) I N T E G E R B A R R A Y I 1 0 ) t F M T / * F7*3*/ I N T E G E R C A R R A Y ( 1 0 ) , F M A / • F 7 o 3 • / C I N I T I A L I Z E V A R I A B L E S Q, * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C0NT=1» NAIV=1 N S I M = 5 0 0 K S = N S I M - 4 0 o TR=0o01 DO 1 5 0 0 J M = 1 , 4 DO 1 4 0 0 N R E V = 1 0 , 8 0 , 1 0 NP£R=NR£V+1 N P E R 1=NPER - 1 GAR=100o R F R E = e 0 6 * { 1 0 « / N R E V ) V A R = 0 o 0 1 8 4 6 * ( 1 0 o / N R E V ) SUM=0« T O T = 0 . T R B L = 0 o 0 N0=0o . C B E G I N C C A L C U L A T I O N OF I N I T I A L P E R I O D CT=NPER-1) C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * S = 0 o 0 STD= S Q R T ( V A R ) F M = o 0 8 * ( l O o / N R E V ) X X = 1 0 0 . RAN(1)=0<,0 T=NREV R P Q R T ( l ) = 1 0 0 . 0 X L = A L 0 G ( X X / 1 0 0 o ) DD1=XL+<RFRE+ . 5 *VAR) *T D D 2= X L + f R F R E - . 5 * V A R ) * T D B = S Q R T ( V A R * ( 2 * T ) ) 0 1 = D D 1 / D B * C - 1 ) D 2 = D D 2 / D B # ( - 1 ) A = E X P ( - R F R E * T ) O P V A L ( 1 ) = < ( X X * E R F C ( D I ) ) - ( 1 0 0 * A * E R F C ( D2 ) I ) * . 5 X L A B ( 1 ) = 1 0 0 * £ X P ( - R F R E * T ) + O P V A L ( 1 ) W L T H ( l ) = 1 0 0 o O * E X P { - R F R E * T ) + O P V A L ( l ) X ( l ) = R P 0 R T ( l ) * E R F C ( D l ) * o 5 I R F = W L T H ( 1 ) - X ( l ) W L T H U ) = W L T H { l ) - ( X< 1 J * T R J 84 C C A L C U L A T I O N OF SECOND STAGE <T=NPER-2 TO 1) DO 1000 K = 1 , N S I M L=NPER DO 1 0 0 J = 2 , N P E R 1 T=NREV Z-RANDLlS»FM»STD) R A N ( J ) =Z J J = J - 1 R P O R T ( J ) = R P O R T ( J J ) * Z W L T H ( J ) = ( ( W L T H ( J J ) - X ( J J ) ) * E X P ( R F R E ) ) + X { J J ) * Z C R E V I S I O N Q * # * * * * * * * * * * * * * * * * * * * * * * * * * # # T = T - J J I F ( N A I V » E Q o 2 ) GO TO 1 2 3 4 X C = A L O G ( R P 0 R T ( J ) / 1 0 0 } A = E X P ( - R F R E * T ) D l = ( X C + ( ( R F R E + « 5 * V A R ) * T ) ) / S Q R T ( VAR* ( 2 * T ) ) # ( - l ) D 2 = ( X C * ( ( R F R E - . 5 * V A R ) * T ) ) / S Q R T ( V A R # ( 2 * T ) ) * < - l I O P V A L ( J ) = ( ( R P O R T ( J ) * E R F C ( D I ) ) - ( 1 0 0 * A # E R F C ( D2 > ) ) * , 5 X{ J ) = RPORT ( J ) * ( E R F C ( D 1 ) ) * o 5 GO TO 3 4 5 6 1 2 3 4 X ( J ) = X ( J J ) * Z O P V A L ( J ) = R P Q R T ( J ) - 1 0 0 o I F ( O P V A L { J ) . L T . O „ ) O P V A H J ) = 0 » 0 3 4 5 6 T R C S T = T R * A B S ( X ( J ) - X ( J J ) ) W L T H ( J ) = W L T H ( J ) - T R C S T C C A L C U L A T E L I A B I L I T Y X L A B ( J ) = 1 0 0 * £ X P ( - R F R E ^ T ) + 0 P V A L { J ) 1 0 0 CONTINUE C C A L C U L A T I O N OF 3RD STAGE (T=0) C ^sse * # # # # * * * * * * * * * * * * * * * * * * * * * * Z=RANDL(S»FM,STD) R A N ( L ) = Z L L = L - 1 W L T H ( L ) = ( ( V I L T H ( L L ) - X ( L L ) ) * E X P ( R F R E ) ) + X ( L L )*Z W L T H ( L ) = W L T H ( L ) - ( ( X ( L L ) * Z ) * T R ) RPORT( L ) = RPORT ( L L ) * Z O P V A L U ) = R P O R T ( L ) - 1 0 0 . I F ( O P V A L ( L J o L T . O o ) O P V A L ( L ) = 0 . 0 X ( L ) = 0 o 0 X L A B ( L ) = 1 0 0 o + 0 P V A L ( L ) I F ( C 0 N T o E Q o l o ) GO TO 97 I F ( K . G T o 4 ) GO TO 97 W R I T E ( 6 , 9 1 ) 91 FORMAT(6X»'INTERMEDIATE C A L C U L A T I O N S • , / / 6 X , • X - J • , 6 X , » R P 0 R T ' » 16X,«WLTH» » 6 X , • R A N D - Z • f 4 X , ' O P V A L ' » 5X t * L I A B *) DO 9 6 J = 1 , N P E R W R I T E ( 6 » 9 5 ) X ( J ) , RPORT ( J ) »WLTH( J ) , R A N ( J ) , OPVAL ( J ) , XL A B U ) 95 F O R M A T ( I X , 6 F 1 0 . 2 ) 9 6 CONTINUE 9 7 CONTINUE 85 PROFIT AND AVE. PROFIT CALCULATIONS. IF(RPORT(NPER)0LE0IOO) GO TO 110 PROF(K) = WLTH(NPER)-RPORT(NPER) GO TO 990 110 PR0F(K)=WLTH(NPER)-100. TRBL=TRBL+lo N0=N0+1 XL OSS(NO) = PROF(K) 990 SUM=SUM*PROF{K) CSUM(K)=SUM AVE(K)=CSUM(K)/K IF(CONT.EQ.1.) GO TO 993 IF(K.GTo4) GO TO 993 WRITE(6,991)PR0F(K) 991 FORMAT C * 0•» 5Xt * VALUE OF PROF IT•,F10.4) WRITE(6,992) 992 FORMAT(*0' » 4 X » 1 4 ( ' * *• ) ) 993 RPORTI(K)=RPORT(L) TOT=T0T+RPORTI<K) AVEPRT(K)=TOT/K Q=0. XY = 0. DO 995 JJ=1,K I F ( K . E Q o l ) GO TO 996 Q = Q + ( ( R P O R T I { J J ) - A V E P R T ( K ) ) * * 2 ) / { K - l ) 995 X Y = X Y + ( ( P R O F ( J J ) - A V E ( K > ) * * 2 ) / ( K - l ) 996 VARPRT(K)=SQRT(Q) VARPFTt K) = SQRT(XY) 1000 CONTINUE IFIC0NT.EQ.2) GO TO 1400 WRITE(6,1549) 1549 FORMAT( • 1') WRITE(6,1300) WRITE(6,5300) 1300 F O R M A T ( 7 X , • P R O F I T • , 5 X , « A V E « , 7 X , ' S T D ' , 5 X , ' R P O R T I ' , 5 X , ' A V E ' , 7 X 2,•STD') 5300 F0RMATC7X, * ',5X,• • , 7X , • » ,5X,« • ,5X,« ' ,7X 2,» • ) DO 1010 KK=KS,NSIM MR ITE t6t1011JPROF (KK ) f AVE *KK ) ,VARPFT(KK) ,RPORTI(KK),AVEPRTi KK),V 1PRT(KK) 1011 F0RMAT(2X,6F10.2) 1010 CONTINUE TRBLP=(TRBL/NSIM)*100. WRITE(6 t1012)TR,NREV,NPER,RFRE,VAR,FM 1012 FORMAT(•0*,•TRANSo COST•,8X,FlOo6,6X,<N0. OF REVISIONS*,8X,110,/ 1N0. OF P E R I O D S ' » 5 X , I 1 0 , 6 X » ' R I S K FREE R A T E • » 1 0 X , F 1 0 . 6 , / , » V A R I A N C E 2 U X , F 1 0 . 6 , 6 X , » M £ A N ' ,20X,F10.6) WRITE (6,1100 )TRBL,TRBLP 1100 F O R M A T ( » 0 « , ' D I S A S T E R ' , 1 1 X , F 1 0 « 0 , 6 X , > P E R C E N T ' , 1 7 X , F 1 0 « 6 ) 86 C PLOT R O U T I N E £ * * * * * * * * * * * 7>t * * * * * * * * * * # DO 1 3 0 1 ! J = 1 , N S I M D U M 1 ( I J ) = R P O R T I ( I J ) 1301 DUM{ I J ) = P R O F U J ) C A L L S S 0 R T ( D U M , N S I M , 3 ) C A L L S S 0 R T 1 D U M 1 , N S I M , 3 ) N = D U M ( 1 ) - 1 M = D U M 1 ( 1 ) - 1 XMIN=N YMIN=M DX=2»0 D Y = 4 0 o NX=NSIM NY=NSIM C A L L H I S T G M ( X M I N , D X , 1 0 , B A R R A Y , N X , P R O F , 6 . 5 , F M T , 7 ) W R I T E ( 6 , 1310) BARRAY 1310 F O R M A T ( / • THE D I S T R I B U T I O N I S « / 7 X , 1 0 I 6 ) C A L L H I S T G M ( Y M I N »DY,10» C A R R A Y , N Y » RPORT 1 , 6 . 5 , F M A t 7 ) W R I T E ( 6 » 1 3 1 2 ) CARRAY 1312 FORMAT (/ * THE D I S T R I B U T I O N IS ( / 7 X T .1016 ) C * * * * * * * * * * * * * * * * * * * * * * * * * C A D D I T I O N S TO MAINo - D I S A S T E R L E V E L C A L C U L A T I O N S SAM=0. 0 NE=NO 11 = 0 I F ( N O e E Q . O ) GO TO 1 3 6 0 DO 1 3 4 0 J = 1 , N Q 11 = 11+1 1 3 4 0 SAM=SAM+XLOSS( J ) SAMMN=SAM/I1 SVAR=0o0 12 = 0 DO 1 3 4 5 1=1, NE 12=12+1 1345 S V A R = S V A R + ( ( S A M M N - X L O S S ( I ) ) * * 2 ) I F ( N O o E Q o l ) 12=2 S V A R = S Q R T ( S V A R / ( I 2 - 1 „ ) ) W R I T E 1 6 , 1 3 4 6 ) 1 3 4 6 F 0 R M A T P 1 ' , ' D I S A S T E R L O S S E S o ' ) W R I T E ( 6 , 1 5 5 0 ) 1 5 5 0 F O R M A T ! ' 0 ' ) W R I T E ( 6 , 1 3 5 0 ) ( X L O S S ( K ) , K = 1 , 1 2 ) 1 3 5 0 FORMAT! 5F1 Oo 5) W R I T E ( 6 , 1 3 5 5 ) S A M M N , S V A R 1355 FORMAT C O ' , ' MEAN L O S S ' , 1 0 X , F 1 0 . 2 , 2 X , / / » STANDARD D E V I A T I O N ' 1 , 1X,F 1 0O2 ) 87 GO TO 1 3 6 5 1 3 6 0 W R I T E ( 6 , 1 3 6 1 J 1361 FORMAT ! ' 0 ' , ' N O D I S A S T E R L O S S E S ' ) 1 3 6 5 W R I T E ( 6 , 1 5 5 0 ) W R I T E ! 6 , 1 3 7 0 ) ( D U M { J ) , J = 1 , 2 5 ) 1370 F O R M A T ! * 0 ' , 2 X , ' L A R G E S T L O S S E S - U P TO 5% OF T O T A L ' , / / , ( 5 F 8 o 2 ) ) A D l = 0 o 0 DO 1371 K = l , 2 5 1 3 7 1 AD1=AD1+DUM!K) A D M l = A D l / 2 5 „ V 1 = 0 . 0 DO 1372 K = l , 2 5 1372 V 1 = V 1 + ! ! D U M ! K ) - A D M 1 ) * * 2 ) V S D 1 = S Q R T ! V l / 2 4 o ) W R I T E ! 6 , 1 3 7 3 J A D M l , VSD1 1 3 7 3 FORMAT!* 0 * , 2 X , • M E A N L O S S • , 1 0 X , F 1 0 . 2 , / / , 2 X , • STANDARD D E V I A T I O N ' 1 F 1 0 » 2 ) AD2=AD1 V2 = V1 DO 1 3 7 5 J = 2 6 , 5 0 1375 A D 2 = A D 2 + D U M i J ) ADM2=AD2/50o DO 1 3 7 6 J = 2 6 , 5 0 1 3 7 6 V 2 = V 2 + ( I D U M ! J ) - A D M 2 ) * * 2 ) V S D 2 = S Q R T ! V 2 / 4 9 . ) W R I T E ( 6 , 1 5 5 0 ) W R I T E ! 6 , 1 3 7 7 ) ! D U M ! J ) , J = l , 5 0 ) 1 3 7 7 F O R M A T ( ' 0 * , 2 X , * LARGEST L O S S E S - U P TO 10S OF T O T A L * » / / » ! 5 F 8 o 2 ) ) W R I T E ! 6 , 1 3 7 3 ) A D M 2 , V S 0 2 W R I T E ! 6 , 1 3 1 1 ) 1311 FORMAT I • 1 ' ) 1 4 0 0 CONTINUE T R = T R + 0 . 0 0 5 1 5 0 0 CONTINUE STOP END * * * SUBPROGRAM DICTIONARY * * * 88 NAME ATTRIBUTES REFERENCES ABS 72 ALOG 34 62 ERFC 41 44 66 67 EXP 40 42 43 57 HISTGM 165 168 RANDL 53 79 SQRT 28 37 64 6 5 210 221 SSORT 155 156 <EXIT> 231 *** VARIABLE DICTIONARY * * * 63 126 75 127 82 187 NAME ATTRIBUTES REFERENCES A 4 0 * 41 6 3 * 66 ADM 1 20 6* 209 211 ADM2 218 * 220 225 ADI 203 * 205 * 206 214 AD2 214* 217 * 218 AVE ID ) 1 110 * 125 139 AVEPRT (D ) 1 119* 124 139 BARRAY 1*4 <D ) 4 165 166 CARRAY 1*4 iD ) 5 168 169 CONT 8* 89 111 129 CSUM to ) 1 109* 110 DB 3 7 * 38 39 DDI 3 5* 38 DD2 36 * 39 DUM (D ) 1 217 154* 220 155 223 157 201 205 209 DUM1 ID ) 1 1 53 * 156 158 DX 161* 165 DY 162* 168 V DI 3 8 * 41 44 5 4 * 66 67 D2 39 * 41 65 * 66 FM 29 * 53 79 144 FMA 1*4 5 168 FMT 1*4 4 165 GAR 17 * I 183 * 185 I J 15 2* 153 154 IRF 4 5 * 11 175* 178* 180 12 18 2* 184* 186 * 187 192 J 5 1 * 54 55 56 57 62 66 67 69 70 71 72 73 75 95 177* 179 201 216* 217 219 * 220 223 J J 5 5 * 56 57 60 69 72 122* 124 125 JM 13* K 49* 90 110 112 123 124 205 208* KK 138* 139 KS 11* 138 L 50* 80 86 8 7 LL 81* 82 M 15 8* 160 N 157* 159 NAI V 9* 61 NE 174* 183 NO 23* 106* NPER 15* 16 144 NPERl 16* 51 NREV 14* 15 144 NSIM 10* 1 1 156 163 NX 163* 165 NY 164* 168 OPVAL {D ) 1 41* 75 85* PROF (D ) 1 102* 139 154 0 120* 124* RAN CD ) 1 3 1 * RFRE 18* 35 63 64 RPORT (D ) 1 33* 70 84* RPORTI (D ) 1 117* S 27* 53 SAM 173* 179* SAMMN 180* 185 STD 28* 53 SUM 20* 108* SVAR 181* 185* T 32* 35 52* 60* TOT 21* 118* TR 12* 46 TRBL 22* 105* TRBLP 143* 148 TRCST 72* 73 VAR 19* 28 144 VARPFT <D ) 1 127* VARPRT (D ) 1 126* VSD1 210* 211 VSD2 221* 225 VI 20 7* 209* V2 215* 220* 89 102 104 107 108 109 113 117 118 119 122 125 126 127 192 204-209 81 82 83 84 85 88 117 83 84 107 174 176 177 186 50 94 101 102 104 18 19 29 32 52 49 138 143 152 155 164 42 43 66* 70* 71* 86* 88 95 104* 107 108 113 125 165 126 54* 80* 95 36 40 42 43 57 65 75 82 144 44 56* 62 66 67 85 95 101 102 117 118 124 139 153 168 79 180 194 79 109 187* 194 36 37 40 42 43 63 64 65 75 119 72 83 144 229* 143 148 35 36 37 64 65 139 139 210 215 221 90 WITH { D ) 1 43* 45 46* 57* 73* 82* 83* 95 102 104 X { D ) 1 44* 45 46 57 67* 69* 72 82 83 87* 95 XC 62* 64 65 XL 34* 35 36 XLAB (0 ) 1 42* 75* 88* 95 XLOSS CD J 1 107* 179 185 192 XMIN 159* 165 XX 3 0* 34 41 XY 121* 125* 127 YMIN 160* 168 Z 53* 54 56 57 69 79* 80 82 83 84 *** STATEMENT LABEL DICTIONARY *** LABEL DEF * N REFERENCES 91 92 91 95 96 95 96 97 94 97 98 89 100 76 51 110 104 101 990 108 103 991 114 113 992 116 115 993 117 111 995 125 122 996 126 123 1000 128 49 1010 142 138 1011 141 139 1012 145 144 1100 149 148 1234 69 61 1300 134 132 1301 154 15 2 1310 167 166 1311 227 226 1312 170 169 1340 179 177 1345 185 183 1346 189 188 1350 193 192 13 55 195 194 1360 198 176 1361 199 198 1365 200 197 1370 2 02 201 1371 205 204 1372 209 208 1373 212 211 1375 217 216 225 91 1376 220 219 1377 224 223 1400 228 14 129 1500 230 13 1549 131 130 15 50 191 190 200 3456 72 68 5300 136 133 * * * L O G I C A L I/O U N I T S D I C T I O N A R Y * * * U N I T R E F E R E N C E S 6 91 95 113 115 130 132 133 139 144 148 166 169 188 190 192 194 198 200 201 211 222 223 225 226 / 92 Appendix B Summary S t a t i s t i c s : Intermediate C a l c u l a t i o n s The f o l l o w i n g t a b l e s provide examples o f the more important intermediate values c a l c u l a t e d under v a r i o u s r e v i s i o n s t r a t e g i e s . For a l l the t a b l e s a one percent t r a n s a c t i o n cost i s assumed. R e v i s i o n P o l i c y : Table 1 Annual r e v i s i o n Table 2 Every s i x months Table 3 Every f o u r months Label D e f i n i t i o n s : X-J: The a c t u a l amount i n v e s t e d i n the reference p o r t f o l i o a t time t , as given by equation (3-19). RPORT: The value o f the reference p o r t f o l i o a t time t , as given by (3-22). WLTH: The wealth p o s i t i o n o f the company at time t , as g i v e n by (3-23). RAND-Z: The simulated r e t u r n on the reference p o r t f o l i o f o r the p e r i o d under c o n s i d e r a t i o n . 93 OPVAL: The value o f the o p t i o n a t time t , as d e f i n e d by the g e n e r a l i z e d v e r s i o n o f (3-17). LIAB: The l i a b i l i t y o f the company a t time t , as giv e n by the g e n e r a l i z e d v e r s i o n o f (3-18), or by the c o n d i t i o n s d e s c r i b e d i n equations (3-27) to (3-29). The v a l u e o f p r o f i t g i v e n at the c o n c l u s i o n o f each s i m u l a t i o n i s the amount the company must charge the i n v e s t o r i n order to break even, i f the value i s negative. The d e r i v a t i o n o f the p r o f i t i s g i v e n by equations (3-30) and (3-31). I t should be noted t h a t the i n i t i a l values o f the v a r i a b l e s are i d e n t i c a l f o r a l l the s i m u l a t i o n s . The t e r m i n a l v a l u e o f the investment i n the reference p o r t f o l i o ( i e . X*J ) i s always zero because o f the assumption t h a t the p o r t f o l i o must always be l i q u i d a t e d at the t e r m i n a t i o n of the c o n t r a c t . F i n a l l y , the o p t i o n v a l u e , OPVAL, must always be g r e a t e r than or equal to zero. 94 I N T E R M E D I A T E C A L C U L A T I O N S X - J RPORT WLTH R A N D - Z OPVAL L I A B 9 4 . 6 4 1 0 0 . 0 0 1 0 0 . 2 1 0 . 0 4 6 . 2 8 1 0 1 . 1 6 1 0 6 . 8 7 1 1 0 o 9 9 1 1 0 . 8 4 l o l l 5 3 . 50 1 1 1 . 7 8 1 0 5 . 4 . 2 1 1 0 . 3 6 1 1 0 . 4 6 0 . 9 9 4 9 . 3 9 1 1 1 . 2 7 1 2 5 . 50 1 2 8 . 1 6 1 2 7 . 5 7 1 . 1 6 6 2 . 8 6 1 2 8 . 5 6 1 4 0 . 8 2 1 4 2 o 3 2 1 4 1 . 4 1 1 . 11 7 2 . 74 1 4 2 . 51 1 7 8 . 9 7 1 7 9 . 1 7 1 7 7 . 5 3 1 . 2 6 1 0 5 . 1 0 1 7 9 . 1 9 1 9 2 . 6 5 1 9 2 . 7 1 1 9 0 . 8 3 1. 08 1 1 4 . 0 5 1 9 2 . 7 1 1 7 8 . 1 3 1 7 8 . 2 1 1 7 6 . 0 7 0 . 9 2 9 4 . 68 1 7 8 . 2 1 1 6 7 . 7 6 1 6 7 . 8 1 1 6 5 . 4 5 0 . 9 4 7 9 . 12 1 6 7 . 8 1 1 5 0 . 0 4 1 5 0 . 0 8 1 4 7 . 4 1 0 . 8 9 5 5 o 9 0 1 5 0 . 0 8 OoO 1 5 4 e 5 2 1 5 0 o 1 4 1 . 0 3 5 4 . 5 2 1 5 4 . 5 2 VALUE OF P R O F I T - 4 . 3 8 2 3 * ** ## JJCJSS jfcjje ** * I N T E R M E D I A T E C A L C U L A T I O N S X - J RPORT WLTH R A N D - Z OP VAL L I A B 9 4 . 6 4 1 0 0 . 0 0 1 0 0 . 2 1 0 . 0 4 6 . 2 8 1 0 1 . 1 6 1 2 3 . 6 4 1 2 5 . 9 2 1 2 4 . 8 0 1 . 2 6 6 8 . 0 3 1 2 6 . 3 1 1 4 2 o 4 1 1 4 3 . 6 5 1 4 2 . 0 8 l o 1 4 8 1 . 9 5 1 4 3 . 82 1 3 7 . 8 8 1 3 9 . 4 8 1 3 7 . 8 9 0 . 9 7 7 4 . 0 0 1 3 9 . 7 1 1 3 0 . 7 5 1 3 3 . 0 9 1 3 1 . 5 0 0 . 9 5 6 3 . 6 4 1 3 3 . 4 0 1 2 9 . 1 6 1 3 1 . 8 3 1 3 0 o 2 9 0 o 9 9 5 8 o 0 8 1 3 2 . 16 1 4 3 . 7 3 1 4 4 . 9 7 1 4 3 . 0 9 1 . 1 0 6 6 . 4 3 1 4 5 . 0 9 1 3 9 . 4 0 1 4 0 . 7 7 138o 85 0 . 9 7 5 7 . 3 6 1 4 0 . 8 9 1 8 2 . 6 5 1 8 2 . 6 6 1 7 9 0 8 6 1 . 3 0 9 3 . 9 6 1 8 2 . 6 6 1 5 4 . 75 1 5 4 . 7 6 1 5 1 . 5 2 0 . 8 5 6 0 . 5 9 1 5 4 . 7 7 OoO 1 6 0 . 7 7 1 5 5 . 7 1 1 . 0 4 6 0 . 77 1 6 0 . 7 7 VALUE OF P R O F I T - 5 . 0 5 7 1 ** ** I N T E R M E D I A T E C A L C U L A T I O N S X - J RPORT WLTH R A N D - Z OPVAL L I A B 9 4 . 6 4 1 0 0 . 0 0 1 0 0 . 2 1 0 . 0 4 6 . 28 1 0 1 . 16 8 6 . 9 7 9 4 . 6 9 9 5 . 4 6 0 . 9 5 38 . 1 3 9 6 . 4 0 1 0 8 o 7 2 1 1 3 . 1 3 1 1 2 o 7 0 1 . 1 9 5 2 . 05 1 1 3 . 9 2 8 6 . 6 9 9 6 . 7 9 9 7 o 0 2 0 . 8 6 3 3 . 1 2 9 8 . 8 2 1 1 2 . 4 4 1 1 7 . 3 6 1 1 5 „ 8 2 1 . 2 1 4 8 . 3 4 1 1 8 . 1 1 8 8 . 6 8 1 0 0 . 9 1 1 0 0 . 0 3 0 . 8 6 2 8 . 93 1 0 3 . 0 1 6 7 . 1 0 9 0 . 4 4 9 1 . 3 2 0 . 9 0 1 6 . 2 0 9 4 . 8 6 9 2 . 3 2 1 0 6 . 01 1 0 4 . 1 2 l o 17 2 4 . 2 7 1 0 7 . 8 0 7 5 . 2 7 9 9 . 5 1 9 9 . 0 1 0 o 9 4 1 3 . 86 1 0 2 . 5 5 8 1 . 2 3 1 0 3 . 7 7 1 0 3 . 6 5 1 . 0 4 1 1 . 4 6 1 0 5 . 6 4 OoO 9 1 . 9 7 9 5 . 0 8 0 . 8 9 0 . 0 1 0 0 . 0 0 VALUE OF P R O F I T - 4 . 9 2 1 0 * #* ** ** ** # 9 5 I N T E R M E D I A T E C A L C U L A T I O N S X - J RPORT WLTH R A N D - Z OPVAL L IAB 9 4 . 6 4 1 0 0 . 0 0 1 0 0 . 2 1 0 . 0 4 6 . 2 8 1 0 1 . 1 6 8 8 . 5 4 9 5 . 4 4 9 6 . 0 0 0 . 9 5 4 0 . 4 2 9 6 . 9 7 6 6 . 5 8 7 9 . 7 5 8 1 . 4 6 0 . 8 4 2 4 . 9 6 8 3 . 2 3 5 0 o 7 9 7 0 . 2 1 7 3 . 7 9 0 . 88 1 6 . 08 7 6 . 1 3 6 5 . 5 3 8 0 . 7 3 8 1 . 9 5 1 . 1 5 2 2 . 7 5 8 4 . 6 3 8 7 . 2 4 9 6 . 56 9 5 . 08 1 . 2 0 3 4 . 7 2 9 8 . 4 8 8 6 . 9 5 9 6 . 9 7 9 5 . 6 9 1 . 0 0 3 3 . 2 8 9 8 . 9 9 8 3 . 3 8 9 5 . 1 7 9 4 . 3 0 0 . 9 8 2 9 . 8 3 9 7 . 5 4 9 6 . 0 3 1 0 4 . 7 1 1 0 2 . 8 7 1 . 1 0 3 6 . 4 6 1 0 6 . 2 2 1 0 8 . 7 9 1 1 4 . 7 5 1 1 2 . 1 5 1 . 1 0 4 3 . 76 1 1 5 . 6 5 8 9 . 3 8 1 0 1 . 3 7 9 9 . 3 8 0 . 8 8 2 9 . 33 1 0 3 . 4 1 1 1 5 . 4 0 1 2 0 . 4 7 1 1 6 . 2 7 1 . 1 9 4 4 . 80 1 2 1 . 1 4 1 2 2 . 2 5 126.11 1 2 1 . 6 2 1 . 0 5 4 7 . 8 9 1 2 6 . 5 6 10 3 . 18 1 1 2 . 1 6 1 0 7 . 8 9 0 . 8 9 32 . 25 1 1 3 . 3 1 1 1 3 . 3 6 1 1 9 . 4 3 1 1 4 . 6 2 1 . 0 6 3 6 . 5 5 1 2 0 . 0 8 1 2 3 . 6 1 1 2 7 . 0 8 1 2 1 . 8 2 1 . 0 6 4 1 . 3 2 1 2 7 . 3 9 1 4 8 . 4 3 1 4 8 . 8 2 1 4 2 . 6 7 1 . 1 7 6 0 . 16 1 4 8 . 85 1 5 4 . 1 3 1 5 4 . 2 2 1 4 7 . 8 2 1 . 0 4 6 2 . 83 1 5 4 . 2 3 1 4 9 . 8 2 1 4 9 . 8 5 1 4 3 . 2 2 0 . 9 7 5 5 . 6 8 1 4 9 . 8 6 1 3 3 . 6 2 1 3 3 . 6 7 1 2 6 . 6 7 0 . 89 3 6 . 6 3 1 3 3 . 6 7 0 . 0 1 3 9 . 2 2 1 3 0 . 6 2 1 . 0 4 3 9 . 2 2 1 3 9 . 2 2 VALUE OF P R O F I T - 8 . 6 0 1 4 * *# ** j j t * * 5 ) C ** * I N T E R M E D I A T E C A L C U L A T I O N S X - J RPORT WLTH R A N D - Z OPVAL L I A B 9 4 . 6 4 1 0 0 . 0 0 1 0 0 . 2 1 0 . 0 4 6 . 28 1 0 1 . 16 1 0 9 . 7 4 1 1 3 . 2 5 1 1 2 . 7 7 1 . 1 3 5 7 . 3 6 1 1 3 . 9 1 1 2 3 . 5 5 1 2 5 . 8 4 1 2 4 . 9 3 1. 11 6 7 . 9 6 1 2 6 . 2 3 1 2 8 . 0 7 1 3 0 . 1 2 1 2 9 . 1 2 1 . 0 3 7 0 . 4 0 1 3 0 . 4 5 1 4 3 . 6 7 1 4 4 . 8 4 1 4 3 . 4 9 1 .11 8 3 . 1 3 1 4 5 . 0 1 1 3 6 . 7 0 1 3 8 . 3 1 1 3 6 . 9 4 0 . 9 5 7 4 . 78 1 3 8 . 5 5 1 5 1 . 7 0 1 5 2 . 5 8 1 5 0 . 9 0 1 . 1 0 8 6 . 9 9 1 5 2 . 7 0 1 6 1 . 8 0 1 6 2 . 3 6 1 6 0 . 5 0 1 . 0 6 9 4 . 7 3 1 6 2 . 4 3 1 7 2 . 9 5 1 7 3 . 2 7 1 7 1 . 2 2 1 . 0 7 1 0 3 . 5 4 1 7 3 . 3 1 2 0 0 . 8 7 2 0 0 . 9 4 1 9 8 . 5 0 1 . 1 6 1 2 9 . 0 6 2 0 0 . 9 5 1 8 1 . 58 1 8 1 . 7 6 1 7 9 . 0 6 0 . 9 0 1 0 7 . 6 9 1 8 1 . 7 7 1 5 6 . 5 8 1 5 7 . 2 2 1 5 4 . 2 1 0 . 8 6 8 0 . 9 4 1 5 7 . 28 1 5 1 . 5 7 1 5 2 . 3 4 1 4 9 . 2 4 0 . 9 7 7 3 . 7 5 15 2 . 4 2 1 6 1 . 4 8 1 6 1 . 8 3 1 5 8 . 5 1 1 . 0 6 8 0 . 80 1 6 1 . 8 6 1 7 9 . 0 0 1 7 9 . 0 7 1 7 5 . 4 5 1 . 1 1 9 5 . 5 5 1 7 9 . 0 8 1 8 4 . 7 6 1 8 4 . 7 8 1 8 0 . 9 9 1 . 0 3 9 8 . 7 1 1 8 4 . 7 8 2 1 0 . 2 7 2 1 0 . 2 7 2 0 6 . 1 1 1 . 1 4 1 2 1 . 5 8 2 1 0 . 2 7 2 2 4 . 4 0 2 2 4 . 4 0 2 1 9 . 9 6 1 . 0 7 1 3 3 . 0 1 2 2 4 . 4 0 2 6 0 . 6 6 2 6 0 . 6 6 2 5 5 . 7 3 1 . 1 6 1 6 6 . 4 9 2 6 0 . 6 6 2 5 9 . 0 5 2 5 9 . 0 5 2 5 3 . 9 5 0 . 9 9 1 6 2 . 0 1 2 5 9 . 0 5 0 . 0 2 3 6 . 3 7 2 2 8 . 7 5 0 . 9 1 1 3 6 . 3 7 2 3 6 . 3 7 VALUE OF P R O F I T - 7 . 6 1 9 3 * tfif. ** ** ** i£s}s #3js # XA.fiLf._3. I N T E R M E D I A T E C A L C U L A T I O N S X - J RPORT WLTH 9 4 . 6 4 1 0 0 . 0 0 1 0 0 . 2 1 1 0 5 . 8 8 1 0 9 . 7 8 1 0 9 . 4 7 1 0 0 . 6 5 1 0 5 . 4 9 1 0 5 . 3 5 1 0 2 . 6 1 1 0 7 . 3 6 1 0 7 . 2 1 9 1 . 3 3 9 8 . 4 2 9 8 . 6 5 9 8 . 9 1 1 0 4 . 7 7 1 0 4 . 6 1 9 2 . 5 9 1 0 0 o 0 5 1 0 0 . 2 1 1 0 3 . 5 3 1 0 9 . 0 3 1 0 8 . 5 6 1 1 7 . 2 2 1 2 0 . 75 1 1 9 . 6 6 1 1 7 . 7 0 1 2 1 o 3 1 1 2 0 . 2 4 1 0 6 . 5 5 1 1 2 . 1 8 1 1 1 . 3 2 1 1 6 . 1 4 1 2 0 . 2 6 1 1 8 . 9 9 1 2 9 . 1 3 1 3 1 . 6 3 1 2 9 . 9 1 1 4 8 . 1 4 1 4 9 . 2 1 1 4 6 . 9 7 1 6 0 . 5 4 1 6 1 . 1 1 1 5 8 . 64 1 6 1 . 5 8 1 6 2 . 0 9 1 5 9 . 5 8 1 6 8 . 0 5 1 6 8 . 3 9 1 6 5 . 7 5 1 7 5 . 3 9 17 5 . 59 1 7 2 . 8 2 1 6 8 . 5 4 1 6 8 . 8 2 1 6 5 . 9 3 1 6 9 . 3 4 1 6 9 . 5 6 1 6 6 . 61 1 4 8 . 0 7 1 4 8 . 9 1 1 4 5 . 7 2 1 5 2 . 9 0 1 5 3 . 4 3 1 5 0 . 1 1 1 3 9 . 6 5 1 4 0 o 8 8 1 3 7 . 4 2 1 4 0 . 6 2 1 4 1 . 6 2 1 3 8 . 1 0 1 3 5 . 1 3 1 3 6 . 45 1 3 2 . 86 1 5 0 . 1 4 1 5 0 . 3 6 1 4 6 . 4 4 1 6 0 . 9 7 1 6 1 . 0 0 1 5 6 . 8 8 1 6 9 . 0 2 1 6 9 . 0 2 1 6 4 . 74 1 6 6 . 5 6 1 6 6 . 5 6 1 6 2 . 1 7 1 5 0 . 3 2 1 5 0 . 3 2 1 4 5 . 67 OoO 1 5 2 . 4 1 1 4 6 . 1 5 VALUE OF PROFIT - 6 . 2 6 4 3 R A N D - Z OPVAL L I A B 0 . 0 4 6 . 2 8 1 0 1 . 1 6 l o 10 5 4 . 5 6 1 1 0 . 55 0 . 9 6 4 9 . 3 5 1 0 6 . 4 7 l o 0 2 5 0 . 02 1 0 8 . 3 0 0 . 9 2 4 0 . 4 7 9 9 . 9 2 1 . 0 6 4 5 . 2 6 1 0 5 . 9 2 0 . 9 5 3 9 . 6 8 1 0 1 . 5 6 1 . 0 9 4 6 . 9 1 1 1 0 . 0 4 1 .11 5 6 . 9 2 1 2 1 . 3 3 1 . 0 0 5 6 . 1 7 1 2 1 . 8 8 0 . 9 2 4 6 . 0 9 1 1 3 . 1 2 1 . 0 7 5 2 . 50 12 0 o 8 9 1 . 0 9 6 2 . 2 1 1 3 1 . 9 8 1 . 1 3 7 8 . 1 5 1 4 9 . 3 3 1 . 0 8 8 8 . 55 1 6 1 . 1 7 1 . 0 1 8 8 . 0 6 16 2..15 1 . 0 4 9 2 . 8 4 1 6 8 . 4 2 1 . 0 4 9 8 . 50 1 7 5 . 6 1 0 . 9 6 9 0 . 18 1 6 8 . 8 4 l o O O 8 9 . 33 1 6 9 o 5 8 0 . 8 8 6 7 . 1 1 1 4 8 . 9 8 1 . 0 3 6 9 . 9 4 1 5 3 . 4 7 0 . 9 2 55o 76 1 4 0 . 9 7 1 . 0 1 5 4 . 7 5 1 4 1 . 6 9 Oo 96 4 7 . 8 5 1 3 6 . 5 4 1.1.0 5 9 . 8 8 1 5 0 . 3 7 1 . 0 7 6 8 . 6 3 1 6 1 . 0 0 1 . 0 5 7 4 . 35 1 6 9 o 0 2 0 . 9 9 7 0 . 4 8 1 6 6 . 5 6 0 . 9 0 5 2 . 3 0 1 5 0 . 3 2 1 . 0 1 5 2 . 41 1 5 2 . 4 1 * * * #J§£ Appendix C Summary- S t a t i s t i c s : Naive Strategy This appendix provides the summary s t a t i s t i c s f o r the naive s t r a t e g y . This s t r a t e g y simply assumes t h a t the i n i t i a l premium o f $100.00 i s i n v e s t e d i n a p o r t f o l i o o f s e c u r i t i e s and h e l d f o r the d u r a t i o n of the c o n t r a c t . No p o r t f o l i o r e v i s i o n occurs during t h i s p e r i o d . The l a b e l d e f i n i t i o n s of the previous appendices apply to the t a b l e s e x h i b i t e d . The number o f peri o d s i s synonymous t o the r e v i s i o n periods o f the previous t a b l e s . 9 8 TABLE 1 TRANSACTIONS COST 02. NAIVE STRATEGY NUMBER AVERAGE STANDARD AVERAGE STANDARD PERIODS LOSS DEVIATION PORTFOLIO DEVIATION 10 -0.33 2.64 255.03 115.98 20 -0.59 3.78 242.45 115.00 30 -0.35 2.98 242.58 108.42 40 -0.40 2.78 236.37 103.15 50 -0.57 3.67 246.58 115.07 60 -0.58 3.95 245.90 113.61 70 -0.77 4.79 240.38 113.66 80 -0.44 3.08 246.09 113.28 99 TABLE 2 TRANSACTIONS COST 0%o DISASTER LOSSES; NAIVE STRATEGY NUMBER NUMBER PERCENT AVERAGE STANDARD PERIODS LOSS - LOSS LOSS DEVIATION 10 15 3.00 -11.15 10.95 20 22 4.40 -13.3.6 12.69 30 10 2.00 -17.43 12.69 40 - 13 2.60 -15.38 8.52 50 16 3.20 -17.74 11.15 60 14 2.80 -20.66 12.35 70 21 4.20 -18.41 15.20 80 16 3.20 -13.65 11.11 ioo Appendix D Summary S t a t i s t i c s : O v e r a l l Losses  Label D e f i n i t i o n s AVERAGE LOSS: The mean l o s s i n c u r r e d by the company over f i v e hundred s i m u l a t i o n s . This i s the average amount the company must charge the i n s u r e d i n order to break even under the p a r t i c u l a r r e v i s i o n s t r a t e g y . AVERAGE PORTFOLIO: The mean value o f the reference p o r t f o l i o over f i v e hundred s i m u l a t i o n s , given the p a r t i c u l a r r e v i s i o n s t r a t e g y . Table 5 i s the s p e c i a l case o f zero t r a n s a c t i o n c o s t s . I t c l a r i f i e s the e f f e c t o f i n c r e a s i n g the number of r e v i s i o n s by e l i m i n a t i n g the accumulating e f f e c t o f the t r a n s a c t i o n c o s t s . 101 TABLE 1 TRANSACTIONS COST 1%0 NUMBER AVERAGE STANDARD AVERAGE STANDARD REVISIONS LOSS DEVIATION PORTFOLIO DEVIATION 10 -6.98 3.34 255.03 115.98 20 -8.03 2.81 242.45 115.00 30 -8.84 2.68 242.58 108.42 40 -9.05 2.71 236.37 103.15 50 -9.96 2.86 246.58 115.07 60 -10.62 2.73 245.90 113.61 70 -10.84 3.00 240.38 113.66 80 -11.41 2.97 246.09 113.28 102 TABLE 2 TRANSACTIONS COST l„5% NUMBER AVERAGE STANDARD AVERAGE STANDARD REVISIONS LOSS DEVIATION PORTFOLIO DEVIATION 10 -10.44 4.00 250.16 109.50 20 -11.66 3.67 240.87 107.05 30 -13.05 3.82 243.04 113.01 40 -13.81 3.52 237.12 101.81 50 -14.81 3.78 245.05 104.95 60 -15.63 4.15 245.48 113.91 70 -16.66 3.95 247.31 109.37 80 -17.26 4.02 246.39 104.39 103 TABLE 3 TRANSACTIONS COST 2 £ . NUMBER AVERAGE STANDARD AVERAGE STANDARD REVISIONS LOSS DEVIATION PORTFOLIO DEVIATION 10 - 1 3 o 6 6 5 . 2 4 2 4 5 . 4 3 1 1 5 . 2 4 2 0 - 1 5 . 2 9 5 . 3 2 2 3 2 . 9 2 1 1 3 . 9 5 30 - 1 7 . 4 9 4 . 7 3 2 4 7 . 0 0 1 1 0 . 1 3 4 0 - 1 8 . 8 6 4 . 9 6 2 4 5 . 9 0 1 1 0 . 5 8 50 - 2 0 . 0 2 4 . 9 2 2 4 6 . 8 8 1 0 5 . 7 7 60 - 2 0 . 8 6 4 . 9 5 2 4 4 . 2 0 1 0 6 . 8 1 70 - 2 2 . 0 3 5 . 1 1 2 1 8 . 0 8 1 0 6 . 2 3 80 - 2 3 . 0 2 5 . 6 6 2 5 0 . 3 8 1 1 5 . 7 6 104 TABLE 4 TRANSACTIONS COST 2*5% NUMBER AVERAGE STANDARD AVERAGE STANDARD REVISIONS LOSS DEVIATION PORTFOLIO DEVIATION 10 -16.88 6.01 240.07 110.74 20 -19.33 5.62 238.75 104.19 . 30 -21.57 5.69 240.41 106.14 40 . -23.26 6.37 240.69 115.24 50 -25.34 6.36 256.88 114.91 60 -25.52 5.59 233.24 99.44 70 -2 6.90 6.39 237.8 5 107.96 80 -28.49 6.55 240.84 108.64 105 TABLE 5 TRANSACTIONS COST 01. NUMBER AVERAGE STANDARD AVERAGE STANDARD REVISIONS LOSS DEVIATION PORTFOLIO DEVIATION 10 -0.05 1.89 255.03 115.98 20 -0.21 1.51 242.45 115.00 30 -0.17 1.14 242.58 108.42 40 0.08 1.01 236.37 103.15 50 0.00 1.02 246.58 115.07 60 -0.10 0.83 245.90 113.61 70 0.02 0.88 240.38 113.66 80 0.02 0.80 246.09 113.28 106 Appendix E Summary S t a t i s t i c s : D i s a s t e r Losses The f o l l o w i n g t a b l e s summarize the l o s s e s which the company in c u r s i f the guarantee i s e x e r c i s e d . Label D e f i n i t i o n s NUMBER LOSS: The number o f times the guarantee was e x e r c i s e d over 500 s i m u l a t i o n s , per r e v i s i o n s t r a t e g y . PERCENT LOSS: The number of l o s s e s as a percent o f the t o t a l number of s i m u l a t i o n s . AVERAGE LOSS: This i s the average d o l l a r l o s s i n c u r r e d by the company as a r e s u l t o f the guarantee being e x e r c i s e d . Table 5 i s the s p e c i a l case o f no t r a n s a c t i o n c o s t s . 107 TABLE 1 TRANSACTIONS COST 1%. DISASTER LOSSES NUMBER NUMBER PERCENT AVERAGE STANDARD REVISIONS LOSS LOSS LOSS DEVIATION 10 1 5 3 . 0 0 - 4 . 1 1 5 o 5 5 20 22 4 . 4 0 - 5 o 8 0 3 . 5 8 3 0 10 2 . 0 0 - 7 . 3 8 2 . 8 2 40 13 2 . 6 0 - 6 . 7 0 2 . 9 1 50 16 3 . 2 0 - 6 . 4 7 2 . 6 0 60 14 2 . 8 0 - 7 . 7 7 2 . 1 8 7 0 21 4 . 2 0 - 7 . 2 3 2 . 9 4 80 16 3 . 2 0 - 7 . 5 5 2 . 0 0 108 TABLE 2 TRANSACTIONS COST 1.5* DISASTER LOSSES NUMBER NUMBER PERCENT AVERAGE STANDARD REVISIONS LOSS LOSS LOSS DEVIATION 10 12 2.40 -8.63 6.64 20 20 4.00 -7.88 3.76 30 19 3.80 -8.68 2.91 40 19 3.80 -10.44 2.41 50 13 2.60 -10.96 3.80 60 13 2.60 -9.95 2.93 70 22 4.40 -12.54 3.08 80 13 2.60 -12.87 3.43 109 TABLE 3 TRANSACTIONS COST 2%* DISASTER LOSSES NUMBER NUMBER PERCENT AVERAGE STANDARD REVISIONS LOSS LOSS LOSS DEVIATION 10 21 4.20 -9o03 4.47 20 23 4.60 -10o94 3.89 30 11 2.20 -11.63 2.70 40 15 3.00 -12.14 2.10 50 8 1. 60 -14.55 2.29 60 9 1.80 -15.06 3.37 70 17 3.40 -15.92 4.11 80 16 3.20 -15.48 2.34 110 TABLE 4 TRANSACTIONS COST 2.53. DISASTER LOSSES NUMBER NUMBER PERCENT AVERAGE STANDARD REVISIONS LOSS LOSS LOSS DEVIATION 10 18 3.60 -10.08 6.19 20 15 3.00 -11.91 3.31 30 9 1.80 -14.47 3.28 40 14 2.80 -15.09 2.83 50 10 2.00 -15.69 2.52 60 16 3.20 -18.95 4.54 70 19 3.80 -19.36 4.79 80 17 3.40 -21.76 3.47 I l l TABLE 5 TRANSACTIONS COST 0%o DISASTER LOSSES NUMBER NUMBER PERCENT AVERAGE STANDARD REVISIONS LOSS LOSS LOSS DEVIATION 10 1 5 3»00 0 o l 6 5 . 5 0 20 2 2 4 . 4 0 - 0 o 6 2 3 „ 3 1 3 0 10 2 o 0 0 - 1 . 6 5 2 . 1 6 40 13 2 . 6 0 - 0 . 8 8 2 . 4 6 50 16 3 . 2 0 0 . 0 0 2 . 2 6 6 0 14 2 . 8 0 - 0 . 8 2 1 . 6 8 70 21 4 . 2 0 - 0 . 1 1 2 . 6 6 80 16 3 . 2 0 0 . 1 6 1 . 4 7 112 Appendix F Summary S t a t i s t i c s : Largest Losses These t a b l e s summarize the magnitude o f l o s s e s the company in c u r s given the r e v i s i o n p o l i c y and the t r a n s a c t i o n c o s t s . Table 5 i s again the s p e c i a l case of no t r a n s a c t i o n c o s t . Label D e f i n i t i o n s : AVERAGE LOSS: FIVE PERCENT The mean of the 25 l a r g e s t l o s s e s o c c u r i n g over 500 s i m u l a t i o n s , under each r e v i s i o n s t r a t e g y . AVERAGE LOSS: TEN PERCENT The same as above, except the l a r g e s t 50 l o s s e s are considered. I t should be noted t h a t these l o s s e s i n c l u d e e x e r c i s i n g the guarantee, as w e l l as the loss e s c r e a t e d by t r a n s a c t i o n c o s t s . 113 TABLE 1 TRANSACTIONS COST l%0 LARGEST LOSSES FIVE PERCENT TEN PERCENT NUMBER AVERAGE STANDARD AVERAGE STANDARD REVISIONS LOSS DEVIATION LOSS DEVIATION 10 -14.95 2.01 -13.41 1.83 20 -14.53 1.79 -13.35 1.53 30 -15.60 2.04 -14.26 1.73 40 -15.67 2.46 -14.34 1.98 50 -16.83 1.98 -15.52 1.69 60 -17.80 1.89 -16.23 1.77 70 -19.06 2.92 -17.07 2.51 80 -19.16 2.93 -17.43 2.41 114 TABLE 2 TRANSACTIONS COST 1.5$. LARGEST LOSSES FIVE PERCENT TEN PERCENT NUMBER AVERAGE STANDARD AVERAGE STANDARD REVISIONS LOSS OEVIATION LOSS DEVIATION 10 -20.18 2.14 -18.33 2.04 20 -20.47 2.08 -18.82 1.91 30 -23.20 2.95 -21.10 2.60 40 -22.66 2.19 -21.00 2.00 50 -24.74 2.82 -22.71 2.48 60 -27.04 4.46 -24.40 3.69 70 -27.13 3.54 -24.94 2.98 80 -27.86 4.22 -25.59 3.40 115 TABLE 3 TRANSACTIONS COST 2%0 LARGEST LOSSES FIVE PERCENT TEN PERCENT NUMBER AVERAGE STANDARD AVERAGE STANDARD REVISIONS LOSS DEVIATION LOSS DEVIATION 10 -27o00 3o93 - 2 4 o l 9 3.46 20 -28.83 9.29 -25.33 6.97 30 -30.32 5.86 -27.16 4.70 40 -31.25 3.67 -28.93 3.08 50 -33.75 3.61 -30.73 3.41 60 -34.17 2.99 -31.70 2.80 70 -36.09 4.67 -33.09 3.97 80 -39.26 5.41 -35.58 4.69 116 TABLE 4 TRANSACTIONS COST 2 0 5 l 0 LARGEST LOSSES FIVE PERCENT TEN PERCENT NUMBER AVERAGE STANDARD AVERAGE STANDARD REVISIONS LOSS DEVIATION LOSS DEVIATION 10 - 3 2 » 2 5 3o06 -29.07 3 o l 8 20 -34.36 3.77 -31.29 3.50 30 -37.94 5.46 -33.94 4.85 40 -41.48 8.19 -36.82 6.70 50 -43.06 6.26 -38.85 5.36 60 -39.95 4.06 -37.09 3.59 70 -43.74 4.92 -40.53 4.20 80 -47.20 7.54 -43.05 6.09 117 TABLE 5 TRANSACTIONS COST 02. LARGEST LOSSES FIVE PERCENT TEN PERCENT NUMBER AVERAGE STANDARD AVERAGE STANDARD REVISIONS LOSS DEVIATION LOSS DEVIATION 10 -4.99 2.33 -3.68 1.89 20 -4.49 1.86 -3.27 1.58 30 -3.34 1.38 -2.51 1.14 40 -2.38 1.17 -1.71 0.95 50 -2.62 1.03 -1.84 0.92 60 -2.28 0.73 -1.77 0.64 70 -2.15 0.81 -1.58 0.71 80 -1.69 0.67 -1.29 0.56 

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