UBC Theses and Dissertations

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UBC Theses and Dissertations

Discrete hedging in insurance risk management Sator, Imre Emil 1976

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D I S C R E T E HEDGING I N INSURANCE R I S K MANAGEMENT by IMRE E M I L 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 T H E S I S SUBMITTED I N P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF . MASTER OF S C I E N C E I N BUSINESS  ADMINISTRATION in  THE FACULTY OF COMMERCE AND BUSINESS  We a c c e p t t h i s  thesis  to the required  as  ADMINISTRATION  conforming  standard  THE U N I V E R S I T Y OF B R I T I S H COLUMBIA July, (T)  1976  Imre B m i l S a t o r ,  1976  In  presenting  an  advanced  the I  Library  further  for  his  of  this  written  thesis  degree shall  agree  scholarly  by  this  at  the U n i v e r s i t y  make that  it  purposes  for  freely  permission may  representatives. thesis  in p a r t i a l  is  financial  British  2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  Columbia,  British  by  for  gain  Columbia  shall  the  that  not  requirements I  agree  r e f e r e n c e and copying  t h e Head o f  understood  Depa r t m e n t  of  of  for extensive  permission.  The U n i v e r s i t y  of  available  be g r a n t e d  It  fulfilment  of  be a l l o w e d  or  that  study.  this  thesis  my D e p a r t m e n t  copying  for  or  publication  without  my  1  ABSTRACT  Based upon the 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 ,  Schwartz  developed an 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 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 w i t h an asset value guarantee. of t h i s contract, reference  Under the  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  portfolio of securities  whichever i s greater.  the value of  o r a minimum g u a r a n t e e d  In t h i s sense,  the contract  is  conditions a  amount,  synonymous  to  investment i n a m u t u a l f u n d and a term 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 risk.  p r o v i s i o n , however,  gives r i s e to nondiversifiable  In the event of a general market c o l l a p s e ,  simultaneously l i a b l e f o r the guarantee  on a l l mature c o n t r a c t s .  an e v e n t u a l i t y c o u l d p r o v e t o be d i s a s t e r o u s . proposes  A t any p o i n t i n t i m e , t h e b e n e f i t s  be v i e w e d as t h e p r e s e n t v a l u e o f o p t i o n on the r e f e r e n c e  s t a t e d as t h e p r e s e n t v a l u e o f the put o p t i o n .  the guarantee  portfolio.  Since the c a l l  option is  p l u s the value of it  can also  portfolio.  a be  p o r t f o l i o plus the value  sold short,  a fully  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 reference  this  o f t h e c o n t r a c t may  Conversely,  the reference  Such  The e q u i l i b r i u m m o d e l  a hedging strategy which eliminates the p r o b a b i l i t y of  type o f a l o s s .  call  t h e company b e c o m e s  of  hedged the  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 no  gains  ii  or losses w i l l  occur.  In p r a c t i s e , however, c o n t i n u o u s hedging i s because o f t r a n s a c t i o n c o s t s .  impossible  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 g a i n s Exposure to r i s k , t h e r e f o r e , i s n o t  and  losses.  eliminated.  T h i s d i s s e r t a t i o n d e a l s w i t h the problem o f 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 s t r a t e g y . of simulation and  techniques,  Through the  the impact o f v a r i o u s  revision strategies  t r a n s a c t i o n c o s t l e v e l s on the l o s s e s i s examined.  o f comparison, a n a i v e the n a i v e  s t r a t e g y was  employment  a l s o developed.  As  a basis  That i s , under  s t r a t e g y , the company buys the market p o r t f o l i o w i t h  premium and h o l d s i t u n t i l m a t u r i t y .  The  case under  consideration  i s t h a t o f the s i n g l e premium c o n t r a c t w i t h a known m a t u r i t y  The  the  date,  r e s u l t s o f the a n a l y s i s e s t a b l i s h the dominance o f  p e r i o d i c r e v i s i o n s t r a t e g y over the n a i v e .  Furthermore, f o r lower  t r a n s a c t i o n c o s t l e v e l s , the d i s p e r s i o n o f l o s s e s i s reduced as number o f r e v i s i o n s i s i n c r e a s e d . not p r o v i d e  an o p t i m a l  establishment  s o l u t i o n , i t does p r o v i d e  of acceptable  risk.  s t r a t e g y w i l l f i n d acceptance not  the  A l t h o u g h the s i m u l a t i o n model does the framework f o r  o f a management s t r a t e g y which i s c o n s i s t e n t w i t h  firm's perception  the  the  the  I t i s hoped t h a t the proposed  o n l y by the  insurance  a l s o i n the a r e a s o f mutual and p e n s i o n f u n d management.  industry V  but  i i i  TABLE OF CONTENTS  CHAPTER  1.  2.  3.  4.  .  PAGE  INTRODUCTION ^  1  OVERVIEW  5  1.1  Background  5  1.2  Risk  7  1.3  L i t e r a t u r e Overview * r  — *  THE E Q U I L I B R I U M MODEL  9 14-  2.1  The P r i c i n g o f a n 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  25  2.5  Summary  Premium Case  31  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  33  3.3  The R e t u r n o n 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 o g r a m  39  Composition  A N A L Y S I S OF RESULTS ~  *  52  4.1  Intermediate Results  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 Losses  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 Losses  60  4.5  The R e v i s i o n S t r a t e g y :  Largest Losses  4.6  Summary  >-  65 67  iv  CHAPTER  5.  PAGE  CONCLUSIONS BIBLIOGRAPHY  7  0  77  V  APPENDIX  PAGE  A.  S i m u l a t i o n Program  B.  Summary  C.  Statistics:  81 Intermediate Calculations  92  Table 1  94  Table 2  95  Table 3  96  Summary  S t a t i s t i c s : Naive Strategy  97  Table 1  98  Table 2 D.  E.  F.  „  99  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%!  T a b l e 3. T r a n s a c t i o n C o s t  2%  T a b l e 4. T r a n s a c t i o n Cost  2hi  T a b l e 5. T r a n s a c t i o n C o s t  0% —  •*  102 103  Summary S t a t i s t i c s : D i s a s t e r  1  0  4  105  Losses  106  T a b l e 1. T r a n s a c t i o n C o s t  1%  T a b l e 2. T r a n s a c t i o n C o s t  lh%  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 Cost  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  Summary  S t a t i s t i c s : Largest  107 —  Losses  1°  -~  —  112  T a b l e 1. T r a n s a c t i o n C o s t 1% ^ T a b l e 2. T r a n s a c t i o n C o s t  lh%  T a b l e 3. T r a n s a c t i o n C o s t  2%  T a b l e 4. T r a n s a c t i o n C o s t  2h%  T a b l e 5. T r a n s a c t i o n C o s t  0%  8  H3 *  H  4  115 •  1  1  6  2  1  7  vi  ACKNOWLEDGEMENTS  The most d i f f i c u l t t a s k f a c i n g t h e s t u d e n t o f any d i s c i p l i n e i s t o a d e q u a t e l y e x p r e s s h i s g r a t i t u d e and i n d e b t e d n e s s i n d i v i d u a l s whose guidance  t o those  and encouragement was a p r e r e q u i s i t e t o t h e  c o m p l e t i o n o f h i s academic u n d e r t a k i n g . A s p e c i a l e x p r e s s i o n o f g r a t i t u d e must be extended t o P r o f e s s o r s M i c h a e l J . Brennan and Eduardo S. Schwartz o f t h e F a c u l t y o f Commerce and B u s i n e s s A d m i n i s t r a t i o n o f t h e 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 c o u n s e l throughout  the course o f study.  Dr. Brennan, as c h a i r m a n o f t h e d i s s e r t a t i o n committee, n o t o n l y d i r e c t e d t h e c o u r s e o f t h e a n a l y s i s , b u t 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 p r o v i d e d c o n s t a n t guidance  throughout  t h e development o f  the a n a l y s i s , a l o n g 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 , w h i c h 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  t o P r o f e s s o r s P. B o y l e and P. L a r k e y , who c o n s e n t e d manuscript  to reading the  and a c t e d as members o f t h e d i s s e r t a t i o n committee.  I t i s also  a p r i v i l e g e t o r e c o g n i z e t h e y e a r s 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. S t a n b u r y , R. W. White and W. F. J . Wood, f o r w h i c h 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 , t o t h e t y p i s t , my w i f e , a p e r s o n a l n o t e o f thanks f o r e n d u r i n g t h e r e v i s i o n s and more.  1  Introduction  W i t h the emergence 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 i n Canada and t h e U n i t e d Kingdom as a v i a b l e a l t e r n a t i v e t o s t r a i g h t i n s u r a n c e , a number o f i m p o r t a n t i s s u e s a r i s e w i t h r e s p e c t t o t h e v a l u e and t h e r i s k s a s s o c i a t e d w i t h t h i s t y p e o f a c o n t r a c t .  I t can be a r g u e d  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 t y p e o f a c o n t r a c t i s n o t r e a l l y a l i f e i n s u r a n c e i n s t r u m e n t , b u t a s e c u r i t y o r a s s e t 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 i n v e s t m e n t p e r i o d i c premium c o n t r a c t .  s c h e d u l e i n the c a s e o f t h e  T h i s c o n t r o v e r s y has o n l y 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 t h e S e c u r i t i e s Exchange Commission e s t a b l i s h i n g t h e p r o c e d u r e s f o r the r e g i s t r a t i o n o f the s a l e o f instruments.  To J a n u a r y  1976,  these  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, 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 may  be v i e w e d as an i n s t r u m e n t w i t h a f i x e d t e r m i n a l a s s e t v a l u e , p l u s  a premium, t h e magnitude o f w h i c h depends on t h e p e r f o r m a n c e o f a reference p o r t f o l i o of s e c u r i t i e s .  The  i n s u r a n c e component i s t h e  fixed  t e r m i n a l a s s e t v a l u e , o r a g u a r a n t e e by t h e company t h a t i n t h e event o f a c o n s i d e r a b l e s t o c k market d e c l i n e , t h e 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. is zero.  Obviously, the lower bound of the premium  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 i s 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 i t s 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 i s 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 i n 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 p o i n t of "view of the insurance company,  however, n e i t h e r of these a l t e r n a t i v e s seem s a t i s f a c t o r y because of the guaranteed amount.  In the event of a market c o l l a p s e , the b e n e f i c i a r y  of the contract w i l l obviously e l e c t to receive the guaranteed amount, as he would i n a l l cases where t h 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 d i s a s t e r occurs  only i f the value of the reference p o r t f o l i o p l u s the amount invested i n the r i s k f r e e asset i s l e s s than the guaranteed amount.  The o b j e c t i v e  f u n c t i o n 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 t h e o r e t i c a l framework f o r such a strategy has been proposed by E. Schwartz i n h i s d o c t o r a l d i s s e r t a t i o n .  In a very general sense,  he shows t h a t the v a l u a t i o n of the equity l i n k e d l i f e insurance i s c l o s e l y r e l a t e d to the option p r i c i n g problem. gives r i s e to a hedging strategy.  contract  This i n t e r p r e t a t i o n  T h e o r e t i c a l l y , i f a f u l l y hedged  p o s i t i o n i s maintained by continuous adjustment and the appropriate p r i c e i s charged f o r the c o n t r a c t , 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 f e a s i b l e because of t r a n s a c t i o n costs.  A d i s c r e t e 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 d i s a s t e r losses discussed p r e v i o u s l y , may be reduced. The o b j e c t i v e o f t h i s t h e s i s i s to examine the hypothesis  that  the p r o b a b i l i t y of d i s a s t e r losses may be reduced by adopting a d i s c r e t e hedging p o l i c y i n the management of equity l i n k e d l i f e insurance c o n t r a c t s .  4  A l t h o u g h numerous a l t e r n a t i v e s  exist,  the a n a l y s i s w i l l  focus  entirely  on the s i n g l e p e r i o d , s i n g l e premium c o n t r a c t .  In order to test  hypothesis,  adopted.  computer s i m u l a t i o n techniques were  The d e v e l o p m e n t outline.  of  Chapter 1 provides  equity linked l i f e  the a n a l y s i s  sections  following  A r e v i e w o f some o f  topic i s also presented  Chapter 2 focuses mainly on the Black-Scholes and the r e l e v a n t  to the  a b r i e f d i s c u s s i o n of the e v o l u t i o n  insurance contracts.  l i t e r a t u r e pertinent to this  conforms  o f the  Schwartz d i s s e r t a t i o n , which i n  o f s i m u l a t i o n and the s i m u l a t i o n model employed.  Chapter 4  of this analysis.  the f i n d i n g s o f the t h e s i s analysis. conclusion.  Chapter  and p r o v i d e s recommendations  5 i s a restatement  for  a  effect  discussion  summarizes further  o f the p r o b l e m and a b r i e f  An appendix has been p r o v i d e d f o r  and the s i m u l a t i o n program.  section.  option valuation formula  3 contains  the foundations  of  the  in this  Chapter  forms  the  the r e l e v a n t  statistics  5  Chapter  1.1  1  Background  I n t e r e s t i n g l y enough, the f i r s t marketed by to  companies, were i n f a c t term  whole l i f e  instruments.  somewhat u n c l e a r , i t h a s  Although  life  insurance contracts  insurance p o l i c i e s ,  the reason  as  opposed  f o r t h i s phenomenon i s  been argued that a general misunderstanding  the theory o f insurance or the i n f a n t s t a t e of the theory caused to  take t h i s p o s i t i o n .  B r i e f l y , term  i n s u r a n c e , as  provides coverage f o r a s p e c i f i e d p e r i o d . is  concerned  t h e name  In t h i s sense,  term  insurance  p r i m a r i l y w i t h the p r o b a b i l i t y of a c c i d e n t a l death  As  such,  i t may  and eventual  b e v i e w e d a s a b e t b e t w e e n t h e company  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 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  The  whole l i f e  e v e n t u a l i t y of death for  c o n t r a c t , on  as a c e r t a i n t y .  a reserve during the l i f e  will  the o t h e r hand, r e c o g n i z e s The  and  not.  company a s s u m e s t h e  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  b u i l d up  companies  implies,  n a t u r a l death w i t h i n the s p e c i f i e d p e r i o d o f time, not w i t h the certainty.  of  the  liability  t h e r e f o r e must  o f the c o n t r a c t i n o r d e r t o meet i t s  obligation at i t s termination.  The  premiums a s s o c i a t e d w i t h t h e s e  Term i n s u r a n c e p r e m i u m s t e n d t o b e t h e f a c t t h a t t h e company may c o n t r a c t , and  the r e a l i t y  lower  n o t be  contracts vary considerably.  than whole l i f e ,  reflecting  l i a b l e f o r the face value of  t h a t aged i n d i v i d u a l s are excluded  from  the this  6  type of an instrument. With the evolution of nonforfeiture clauses i n 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 i s 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, ( i e . within the specified time of the contract ).  If the  insured i s alive at the termination of the contract, the insured sum is s t i l l paid by the company. savings plan.  In essence then, i t i s simply an insured  7  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 r a t e s s i n c e 1945  and t h e s h a r p  f l u c t u a t i o n s have caused p o l i c y h o l d e r s and f o r t h a t m a t t e r  insurance  companies t o r e v i e w t h e 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 to y i e l d .  purport  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 a n y t h i n g b u t s a t i s f a c t o r y , g i v e n 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 p e r y e a r .  Those on f i x e d incomes, o r i n d i v i d u a l s  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 spiraling i n f l a t i o n rates.  cashing to the  I t can be argued t h a t i n s u r a n c e companies  s h o u l d have done v e r y w e l l , r e p a y i n g u n i n f l a t e d d o l l a r s w i t h  inflated  ones, b u t t h e r e t u r n s d e c l a r e d b y companies does n o t seem t o r e f l e c t  this.  I t remains t o be seen what impact c u r r e n t e f f o r t s t o r e d u c e 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 s i g n e d i n the l a s t f i v e o r s i x y e a r s .  1.2  Risk  Although  the r i s k o f death i s o f t e n a l u d e d t o by  insurance  salesmen as a j u s t i f i c a t i o n f o r t h e premiums c h a r g e d , 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 . i s synonymous t o p o r t f o l i o t h e o r y . implies that the unsystematic case may  be d i v e r s i f i e d away.  the i n d u s t r y r e m a i n s .  B a s i c a l l y , the t h e o r y o f  insurance  The p o o l i n g o f independent r i s k  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 Consequently, only the r i s k inherent i n  The key t o t h e 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 o b s e r v e d , t h e n the number o f deaths p e r p e r i o d 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 a c c u r a c y ,  assuming t h a t t h e r e i s a  may  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 r e m a i n s , t h e r e f o r e ,  i s t h e m a t c h i n g o f t h e l i q u i d a t i o n o f i n t e r e s t e a r n i n g a s s e t s and a g a i n s t t h e company, as t h e y become due.  claims  I f the theory d i d not h o l d ,  t h e n t h e r e 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 c l a u s e s s u c h as war, n a t u r a l d i s a s t e r s , a c t o f God, In essence,  etc., included i n a l l p o l i c i e s .  these phenomena v i o l a t e t h e independence a s s u m p t i o n .  As the above argument i m p l i e s , under t h e c o n d i t i o n s o f the p o l i c i e s d i s c u s s e d thus f a r , the i n s u r a n c e company assumes a l l t h e r i s k . P r i m a r i l y because o f t h e 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 g e n e r a l d i s s a t i s f a c t i o n o f p o l i c y h o l d e r s w i t h the subsequent d e c l i n e i n the v a l u e o f t h e p o l i c y , i n s u r a n c e companies began t o market an e q u i t y l i n k e d i n s t r u m e n t f i r s t i n the N e t h e r l a n d s B r i t a i n and Canada, and now  i n the e a r l y f i f t i e s , t h e n l a t e r i n  i n the United States.  The p r i m a r y d i f f e r e n c e  between t h i s t y p e o f p o l i c y and t h e c o n v e n t i o n a l i n s t r u m e n t p o r t i o n o f the r i s k i s b o r n e by the i n s u r e d . t h e b e n e f i t s o f the c o n t r a c t may  i s that a  As d i s c u s s e d p r e v i o u s l y ,  be v i e w e d as the 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 g u a r a n t e e d amount, w h i c h e v e r i s g r e a t e r . The t h e o r y b e h i n d 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 t o assume a p a r t o f t h e r i s k because o f t h e 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 a g a i n s t i n f l a t i o n .  S i n c e 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 f r o m  the c o n v e n t i o n a l approach, t h e v a l u a t i o n o f t h i s i n s t r u m e n t and  the  d e t e r m i n a t i o n o f the premium s t r u c t u r e p r e s e n t s some u n i q u e p r o b l e m s . In a narrow s e n s e , the p o l i c y i s s i m i l a r t o b u y i n g a m u t u a l f u n d and term i n s u r a n c e as a complement, f r o m t h e p o i n t o f v i e w o f t h e i n s u r e d .  It  9  s h o u l d be n o t e d t h a t t h i s t y p e o f i n s t r u m e n t p r o v i d e s t h e p o i n t from g e n e r a l l y accepted insurance theory. the a s s e t v a l u e guarantee  departure  The r i s k a s s o c i a t e d w i t h  can not be d i v e r s i f i e d away.  A g e n e r a l market  d e c l i n e w i l l r e s u l t i n a c a t a s t r o p h e as t h e guarantee w i l l be e x e r c i s e d under a l l m a t u r i n g c o n t r a c t s .  A l t h o u g h a c o n s i d e r a b l e number o f p a p e r s have been w r i t t e n on t h e 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 i n s u r a n c e p o l i c i e s , most o f them p u r s u e what may  be d e f i n e d as a n a i v e approach.  That i s , most o f them f o c u s  on  t h e p r o b l e m o f e s t a b l i s h i n g adequate r e s e r v e s , w i t h o u t c o n s i d e r i n g a l l the parameters of the problem.  I n t h e t r a d i t i o n a l s e n s e , t h i s meant  average o r mean r e s e r v e r e q u i r e m e n t s .  I f companies e n j o y e d t h e same  degree o f e x p e r i e n c e w i t h t h e e q u i t y l i n k e d p r o d u c t s as t h e y do w i t h whole l i f e c o n t r a c t s , t h e n such a s t r a t e g y may c e r t a i n extent.  1.3  w e l l be a c c e p t a b l e t o a  T h i s , however, i s n o t t h e c a s e .  L i t e r a t u r e Overview  Squires, ( 1 6 )  i n a 1974 p a p e r c o r r e c t l y i d e n t i f i e s  the  i n a d e q u a c i e s o f e x i s t i n g models a t t e m p t i n g t o e x p l a i n s t o c k market behavior.  U n f o r t u n a t e l y , h i s assumptions about t h e market a r e a l s o open  to c r i t i c i s m , w h i c h t e n d t o n e g a t e 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 t h a t i n t h e case o f 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 t h e 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, t h e n t h e company i s  10  s u b j e c t e d t o s u b s t a n t i a l r i s k s d u r i n g t h e subsequent t r o u g h .  This  a s s e r t i o n c l e a r l y i g n o r e s t h e random w a l k o r e f f i c i e n t market h y p o t h e s i s . I f t h e random w a l k i s an a c c u r a t e d e s c r i p t i o n o f p r i c e b e h a v i o r , o f w h i c h t h e r e i s c o n s i d e r a b l e e v i d e n c e , t h e n a "market peak" can o n l y be i d e n t i f i e d i n • retrospect.  P u t a n o t h e r way,  h o l d s , t h e n i t i s i m p o s s i b l e t o determine  i f t h e random w a l k h y p o t h e s i s  i f today's s t o c k p r i c e i s a t  i t s "peak" because i n o r d e r t o a c c o m p l i s h t h a t , tomorrow's p r i c e must be known.  T h i s i s n o t p o s s i b l e because o f t h e c o n d i t i o n : P  (  X> t  0 ) =  P (  X< t  0 )  =  .5  t h a t i s , t h e 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 e q u a l s t h e p r o b a b i l i t y o f a n e g a t i v e p r i c e change ( i e . 0.5  ).  I f such i s t h e c a s e , t h e n t h e  b e s t e s t i m a t e o f tomorrow's p r i c e must be today's p r i c e . d e r i v e d by a l g e b r a .  T h i s can  be  I n e f f e c t t h e n , s i n c e tomorrow's p r i c e i s n o t 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 t o  n o t e however, i s t h a t h i s approach c o m p l e t e l y i g n o r e s t h e i m p l i c a t i o n s o f the hedging s t r a t e g y . The m a j o r c o n t r i b u t i o n s t o t h e u n d e r s t a n d i n g o f t h e n a t u r e 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 were s i m u l a t i o n models by T u r n e r  (18), DiPaolo  (6 ) , and Kahn (8 ) .  T h e i r work may  developed  be v i e w e d  as t h e p o i n t o f d e p a r t u r e f r o m t h e c o n v e n t i o n a l i n t e r p r e t a t i o n o f t h e 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  a g g r e s s i v e , whereas t h e t r a d i t i o n a l p o s i t i o n i s d e f e n s i v e .  As T u r n e r  (17 )  s t a t e s , i n r e f e r e n c e t o a p a p e r by S i d n e y Benjamin, " t h e s t a t e d approach  11  to v a l u a t i o n , t h a t i s , t o 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,;; w h i c h , i n t h e o p i n i o n o f t h e a c t u a r y , w o u l d be r e q u i r e d c o n s i d e r i n g t h e n a t u r e o f t h e guarantees p r o v i d e d a n d t h e f i n a n c i a l s i t u a t i o n a t that time".  But c l e a r l y , t h e most i m p o r t a n t element o f  t h e 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 , t h e e x p e c t e d performance o f t h e r e f e r e n c e p o r t f o l i o , o r i n a more g e n e r a l sense, t h e e x p e c t e d performance o f t h e m a r k e t .  Since the b a s i c theory  o f t h e e q u i t y l i n k e d p r o d u c t i s t h a t t h e i n v e s t o r i s w i l l i n g t o assume a p o r t i o n o f t h e r i s k i n o r d e r t o a t t a i n a h i g h e r 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. requirement,  The r e s e r v e  t h e r e f o r e , must be a f u n c t i o n o f t h e e x p e c t e d r a t e o f  r e t u r n on t h e m a r k e t , and t h e 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 t h a n t h e amount o f t h e g u a r a n t e e . the v a r i o u s c r i t i q u e s o f t h e a f o r e m e n t i o n e d  As p o i n t e d o u t i n :  papers, there i s a general  r e l u c t a n c e on t h e p a r t o f a c t u a r i e s t o a c c e p t t h e c u r r e n t t h e o r y o f c a p i t a l asset p r i c i n g .  T h i s may account f o r t h e h e s i t a t i o n  observed  i n v i e w i n g t h e i n s t r u m e n t as an o p t i o n p r i c i n g p r o b l e m , as opposed t o a r e s e r v e problem. Turner  (17 ) b a s i c a l l y v i e w s t h e p r o b l e m i n t h r e e s t a g e s .  F i r s t , he r e c o g n i z e s t h a t some c o n c l u s i o n s must be made c o n c e r n i n g t h e nature o f the p r o b a b i l i t y density f u n c t i o n of s e c u r i t y returns.  Second,  he f o c u s e s on t h e e v a l u a t i o n o f t h e n e t r i s k premium o f an a s s e t v a l u e guarantee  a t t h e end o f t h e c o n t r a c t p e r i o d , g i v e n t h e e q u i t y l i n k e d  12  instrument.  T h i r d l y , he a n a l y z e s  the s e n s i t i v i t y o f the n e t  risk  premium t o changes i n u n d e r l y i n g p a r a m e t e r s , such as i n v e s t m e n t p e r i o d , charges a g a i n s t t h e r e t u r n on e q u i t y , t a x e s , and decrements i n m o r t a l i t y and w i t h d r a w a l s .  The  o v e r a l l i m p l i c a t i o n s o f 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 o f a framework w h i c h f o r c e s a c t u a r i e s i n t o v i e w i n g  the  p r o b l e m o f 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 v a l u e g u a r a n t e e i n a more q u a n t i t a t i v e o r a n a l y t i c s e t t i n g t h a n t h e t r a d i t i o n a l approach. DiPaolo  r e l i e s on Monte C a r l o t e c h n i q u e s t o g e n e r a t e o r  simulate s e c u r i t y trends, which are then u t i l i z e d to evaluate  the  adequacy o f t h e i n v e s t m e n t r i s k premium c h a r g e d , f o r an e q u i t y b a s e d endowment p o l i c y .  H i s b a s i c a s s u m p t i o n i s t h a t a r i s k premium i s  deemed t o be adequate i f t h e p r o b a b i l i t y o f t h e r i s k f u n d b e i n g s t a t e o f r u i n i s s m a l l a f t e r t h e l a s t c o n t r a c t matures.  in a  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 the l a s t c o n t r a c t .  As  i s t h e c a s e w i t h most s i m u l a t i o n models, h i s does  n o t p r o v i d e an o p t i m a l 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 t h e outcomes.  Nevertheless  of  t h e model does r e c o g n i z e  various  t h e f a c t t h a t the  rate  o f r e t u r n on t h e 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 o f the funds i n v o l v e d .  Kahn's approach t o the p r o b l e m i s s i m i l a r t o t h a t o f T u r n e r and D i P a o l o .  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 t e c h n i q u e s t o  g e n e r a t e a market t r e n d o r r e t u r n and uses t h e r e s u l t s t o p r o j e c t various insurance  alternatives.  I n g e n e r a l , h i s f i n d i n g s show t h e  13  extreme s e n s i t i v i t y o f e a r n i n g s o f a v a r i a b l e l i f e i n s u r a n c e w i t h respect to investment performance.  company  He a l s o shows t h a t the  cost  o f a minimum d e a t h b e n e f i t g u a r a n t e e v a r i e s w i d e l y w i t h i n v e s t m e n t performance. I n g e n e r a l , the s i g n i f i c a n c e o f t h e s e a n a l y s e s 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 of the problem.  lies  i n the  As s t a t e d p r e v i o u s l y ,  the d e p a r t u r e f r o m t h e t r a d i t i o n a l d e f e n s i v e p o s i t i o n t h a t t h e o v e r r i d i n g f a c t o r i s the e s t a b l i s h m e n t a breakthrough.  o f r e s e r v e s , must be v i e w e d as  A l l three authors e x p l i c i t l y recognize  that  the  c r i t i c a l i s s u e i s t h e p e r f o r m a n c e o f the i n v e s t m e n t 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 o f the p r o b a b i l i t y o f a s u s t a i n e d market d e c l i n e o r a g e n e r a l The  collapse.  f o l l o w i n g c h a p t e r w i l l show t h 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 t o v i e w the p r o b l e m i n such a framework t h a t the p r o b a b i l i t y o f r u i n may  be c o m p l e t e l y  e l i m i n a t e d through a process o f hedging.  I n o r d e r t o a c c o m p l i s h t h i s , a comprehensive o v e r v i e w o f t h e  Black-  S c h o l e s o p t i o n p r i c i n g model w i l l be p r e s e n t e d , f o l l o w e d by a d i s c u s s i o n o f t h e m a j o r i s s u e s o f the Schwartz d i s s e r t a t i o n ; T h i s w i l l  provide  the n e c e s s a r y background f o r t h e s i m u l a t i o n model employed i n t h i s analysis.  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 O p t i o n  I n t h e most g e n e r a l sense, an o p t i o n may be d e f i n e d as t h e r i g h t t o buy o r s e l l an a s s e t , s u b j e c t t o 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 o f time.  The p r i c e p a i d f o r t h e 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 t o as t h e s t r i k i n g o r e x e r c i s e p r i c e .  The l a s t day on w h i c h  i t c a n be e x e r c i s e d i s t h e m a t u r i t y o r e x p i r a t i o n d a t e .  There a r e  b a s i c a l l y two t y p e s o f o p t i o n s ; t h e European, and t h e A m e r i c a n .  An  A m e r i c a n o p t i o n may be e x e r c i s e d any t i m e up t o and i n c l u d i n g t h e m a t u r i t y d a t e , whereas t h e European c a n o n l y be e x e r c i s e d on t h e specified future The  date.  s i m p l e s t k i n d o f o p t i o n i s t h e r i g h t t o buy a s i n g l e share  o f common s t o c k , o r c a l l o p t i o n .  I t s counterpart, the put option, i s  t h e r i g h t t o s e l l one s h a r e o f common t o a n o t h e r p a r t y .  I t c a n be  r e a d i l y s e e n t h a t a number o f c o m b i n a t i o n s o f t h e two b a s i c o p t i o n t y p e s a r e p o s s i b l e , depending on t h e o b j e c t i v e s o f t h e i n d i v i d u a l . For t h e p u r p o s e s o f t h i s p o r t i o n o f t h e a n a l y s i s , however, t h e f o c u s w i l l be on t h e c a l l  option.  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 t h e v a l u e o f t h e o p t i o n and t h e p r i c e o f t h e u n d e r l y i n g s e c u r i t y .  I t c a n be e x p e c t e d  t h a t t h e h i g h e r t h e p r i c e o f t h e s t o c k , t h e g r e a t e r s h o u l d be t h e v a l u e of the option.  I f the stock p r i c e i s considerably greater than the  15  e x e r c i s e p r i c e , t h e n , t h e o p t i o n w i l l p r o b a b l y be e x e r c i s e d . f o r m a l l y , a t t h i s p o i n t , t h e v a l u e o f t h e o p t i o n w i l l be  More  approximately  e q u a l t o t h e p r i c e o f t h e s t o c k minus t h e p r i c e o f a p u r e d i s c o u n t bond t h a t matures on t h e same date as t h e o p t i o n , and has a f a c e v a l u e 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 .  ( 2-1 )  V0  t  =  PE  -  B ( e"  That i s :  r t  *  )  Where VO^ i s t h e v a l u e o f t h e o p t i o n a t time t ; PE  the p r i c e o f the  -rt* s e c u r i t y a t t i m e t ; and t h e e x p r e s s i o n B ( e d i s c o u n t bond.  ) the p r i c e o f the  S i n c e t h e p r o b a b i l i t y o f e x e r c i s i n g t h e o p t i o n becomes  v e r y h i g h as t h e m a t u r i t y date approaches ( i e . as p e r t h e above a s s u m p t i o n ) , t h e p r o c e s s may be v i e w e d as a d e f e r r e d p u r c h a s e p l a n . The v a l u e o f t h e d i s c o u n t bond a t t h e c r e a t i o n o f t h e o p t i o n r e p r e s e n t s the amount w h i c h must be 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 n o r d e r t o i n s u r e t h a t s u f f i c i e n t funds a r e a v a i l a b l e t o e x e r c i s e t h e o p t i o n a t maturity.  I n essence,  t h i s i s t h e same as a d e f e r r e d p l a n .  On t h e o t h e r hand, i f t h e s t o c k p r i c e , PE i s c o n s i d e r a b l y l e s s t h a n t h e s t r i k i n g p r i c e , t h e o p t i o n w i l l p r o b a b l y e x p i r e , so t h a t i t s v a l u e s h o u l d be n e a r z e r o .  Furthermore, i f the e x p i r a t i o n  date i s f a r o f f , t h e n t h e p r i c e o f t h e d i s c o u n t bond w i l l be l o w , i m p l y i n g t h a t t h e o p t i o n v a l u e w i l l be a p p r o x i m a t e l y stock price.  t h e same as t h 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 v a l u e  should approximately  e q u a l t h e d i f f e r e n c e between t h e s t o c k p r i c e  16  and t h e e x e r c i s e p r i c e , o r z e r o i f t h e s t o c k p r i c e i s l e s s t h a n t h e striking price.  N o r m a l l y , i t c a n be e x p e c t e d t h a t t h e v a l u e o f t h e  o p t i o n s h o u l d d e c l i n e , i f t h e r e i s no change i n t h e s t o c k p r i c e . W i t h i n t h i s framework i t c a n be e x p e c t e d t h a t t h e 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 g i v e n p e r c e n t change i n t h e  p r i c e o f t h e s t o c k , a l a r g e r p e r c e n t change w i l l o c c u r i n t h e v a l u e of the option, given that maturity i s h e l d constant.  I t s h o u l d be  n o t e d , however, t h a t t h e r e l a t i v e v o l a t i l i t y o f t h e o p t i o n i s n o t c o n s t a n t as i t depends on s t o c k p r i c e and m a t u r i t y .  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  The o r i g i n s o f t h e 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 may be found i n t h e works o f S p r e n k l e etc.  (1961), A y r e s  (1963), Samuelson (1965),  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 t h e v a l u a t i o n o f  w a r r a n t s , b u t f o r a l l i n t e n t s and p u r p o s e s , applicable t o other options.  the theory i s e q u a l l y  The m a j o r p r o b l e m w i t h t h e s e  earlier  f o r m u l a t i o n s i s t h e f a c t t h a t some o f t h e p a r a m e t e r s were l e f t  undefined.  The k e y a s s u m p t i o n t h a t t h e y do u t i l i z e , however, i s a conclusiQn;..of:.the work o f Thorpe and Kassouf  (1967).  They n o t e t h a t t h e v a l u a t i o n o f  the w a r r a n t h i n g e s upon t h e r a t i o o f s t o c k o p t i o n s t o s h a r e s needed t o c r e a t e a hedged p o s i t i o n by g o i n g s h o r t i n one s e c u r i t y and l o n g i n the other.  As B l a c k and S c h o l e s p o i n t o u t , what t h e y f a i l e d t o r e c o g n i z e  17  was t h e f a c t t h a t the e x p e c t e d r a t e o f r e t u r n on s u c h a p o s i t i o n must be e q u a l t o t h e 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 t o 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.  B e f o r e p r o c e e d i n g t o t h e i r model, however, a r e v i e w 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 a r e assumed f o r t h e  s t o c k and t h e o p t i o n .  a)  More s p e c i f i c a l l y :  The s h o r t t e r m i n t e r e s t r a t e i s assumed t o be known and c o n s t a n t  through time.  T h i s may be r e l a x e d under  certain conditions. b)  S t o c k p r i c e s f o l l o w a random w a l k i n c o n t i n u o u s t i m e 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 t o t h e square o f t h e stock p r i c e .  That i s , t h e d i s t r i b u t i o n o f s t o c k p r i c e s  i s l o g normal and t h e v a r i a n c e r a t e o f t h e r e t u r n on the s t o c k i s c o n s t a n t . c)  There a r e no d i v i d e n d s  d)  The o p t i o n i s European,  o r any o t h e r d i s t r i b u t i o n s . ( i e . exercisable at maturity  only).  e)  No t r a n s a c t i o n s c o s t s i n b u y i n g o r s e l l i n g t h e s t o c k or the option. analysis).  ( T h i s w i l l be r e l a x e d i n t h e subsequent  18  £)  I t i s p o s s i b l e t o borrow a t t h e s h o r t t e r m 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 s h o r t  selling.  I t c a n be r e a d i l y s e e n t h a t under t h e s e a s s u m p t i o n s , t h e v a l u e o f t h e o p t i o n depends o n l y on t h e s t o c k p r i c e and t i m e , and on parameters w h i c h a r e t a k e n as known.xonstants. Under s u c h  circumstances  i t i s p o s s i b l e t o f o r m a hedged p o s i t i o n b y s h o r t i n g t h e 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 that the value o f the o p t i o n w i l l n o t depend on t h e s t o c k p r i c e b u t on time and t h e c o n s t a n t s . That i s , more f o r m a l l y t h e v a l u e o f t h e o p t i o n may be e x p r e s s e d a s :  ( 2-2 )  w ( x, t )  or as a f u n c t i o n o f t h e s t o c k p r i c e x and t i m e t . To f o r m t h e hedged p o s i t i o n , t h e number o f o p t i o n s t h a t must be s o l d a g a i n s t one s t o c k may be w r i t t e n a s : ( 2-3 ) The  1/  W l  ( x, t )  s u b s c r i p t denotes t h e p a r t i a l w i t h r e s p e c t t o t h e f i r s t argument.  To show t h a t t h e v a l u e o f t h e hedged p o s i t i o n does n o t depend o n s t o c k p r i c e , n o t e t h a t w^  (' x , t ) i s t h e r a t i o o f t h e change i n t h e  o p t i o n v a l u e t o t h e change i n t h e s t o c k p r i c e .  That i s , i f x changes  by A x , t h e o p t i o n p r i c e w i l l change b y w., ( x , t ) Ax, so t h a t t h e  19  v a l u e o f l/w^ ( x , t ) o p t i o n s w i l l change b y A x .  T h e r e f o r e a change  i n t h e v a l u e o f t h e l o n g p o s i t i o n i n x w i l l be a p p r o x i m a t e l y by t h e change i n t h e v a l u e o f t h e s h o r t p o s i t i o n i n 1/w^  offset  options.  I f c o n t i n u i t y i s assumed, t h e n i t c a n be shown t h a t t h e a p p r o x i m a t i o n s become e x a c t and t h e r e t u r n on t h e hedged p o s i t i o n i s c o m p l e t e l y independent o f t h e changes i n t h e v a l u e o f t h e s t o c k .  That i s , t h e  r e t u r n on t h e hedged p o s i t i o n becomes c e r t a i n .  I t i s i m p o r t a n t t o n o t e t h a t t h e 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 t h e o r y .  Under t h e random w a l k and  c o n s t a n t v a r i a n c e r a t e assumptions t h e c o v a r i a n c e between t h e r e t u r n s on t h e e q u i t y and t h e s t o c k w i l l be z e r o . to  t h e market p o r t f o l i o c o n c e p t .  The same argument a p p l i e s  Consequently,  under a c o n t i n u o u s  adjustment p o l i c y , t h e r i s k i n t h e hedged p o s i t i o n i s z e r o .  Even i f  c o n t i n u o u s a d j u s t m e n t does n o t o c c u r , i t i s e x p e c t e d t h a t t h e r i s k be s m a l l .  will  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 c o m p l e t e l y  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 t h e r e s t o f t h i s analysis. The v a l u e o f t h e e q u i t y i n t h e p o s i t i o n , g i v e n one s h a r e l o n g and 1/w^ ( 2-4 )  options short i s defined as: Ve  =  x -  w/w.  20  and t h e change i n Ve o v e r a s h o r t i n t e r v a l ( 2-5 );  AVe  = Ax -  t as:  Aw/*^  Under t h e a s s u m p t i o n o f c o n t i n u o u s a d j u s t m e n t , Aw c a n be expanded t h r o u g h s t o c h a s t i c c a l c u l u s and be shown t h a t t h e v a l u e o f t h e e q u i t y becomes: 2 (2-6  - ( h^Y\_  )  v  x  2 +  w  2  -*  A t  / i W  2 where t h e 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 v a r i a n c e r a t e o f r e t u r n on t h e s t o c k .  i s the  F u r t h e r m o r e , s i n c e t h e r e t u r n on  the e q u i t y i s k n o w n f o r c e r t a i n t o be r A t , t h e change i n t h e e q u i t y can be e x p r e s s e d a s : ( 2-7 )  ( x - w/w.^  rAt  E q u a t i n g (2-6) and ( 2 - 7 ) , d r o p p i n g t and r e a r r a n g i n g g i v e s t h e f o l l o w i n g d i f f e r e n t i a l equation  ( 2-8 )  f o r the value o f the option:  w  2  2 2 = rw - rxw^ - h (v x w ^  )  Assuming t * t o be t h e m a t u r i t y d a t e and c t h e e x e r c i s e p r i c e and t h e f o l l o w i n g boundary c o n d i t i o n s , ( 2-9 ) and  w( x , t * ) = x - c  for x ^ c  w( x , t * ) = 0  for x < c  and  s o l v i n g . (2-8)  formula. ( 2 ) .  ( 2-10  subject  The  to  (2-9)_ r e s u l t s i n the o p t i o n  f o r m u l a may  be s t a t e d  )  w(x,t) = x N  )  d  (d ) 1  valuation  as:  - ce "^"**) N  (d )  1  2  where ( 2-11  = ln(x/c) + (r + % v ) ( t * - t ) 2  v(t*-t)  %  lnfx/c) + (r - h v ) ( t * - t ) 2  A  =  2 v(t*-t)  and N(d)  %  i s the c u m u l a t i v e n o r m a l d e n s i t y  function given  d ( 2-12  )  N(d)  = 1/.(2TT )  2  , e""  -,2  r  dx  2 W  ) with respect to  T a k i n g t h e p a r t i a l d e r i v a t i v e o f e q u a t i o n (2-10 the f i r s t argument and  ( 2-13  )  by:  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  w  1  (x,t) = N  (d ) 1  w h i c h i s o f p a r t i c u l a r i m p o r t a n c e t o t h i s a n a l y s i s , as the defines  expression  the r a t i o o f s t o c k t o o p t i o n s i n t h e hedged p o s i t i o n .  22  B l a c k and S c h o l e s c o n t i n u e by p o i n t i n g out t h a t as c a n seen f r o m e x p r e s s i o n  ( 2-10  ),  be  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 t h e e x p e c t e d r e t u r n on the s t o c k .  T h i s , however, does n o t n e g a t e t h e  p r o p o s i t i o n t h a t t h e e x p e c t e d r e t u r n on t h e o p t i o n i s a f u n c t i o n o f the e x p e c t e d r e t u r n on the s t o c k . confirms  t h a t t h e p r i c e o f t h e 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 functions. ( 2-10  In e f f e c t , then,the f o r m u l a t i o n  ) and  ( 2-13  As p r e s e n t e d  i n a previous  argument, f r o m  ) i t can be shown t h a t the v o l a t i l i t y o f t h e  i s always g r e a t e r t h a n t h e v o l a t i l i t y o f t h e s t o c k . the  equations option  That i s , s i n c e  ratio  ( 2-14  )  xw  i s always g r e a t e r t h a n one,  1  / w  i t i m p l i e s t h a t the r e l a t i v e  volatilities  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 s u c h t h a t the v o l a t i l i t y o f t h e  option  w i l l always exceed the v o l a t i l i t y o f t h e s e c u r i t y . B l a c k and S c h o l e s a l s o show t h a t e q u a t i o n  ( 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 a s s e t p r i c i n g model,but f o r t h e p u r p o s e s o f t h i s a n a l y s i s , the p r e v i o u s  development w i l l s u f f i c e .  It is  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  B l a c k and S c h o l e s when t h e y compared t h e 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 t h e o b s e r v e d v a l u e s  t e n d t o d e v i a t e f r o m the p r e d i c t e d v a l u e s i n a s y s t e m a t i c manner.  23  The p u r c h a s e r s o f c a l l o p t i o n s t e n d t o o v e r p a y , b u t t h e w r i t e r s o f the o p t i o n r e c e i v e a p p r o x i m a t e l y According costs. for  t o B l a c k and S c h o l e s ,  what t h e v a l u a t i o n f o r m u l a p r e d i c t s . t h e d i f f e r e n c e must be t h e t r a n s a c t i o n s  They a l s o n o t e t h a t t h e o b s e r v e d d i f f e r e n c e s t e n d t o be g r e a t e r  o p t i o n s on low r i s k s t o c k s t h a n h i 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  c o s t s , h o w e v e r , remove t h e 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 .  S i n c e i t s d e r i v a t i o n , t h e o p t i o n v a l u a t i o n f o r m u l a has r e c e i v e d a c o n s i d e r a b l e amount o f a t t e n t i o n from academics.  As s t a t e d i n t h e  a s s u m p t i o n s , B l a c k and S c h o l e s r e s t r i c t e d t h e a n a l y s i s t o paying s e c u r i t i e s .  non-dividend  M e r t o n ( 1 0 ) extends t h e model and shows t h a t t h i s  a s s u m p t i o n may be r e l a x e d t o s t o c k s p a y i n g c o n t i n u o u s d i v i d e n d s .  He  a l s o shows t h a t i f t h e r e a r e no d i v i d e n d s , t h e n i t w i l l n e v e r pay t o e x e r c i s e an A m e r i c a n o p t i o n b e f o r e t h e m a t u r i t y d a t e , i m p l y i n g  that  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  r e s p e c t t o s e n s i t i v i t y , he n o t e s t h a t t h e v a l u e o f t h e o p t i o n w i l l increase continuously t o the extent that the maturity t , the r i s k free 2 rate r , or the variance rate v  increases.  The upper bound, he  concludes,  must be t h e s t o c k p r i c e . B r i e f l y , t h e v a l u a t i o n f o r m u l a must be c o n s i d e r e d as one o f the most i m p o r t a n t b r e a k t h r o u g h s i n f i n a n c e .  I t not only 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 t o a r a t h e r complex p r o b l e m , b u t 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.  F o r example, i n t h e  a r e a o f c o r p o r a t e f i n a n c e i t c a n be shown r a t h e r e a s i l y t h a t t h e s t o c k  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 t h e a s s e t s o f t h e f i r m assuming bonds t o be o u t s t a n d i n g .  Furthermore,  be v i e w e d as an o p t i o n c o n t r a c t .  each bond i n t e r e s t payment  may  I n g e n e r a l , i t can be shown t h a t t h e  t o t a l number o f o p t i o n s i n t h i s c o n t e x t i s e q u a l t o n + 1, where n e q u a l s the number o f i n t e r e s t payments t o be made. may  C l e a r l y , t h e arguments  be extended t o c o n v e r t i b l e " i n s t r u m e n t s , e t c . , b u t t h e s e become  rather complicated. Having d e v e l o p e d  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 n e c e s s a r y t o i n t e g r a t e i t i n t o t h e e q u i t y l i n k e d i n s u r a n c e framework.  life  I n order t o achieve t h i s , the next s e c t i o n w i l l  f o c u s on t h e 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 t h e p r e v i o u s c h a p t e r .  2.3  The Schwartz D i s s e r t a t i o n  T h i s d i s s e r t a t i o n o f f e r s one o f t h e most i m p o r t a n t c h a l l e n g e s to  the l i f e insurance i n d u s t r y .  I n essence,  to  c o m p l e t e l y r e - e v a l u a t e i t s p o s i t i o n w i t h r e s p e c t t o t h e management  of equity l i n k e d l i f e insurance contracts.  i t c a l l s upon the i n d u s t r y  I n 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 of the concept o f r i s k . The d i s s e r t a t i o n may One,  be d i v i d e d i n t o two d i s t i n c t a r e a s .  t h e p r e v i o u s l y c i t e d problems c o n c e r n i n g t h e v a l u a t i o n o f c e r t a i n  o p t i o n s and two, t h e i n t e r p r e t a 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  insurance  25  instrument  i n t h e 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 ,  a u t h o r d e v e l o p s n u m e r i c a l methods t o e v a l u a t e o p t i o n s on paying d i s c r e t e dividends.  the  securities  He p r o c e e d s t o 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 t o e x e r c i s e A m e r i c a n o p t i o n s p r i o r t o t h e d a t e ( i e . the s e c u r i t y pays d i s c r e t e d i v i d e n d s  ).  maturity  B r i e f l y , he shows  t h a t a " c r i t i c a l s t o c k p r i c e " can be d e t e r m i n e d , above w h i c h 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 o f the d i s s e r t a t i o n d e a l s w i t h e q u i t y l i n k e d  instruments  w i t h an a s s e t v a l u e g u a r a n t e e , i n t h e 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 o p t i m a l i n v e s t m e n t s t r a t e g y o f  the f i r m i s d e v e l o p e d . equations  A l o n g the same l i n e s , p a r t i a l  differential  a r e g i v e n f o r the o p t i o n components o f the c o n s t a n t ,  continuous  premium c o n t r a c t , t h e p e r i o d i c premium c o n t r a c t and t h e s i n g l e premium contract.  F o r t h e p u r p o s e s 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 c o n t r a c t w i t h an a s s e t v a l u e g u a r a n t e e , i s an  instrument  insurance  providing  the  b e n e f i t o f e i t h e r the 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 g u a r a n t e e d amount, w h i c h e v e r i s g r e a t e r .  A c a l l option  permits  the h o l d e r t o p u r c h a s e an a s s e t a t a g i v e n p r i c e , whereas t h e p u t  option  26  i s t h e r i g h t t o s e l l an a s s e t f o r a c e r t a i n o r p r e d e t e r m i n e d amount. W i t h i n t h i s framework, t h e c o n t r a c t may be d e s c r i b e d as f o l l o w s . I f , upon m a t u r i t y , 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 exceeds t h e g u a r a n t e e d amount, t h e 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 t h e reference p o r t f o l i o . amount.  I n t h i s sense, t h e e x e r c i s e p r i c e i s t h e guaranteed  The v a l u e o f t h e o p t i o n must t h e r e f o r e be t h e d i f f e r e n c e  between t h e g u a r a n t e e and t h e p o r t f o l i o v a l u e .  The c a l l o p t i o n , o b v i o u s l y  w i l l n o t be e x e r c i s e d i f t h e v a l u e o f t h e g u a r a n t e e exceeds t h e v a l u e of the p o r t f o l i o . P u t a n o t h e r way, t h e v a l u e o f t h e p u t o p t i o n p l u s 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 must e q u a l t h e v a l u e o f t h e g u a r a n t e e p l u s the v a l u e o f t h e 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 v i e w e d as t h e  benefits o f the contract. I t s h o u l d be n o t e d t h a t t h e p r e v i o u s l y s t a t e d assumptions o f the Black-Scholes i n non-dividend  model, i n p a r t i c u l a r ,  t h a t t h e i n v e s t m e n t i s made  p a y i n g s e c u r i t i e s , s t i l l a p p l y t o t h e above argument.  A l s o , g i v e n t h a t t h 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 i s made a t t h e t i m e t h e c o n t r a c t i s p u r c h a s e d , Schwartz shows t h a t t h e v a l u e o f the o p t i o n a t any p o i n t i n t i m e c o r r e s p o n d s e x a c t l y t o t h e B l a c k - S c h o l e s c a l l option formulation.  That i s , t h e p a r t i a l d i f f e r e n t i a l  may be e x p r e s s e d a s :  ( 2-15 )  2 h v  2 x  w,,' + r x w  n  - rw ^ w  ?  =  0  equation  27  w i t h t h e boundary c o n d i t i o n s :  ( 2-16 )  w ( x , t ) = max  ( 0, x-g )  T h e r e f o r e , a t any time T , t h e v a l u e o f t h e o p t i o n may be s t a t e d a s :  ( 2-17 )  w ( x ( ) , t - T- , g ( t ) ) = X(T )  ( 2-18 )  where d  N(d ) - g ( t ) e "  T  x  r ( t  "  T }  N(d )  = ( l n ( x ( T ) / g ( t ) + ( r + % v ) ( t - x )/v ( t - x ) 2  1  2  d  C 2-20 )  N(d) = 1/C2T.  2  = d  - v (t-x  ( 2-19 )  x  )  h  h  2  ) / %  o o  e"  %  ( x )  dx  and N(d) i s t h e c u m u l a t i v e normal d e n s i t y f u n c t i o n , as b e f o r e . the a s s e t v a l u e g u a r a n t e e ,  g, has b e e n s u b s t i t u t e d f o r t h e e x e r c i s e  p r i c e , c, t h e s e e q u a t i o n s a r e i d e n t i c a l t o e q u a t i o n s respectively.  Although  ( 2-10 ) t o ( 2-12 ) ,  The b a s i c c o n c l u s i o n s t h a t t h e v a l u e o f t h e o p t i o n a t  any time t , can be e x p r e s s e d  i n terms o f t h e c u r r e n t p r i c e o f t h e r e f e r e n c e  p o r t f o l i o , t h e v a r i a n c e r a t e o f r e t u r n on t h e p o r t f o l i o , t h e t i m e t o m a t u r i t y and t h e r a t e o f i n t e r e s t , s t i l l h o l d s . i s unobservable,but  Only the v a r i a n c e r a t e  t h i s c a n be e s t i m a t e d f r o m h i s t o r i c a l  data.  28  Schwartz goes on t o p o i n t o u t t h a t i f e q u a t i o n ( 2-17 ) d i d n o t h o l d i n t h e s e n s e t h a t t h e c a l c u l a t e d v a l u e was n o t t h e 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, t h e t o t a l v a l u e o f t h e c o n t r a c t t o t h e i n s u r e d , a t t h e t i m e t h e i n s t r u m e n t i s c r e a t e d , c a n be e x p r e s s e d as 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 d amount p l u s t h e v a l u e o f t h e c a l l option:  ( 2-21 )  where  PV  n  Cb(tO) = g ( t ) e "  r t  4w(x(0), t , g(t))  P V Q ( b ( t ) ) i s t h e present v a l u e o f the b e n e f i t s .  Similarly,  the p r e s e n t v a l u e o f t h e b e n e f i t s may be e x p r e s s e d i n terms o f 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 and t h e v a l u e o f t h e p u t o p t i o n :  ( 2-22 )  PV  0  (b(t)) = PV  Q  (x(t)) * p(x(0), t , g ( t ) )  S i n c e t h e v a l u e o f t h e c a l l o p t i o n i s g i v e n b y e q u a t i o n ( 2-17 ) , t h e v a l u e o f t h e p u t may be c a l c u l a t e d f r o m ( 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  - PV (x(t)) Q  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 t h e s t a t e d assumptions o f t h e p r e v i o u s s e c t i o n , t h e p r e s e n t v a l u e o f t h e b e n e f i t s may v i e w e d as t h e premium charged f o r t h e c o n t r a c t .  be  I t i s important  to  note t h a t w i t h i n t h i s framework, t h e v a l u e o f t h e c a l l o p t i o n must be d e t e r m i n e d b e f o r e t h e premium can be e s t a b l i s h e d . S i n c e a l l t h e parameters o f t h e v a l u a t i o n f o r m u l a , ( 2-17  ) , a r e known, o r can  e s t i m a t e d , t h e p r o c e d u r e becomes m e c h a n i c a l .  be  I t s h o u l d be r e c o g n i z e d ,  however, t h a t 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 most r e l e v a n t c a l c u l a t i o n i s the d e t e r m i n a t i o n o f the v a l u e o f the put o p t i o n .  In  a s t r i c t sense, t h i s amount r e p r e s e n t s t h e charge f o r t h e  guarantee,  o r t h e amount w h i c h t h e company r e c e i v e s f o r assuming t h e  investment  risk. I t was  i n d i c a t e d i n t h e d i s c u s s i o n o f the  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 no g a i n s o r l o s s e s w i l l be encountered p o l i c y i s observed.  be formed, such t h a t  as l o n g as a c o n t i n u o u s  T h i s hedged p o s i t i o n was  t h e s e c u r i t y and s h o r t i n the c a l l o p t i o n . t o t h e i n s u r a n c e case i n q u e s t i o n .  Black-Scholes  revision  formed by g o i n g l o n g i n  The same l o g i c i s a p p l i c a b l e  S i n c e t h e company has s o l d a  call  o p t i o n on t h e r e f e r e n c e p o r t f o l i o s h o r t , i n o r d e r t o e l i m i n a t e t h e a s s o c i a t e d r i s k , i t must t a k e a l o n g p o s i t i o n i n t h e p o r t f o l i o .  In  o r d e r t o m a i n t a i n t h e hedged p o s i t i o n , i t must r e v i s e t h e l o n g p o s i t i o n continuously.  That i s from e q u a t i o n ( 2-14  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)  ) i t can be shown t h a t  30  where x i s 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 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 t h e f i r s t arguement.  T h i s 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  w h i c h a c u t a l l y must be i n v e s t e d i n the r e f e r e n c e p o r t f o l i o .  From  t h i s i t f o l l o w s t h a t n o t a l l t h e funds have t o 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 .  T h i s can be shown by t h e f o l l o w i n g arguments.  F i r s t l y , t h e v a l u e o f the o p t i o n must always be l e s s t h a n o r e q u a l t o the v a l u e o f the p o r t f o l i o ,  ( i e . w-<  x ).  Secondly,  i t must exceed  o r be e q u a l t o t h e d i f f e r e n c e between t h e p o r t f o l i o v a l u e and e x e r c i s e p r i c e ( i e . wr> concave upward.  x-e).  the  T h i r d l y , i t i s assumed t h a t w ( x , t ) i s  G i v e n t h e s e 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 r e s p e c t t o 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.  I n f a c t the range o f t h e 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 e q u a l z e r o i f (x-e) i s l e s s than o r e q u a l t o z e r o . hand, w^  ( x , t ) can o n l y e q u a l one i f x becomes i n f i n i t e ( i e . w^( «>,t)=l).  From ( 2-25  ( 2-26  On t h e o t h e r  )  ) i t follows that:  x w  n  ( 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  n e c e s s a r y t o f o l l o w the  Schwartz d i s s e r t a t i o n v e r y c l o s e l y , as i t p r o v i d e s framework f o r the subsequent s i m u l a t i o n problem. t h i s framework t h a t p r o v i d e d  the t h e o r e t i c a l In essence, i t i s  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 h y p o t h e s i s ,  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  o f t r a n s a c t i o n s c o s t s , a c o n t i n u o u s model i s n o t p r a c t i c a l .  Because  o f t h i s l i a b i l i t y , c e r t a i n assumptions must be made c o n c e r n i n g  the  d i s t r i b u t i o n o f r e t u r n s on the market o r p o r t f o l i o , about t h e s i z e o f t h e t r a n s a c t i o n s c o s t s p e r r e v i s i o n and the r e v i s i o n s c h e d u l e  itself.  The n e x t c h a p t e r o f t h e a n a l y s i s d e a l s w i t h t h e s e a s s u m p t i o n s , the foundations hypothesis  o f s i m u l a t i o n and t h e subsequent model d e v e l o p e d f o r t h e test.  I t also considers  alternatives tested.  the r e s u l t s o f t h e  various  32  Chapter 3 Development o f t h e 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  I n t h e most g e n e r a l s e n s e , s i m u l a t i o n may b e s t be d e s c r i b e d as t h e p r o c e s s  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  o f a f o r m a l model d e s i g n e d  t o r e p r e s e n t o n l y those f e a t u r e s o f t h e s y s t e m  under s t u d y w h i c h a r e b e l i e v e d t o be s i g n i f i c a n t i n v i e w o f t h e 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 .  I n o t h e r w o r d s , i t i s t h e s y n t h e s i s and  a n a l y s i s o f a system w i t h t h e f u n c t i o n i n g o f t h e r e a l system b e i n g represented. must e x i s t .  T h i s does n o t , however, mean t h a t t h e system b e i n g  modeled  F o r example, c e r t a i n p h y s i c a l phenomena s i m p l y t a k e t o o  l o n g f o r t h e a n a l y s t t o o b s e r v e , whereas a s i m u l a t i o n program c a n reduce t h e t i m e f a c t o r s u c h t h a t i t becomes v e r y s i m p l e t o s t u d y t h e p r o b l e m . Simulation i s n o t intended t o provide optimal s o l u t i o n s t o the problem, r a t h e r i t p e r m i t s t h e a n a l y s t t o employ a n a l g o r i t h m , t h e p a r a m e t e r s o f w h i c h may be a l t e r e d b y t h e a n a l y s t so t h a t a range o f s o l u t i o n s may be g e n e r a t e d .  I n t h i s s e n s e , t h e model i s p r e d i c t i v e , g i v e n t h a t  c e r t a i n assumptions about t h e r e l e v a n t p a r a m e t e r s have been made.  I n t h e case 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 , g i v e n t h e c o m p l e x i t i e s o f t h e p r o b l e m and t h e 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 c o n s i d e r a b l e amount o f r e s e a r c h and a n a l y s i s must be p e r f o r m e d b e f o r e a company s h o u l d u n d e r t a k e such a p r o p o s a l .  S i n c e t h e system  d i s c u s s e d i n t h e p r e v i o u s c h a p t e r does n o t e x i s t i n t h e r e a l  world,  33  and t h e r e f o r e i s u n o b s e r v a b l e , t h e a n a l y s t must b u i l d a model o f t h e 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 w h i c h  can  a f f e c t the model i n o r d e r t o determine t h e s e n s i t i v i t y o f t h e model t o these e x t e r n a l i t i e s .  T h i s i m p l i e s t h a t i t i s n o t enough t o q u a n t i f y  t h e model p a r a m e t e r s , b u t a l s o t h e 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 n o t , t h e n 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 a r e the t o p i c s o f d i s c u s s i o n i n t h e n e x t s e c t i o n .  3.2  The R e l e v a n c e o f P o r t f o l i o  Composition  I n t h e p r e v i o u s c h a p t e r , under t h e d i s c u s s i o n o f t h e  theoretical  model, c o n s t a n t r e f e r e n c e was b e i n g made t o t h e r e f e r e n c e p o r t f o l i o of s e c u r i t i e s ; .  The a c t u a l c o m p o s i t i o n o f t h e p o r t f o l i o was  c o n s i d e r e d a t a l l however.  not  The o b v i o u s q u e s t i o n t h e r e f o r e , i s t h e  r e l e v a n c e o f t h e c o m p o s i t i o n o f t h e p o r t f o l i o t o t h e model b e i n g 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 s i m p l y  buy t h e market p o r t f o l i o and r e v i s e i t o n l y t o m a i n t a i n the r a t i o s . That i s , g i v e n t h e amount o f w e a l t h a v a i l a b l e f o r i n v e s t m e n t i t s h o u l d be d i s t r i b u t e d  among t h e s e c u r i t i e s i n s u c h a way  t h a t the r a t i o o f  t h e amount i n v e s t e d i n each s e c u r i t y t o t h e t o t a l w e a l t h i s t h e same as t h e r a t i o o f the v a l u e o f t h e s e c u r i t i e s J o f each company t o t h e t o t a l v a l u e o f t h e market p o r t f o l i o .  I f t h e r a t i o s i n t h e market  p o r t f o l i o change r e s u l t i n g from t h e r e i n v e s t m e n t o f d i v i d e n d s , new i s s u e s , e t c . , t h e n t h e i n v e s t m e n t p o r t f o l i o s h o u l d be a l t e r e d so t h a t t h e r a t i o s remain t h e same.  34  I f t h e p o l i c y o f c o n t i n u o u s r e v i s i o n c o u l d be p u r s u e d , ( i e . no t r a n s a c t i o n s c o s t s ) , t h e n t h e c o m p o s i t i o n o f t h e  reference  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 t h e 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 v a l u e o f t h e p o r t f o l i o , n o t on an r i s k r e t u r n r e l a t i o n s h i p w h i c h has been e s t a b l i s h e d .  implicit  Looking at i t  from a n o t h e r a n g l e , s i n c e the o b j e c t i v e under the c o n t i n u o u s r e v i s i o n s t r a t e g y i s t o be f u l l y hedged a t a l l t i m e s , i t becomes i r r e l e v a n t what t h e v a l u e o f t h e p o r t f o l i o i s , o r what t h e r e t u r n on t h e p o r t f o l i o has been.  T h i s must be t r u e , because as i t was  stated before, i f a  f u l l y hedged p o s i t i o n i s m a i n t a i n e d , t h e n the i n s u r a n c e  company w i l l  not e x p e r i e n c e l o s s e s o r g a i n s as i t has n o t assumed any r i s k .  In  t h i s s e n s e , o n l y t h e v a r i a n c e o f the r e f e r e n c e p o r t f o l i o i s o f i m p o r t a n c e because the v a l u e o f t h e c a l l o r p u t o p t i o n depends on variance  the  rate.  Since i n a p r a c t i c a l s i t u a t i o n , only 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 p u r s u e d , because o f t r a n s a c t i o n s c o s t s , company w i l l be exposed t o some r i s k .  In t h i s sense, the composition  o f 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 portfolio.  The  the  the  e x t e n t t o w h i c h 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 t h e company c a n u n d e r t a k e d u r i n g t h e l i f e o f t h e  contract.  That i s , assuming t h a t the e x p i r a t i o n d a t e i s known f o r c e r t a i n , the company i s s u b j e c t t o p o r t f o l i o r e l a t e d r i s k f r o m the l a s t r e v i s i o n p o i n t t o t h e e x p i r a t i o n d a t e , g i v e n t h a t a market c o l l a p s e has occurred  i n the previous  periods.  not  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 ( i e . 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 i s 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. the simulation model.  This is the position taken with respect to  36  S i n c e most random number g e n e r a t o r s f r o m a lognormal become n e c e s s a r y .  do not g e n e r a t e d i r e c t l y  d i s t r i b u t i o n , but from a standard normal, transformations 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 p r e s e n t e d  by  r e v i e w i n g 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 n o r m a l l y 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 , t h e n i t can be 2 s a i d t h a t X i s l o g n o r m a l l y 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 c o r r e s p o n d i n g l y Y i s N(u, v ) . The d i s t r i b u t i o n o f X i s c o m p l e t e l y 2 s p e c i f i e d by t h e two parameters u and v . simplest natural d e f i n i t i o n .  O b v i o u s l y , t h i s i s the  I t i s e v i d e n t , however, t h a t X cannot  assume zero v a l u e s as t h e t r a n s f o r m a t i o n Y = I n X i s n o t d e f i n e d f o r X = 0.  S i n c e X and Y have t h e 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 a r e r e l a t e d by: ( 3-2  )  A(x) = N(lnx)  (x>  ( 3-3  )  A (x) = 0  (x <0)  ( 3-4  ) and  0)  d A (x)=£L/xv(2 ^ ) ) e x p ( - ( l / 2 v ) ( l n x - u ) ) dx J s  2  2  (x>0)  37  which describes the frequency curve with a single mode a t : -2 r  -7 r  U  -\  ( 3-5 )  - V  x = e  The mean may be defined as: ( 3-6 ) x = e  u +  ^  2  and the variance:  r , (3-7) 7  ?  ^ 2 2u + v ( e - 1) v = e ^  A  2  v  v  -2  ( 3-8 )  =  x  2 z  n  where n i s the c o e f f i c i e n t of v a r i a t i o n of the d i s t r i b u t i o n . median i s simply e . u  The  I t should be noted that the two-parameter lognormal  d i s t r i b u t i o n does possess reproductive properties, which i s the j u s t i f i c a t i o n f o r the assumption that the returns on s e c u r i t i e s i s distributed lognormally.  That i s , i f the return R on s e c u r i t y j , at  time t , i s given by:  ( 3-9 )  R. t 3  = (Pt 3  " P)/ Pt-1 t ^ l J  J  38  where P i s t h e p r i c e o f t h e s e c u r i t y , a n d 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 f r o m t h e c o r o l l a r y t h a t :  ( 3-10)  l n X - lnX 1  2  =  l n ^ / X ^  i m p l y i n g t h a t t h e l o g n o r m a l 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  reproductive  properties.  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 s t a n d a r d transformation  i s necessary:  ( 3-11)  ( 3-12 )  d e v i a t i o n SX, o n l y t h e f o l l o w i n g  0  v = (In(1.0 * X S / X M 2  and  x = Tri (XM) -  2  h (v ) 2  Since t h e generator s e l e c t s a value s from S which i s N(0,1), the following evaluation  ( 3-13 )  occurs:  x = exp (X + v ; s )  I n t h e s i m u l a t i o n program, t h e mean r e t u r n was s p e c i f i e d as 8% and the v a r i a n c e r a t e on t h e 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  g e n e r a t o r i s a t b e s t a pseudo-random number g e n e r a t o r . t h i s may  The b i a s w h i c h  i n t r o d u c e , ' i s p r o b a b l y so m i n i m a l t h a t i t i s n o t w o r t h w h i l e  t o p u r s u e the e f f e c t by p e r f o r m i n g randomness t e s t s .  3.4  The  S i m u l a t i o n Program  G i v e n t h a t t r a n s a c t i o n s c o s t s a r e 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 l o n g e r v a l i d f r o m a p r a c t i c a l p o i n t o f v i e w t o assume t h a t c o n t i n u o u s adjustment o f t h e r a t i o o f the l o n g p o s i t i o n i n the r e f e r e n c e p o r t f o l i o 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 is possible.  S i n c e , as s t a t e d p r e v i o u s l y , t h e o b j e c t i v e o f t h i s  a n a l y s i s i s t o examine the p o t e n t i a l l o s s e s w h i c h 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 n e c e s s a r y t o determine o r d e f i n e the t y p e s o f l o s s e s w h i c h may  Reviewing the previous t y p e s o f p o t e n t i a l l o s s e s may  arguments, i t becomes e v i d e n t t h a t  occur.  The  f i r s t t y p e may  simply a d d i t i o n a l costs of conducting business, t h a t t r a n s a c t i o n s c o s t s a r e now  occur.  relevant.  two  be viewed as  a r i s i n g from the f a c t  I n t h i s sense the word l o s s  i s a misnomer as t h e company w i l l s i m p l y charge the p o l i c y h o l d e r  for  t h e s e a d d i t i o n a l c o s t s ; b u t f o r t h e sake o f s i m p l i c i t y , the a f o r e m e n t i o n e d terminology  w i l l be adhered t o .  The magnitude o f t h e s e l o s s e s o v e r  t h e l i f e o f t h e c o n t r a c t w i l l be r e l a t e d t o the s i z e o f the imbalances  40  i n t h e h e d g e d 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 a n d t h e number o f revisions planned f o r , during the contract period. be p u r s u e d f u r t h e r i n a n o t h e r  The s e c o n d  This point w i l l  section.  t y p e o f l o s s may b e d e f i n e d a s a " d i s a s t e r  loss".  Although i t i s expected that the occurence o f t h i s types o f loss should b e 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, these are very important. c o l l a p s e i n the market.  T h i s type o f l o s s can occur from a g e n e r a l Looking a t i tfrom another angle, i f the  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 a m o u n t , t h e company m u s t have t h e a b i l i t y t o borrow, o r f a c e bankruptcy.  either  I t s h o u l d be n o t e d  that the c r i t e r i a o f the reference p o r t f o l i o being g r e a t e r than the 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 b y t h e amount i n the r i s k free asset.  As s t a t e d p r e v i o u s l y ,  invested  i t i s not necessary, 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 reference p o r t f o l i o .  Consequently,  t h e excess can 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 requirements.  The i n i t i a l p a r a m e t e r s arbitrarily.  More  Market  specifically:  return  Variance  o f the s i m u l a t i o n program were s e t  8% 0.01846  41  Risk free rate  6%  Contract p e r i o d  10 y e a r s  Number o f r e v i s i o n s  10  Transactions  1%  costs  Guaranteed amount  $100.00  I n i t i a l value, Ref. Port.  $100.00  Number o f 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 , g u a r a n t e e d amount and i n i t i a l 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 were h e l d c o n s t a n t f o r a l l the simulations.  The number o f s i m u l a t i o n s were v a r i e d f r o m 500  t o 2,000 i n o r d e r t o e s t a b l i s h t h e s t a b i l i t y o f t h e r e s u l t s .  In general,  i t was found t h a t about a one one-hundredth c e n t change o c c u r r e d i n the mean l o s s e s i f t h e number o f s i m u l a t i o n s were expanded f r o m 500 t o 2,000.  About t h e same magnitude change o c c u r r e d i n t h e s t a n d a r d  d e v i a t i o n o f these l o s s e s . for  C o n s e q u e n t l y , 500 s i m u l a t i o n s were adopted  a l l t h e r u n s , as t h e changes d e s c r i b e d were deemed t o be i n s i g n i f i c a n t .  I n i t i a l l y , t h e t r a n s a c t i o n s c o s t s were p e r m i t t e d t o v a r y 1% t o 2.5% by i n c r e m e n t s o f 0.5%.  This, i n effect,results i n a cost  o f 2 t o 5% t o g e t i n and o u t o f t h e m a r k e t , w h i c h i s f a i r l y of r e a l i t y .  from  representative  These c o s t s may be d i v i d e d i n t h e f o l l o w i n g manner:  a) The c o s t o f b u y i n g t h e i n i t i a l  portfolio.  b) R e v i s i o n c o s t s . c) The c o s t o f s e l l i n g t h e 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 a t t h e end o f t h e c o n t r a c t , t h e r e f e r e n c e p o r t f o l i o must be l i q u i d a t e d .  I t s h o u l d a l s o be n o t e d t h a t  transactions costs only apply t o the reference p o r t f o l i o , not t o the investment  i n the r i s k f r e e asset.  The l a t t e r i s assumed t o be t h e  e q u i v a l e n t o f a s a v i n g s a c c o u n t , w h i c h t y p i c a l l y does n o t i n c u r r transactions costs.  As s t a t e d p r e v i o u s l y , t h e mean r e t u r n o n t h e p o r t f o l i o was assumed t o be 8% w i t h a v a r i a n c e o f 0.01846. constant f o r the runs.  T h i s was a l s o h e l d  V a r i o u s arguments may be made about t h e  a p p r o p r i a t e n e s s o f t h e assumed mean r e t u r n , b u t i t s h o u l d be n o t e d t h a t t h e c i r t i c a l assumption  i s the variance r a t e .  The r e v i s i o n p a r a m e t e r was p e r m i t t e d t o v a r y a c r o s s t h e simulations.  I n i t i a l l y , annual r e v i s i o n s were adopted as t h e p o l i c y  o f t h e i n s u r a n c e company, b u t i n subsequent s i m u l a t i o n s i t was changed t o s i x month, f o u r month and t h r e e month i n t e r v a l s .  I n essence t h e  s i m u l a t i o n paramenters may be d e f i n e d as t h e t r a n s a c t i o n c o s t range and t h e r e v i s i o n p o l i c y range, as p e r t h e p r e d e f i n e d  increments.  I n o r d e r t o f a c i l i t a t e an e a s i e r u n d e r s t a n d i n g o f t h e computer program p r o v i d e d i n A p p e n d i x ( A ) , some o f t h e n o t a t i o n o f t h e p r e v i o u s c h a p t e r s w i l l be a l t e r e d t o conform t o t h a t o f t h e program. S i n c e some o f t h e mechanics o f t h e program a r e n o t r e l e v a n t t o t h i s d i s c u s s i o n , o r may be summarized i n one e q u a t i o n , a d i c t i o n a r y o f t h e  43  v a r i a b l e s and f u n c t i o n s has been p r o v i d e d i n order to c l a r i f y the  a t t h e end o f the program  logic.  The m a i n p o r t i o n o f t h e program may a) The  i n i t i a l period.  be v i e w e d i n f o u r  T h i s i s synonymous t o the  stages:  creation  o f the c o n t r a c t and t h e subsequent v a l u a t i o n o f  the  benefits. b) The r e v i s i o n p o l i c y . T h i s p o r t i o n d e a l s w i t h the r e v i s i o n v  o f the hedged p o s i t i o n , g i v e n t h e h y p o t h e t i c a l market r e t u r n f o r t h e 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) T e r m i n a t i o n o f the c o n t r a c t .  This s e c t i o n determines  t h e f i r m ' s p e r f o r m a n c e w i t h r e s p e c t t o t h e c o n t r a c t under consideration.  I n e f f e c t , i t e s t a b l i s h e s the  firm'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  v i e w e d as an e v a l u a t i o n o f t h e p e r f o r m a n c e 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 r e s p e c t 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 ) .  G i v e n t h e a s s u m p t i o n t h a t a t the t i m e o f t h e i n i t i a t i o n  of  the c o n t r a c t , the 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 X, i s e q u a l t o  the  g u a r a n t e e d amount, f r o m e q u a t i o n s ( 2-17  of  ) t o ( 2-20  ) , the v a l u e  the o p t i o n may be d e t e r m i n e d as f o l l o w s :  ( 3-14 )  since  ln(X(0)/g(t) = 0  from  ln(100/100)  ( 3-15 )  d^ becomes  d  ( 3-16 )  and  d  = 0  x  = ((r + v )  2  = d -  (t))/v(t)  2  v(t)  1  i 2  %  so t h a t t h e v a l u e o f t h e o p t i o n a t time z e r o becomes:  ( 3-17 )  OPT(O) = X ( 0 ) N ( d ) - g ( t ) e " 1  where t h e p r e v i o u s v a r i a b l e d e f i n i t i o n s a p p l y .  r t  N(d ) 2  Having determined t h e  v a l u e o f t h e o p t i o n , t h e 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 a s :  ( 3-18 )  L{0) = g ( t ) e "  r t  +  OPT(O)  and t h e a c t u a l 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 x, a s :  ( 3-19 )  x (0) = X(0) • NCcy  45  S i n c e t h e i n i t i a l w e a l t h p o s i t i o n must e q u a l t h e i n i t i a l  liability  o f t h e f i r m , a f t e r t h 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 i s made, the w e a l t h p o s i t i o n becomes:  ( 3-20 )  W(0) = L ( 0 )  -  ( x ( 0 ) * TR)  where TR i s t h e 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 p e r c e n t a g e . The 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 becomes:  ( 3-21 )  RF(0) = W(0) - x ( 0 )  These e q u a t i o n s summarize t h e company's p o s i t i o n a t t h e c r e a t i o n o f t h e contract.  I t s h o u l d be n o t e d t h a t s i n c e i t i s assumed t h a t t h e i n i t i 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 e q u a l s t h e g u a r a n t e e d amount, t h e i n i t i a l 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 w i l l be i d e n t i c a l f o r a l l the simulations.  The i n i t i a l w e a l t h w i l l depend on t h e s i z e o f  t h e t r a n s a c t i o n s c o s t s i n t h e 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) a r e constant  i n equation  (3-20).  S i m i l a r l y , t h e amount  invested  i n t h e r i s k f r e e a s s e t depends on t h e v a l u e o f W(0) i n e q u a t i o n The second s t a g e o f t h e program e v a l u a t e s  (3-21).  t h e 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 t i m e , as d e f i n e d by t h e 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 g e n e r a t e d , so t h a t 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 a t t i m e t is:  ( 3-22  )  X(t) = X ( t - l ) * Z ( At)  where Z i s t h e s i m u l a t e d r e t u r n on t h e p o r t f o l i o .  That i s , i f Z i s  g r e a t e r t h a n one, t h e 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 t h e v a l u e o f the p o r t f o l i o .  I f i t i s l e s s t h a n one, a l o s s o r r e d u c t i o n i n t h e  value i s implied.  Since the value of the reference p o r t f o l i o  changed, t h e hedged p o s i t i o n has a l s o changed, t h e r e f o r e t h e i n the p o r t f o l i o must be a l t e r e d . t h e v a l u e o f t h e o p t i o n may t h e hedged p o s i t i o n may  From e q u a t i o n s  has investment  (3-14) t o (3-17)  be c a l c u l a t e d f o r time t , and f r o m (3-19)  be r e - e s t a b l i s h e d .  The new w e a l t h p o s i t i o n  a t time t c a n be d e f i n e d a s :  ( 3-23  )  W(t) = ( W ( t - l ) - x ( t - l ) ) »• e  r  A  Z  \  -((x(t) - x(t-l)))  + x ( t - l ) ? Z (A t ) TR  The f i r m ' s l i a b i l i t y a t time t i s d e t e r m i n e d f r o m e q u a t i o n (3-18). The t h i r d s e c t i o n o f t h e program determines  the  final  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 r e s p e c t t o t h e c o n t r a c t .  The  w e a l t h p o s i t i o n i s determined by e q u a t i o n ( 3 - 2 3 ) , b u t i t s h o u l d 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  final  t h a n f o r t h e r e v i s i o n s , as t h e 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 t h e p o i n t , t h e f i n a l w e a l t h 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 )  r((x(t*-l)  where t * i s t h e m a t u r i t y date.  ( 3-25 )  e  r  A t  ;  + x ( t * - l ) •. Z)  Z) • TR)  The v a l u e o f t h e o p t i o n a t m a t u r i t y  OPT(t*) = X ( t * ) * g  f o r X ( t * ) >g, o r :  ( 3-26 )  f o r X ( t * ) <g.  OPT(t*) = 0  The company's l i a b i l i t y , t h e r e f o r e , e q u a l s :  C 3-27 )  L ( t * ) = g + OPT(t*)  The l i a b i l i t y may a l s o be e x p r e s s e d a s :  ( 3-28 )  f o r X.(t*) >g,  ( 3^29 )  L(t*) = X(t*)  or  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 greater  than  the guarantee, then t h e p r o f i t t o the f i r m i s :  ( 3-30 )  P R ( t * ) = W(t*) « X ( t * )  otherwise i t i s :  ( 3-31 )  P R ( t * ) = WCt*) - g  S i n c e 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 c o s t s a t t h e c r e a t i o n of the contract, the p r o f i t represents  t h e amount t h e company must  charge t h e i n s u r e d a t t h e c r e a t i o n o f t h e c o n t r a c t , o v e r and above the o t h e r c o s t s , i n o r d e r t o b r e a k even. i s negative.  I n t h i s sense, t h e p r o f i t  I t s h o u l d be n o t e d , however, t h a t i n v e r y u n i q u e c a s e s  t h e 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, b u t the wealth p o s i t i o n i s greater.  This i s possible since a p o r t i o n o f the  premium i s 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 , s o t h a t i n c e r t a i n i n s t a n c e s , t h e 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 R F ( t * ) r e p r e s e n t s  the t e r m i n a l v a l u e o£ t h e i n v e s t m e n t i n t h e  r i s k f r e e a s s e t , so t h a t W(t*) amount.  i s a c t u a l l y g r e a t e r t h a n the g u a r a n t e e d  S i n c e the outcome d e s c r i b e d by e q u a t i o n  (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 g i v e n by The  (3-31) t o the company.  s c a r c i t y o f t h i s phenomenon i s e v i d e n c e d by t h e f a c t t h a t i t o n l y  o c c u r r e d under the c o n d i t i o n o f a n n u a l r e v i s i o n and one transaction cost  criteria.  Appendix B provides  examples o f the i n i t i a l ,  and t e r m i n a l v a l u e s d e r i v e d f r o m t h e p r e v 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 . appendix p r o v i d e s  percent  intermediate  equations, The  f o r various  summary page o f  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  the  a  r e c o n c i l i a t i o n of 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 , t h e arguments p r e s e n t e d have d e a l t w i t h 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 . on t h e a c c u m u l a t i o n and p r o c e s s i n g the whole s i m u l a t i o n .  The  only  r e s t o f t h e program f o c u s e s  o f the r e l e v a n t i n f o r m a t i o n o v e r  F o r each s i m u l a t i o n t h e program s t o r e s  t e r m i n a l v a l u e o f the r e f e r e n c e p o r t f o l i o , t h e l o s s ( p r o f i t )  the associated  w i t h i t , and t h e c l a s s i f i c a t i o n o f the l o s s i n t o t r a n s a c t i o n c o s t d e r i v e d , o r d i s a s t e r l o s s , as p e r t h e c r i t e r i o n d i s c u s s e d  previously.  That i s , i f t h e l o s s o c c u r s because t h e v a l u e o f t h e r e f e r e n c e  portfolio  i s l e s s t h a n the 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 b e n e f i c i a r y  50  e x e r c i s i n g the guarantee, t h e l o s s i s deemed t o be d i s a s t e r l o s s . Otherwise, t h e l o s s i s assumed t o be t h e r e s u l t o f t h e t r a n s a c t i o n s c o s t s i n c u r r e d because o f t h e r e v i s i o n p o l i c y under  consideration.  Upon t h e c o m p l e t i o n o f the s i m u l a t i o n under t h e e x i s t i n g parameter values,  t h e program c a l c u l a t e s t h e mean and s t a n d a r d  deviation of  the l o s s e s and o f the 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 and summarizes the s t a t i s t i c s  i n the form o f t a b l e s .  (Refer t o Appendix D ) .  are a l s o produced f o r t h e mean and s t a n d a r d losses.  ( Appendix E ) .  In order  Tables  d e v i a t i o n o f the d i s a s t e r  t o a n a l y s e t h e magnitude o f  o r d i n a r y o r t r a n s a c t i o n c o s t l o s s e s , t h e program a l s o produces t a b l e s o f t h e a c t u a l v a l u e s , mean and s t a n d a r d f i v e percent  and t e n p e r c e n t  d e v i a t i o n o f the l a r g e s t  losses incurred, respectively.  These  t a b l e s may be found i n Appendix F.  The  f i n a l s e c t i o n o f t h e program summarizes t h e above  s t a t i s t i c s across  a l l the simulations.  the mean and s t a n d a r d under t h e v a r i o u s policies.  The t a b l e s g e n e r a t e d  d e v i a t i o n o f t h e above d e s c r i b e d  loss  represent categories  combinations o f t r a n s a c t i o n c o s t l e v e l s and r e v i s i o n  These may be r e f e r r e d t o i n Appendix E.  With s l i g h t m o d i f i c a t i o n t o t h e program, t h e n a i v e was a l s o t e s t e d .  T h i s a l t e r n a t i v e may simply  strategy  be viewed as t h e f o r m a t i o n  of the reference p o r t f o l i o a t the c r e a t i o n o f the contract, w i t h the assumption t h a t no r e v i s i o n s w i l l take p l a c e d u r i n g  its life.  Under  t h i s p o l i c y the o n l y a d d i t i o n a l c o s t s i n c u r r e d a r e t h e i n i t i a l terminal transactions costs. i n A p p e n d i x C.  The  and  r e s u l t s of t h i s run are presented  52  Chapter 4  Analysis of Results  The  a n a l y s i s o f t h e r e s u l t s proceeds f r o m a n i n d e p t h  e x a m i n a t i o n o f the i n t e r m e d i a t e the s i m u l a t i o n o u t p u t .  relationships to theevaluation of  Because o f t h e volume o f o u t p u t  provided  by t h e s i m u l a t i o n , o n l y t h e 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 , a n  e x a m i n a t i o n o f each o f the appendices i s p r e s e n t e d .  4.1  Intermediate  Results  Appendix B provides for  the r e s u l t s o f intermediate c a l c u l a t i o n s  one p e r c e n t t r a n s a c t i o n s c o s t s and a n n u a l , s i x and f o u r month  revisions, respectively.  I t s h o u l d be n o t e d t h a t t h e i n i t i a l  i n the reference p o r t f o l i o , X-J, the value o f the reference  investment  portfolio,  RPORT, t h e i n i t i a l w e a l t h WLTH, the v a l u e o f t h e o p t i o n , OPVAL,and t h e liability  t o the f i r m , LIAB, i s t h e 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 t h e r e f e r e n c e 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  first  beneficiary w i l l  two examples o f T a b l e 1 c l e a r l y i n d i c a t e t h a t t h e  n o t e x e r c i s e the guarantee, b u t e l e c t t o r e c e i v e 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 , s i n c e i n b o t h cases t h i s  53  exceeds t h e v a l u e o f t h e g u a r a n t e e . t h e guarantee  I n t h e t h i r d example, however,  i s e x e r c i s e d , as 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 l e s s t h a n t h e guarantee.  T h i s , i n e f f e c t , i s an example o f t h e  disaster losses discussed i n the previous chapters.  This table also  shows t h e impact o f t h e d e c l i n e i n 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 w i t h r e s p e c t t o t h e a c t u a l investment i n t h e p o r t f o l i o .  The  f l u c t u a t i o n s i n t h e r e t u r n s and subsequent l o s s e s on t h e r e f e r e n c e p o r t f o l i o r e s u l t i n t h e v a l u e o f t h e o p t i o n becoming z e r o , and t h e i n v e s t m e n t i n t h e p o r t f o l i o b e i n g reduced c o n s i d e r a b l y .  On t h e o t h e r  hand, when p o s i t i v e r e t u r n s a r e e x p e r i e n c e d , as g i v e n i n t a b l e s two and t h r e e , t h 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 i s a l m o s t to t h e v a l u e o f t h e p o r t f o l i o .  equal  That i s , t h e q u a n t i t y X ( t ) •. N(d-^)  i s a l m o s t e q u a l t o one, and i n f a c t becomes one i n some o f t h e c a s e s . S i n c e i n t h e s e cases t h 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 i s g r e a t e r t h a n t h e a c t u a l w e a l t h p o s i t i o n o f t h e company, s m a l l amounts o f b o r r o w i n g must o c c u r i n o r d e r t o a r r i v e a t t h e 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 b o r r o w i n g i s  a c c o m p l i s h e d a t zero c o s t f o r s i m p l i c i t y .  The i n c r e a s e i n t h e l o s s e s ,  ( i e . t h e n e g a t i v e p r o f i t v a l u e s ) , c a n be a t t r i b u t e d t o t h e i n c r e a s e 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 t o recognize t h a t doubling o r t r i p l i n g the number o f r e v i s i o n s does n o t n e c e s s a r i l y r e s u l t i n d o u b l i n g 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 , t h e l o s s e s i n c u r r e d , as g i v e n i n  54  T a b l e 3, are l e s s t h a n those i n d i c a t e d i n T a b l e 2, w h i c h r e s u l t f r o m ten l e s s r e v i s i o n s .  W h i l e the c o m p a r i s o n o f the r e s u l t s o f  s i m u l a t i o n s i s i n a d e q u a t e t o draw c o n c r e t e 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 p r e s e n t  conclusions  two  from,  t h a t the l a r g e r the number  o r r e v i s i o n s , the l e s s t h e company i s exposed t o abnormal g a i n s losses.  or  F u r t h e r m o r e , i t f o l l o w s t h a t t h e 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 t h e r e q u i r e d change i n t h 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 , i n o r d e r t o r e - e s t a b l i s h t h e 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 t h e s e a s s e r t i o n s can be t h e n t h e f i r m ' s exposure t o abnormal l o s s e s may  be r e d u c e d by f o l l o w i n g  the h e d g i n g s t r a t e g y , w h i c h i n e f f e c t i s t h e h y p o t h e s i s  The  substantiated,  o f the a n a l y s i s .  i n t e r m e d i a t e r e s u l t s a l s o s u b s t a n t i a t e t h e argument  t h a t t h e v o l a t i l i t y o f t h e o p t i o n i s always g r e a t e r t h a n t h e o f t h e s e c u r i t y or r e f e r e n c e p o r t f o l i o .  volatility  F o r example, an 11% change  i n 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 n T a b l e 1, r e s u l t s i n a 15.6%  change i n t h e v a l u e o f 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 v a l u e causes a 40% d e c l i n e i n t h e v a l u e o f t h e o p t i o n . I t s h o u l d be n o t e d 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 t h e o p t i o n i s dependent n o t o n l y on the 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 , b u t on the m a t u r i t y p e r i o d .  also  55  4.2  The N a i v e  Strategy  Under t h e n a i v e s t r a t e g y i t i s assumed t h a t t h e company s i m p l y i n v e s t s t h e t o t a l premium i n t h e r e f e r e n c e p o r t f o l i o and does n o t 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 contract  ( Appendix C ) . The i n i t i a l i n v e s t m e n t i n t h e p o r t f o l i o  i s assumed t o be $100.00 w h i c h i s o n l y 21 c e n t s l e s s t h a n t h e c a l c u l a t e d premium under t h e r e v i s i o n s t r a t e g y o p t i o n .  S i n c e 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 c a n be i n c u r r e d under t h i s s t r a t e g y , t h e y are o m i t t e d  from t h e c a l c u l a t i o n s f o r t h e sake o f s i m p l i c i t y .  The  number o f s i m u l a t i o n s were s e t a t 500, as f o r t h e r e v i s i o n s t r a t e g y option. concept.  The number o f p e r i o d s  i s synonymous t o t h e r e v i s i o n p e r i o d  That i s , f o r example, 10 p e r i o d s  r e t u r n on t h e p o r t f o l i o a n n u a l l y ,  implies the c a l c u l a t i o n of the  20 p e r i o d s , s e m i - a n n u a l l y , e t c .  T a b l e 2 summarizes t h e d i s a s t e r l o s s e s t h e company may incurby f o l l o w i n g t h i s s t r a t e g y .  The number o f l o s s e s r e f e r s t o t h e number o f  times t h e g u a r a n t e e i s e x e r c i s e d o v e r 500 s i m u l a t i o n s .  The p e r c e n t a g e  s i m p l y r e - e x p r e s s e s t h e number o f l o s s e s i n terms o f t h e number o f simulations.  The average l o s s i s t h e mean o f t h e d i s a s t e r l o s s e s  under t h e p a r t i c u l a r s t r a t e g y , t h e s t a n d a r d  deviation, the deviation  from t h a t mean. The outcomes under t h e v a r i o u s o p t i o n s o f t h e n a i v e s u g g e s t t h a t t h e p r o c e s s i s random. many p e r i o d s  are considered  strategy  That i s , i t i s i r r e l e v a n t how  o v e r t h e number o f 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 n o t d i s p l a y any t r e n d .  It is interesting  t o n o t e , however, t h e magnitude o f the d e v i a t i o n s i n r e l a t i o n t o magnitude o f the mean l o s s e s .  T h i s can be a t t r i b u t e d p a r t l y t o  l i m i t e d number o f o b s e r v a t i o n s  under c o n s i d e r a t i o n .  l o s s magnitudes a r e c o n s i d e r e d , and the s t a n d a r d  the the  When t h e i n d i v i d u a l  t h e y appear t o be random.  The mean  d e v i a t i o n o f the l o s s e s become c r i t i c a l , as t h e  naive  s t r a t e g y s t a t i s t i c s a r e u s e d as the benchmark t o compare the r e v i s i o n strategy  against.  The  . f i r s t : . T a b l e summarizes t h e l o s s e s i n terms o f  the  t o t a l number o f s i m u l a t i o n s p e r s t r a t e g y p e r f o r m e d , and t h e c o r r e s p o n d i n g mean v a l u e s o f t h e r e f e r e n c e p o r t f o l i o .  S i n c e under the n a i v e  strategy  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 t h e r e i s no i n v e s t m e n t 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 w e a l t h e x c e e d i n g t h e g u a r a n t e e d amount when 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 l e s s t h a n t h e g u a r a n t e e 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 o n l y o c c u r i f a h e d g i n g p o l i c y i s adopted. As e x p e c t e d , the average l o s s p e r 500 close to zero. -0.33  t o -0.77  simulations i s very  I n f a c t the range o v e r the p e r i o d s c o n s i d e r e d dollars.  The  large standard  i s from  d e v i a t i o n s and the f a c t  t h a t 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 ) , i m p l y the d i s t r i b u t i o n i s v e r y h i g h l y skewed t o t h e l e f t . standard  deviation provides  that  I n e f f e c t , the  a good i n d i c a t i o n o f t h e magnitude o f  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 t o the average.  the  57  The  average v a l u e s  number o f p e r i o d s  o f the reference p o r t f o l i o over the  i s also w i t h i n expectations.  The range o f t h e averages  i s f r o m $236.67 t o $255.03 w h i c h c a n be t r a n s l a t e d i n t o compounding $100.00 i n v e s t m e n t a t a c o n t i n u o u s r a t e o f f r o m 8.5% t o about 9.5%. T h i s i s r e a s o n a b l e as t h e mean r e t u r n p a r a m e t e r on t h e r e f e r e n c e p o r t f o l i o was s e t a t 8%.  One s t a n d a r d  d e v i a t i o n f r o m t h e mean t r a n s l a t e s  i n t o a range o f about 4% t o 13%, w h i c h i s a c c e p t a b l e  i n terms o f t h e  v a r i a n c e r a t e assumed. I n g e n e r a l , i t may be c o n c l u d e d t h a t a l t h o u g h t h e number o f d i s a s t e r l o s s e s compared t o t h e number o f s i m u l a t i o n s i s n o t v e r y  high,  ( i e . f r o m 2 t o 4.2%), g i v e n t h e average s i z e and d e v i a t i o n o f t h e l o s s e s , t h e s t r a t e g y cannot be c o n s i d e r e d The  as v e r y e f f e c t i v e .  r e s i t s , o f t h e a n a l y s i s i s c o n c e r n e d w i t h comparing t h e  r e v i s i o n s t r a t e g y t o t h e n a i v e p r o c e d u r e , and showing t h a t t h e f o r m e r i s dominant o v e r t h e l a t t e r .  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  A p p e n d i x D summarizes t h e o v e r a l l l o s s e s i n c u r r e d by t h e company under t h e 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 .  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 t h e 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 2.5%,  t a b l e s p r e s e n t summaries o f t r a n s a c t i o n s c o s t s r a n g i n g  f r o m 1% t o  as w e l l as t h e case i n w h i c h no t r a n s a c t i o n s c o s t s a r e 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 , e x c e p t t h a t t h e i n v e s t m e n t i n the r e f e r e n c e p o r t f o l i o i s r e v i s e d a t the end o f each p e r i o d . As e x p e c t e d , a s : t h e t r a n s a c t i o n c o s t s a r e i n c r e a s e d 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 t h e number o f r e v i s i o n s The  standard  the constant.  d e v i a t i o n s a s s o c i a t e d w i t h t h e s e means do n o t , however,  increase at a proportional rate.  That i s , d o u b l i n g  c o s t does n o t r e s u l t i n d o u b l i n g t h e s t a n d a r d  the  deviation.  transaction F o r example,  d o u b l i n g the r a t e from 1% t o 2% i n c r e a s e s the average l o s s from t o -13.66 i n t h e c a s e o f a n n u a l r e v i s i o n s , b u t the s t a n d a r d o n l y changes f r o m 3.34  t o 5.24  -6.98  deviation  respectively.  I f t r a n s a c t i o n costs are h e l d constant,  a comparison o f  average l o s s e s o v e r 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 averages t e n d t o i n c r e a s e a t a d e c r e a s i n g i s increased ten periods a t a time.  the the  r a t e , as t h e r e v i s i o n p e r i o d  This tends to h o l d f o r a l l the  cases e x c e p t t h e 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 . the average l o s s i s v e r y c l o s e t o z e r o .  In t h i s case,  F u r t h e r m o r e , comparing t h e  l a t t e r case t o t h e n a i v e 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 o v e r the n a i v e , i n terms o f average l o s s e s . c o s t s are c o n s i d e r e d ,  I f positive transaction  t h e n t h i s c o m p a r i s o n cannot be made.  59  The s t a n d a r d d e v i a t i o n s o f t h e average l o s s e s under t h e various strategies also provide valuable information. t r a n s a c t i o n costs constant and'increasing  Holding  t h e number o f 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 t h e s t a n d a r d d e v i a t i o n , b u t as t h e number o f r e v i s i o n s reaches f o r t y t o 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 t o r i s e .  F o r t h e 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 t h e d e v i a t i o n i s about t h e same o r l o w e r t h a n f o r t e n r e v i s i o n s , b u t f o r t h e h i g h e r l e v e l s , i t exceeds t h e d e v i a t i o n a f t e r ten r e v i s i o n s .  I f t h e t r a n s a c t i o n c o s t s a r e e l i m i n a t e d ( i e . s e t a t 0%) ,  t h e n i t becomes c l e a r t h a t t h e more o f t e n t h e 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 o f the l o s s e s . consistent with expectations.  This i s e n t i r e l y  Comparison o f t h e s e d e v i a t i o n s t o t h o s e  d e r i v e d under t h e n a i v e s t r a t e g y l e a d s t o t h e c o n c l u s i o n t h a t t h e 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 a r e n o t o n l y d e c l i n i n g ,  but i n magnitude t h e y a r e c o n s i d e r a b l y s m a l l e r t h a n t h e ones d e r i v e d under t h e n a i v e p l a n .  As d i s c u s s e d p r e v i o u s l y , under t h e n a i v e  plan  the d e v i a t i o n s do n o t d i s p l a y any t r e n d , b u t suggest t h a t t h e y a r e random.  The average 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 , as g i v e n by the v a r i o u s t a b l e s , a r e w i t h i n e x p e c t a t i o n s . t r a n s l a t e i n t o a 7.5 t o 9.5%  The d o l l a r  values  a n n u a l r e t u r n o v e r t e n y e a r s on an i n i t i a l  i n v e s t m e n t o f about a hundred d o l l a r s .  This i s reasonable,  given the  o r i g i n a l assumptions about t h e mean r e t u r n and v a r i a n c e on t h e market  60  portfolio.  I t s h o u l d be n o t e d t h a t t h e average v a l u e and d e v i a t i o n s  o f t h e r e f e r e n c e p o r t f o l i o a r e t h e same f o r t h e n a i v e s t r a t e g y and t h e zero and one p e r c e n t r e v i s i o n s t r a t e g i e s because t h e same sequence o f random numbers were g e n e r a t e d i n t h e s e 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 t h e 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 t h e r e v i s i o n strategy i s superior t o that of the naive plan, since the same p e r c e n t  losses a r e generated.  The d i f f e r e n c e i n t h e r e s u l t s must  t h e n be a t t r i b u t e d t o t h e e f f e c t s o f f o l l o w i n g t h e h e d g i n g s t r a t e g y . The  average v a l u e s f o r t h e r e f e r e n c e p o r t f o l i o s o f t h e o t h e r s t r a t e g i e s  a r e d i f f e r e n t because a d i f f e r e n t sequence o f random numbers a r e g e n e r a t e d due t o t h e changes i n t h e 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 a c c e p t a b l e , however, i n terms o f t h e o r i g i n a l p a r a m e t e r s .  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  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 o c c u r i f t h e g u a r a n t e e i s e x e r c i s e d by t h e b e n e f i c i a r y because o f a g e n e r a l market d e c l i n e and t h e subsequent p o o r performance o f t h e r e f e r e n c e p o r t f o l i o i n r e l a t i o n to t h e guarantee.  A p p e n d i x E summarizes t h e s e l o s s e s 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 s and r e v i s i o n s t r a t e g i e s .  As n o t e d w i t h t h e o v e r a l l l o s s e s , t h e average d i s a s t e r l o s s i n c u r r e d under t h e 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 , ( h o l d i n g t r a n s a c t i o n costs constant), tend t o 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 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 s a r e i n c 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 n o t c o n f l i c t w i t h t h e above c o n c l u s i o n . I t i s n o t p o s s i b l e however, t o i s o l a t e t h e i n c r e m e n t a l  losses 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 costs, ( i e . holding r e v i s i o n periods c o n s t a n t ) , w i t h i n t h e e x i s t i n g s i m u l a t i o n program.  In order t o accomplish  t h i s , a c o n s i d e r a b l e amount o f r e p r o g r a m i n g i s n e c e s s a r y . I f zero t r a n s a c t i o n c o s t s a r e c o n s i d e r e d ,  (Table 5 ) , t h e n b o t h  g a i n s and l o s s e s appear under t h e average l o s s c a t e g o r y , b u t t h e y t e n d t o be c l o s e t o zero i n magnitude. investment i n the r i s k f r e e asset.  The g a i n s c a n be a t t r i b u t e d t o t h e That i s , a l t h o u g h  the b e n e f i c i a r y  e l e c t s t o r e c e i v e t h e g u a r a n t e e d amount because t h e v a l u e o f t h e reference p o r t f o l i o i s l e s s than t h e guarantee, the t o t a l  wealth  p o s i t i o n o f t h e company i s n o t . The d i f f e r e n c e between t h e v a l u e o f the r e f e r e n c e p o r t f o l i o and t h e w e a l t h p o s i t i o n l i s t h e amount i n v e s t e d i n the r i s k free asset.  Comparison o f t h e s e r e s u l t s t o t h o s e d e r i v e d under t h e n a i v e 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 c o s t s o f 01 t o 1%! t h e r e v i s i o n s t r a t e g y i s dominant o v e r t h e n a i v e , i n terms o f average l o s s e s .  The  r e s u l t s t e n d t o 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 c o s t s o f 2 and 2%%, i n terms o f average l o s s e s .  I t may be c o n c l u d e d , however, t h a t i f t h e  number o f r e v i s i o n s a r e k e p t t o t e n p e r c o n t r a c t p e r i o d , t h e n 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 t o the naive.  62  A n a l y s i s o f t h e s t a n d a r d d e v i a t i o n s o f t h e average l o s s e s r e v e a l s t h e a c t u a l impact o f t h e r e v i s i o n s t r a t e g y .  At a l l levels  o f t r a n s a c t i o n c o s t s t h e d e v i a t i o n s t e n d t o improve as t h e number o f r e v i s i o n s i s increased.  The improvement may be v i e w e d as a d e c l i n e  i n t h e magnitude o f t h e d e v i a t i o n as t h e number o f r e v i s i o n s a r e increased,  o r i n terms o f t h e r a t i o o f t h e mean l o s s t o t h e s t a n d a r d  deviation associated with the loss. deviation increases  That i s , a l t h o u g h t h e s t a n d a r d  i n some c a s e s as t h e number o f r e v i s i o n s a r e  i n c r e a s e d , t h e r e l a t i o n s h i p o f t h e average l o s s t o t h e d e v i a t i o n s h o u l d be c o n s i d e r e d  i n s t e a d o f t h e a b s o l u t e magnitude o f t h e 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 o f t h e 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, a t \% 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 t h e d e v i a t i o n i s 5.55, a t 80 r e v i s i o n s i t i s 2.00.  Under t h e n a i v e s t r a t e g y , however, t h e  r e s p e c t i v e d e v i a t i o n s a r e 10.95 and 11.11.  F o r t h e c a s e o f no t r a n s a c t i o n  c o s t , t h e d e c l i n e ' i s even more d r a m a t i c , r a n g i n g f r o m 5.50 t o 1.47, respectively.  Even a t t h e h i g h e s t  cost l e v e l considered,  r e s p e c t i v e d e v i a t i o n s a r e 6.19 and 3.47 w h i c h a r e s t i l l  ( i e . 2%%), the considerably  below t h e n a i v e r e s u l t s .  G e n e r a l i z i n g , t h e i m p l i c a t i o n s o f t h e average l o s s e s and t h e r e s p e c t i v e s t a n d a r d d e v i a t i o n s a r e t h a t b y i n c r e a s i n g t h e number o f r e v i s i o n s , t h e company reduces t h e d i s p e r i s o n o f t h e l o s s e s , w h i c h c a n  63  be v i e w e d 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 t h e 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 o t h e r hand, can i n t e r p r e t e d as the c o s t o f r e d u c i n g may  the r i s k .  The measure o f r i s k  a l s o be i n t e r p r e t e d as the magnitude o f l o s s e s i n c u r r e d by  company, b u t the m a j o r weakness o f c o n s i d e r i n g a b s o l u t e s component. considered  be  the  i s t h e random  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 as an i n d i c a t o r o f t h e 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 ,  t h e n t h e random component must be i s o l a t e d i n o r d e r t o d e t e r m i n e t h e 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 h i s t o g r a m s summarize t h e d i s a s t e r l o s s e s  e x p e r i e n c e d under t h e n a i v e s t r a t e g y and t h e r e v i s i o n s t r a t e g y f o r t h e 20 p e r i o d a l t e r n a t i v e .  T h i s i s synonymous t o r e v i s i n g t h e p o r t f o l i o  e v e r y s i x months, o r i n t h e case o f t h e n a i v e s t r a t e g y , g e n e r a t i n g r e t u r n on t h e p o r t f o l i o s e m i - a n n u a l l y .  a  I t i s evident from the  h i s t o g r a m s t h a t n o t o n l y a r e t h e mean and s t a n d a r d but a l s o t h e a b s o l u t e magnitude o f the . l o s s e s .  d e v i a t i o n s reduced,  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 $29.22 ( i e . f r o m 42.33 t o 13.11). A n a l y s i s o f the b e h a v i o r o f the l a r g e s t l o s s i s n o t p u r s u e d , however, because o f t h e changes i n the r e t u r n g e n e r a t i n g i n c r e a s i n g t h e number o f r e v i s i o n s . r e s u l t s i n doubling  D o u b l i n g the number o f r e v i s i o n s  t h e number o f random r e t u r n s g e n e r a t e d .  t u r n can r e s u l t i n t h e g e n e r a t i o n  This, i n  o f an e x t r e m e l y l a r g e l o s s .  o f t h i s p o s s i b i l i t y , the more c o n s e r v a t i v e means and s t a n d a r d  sequence r e s u l t i n g f r o m  Because  approach o f examining t h e  d e v i a t i o n s o f the l o s s e s i s adopted.  64  NAIVE STRATEGY  TRANSo COST- 0% NOo PERIODS- 2 0 MEAN LOSS-13*36 STOo DEVo12o69  20  15  10  * * *  * ** * ** 0  LOSS {$) FREQUENCY  . / 42  * o / 36  *  , / 30  # * *  * ** *  / 24  /  /  18  12  /  REVISION STRATEGY DISASTER LOSSES  20  TRANSo COST- 1 ? NOo PERIODS- 2 0 MEAN LOSS-5o80 STD« DEV.3.58  _  15  o o  #  10  ' ^ t  o  o 5  *  *  „ 0  _ o o o / oo LOSS ( $ ) 42 FREQUENCY 0 0  o / o o • ./ 36 30 0  000/ 0  24  #  *  $ I  4=  *  *  *  *  *  *  * * *  * * *  6  o / 0  000/ oo  o o o / o o o / 18 12 0 2 8  #  *  *  *  * ***** * *  #  *  *  0  0  4  #  12  65  4.5  The R e v i s i o n S t r a t e g y :  Largest  Losses  The f i n a l s e t o f t a b l e s , p r e s e n t e d i n A p p e n d i x F, summarize t h e l a r g e s t 5 and 10% l o s s e s o v e r t h e 500 s i m u l a t i o n s p e r r e v i s i o n s t r a t e g y and t h e v a r i o u s l e v e l s o f t r a n s a c t i o n c o s t s . losses represent  The average  t h e means o f t h e 25 and 50 l a r g e s t l o s s e s  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 .  incurred  I n essence i t i s an  a n a l y s i s o f t h e t a i l end o f t h e t o t a l l o s s d i s t r i b u t i o n . As i n t h e p r e v i o u s  t a b l e s , t h e average l o s s e s i n c r e a s e  with  the number o f r e v i s i o n s , b u t t h e d e v i a t i o n s f r o m t h e mean do n o t increase l i n e a r l y .  As t h e t r a n s a c t i o n c o s t s i n c r e a s e f r o m 1 t o 2%%  t h e d e v i a t i o n s do i n c r e a s e more r a p i d l y as t h e number as t h e number of r e v i s i o n s i s increased.  Even a t 2%%, t h e d e v i a t i o n s o n l y double  i f t h e number o f r e v i s i o n s i s i n c r e a s e d from 10 t o 80.  F o r t h e case  o f zero t r a n s a c t i o n c o s t , t h e average l o s s and c o r r e s p o n d i n g d e v i a t i o n d e c l i n e s as t h e number o f r e v i s i o n s i s i n c r e a s e d .  standard In  essence t h e i m p l i c a t i o n s o f t h e s e t a b l e s i s t h a t i f t h e company i s a b l e t o reduce t h e t r a n s a c t i o n c o s t l e v e l ( i e . t h e p e r c e n t 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  with  the l o s s e s .  These r e s u l t s may be examined f r o m a n o t h e r p o i n t o f v i e w . the company adopts a p o l i c y o f c h a r g i n g  If  t h e i n v e s t o r f o r t h e average o f  the 25 o r 50 l a r g e s t l o s s e s , p e r c o n t r a c t , a c o n s i d e r a b l e  decrease  66  occurs  i n the r i s k  associated w i t h the p o r t f o l i o of contracts.  following table  s u m m a r i z e s t h e m i n i m u m a n d maximum n u m b e r o f  ( i n percentage  terms ) w h i c h o c c u r over the v a r i o u s  levels,  g i v e n t h e above  The losses  transaction  strategy.  Percent  Losses  5%  10%  Trans. Cost  Min. Loss  Max. Loss  Min. Loss  Max. Loss  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%  That i s ,  f o r example,  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  of  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 5% ) ,  ( ie.  irregardless  of which r e v i s i o n strategy  minimum number o f c o n t r a c t s 1.6%  o r 8 , t h e maximum, 2.6%  strategy, for  the percentage  levels  on which i t w i l l o r 13.  level.  Clearly, for a specific  increase  the  revision  not exceed these  I t s h o u l d a l s o be n o t e d  losses  be  limits  that  are r e l a t i v e l y constant f o r the v a r i o u s t r a n s a c t i o n  (although the average losses  increases).  i t pursues,  incur a loss w i l l  can vary, but i t w i l l  a given transaction cost  the l i m i t s  cost  as t h e t r a n s a c t i o n  cost cost  67  I f zero t r a n s a c t i o n c o s t s a r e assumed, t h e t r e n d s r e s u l t s are very s i m i l a r t o those obtained disaster losses.  i n the analysis of the  As t h e number o f r e v i s i o n s i s i n c r e a s e d , t h e average  l o s s d e c l i n e s and t h e s t a n d a r d smaller.  i n the  d e v i a t i o n o f t h e l o s s e s becomes  I n e f f e c t , t h e d i s t r i b u t i o n o f t h e l o s s e s a t t h e f i v e and  t e n p e r c e n t l e v e l becomes t i g h t as t h e number o f r e v i s i o n s a r e increased.  4.6  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 .  Summary  I n t h i s s e c t i o n an attempt has been made t o a n a l y s e t h e l o s s e s w h i c h o c c u r when a h e d g i n g s t r a t e g y i s a d o p t e d by t h e company. As a b a s i s o f c o m p a r i s o n , t h e r e s u l t s o f t h e n a i v e o r buy and h o l d s t r a t e g y have a l s o been p r e s e n t e d . considered  A l t h o u g h t h e l o s s e s have been  from a number o f d i f f e r e n t p o i n t s o f v i e w , from t h e r e s u l t s  i t i s evident  t h a t t h e h e d g i n g s t r a t e g y i s dominant o v e r t h e n a i v e .  The average l o s s e s may be v i e w e d as t h e c o s t o f r e d u c i n g  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 t h e r e v i s i o n s t r a t e g y under consideration.  W i t h t h e n a i v e s t r a t e g y , however, t h e r e i s no a t t e m p t  made t o reduce o r 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 t h e 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 o v e r t h e 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 l o s s e s and t h e i r r e s p e c t i v e s t a n d a r d d e v i a t i o n s may a t t r i b u t e d t o t h e change i n the t r a n s a c t i o n c o s t .  be  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 c o n c l u d e d f r o m t h e e v i d e n c e s u p p l i e d by the program, t h e changes a r e c o n s i s t e n t enough t o s u p p o r t the above conclusion.  From the r e s u l t s i t i s e v i d e n t t h a t i t i s i n the i n t e r e s t  o f t h e company t o m i n i m i z e the t r a n s a c t i o n c o s t p e r s h a r e , t o t h e extent that t h i s i s p o s s i b l e .  A l t h o u g h t h e p o i n t was  not pursued i n  t h e a n a l y s i s , s a v i n g s r e s u l t i n g f r o m a r e d u c t i o n i n the t r a n s a c t i o n c o s t p e r s h a r e p r o v i d e the o p p o r t u n i t y t o i n c r e a s e t h e number o f r e v i s i o n s and c o n s e q u e n t l y reduce the r i s k o f l o s s e s . 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 based on the o b s e r v e d t r e n d s .  This i s not hypothesis  T h i s a s p e c t c o u l d be c o n s i d e r e d  in  subsequent a n a l y s e s .  I n t h e case o f l o w e r t r a n s a c t i o n c o s t s p e r s h a r e , 1%%), the e v i d e n c e s u p p o r t s  the h y p o t h e s i s  ( i e . 0 to  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 , t h e d e v i a t i o n o r r i s k o f l o s s i s reduced.  If  the t r a n s a c t i o n c o s t s a r e i n t h e range o f 2 t o 2%% p e r s h a r e t r a d e d , the t r e n d i s towards a d e c l i n e i n t h e d e v i a t i o n s as t h e r e v i s i o n s a r e i n c r e a s e d , b u t t h e f l u c t u a t i o n s do cause c o n c e r n .  The f l u c t u a t i o n s  t e n d t o o c c u r 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 t o about 60 t o 70 p e r p e r i o d .  Up t o t h e s e l e v e l s , and a f t e r , t h e 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 i s , by maintaining a fully hedged position between the investment in the reference portfolio and the c a l l 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 w i l l 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 , t h e o b j e c t i v e o f t h i s d i s s e r t a t i o n was t o p r o v e t h a t by a d o p t i n g  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 l o s s e s w h i c h can a r i s e as r e s u l t o f a g e n e r a l market d e c l i n e .  As a b a s i s o f c o m p a r i s o n , i t was  assumed t h a t t h e market p o r t f o l i o w o u l d be bought and h e l d . a n o t h e r way,  a  Put  i f management r e j e c t s t h e 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 t h e p r o b l e m , t h e n i t i s n o t u n r e a s o n a b l e t o assume b u y i n g 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 .  the  I n a d d i t i o n , an a t t e m p t has  been made t o examine t h e 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 t h a t t h e h e d g i n g concept i s v a l i d  w i t h i n t h i s framework as t h e company t a k e s a l o n g p o s i t i o n i n the r e f e r e n c e p o r t f o l i o and a s h o r t p o s i t i o n i n the c a l l o p t i o n on portfolio.  According  t o the B l a c k - S c h o l e s  form t h e hedged p o s i t i o n 1/w-^ share held.  the  formulation, i n order  o p t i o n s must be s o l d s h o r t a g a i n s t  With respect to the insurance  c o n t r a c t , s i n c e one  to each  option  on t h e r e f e r e n c e p o r t f o l i o i s s o l d s h o r t , t o f o r m the hedged p o s i t i o n , only the determination required.  ( 5-1  The  )  T h i s was  o f the necessary investment i n the p o r t f o l i o i s  g i v e n by t h e  xCt) = X ( t ) •  expression:  N(d ) x  s i m u l a t i o n model g e n e r a t e d a r a t e o f r e t u r n on t h e p o r t f o l i o a t  s p e c i f i e d t i m e s , w h i c h caused t h e 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 t o be no l o n g e r v a l i d .  I n order to r e - e s t a b l i s h the  72  h e d g e d 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 p o r t f o l i o o r buy a d d i t i o n a l shares, or negative not  only  r e t u r n was g e n e r a t e d .  initial  a portion of the  depending on whether a p o s i t i v e I n t h i s w a y , t h e company i n c u r r e d  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  of r e v i s i n g the p o r t f o l i o .  Analysis o f the intermediate  calculations indicated that i f  p o s i t i v e r e t u r n s were generated on t h e reference percent  p o r t f o l i o , the  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 case 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 declined considerably.  This  a n o t h e r way, i f t h e v a l u e over time, portfolio.  i s entirely w i t h i n expectations.  of the reference  portfolio  t h e n t h e company s h o u l d b e a l m o s t f u l l y  invested Put  i s increasing  invested i nthe  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 stands t o  reason t h a t t h e investment should be reduced.  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 b e t w e e n 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 wealth  i s always i n v e s t e d i n a r i s k f r e e  asset.  This p r i n c i p l e accounts f o r t h e f a c t t h a t even though 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 the t e r m i n a l value  o f the reference  show a p r o f i t .  p o r t f o l i o may b e 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 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  show a p r o f i t  exercised, but  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. t h e company w i l l  That i s ,  Under such  circumstances  i n s p i t e of the f a c t that the value  of  73  the r e f e r e n c e p o r t f o l i o d i d n o t exceed the guaranteed amount. During the c o u r s e o f t h e s i m u l a t i o n s , t h i s s i t u a t i o n o n l y for  low t r a n s a c t i o n c o s t  occurred  levels.  I n the c o u r s e o f t h e a n a l y s i s , two types o f l o s s e s were examined.  The f i r s t may be d e f i n e d as t r a n s a c t i o n c o s t d e r i v e d l o s s e s .  These a r e s i m p l y  t h e r e s u l t o f 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  c o s t s , as w e l l as 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 . words, t h e s e a r e t h e c o s t s o f a d o p t i n g  In o t h e r  the hedging s t r a t e g y .  The  t e r m i n a l v a l u e o f t h e l o s s p e r s i m u l a t i o n i s t h e amount which t h e i n s u r e d must be charged f o r , i n o r d e r  t o ensure t h a t t h e company  breaks even a t t h e t e r m i n a t i o n o f t h e c o n t r a c t . l o s s was d e f i n e d as a d i s a s t e r l o s s .  The second type o f  T h i s type o f l o s s occurs when  the b e n e f i c i a r y e x e r c i s e s t h e guarantee, because 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 l e s s than t h i s amount. the company e x p e r i e n c e s  As mentioned above,  a l o s s under t h e s e c o n d i t i o n s o n l y i f t h e  v a l u e o f t h 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 does n o t exceed t h e v a l u e o f the guarantee.  T h i s type o f a l o s s w i l l  o c c u r o n l y i f t h e r e i s a g e n e r a l d e c l i n e i n t h e market. the m i n i m i z a t i o n  I n essence,  o f the d i s p e r s i o n o f t h i s type o f a l o s s i s the  o b j e c t i v e f u n c t i o n o f t h e hedging s t r a t e g y .  Within  t h e framework o f t h e s e d e f i n i t i o n s , t h e l o s s e s were  examined from a number o f d i f f e r e n t p o i n t s o f view.  Firstly,  consideration  was g i v e n t o t h e average l o s s and i t s d i s p e r s i o n over t h e 500 s i m u l a t i o n s  74  per s t r a t e g y .  S e c o n d l y , t h e d i s a s t e r l o s s e s were i s o l a t e d and a n a l y s e d  i n terms o f t h e mean and s t a n d a r d  d e v i a t i o n , p e r case.  T h i r d l y , the  l a r g e s t 5 and 10% o f t h e l o s s e s p e r c a s e were examined, i n terms o f t h e 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  c o s t s were s i m u l a t e d f o r each c a s e .  The c r i t e r i o n f o r t h e a c c e p t a n c e  o f t h e h e d g i n g s t r a t e g y o v e r t h e n a i v e was e s t a b l i s h e d i n terms o f the b e h a v i o r o f t h e d e v i a t i o n o f t h e l o s s e s o v e r t h e v a r i o u s a l t e r n a t i v e s . That i s , t h e d e v i a t i o n s o f t h e l o s s e s h a d t o be n o t o n l y l e s s t h a n t h o s e g i v e n under t h e n a i v e s t r a t e g y , b u t t h e y s h o u l d a l s o d e c l i n e as the number o f r e v i s i o n s was  increased.  The r e s u l t s o f t h e a n a l y s i s i n d i c a t e t h a t t h e h e d g i n g s t r a t e g y i s dominant o v e r t h e n a i v e .  W h i l e t h e average l o s s e s i n c r e a s e as  t h e number o f r e v i s i o n s i s i n c r e a s e d , t h e d i s p e r s i o n o f t h e l o s s e s decreases.  I t i s e x p e c t e d t h a t t h e average l o s s s h o u l d i n c r e a s e 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 , because o f t h e a d d i t i o n a l transaction costs.  The s m a l l e r d e v i a t i o n s i m p l y t h a t t h e d i s t r i b u t i o n  o f t h e l o s 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 t h e r e v i s i o n s a r e i n c r e a s e d . I n t h i s s e n s e , t h e r i s k a s s o c i a t e d w i t h t h e s t r a t e g y i s reduced. i s , t h e average l o s s may be v i e w e d as t h e c o s t o f r e d u c i n g t o t h e l e v e l i n d i c a t e d by t h e s t a n d a r d  deviation.  That  the r i s k  P u t a n o t h e r way,  t h e 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 t h e number o f r e v i s i o n s i s the c o s t o f reducing incremental  t h e r i s k by t h e amount i n d i c a t e d by t h e  change i n t h e s t a n d a r d  deviation.  75  From t h e r e s u l t s o f t h e 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.  F o r example, i n t h e l a s t  chapter  c o n s i d e r a t i o n was g i v e n t o t h e s t r a t e g y o f c h a r g i n g t h e i n s u r e d t h e average o f t h e l a r g e s t 5% l o s s e s o v e r and above t h e v a l u e o f t h e o p t i o n and 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 . t h a t t h i s p r o p o s a l i s s i m p l y an a l t e r n a t i v e .  I t s h o u l d be n o t e d No e f f o r t has been made  i n t h e c o u r s e o f t h i s a n a l y s i s t o examine t h e m a r k e t a b i l i t y o f t h e instrument, g i v e n these a d d i t i o n a l c o s t s .  Furthermore,  t h e degree  o f r i s k w h i c h a company may assume depends on t h e r i s k a v e r s i o n o f management, n o t o n a n o p t i m a l s o l u t i o n w h i c h one may e x p e c t . c o n s t r a i n t s , i n t h i s sense a r e exogenous t o t h e o u t l i n e d  The  procedure.  U t i l i t y f u n c t i o n s have been i g n o r e d i n t h e c o u r s e o f t h e a n a l y s i s because a t any p o i n t i n t i m e , t h e r e q u i r e d i n v e s t m e n t  i n the reference  p o r t f o l i o depends n o t o n i t s e x p e c t e d v a l u e , b u t t h e c u r r e n t v a l u e . The e x p e c t e d r e t u r n on t h e i n s t r u m e n t , does however, depend on t h e expected performance o f t h e r e f e r e n c e p o r t f o l i o .  Furthermore, t h e  l e v e l o f a c c e p t a b l e 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 t o r e i n f o r c e the contention that 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 equity 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 guarantee c o r r e c t i n t e r p r e t a t i o n o f t h e problem. shown t h a t t h e a d o p t i o n o f t h e h e d g i n g  Furthermore,  i s the  i t has been  strategy yields superior results  t o t h o s e g i v e n b y t h e c o n v e n t i o n a l buy and h o l d o p t i o n .  The p r o b l e m  76  of mortality has been excluded from the analysis, p r i m a r i l y to simplify the r e s u l t s . I t i s recommended that i n subsequent analyses of the problem, this variable be included to determine i t s impact on the r e s u l t s .  Because o f the exclusion of the mortality problem,  t h i s analysis may be viewed also as an investment i n a mutual fund and a term insurance p o l i c y on that investment. This analysis has provided a v i a b l e a l t e r n a t i v e to the management o f r i s k within the e x i s t i n g framework,  i t remains to be  seen whether the necessary l e g a l conditions w i l l be brought about i n 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 e x i s t i n g l e g i s l a t i o n governing the management of r i s k .  77 BIBLIOGRAPHY  Aitchison,  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.  B l a c k , F. and S c h o l e s , M.J., The P r i c i n g o f O p t i o n s 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.  B l a c k , F. and S c h o l e s , M.J., The V a l u a t i o n o f O p t i o n C o n t r a c t s and a T e s t o f Market E f f i c i e n c y .  J o u r n a l o f F i n a n c e , V o l . 27,  May 1972.  Brennan, M.J., S c h w a r t z , E.S., The P r i c i n g 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 P o l i c i e s w i t h an A s s e t V a l u e  Guarantee.  J o u r n a l o f F i n a n c i a l Economics, June 1976.  B u r i n g t o n , R.S. and May, D.C, S t a t i s t i c s w i t h Tables.  Handbook o f P r o b a b i l i t y and  M c G r a w - H i l l Co., New Y o r k , Second  E d i t i o n , 1970.  D i P a o l o , F., An A p p l i c a t i o n  o f S i m u l a t e d S t o c k Market Trends  t o I n v e s t i g a t e a R u i n Problem. A c t u a r i e s , V o l . X X I , 1969.  Transactions Society of  E l d e r t o n , W.P. Curves,  and Johnson,  N,L,,  Systems o f  Cambridge U n i v e r s i t y P r e s s ,  Kahn, P.M.,  Frequency  1969,  Projections of Variable Life  Insurance  Transaction Society of Actuaries, V o l . XXIII,  Leckie, in  S,H.,  Variable Annuities  the United States  of Actuaries  M e r t o n , R.C,,  J o u r n a l of the  S o c i e t y , V o l , XX,  Part  Insurance Institute  2, O c t o b e r  The T h e o r y o f R a t i o n a l O p t i o n P r i c i n g .  J o u r n a l o f Economics Spring  1971,  and V a r i a b l e L i f e  of America.  Student's  Operations.  a n d Management  1972.  Bell  S c i e n c e , V o l , 4, No.  1,  1973,  Schwartz,  E.S.,  Generalized Option P r i c i n g Models:  Solutions  a n d t h e P r i c i n g o f a New L i f e  U n p u b l i s h e d Ph.D.  Dissertation,  Insurance  Numerical Contract.  F a c u l t y o f Commerce  Business A d m i n i s t r a t i o n , the U n i v e r s i t y  of B r i t i s h  and Columbia,  September,1975.  S c o t t , W.F,,  A Reserve  B a s i s f o r M a t u r i t y Guarantees  U n i t - L i n k e d L i f e Assurance. University,  Edinburgh,  Working  October  1975.  Paper,  in  Meriot'Watt  Sharpe, W.F.,  P o r t f o l i o Theory and C a p i t a l  M c G r a w - H i l l B o o k Company, New Y o r k  Shook, R.C.  Markets.  1970.  and H i g h l a n d , H . J . , P r o b a b i l i t y Models  Business A p p l i c a t i o n s ,  R.D,  Irwin,  Inc.,  with  Homewood,  Illinois,  1969.  S m i t h , K.V., " P o r t f o l i o Management, New Y o r k  Holt, Rinehart  Winston,  Inc.,  Squires,  R.T. , U n i t - L i n k e d A s s u r a n c e : O b s e r v a t i o n s  1971,  x  Propositions. P a r t 1, No.  Turner,  416,  Transactions  Based  Society of A c t u a r i e s , V o l . XXI,  Insurance  Inc.  1969.  i n the United  Transactions Society of Actuaries,  Vol. XXIII,  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 . Hall  101,  1974.  S,H. ^ " E q u i t y Based L i f e  Kingdom.  and  J o u r n a l of the I n s t i t u t e of A c t u a r i e s , V o l .  Turner, S . H . ^ A s s e t V a l u e Guarantees Under E q u i t y Products.  and  1971.  Prentice-  Second E d i t i o n , 1971.  Van H o m e , J .C., "of Readings.  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  R.D.  Irwin,  Inc.,  Homewood, I l l i n o i s ,  1967.  Book  80  21)  Weston, T.F. and Brigham, E. F., Managerial Finance.  Holt,  Rinehart and Winston, Inc. Fourth Edition, 1972,  i  81  Appendix A S i m u l a t i o n Program  T h i s appendix c o n t a i n s t h e s i m u l a t i o n program w h i c h the s u p p o r t i v e s t a t i s t i c s f o r t h e a n a l y s i s . appendices  generates  The t a b l e s o f t h e f o l l o w i n g  a r e e d i t e d from t h e o u t p u t o f t h i s program.  I t s h o u l d be  n o t e d t h a t i f C0NT=2 t h e n t h e r e s u l t s o f i n t e r m e d i a t e c a l c u l a t i o n s a r e a l s o p r o d u c e d as o u t p u t  ( i e . as p e r A p p e n d i x B ) , o t h e r w i s e , i f C0NT=1,  o n l y t h e t e r m i n a l v a l u e s a r e p r o v i d e d as o u t p u t .  S i m i l a r l y , i f NAIV=2,  then t h e s i m p l e o r n a i v e buy and h o l d p o l i c y i s e v a l u a t e d ( i e . no r e v i s i o n s ) , o t h e r w i s e , i f N A i V = l , t h e g e n e r a l model w i t h a p p r o p r i a t e r e v i s i o n s i s e v a l u a t e d and w r i t t e n .  When e v a l u a t i n g t h e n a i v e s t r a t e g y ,  t h e f o l l o w i n g changes must be made:  1)  Initialize: TR=0.0 DO 1500 JM=1,1 DO 1400 NREV= 10, 10, 10  2) Add b e f o r e t h e comment c a r d 'BEGIN': IF  (NAIV. EQ. 2) GO TO 1  3) Add a t t h e bottom o f page one o f t h e 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 c o s t s a r e c a l c u l a t e d e x t e r n a l l y because c o n s i d e r a b l e changes must be made i n t h e program i n o r d e r to accomplish  this  internally.  83  DIMENSION RPORT(99),WLTH(99),OPVAL(99),X(99),RAN{99),XLAB(99), 1CSUM1500),AVE(500),PROF{500),RP0RT1{500) AVEPRT(500),VARPFT(500), 2VARPRT(500),DUM(500),0UM1(500),XLOSS(500) I N T E G E R B A R R A Y I 1 0 ) t F M T / * F7*3*/ INTEGER C A R R A Y ( 1 0 ) , F M A / • F 7 o 3 • / I N I T I A L I Z E VARIABLES t  C Q,  *  *  *  *  *  *  *  *  e  C C C  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  C0NT=1» NAIV=1 NSIM=500 KS=NSIM-40o 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 NPER1=NPER-1 GAR=100o RFRE= 06*{10«/NREV) VAR=0o01846*(10o/NREV) SUM=0« TOT=0. TRBL=0o0 N0=0o . BEGIN 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 C T = N P E R - 1 ) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * S = 0o0 STD= S Q R T ( V A R ) FM=o08*(lOo/NREV) XX=100. RAN(1)=0<,0 T=NREV RPQRT(l) =100.0 XL=AL0G(XX/100 ) DD1=XL+<RFRE+.5*VAR)*T DD2=XL+fRFRE-.5*VAR)*T DB=SQRT(VAR*(2*T)) 01=DD1/DB*C-1) D2=DD2/DB#(-1) A=EXP(-RFRE*T) OPVAL(1)=<(XX*ERFC(DI))-(100*A*ERFC( D2 ) I ) * . 5 XLAB(1)=100*£XP(-RFRE*T)+OPVAL(1) WLTH(l)=100oO*EXP{-RFRE*T)+OPVAL(l) X(l)=RP0RT(l)*ERFC(Dl)*o5 IRF=WLTH(1)-X(l) W L T H U ) = W L T H { l ) - ( X< 1 J * T R J o  84  C  C A L C U L A T I O N OF SECOND STAGE  C  DO 1 0 0 0 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 JJ=J-1 RPORT(J)=RPORT(JJ)*Z WLTH(J)=((WLTH(JJ)-X(JJ))*EX P(RFRE))+X{J J ) * Z REVISION  Q  <T=NPER-2  TO 1)  * # * * * * * * * * * * * * * * * * * * * * * * * * * # #  T=T-JJ I F ( N A I V » E Q o 2 ) GO TO 1 2 3 4 XC=ALOG(RP0RT(J)/100} A=EXP(-RFRE*T) D l = ( X C + ( ( R F R E + « 5 * V A R ) * T ) ) / S Q R T ( VAR* ( 2 * T ) ) # ( - l ) D2=(XC*((RFRE-.5*VAR)*T))/SQRT(VAR#(2*T))*<-lI 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 1234 X ( J ) = X ( J J ) * Z OPVAL(J)=RPQRT(J)-100o I F ( O P V A L { J ) . L T . O „ ) O P V A H J)=0»0 3456 T R C S T = T R * A B S ( X ( J ) - X ( J J ) ) WLTH(J)=WLTH(J)-TRCST C CALCULATE L I A B I L I T Y XLAB(J)=100*£XP(-RFRE^T)+0PVAL{J) 100 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) RAN(L)=Z LL=L-1 WLTH(L)=( ( V I L T H ( L L ) - X ( L L ) ) * E X P ( R F R E ) ) + X ( L L )*Z WLTH(L)=WLTH(L)-((X(LL)*Z)*TR) RPORT( L ) = RPORT ( L L ) * Z OPVALU)=RPORT(L)-100. IF(OPVAL(L JoLT.Oo) OPVAL(L)=0.0 X(L)=0o0 XLAB(L)=100o+0PVAL(L) I F ( C 0 N T o E Q l o ) GO TO 9 7 I F ( K . G T o 4 ) GO TO 9 7 WRITE(6,91) FORMAT(6X»'INTERMEDIATE CALCULATIONS•,//6X,•X-J•,6X,»RP0RT'» 16X,«WLTH» » 6 X , • R A N D - Z • 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 ) FORMAT(IX,6F10.2) CONTINUE CONTINUE o  91  f  95 96 97  85  PROFIT  AND  AVE.  PROFIT  CALCULATIONS.  I F ( R P O R T ( N P E R ) 0 L E 0 I O O ) GO TO 110 PROF(K) = WLTH(NPER)-RPORT(NPER) GO TO 9 9 0 110 PR0F(K)=WLTH(NPER)-100. TRBL=TRBL+lo N0=N0+1 XL O S S ( N O ) = P R O F ( K ) 990 SUM=SUM*PROF{K) CSUM(K)=SUM AVE(K)=CSUM(K)/K I F ( C O N T . E Q . 1 . ) GO TO 9 9 3 I F ( K . G T o 4 ) GO TO 9 9 3 WRITE(6,991)PR0F(K) 991 FORMAT C * 0•» 5Xt * VALUE OF PROF I T • , F 1 0 . 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 9 9 5 J J = 1 , K I F ( K . E Q o l ) GO TO 996 Q=Q+((RPORTI{JJ)-AVEPRT(K))**2)/{K-l) 995 XY=XY+((PROF(JJ)-AVE(K>)**2)/(K-l) 996 VARPRT(K)=SQRT(Q) V A R P F T t K) = S Q R T ( X Y ) 1000 CONTINUE I F I C 0 N T . E Q . 2 ) GO TO 1 4 0 0 WRITE(6,1549) 1 5 4 9 FORMAT( • 1') WRITE(6,1300) WRITE(6,5300) 1300 FORMAT(7X,•PROFIT•,5X,«AVE«,7X,'STD',5X,'RPORTI',5X,'AVE',7X 2,•STD') 5 3 0 0 F0RMATC7X, * ',5X,• • , 7X , • » ,5X,« • ,5X,« ' ,7X 2,» • ) DO 1 0 1 0 KK=KS,NSIM MRITEt6t1011JPROF(KK) AVE*KK) ,VARPFT(KK) ,RPORTI(KK),AVEPRTi KK),V 1PRT(KK) 1011 F0RMAT(2X,6F10.2) 1010 CONTINUE TRBLP=(TRBL/NSIM)*100. WRITE(6 1012)TR,NREV,NPER,RFRE,VAR,FM 1012 FORMAT(•0*,•TRANSo C O S T • , 8 X , F l O o 6 , 6 X , < N 0 . OF R E V I S I O N S * , 8 X , 1 1 0 , / 1 N 0 . OF P E R I O D S ' » 5 X , I 1 0 , 6 X » ' R I S K F R E E 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 , 1 1 0 0 ) T R B L , T R B L P 1100 FORMAT(»0«,'DISASTER',11X,F10«0,6X, PERCENT',17X,F10«6) f  t  >  86  C  PLOT  £  *  C  DO 1 3 0 1 !J=1,NSIM DUM1(IJ)=RPORTI ( I J ) DUM{ I J ) = P R O F U J ) CALL SS0RT(DUM,NSIM,3) CALL SS0RT1DUM1,NSIM,3) N=DUM(1)-1 M=DUM1(1)-1 XMIN=N YMIN=M DX=2»0 DY=40 NX=NSIM NY=NSIM CALL HISTGM ( 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 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 » D Y , 1 0 » 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 FORMAT (/ * T H E D I S T R I B U T I O N IS / 7 X .1016 ) * * * * * * * * * * * * * * * * * * * * * * * * *  1301  *  ROUTINE *  *  *  *  *  *  *  *  *  7>t  *  *  *  *  *  *  *  *  *  *  #  o  1310 1312  C  1340  1345  1346 1550 1350 1355  (  T  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 SAM=SAM+XLOSS( J ) SAMMN=SAM/I1 SVAR=0o0 12 = 0 DO 1 3 4 5 1=1, NE 12=12+1 SVAR=SVAR+((SAMMN-XLOSS(I))**2) IF(NOoEQol) 12=2 SVAR=SQRT(SVAR/(I2-1„)) WRITE16, 1346) F0RMATP1',' DISASTER LOSSESo') WRITE(6,1550) FORMAT!'0') WRITE(6,1350)(XLOSS(K),K=1,12) F O R M A T ! 5F1 Oo 5) W R I T E ( 6 , 1 3 55)SAMMN,SVAR FORMAT C O ' , ' MEAN L O S S ' , 1 0 X , F 1 0 . 2 , 2 X , / / » STANDARD 1,1X,F10O2)  DEVIATION'  87  1360 1361 1365 1370 1371  1372 1373  1375 1376  1377 1311 1400 1500  GO TO 1 3 6 5 W R I T E ( 6 , 1 3 61J FORMAT ! ' 0 ' , ' N O D I S A S T E R L O S S E S ' ) WRITE(6,1550) W R I T E ! 6 , 1 3 7 0 ) ( D U M { J ) ,J=1 ,25) 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 ) ) ADl=0o0 DO 1 3 7 1 K=l,25 AD1=AD1+DUM!K) ADMl=ADl/25„ V1=0.0 DO 1372 K=l,25 V1=V1+!!DUM!K)-ADM1)**2) VSD1=SQRT!Vl/24o) W R I T E ! 6 , 1 3 7 3 J A D M l , VSD1 F O R M A T ! * 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 ' 1F10»2) AD2=AD1 V2 = V1 DO 1 3 7 5 J=26,50 AD2=AD2+DUMiJ) ADM2=AD2/50o DO 1 3 7 6 J=26,50 V2=V2+(IDUM!J)-ADM2)**2) VSD2=SQRT!V2/49.) WRITE(6,1550 ) WRITE!6,1377)!DUM!J),J=l,50) 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 10S OF T O T A L * » / / » ! 5 F 8 o 2 ) ) WRITE!6,1373)ADM2,VS02 WRITE!6,1311) FORMAT I • 1 ' ) CONTINUE TR=TR+0.005 CONTINUE STOP END  *** NAME  SUBPROGRAM  ATTRIBUTES  ABS ALOG ERFC EXP RANDL SQRT  SSORT <EXIT>  *** NAME  A ADM 1 ADM2 ADI AD2 AVE AVEPRT BARRAY 1*4 CARRAY 1*4 CONT CSUM DB DDI DD2 DUM DUM1 DX DY DI D2 FM FMA FMT GAR I IJ IRF 11 12 J  JJ  1*4 1*4  ID (D <D  62 44 42 168 79 37 221 156  VARIABLE  ATTRIBUTES  iD  ) ) ) )  to  )  (D  )  ID  )  88  ***  REFERENCES 72 34 41 40 165 53 28 210 155 231  HISTGM  DICTIONARY  66 43  67 57  63  75  82  64  65  126  127  187  201  205  209  66  67  187 56 71 179  192 57 72 201  62 73 216*  66 75 217  60  69  72  122*  DICTIONARY  ***  REFERENCES 40* 20 6* 218* 203* 214* 1 1 4 5 8* 1 37* 3 5* 36* 1 217 1 161* 162* 38* 39* 29* 5 4 17* 183* 15 2 * 45* 175* 18 2 * 51* 67 219* 55*  41 209 220 205* 217* 110* 119* 165 168 89 109* 38 38 39 154* 220 153* 165 168 41 41 53 168 165  63* 211 225 206 218 125 124 166 169 111 110 39  66 214 139 139 129  155 223 156  158  44 65* 79  5 4* 66 144  185 153  154  178* 184* 54 69 95 220 56  180 186* 55 70 177* 223 57  157  V  89  JM K  KK KS L LL M N NAI V NE NO NPER NPERl NREV  NSIM NX NY OPVAL  {D  )  PROF  (D  )  0 RAN RFRE  CD  )  RPORT  (D  )  RPORTI S SAM SAMMN STD SUM SVAR  (D  )  T  TOT TR TRBL TRBLP TRCST VAR VARPFT VARPRT VSD1 VSD2 VI V2  <D (D  ) )  124 13* 49* 110 123 205 138* 11* 50* 86 81* 15 8* 157* 9* 174* 23* 15* 144 16* 14* 144 10* 156 163* 164* 1 75 1 139 120* 1 18* 63 1 70 1 27* 173* 180* 28* 20* 181* 32* 52* 21* 12* 22* 143* 72* 19* 144 1 1  210* 221* 20 7* 215*  125 90 112 124 208* 139 138 80 87 82 160 159 61 183 106* 16 51 15 1 1  163 165 168 41* 85* 102* 154 124* 31* 35 64 33* 84* 117* 53 179* 185 53 108* 185* 35 60* 118* 46 105* 148 73 28 127* 126* 211 225 209* 220*  102 113 125 209  104 117 126  107 118 127  108 119 192  109 122 204-  81 88 83  82 117 84  83  84  85  107 50  174 94  176 101  177 102  186 104  18  19  29  32  52  49 164  138  143  152  155  42 86* 104* 165 126 54* 36 65 44 85  43 88 107  66* 95 108  70*  71*  113  125  80* 40 75 56* 95 124  95 42 82 62 101 139  43 144 66 102 153  57  194 37 64  40 65  42 75  43  83 148  144  229*  36  37  64  118 79 180 194 79 109 187* 36 63 119 72 143  35 139 139  210 221  215  67 117 168  65  90 WITH  {D  )  X  {D  )  (0 CD  J  XC XL XLAB XLOSS XMIN XX XY YMIN Z  *** LABEL  91 95 96 97 100 110 990 991 992 993 995 996 1000 1010 1011 1012 1100 1234 1300 1301 1310 1311 1312 1340 1345 1346 1350 13 55 1360 1361 1365 1370 1371 1372 1373 1375  )  1 83* 1 72 62* 34* 1 1 159* 3 0* 121* 160* 53* 82  STATEMENT  43* 95 44* 82 64 35 42* 107* 165 34 125* 168 54 83  LABEL  56 84  225  57*  73*  82*  57 95  67*  69*  88* 185  95 192  57  69  79*  80  ***  REFERENCES  91 95 94 89 51 101 103 113 115 111 122 123 49 138 139 144 148 61 132 15 2 166 226 169 177 183 188 192 194 176 198 197 201 204 208 211 216  46* 104 46 87*  41 127  DICTIONARY  DEF * N  92 96 97 98 76 104 108 114 116 117 125 126 128 142 141 145 149 69 134 154 167 227 170 179 185 189 193 195 198 199 200 2 02 205 209 212 217  45 102 45 83 65 36 75* 179  91  1376 1377 1400 1500 1549 1 5 50 3456 5300  220 224 228 230 131 191 72 136  219 223 14 13 130 190 68 133  * * *  U N I T  6  L O G I C A L  129  200  I/O  U N I T S  D I C T I O N A R Y  * * *  R E F E R E N C E S  91 139 192 223  95 144 194 225  113 148 198 226  115 166 200  130 169 201  132 188 211  133 190 222  /  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 p r o v i d e examples o f t h e more i m p o r t a n t i n t e r m e d i a t e v a l u e s 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 . F o r a l l t h e t a b l e s a one p e r c e n t t r a n s a c t i o n c o s t i s assumed.  Revision Policy:  Table 1  Annual r e v i s i o n  Table 2  E v e r y s i x months  Table 3  E v e r y 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 t h e r e f e r e n c e p o r t f o l i o a t time t , as g i v e n b y e q u a t i o n ( 3 - 1 9 ) .  RPORT:  The 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 a t t i m e t , as g i v e n b y (3-22).  WLTH:  The w e a l t h p o s i t i o n o f t h e company a t t i m e t , as g i v e n by (3-23).  RAND-Z:  The s i m u l a t e d r e t u r n o n t h e r e f e r e n c e p o r t f o l i o f o r t h e p e r i o d under c o n s i d e r a t i o n .  93  OPVAL:  The v a l u e o f t h e o p t i o n a t t i m e t , as d e f i n e d by t h e 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 t h e company a t t i m e t , as g i v e n by t h e g e n e r a l i z e d v e r s i o n o f ( 3 - 1 8 ) , o r by t h e c o n d i t i o n s d e s c r i b e d i n e q u a t i o n s to  (3-27)  (3-29).  The v a l u e o f p r o f i t g i v e n a t t h e 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 t h e amount t h e company must charge t h e i n v e s t o r i n o r d e r t o b r e a k even, i f t h e v a l u e i s n e g a t i v e . by e q u a t i o n s  The d e r i v a t i o n o f t h e p r o f i t i s g i v e n  (3-30) and (3-31).  I t s h o u l d be n o t e d t h a t t h e i n i t i a l v a l u e s o f t h e 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 . investment i n the reference p o r t f o l i o  The t e r m i n a l v a l u e o f t h e ( i e . X * J ) i s always z e r o  because o f t h e a s s u m p t i o n t h a t t h e 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 o f the c o n t r a c t .  F i n a l l y , t h e o p t i o n v a l u e , OPVAL,  must always be g r e a t e r t h a n o r e q u a l t o z e r o .  94 INTERMEDIATE  CALCULATIONS  X-J 94.64 106.87 105.4.2 1 2 5 . 50 140.82 178.97 192.65 178.13 167.76 150.04 OoO  RPORT 100.00 110o99 110.36 128.16 142o32 179.17 192.71 178.21 167.81 150.08 154 52  WLTH 100.21 110.84 110.46 127.57 141.41 177.53 190.83 176.07 165.45 147.41 150 14  VALUE  OF P R O F I T  -4.3823  e  RAND-Z 0.0 loll 0.99 1.16 1.11 1.26 1 . 08 0.92 0.94 0.89 1.03  o  OPVAL 46.28 5 3 . 50 49.39 62.86 7 2 . 74 105.10 114.05 9 4 . 68 7 9 . 12 55o90 54.52  * **  JJCJSS jfcjje ## INTERMEDIATE CALCULATIONS  X-J 94.64 123.64 142o41 137.88 130.75 129.16 143.73 139.40 182.65 1 5 4 . 75 OoO  RPORT 100.00 125.92 143.65 139.48 133.09 131.83 144.9 7 140.77 182.66 154.76 160.77  VALUE  OF P R O F I T  WLTH 100.21 124.80 142.08 137.89 131.50 130o29 143.09 1 3 8 o 85 179 86 151.52 155.71  ** * RAND-Z 0.0 1.26 lo 14 0.97 0.95 0o99 1.10 0.97 1.30 0.85 1.04  0  LIAB 101.16 126.31 1 4 3 . 82 139.71 133.40 1 3 2 . 16 145.09 140.89 182.66 154.77 160.77  **  CALCULATIONS  X-J 94.64 86.97 108o72 86.69 112.44 88.68 67.10 92.32 75.27 81.23 OoO  RPORT 100.00 94.69 113.13 96.79 117.36 100.91 90.44 1 0 6 . 01 99.51 103.77 91.97  WLTH 100.21 95.46 112o70 97 02 115„82 100.03 91.32 104.12 99.01 103.65 95.08  VALUE  OF P R O F I T  -4.9210  #*  **  *  OP VAL 46.28 68.03 81.95 74.00 63.64 58o08 66.43 57.36 93.96 60.59 6 0 . 77  -5.0571  ** INTERMEDIATE  LIAB 101.16 111.78 111.27 128.56 1 4 2 . 51 179.19 192.71 178.21 167.81 150.08 154.52  RAND-Z 0.0 0.95 1.19 0.86 1.21 0.8 6 0.90 l o 17 0o94 1.04 0.89  o  **  OPVAL 4 6 . 28 38 . 1 3 5 2 . 05 33.12 48.34 2 8 . 93 16.20 24.27 1 3 . 86 11.46 0.0  LIAB 1 0 1 . 16 96.40 113.92 98.82 118.11 103.01 94.86 107.80 102.55 105.64 100.00  ** #  9 5  INTERMEDIATE  CALCULATIONS  X-J 94.64 88.54 66.58 50o79 65.53 87.24 86.95 83.38 96.03 108.79 89.38 115.40 122.25 1 0 3 . 18 113.36 123.61 148.43 154.13 149.82 133.62 0.0  RPORT 100.00 95.44 79.75 70.21 80.73 9 6 . 56 96.97 95.17 104.71 114.75 101.37 120.47 126.11 112.16 119.43 127.08 148.82 154.22 149.85 133.67 139.22  WLTH 100.21 96.00 81.46 73.79 81.95 9 5 . 08 95.69 94.30 102.87 112.15 99.38 116.27 121.62 107.89 114.62 121.82 142.67 147.82 143.22 126.67 130.62  VALUE  OF P R O F I T  -8.6014  *  *#  **  *  OPVAL 46.28 4 0 . 42 24.96 1 6 . 08 22.75 34.72 33.28 29.83 36.46 4 3 . 76 2 9 . 33 4 4 . 80 47.89 32 . 25 36.55 41.32 6 0 . 16 6 2 . 83 55.68 36.63 39.22  jjt*  INTERMEDIATE  *5)C  L IAB 101.16 96.97 83.23 76.13 84.63 98.48 9 8 . 99 97.54 106.22 115.65 103.41 121.14 126.56 113.31 120.08 127.39 1 4 8 . 85 154. 23 149.86 133.67 139.22  **  *  CALCULATIONS  RPORT 100.00 113.25 125.84 130.12 144.84 138.31 152.58 162.36 173.27 200.94 181.76 157.22 152.34 161.83 179.07 184.78 210.27 224.40 260.66 259.05 236.37  WLTH 100.21 112.77 124.93 129.12 143.49 136.94 150.90 160.50 171.22 198.50 179.06 154.21 149.24 158.51 175.45 180.99 206.11 219.96 255.73 253.95 228.75  V A L U E OF P R O F I T  -7.6193  X-J 94.64 109.74 123.55 128.07 143.67 136.70 151.70 161.80 172.95 200.87 1 8 1 . 58 156.58 151.57 161.48 179.00 184.76 210.27 224.40 260.66 259.05 0.0  RAND-Z 0.0 0.95 0.84 0 . 88 1.15 1.20 1.00 0.98 1.10 1.10 0.88 1.19 1.05 0.89 1.06 1.06 1.17 1.04 0.97 0 . 89 1.04  tfif.  **  **  RAND-Z 0.0 1.13 1. 11 1.03 1.11 0.95 1.10 1.06 1.07 1.16 0.90 0.86 0.97 1.06 1.11 1.03 1.14 1.07 1.16 0.99 0.91  **  OPVAL 4 6 . 28 57.36 67.96 70.40 83.13 7 4 . 78 86.99 94.73 103.54 129.06 107.69 80.94 73.75 8 0 . 80 95.55 98.71 121.58 133.01 166.49 162.01 136.37  i£s}s  LIAB 1 0 1 . 16 113.91 126.23 130.45 145.01 138.55 152.70 162.43 173.31 200.95 181.77 1 5 7 . 28 15 2 . 4 2 161.86 179.08 184.78 2 1 0 . 27 224.40 260.66 259.05 236.37  #3js  #  XA.fiLf._3.  INTERMEDIATE  X-J  RPORT 100.00 109.78 105.49 107.36 98.42 104.77 100o05 109.03 1 2 0 . 75 121o31 112.18 120.26 131.63 149.21 161.11 162.09 168.39 1 7 5 . 59 168.82 169.56 148.91 153.43 140o88 141.62 1 3 6 . 45 150.36 161.00 169.02 166.56 150.32 152.41  VALUE OF  PROFIT  94.64 105.88 100.65 102.61 91.33 98.91 92.59 103.53 117.22 117.70 106.55 116.14 129.13 148.14 160.54 161.58 168.05 175.39 168.54 169.34 148.07 152.90 139.65 140.62 135.13 150.14 160.97 169.02 166.56 150.32 OoO  *  CALCULATIONS  * *  WLTH 100.21 109.47 105.35 107.21 98.65 104.61 100.21 108.56 119.66 120.24 111.32 118.99 129.91 146.97 1 5 8 . 64 159.58 165.75 172.82 165.93 1 6 6 . 61 145.72 150.11 137.42 138.10 1 3 2 . 86 146.44 156.88 1 6 4 . 74 162.17 1 4 5 . 67 146.15 -6.2643 #J§£  RAND-Z 0.0 lo 10 0.96 lo02 0.92 1.06 0.95 1.09 1.11 1.00 0.92 1.07 1.09 1.13 1.08 1.01 1.04 1.04 0.96 loOO 0.88 1.03 0.92 1.01 Oo 96 1.1.0 1.07 1.05 0.99 0.90 1.01  OPVAL 46.28 54.56 49.35 50. 02 40.47 45.26 39.68 46.91 56.92 56.17 46.09 5 2 . 50 62.21 78.15 8 8 . 55 88.06 92.84 9 8 . 50 9 0 . 18 8 9 . 33 67.11 69.94 55o 76 54.75 47.85 59.88 68.63 7 4 . 35 70.48 52.30 5 2 . 41  LIAB 101.16 1 1 0 . 55 106.47 108.3 0 99.92 105.92 101.56 110.04 121.33 121.88 113.12 12 0 o 8 9 131.98 149.33 161.17 16 2..15 168.42 175.61 168.84 169o58 148.98 153.47 140.97 141.69 136.54 150.37 161.00 169o02 166.56 150.32 152.41  Appendix 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  T h i s a p p e n d i x p r o v i d e s t h e summary s t a t i s t i c s f o r t h e naive strategy.  T h i s s t r a t e g y s i m p l y assumes t h a t t h e 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 t h e duration of the contract.  No p o r t f o l i o r e v i s i o n o c c u r s d u r i n g t h i s  period. The l a b e l d e f i n i t i o n s o f t h e p r e v i o u s appendices a p p l y t o the t a b l e s e x h i b i t e d .  The number o f p e r i o d s i s synonymous t o t h e  r e v i s i o n periods o f the previous tables.  98  TABLE  TRANSACTIONS  NAIVE  NUMBER PERIODS  AVERAGE LOSS  1  COST  02.  STRATEGY  STANDARD DEVIATION  AVERAGE PORTFOLIO  STANDARD 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  TRANSACTIONS  DISASTER  NUMBER PERIODS  NUMBER LOSS  -  2  COST  LOSSES; NAIVE  PERCENT LOSS  0%  o  STRATEGY  AVERAGE LOSS  STANDARD 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  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  40  -  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 t h e company o v e r f i v e hundred s i m u l a t i o n s .  T h i s i s t h e average amount  t h e company must charge t h e i n s u r e d i n o r d e r t o b r e a k even under t h e 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 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 o v e r f i v e hundred s i m u l a t i o n s , g i v e n t h e p a r t i c u l a r revision  strategy.  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 .  It  c l a r i f i e s t h e e f f e c t o f i n c r e a s i n g t h e number o f r e v i s i o n s by e l i m i n a t i n g t h e a c c u m u l a t i n g e f f e c t o f t h e t r a n s a c t i o n costs.  101  TABLE  TRANSACTIONS  NUMBER REVISIONS  AVERAGE LOSS  STANDARD DEVIATION  1  COST  1%  0  AVERAGE PORTFOLIO  STANDARD 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  TRANSACTIONS  NUMBER REVISIONS  AVERAGE LOSS  STANDARD DEVIATION  2  COST  l„5%  AVERAGE PORTFOLIO  STANDARD 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  TRANSACTIONS  NUMBER REVISIONS  AVERAGE LOSS  STANDARD DEVIATION  3  COST  2£.  AVERAGE PORTFOLIO  STANDARD DEVIATION  10  -13o66  5.24  245.43  115.24  20  -15.29  5.32  232.92  113.95  30  -17.49  4.73  247.00  110.13  40  -18.86  4.96  245.90  110.58  50  -20.02  4.92  246.88  105.77  60  -20.86  4.95  244.20  106.81  70  -22.03  5.11  218.08  106.23  80  -23.02  5.66  250.38  115.76  104  TABLE  TRANSACTIONS  NUMBER REVISIONS  AVERAGE LOSS  STANDARD DEVIATION  4  COST  2*5%  AVERAGE PORTFOLIO  STANDARD 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  -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  40  .  105  TABLE  TRANSACTIONS  NUMBER REVISIONS  AVERAGE LOSS  STANDARD DEVIATION  5  COST  01.  AVERAGE PORTFOLIO  STANDARD 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 t h e l o s s e s w h i c h t h e company i n 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 t h e g u a r a n t e e was e x e r c i s e d o v e r 500 s i m u l a t i o n s , p e r r e v i s i o n s t r a t e g y .  PERCENT LOSS:  The number o f l o s s e s as a p e r c e n t o f t h e t o t a l number o f s i m u l a t i o n s .  AVERAGE LOSS:  T h i s i s t h e average d o l l a r l o s s i n c u r r e d by t h e company as a r e s u l t o f t h e g u a r a n t e e b e i n g exercised.  T a b l e 5 i s t h e 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  TRANSACTIONS  DISASTER  NUMBER REVISIONS  1  COST  1%.  LOSSES  NUMBER LOSS  PERCENT LOSS  AVERAGE LOSS  10  15  3.00  -4.11  5o55  20  22  4.40  -5 80  3.58  30  10  2.00  -7.38  2.82  40  13  2.60  -6.70  2.91  50  16  3.20  -6.47  2.60  60  14  2.80  -7.77  2.18  70  21  4.20  -7.23  2.94  80  16  3.20  -7.55  2.00  o  STANDARD DEVIATION  108  TABLE  TRANSACTIONS  DISASTER  2  COST  1.5*  LOSSES  NUMBER REVISIONS  NUMBER LOSS  PERCENT LOSS  AVERAGE LOSS  STANDARD 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  TRANSACTIONS  DISASTER  3  COST  2%*  LOSSES  NUMBER REVISIONS  NUMBER LOSS  PERCENT LOSS  AVERAGE LOSS  STANDARD 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. 6 0  -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  TRANSACTIONS  DISASTER  4  COST  2.53.  LOSSES  NUMBER REVISIONS  NUMBER LOSS  PERCENT LOSS  AVERAGE LOSS  STANDARD 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  Ill  TABLE  TRANSACTIONS  DISASTER  NUMBER REVISIONS  NUMBER LOSS  15  3»00  20  22  30  COST  0%o  LOSSES  PERCENT LOSS  10  5  AVERAGE LOSS  STANDARD DEVIATION  0ol6  5.50  4.40  -0 62  3„31  10  2o00  -1.65  2.16  40  13  2.60  -0.88  2.46  50  16  3.20  0.00  2.26  60  14  2.80  -0.82  1.68  70  21  4.20  -0.11  2.66  80  16  3.20  0.16  1.47  o  112  Appendix F Summary S t a t i s t i c s : L a r g e s t Losses  These t a b l e s summarize t h e magnitude  o f l o s s e s t h e company  i n c u r s g i v e n t h e r e v i s i o n p o l i c y and t h e t r a n s a c t i o n c o s t s .  Table 5  i s a g a i n t h e 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 .  Label D e f i n i t i o n s :  AVERAGE LOSS: FIVE PERCENT The mean o f t h e 25 l a r g e s t l o s s e s o c c u r i n g o v e r 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 a s above, e x c e p t t h e l a r g e s t 50 l o s s e s a r e considered.  I t s h o u l d be n o t e d t h a t t h e s e l o s s e s i n c l u d e e x e r c i s i n g t h e g u a r a n t e e , as w e l l as t h e l o s s 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  TRANSACTIONS  LARGEST  FIVE  NUMBER REVISIONS  AVERAGE LOSS  1  COST  l%  0  LOSSES  PERCENT  TEN  STANDARD DEVIATION  AVERAGE LOSS  PERCENT  STANDARD 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  TRANSACTIONS  LARGEST  FIVE  NUMBER REVISIONS  AVERAGE LOSS  2  COST  1.5$.  LOSSES  PERCENT  TEN  STANDARD OEVIATION  AVERAGE LOSS  PERCENT  STANDARD 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  TRANSACTIONS  LARGEST  FIVE  NUMBER REVISIONS  AVERAGE LOSS  3  COST  2%  0  LOSSES  PERCENT  TEN  STANDARD DEVIATION  AVERAGE LOSS  PERCENT  STANDARD DEVIATION  10  -27o00  3o93  -24ol9  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  TRANSACTIONS  LARGEST  FIVE  NUMBER REVISIONS  AVERAGE LOSS  4  COST  2 5l 0  0  LOSSES  PERCENT  TEN  STANDARD DEVIATION  AVERAGE LOSS  PERCENT  STANDARD DEVIATION  10  -32»25  3o06  -29.07  3ol8  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  TRANSACTIONS  LARGEST  FIVE  NUMBER REVISIONS  AVERAGE LOSS  5  COST  02.  LOSSES  PERCENT  TEN  STANDARD DEVIATION  AVERAGE LOSS  PERCENT  STANDARD 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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