A MATHEMATICAL PROCEDURE FOR SELECTING AMONG ALTERNATIVE UTILITY FUNCTIONS by GERRIT PAUL VAN DER SCHOOT B.Sc, D e l f t U n i v e r s i t y o f Technology, 1974 M.Sc., D e l f t U n i v e r s i t y o f Technology, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUSINESS ADMINISTRATION in THE FACULTY OF GRADUATE STUDIES (The 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 ) We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA March 1981 © G e r r i t P a u l v a n der Schoot, 1981 In p r e s e n t i n g this thesis i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t it the L i b r a r y s h a l l make f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I further agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s thesis f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . understood t h a t copying or p u b l i c a t i o n f o r f i n a n c i a l gain The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 n r _ C I O /no \ Hou^eM ^T, of t h i s thesis s h a l l n o t be a l l o w e d without my permission. Date It i s V<^%\. Columbia written - i - Abstract This thesis presents a mathematical procedure, c a l l e d the R^-method, for selecting among alternative u t i l i t y functions to represent a decision maker's r i s k preference. A general class of u t i l i t y functions i s introduced and for f i v e a l t e r n a t i v e members of this class, the absolute r i s k aversion at the i n i t i a l wealth w , i . e . R.(w ), i s expressed as a function of: o A o (i) the parameters of a nondegenerate gamble z; and ( i i ) the decision maker's response to that gamble ( i n terms of r i s k premium, or c e r t a i n t y equivalent, or p r o b a b i l i t y equivalent, or gain equivalent). Mathematical r e s u l t s are obtained for two d i f f e r e n t gambles. calculates the values of R^ for several responses to d i f f e r e n t The R^-method reference gambles, and then selects the u t i l i t y function with the least r e l a t i v e standard deviation over the R^ values. The procedure i s based on the fact, that f o r the decision maker's actual u t i l i t y function, R^ must t h e o r e t i c a l l y a t t a i n the same value at W, q assess R^. namely R^( )> W n o D Suggestions are made for extending matter what gamble i s used to the R -method to incorporate A r i s k proneness as well as attitudes which are r i s k averse over one part of the domain and r i s k seeking over another part. matical extensions F i n a l l y , a chapter on mathe- i s provided i n order to improve the R^-method by including a larger set of a l t e r n a t i v e u t i l i t y functions. Contents Abstract Contents 1. Introduction 2. Notation and Preliminaries 3. Mathematical Results f o r R.(w ) A o 3.1. The idea behind the mathematical r e s u l t s 3.2. A generalized class of u t i l i t y functions 3.3. Risk aversion R^ as a function of certainty equivalent or r i s k premium for the gamble z = <h,p;-h> 3.A. Risk aversion R^ as a function of p r o b a b i l i t y equivalent for the gamble z = <h,p;-h> 3.5. Risk aversion R^ as a function of gain equivalent for the gamble z = <h,p;-h> 3.6. Risk aversion R^ as a function of certainty equivalent or r i s k premium for the gamble z = <h,p;0> 3.7. Risk aversion R as a function of p r o b a b i l i t y equivalent . f o r the gamble z = <h,p;0> 3.8. Risk aversion R as a function of gain equivalent for the gamble 2 = <h,p;0> 4. The R -method A 4.1. The R -method for r i s k averse decision makers A 4.2. The R^-method including r i s k proneness and r i s k neutrality - iii - 5. Mathematical Extensions : 85 5.1. Extended results for u(w;£,a,T,g,n) : 87 5.2. Extended results f o r the r e l a t i v e r i s k aversion ^(w^^) : 93 5.3. Extended results for other gambles : 96 5.4. The results using Taylor series expansions 104 5.5. The sum-of-exponentials u t i l i t y 107 function 5.6. Summary References •' H I : 113 - iv - Acknowledgement The. author Is grateful to his; supervisor, Professor D, Wehrung, for many valuable comments and stimulating discussions throughout the preparation of t h i s thesis-. He would also l i k e to thank the other committee members-, Professors B., Schwab and L.G. Mitten, for their assistance. Business F i n a l l y , suggestions- of Professor J.W. Pratt of Harvard School have been h e l p f u l i n determining knowledge i n the f i e l d . the current state of - 1 - 1 Introduction Decision normative and theory i s o f t e n c l a s s i f i e d under two descriptive. Normative d e c i s i o n theory i s concerned with p r e s c r i p t i o n of courses of a c t i o n that maker's p r e f e r e n c e s . describe the The i n t e r r e l a t e d headings, conform most c l o s e l y to the the decision purpose of d e s c r i p t i v e d e c i s i o n t h e o r y i s to d e c i s i o n maker's p r e f e r e n c e s and preferences into his decisions. he incorporates these In other words, the normative theory guides the d e c i s i o n maker i n what he t h e o r y f o c u s s e s on what the the way should do, whereas the d e c i s i o n maker does do. descriptive Even though both l theories logical are concerned w i t h b e h a v i o r a l a s p e c t s , i t appears t h a t concepts are m a i n l y used when d e s c r i b i n g d e c i s i o n making. approach and The normative theory u s u a l l y i s i n the (and follows psycho- predicting) an human axiomatic l i t e r a t u r e a l s o known under the heading decision 2 analysis. As the main r e s u l t of t h i s r e s e a r c h d e c i s i o n making, our psychological focus w i l l be a s p e c t s w i l l be i s a normative t o o l f o r 3 on d e c i s i o n a n a l y s i s , but covered as we go behavioral along. 1. Reviews o f b e h a v i o r a l d e c i s i o n t h e o r y i n c l u d e Edwardst1954,1961), Luce and SuppesC1965), Becker and M c C l i n t o c k ( 1 9 6 7 ) , S l o v i c and L i c h t e n s t e i n ( 1 9 7 1 ) , Rapoport and W a l l s t e n ( 1 9 7 2 ) , F i s c h e r and Edw a r d s ( 1 9 7 3 ) , and S l o v i c , F i s c h h o f f and L i c h t e n s t e i n ( 1 9 7 7 ) . Most o f t h e s e r e f e r e n c e s are m a i n l y o r i e n t e d towards p s y c h o l o g i c a l f a c e t s . 2. See, f o r example, Keeney(1978): " D e c i s i o n A n a l y s i s p r o v i d e s a n o r m a t i v e t h e o r y which p r e s c r i b e s how a d e c i s i o n maker s h o u l d behave i n o r d e r t o be c o n s i s t e n t w i t h h i s judgments and p r e f e r e n c e s . I t does not p r o v i d e a method f o r d e s c r i b i n g how, i n f a c t , i n d i v i d u a l s do behave." 3. I n t r o d u c t o r y r e a d i n g s i n d e c i s i o n a n a l y s i s i n c l u d e P r a t t , R a i f f a and S c h l a i f e r ( 1 9 6 4 ) , SwaltnC1965), Hammond( 1967), How.ird(1968,1980) and Keeney( 1978). S p e c i a l i s s u e s on d e c i s i o n a n a l y s i s appeared i n IEEE T r a n s a c t i o n s on SSC (see lloward(1968)) and O p e r a t i o n s Research (see K i r k w o o d ( 1 9 8 0 ) ) . Many t e x t books i n d e c i s i o n a n a l y s i s — a t d i f f e r e n t l e v e l s — are a v a i l a b l e , such as R a i f f a ( 1 9 6 8 ) , S c h l a i f e r ( 1 9 6 9 ) , Brown, Kahr and P e t e r s o n t 1 9 7 4 ) , Keeney and R a i f f a ( 1 9 7 6 ) , J o n e s ( 1 9 7 7 ) , L a V a l l e ( 1 9 7 8 ) , and Holloway (1979). and - 2 - Decision analysis i s a d i s c i p l i n e providing a normative, l o g i c a l framework for decision making under uncertainty. Within this framework, the decision maker's preferences for the possible consequences of any action are a key element i n the problem description. analysis i s provided by a set of axioms. we may The logic of decision For the sources of these axioms actually have to go back to Bernoulli(1738) who of maximizing u t i l i t y . set forth the idea However, the axiomization of u t i l i t y was rigorously developed with the work of Von Neumann und In an economic, game-theoretic first Morgenstern(1947). context they postulated the set of axioms which we w i l l adopt i n this research. Essential i s the idea that i f an appropriate u t i l i t y (value) i s assigned to each consequence and the expected u t i l i t y of each alternative i s calculated, then the best course of action i s the alternative with the highest expected u t i l i t y . This so-called expected u t i l i t y hypothesis has been the basis for most of the research i n u t i l i t y or if value theory. The Von Neumann and Morgenstern(1947) u t i l i t y notion was the foundation for the work of Marschak(1950), Savage(1954), Luce and Raiffa (1957), Pratt, R a i f f a and Schlaifer(1965), and Fishburn(1970), who came up with d i f f e r e n t sets of axioms, which a l l prescribe expected u t i l i t y as an appropriate guide for decision making. The axioms of u t i l i t y theory assume that either numerical, objective p r o b a b i l i t i e s e x i s t , or that both (subj e c t i v e ) p r o b a b i l i t i e s and u t i l i t i e s can be j o i n t l y derived. At this point we w i l l not dwell on the problems associated with subjective p r o b a b i l i t i e s , but instead assume that the decision maker knows the likelihoods of the possible consequences. Hence, we w i l l b a s i c a l l y follow Luce and Raiffa(1957) 4. Fishburn(1968) provides a review a r t i c l e on u t i l i t y theory with an extensive bibliography up to 1968. A more recent review i s given by FishburnC1978). - 3- in this respect. The next important step then i s to assess the decision maker's preferences for each of the possible consequences. We have noted that the decision maker's preferences for the various consequences i s an essential ingredient i n the analysis of decisions under uncertainty. These preferences can be represented by u t i l i t y functions i n the sense of Von Neumann and Morgenstern(1947). for Many different evaluating u t i l i t y functions have been proposed. techniques See, for example, t Mosteller and Nogee(1951), Davidson, Suppes and Siegel(1957), Becker, DeGroot and Marschak(1964), Meyer and Pratt(1968), Schlaifer(1969), and Hammond(1974). Basic to most assessment procedures are the well known reference gambles and a c l a s s i f i c a t i o n of attitudes towards r i s k . Schlaifer (1971) published a book of computer programs for decision analysis including various assessment techniques. The preceding discussion together with the 5 many applications of decision analysis indicate the need to have e f f e c t i v e methods for assessing u t i l i t y functions. The major contribution of this research is a suggested procedure for selecting a u t i l i t y function from among a l t e r n a t i v e s . procedure i s the absolute r i s k aversion R.(w) Basic to the new over wealth, introduced by Pratt(1964) and Arrow(1965,1971). independently For each of a number of u t i l i t y functions we w i l l express the absolute r i s k aversion at the wealth l e v e l w , i . e . R,(w ), as a function of: o A o. initial (i) the parameters of a nondegenerate gamble z, and (ii) the certainty equivalent, or the r i s k premium, or the p r o b a b i l i t y equivalent, or the gain equivalent of that gamble. 5 . We m e n t i o n o n l y a f e u c o n t r i b u t i o n s t o t h e p u b l i s h e d l i t e r a t u r e : G r a y s o n f 1 9 6 0 ) , Swalm(1966), S p e t z l e r ( 1 9 6 8 ) , H o w a r d , M a t h e s o n a n d N o r t h ( 1 9 7 2 ) , d e N e u f v i l l e a n d K e e n e y ( 1 9 7 2 ) , a n d Hauser and Urban(1979). - 4 - The u t i l i t y functions include the l i n e a r function and f i v e functions within the well known Hyperbolic Absolute Risk Ayersipn (HARA) c l a s s . After obtaining the decision maker's responses for two or more reference gambles based on ( i ) and ( i i ) , the new procedure (called R.-method) i s then able to select the u t i l i t y function from among the alternatives that conforms most closely to the decision maker's preferences. The R - method seems quite appealing i n several ways. It enables us to use d i f f e r e n t response modes, thus avoiding problems inherent i n using only one response mode . 6 Also, the R^-method has the advantage of requiring knowledge about the decision maker's i n i t i a l endowment only a f t e r responses for the reference gambles have been obtained. In spite of the sometimes i n t r i c a t e mathematical derivations, the r e s u l t s are not complicated hence the R^-method can be applied i n a straightforward manner. and The R - method i s described for a limited class of u t i l i t y functions, but section and Chapter 5 w i l l give various extensions to how one may 4.2, and provide several directions as proceed to enlarge the usefulness of the R^-method. F i n a l l y , the basic idea to express R. = R.(w ) as a function of (i) and A A o ( i i ) seems to be a clear basis for extended research. The outline of the thesis i s as follows. After the necessary notation, d e f i n i t i o n s , lemmas and other preliminaries are given i n Chapter 2, we present i n Chapter 3 our r e s u l t s for R (w ) expressed as function of ( i ) and A (ii). will O The f i r s t section of Chapter 4 w i l l introduce and discuss the R A method, assuming that the decision maker conforms to one averse u t i l i t y function. (unknown) r i s k Section 4.2. w i l l discuss the case where this 6 E x a m p l e s o f s u c h p r o b l e m s c a n be f o u n d i n L i c h t e n s t e i n a n d S l o v i c ( 1 9 7 1 ) , S l o v i c ( 1 9 7 2 ) , s t e i n and S l o v i c < 1 9 7 3 ) , C r e t h e r and P l o t t ( 1 9 7 9 ) , and Kahneman and T v e r s k y ( 1 9 7 9 ) . Lichten- assumption i s violated. Hence, i n t h i s section we w i l l c e r t a i n l y consider r i s k proneness and r i s k n e u t r a l i t y . mathematical extensions and should serve as a d i r e c t i o n for extending the use of the R.-method. A included. Chapter 5 w i l l be a c o l l e c t i o n of At the end of the thesis our reference l i s t i s - 2 6 - Notation and Preliminaries A gamble z, which r e s u l t s i n a net change of h^ w i t h p r o b a b i l i t y (0 < p < 1) and i s denoted by a net change of h z = <h^,p;h2> . e x p l i c i t l y w i t h i n the b r a c k e t s , values p w i t h the complementary p r o b a b a i l i t y 1-p, Note t h a t the p r o b a b i l i t y i s the l i k e l i h o o d that appear w i t h i n the b r a c k e t s . degenerate i f 1 The p, which appears of the f i r s t gamble z i s s a i d of the to be two non- 0 < p < 1 . U s u a l l y gambles are presented either by branch diagrams ( F i g u r e 2.1.) or by c h a r t s ( F i g u r e 2.2.): Figure z The 2.1. gamble presented Figure The z z = <h p;h > l5 2 as a branch diagram. 2.2. gamble presented z = <h^,p;h2 > as a c h a r t , where the shaded area i s a p of the t o t a l fraction area. 1. Wc a s s u m e t h e r e a d e r t o b e q u i t e f a m i l i a r w i t h t h e i d e a s a n d d e f i n i t i o n s g i v e n b y P r a t t ( 1 9 6 4 ) a n d Schlaifer(1969). T h i s c h a p t e r draws h e a v i l y on these r e f e r e n c e s . A v e r y good e x p o s i t i o n i n t h i s c o n t e x t i s p r o v i d e d by Keeney and R a i f f a ( 1 9 7 6 ) . The s p e c i a l case when p = \ i s c a l l e d a f i f t y - f i f t y gamble and w i l l be denoted if by z = <h j ^ h ^ = < ' 1'^2 1 > ' T h e 8 ble a m i t s expected v a l u e i s z e r o , i . e . E ( z ) = 0 . gamble can be r e p r e s e n t e d by <h;-h> 2 *- s s a ^ t t> o e fair Hence, a f a i r f i f t y - f i f t y , which has a s t a n d a r d d e v i a t i o n equal t o h . Let now u(w) be a c o n t i n u o u s , u n i d i m e n s i o n a l , m o n o t o n i c a l l y increasing 2 utility f u n c t i o n over w £ R, where the a t t r i b u t e i s chosen to be wealth. We w i l l not c o n s i d e r d e c r e a s i n g f u n c t i o n s , however the d e r i v e d r e s u l t s be q u i t e s i m i l a r f o r m o n o t o n i c a l l y d e c r e a s i n g u t i l i t y f u n c t i o n s . the d e c i s i o n maker's i n i t i a l will Assume endowment to be w o D e f i n i t i o n 2.1. A c e r t a i n t y e q u i v a l e n t o f a gamble z i s an amount z = z(w , z ) , such t h a t the d e c i s i o n maker i s i n d i f f e r e n t o between z and the amount z f o r s u r e . Hence, z = 2(w o ,z) i s d e f i n e d by u(w +z) = E[u(w + Z ) ] . o o (2.1.) As we have chosen the a t t r i b u t e w to be wealth, we may substitute cash 3 equivalent of for certainty equivalent. Note t h a t , of course, z i s a f u n c t i o n the gamble z i t s e l f , but a l s o o f the i n i t i a l endowment w . In other words, i t i s assumed t h a t the d e c i s i o n maker may behave d i f f e r e n t l y at 2. T h e c o n c e p t s are equally 3. C a s h e q u i v a l e n t s valid f o r nonmonetary a t t r i b u t e s . are extensively discussed i n L a V a l l e t 1968). - various wealth l e v e l s . 8 - Except for the case of a linear u t i l i t y function, a decision maker i s assumed to state d i f f e r e n t cash equivalents at different i n i t i a l wealth positions for the same nondegenerate gamble. Let us i l l u s t r a t e this point with the following example. Example 2.1. A decision maker i s assumed to behave according to the u t i l i t y 2 function the fair u^(w) = w-(.25)w , where f i f t y - f i f t y gamble z = 0 ^ w ^ 1.5 . <0.5;-0.5> calculate that at an i n i t i a l wealth of i s z(w',z) « -0.081, as i t holds that o Suppose he i s faced with . Using (2.1.) i t i s easy to w^ = 0.5 h i s certainty equivalent u(0.5-0.081) « J>u(0) + ^u(l) . Thus, the decision maker i s i n d i f f e r e n t between the gamble and losing a sure amount of ,.0.081 . At an i n i t i a l wealth of is 2(w",z) w -0.118, as o w^ = 1 his certainty equivalent u(l-0.118) « %u(0.5) > %u(l.5) . Hence, z(w',£) o f 2(w",z) . o D e f i n i t i o n 2.2. The r i s k premium TT = TT(W , z) of a gamble z i s i t s expected t Q value minus i t s certainty equivalent. Hence, TT = TT(W . z) o (2.2.) i s defined by TT = E(z) - 2 , From (2.2.) we see that z = E(z) - TT , which after substitution i n (2.1.) results i n the following fundamental relationship: (2.3'.) u[w + E(z~) o where z = 2(w ,z) . o - TT] = E[u(w +z) ] o - 9 - or, more formally (2.3".) uLw o + E(a) - TT(W ,Z)] = E[u(w + 2 ) ] o o which reduces for a f a i r gamble to (2.4.) utw o Example 2.2. - TT(W ,z)] = Etu(w +z)] . o o Referring to Example 2.1. i t can be calculated, that, using (2.3.), the decision maker's r i s k premium at the i n i t i a l wealth level of w 1 o ly, = 0.5 i s TT( ',z) « 0.081, as o w ^(w^z) ^ 0.118 . u(0.5+0-0.081) « ^u(0) + ^ u ( l ) . Similar- Of course, these results could have been obtained more d i r e c t l y i n this case. Clearly, for a f a i r gamble (2.2.) becomes '"' = -£, which proves the following lemma. Lemma 2.1. The r i s k premium of a f a i r gamble z i s the negative of the certainty equivalent of that gamble. By D e f i n i t i o n 2.2. we have f o r any constant c (2.5.) TT(W ,z) = "T(w +c,z-c) o . o It i s often feasible and i l l u s t r a t i v e to present a gambling situation with the corresponding r i s k premium i n a graph of the u t i l i t y function. This point w i l l be explained through the following example. - 10 - Example 2.3. Figure 2.3. provides a picture of the u t i l i t y function 2 u^(w) = w-(.25)w (0 £ w ^ 1.5) as introduced i n Example 2.1. We w i l l leave i t for the reader to v e r i f y the resulting graph, as presented for w" = 1. u. (w) *• w Figure 2.3. 2 A decision maker with u (w) = w-(.25)w , facing a f a i r f i f t y - f i f t y gamble z = <0.5;-0.5> states for h i s r i s k premium IT = TT (w ,z) = 0.118 at h i s i n i t i a l wealth level w" = 1 . D e f i n i t i o n 2.3. The insurance premium u =y ( >z) of a gamble z i s the w Q negative of the certainty equivalent of that gamble. Hence, y (w ,"z) i s defined by - 11 - (2.6.) u(w ,"2;) = - z(w ,z) . o o From Lemma 2.1. we derive, that for a f a i r gamble the r i s k and insurance premium coincide. D e f i n i t i o n 2.4. An amount h such that the decision maker i s indifferent between the status quo (his i n i t i a l wealth w ) and the o gamble z = <h,p;-h> i s called the gamble's gain equivalent and i s denoted by Hence, h = h(w ,p,h) . h = h(w ,p,h) i s defined by Q u(w ) = E[u(w +z)] = p.u(w +h) + (l-p).u(w -h) . o o o o (2.7.) In order to have the expression gain equivalent make sense, we w i l l obviousl y require h to be positive. D e f i n i t i o n 2.5. A probability p .such that the decision maker i s i n d i f f e rent between the status quo w and the gamble is called the gamble's probability equivalent and i s denoted by p = p(w ,h) . o Hence, (2.8.) z = <h,p;-h> p = p(w ,h) i s defined by u(w ) = E[u(w +z)] = p.u(w +h) + (l-p).u(w -h) o o o o - 12 - U t i l i t y functions that have i d e n t i c a l implications for action are called s t r a t e g i c a l l y equivalent. This notion can be formalized through the following d e f i n i t i o n . D e f i n i t i o n 2.6. Two u t i l i t y functions, u^(w) and u^Cw), are s t r a t e g i c a l l y equivalent, written —* u,(w) ^ u„(w), i f there exist con1 2 stants a and $>0 such that u^(w) = a + B.u^Cw) . It i s now straightforward to show that s t r a t e g i c a l l y equivalent utility functions imply the same preference ranking for any two gambles. Also, i f u^(w) ^ u^(w), then both u^(w) and u^(w) give the same cash equivalents for a gamble. For example, l e t u (w) = a + B.u (w) (2.9.) 1 and consider z = (2.10.) where 2 <h^,p;h2> . 8>0 Then, we have by (2.1.) u,(w +z,) = p.u. (w +h_) + (1-p).u. (w +h ) l o l 1 o 1 1 o 2 r r and (2.11.) u.(w +z„) = p.u.(w +h.) + (l-p).u (w +h ) 0 l o l I o l l so that substitution of (2.9.) i n (2.10.) yields p.[ a + 3.u„(w +h,)] + (1-p).[a + B.u„(w +h„)] result: o \ l o l o l a + r3.u (w +2^) = 2 which gives the following - 13 - u„(w + z ) = p.u„(w + h ) + ( l - p ) . i i (w +h ) . (2.12.) 2 o l Hence, equating 2 o l 2 o 2 (2.11.) and (2.12.), we find that z, - 2„ . If u^(w) and u^(w) are s t r a t e g i c a l l y equivalent, we can also say that u^(w) determines u^(w) up to p o s i t i v e linear transformations. It w i l l appear to be quite important to categorize the attitude of a decision maker with respect to r i s k . Or, the question arises whether a decision maker i s a r i s k lover or a r i s k avoider. This i s not to say that a decision maker w i l l always exhibit the same r i s k attitude over the entire region. In fact, many researchers believe that often decision makers are r i s k averse for gains while being r i s k prone for negative outcomes. For theoretical purposes, however, D e f i n i t i o n 2.7. assumes ho change of r i s k attitude over the entire domain. D e f i n i t i o n 2.7. A decision maker i s said to be r i s k averse i f u[E(w +z)] > E[u(w +£)], o o r i s k neutral i f u[E(w +z)] = E[u(w +z)], and o o r i s k prone i f u[E(w +z)] < E[u(w +z)] o o for a l l nondegenerate gambles z . This research w i l l focus on the r i s k averse and r i s k neutral decision maker. A r i s k averse decision maker w i l l prefer the expected consequence of any nondegenerate gamble to the gamble i t s e l f . the most common s i t u a t i o n i n practice. And i t i s believed that this i s The r i s k neutral case w i l l be i n - U. S e e , f o r e x a m p l e , S w a l r a C 1 9 6 6 ) a n d K a h n e m a n a n d T v e r s k y ( 1 9 7 9 ) . - 14 \ eluded f o r completeness sake. The r i s k prone decision maker i s not con- sidered here, even though our results can quite e a s i l y be extended i n that direction. Lemma 2.2. A decision maker i s r i s k averse i f and only i f his u t i l i t y function i s concave. The proof of Lemma 2.2. can be found i n Keeney and Raiffa(1976). Note that the u t i l i t y function i n Example 2.1. i s concave over the defined range. Therefore, this decision maker i s r i s k averse. And indeed, consistent with D e f i n i t i o n 2.7., i t holds that choosing w = 0.5 o 1 u[E(w'+z)] > E[u(w'+z)]. For example, o o and z = <0.5.-0.5> , we have u[E(w'+z)] = u[E(w')] = o o 7/16 > 6/16 = JJU(W'-0.5) + %u(w*+0.5) = E[u(w'+z)] . o o o Lemma 2.3. A decision maker with an increasing u t i l i t y function i s r i s k averse i f and only i f for any nondegenerate gamble 1 i t holds that: (2.13.) E(z) > 2 . For the proof, we f i r s t assume r i s k aversion, so that by D e f i n i t i o n 2.7. we have u[E(w +z)] > E[u(w +z)] o o . Using (2.1.) we see that ULE(W o +Z)] u(w +2), which for an increasing function y i e l d s o (2.14.) Noting that E(w +z) > w +2 . o o E(w +z) = w + E(z) , we see that (2.13.) follows d i r e c t l y from - 15 - (2.14.). Second, assuming (2.13.) gives d i r e c t l y (2.14.), so that U[E(W +Z)] Q > u(w +z) = E[u(w +z)], as the u t i l i t y function i s increasing. o o / decision maker i s r i s k averse. Hence, the This completes the proof of Lemma 2.3. • Using Lemma 2.3. and D e f i n i t i o n 2.2. we e a s i l y arrive at the following Lemma 2.4. A decision maker with an increasing u t i l i t y function i s r i s k averse i f and only i f h i s r i s k premium i s positive for a l l nondegenerate gambles. For the r i s k neutral decision maker similar lemmas can be derived. We w i l l state these lemmas without proofs as these proofs are similar to those of the preceding lemmas. Lemma 2.5. A decision maker with an increasing u t i l i t y function i s r i s k neutral i f and only i f for any nondegenerate gamble z i t holds that (2.15.) E(z) = z . Lemma 2.6. A decision maker with an increasing u t i l i t y function i s r i s k neutral i f and only i f h i s r i s k premium i s zero for a l l nondegenerate gambles. Now assume u(w) to be l i n e a r , i . e . u(w) = a + Bw ^ w , where 6 > 0 . We see that E[u(w +z)] = Eta + B ( w + z)] = a + 8[w + E ( z ) ] and o o o u[E(w +z)] = u[w + E(z)] = a + B t w + E(z)] , so that o o o u[E(w +£)] = E[u(w + Z ) L o o tl - 16 - Hence, according to D e f i n i t i o n 2.7., the decision maker i s r i s k neutral, which proves the following lemma. Lemma 2.7. A decision maker with a linear u t i l i t y u(w) We or not. are now (8 = a + $w > 0) function i s r i s k neutral. able to investigate whether a decision maker i s risk averse What we would l i k e to do next i s to state a measure of r i s k aversion so that an indication i s provided when one decision maker is more r i s k averse than another. The d e f i n i t i o n we are about to give here stems from Pratt (1964) and Arrow(1965,1971), and i t assumes u(w) i s continuous and twice differentiable. D e f i n i t i o n 2.8. The ^ — ^ — — — — i ^ — — — — — absolute r i s k aversion at w, R (w), of a u t i l i t y • • function u(w), (2.16.) R A ( W A i s defined by: ) . _ « ^ L . u'(w) The absolute r i s k aversion R (w) i s a function of w, A to be a measure with many desirable properties. and i t c e r t a i n l y seems Note that u'(w) positive for monotonically increasing functions and that u"(w) i s always i s negative for concave functions, so that R (w) i s positive for r i s k averse decision makers. S i m i l a r l y , i t can be shown that R.(w) = 0 for r i s k neutral decision A makers and R^(w) Two < 0 for r i s k prone decision makers. related lemmas follow: - 17 - Lemma 2 . 8 . Two s t r a t e g i c a l l y e q u i v a l e n t u t i l i t y f u n c t i o n s have the same a b s o l u t e r i s k a v e r s i o n R,(w) A Proof: For u^(w) = a + Bu^Cw) , where B > 0 , we have ui'Cw) (2.17.) R A 1 Lemma 2 . 9 . If = Bu'(w) then they are u"(w) d — R.(w) = , , . = —[log u'(w)] A u (w) dw after -JR u2(w) A ( w ). ' Z strategically J — R^( w ) dw = l o g u'(w) + risk equivalent, so that by i n t e g r a t i o n e of (w)dw (with C. 1 , which (2.18.) gives J e C = A A 1 (with C (w)dw -JR e dw = u'(w) . an i n t e g r a t i o n e C 1 Now, R.(w) determines u(w) up to p o s i t i v e u'(w) u(w) = a + Btje constant) + C0 l i n e a r transformations, that (2.20.) = R exponentiating (2.18) Integration u''(w) = - JL two u t i l i t y f u n c t i o n s have the same a b s o l u t e b e i n g an i n t e g r a t i o n c o n s t a n t ) (2.19.) S uj(w) a v e r s i o n R^(w), yields Bu"(w) (w)=--^ A > 1 Proof: . —JR (w)dw A dw] by n o t i n g - 18 -c where a = — -c and 8 = e e > 0 are two constants. Finally, i f R ,(w) = R „(w), then i t follows from (2.20.) and Definition A,l A,2 the corresponding u^(w) and u^(w) are s t r a t e g i c a l l y 2.6., that equivalent, i . e . u^(w) ^ u (w) . • 2 Related to the absolute r i s k aversion R,(w) are two other measures of A r i s k aversion. One i s the r e l a t i v e r i s k aversion R (w) as defined by Pratt R (1964) and Arrow(1965,1971); the other i s the p a r t i a l r e l a t i v e r i s k aversion - 5 Rp(w;w ), as introduced by Menezes and Hanson(1970) . We w i l l now provide o the definitions Definition of R„(w) and R (w;w ) . R P o 2.9. The r e l a t i v e r i s k aversion at w, R (w), of a u t i l i t y R function u(w), i s defined by n (2.21.) / U"(W) N R (w) = - w u'(w) R Definition 2.10. The p a r t i a l r e l a t i v e r i s k aversion R (w;w ) of a u t i l i t y p Q function u(w), i s defined by (2.22.) R (w;w ) = — w y ° where w o u"(w + w) o u'(w + w) i s some fixed wealth level, Zeckhauser and Keeler(1970) established a direct relationship between R (w) R (w), and R (w;w ), which we w i l l present through the following lemma. R P o 5. T h e s e a r e n o t t h e o n l y m e a s u r e s o f r i s k a v e r s i o n . Especially i n the finance l i t e r a t u r e different m e a s u r e s o f r i s k a v e r s i o n h a v e b e e n i n t r o d u c e d , d i s c u s s e d , a n d a p p l i e d ,• e. g. Y a a r i ( 1 9 6 9 ) , M a y s l i a r (1975), M i 1 1 f r ( 1975), K u b i n s t e i n ( 1 9 7 6 ) , and K a l l b e r g and Ziemba(1978). A - 19 - Lemma 2.10. R^Cwjw ) = R_(w P o R o + w) — w .R.(w + w) . 0 A 0 The proof of Lemma 2.10. i s obvious. Let us now i l l u s t r a t e the idea of a r i s k aversion measure with the following example. Example 2.4. Consider two decision makers with two different functions: u^w) = w - (l/4)w utility (see Examples 2.1. to 2.3.) and u (w) = 2 2 w - (l/3)w , both defined over the range 0 <^ w £ 3/2 . Using D e f i n i t i o n 2.8. we calculate R, . (w) = ~ — and R „(w) = -r-nr , so that R (w) A,l 2 - w A,2 3/2 - w A,l < R „(w) when 0 < w < 3/2 . Thus, over the defined wealth range, the A,2 f i r s t decision maker i s less r i s k averse than the second decision maker. But what does this actually mean? The answer i s that one decision maker i s less r i s k averse than another implies that for a l l nondegenerate gambles within the defined wealth range his r i s k premium ois smaller than that of the other decision maker. For example, assuming an i n i t i a l wealth level of w" = 1 , we have R, .(w") = 1 < 2 = R. _(w") . Accordingly, for the gamble o A,l o A,2 o z = <h:-h> we have TT,(W",Z) « 0.118 , which i s less than 1 o TT (w",z) « 0.207 . I o . In fact, we introduced i n Example 2.4. some kind of comparative r i s k aversion. The following lemma has actually been i l l u s t r a t e d i n the example. The proof of Lemma 2.11. can be found i n Pratt(1964). Lemma 2.11. v I t holds that TT^W.Z) < TT^CW.Z) for a l l w and z, i f and only i f R (w) < R _(w) for a l l w . A,l >^ A - 20 - In order to make e f f e c t i v e use of the absolute r i s k aversion R (w) A we state another important lemma: Lemma 2.12. R (w) i s positive (negative) for a l l w i f and only i f the A decision maker i s r i s k averse (risk prone). Proof: We show the v a l i d i t y of the r i s k averse case. The proof for the r i s k prone case uses similar arguments and i s therefore omitted. now ^ ( ) R u"(w) w > Assume 0 . Since u'(w) > 0 , because u(w) i s increasing, we have = - R (w).u'(w) < 0 , so that u(w) i s concave. follows that the decision maker i s r i s k averse. Using Lemma 2.2. i t The converse i s straight- forward and has been stated e a r l i e r i n this chapter. In addition, i t i s clear, using Lemma 2.7., Lemma 2.13. I f R.(w) = 0 | that the following lemma holds: , the decision maker i s r i s k neutral. The converses of Lemma 2.7. and Lemma 2.13. can be proved, though the proofs w i l l be lengthy and not quite relevant. Hence, this i s not discussed here. We have seen, that TT(W ,z) i s a function of both w and the gamble z . o o Now we would l i k e to discuss what happens to TT(W ,Z) as w varies. I t i s o o believed that many decision makers w i l l pay a smaller r i s k premium as their ( i n i t i a l ) wealth increases. The reason being that one can better afford to take a certain r i s k as one becomes richer. Of course, this i s not a general - 21 - rule, i t only provides motivation decreasingly, to investigate whether a decision maker i s constantly, or increasingly r i s k averse. Pratt(1964) provides several d e f i n i t i o n s and theorems i n that d i r e c t i o n , the major d e f i n i t i o n being: , D e f i n i t i o n 2.11. A r i s k averse decision maker i s decreasingly (constantly; increasingly) r i s k averse i f w(w^,z) for any gamble z decreases ( i s constant; decreases) as w increases. o Menezes and Hanson(1970) extended Pratt's theorems, while stating and proving the following lemma. Lemma 2.14. Let the r i s k premium for a gamble z be TT(W , Z ) and A be l t i p l i c a t i v e factor such that P(w +Az<0) = 0 , then: a multip (2.24.) (2.25.) (2.26.) | — [TT(W , 2 ) ] dw o 8_ TT(W J A Z ) o 8A 9_ =0 (i) < ^ 0 < (ii) TT(AW , A Z ) o 8A ^ 0 < (iii) i f the corresponding function ( i ) R.(w ) , ( i i ) A Rp(w ;w ) , o Q (iii) R^^) constant, or decreasing O *- respectively increasing, s in w . - 22 - A very useful application of the absolute r i s k aversion R^(w) and the r e l a t i v e r i s k aversion R (w) i s provided by.the following two lemmas. R Lemma 2.15. I f the absolute r i s k aversion i s constant, say R^( ) w > = c then: (2.27.) u(w) ^ w i f and only i f R (w) = 0 , A (2.28.) u(w) ^ - e ~ A C W i f and only i f R,(w) = c > 0 . A u" (w) Assume f i r s t u(w) ^ w , so that R (w) = - ,, ( = 0 . Conversely, A u (w) Proof: A 1 assuming R.(w) = 0, (2.20.) gives u(w) = a + Bw , where 6 > 0, so that u(w) ^ w . This proves (2.27.). Now assume u(w) ^ - e C W , so that R^( ) w = u"(w) - ^ , = a - 8e C W c . Conversely, assuming ^ ( ) = c > 0, (2.20.) gives u(w) = R w , where 8 > 0, so that u(w) 'v* - e C W . This proves (2.28.) and completes the proof of Lemma 2.15. Lemma 2.16. I f the r e l a t i v e r i s k aversion i s constant, say R (w) = c , t) then: (2.29.) u(w) % w 1 _ C i f and only i f 0 < R (w) = c < 1 , D R (2.30.) u(w) ^ log w i f and only i f R (w) = c = 1 , D R (2.31.) u(w) a- - w 1-c i f and only i f R (w) = c > 1 . The proof of Lemma 2.16. i s similar D R to the proof of Lemma 2.15. and can be found i n Pratt(1964). We w i l l f i n i s h this chapter with the following example: - 23 - Example 2.5. Following Lemma 2.14. we see that with (2.24.) i f R.(w ) i s constant i n w , so that A o o (2.32.) TT(W',2) = TT(W",2) 0 where 0 o w 1 o i f R.(w ) i s constant as w varies. A o o that (2.32.) holds i f u(w) ^ - e ° 3 ~ |_[Tr(w ,2) ]= 0 o W f w" o Then, according to Lemma 2.15. we have . The l a t t e r i s graphically shown i n Figure 2.4. below. u(w) o Figure 2.4. A decision maker with u(w) a, - e w i l l state the same r i s k premium for a nondegenerate gamble z at any ( i n i t i a l ) wealth l e v e l . C W - 24 - 3 Mathematical T h i s chapter Results w i l l derive ^ ( w 0 ^> for R (w ) A Q b r i e f l y denoted by R^, as a function of: (i) the parameters o f a nondegenerate gamble z; (ii) the c e r t a i n t y e q u i v a l e n t , or the r i s k premium, or the p r o b a b i l i t y e q u i v a l e n t , or the g a i n e q u i v a l e n t Some o f the d e r i v e d R^(WQ) f u n c t i o n s are not i n terms of ( i ) and of p and ( i i ) ; e.g. the parameters of We will introduce f i r s t discuss gives the gamble. implicit functions of i s i m p l i c i t l y expressed i n terms z. the i d e a behind the mathematical r e s u l t s , a g e n e r a l i z e d c l a s s of u t i l i t y t h i s chapter of that e x p l i c i t but R^ and function R functions, while w i t h i n a subclass then the remainder of of the generalized A c l a s s of u t i l i t y r e s u l t s by functions. i n t r o d u c i n g a new Chapter 4 w i l l g i v e an a p p l i c a t i o n of method f o r s e l e c t i n g a u t i l i t y mathematical r e s u l t s are extended i n Chapter 5, results f o r other u t i l i t y functions. \ the function. f o r i n s t a n c e by g i v i n g The - 25 - 3.1. The idea behind the mathematical results. Let u(w) be a continuous, twice d i f f e r e n t i a b l e , and increasing u t i l i t y function over wealth w. r i s k aversion R.(w) Pratt(1964) then interprets the absolute by considering a decision maker's r i s k premium for a A small, f a i r gamble z. variance a 2 Thus, l e t z be a gamble with E(z) = 0 and 2 . We w i l l now = O . z (2.4.), which gives expand u(w o ) around w o small on both sides of 2 (3.1.) u(w -TT) = u(w o ) - TTU'(W ) + J - o o u"(w) - 2 and ~2 (3.2.) E[u(w + 5 ) ] = E[u(w o ) + zu'(w o ) + o u"(w 2 ) + ] o 2 = u(w Equating ) + ~ o 2 u"(w ) + o (3.1.) and (3.2.) and neglecting higher-order terms (as we are actually considering i n f i n i t e s i m a l gambles) gives us (3.3.) u(w ) - TTU'(W o o ) o (=> u(w ) + %a u"(w ) o 2 from which i t follows that (w)«4 H (3.4.) R sR A i.e. R A ° a V- 2TT(W 2 G ~ ,Z) 2 i s twice the r i s k premium per unit of variance for i n f i n i t e s i m a l - 26 - gambles. S i m i l a r l y , Pratt(1964) shows, that for z being a nonfair gamble, i t holds that (3.5.) 2TT(W ,Z) o R.[w +E(z")] « a- 2 z and for z = <h;-h> we have (3.6.) R s R ( ) « |(2p - 1) A A o h where p = p(w ,h), o W and i n both cases z i s assumed to be i n f i n i t e s i m a l . Pratt's interpretation of the absolute r i s k aversion i s certainly useful, which can be i l l u s t r a t e d through the following example. Example 3.1. The decision makers i n Example 2.3. were assumed to behave according to the u t i l i t y functions u^ = w - (l/4)w (0 <^ w <^ 3/2) respectively. that R ^w") A,lo 2 and u^ w - (l/3)w = At an i n i t i a l wealth of w^ = 1 i t was shown = 1, R (w") = 2, TT (w",z) = 0.118 and TT (W",5) = 0.207, A,2o l o 2 o 0 0 where z = <0.5;-0.5>. Using (3.4.) we can estimate R by R where 2TT (w"s2) (3.7.) R, .(w") = A, 1 o 1 ° 2 z = 0.944 (R. . = 1) A, 1 1.656 (R and A ft ^ ( 3 - 8 , ; 6 A,2(w") = R 2 2TT (w" 5 ) —^—2 2 = a~ z A A „ = 2). '2 E s p e c i a l l y i n the l a t t e r case the estimation error seems quite large. - 27 - We can formalize the estimation error by the following D e f i n i t i o n 3.1. d e f i n i t i on. The R -percentage error i s defined by A . R (w ) - R (w ) A o A o A (3.9.) R % ESTIMATION ERROR = A A .100 R.(w ) A o An i l l u s t r a t i o n i s given by the following example. Example 3.2. Referring to Example 3.1. we find that by (3.9.) the R ,% ESTIMATION ERROR = 5.6 and the R. J„ ESTIMATION ERROR = 17.2 A,l A,2 Altogether, using Pratt's interpretation of R as a formula i s limited A in the sense that the formula i s an approximation and i t requires ranged gambles. function. small- On the other hand, the formula applies to any u t i l i t y The question now arises whether we are able to come up with exact results and not only approximations or estimations. The answer has c l e a r l y proven to be 'yes', however at the price of specifying the shape of the u t i l i t y function (not i t s essential parameter). This has led to the f i r s t R^ as a function of the r i s k premium and the parameter of a f a i r THEOREM 3.1. For a decision maker with a quadratic u t i l i t y gamble: function u(w) ^ w - cw , where c > 0 and w < — , i t holds that 2c 2TT (3.10.) R A 2 2 where TT = Tr(w ,z) and z a f a i r gamble with standard o deviation a = 0~. - 28 - Proof: Noting that u u(w -TT) = u(w ) o o (m) TTU'(W = 0 for m = 3,4, , (3.1.) becomes , m dw 2 ~ %TT u"(w ) and (3.2.) becomes E[u(w +z)] = o o (w) = ) + o d u ( w ) 2 u(w^) + ho~ u"(w ) . Equating the last two equations according to (2.4.) o 2 2 - im'Cw ) + %TT U"(W ) = %C~ u"(w ), from which the desired result o o z o gives (3.10.) follows d i r e c t l y . Notice that i f one uses (3.4.) to estimate for the u t i l i t y function u(w) ^ w - cw , the R % ESTIMATION ERROR becomes A 2 2TT R.% ESTIMATION ERROR = A a 2 2TT 2 - TT a a2 2TT 2 2 .100 = ^ - (100), 2 where a a 2 = a~ . z 2 - TT Example 3.3. Using the above expression, we are now able to check the results i n Example 3.2.: The R^ ^% ESTIMATION ERROR = the R % ESTIMATION ERROR = A,2 0 ^'^VX = 17.14. (.1/4; ( 11 ft) 2 =5.57 and The differences are due to rounding-off errors. Theorem 3.1. has provided the stimulus f o r a search for similar results with other u t i l i t y functions. considered i s addressed The question of which functions can be i n the following section. - 29 - 3.2. A generalized class of u t i l i t y In this section we w i l l functions. introduce a generalized class of u t i l i t y functions. We w i l l THEOREM 3.2. f i r s t state: I f U(w) and u(w) have absolute r i s k aversion R (w) A,U and R n for f p K £I22E- some d i f f e r e n t i a b l e function f(w) ^ 0, i t holds R „(w) = R (w) - ^ T T • A,U A,u f(w) (3.11.) T. (w) respectively, then i f U'(w) = f(w).u'(w) j U A y ( ^ = - " U w ; ( w ) = _ f(w).u"(w) + f'(w).u'(w) _ _ u"(w) _ f'(w f(w).u'(w) u'(w) f(w u'(w) - » ( ^ = R. (w) A,u f'(w) —s- . f(w) Now l e t us define a generalized class of u t i l i t y functions u(w) by (3.12.) u'(w) ^ f(w), r Bwa + —- n 6w a with w > 0, B > 0, a > 0, T + 0, + n > 0, and T) = 1 i f x -»• °°. Using Theorem 3.2. i t i s easy to v e r i f y that the absolute r i s k aversion R.(w) corresponding to (3.12.) i s : (3.13.) p ( , _ u"(w) _ aBw 3 R A (w) - - ~ T f'(w) ^ +n - 30 - we define u(w) = u ( w ; £ , a , T , B , T | ) b y Choosing f(w) = u*(w) = u ' C w j ^ . a . T . B . n ) ~ w (3.14.) ? According : (3.15.) R(w) A 5 (3.16.) +n !BwC R (w) E, = 0, we o b t a i n a to (3.14.) i s by D e f i n i t i o n 2.9, - e. R Selecting +n class of u t i l i t y functions u(w;0,a,T,8>n) as by r (3.17.) u' ( w ; 0 , a , T , 8 , r i ) ^ Bwa — ^ + n Multiplying the right hand side of (3.17.) by (3.18.) corresponding w Bwc 5W _ TIT - +n a-l _ aBw' The r e l a t i v e r i s k aversion corresponding c -T to (3.13.) we have for the absolute r i s k aversion to u ( w ; £ a , T , B 5 r i ) defined 'Bwa T and substituting gives u ' ( w ) 'V/ ( w 3 + c) T so that both (3.14.) and (3.17.) generalize the class of u t i l i t y functions - 31 - (3.18.), which class was proposed by Pratt(1964). Integration of (3.17.) for a = 1 gives l-T (3.19.) u(w;0,l,T,B,n) ^ (T B(1-T)| T t 1) which i s the well known Hyperbolic Absolute Risk Aversion (HARA-) class of u t i l i t y functions as defined by Merton(1971). Hence, (3.14.) and (3.17.) are also generalizations of the HARA-class. Not for a l l functions within the HARA-class we need to use a l l parameters e x p l i c i t l y , for some i t w i l l TIT suffice to substitute c = —^ , so that by (3.15.) we have (3.20.) R A ( ) = _ L _ w +c w which d i r e c t l y explains the name of the HARA-class. We w i l l now continue to give c r i t e r i a for (3.14.) to be s t r i c t l y decreasingly r i s k averse. THEOREM 3.3. The class of u t i l i t y functions u(w;£,a,T,6,n) as defined by (3.14.) i s s t r i c t l y decreasingly r i s k averse i f : (3.21.) w > d, T > 0 and f i n i t e , and 0 <. £ <_ xa, where (3.22.) d = < max 0 where - ^ 8 ' (3(xa-0 - 32 - 2? i(a-l) + (3.23.) K = 26 According n 2 2 h T ; - a T Proof: ^\ ^ M\ a ((a-1) +.a {—M 2 - to Lemma 2.12. u(w; E,, a, T , 3 >n) i s r i s k averse i f ^ ( ) i s R w p o s i t i v e , so that with (3.15.) we have aBw a-1 (3.24.) Bw + ^ w > 0 n which gives (3.25.) V. w> K ± when B(Ta-c-) Ta. Furthermore, by D e f i n i t i o n 2.11. and Lemma 2.14. i t follows that the u t i l i t y function u(w;E,,a,T,B,r|) i s s t r i c t l y decreasingly r i s k averse when i n addition i t holds that ^-[R,(w)] < 0, or: dw A fBw a 1 a(a-l)(Bwa (3.26.) We w i l l 2 ) - Bw r o apw • T a - 1 i (aBw a V w solve (3.26.) for w by finding the roots of — [R.(w)] = 0 , i . e . the • > ° dw A l e f t hand side of (3.26.) set equal to zero. We now multiply the LHS of 2 3 (3.26.) by -|— and set this equal to zero. find a quadratic equation i n w , namely After reordering the terms, we - 33 - (3.27.) (f - a)|-(w ) a + 2 [ 1 | + a(a-l)]$nw + fy\ a 2 = 0 which gives the following two roots for w c a(a-l) + (3.28.) w1 ± n a (a-l) T 2 23. i T 1% + a .-^ = K. a T Note that 'the LHS of (3.27.) has a negative l i m i t i n g value for w ->- , as r ft (— - ) — < 0 for 6,T > 0 and 0 < £ < T a . Assume now n > 0, so that 00 2 a T K - T > K = for 8,T > 0 , as — ( - - a) < 0 + T T (3.27.) i s negative for w 3 > K and w and < K 3 ,Va , w > [K_] i s one solution of (3.26.). 2, .»2 2 4^1 Va (a-l) + a • — [a(a-l) + % T 2 & < a. i n this case, so that w 3 • ^f(ax-C)! a k > a(a-l) + = > K_ w i l l > 0 which i s included i n (3.21.) and (3.22.). w > [K ] ^ 3 and the i n i t i a l conditions •^ — + n > 0 W T > where again c = . 2E T or <_ 0, as for always be overruled (or: w > [- ^ ] ^ a ) and p (3.21.) to (3.23.) have been obtained. For u(w;0,a,T,6>i~|) i t I (3.29.) > K Combining now (3.25.), T w > 0, the conditions 3 Assuming n < 0 gives K_ £ 0 <. £ < T a , so that for n <_ 0 the condition w by w Hence, the LHS of T 0 and s • easy to see that (3.21.) to (3.23.) become w > max[0, -c, c ( a - l ) ] The conditions V* (3.29.) were e a r l i e r derived by Pratt(1964) P for his class of u t i l i t y functions averse. (3.18.) to be s t r i c t l y decreasingly risk - 34 - For u(w;0,l,T,B,n) we find that (3.21.) to (3.23.) reduce to: (3.30.) T > 0 and w > max 0 - 21 6 The remainder of this chapter w i l l be r e s t r i c t e d to the function R A I for u(w;0,l , T , 8 , T ) ) , where T = -1, \, 1, 2 and u(w;0,0,T,8,ri). twofold. 00 respectively, and for The reason behind this selection of u t i l i t y functions i s F i r s t , for these u t i l i t y functions a l l desired functions R be obtained, so that a very complete presentation results. could Second, for the purpose of the new assessment procedure, as discussed i n Chapter 4, these functions s u f f i c e i n explaining and i l l u s t r a t i n g the method. However, i n Chapter 5 we w i l l give extended results for other functions within the class defined by (3.17.). Table 3.1.* l i s t s the above six u t i l i t y functions. derivation of the contents of Table 3.1. can be found i n Pratt(1964) and Merton(1971). The Table 3.1. Utility Function: Name: E x p l i c i t Expression with FIT c = -5— ( E x p o n e n t i a l : c = 6 ) P Absolute r i s k as a f u n c t i o n o f w : u(w;0,0,T,8,n) linear w constant u(w;0,1,-1,8,n) quadratic - (w + c) increasing u(w;0,l,%,B,n) square r o o t V decreasing u(w;0,l,l,8,n) logarithmic log u(w;0,l,2,8,n) hyperbolic u(w;0,l,°°,c,l) exponential w + c (w + c) 1 w + c - e -cw aversion decreasing decreasing constant - 36 - 3.3. Risk aversion as a f u n c t i o n o f c e r t a i n t y e q u i v a l e n t or r i s k premium f o r the gamble z = <h,p;-h>. Consider the gamble z = <h,p;-h>, where E ( z ) = h(2p-l) : + h F i g u r e 3.1. (3.31.) z = <h,p;-h> (1-p)^^- h so that by (2.1.) (3.32.) u(w +z) = p.u(w +h) + (l-p).u(w -h) o o o Hence, f o r u(w;0,1,T,6,n) w i t h T f 1, (3.32.) r e s u l t s i n : rS(w +z) l-T +n 3(1-T) (3.33.) 6(w +h) o B(l-x) Using (3.20.) and f u r t h e r r e d u c i n g hR.l-T hR^l-T A l 1+ l-T +n +(l-p) (3.33.) p r o v i d e s f o r T t 1 zR l-T (3.34.) 6(w -h) l-T +n = P 1+ + (1-p) 1 - S u b s t i t u t i o n o f z = h(2p-l) - TT p r o v i d e s e a s i l y a r e l a t i o n s h i p c o n t a i n i n - 37 - TT = TT(W ,Z). O R as a function of z and TT for u(w;0,l , x , B , n ) with x ^ 1 A can be derived from (3.34.). The function R for u(w;0,1,1,8,n) and A u(w;0,0,x,6,Tl) w i l l be derived d i f f e r e n t l y . We continue by specifying the functions R^ for the u t i l i t y functions of Table 3.1. Again, we w i l l denote R (w ) by R . A o A A 1 . J For u(w;0,1,-1,8,11) ^ - (w+c) , with c = - the relationship (3.34.) p becomes: (3.35.) (1-2RJ = p(l-hR) 2 A 2 + (l-p)(l+hR ) 2 A A A which gives (3.36.) - 2[h(2p-l) - 2j _ R h A - 2 2 2TJ_ 4p(l-p)h 2 2 + 2h(2p-l)TT - TT 2 / and for the f a i r f i f t y - f i f t y gamble z = <h;-h> the result becomes quite simple: (3.37.) R A = _22 i h v ' 2°_. For u ( w ; 0 , l , % , 8 , n ) (3.38.) = ~ 2 2 - z _2TT v h - TT s/w+T , with ^ y l + 22R = A f o r z = <h;-h>. 2 2 c = -~, (3.34.) gives <£p p V l + 2hR + ( l - p ) V l - 2hR A • A - 38 - so that (3.39.) 2p(l-p)[h(2p-l) - z] ,„ 2 „ , 2. 2 .2 (2p -2p+l) h - 2h(2p-l)z + z = A x 2p(l-p)ir . 2,, ,22 2 4p (1-p) h + TT and for the f a i r f i f t y - f i f t y gamble z = <h;-h> this gives: (3.40.) R A = -22 ,2 ,.2 h + 4z x = 2TT for z = <h:-h>. 2 2 h + 4TT 3°. For u(w;0,l,l,B,n) ^ log (w + c) , with c = §, (3.34.) cannot be P applied. However, applying (3.32.) gives (3.41.) log(w +c+z) = p.log(w +c+h) + (1-p).log(w +c-h) o o o which gives, after noting that according to (3.20.) W q + c = [R ] following r e s u l t : *r 2 5- + h(2p-l) - TT = R A (3.42.) 1-p +h ^- - h so that for the f a i r f i f t y - f i f t y gamble z = <h;-h> we have , the - 39 - (3.43.) 4^. v ~ = A 2 G 2 = ,2 „2 h + z For u(w;0,1,2,3,11) ^ - + w (3.44.) 1 1 + %zR A TT c > with c = p 1 + %hR A = for .2 2 h + TT + z = <h;-h>, (3.34.) gives 1-p 1 - %hR. A so that ^ (3.45.) 2[h(2p-l) - z] = h A 2TT - h(2p-l)z 2 4p(l-p)h + h(2p-l)TT and f o r the f a i r f i f t y - f i f t y gamble z = <h;-h> this gives (3.46.) 5^. R -2z 2 ti For u ( w ; 0 , l , c o jC 2TT = A = h ,1) ^ - e W z = <h;-h>. (3.34.) provides 1-T zR. lim C for 2 1 + hR. 1-T p. lim 1 + X-Ko (3.47.) hR. + (1-p).lim 1 - or, using lim 1 + X-x» (3.48.) 1-T -zR„ A i _ e = k . , this becomes -hR, A p.e , + hR, x A (l-p).e 1-T - 40 - Hence, -zR e A _ = e -[h(2p-l) - Tr]R A 4 (3.49.) hR = where the hyperbolic For the f a i r 2p.cosh(hR) A + (l-2p).e cosine of x i s defined by cosh(x) = % ( e + e ) X X f i f t y - f i f t y gamble z = <h;-h> the following result then e a s i l y obtained: -zR (3.50.) e TTR = e A = cosh(hR) A for I = <h;-h>. we w i l l of course have R = 0, as the A decision maker i s r i s k neutral with a l i n e a r u t i l i t y function. 6 . For u(w;0,0,T,$,ri) ^ w A summary of the results i n this section i s provided through Tables 3.2. and 3.3., which appear after section 3.5. - 41 - 3.4. Risk aversion as a f u n c t i o n of p r o b a b i l i t y e q u i v a l e n t f o r the gamble z = <h,p;-h>. Consider the gamble z = <h,p;-h>, so t h a t by D e f i n i t i o n 2.5. and (2.8.) we have f o r u(w;0,1,x, 8,n) with 6w l-T ^ o x ? 1 : B(I-T) (3.51.) rB(w o B(1-T) Using + n (3.20.) and f u r t h e r r e d u c i n g 1 = + (1-P) (3.51.) p r o v i d e s p 1 + + n for x ^ 1 hR. + l-T B(w -h) l-T hR. (3.52.) l-T +h) l-T (1-p) 1 - Hence, f o r p = p(w ,p,h) i t h o l d s o hR. 1 (3.53.) P = hR. - l-T ~ 1 - l-T 1 + 1 - The d e s i r e d f u n c t i o n s R^ f o r u(w;0,1 , x , B > n ) vith (3.52.) o r (3.53.) by s p e c i f y i n g x more d i r e c t way. l-T hR. x 1 can be d e r i v e d F o r x = 1 we w i l l d e r i v e a r e s u l t from in a - 42 - 1°. For u(w;0,l,-l,8,ri) * ~ (w + c) , with c = - ^, (3.52.) gives (3.54.) 1 = p(l - hR ) + 2 A (l-p)(l + hR ) 2 A which gives (3.55.) R = -(2p - 1) A h 2 so that (3.56.) p = khR A + k which can be v e r i f i e d through (3.53.) as well. 2° . For u(w;0,l,%,8,n) ^ v*w + c , with c= jg, (3.57.) 1 = pVl + 2hR (3.52.) gives + ( l - p ) v l " 2hR A A which gives, after taking both sides to the power two and after some rearranging: (3.58.) (1 - 2p)hR + p(l-p) A so that (3.59.) R, A -2^2P-D(P-1) h(l-2p+2p ) 2 2 = p(l-p)vl - 4h R 2 2 A - 43 - whereas (3.53.) w i l l give an e x p l i c i t expression for p. 3^. For u(w;0,l,l,B,n) ^ log (w + c) , with c = §, (3.52.) and (3.53.) p cannot be applied. (3.60.) However, using (2.8.), we have log(w +c) = p.log(w +h+c) + (l-p).log(w -h+c) o o o which gives, after noting that according to (3.20.) w + c = [R ] , o A 1 log(l-hR ) A ( 3 ' £. 6 1 - ) P log(l-hR ) - log(l+hR ) ' = A For u(w;0,l,2,6,n) ^ - (3.62.) 1 = w A ]_ c , with c = p [ l + %hR ] 1 + ( l - p ) [ l - %hR ] i\ so that (3.63.) R = A 2 ( 2 P ~ h l ) yielding (3.64.) p = %jhR + % A which can be v e r i f i e d (3.52.) gives through (3.53.) as well. A 1 - 44 - 31. For u(w;0,l,°°,c,l) ^ - e (3.65.) C w (3.52.) provides hR. 1 = p. 1 im 1 + lim 1 * + = e T (3.66.) 6°. R A = l. l o g hR. + l-T which gives, using l-T l-T (l-p).lim 1 - -k _£_ For u(w;0,0,T,6,ri) ^ w we w i l l simply have R = 0, so that p A summary of the results i n this section i s provided through Table 3.2., which appears after section 3.5. - 3.5. Risk aversion 45 - as a f u n c t i o n of g a i n e q u i v a l e n t f o r the gamble z = < K , p ; - h > . Consider the gamble z = <fi,p;-h> , so that by D e f i n i t i o n 2 . 4 . and v by ( 2 . 7 . ) we have f o r u ( w ; 0 , 1 , T , B , n ) , w i t h T ? 1: 1-T Bw B(l-T) (3.67.) B(w +h) __£_ B(1-T) T + n + _ ( 1 U s i n g ( 3 . 2 0 . ) and f u r t h e r r e d u c i n g ( 3 . 6 7 . ) p r o v i d e s (3.68.) 1 = p 1 •+ hR. r p ) B(w -h) |__£_ + ^ T n f o r T / 1: 1-T + (1-p) 1 - h R A) 1 _ T / which we w i l l use i n most of the f o l l o w i n g d e r i v a t i o n s o f R. as a f u n c t i o n A of h f o r the u t i l i t y f u n c t i o n s l i s t e d i n T a b l e 3 . 1 . 1°• For u ( w ; 0 ) l , - l , g , n ) (3.69.) - (w + c) 1 = p(l - hRA)2 + (l-p)(l so that (3.70.) , with c = - J = R A 2ph - 2(l-p)h ph2 + (l-p)h2 P + hRA)2 , (3.68.) gives - 46 - which becomes for p = \ : (3-71.) R = A -P^n h +h for z = <n>h>. For u(w;0,l,%,B,n) ^ W + c , w i t h c = -§ , (3.68.) g i v e s 2p 2J\ (3.72.) 1 = [ i + 2hR ]^ p A + (l-p)[l - 2hR ] % A which gives after several calculations: 7 o x „ _ 2p(l-p)[ph+(p-l)h] A For p = \ (3.74.) (3.73.) becomes R. = A 3^. tp h + (l-p)^h] For u(w;0,l,l,B,ri) - H) (h + h ) 4 ( H for z = <h;-h>, 2 ^ log(w + c) , with c = ^ , (3.68.) does not apply. However, using (2.7.), we have (3.75.) log(w +c) = p-log(w +c+h) + (l-p)-log(w +c-h) o o o which gives, after noting that according to (3.20.) (3.76.) . 1 = [1 + h R j [ l - h R j A A P 1P . w + c = [R 1 o A 1 - 47 - Formula (3.77.) (3.76.) becomes for p ^ h R. A 4°. = hh ^ For u(w;0,l,2,B,n) ^ (3.78.) for z = <h;-h>. , with c = ^ — w + c 1 = p[l + %hR ] A , (3.68.) gives p _ 1 + ( l - p ) [ l - ^hR ] 1 A so that (3.79.) = 2[ph - (l-p)h] hh which for p = % becomes (3.80.) 5°. R. = h -h for z = <h;-h> hh For u(w;0,l,°°,c,l) % - e ° W (3.68.) provides l-T (3.81.) hR. 1 = p•1im 1 + + (l-p)-lim 1 -r-Ko l-T which gives, using lim 1 + - ~ A 1 = p•e hR (3.82.) For p = h (3.82.) becomes -k h R + (1-p)-e A l-T - 48 - — 83.) 1 = 2 hR e For u(w;0,0,T,8,n) ^ w A ^ + hR • e A we simply have n" = 1-P h -and R. = 0 A . P . A summary o f the r e s u l t s i n t h i s s e c t i o n i s p r o v i d e d through es 3.2. and 3.3., which appear on the f o l l o w i n g pages. T a b l e 3.2. Summary t a b l e o f the r e s u l t s f o r the gamble z = <h,p;-h>. Absolute Utility Function: Risk Aversion Certainty Equivalent 2 R, A as a function of : Probability Equivalent p R i s k Premium IT Gain Equivalent h Linear: R u(w;0,0,T,|3,ri) A = 0 R R 1 Quadratic: u(w;0,l,-l,B,n) „ R = 0 h 2 -,2 2TT 4p(l-p)h +2h(2p-l)-Tr 2 A = 0 R R A - i ( 2 p - 1) = 0 A 2 „ _ 2ph - 2(l-p)h A - 2 pli + 2 2 (l-p)h A Z A = 2 p ( l - p ) t h ( 2 p - l ) - 2] u(w;0,l,%,6,ri) 2 _ R (2p -2p+l) h -2h(2p-l)2+2 2 2 2 2p(l-p)TT 4p (l-p) h A 2 2 JJ + IT 2 2 h ( 2 p - l ) - ir = ' 1 fl l fl 1 2 = - ~ + £ +h i - h A '•A J '•A J P 1 _ P 1 h[l-2p+2p ] 2 2 2p(l-p)[ph+(p-l)h] [p h+ A (l-p) h] 2 2 2 P = l fl fl JJ _ _ -2p(2j5-l)(f5-l) A Logarithmic: u(w;0,l,l,8,n) R A = _ 2 [ h ( 2 p - l ) - 2] " Square Root: A l p 1 _ log(l-hR ) p A 1= log(l-hR )-log(l+hR ) A (l+hR ) (l-hR ) P A 1 _ P A A Hyperbolic: u(w;0,l,2,6,n) _ 2 [ h ( 2 p - l ) - 2] R h A -2R Exponential: u(w;0,l,°°,c,l) e 2 r - - h(2p-l)2 2 7 r R 4p(l-p)h -[h(2p-D A = +h(2p-l)TT - TT]R e A K _ 2[ph - ( l - p ) h ] _ • * • hh = hR. A „ = 1 ( 2 * - 1) A hR 2p.cosh(hR ) + ( l - 2 p ) e A 2p.cosh(hR )+(l-2p)e A A -hR 1 = pe hR A + (l-p)e A Table 3.3. Summary table of the results for the gamble z = <h;-h>. Absolute Utility Function: Name: Certainty Equivalent 2 R. as a f u n c t i o n o f : A Gain Risk Premium TT Equivalent h u(w;0,0,T,6,n) Linear R =0 R = 0 u(w;0,l,-l,B,n) Quadratic u(w;0,l,%,B,n) u(w;0,l,l,B,n) u(w;0,l,2,B,n) 5 ' Square Root r A R r Logarithmic " A ~ r " .2 . .2 h + 4z 2 2 - ~ r 2 z .2 .2 h +z + " 2 2 R 2 h -2R - A u - A - A = A A = cosh(hR.) A e A 21T A 2 2 h - TT 27T „ , 2 . 2 h + 4TT 27T . 2 2 h + TT H 2 h uR A e r 2 2 .2 2 h -2 + A R A a R = A Hyperbolic Exponential Aversion A A u(w;0,l,°° c,l) Risk A = 0 _ 2(h-h) 2 .2 h +h c _ A ~ h r h -h ~ hh \ -V A 2 (h + A R 4(h-h) " h hh , r -hR, hR , 1 A A 1 = - e +e l1 A = cosh(hR.) A - 51 - 3.6. Risk aversion as a f u n c t i o n o f c e r t a i n t y e q u i v a l e n t o r r i s k premium f o r the gamble z = <h,p;0>. - Consider the gamble z = <h,p;0>, where E ( z ) = ph : F i g u r e 3.2. (3.84.) z = <h,p;0> so t h a t by (3.85.) (2.1.) u(w +z) = p.u(w +h) + (l-p).u(w ) o o o r Hence, f o r u(w;0,1,T,8,n) w i t h T ± 1, (3.85.) r e s u l t s i n : rB(w +z) x l-T +n .6(1-T) (3.86.) B(W +h) o 8(1-T) Using — = — (3.20.) and f u r t h e r r e d u c i n g ^ + n l-T Bw + (l-p) + n (3.86.) g i v e s f o r T ^ 1 l-T (3.87.) l-T = P 1+ hR , A l-T + (1 - p) S u b s t i t u t i o n o f z = ph - TT i n (3.87.) p r o v i d e s e a s i l y a r e l a t i o n s h i p - 52 - containing TT = TT(W ,Z). R o A as a function of 2 and TT for u ( w ; 0 , l , T , $ , r i ) with T f 1 can be derived from (3.87.). The case T = 1 w i l l be derived differently. ll. For u(w;0,l,-l,B,n) ~ - (w + c ) , with c = - §, (3.87.) gives 2 P (3.88.) (1-2R ) = 2 p(l-hR ) A + (1-p) 2 A which gives (3.89.) {V -*\ = ,2 „2 R. A 2 I..2 .. h ph - z 2 ; r p(l-p)h 2 + 2phTT - TT and for the f i f t y - f i f t y gamble z = <h;0> (3.90.) R = = A h 2^. - 22 For u ( w ; 0 l % g r ) ^ (3.91.) J ) J ) - for S = <h;0>. 2 h + 4hfT - 4TT + c , with c = 1 V l + 22R A = pVl + 2hR A (3.87.) gives + (1-p) so that (3.92.) R A = ~2p(l-p)(z-ph) [2-hp ] 2 2 = 2p(l-p)TT [p(l-p)h - TT] 2 and ex for the f i f t y - f i f t y gamble z = <h;0> we obtain : no \ (3.93.) rt 4h-82 R 8TT = , = (42-hr A N for z = <h;0>. (h-4TT) Z For U(W;0,1,1,B,TI) ^ log (w + c) , with c = §, (3.87.) cannot be P applied. However, using (3.85.) gives us: (3.94.) log(w +c+z) = p.log(w +c+h) + (l-p).log(w +c) o o o which gives, after noting that according to (3.20.) the w + c = [R ] o A following r e s u l t : (3.95.) 1 + 2R A = 1 + (ph - TT)R A A = [1 + hR ] A P A or log(l+zR ) A (3.96.) p= = log(l+hR ) A A l o g [ l + (ph—TT)R ] _ l o g ( l + hR ) A A Hence, for the f i f t y - f i f t y gamble z = <h;0>, we have: (3.97.) R = ^ - ^ i 2 = ^ (h - 2TT) For u(w;0,l,2,B,n) ^ " — J — » w + c w i t h c f o r 2 = < h = ^5 , (3.87.) gives p ; 0 > , - 54 - (3.98.) 1 + %zR 1 + %hR A + (1-p) A so that (3.99.) R 2(ph-z) (l-p)h2 = A 2jn_ (l-p)h(ph-Tr) = and for the f i f t y - f i f t y gamble z = <h;0> we have (3.100.) R A 2h - 4z hz = For u(w;0,l,°°,c,l) i> - e (3.101.) lim 1 + zR. C 8TT 1-T 1-T -zR (3.102.) = e 1-T + (1-p) -k -(ph-Tr)R A e - 1 -hR. e - 1 P = hR. p. lim 1 + T z = <h;0>. (3.87.) provides w which gives, using lim 1 + X"KO for h(h-2Tr) e -hR. e -1 and for the f i f t y - f i f t y gamble z = <h;0> (3.103.) 6 . —zR A For u(w;0,0,T,6,n) ^ w -(%h-Tr)R, - ; hR e we w i l l again have A +1 R for z = <h = 0. A summary of the results i n this section i s provided through Tables 3.4. and 3.5., which appear after section 3.8. - 55 - 3.7. Risk aversion R as a function of p r o b a b i l i t y equivalent for the gamble z = <h,p;0>. Consider the gamble z = <h,p;0>, and note that D e f i n i t i o n 2.5. does not apply for this gamble. D e f i n i t i o n 2.5. i s the d e f i n i t i o n as known i n the l i t e r a t u r e , hence i t i s included under the preliminaries. For the purpose of this section we w i l l need: D e f i n i t i o n 3.2. A p r o b a b i l i t y p such t h a t the d e c i s i o n maker is i n d i f f e r e n t between z = W + Q %h a n d the gamble <h,p;0> i s c a l l e d the gamble's p r o b a b i l i t y and i s denoted by equivalent p = pCw^jh). Hence, p = p(w^,h) i s defined by (3.104.) u(w +^h) = p.u(w +h) + (l-p).u(w ) o o . o r so that f o r u(w;0,1,T,3,n) T B(1-T) r with T f 1 B(w +hh) ^ ° . — - — + we have l-T n (3.105.) B(I-T) B(W +h) o 1-T + n 1-T Bw + (l-p) Using (3.20.) and further reducing (3.105.) gives for T ^ 1 + Tl - 56 - l-T (3.106.) 1 + l-T hR. 1 + 2T + (1-p) Hence l-TL X 1 (3.107.) P = l-T 1 - T We w i l l now specify our results for the gamble z = <h,p;0> for the u t i l i t y functions i n Table 3.1. lj\ For u(w;0,l,-l,B,n) * - (w + c) , with c = (3.108.) (1.- % h R ) = 2 A p(l - hR ) 2 A + - j, (3.106.) gives (1-p) wh i ch g ive s (3.109.) R A = *<1 - P> h ( l - 4p) 2 so that 4 - hR. (3.110.) 4(2 - hR ) A which can be v e r i f i e d through (3.107.) as well, V - 57 - 2^. For u(w;0,l,%,B,n) ^ Vw + c , with c = jg, V I + hR A (3.111.) p V l + 2hR A = (3.106.) gives +( 1 - p ) r which gives . -4p(2p-l)(p-l) (3.112.) R. = [2p -l] h A 2 2 Formula (3.112.) does not permit a way of expressing p as a function of h R , however applying (3.107.) gives (3.113.) p = 1 - V I + hR A • 1 - V I + 2hR A 3^. For u(w;0,l,l,6,n) ^ log (w + c) , with c = (3.106.) and (3.107.) P However, applying (3.104.) gives us cannot be applied. (3.114.) p.log(w +h+c) + o (1-p).log(w +c) o which gives, after noting that according to (3.20.) w + £ = [R ]-1 o A (3.115.) log(w +%h+c) o p = = l o g ( l + %hR ) 2- . l o g ( l + hR ) A 4^. For u(w;0,l,2,B,n) ^ - £-^r , with c = ^ , (3.106.) gives - 58 - (3.li6.) (1 + %jhR ) A = 1 p ( l + %hR ) A 1 A + (1-p) which gives 2(2p - 1) (3.117.) R A = (1 - p)h so that 2 + hR. (3.118.) 4 + hR. which can v e r i f i e d through (3.107.) as,well, 5jL For u(w;0,l,°°,c,l) % - e (3.119.) hR *\ A lim 1 + 2T C W , (3.106.) provides 1-T = p.lim 1 + X"K» 1-T which gives, using lim 1 + X-K» (3.120.) R A = e -k h ' S 11 " P. = lo so that —%hR 1 - e (3.121.) P = -hR. 1 - e which can be v e r i f i e d through (3.107.) as well. hR. 1-T + (1-p) - 59 - 6°. For u(w;0,0,T,B,r|) ^ w we simply have A summary of the results i n this section Table 3.4., which appears after section 3.8. R = 0 and i s provided p s h. through - 60 - 3.8. Risk aversion R as a function of gain equivalent for the gamble z = <h,p;0>. Consider the gamble z = <h,p;0> , and note that D e f i n i t i o n 2.4. does not apply for this gamble. D e f i n i t i o n 2.4. concerning gain equivalents i s best known, hence included under the preliminaries. For the purpose of th section we w i l l now state: D e f i n i t i o n 3.3. An amount h such that the decision maker i s indifferent between the status quo W plus p-h, and the gamble z = q <fi,p;0> i s called the gamble's gain equivalent and i s denoted by h = h(w p,h). o> Hence, h = h(w ,p,h) i s defined by o (3.122.) u(w o + ph) = p-u(w o + h) + (l-p)-u(w ) o so that for u(w;0,1,T,B,n) with T f 1 we have T BU-T) B(w +ph) o _ — ^ 1-T + r, (3.123.) r B(1-T) B(w +h) i 1-T rBw + (1-p) V. J Using (3.20.) and furhter reducing (3.123.) gives for T ± 1: 1-T - 61 - l-T phR (3.124.) l-T 1 + = P 1 + + (1 " p) • We w i l l now specify R^ as a function of h for z = <h,p;0> for the u t i l functions l i s t e d i n Table 3.1. For u ( w ; 0 , l , - l , B , T i ) ^ - (w + c) , with (3.125.) (1 - p h R ) = 2 A p(l - hR ) A c 2 ~ g » (3.124.) gives = + (1- ) P which gives - 2(h - h) (3.126.) v so that for p= % (3.126.) becomes (3.127.) R = A 2^. 4(h - h) " 2h - h 2 for z = <h;0>. 2 For u(w;0 l,%,6,n) ^ Vw + c , with c = (3.128.) 5 t l + 2phR = p [ l + 2hR which gives after several calculations (3.129.) R A = 2(l-p)(h-h) " (h - p h ) 2 , + (3.124.) gives (1-p) - 62 - Hence, for the f i f t y - f i f t y gamble z=<h;0>, i . e . p = % , (3-130.) R = M 3^. for I ' ^ h (2h A we have = <h;0>. h r - For u(w;0,l,l,g,n) ^ log(w + c) , with c = ^ (3.124.) does not P apply. However, using (3.122.), we have (3.131.) log(w +c+ph) = p-log(w +c+h) + (1-p)•log(w +c) Q Q Q which gives, after noting that according to (3.20.) (3.132.) [1 + hR.] A = P 1 + phR A w + c = [R ]-1 o A . Hence, for p = h this gives (3.133.) R = 4 ( f i A 4i. I h For u(w;0,l,2,B,n) ^ " (3-134.) ]— W [1 + % hR ] 1 P so that (3.135.) R A = for z = <h;0>. h ) 2(h-h) (l-p)hh + , with c = = p[l + %h , (3.124.) gives K C R ] 1 ° + (i- ) p - 63 - Hence, f o r p = % (3.136.) IV this gives 4(h - h) R A = F o r u(w;0,l,°°,c,l) ^ - e phR (3.137.) f o r z = <h:0>. hh C (3.124.) p r o v i d e W l-T hR. lim 1 + p • lim 1 + -r-X» + (1-p) •r-X» l-T which g i v e s , u s i n g l i m 1 + -hR (3.138.) l-T = A p-e = -phR e e -k A - (1-p) so t h a t f o r p = % (3.139.) 6° . e -hR. A F o r u(w;0,0 , T , 8 5 r)) —%hR „ A 2e = ^ w we w i l l , -1 again f o r z = <h;0>. simply state h A summary o f the r e s u l t s i n t h i s s e c t i o n i s p r o v i d e d T a b l e s 3.4. and 3.5., which appear on the f o l l o w i n g pages. = h and R through = 0. Table 3.4. Summary table of the results for the gamble z = <h,p;0>. Absolute Utility Function: Risk Aversion R, A as a function Probability Equivalent p R i s k Premium ir Certainty Equivalent z of : Gain Equivalent h Linear: u(w;0,0,T,6,Tl) R = 0 A R = 0 A R A = 0 R A * ° v Quadratic: u(w;0,1,-1,6,n) „ _ A 2(ph-2) u R 7 2 ph - g - ' 2 1 1 p(l-p)h +2phTT-ir A 2 A 2 _ 4(1-20) h(l-4§) „ A _ 2(h - h) -2 ,2 h - ph Square Root: _ uU.O.l.if.B.n) 2 Logarithmic: u(w;0,l,l,i3,n) -2p(l-p)(z-ph) r - K 1J Iz-hp A log(l A n log(l P + hR ) 2p(l- )T7 P - log[l + (ph-TT)R A 2 A ] l o g ( l + hR.) A A _ -4p(2p-l)(p-l) R [p(l-p)h - TT] A + zR ) p r _ R 2 h[2f> -l] 2 2 A log(l P jy _ A + lshR ) A " log(l + hR ) 2(l-p)(h"h) - 2 [h-ph] 2 A 1 + phR = [1 + hR J A A P A Hyperbolic: u(w;0,l,2,B,n) A _ 2(ph-z) (l-p)hz R A - 2 T (l-p)h(ph-TT) „ A K - 2(2p-l) h(l-p) 2(h-h) * (l-p)hh A A Exponential: u(w;0,l, ,c,l) -(ph-TT)R O T e = 1 + p e A - l -1+p , -hR, i e -1 -hR pe "PhR. = e -(1-p) A ( Table 3.5. Summary ruble of the results for the gamble 1 • <h;0>. Absolute Utility Function: u(w;0,l,-l,B,n) u(w;0,l,%,B,ri) U(W;0,1,1,B,TI) Name: Quadratic Square Root Logarithmic u(w;0,l,2,6,n) Hyperbolic u(w;0,l,°°,c,l) Exponential Risk Certainty ^ Equivalent z R 2h - 42 . 2 h~ - 2 z A R. = A | A 2 4h - 82 A = k p mium TT ," A . i f / ™ * re 8TT 2 . ± h +4hTr 4TT A R. = A + x Gain Equivalent h R - 4(h - h) A " -2 ,2 2h - h A 4(h - h) 8TT (2h - h ) (h - 4TTV R. = 2R of : — A h2 r 2 g R. = (42 - hV 2h - 42 R i A T2 = R R. = A h - 22 R R^ as a f u n c t i o n Aversion 4(h - h) 8TT R (h - 2irV 8rr h(h - 2TT) -(^h-TT)RA 2 A = 4(h - h) R A -hR. hh —%hR = 2e " - 1 - 66 - 4 The R -method A In t h i s c h a p t e r we w i l l function e s t a b l i s h a new procedure to s e l e c t a u t i l i t y from among a l t e r n a t i v e s . The procedure i s based on the mathematical r e s u l t s i n Chapter 3 and i s c a l l e d the R^-method. alternative u t i l i t y functions l i s t e d 4.1. the f u n c t i o n s a r e assumed to be the f i v e r i s k a v e r s e u t i l i t y i n T a b l e 3.1. the R^-method t o i n c l u d e over t h e i r e n t i r e In s e c t i o n region. utility The remainder o f t h i s c h a p t e r w i l l f u n c t i o n s t h a t a r e not s t r i c t l y extend r i s k averse - 67 4.1. - The R - method for r i s k averse decision makers. A— • • '• •— 1 : In t h i s section we w i l l assume that the decision maker behaves according to a s i n g l e r i s k averse u t i l i t y function over the entire wealth region., In the next section we w i l l eliminate this assumption. mathematical extensions Also, the i n Chapter 5 w i l l give d i r e c t i o n as to how one may expand the set of u t i l i t y functions from which one i s chosen. From a notational point of view we w i l l c a l l A the set of r i s k averse u t i l i t y functions l i s t e d i n Table 3.1. D e f i n i t i o n 4.1. The class A of r i s k averse u t i l i t y functions contains utility (4.1.) Hence: functions u(w;0,l,T,B,n) where and x e T, T = (-l,^,l,2,oo) with c = (4.2.) i f T < 00 a n d 6 c = 3 and n = 1 if T ->• °° . The R -method w i l l select a u t i l i t y function within class A to which A the decision maker conforms most c l o s e l y . The s e l e c t i o n procedure starts off by obtaining the decision maker's response to two or more reference f gambles. We suggest taking at least three d i f f e r e n t gambles, and the result i of the R^-method w i l l only benefit by taking more gambles. responses may The required be chosen from the four indicated i n Chapter 3, i . e . the r i s k premium, the c e r t a i n t y equivalent, the p r o b a b i l i t y equivalent, or the gain equivalent. The question as to how we may be able to a r r i v e at the decision maker's exact response w i l l be deferred u n t i l l a t e r ; for the moment we will - 68 - assume that the responses are assessed without error, and that the decision maker responds consistently according to one (unknown) r i s k averse u t i l i t y function. Now, l e t us suppose, f o r example, that the decision maker provides us with TT^, ^ 2 a n d TT^ as h i s r i s k premiums f o r the gambles z^ = <h^,p;-h^>, 2 2 = < ^ 2' ' " 2 ' 1 P _ 1 > a m * 3 2 = < h 3»P' ^3 _ > respectively, where ? £ • We are then able to calculate the value of R. f o r the u t i l i t y functions A class A, according to the results l i s t e d special case when p = %) . within i n Table 3.2. (or Table 3.3. f o r the This way we obtained for each u t i l i t y function within A three values f o r the function R , which may or may not be equal. A If a l l three responses lead to the same value for R^ f o r one u t i l i t y function within A, then these responses are consistent with this u t i l i t y function. Hence, this w i l l be the function selected as the one to which the decision maker conforms because R. i s the value of the absolute r i s k aversion A at the i n i t i a l wealth l e v e l w . That i s , R. = R.(w ) has a s p e c i f i c value no o A A o r matter what gamble the decision maker i s faced with. Let us c l a r i f y this point with the following example, since i t i s an e s s e n t i a l facet of the R method. Example 4.1. (4.3.) A decision maker i s assumed to behave according to u(w;0,l,2,B,n) ^ - — r — - w T where c = --£ 2 c P He states 25 and 16 as the r i s k premiums f o r 2^ = <50;-50> and 2^ = respectively. Is this consistent with the function R <40;-40> taking on the same - 69 - value at w^? Yes, according to (3.46.) we find that indeed 2TT (4.4.) A, 1 o 1 = 2(25) = 2(16) (50) 2 (40) :TT 1 2 h 2 More often none of the u t i l i t y functions within A w i l l a t t a i n the same values for R^ at W from several responses. q In this case we cannot d i r e c t l y determine which u t i l i t y function should be selected to represent the decision maker's preferences. We can s t i l l select a u t i l i t y function within A to • which the decision maker conforms most closely. In other words, we w i l l select the u t i l i t y function within A that represents "best" the decision maker's behavior. Certainly, the answer to the question as to what i s "best" i s highly subjective and dependent on the choice of an appropriate c r i t e r i o n . However, i t does seem to make sense to select the u t i l i t y function for which the three R. values have the least r e l a t i v e standard deviation. What we mean A by ' r e l a t i v e ' w i l l be explained i n the formal description of the R^-method. We w i l l now present t h i s stepwise description of the R -method for r i s k averse u t i l i t y functions. For the time being we w i l l l i m i t ourselves to the use of r i s k premiums and the gamble z = <h,p;-h> to aid understanding. Step 1: Present the decision maker with n.^ 2 d i f f e r e n t gambles 2 <h^,p^;-tu> ( i = l,2,...,n) where h_^ > 0. terms of r i s k premiums i s W q = The decision maker's response i n = TT^(W ,Z\) ( i = l,2,...,n) respectively, where O i s the decision maker's (yet unknown) i n i t i a l endowment. It i s assumed that the assessment of the r i s k premiums i s completed without errors, and that the decision maker responds consistently according utility Step 2: to a single (unknown) function. Check the r i s k premiums to determine whether the decision maker i s r i s k averse, i . e . TT^ > 0 for a l l i . (If not, then refer to section 4.2. where r i s k proneness i s considered.) Step 3: Calculate the values f o r the functions according to the r e s u l t s i n Table 3.2. for a l l u t i l i t y functions within A and for a l l i . We w i l l use the following notation: / 2 1 2 u(w;0,l,-l,B,n) R (w ,-l,l) u(w;0.1,3s,g,n) R (w ,^,l) u(w;0,l,°°,c,l) A A o R (w ,-l,2) A R (w ,-l,n) , o A A • R (w ,~,l) R (w ,-,2) o A o o S • • i . e . R.(w ,x,i) i s the value of the function R, according A o A Table 3.2. f o r the gamble z o R (w ,J ,n) o • A z n 2 R (w ,»,n) A o to the r e s u l t s i n and f o r the u t i l i t y function within A for which x G T. Step 4: Let y(R ,x) and a(R ,T) be respectively the mean and standard deviation of the n R -values R,(w ,x,i) where i = l,2,...,n and x £ T. A A o Identify x* £ T such that i t s r e l a t i v e standard deviation i s - 71 - ,x*) a(R (4.5.) — = Min |U(R ,T*)| xeT A a(R ,x) - . |p(R ,x)| A Select u(w;0,l,x*,8,n) as the u t i l i t y function within A that conforms most c l o s e l y to the decision maker's behavior. Step 5: * Choose as an estimation of R.(w ) : A o (4.6.) R (W ) = u(R ,x*) a Q A Step 6: Obtain the decision maker's status quo W. Step 7: I f x* Q R. (w ) = c . A o (4.7.) °°, then we select the parameter c = R A ( W ) > a Q s i t : holds that I f x* < , we e a s i l y derive from (3.20.) 00 c = |- - w A ° R T so that we w i l l select the parameter c = - w . We w i l l i l l u s t r a t e the R -method with the following example. Example 4.2. 2 1 A decision maker i s presented with the following three gambles: = <1,.75;-1>, z 2 = <2;-2> and = <3;-3>, for which he indicates - 72 - respectively TT^ = .2, TT^ = 1 and TT^ = 2 as the r i s k premiums. Lemma 2.4. i = 1,2,3. According to his. responses are consistent with r i s k aversion, as TT_^ > 0 for The calculations i n Steps 3 and 4 can now results l i s t e d i n Tables 3.2. shown i n Table 4.1. and 3.3. on the next page. be performed using the The r e s u l t s of the calculations are We see from Table 4.1., that T* = 2, so that we w i l l select u(w;0,l,2,g,n) as the u t i l i t y function within A that conforms most c l o s e l y to the decision maker's r i s k attitude. with Step 5, we choose R. (w ) = U(R,,T*) = .472 A o A In accordance as an estimation of R.(w ). o A 2 Suppose further, that W q = 9, so that we have c = —2JT72" ~ 9 = - 4.763 by (4.7.) Hence, the selected u t i l i t y function can be written as: ( 4 - ' 8 ) Table 4.1. U ( W ) - - w - 4~.763 i s shown on page 73. In Step 1 we assumed that the assessment of the decision maker's responses w i l l be performed without error. This thesis w i l l not address the question as to which method should be used for this purpose. the l i t e r a t u r e contains We s u f f i c i e n t directions for this problem and we refer to the suggested method by Becker, DeGroot, and Marschak(1964). discussions i n Grether and Plott(1979), Lichtenstein and Slovic(1972), who f e e l that Also, the Slovic(1973), and actually a l l employed this method, provide a good reference. The c r i t e r i o n we use i n the R -method to determine which u t i l i t y A function i s "best", i s the r e l a t i v e standard deviation, or the absolute value of the c o e f f i c i e n t of v a r i a t i o n . As mentioned before, we f e e l that the v a r i a t i o n Table 4.1. The results of calculations r e f e r r i n g to Example 4.2. U t i l i t y function u(w;0,l,-l,f3,n) R as a function of A the r i s k premium TT 2TT 4p(l-p)h +2h(2p-l)-Tr 2 u(w;O,l,*s,0,n) 2P(1-P)TT 2 2 2 4p (1-p) h + TT h(2p-l) - TT = - (l l fl ) + i +h ^ -h |R J l J p 1 _ R.Cw ,T,2) A O R (w ,t,3) A o V a a l l u .234 .667 .800 .567 .242 .43 .415 .250 .160 .275 .106 .39 .462 .400 .308 .390 .063 .16 .471 .500 .444 .472 .023 .05 .469 .609 .676 .585 .086 .15 2 2 R u(w;0,l,l,B.n) R (w ,T,1) A O A | A p R A u(w;0,l,2,8,ri) A 2TT 4p(l-p)h +h(2p-l)Tr 2 e -[h(2p-l)-Tr]R _ A u(w;0,l,oo, ,l) hR c 2p.cosh(hR )+(l-2p)e A - 74 - of the n R -values should be as small as, possible for the selected function, however allowing greater v a r i a t i o n for a greater mean u of the R^-yalues. This i s exactly what i s implied by using the r e l a t i v e standard deviation as our c r i t e r i o n . Note further that i n the case where for one u t i l i t y function within A a l l R.(w , T , i ) a t t a i n the same value, the r e l a t i v e standard deviation A o v. i n Step 4 reaches i t s minimum at zero. Example 4.3. A decision maker with (yet unknown) i n i t i a l wealth w^ indicates .25 and 1 as h i s r i s k premiums for the gambles z^ = <1;-1> and z^ = <2;-2> respectively. The r e s u l t s of the calculations i n the R^-method are shown i n Table 4.2. below. Table 4.2. U t i l i t y function R (w ,x,l) A o R (W ,T,2) a O u(w;0,l,-l,B,n) .533 .667 u(w;0,l,Js,B,n) .400 .250 u(w;0,l,l,B,n) .471 .400 u(w;0,l,2,3,n) .500 .500 u(w;0,l,°°,c,l) .522 .609 C l e a r l y , the u t i l i t y function u(w;0,l,2,B',n) i s selected. Note that i n the above example not more than one u t i l i t y function within - 75 - class A can have R. (w ,x,l) = R. (w, ,x,2). A o A o forms for R This i s because the functional as a function of the r i s k premium i n Table 3.3. are d i f f e r e n t A and cannot be equal i f the r i s k premiums are not equal. example two gambles 2^ = <h^;-h^> and z^ = " ^ j l ^ , Consider for where h^ ^ h.^ for which the decision maker states TT^ and TT^ as h i s respective r i s k .premiums. Suppose R (W ,2,1) = R (W ,2,2), as was a have by q a q the case i n Example 4.3. We then (3.46.): 2TT 2TT 1 (4.9.) 0 -fh = l h 2 from which i t follows that -n j TT^ as h^ ^ h^. for instance, R (w ,1,1) / A the following. R A ( W We are now able to show that, .1,2), since by (3.43. and (4.9.) we have A O O I f TT^ f 0 ^ n ^ , then 1 R h fw ,1,1) A _ O 2 l 2 *1 + 2TT 2 h 1 1 2IT, 1 ± h 2 1 2 ^ 1 2 2TT„ I 2 1 (4.10.) h A 2 2 2TT 2 as I T . i 4. 1 2"2 I T „ . Hence, R. (w ,1,1) I A o R (W ,1,2) a q If TT. = 0 and ir. 1 z R. (w ,1,2). A o 7 0 or vice versa, then the R -values would be d i f f e r e n t , i . e . R (w ,2,1) ^ R (w ,2,2). A O A Other i n e q u a l i t i e s can be shown s i m i l a r l y . O The functional forms for R^ as a function of the r i s k premium i n Table 3.2. do not d i r e c t l y similar i n e q u a l i t i e s . A Hence, i n theory we may guarantee encounter the same R^-values for more than one u t i l i t y function within A, however we f e e l that i n practice t h i s problem w i l l not a r i s e . In circumstances where t h i s problem does occur, we should increase n (= number of reference gambles) i n the R^-method. - 76 - The functional forms f o r as a function of the probability equivalent in the gamble z <= <h,p;-h> are equal for the quadratic and the hyperbolic functions. A similar case can be seen i n Table 3.3. for R. as a function of A the gain equivalent i n the gamble z = <h;-h> for the logarithmic and the hyperbolic functions. These equalities do not impose any d i f f i c u l t i e s on the R -method based upon p r o b a b i l i t y or gain equivalents, as long as multiple A response modes are employed, as w i l l be discussed i n the next paragraph. The R^-method, as introduced i n this section, employs the r i s k premium for the gamble 2 = <h,p;-h> as the only response mode. I t should be clear, however, that with our results i n Chapter 3 the R -method can employ multiple response modes for both z = <h,p;-h> and z = <h,p;0>. must t h e o r e t i c a l l y a t t a i n a single value at W, q which type of equivalent or gamble are used. This i s because namely R^Cw^), regardless of For example, one reference gamble may ask for the certainty equivalent for the gamble z^ = <h^,p^;-h^>, another for the p r o b a b i l i t y equivalent for might ask for the gain equivalent f o r z^ = ^^yp^^» = <h^;0>. and a t h i r d one The following example i l l u s t r a t e s this point for two reference gambles. Example 4.4. A decision maker indicates that he i s indifferent between the gamble = <h;-100> and the status quo, when the gain equivalent h = h(w ,z^) Q = 180. He indicates that he i s also indifferent between the gamble = <100,fi;0> and winning 50 for sure, when the p r o b a b i l i t y equivalent £ = p(w ,100) Q = .55. See D e f i n i t i o n 3.2. The r e s u l t s of our calculations are summarized in Table 4.3., where we can e a s i l y select u(w;0,l,2,B,n) without even having to use the c r i t e r i o n of r e l a t i v e standard deviation i n Step 4 of the R ~method. A - 77 - Table 4.3. R. aa a f u n c t i o n of A R, as a f u n c t i o n o f A h f o r the gamble f) f o r the gamble u(w;0,l,-l,g,n) .00377 .00333 u(w;0,l, 5.S»T)) .00408 .00635 u(w;0,l,l,g,n) .00444 .00496 u(w;0,l,2,g,n) .00444 .00444 u(w;0,l,°°,c,l) .00432 .00401 Utility function 3 T h i s exmaple t h e r e f o r e the column z^ demonstrates t h a t the e q u a l i t y of the two R^-values i n f o r R^ as a f u n c t i o n of h does not impose any The f o l l o w i n g examples difficulties. p r o v i d e i l l u s t r a t i o n s of s e v e r a l mathematical c a l c u l a t i o n s , m a n i p u l a t i o n s , and r e l a t i o n s h i p s , which a r e a l l based on our r e s u l t s i n Chapter 3. consistency Example 4.5. (4.11.) I t may w e l l be used i n the R^-method, f o r i n s t a n c e checks f o r the assessment of the responses to r e f e r e n c e A d e c i s i o n maker i s assumed to behave a c c o r d i n g u(w;0,l,2,B,n) * - ^ where as gambles. to c = —\} - 78 - He indicates, - 25 as. his. certainty equivalent for the gamble z^ = <50;-50>. What w i l l be his. certainty equivalent for = <20;-20>? Using (3.46.) gives. - 22. (w ,h.) (4.12.) R, A Therefore, f o r i = 1 -2[z^/(20) ] or 2 ^-r —— 5 = h l for i = 1,2. 2 R^ = -2(-25)/(50) = - 4. 2 = .02, and for i = 2 we have .02 = I t i s important to note, that we do not have to know the e x p l i c i t values of the i n i t i a l wealth W q or the parameter c to obtain this result. Example 4.6. Referring to Example 4.5., what w i l l be the decision maker's r i s k premium for the gamble 2^ = <50,. 75;-50>? Expressing T C \ IT^(W ,2^) = o e x p l i c i t l y by (3.45.) results i n 4p(l-p)h R 2 TT . = (4.13.) 3 f o r z. = <h.,p;-h.>. 2-h.(2p-l)R 1 For p = .75, h A = 50 and R A 1 1 1 A = .02, (4.13.) gives for i = 3 : = 25. Note, that the certainty equivalent 2^ = h^(2p-l) - TT^ = 50(.5) - 25 = 0. Example 4.7. A decision maker i s again assumed to behave according to (4.11.) Using (3.46.), we e a s i l y e s t a b l i s h z\ = ^ ^ » ^ < 1 instance (4.14.) _ > f° r 1»2. = TT^ 2 = [h^/h^J TT^» where 3 with By the figures of Example 4.5. we then obtain for 2 = [20/50] (25) = 4, so that TT. = n\ = ^ ( ^ i ^ ) rM2 T T 1 Ih.J TT. i = - 4. In general: - 79 - where TT^ and 2^ are as. defined above. Similar relationships can be derived for other u t i l i t y functions within class A. Example 4.8. A decision maker i s assumed to behave according to the u t i l i t y function u(w;0,l,-l,g,n) 2 - (w + c) , with c = - -j}. ^ p For the gamble z 1 = <h,p;-50>, for what value of p w i l l the decision maker state h = 60, given that h i s r i s k premium for we have = <40;0> i s equal to 5. = 8(5)/[(40) +4(40)(5)-4(5) ] = ^fj 2 2 • According to (3.90.) Then, by (3.70.), we obtain hR + 2h — -^—z2(h+h) - R (h -h ) 2 (4.15.) p = A 2 so that with R = YTK> h = 60, and h = 50, we have p ~ .71. - 80 - 4.2. The R -method including r i s k proneness and r i s k n e u t r a l i t y . A \ The R - method as discussed i n section 4.1. assumed, that the decision A maker conforms to a single r i s k averse u t i l i t y function over the entire wealth region. Actually, t h i s assumption confounds two assumptions (1) that the decision maker i s r i s k averse over the entire domain, and (2) that he conforms to j u s t one u t i l i t y function over the whole region. Let us f i r s t consider according the case where the decision maker s t i l l behaves to one u t i l i t y function, however he i s not r i s k averse over the whole domain. With our r e s u l t s and remarks i n Chapter 3 i t w i l l be s t r a i g h t - forward to discover r i s k n e u t r a l i t y , i n which case the l i n e a r u t i l i t y u(w;0,0,T,B,ri) w i l l be selected. decision maker i s r i s k prone. function I t i s also easy to f i n d out whether the Namely, i f he indicates negative r i s k premiums for reference gambles, then the decision maker i s r i s k prone according to a lemma similar to Lemma 2.4. However, Chapter 3 does not provide a t o o l f o r s e l e c t i n g the "best" function from among a l t e r n a t i v e r i s k prone u t i l i t y functions. Only i f we derive similar mathematical r e s u l t s , can we employ the R^-method f o r r i s k prone u t i l i t y functions. Below we use two examples to give d i r e c t i o n as to how such r e s u l t s may be obtained. Example 4.9. (4.16) Consider the r i s k prone quadratic u t i l i t y function given by 2 u(w) ^ (w + c) over the region w > - c. where c > 0 Note that f o r this u t i l i t y function and the gamble - 81 - z = <h;-h> (2.4.) becomes (4.17.) (w +C-TT) o 2 = ^(w +c+h) + 5g(w +c-h) . p 0 2 2 % u"(w ) o Noting that R. = R (w ) = A A o »/\ u (w ) o = , we can substitute w ; w + c o o +c - - i — i n (4.17.), so that A R [ (4.19.) ~ T~ = -JT A H[ + A h ] 2 + ~R~- h[ h ] 2 A from which we e a s i l y derive the following r e s u l t : (4.19.) R A = - ~ — j h - TT Note that t h i s r e s u l t i s equivalent for z = <h;-h> to the function R, for the r i s k averse A quadratic u t i l i t y function u(w;0,l,-l,3 ,n) ^ - (w + c ) , where c = - -JJ-. 2 P However, according to Lemma 2.4., we know that TT w i l l be p o s i t i v e , whereas i t can be proved that f o r a r i s k prone u t i l i t y function IT w i l l be negative. In other words, (3.37.) and (4.19.) d i f f e r i n that (3.37.) w i l l contain a p o s i t i v e and (4.19.) a negative r i s k premium. Other results for the u t i l i t y function (4.16.) w i l l follow s i m i l a r l y . Example 4.9. (4.20.) Consider the r i s k prone exponential u(w) ^ e u t i l i t y function given by where c > 0. - 82 - For this, u t i l i t y function (2.4.) becomes for the gamble z = <h;-h>: c(w (4.21.) e -TT) = he 0 c(w -h) ° < gives \ TTR e he -c, and some reduction, which after noting that R^ (4.22.) c(w +h) ° + cosh(hR ) A A This r e s u l t i s equivalent to (3.50.) for the r i s k averse exponential utility —cw ^ function u(w;0,l,°°,c,l) ^ - e , however d i f f e r e n t i n the sense that TT w i l l be p o s i t i v e f o r the r i s k averse and negative for the r i s k prone utility exponential function. After deriving complete results for the functions R for a class P of A u t i l i t y functions consisting of (4.16.), (4.20.), and other r i s k prone u t i l i t y functions, we would have the foundation for an R -method based upon the A assumption that the decision maker conforms to a single r i s k prone u t i l i t y function. Let us now consider the case where the decision maker i s r i s k averse (or r i s k prone) over the whole region, however he does not conform to a single r i s k averse (or r i s k prone) u t i l i t y function. conforms to u,(w) for w < w 1 constant. o Suppose the decision maker + b, and to u«(w) for w > w + b, where b i s a z — o We can then employ the R -method for a r i s k averse (or r i s k prone) decision maker i n order to obtain the u t i l i t y function within class A (or - 83 - class P) to which he conforms most closely over the region w < w + b. o S i m i l a r l y , we w i l l use the R -method over the region w >_ w A to assess (w). + b i n order O It should be clear that this process can e a s i l y be extended to the case where the decision maker conforms to three or more r i s k averse (or r i s k prone) u t i l i t y functions. In practice, i t w i l l be d i f f i c u l t to a r r i v e at the value of b, however several reference gambles may be s u f f i c i e n t to provide a good estimation f o r b. Also, we should use a smoothing function around b i n order to connect u^(w) and utility ^(w). An alternative method for assessing the u t i l i t y function of a decision maker that conforms to a single r i s k averse u t i l i t y function over the whole region, or to more than one r i s k averse u t i l i t y function over d i f f e r e n t regions, i s provided by section 5.5., where we discuss the sum-of-exponentials u t i l i t y function. This well known and widely used u t i l i t y function offers great f l e x i b i l i t y and can be assessed by a method based upon our mathematical results i n Chapter 3. Let us f i n a l l y consider the case where the decision maker i s assumed to be r i s k averse over part of his wealth domain and r i s k prone over another part of t h i s domain. We w i l l simplify this case as follows. Suppose the decision maker conforms to the r i s k prone u t i l i t y function u (w) for w < w o + b, and to the r i s k averse u t i l i t y function u (w) for w > w + b. a = o J We w i l l follow the l i t e r a t u r e , e.g. Kahneman and Tversky(1979), by assuming that b w i l l be around zero. That i s , as compared to h i s i n i t i a l wealth W, Q the decision maker i s assumed to be r i s k averse for gains and r i s k prone for losses. For assessing the r i s k averse u t i l i t y function for w >. w^ we can - 84 - employ the R -method using the gamble z J\ = <h ,p ;0>. A. J. However, our r e s u l t s x i n Chapter 3 do not provide us with a s i m i l a r gamble to be used for losses. In other words, we would prefer to have mathematical r e s u l t s for the gamble Iry = <0,p^;-h2> so that the r e s u l t s for this gamble can be used i n an method for assessing the r i s k prone u t i l i t y function u^(w) Fortunately, i t i s not hard to obtain r e s u l t s for z^. equivalent to z^ by using the following (4.23.) h so that by applying 2 = - and u (w) and u (w) we a p 2 = 1 - p^ and 3.5. we will In order to connect the functions should use a smoothing process. finding the best u t i l i t y function u(w) exponentials Q Notice that z^ i s (4.23.) to our r e s u l t s i n Tables 3.4. h i s taken to be r e l a t i v e l y small. W. transformation: p d i r e c t l y obtain the desired r e s u l t s . for w < R^- This could be done by using the gamble 2 = <h,p;-h>, where We can also think of using the sum-of- u t i l i t y function (see section 5.5.) for this purpose. With the remarks and r e s u l t s i n this section we actually gave d i r e c t i o n to a General R -method. This method enables us to assess the u t i l i t y function to which a decision maker conforms. r i s k averse, r i s k prone or even both. The underlying u t i l i t y function may be Also, the General R^-method covers the case where the decision maker conforms to more than one u t i l i t y function over different regions. - 85 - 5 Mathematical Extensions i This chapter w i l l discuss several mathematical extensions r e s u l t s i n Chapter 3, where we expressed R^ = R A ( W Q ) a s a of our function of a response variable and the parameters of a nondegenerate gamble for a class A of u t i l i t y functions defined by D e f i n i t i o n 4.1. The purpose of our r e s u l t s i s to improve the R^-method as discussed i n Chapter 4. extending The following sections cover a wide variety of possible extensions. In section 5.1. we w i l l give some d i r e c t i o n as to how one may be able to obtain the function R^ for u t i l i t y functions beyond the class defined by A. The R^-method w i l l c e r t a i n l y be improved i f the u t i l i t y function can be selected from among a greater number of a l t e r n a t i v e s . In section 5.2. we w i l l note that we could also establish r e s u l t s using R^ = R^Cw^ instead of R^. This may p a r t i c u l a r l y be useful when R^ can only be expressed as a function e x p l i c i t l y including the i n i t i a l wealth. Another extension can be found i n the choice of the reference gamble. In section 5.3. we w i l l show, that i t i s f e a s i b l e to derive similar r e s u l t s for gambles other than 2 = <h,p;-h> and z = <h,p;0>. To be s p e c i f i c , w i l l b r i e f l y discuss the gamble z = <h^,p;ti2> where h^ ^ h^, s o l e l y given by i t s mean and variance, the normally we the gamble distributed gamble, and f i n a l l y the uniformly d i s t r i b u t e d gamble. Section 5.4. focusses on how to derive our r e s u l t s by means of Taylor - 86 - series expansions on both sides of the equation (2.4.) • We f e e l that results f o r other u t i l i t y functions may well be obtained by using this technique. In section 5.5. we w i l l present results similar to those i n Chapter 3 f o r the well known and widely used sum-of-exponentials u t i l i t y function. F i n a l l y , section 5.6. w i l l conclude this chapter with a b r i e f summary of the thesis. - 8,7 - 5.1. Extended r e s u l t s for u(w;g,a,T,g,ri) . In section 3.2. we introduced a class of u t i l i t y functions u(w) = u(w;£,a,T,g,ri) defined by (3.14.). The results of Chapter 3 are confined to s i x member u t i l i t y functions within this class. However, i t should be clear that many other u t i l i t y functions belong to (3.14.) and i t may be possible to a r r i v e at s i m i l a r r e s u l t s for those functions. A subclass of (3.14.) i s formed by (3.18.), of which Pratt(1964) gave several s p e c i a l cases, e.g. (5.1.) and (5.2.) and (5.3.) By s p e c i a l i z i n g (5.1.) one step further we obtain (5.4.) - 88 - where s i n h ^w i s the i n v e r s e h y p e r b o l i c e a s i l y derive (5.5.) the f o l l o w i n g sine f u n c t i o n o f w. From (5.2.) we s p e c i a l case: u(w;0,2,l,8,g) ^ arctan(w) A c t u a l l y , Pratt(1964) introduced h i s s p e c i a l cases (5.1.) to (5.3.) i n - cluding a transformation, the s u b s t i t u t i o n w by w + c. namely This way, (5.4.) and (5.5.) become r e s p e c t i v e l y : (5.6.) u(w) ^ s i n h ^(w + c) and (5.7.) u(w) % arctan(w + c) Other s p e c i a l cases o f u ( w ; £ , a , T , 0 , r i ) i n c l u d e (5.8.) u(w;a-l,a,T,8,ri) ^ f a-lf8s T . 1~ rr, a (l-T)aBI Bw + n ds = T + c 1-T a w a + c where c = jrr B and (5.9.) u(w;0,2,-l,B,n) ^ -w(w 2 + c) , where c = 3n ^ - 89 - To arrive at general and complete results for many of the above u t i l i t y functions, within the class defined by (3.14.), would certainly enhance the application p o s s i b i l i t i e s of the R -method. feel that such results are very hard to obtain. At this moment we However, the following example i l l u s t r a t e s that i t may s t i l l be possible to derive extended results. Consider the u t i l i t y function (5.7.), for which we have by (2.3.) with z = <h;-h>: (5.10.) arctan(w +C-TT) = %arctan(w +c+h) + %arctan(w +c-h) o o o which, after taking the tangent of twice both sides, gives: (5.11.) tan[2arctan(w +C-TT)] = o = tan[arctan(w +c+h) + arctan(w +c-h)] o o Further we have 2(w +C-TT) (5.12.) tan[2arctan(w +C-TT)] = 1 - (w +C-TT) o and (5.13.) tan[arctan(w +c+h) + arctan(w +c-h)] = 2(w +c) 1 - [(w +c) -h ] o 2 Equating (5.12.) and (5.13.) according to (5.11.) gives after cross- 2 - 90 - m u l t i p l i c a t i o n and some reduction: (5.14.) 7T(W + C ) 2 - (TT2+h2)(w o +C) + TT(l+h2) = 0 o which yields 2 (5.15.) TT w + c = o ,2 AI r + h + y [TT .2,2 - h J - 2 2 4TT 2TT though only TT (5.16.) 2 + h 2 ]j[-n - w + c = o 2 - h ] 2 2 - 4TT 2 27T is applicable because of the following: For h > 0 and TT 0, R (w ) should A o A reach zero, so that after noting that (5.17.) u"(w ) o _ R,(w ) = ° u'(w ) o A 2(w +c) o 1 + (w + c ) o i t follows that only (5.16.) should be considered. 2 F i n a l l y , inserting (5.16.) into (5.17.) gives r 2 .2 o (5.18.) A , o \ 2TT[TT +h 44 +h TT u / 2 .2,2 . 2, - VLTT -h J -4TT J , 2.2. /' 2 ^2,2 . 2 (TT +h )V[TT -h 3 -4TT r — so that we derived the absolute r i s k aversion at the i n i t i a l wealth w as a o function of h and the r i s k premium TT for the gamble z = <h;-h> for a decision maker with the u t i l i t y function u(w) y arctan(w + c) - 91 - At the end of this section we would l i k e to have a closer look at the absolute r i s k aversion R as a function of the r i s k premium ir for the A gamble z = <h;-h> and the u t i l i t y functions u(w;0,1 ,T,B,r)), x = -1, h, 1, 2. From Table 3.3. we see that the corresponding functions R^ can be written as 2TT (5.19.) h 2 + a(x) -TT x = - l , h, l , 2, 2 where a ( - l ) = -1, a(%) = 4, a ( l ) = 1, and a(2) = 0. The function a(x) i s given i n Figure 5.1. for the specified values T = -1. %, 1, and 2. O(T) Figure 5.1. The function a(x) i n (5.19.) for T = -1, k, I, and 2. -;i *- -1 From (5.19.) we e a s i l y derive the following: (5.20.) a(T) = 2 T = -1, %, 1, 2, TTR A but, of course, i t would be of much more interest a function of the parameter T . to express a d ) solely as In that way we would have achieved a s i g n i f i c a n t generalization of our previous results. - 92 - Note that we can v i s u a l i z e , f o r instance, a hyperbolic shaped l i n e through the three points i n the f i r s t quadrant of Figure 5.1. Also note that the fourth point ( r , a ) = (-1,-1) i s the mirror image of (r,a) = (1,1) with respect to the l i n e a = - T. I f indeed i t were possible to derive a(T) s o l e l y as a function of T, we would have at once mathematical results f o r a l l functions included by u(w;0,l,T,g,n)• For example, suppose a(-b) to be known, then we d i r e c t l y e s t a b l i s h the function f o r the u t i l i t y function 3/ u(w;0,l,-%,g,n) ^ - (w + c ) through (5.19.) by plugging i n the value of 2 a(-b)results. At t h i s point, however, we have not established such extended - 93 - 5.2. Extended results for the r e l a t i v e r i s k aversion In Chapter 3 we derived R.(w A O R^^)' ) as a function of the parameters of a gamble z, and a response variable of that gamble ( i . e . certainty equivalent, or r i s k premium, or p r o b a b i l i t y equivalent, or gain equivalent). We are also able to derive similar results for the r e l a t i v e r i s k aversion R (w ) R o at the status quo w .. By D e f i n i t i o n 2.9. we have R s R (w ) = w R.(w ) = o R R o o A o w^R^. To be s p e c i f i c now, R^ can be expressed as a function of: (i) the parameters of a nondegenerate gamble z, as a f r a c t i o n of the i n i t i a l wealth w^ (ii) i n case of gains and losses; and the certainty equivalent, or the r i s k premium, or the gain equivalent of that gamble, as a f r a c t i o n of An i l l u s t r a t i o n of the preceding w^. i s a decision maker having a quadratic u t i l i t y function u(w; 0,1 ,--1, B ,Tl) 'v* - (w + c ) where c = - 5. 2 TT he states TT* = — as his r e l a t i v e r i s k o w ° h f a i r , nondegenerate r e l a t i v e gamble z = <h*;-h*> = <—;w o we have: i n i t i a l wealth w (5.21.) R = w R 2TT 2 o , h - 1. Note that h* be like premium for the h i — > . By (3.37.) w o 2TT* TT 2 (h*) 2 - Hence we expressed for this decision maker R^ •ay At his P can be s t a t e d l i k e "one f i f t h " t e n p e r c e n t of my status quo". of the (TT*) = 2 ^R^ ^ d e c i s i o n Baker's & S Q status quo" A ^ and U N C his T ^ O N °^ response - 94 - h* s —— , which i s the standard deviation h of a f a i r f i f t y - f i f t y gamble w TT ° as a f r a c t i o n of w , and TT* * — , which i s the gamble's r i s k premium as a o w o f r a c t i o n of w . The results for the other u t i l i t y functions i n Table 1 o follow s i m i l a r l y . Using enables us also to obtain results for u t i l i t y functions for which results regarding R are d i f f i c u l t — i f not impossible— to obtain. An example i s provided^ by the special exponential u t i l i t y function defined by (5.22.) Consider for u(w) % - e the function n (5.22.): \ (5.23.) As R where c > max (0, 2w) the f a i r f i f t y - f i f t y gamble z = <h;-h>, so that by (2.3.) we have w / c c/ e l = = R (w ) = - w R R o o (5.24.) - -TT ° w +h . O , --e u"(w ) — , v u (w ; o + 2w — = + c w ( w -h 1 --e O , i t follows that o c = w (R - 2) o R Substitution of (5.24.) into (5.23.) yields y (5.25.) which gives R e -* 1 R v 2 - i-e 1 + l 1 v 2 * • 2 i.e ""* 1 as an i m p l i c i t function of h* and TT*. / - 95 - So far we have shown, that for some u t i l i t y functions we might be able to express the r e l a t i v e r i s k aversion (ii). = R^C ) W 0 a s a function of ( i ) and However, we have not e x p l i c i t l y discussed the purpose of doing this. To understand t h i s point, we should note that our results i n Chapter 3 derived the function to be independent from the i n i t i a l wealth W . q This i s an important observation, since therefore we do not require the knowledge of the decision maker's status quo W i n order to obtain his response (e.g. q r i s k premium) to a nondegenerate gamble, nor do we need to know W i n order q to calculate the value of the function R^ for this gamble. So, i f i t i s not feasible to obtain the function R. without w , we may s t i l l be able to arrive A o at a r e s u l t where R^ i s expressed as a function of ( i ) and ( i i ) . That i s , i n t h i s case the knowledge of W would not be required i n order to evaluate q the function R,,. - 96 - 5.3. Extended results for other gambles. In this section we w i l l point out that the results of Chapter 3 can be extended to gambles other than = <h,p;-h> and z^ = results may be obtained for a generalization of z gamble z = <h^,p;h >. 2 <h,p;0>. First, and z^, namely the Case 1 below w i l l center on this gamble. Second, instead of looking at discrete gambles, we may think of considering continuous gambles. Several continuous gambles are i l l u s t r a t e d i n the Cases 2, 3 and 4 below. Case ph 1 1: Consider the gamble z = <h^,p;h > where h^ f h^, and E(z) = + (l-p)h 2 2 = p(h - h ) + h x 2 2 : Figure 5.2. z : The gamble z = <h ,p;h > so that by (2.1.) we have: (5.26.) u(w +z) = p-u(w +h,) + (l-p)-u(w +h„) o o 1 o 2 Hence, for the u t i l i t y function u(w;0,l,T,8,n) with 1 t 1 the following relationship: (5.26.) yields - 97 - r 1-T B(w +z) o +n BU-T) (5.27.) rB(w +h. ) 2—L- B(1-T) 1-T r + 1-T B(w +h ) o 2 + (1-P) n Using (3.20.) and further reducing (5.27.) gives: (5.28.) 1 + 2R. 1-T = p 1+ h 1-T lV (1-p) E x p l i c i t r e s u l t s are given for two cases by specifying T. h 1 + 1-T 2V Setting T - 1 gives by (5.28.) f o r u(w;0,l,-l,8,r,) ^ - (w + c ) , with c = - j 2 (5.29.) (1 - 2R ) = 2 p(l - h R ) x + A (1-P)(1 " h R ) ' 2 A which gives 2[2 - p h - ( l - p ) h l x (5.30) R ~ a 2 2 - ph - (l-p)h 2 2 2 which i s a generalization of respectively (3.36.) and (3.89.), as can d i r e c t l y be shown by selecting By setting c = = h and respectively h 1-P 1 + bh R 1 + 1 + b2R, 2 which gives (5.32.) = - h and h T = 2 i n (5.28.) we obtain for u(w;o,l,2,p\n) ^ - 2n (5.31.) 2 2[2 - p h - ( l - p ) h ] x A h h 2 2 - ph 2 - (1-P)h 2 2 1 A w + c > w = 0. 2 i t h - 98 - which i s a generalization of respectively (3.45.) and (3.99.), as can d i r e c t l y be shown by selecting h^ = h and respectively Case 2: = -h and h^ = 0. Consider a gamble z solely given by i t s mean and variance. We are thinking of I being continuous, although our statements are equally v a l i d for z having a discrete d i s t r i b u t i o n . As no s p e c i f i c a t i o n of the d i s t r i b u t i o n of z has been given, we do not expect to arrive at complete results for many d i f f e r e n t u t i l i t y functions^ However, the example given i n this section should serve as a f i r s t step towards other results possibly to be obtained by good approximations. A gamble s o l e l y given by i t s mean and variance i s e s p e c i a l l y i n t e r e s t i n g from a finance theoretic point of view. For example, the o r i g i n a l Markowitz(1952) theory of p o r t f o l i o selection assumes that the decision maker's (investor's) preferences can be ranked completely i n terms of mean and variance of the p o r t f o l i o return. Also, the discussion on the mean-standard deviation analysis i n Tsiang(1972), Bierwag(1974), Borch(1974), Levy(1974), and Tsiang(1974) i l l u s t r a t e s the interest i n gambles solely given by mean and variance (or standard deviation). We w i l l not further digress 2 on the finance theoretic l i t e r a t u r e of this topic, but instead emphasize that most of the l i t e r a t u r e on gambles solely given by their mean and variance mention the special role of the quadratic u t i l i t y function i n this respect. The reason behind this i s , that for a quadratic u t i l i t y function u(w) = u(w; ,1,-1,B,ri) y - (w + c) , where c = - — , the expected u t i l i t y of a P 2. Closely r e l a t e d references include Biervag(1973), Borch(1969,1973), and Nigro(1972), Hanoch and Levy(1970), and Tobin(1958,1969). Feldstein(1969), G l u s t o f f - 99 - gamble I depends only on the mean u and the variance a hence not on the d i s t r i b u t i o n of the gamble. E[u(z)] = of the gamble, and Namely: E[-(z+c) ] Z - E(2 ) - 2cE(z) - c 2 2 2 2 2 - (y + o ) - 2cu - c (5.33.) 2 2 - (y + c) - a = u(y) - a . For a decision maker with a quadratic u t i l i t y function u(w;0,1,-1, 6,TI) we have by (2.3.): (5.34.) Developing (5.35.) 2 - (w + y - TT + c) o 2 E [ - (w + 1 + c) ] o = the right hand side of (5.34.) gives E[-(w +z+c) ]= -(w + c ) - 2(w +c)y - (y +o ) o o o 2 2 2 2 Substitution of (5.35.) into (5.34.) yields after some rearranging: (5.36.) According 2 2 - 2(w +C)TT = a + 2yrr - TT o to (3.20.) we have = - — ^ — - , which gives with (5.36.) o - 100 - (5.37.) R A - 2 l T a 2 2 + 2yrr - Tr Note that (5.37.) generalizes our i n i t i a l Theorem 3.1. for non-fair gambles. That i s , for a f a i r gamble we have i n (3.10.). y = 0 and hence (5.37.) results d i r e c t l y Also, considering the gamble z = <h;-h>, which has y = 0 and 2 2 . = h , we obtain (3.37.) from (5.37.). S i m i l a r l y , for the gamble z = 2 2 <h;0> with y = ^h and c = fch we easily derive (3.90.) from (5.37.) a Case 3: Consider the normally distributed gamble z with mean y and 2 „ variance o , i . e . P(z<z) = / f^(z)dz , where 2 —00 _ (g-y) i (5.38.) —i=»e f„(z) = N " 2 2 °2 where Tt = 3.14. . OV2TI For this gamble we have by (2.1.): +oo (5.39.) u(w +z) = / u(w +z)f„(z)dz o o N -co Selecting for example the u t i l i t y function u(w;0,1,°°,c,1) ^ - e C W we derive from (5.39.): -c(w +2) -c(w +•£.) - e = - / e f„(z;dz N J —CO (5 40 ) v:>.tu.; i) ^ ~ = _f c w 2 2 o-cy+^c 0 +°° / e 1-;[z-y+co _ 2,2 ] 2o^ d r 5 - 101 - 1.2 2 (5.40.) -cw -cu+^c o - e ° = which easily gives / c / i -v (5.41.) e " c 2 = e -cy+%c a 2 2 so that, using 2 = u - TT , where ir i s the r i s k premium for the gamble z, and c = R , we have: (5.42.) R &LZ». = a = ^ o 2 2 If we select the u t i l i t y function to be the quadratic u(w;0,1 ,-1, (3,ri) 2 x\ - (w + c) , with c = - -pr , we can find i n a similar P (5 43 ) ' U J J K R A = ~ 2 + u 2 ( y 2 a g ) ^2 - z = 2 a way: ?! „ 2 + 2yrr - TT Of course, this r e s u l t also follows d i r e c t l y from (5.37.) i n Case 2, Case 4 : Consider the gamble z uniformly distributed h l + n 2 gamble, where E(z) = (5.44.) u(w +2) o where f , i s defined by TTl UN , = (2.1.) gives / u(w +z)f (2)d2 o UN on [h ,h ] . For this - 102 - h 2 " h hj < z < l h 2 (5.45.) elsewhere Hence, for u(w;0,1,T,8,n) with x ^ 1 r B(w +z) B(I-T) (5.44.) results i n : ^ + n 1-T +00 r -oo ° h — r h— - f u(w + z)dz 2 (5.46.) B(w + h „ ) o "2' >2 + n T r B(w +h,) o (h2-h1)B<£(l-T)(2-T) which gives, using (3.20.), for T (5.47.) ,2-T -- + z A 2 M 1-T T i + n r 2-T — (h2-h )(2-T) 1 R R h + A 1 or zR.^ (5.48.) 1 + 2-T 2-T 1-T A (h2-h )(2-T) 1+ h 2 A h R 1+ l A R 1 By selecting d i f f e r e n t values for T we w i l l be able to derive the results T - 103 - Eor example, l e t us select T = for this uniformly d i s t r i b u t e d gamble. i . e . the quadratic u t i l i t y function. (5.49.) R For this function (5.48.) gives: A - V " 3lh7h7(1-h.R.) 2 A (1 2 -1, 2 3 - (l-h.R ) 1 A 3 A from which i t follows that (5.50.) p 22 - (h +h ) A = Jl 2 l ,2 - 3[h ^ r 2 + h ^ + hj] Of course, this r e s u l t could have been obtained d i r e c t l y from (5.37.) i n Case 2, by noting that y = E(z) = uniformly d i s t r i b u t e d gamble z. \ 1 2^ 2 l^ h +h 2 a n d 0 = 1 12"^ 2~ l-' h h 2 f o r t h e - 5.4. 104 - The results using Taylor series expansions. The idea behind 3 . 1 . , was the mathematical r e s u l t s , as pointed out i n section to expand both sides of the (2.4.) u(w - TT) = E[u(w + o using Taylor series. equation z)] o O r i g i n a l l y we derived our results by expanding (2.4.) e x p l i c i t l y for the functions i n Table 3 . 1 . , i . e . we e x p l i c i t l y calculated (3.1.) and For example, consider the hyperbolic u t i l i t y function (3.2.). u(w;0,1,1,B,n) y - — - — , where c = w + c p simply denoted by u(w), we have tc c i •» (5.51.) so that u (3.1.) (m), \ (w) _ f -1 * w + c = For this function, from now on m. becomes u(w o - TT) = u(w = u(w o TT111/ , m+1 ) - E. — ( - 1 ) m=l m. ) - ? m=l N -u (m), (w ) o (5.52.) and (3.2.) o m , , m+1 (w +c) o N can also be e x p l i c i t l y calculated for the f a i r gamble z = <h;-h>, where again the u t i l i t y function i s taken to be the hyperbolic u t i l i t y function u(w) <v u(w;0,1,1,B,n.) • We derived: - 105 - E[u(w +z)] = ^u(w +h) + %u(w -h) o o o , . 1 » h (m), , 1 92 h ,n+l (m) . = u(w )+•-;. I, — r - u (w ) - -r- h — r (( - l ) -u (w ) o 2 m=l m. o 2 m=l m. o (5.53.) .m u(w ) ~ —• L -• , o 2 m=l , .... \m+l (w +c) o 1. , ? si h m 2 m-1 (w +c)m+1 o m 2m u(w ) o Equating 2m+l m=l (w +c) o (5.52.) and (5.53.) according to (2.4.) gives: 2m (5.54.) m= 1 w + cj v w+c o u"(w ) , , °v = —— u(w) w+c o o 2 so that with R, s R, (w ) = A A o (5.55.) m=l The formula (5.56.) as without (5.57.) = A we obtain S t(%hR ) ] . m=l A 2 1 m A (5.55.) e a s i l y yields: - 1 1 - %TTR, - 1 = 1 - (*>hR V A (%hR V < 1 and A w + c w+c o o loss of generality. Then, (5.56.) gives h 2 < 1 can be assumed - 106 - which was d i f f e r e n t l y d e r i v e d i n (3.46.). an R % Note t h a t t h i s r e s u l t d i s p l a y s ESTIMATION ERROR equal to z e r o . A Other examples are s i m i l a r l y d e r i v e d , e s p e c i a l l y f o r the q u a d r a t i c utility function. The l a t t e r was shown i n the p r o o f o f Theorem 3.1., where i t h o l d s , t h a t u ^ ( w ; 0 , 1 ,-1,B,n) = 0 f o r m = 3, 4, r e s u l t s i n Chapter expansions. C e r t a i n l y , the 3 w i l l not be e a s i e r o b t a i n e d by u s i n g T a y l o r s e r i e s However, we b e l i e v e t h a t the p r e c e d i n g example may basis for deriving results for u t i l i t y f u n c t i o n s other than those l i s t e d i n T a b l e 3.1. or f o r gambles other than those d e a l t w i t h i n Chapter i serve as a 3. - 107 - 5.5. The sum-of-exponentials u t i l i t y function. One of the most used u t i l i t y functions in decision analysis i s the sum-of-exponentials u t i l i t y function, defined by (5.63.) _ -*w u ( w ) e _ . b where a,b,c>0 , a*c _ c w e For this u t i l i t y function we w i l l present i n this section some mathematical r e s u l t s similar to those given i n Chapter 3. Let TT. = TT.(W ,h.), i = 1, 2, 3, be the r i s k premiums for the gambles 1 l o l = <h^;-h.>, i = 1, 2, 3 respectively, for a decision maker who behaves according to the sum-of-exponentials u t i l i t y function (5.63.). assumed that (5.64.) f f h^. According to (2.4.) we e a s i l y obtain u(w - T T . ) O It i s = E[u(w +2.)] o i 1 i = 1, 2, 3, which y i e l d s for (5.63.): -a(w e + 1 (5.65.) 1 2 so -c(w -TT -TT.) ° b-e 0 ) 1 f-a(w+h.) -c(w +h ) -a(w -h ) ° + b-e ° + e ° b-e 1 1 e that after rearranging we obtain the following: 1 + -c(w - 108 - aTT -aw i 1 ah. e -ah. + e 1 1 (5.66.) + b-e -cw CTT. ch. 1 e -eh. l + e I = 0 Thus aTT. (5.67.) -(c-a)w e b-e cosh(ah.) I CTT. i = 1, 2, 3. cosh(ch.) I I f we define aTT. (5.68.) e - cosh(ah.) H. s 1 e CTT. 1 i = 1, 2, 3, - cosh(ch.) I then (5.67.) i s (5.69.) -(c-a)w b-e i = 1, 2, 3, so that the following results: -aw -cw I o ,2 o a e + be e 9 R. s R (w ) = A A o' (5.70.) a _ 2 u"(w ) 2_ u'(w ) o - -(c-a)w 2, o + c be -(c-a)w + cbe -aw ae 2 a 2 o ,, + bee -cw o U - c H. a - cH. I i = 1, 2, 3. - 109 - Hence, we found a (5.71.) 2 2 - c H. 1 a - cH^ = 2 2 a - c H „ ?. = a - cH^ 2 2 a - c H, i a - cH^ U which after cross-multiplications reduces to: (5.72.) H l = H 2 = H 3 One of the above three equations i s redundant, while the other two equations (e.g. H^ = H^ and H^ = H^) provide through existing numerical procedures solutions for a and c. I f the value w of the decision maker's i n i t i a l o wealth i s known, the third parameter b i s easily obtained from (5.69.). In other words, with the preceding we established a procedure to determine the three parameters of (5.63.) i f the decision maker i s known to behave according to this sum-of-exponentials Example: utility function. A decision maker with a sum-of-exponentials u t i l i t y function u(w) —aw — cw . . _ - e - b-e i s known to state the following r i s k premiums for the gambles z^ = <1;-1>, z" = <2;-2>, and z^ = <3;-3>: 2 (5.73.) TTj = . 50 and TT = 1.50 2 and TT = 2.57 3 The values i n (5.73.) are a r t i f i c i a l l y selected inorder to arrive at integer parameter values l a t e r on. The equations (5.72.) with become for these values: defined by (5.68.) - 110 - ' air aiT e (5.74.) - cosh(a) e - cosh(c) e CTT1 air^ 2 - cosh(2a) e - cosh(2c) e CTT_ e - cosh(3a) CTT- A numerically derived solution of (5.74.) y i e l d s : - cosh(3c) a « 1 and c =s 2. One can check: 1.65 - 1.54 _ 4.48 - 3.76 _ 13.09 - 10.07 2.71 - 3.76 20.09 - 27.31 ~ 171.37 - 201.72 % (5.75.) 0.1 = H ( i = 1,2,3) Hence 1 - 4H. (5.76.) R = « 1.167. 1 - 2H. x If i n addition the decision maker's i n i t i a l wealth W q then according to (5.69.) we have (c-a)w (5.77.) b = - e °H ± * 1, so that we f i n a l l y obtain: (5.78.) u(w) - e~ W - e" 2 w i s known to be w c - Ill - 5.6. Summary. The basis for the R^-method i s provided by the mathematical results i n Chapter 3. E s s e n t i a l i s the idea that for a number of u t i l i t y functions, the absolute r i s k aversion at the i n i t i a l wealth w , i . e . R.(w ), can be O A O expressed as a function of (i) the parameters of a nondegenerate gamble; and (ii) the decision maker's response to that gamble, i n p a r t i c u l a r : the r i s k premium, or the certainty equivalent, or the p r o b a b i l i t y equivalent, or the gain equivalent of that gamble. In Chapter 3 we derived the results for f i v e r i s k averse u t i l i t y functions within the general class u(w;£,a,x,B,n)» as introduced and defined by (3.14.). These f i v e alternative u t i l i t y functions also belong to the well known HARAclass. We obtained r e s u l t s r e f e r r i n g to both the gamble 2 = <h,p;-h> and the gamble z = <h,p;0>. The R -method as discussed i n section 4.1. employs these mathematical A r e s u l t s to obtain a procedure for selecting among a l t e r n a t i v e u t i l i t y functions. The basic idea of the R -method i s , that R.(w ) has a s p e c i f i c A A o value, no matter what gamble the decision maker i s faced with. determines The R^-method which u t i l i t y function, chosen from the f i v e r i s k averse u t i l i t y functions, the decision maker conforms to most c l o s e l y . selected should have the least r e l a t i v e standard The function to be deviation for two or more values of the function R, = R.(w ). The values of R. are obtained by the A A o A decision maker's responses for two or more reference gambles, and c a l c u l a t i n g - 112 the corresponding R - values as given i n Chapter 3. Section 4.2. extends A the R^-method to incorporate the r i s k prone and the r i s k neutral cases. Also, the case where the decision maker i s both r i s k averse and r i s k prone over d i f f e r e n t regions was b r i e f l y discussed. Chapter 5 provides good d i r e c t i o n as to how one may method. extend the R - It should be clear that the method w i l l c e r t a i n l y improve by i n - cluding a larger set of alternative u t i l i t y functions. Thereto, i n Chapter 5 we derived and gave d i r e c t i o n to various mathematical extensions of our r e s u l t s i n Chapter 3. -113 - References Arrow, K.J. Lectures, (1965), "Aspects o f the t h e o r y of r i s k b e a r i n g " , Y r j o Jahnsson S a a t i o , H e l s i n k i . Arrow, K.J. 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A mathematical procedure for selecting among alternative utility functions Schoot, Gerrit Paul van der 1981
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Title | A mathematical procedure for selecting among alternative utility functions |
Creator |
Schoot, Gerrit Paul van der |
Date Issued | 1981 |
Description | This thesis presents a mathematical procedure, called the R^-method, for selecting among alternative utility functions to represent a decision maker's risk preference. A general class of utility functions is introduced and for five alternative members of this class, the absolute risk aversion at the initial wealth w[sub o] , i.e. R[sub A](w[sub o] ), is expressed as a function of: (i) the parameters of a nondegenerate gamble z; and (ii) the decision maker's response to that gamble (in terms of risk premium, or certainty equivalent, or probability equivalent, or gain equivalent). Mathematical results are obtained for two different gambles. The R[sub A]-method calculates the values of R[sub A] for several responses to different reference gambles, and then selects the utility function with the least relative standard deviation over the R[sub A] values. The procedure is based on the fact, that for the decision maker's actual utility function, R[sub A] must theoretically attain the same value at w[sub o], namely R[sub A](w[sub o]), no matter what gamble is used to assess R[sub A]. Suggestions are made for extending the R[sub A]-method to incorporate risk proneness as well as attitudes which are risk averse over one part of the domain and risk seeking over another part. Finally, a chapter on mathematical extensions is provided in order to improve the R[sub A]-method by including a larger set of alternative utility functions. |
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Thesis/Dissertation |
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Text |
Language | eng |
Date Available | 2010-03-29 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0094984 |
URI | http://hdl.handle.net/2429/22837 |
Degree |
Master of Science in Business - MScB |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
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