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A mathematical procedure for selecting among alternative utility functions Schoot, Gerrit Paul van der 1981

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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 of Technology, 1974 M.Sc., D e l f t U n i v e r s i t y of Technology, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUSINESS ADMINISTRATION i n THE FACULTY OF GRADUATE STUDIES (The Faculty of Commerce and Business Administration) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1981 © G e r r i t Paul van der Schoot, 1981 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or pub l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Hou^eM T^, V<^%\. n r _ C I O /no \ - i -Abstract This thesis presents a mathematical procedure, called the R^-method, for selecting among alternative u t i l i t y functions to represent a decision maker's risk preference. A general class of u t i l i t y functions is introduced and for five alternative members of this class, the absolute risk aversion at the i n i t i a l wealth w , i.e. R.(w ), is 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 (in terms of risk premium, or certainty equivalent, or probability equivalent, or gain equivalent). Mathematical results are obtained for two different gambles. The R^-method calculates the values of R^ for several responses to different reference gambles, and then selects the u t i l i t y function with the least relative standard deviation over the R^  values. The procedure i s based on the fact, that for the decision maker's actual u t i l i t y function, R^  must theoretically attain the same value at Wq, namely R^( W D)> n o matter what gamble is used to assess R^ . Suggestions are made for extending the RA-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 mathe-matical extensions i s provided in order to improve the R^-method by including a larger set of alternative u t i l i t y functions. Contents Abstract Contents 1. Introduction 2. Notation and Preliminaries 3. Mathematical Results for R.(w ) A o 3.1. The idea behind the mathematical results 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 risk premium for the gamble z = <h,p;-h> 3.A. Risk aversion R^  as a function of probability 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 risk premium for the gamble z = <h,p;0> 3.7. Risk aversion R as a function of probability equivalent . for 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 risk averse decision makers A 4.2. The R^-method including risk proneness and risk neutrality - i i i -5. Mathematical Extensions : 85 5.1. Extended results for u(w;£,a,T,g,n) : 87 5.2. Extended results for the relative risk aversion ^(w^^) : 93 5.3. Extended results for other gambles : 96 5.4. The results using Taylor series expansions 5.5. The sum-of-exponentials u t i l i t y function 5.6. Summary •' H I References : 113 104 107 - i v -Acknowledgement The. author Is grateful to his; supervisor, Professor D, Wehrung, for many valuable comments and stimulating discussions throughout the preparation of this thesis-. He would also like to thank the other committee members-, Professors B., Schwab and L.G. Mitten, for their assistance. Finally, suggestions- of Professor J.W. Pratt of Harvard Business School have been helpful in determining the current state of knowledge in the f i e l d . - 1 -1 Introduction Decision theory i s often c l a s s i f i e d under two i n t e r r e l a t e d headings, normative and d e s c r i p t i v e . Normative decision theory i s concerned with the p r e s c r i p t i o n of courses of action that conform most c l o s e l y to the decision maker's preferences. The purpose of d e s c r i p t i v e decision theory i s to describe the decision maker's preferences and the way he incorporates these preferences into h i s decisions. In other words, the normative theory guides the d e c i s i o n maker i n what he should do, whereas the d e s c r i p t i v e theory focusses on what the decision maker does do. Even though both l theories are concerned with behavioral aspects, i t appears that psycho-l o g i c a l concepts are mainly used when describing (and predicting) human decision making. The normative theory usually follows an axiomatic approach and i s i n the l i t e r a t u r e also known under the heading decision 2 a n a l y s i s . As the main r e s u l t of t h i s research i s a normative tool for 3 decision making, our focus w i l l be on decision analysis, but behavioral and psychological aspects w i l l be covered as we go along. 1. Reviews of b e h a v i o r a l d e c i s i o n theory i n c l u d e Edwardst1954,1961), Luce and SuppesC1965), Becker and McClintock(1967), 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 Wallsten(1972), F i s c h e r and Ed-wards(1973), 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 of these references are mainly 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 provides a normative theory which p r e s c r i b e s how a d e c i s i o n maker should behave i n order to be c o n s i s t e n t w i t h h i s judgments and preferences. I t does not provide 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 readings i n d e c i s i o n a n a l y s i s include 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 issues on d e c i s i o n a n a l y s i s appeared i n IEEE Transactions on SSC (see lloward(1968)) and Operations Research (see Kirkwood(1980)). Many t e x t -books i n d e c i s i o n a n a l y s i s —at 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 Petersont1974), Keeney and R a i f f a ( 1 9 7 6 ) , Jones(1977), LaValle(1978), and Holloway (1979). - 2 -Decision analysis is a discipline providing a normative, logical frame-work for decision making under uncertainty. Within this framework, the decision maker's preferences for the possible consequences of any action are a key element in the problem description. The logic of decision analysis is provided by a set of axioms. For the sources of these axioms we may actually have to go back to Bernoulli(1738) who set forth the idea of maximizing u t i l i t y . However, the axiomization of u t i l i t y was f i r s t rigorously developed with the work of Von Neumann und Morgenstern(1947). In an economic, game-theoretic context they postulated the set of axioms which we w i l l adopt in this research. Essential is the idea that i f an appropriate u t i l i t y (value) is assigned to each consequence and the expected u t i l i t y of each alternative is calculated, then the best course of action is 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 in 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, Raiffa and Schlaifer(1965), and Fishburn(1970), who came up with different 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 probabilities exist, or that both (sub-jective) probabilities and u t i l i t i e s can be jointly derived. At this point we w i l l not dwell on the problems associated with subjective probabilities, but instead assume that the decision maker knows the likelihoods of the possible consequences. Hence, we w i l l basically follow Luce and Raiffa(1957) 4. Fishburn(1968) provides a review article on u t i l i t y theory with an extensive bibliography up to 1968. A more recent review is given by FishburnC1978). - 3 -in this respect. The next important step then is 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 is an essential ingredient in the analysis of decisions under uncertainty. These preferences can be represented by u t i l i t y functions in the sense of Von Neumann and Morgenstern(1947). Many different techniques for evaluating u t i l i t y functions have been proposed. 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 classification of attitudes towards risk. 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 effective 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 alternatives. Basic to the new procedure is the absolute risk aversion R.(w) over wealth, independently introduced by Pratt(1964) and Arrow(1965,1971). For each of a number of u t i l i t y functions we wi l l express the absolute risk aversion at the i n i t i a l wealth level w , i.e. R,(w ), as a function of: o A o. (i) the parameters of a nondegenerate gamble z, and ( i i ) the certainty equivalent, or the risk premium, or the probability 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 : Graysonf1 9 6 0 ) , Swalm(1966), Spetzler ( 1 9 6 8 ), Howard, M a t h e s o n and North (1972 ), de N e u f v i l l e and Keeney(1972), and Hauser and Urban (1979) . - 4 -The u t i l i t y functions include the linear function and five functions within the well known Hyperbolic Absolute Risk Ayersipn (HARA) class. 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) is 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 in several ways. It enables us to use different response modes, thus avoiding problems inherent in using only one response mode6. Also, the R^-method has the advantage of requiring knowledge about the decision maker's i n i t i a l endowment only after responses for the reference gambles have been obtained. In spite of the sometimes intricate mathematical derivations, the results are not complicated and hence the R^-method can be applied in a straightforward manner. The R -method is described for a limited class of u t i l i t y functions, but section 4.2, and Chapter 5 w i l l give various extensions and provide several directions as to how one may proceed to enlarge the usefulness of the R^-method. Finally, the basic idea to express R. = R.(w ) as a function of (i) and ( i i ) seems to A A o be a clear basis for extended research. The outline of the thesis is as follows. After the necessary notation, definitions, lemmas and other preliminaries are given in Chapter 2, we w i l l present in Chapter 3 our results for R (w ) expressed as function of (i) and A O ( i i ) . 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 (unknown) risk averse u t i l i t y function. 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 and S l o v i c ( 1 9 7 1 ) , S l o v i c ( 1 9 7 2 ) , L i c h t e n -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 ) . assumption i s violated. Hence, in this section we w i l l certainly consider risk proneness and risk neutrality. Chapter 5 w i l l be a collection of mathematical extensions and should serve as a direction for extending the use of the R.-method. At the end of the thesis our reference l i s t is A included. - 6 -2 Notation and Prel iminar ies 1 A gamble z, which r e s u l t s i n a net change of h^ with p r o b a b i l i t y p (0 < p < 1) and a net change of h with the complementary p r o b a b a i l i t y 1-p, i s denoted by z = <h^,p;h2> . Note that the p r o b a b i l i t y p, which appears e x p l i c i t l y within the brackets, i s the l i k e l i h o o d of the f i r s t of the two values that appear within the brackets. The gamble z i s said to be non-degenerate i f 0 < p < 1 . Usually gambles are presented either by branch diagrams (Figure 2.1.) or by charts (Figure 2.2.): z Figure 2.1. The gamble z = <h l 5p;h 2> presented as a branch diagram. z Figure 2.2. The gamble z = <h^,p;h2> presented as a chart, where the shaded area i s a f r a c t i o n p of the t o t a l area. 1. Wc assume t h e r e a d e r t o be 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 and d e f i n i t i o n s g i v e n by P r a t t ( 1 9 6 4 ) and S c h l a i f e r ( 1 9 6 9 ) . T h i s c h a p t e r draws h e a v i l y on t h e s e 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 spe 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 by z = <h j ^ h ^ = <' 11'^2 > ' T h e 8 a m b l e 2 *-s s a ^ t o t>e f a i r i f i t s expected value i s zero, i . e . E(z) = 0 . Hence, a f a i r f i f t y - f i f t y gamble can be represented by <h;-h> , which has a standard deviation equal to h . Let now u(w) be a continuous, unidimensional, monotonically increasing 2 u t i l i t y function 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 consider decreasing functions, however the derived r e s u l t s w i l l be quite s i m i l a r for monotonically decreasing u t i l i t y functions. Assume the decision maker's i n i t i a l endowment to be w o D e f i n i t i o n 2.1. A ce r t a i n t y equivalent of a gamble z i s an amount z = z(w ,z), such that the decision maker i s i n d i f f e r e n t o between z and the amount z for sure. Hence, z = 2(w ,z) i s defined by o (2.1.) u(w +z) = E[u(w +Z ) ] . o o As we have chosen the a t t r i b u t e w to be wealth, we may substitute cash 3 equivalent for ce r t a i n t y equivalent. Note that, of course, z i s a function of the gamble z i t s e l f , but also of the i n i t i a l endowment w . In other words, i t i s assumed that the decision maker may behave d i f f e r e n t l y at 2. The c o n c e p t s a r e e q u a l l y v a l i d f o r n o n m o n e t a r y a t t r i b u t e s . 3. C a s h e q u i v a l e n t s a r e e x t e n s i v e l y d i s c u s s e d i n L a V a l l e t 1 9 6 8 ) . - 8 -various wealth levels. Except for the case of a linear u t i l i t y function, a decision maker is assumed to state different cash equivalents at different i n i t i a l wealth positions for the same nondegenerate gamble. Let us illustrate this point with the following example. Example 2.1. A decision maker is assumed to behave according to the u t i l i t y 2 function u^(w) = w-(.25)w , where 0 ^ w ^  1.5 . Suppose he is faced with the fair f i f t y - f i f t y gamble z = <0.5;-0.5> . Using (2.1.) i t is easy to calculate that at an i n i t i a l wealth of w^  = 0.5 his certainty equivalent is z(w',z) « -0.081, as i t holds that u(0.5-0.081) « J>u(0) + ^u(l) . o Thus, the decision maker is indifferent between the gamble and losing a sure amount of ,.0.081 . At an i n i t i a l wealth of w^  = 1 his certainty equivalent is 2(w",z) w -0.118, as u(l-0.118) « %u(0.5) > %u(l.5) . Hence, z(w',£) o o f 2(w",z) . o Definition 2.2. The risk premium TT = TT(W , z) of a gamble z is its expected t Q value minus i t s certainty equivalent. Hence, TT = TT(W . z) is defined by o (2.2.) TT = E(z) - 2 , where z = 2(w ,z) . o From (2.2.) we see that z = E(z) - TT , which after substitution in (2.1.) results in the following fundamental relationship: (2.3'.) u[w + E(z~) - TT] = E[u(w +z) ] o o - 9 -or, more formally (2.3".) uLw + E(a) - TT(W ,Z)] = E[u(w +2)] o o o which reduces for a fa i r gamble to (2.4.) utw - TT(W ,z)] = Etu(w +z)] . o o o Example 2.2. Referring to Example 2.1. i t can be calculated, that, using (2.3.), the decision maker's risk premium at the i n i t i a l wealth level of w1 = 0.5 is TT(w',z) « 0.081, as u(0.5+0-0.081) « ^u(0) + ^u(l) . Similar-o o ly, ^(w^z) ^ 0.118 . Of course, these results could have been obtained more directly in this case. Clearly, for a fair gamble (2.2.) becomes '"' = -£, which proves the following lemma. Lemma 2.1. The risk premium of a fair gamble z is the negative of the certainty equivalent of that gamble. By Definition 2.2. we have for any constant c (2.5.) TT(W ,z) = "T(w +c,z-c) . o o It is often feasible and ill u s t r a t i v e to present a gambling situation with the corresponding risk premium in 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 in Example 2.1. We w i l l leave i t for the reader to verify 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 fa i r f i f t y - f i f t y gamble z = <0.5;-0.5> states for his risk premium IT = TT (w ,z) = 0.118 at his i n i t i a l wealth level w" = 1 . Definition 2.3. The insurance premium u =y ( w Q>z) of a gamble z is the negative of the certainty equivalent of that gamble. Hence, y (w ,"z) is defined by - 11 -(2.6.) u(w ,"2;) = - z(w ,z) . o o From Lemma 2.1. we derive, that for a fair gamble the risk and insurance premium coincide. Definition 2.4. An amount h such that the decision maker is indifferent between the status quo (his i n i t i a l wealth w ) and the o gamble z = <h,p;-h> is called the gamble's gain  equivalent and is denoted by h = h(w ,p,h) . Hence, h = h(wQ,p,h) is defined by (2.7.) u(w ) = E[u(w +z)] = p.u(w +h) + (l-p).u(w -h) . o o o o In order to have the expression gain equivalent make sense, we w i l l obvious-ly require h to be positive. Definition 2.5. A probability p .such that the decision maker is indiffe-rent between the status quo w and the gamble z = <h,p;-h> is called the gamble's probability equivalent and is denoted by p = p(w ,h) . o Hence, p = p(w ,h) is defined by (2.8.) 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 identical implications for action are called strategically equivalent. This notion can be formalized through the following definition. Definition 2.6. Two u t i l i t y functions, u^(w) and u^Cw), are strategically equivalent, written u,(w) ^ u„(w), i f there exist con-—* 1 2 stants a and $>0 such that u^(w) = a + B.u^Cw) . It is now straightforward to show that strategically equivalent u t i l i t y 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, let (2.9.) u1(w) = a + B.u2(w) where 8>0 and consider z = <h^,p;h2> . Then, we have by (2.1.) (2.10.) u,(w +z,) = p.u. (w +h_) + (1-p).u. (w +h ) l o l r 1 o 1 r 1 o 2 and (2.11.) u.(w +z„) = p.u.(w +h.) + (l-p).u 0(w +h ) l o l I o l l o l so that substitution of (2.9.) in (2.10.) yields a + r3.u2(w +2^) = p . [ a + 3.u„(w +h,)] + (1-p).[a + B.u„(w +h„)] which gives the following o \ l o l result: - 13 -(2.12.) u„(w +z) = p.u„(w +h) + (l-p ) . i i (w +h ) . 2 o l 2 o l 2 o 2 Hence, equating (2.11.) and (2.12.), we find that z, - 2„ . If u^(w) and u^(w) are strategically equivalent, we can also say that u^(w) determines u^(w) up to positive linear transformations. It w i l l appear to be quite important to categorize the attitude of a decision maker with respect to risk. Or, the question arises whether a decision maker is a risk lover or a risk avoider. This is not to say that a decision maker w i l l always exhibit the same risk attitude over the entire region. In fact, many researchers believe that often decision makers are risk averse for gains while being risk prone for negative outcomes. For theoretical purposes, however, Definition 2.7. assumes ho change of risk attitude over the entire domain. Definition 2.7. A decision maker is said to be risk averse i f u[E(w +z)] > E[u(w +£)], o o risk neutral i f u[E(w +z)] = E[u(w +z)], and o o risk 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 risk averse and risk neutral decision maker. A risk averse decision maker w i l l prefer the expected consequence of any nondegenerate gamble to the gamble i t s e l f . And i t is believed that this is the most common situation in practice. The risk neutral case will be in-U. See, f o r e x a m p l e , SwalraC1966) and Kahneman and T v e r s k y ( 1 9 7 9 ) . - 14 -\ eluded for completeness sake. The risk prone decision maker is not con-sidered here, even though our results can quite easily be extended in that direction. Lemma 2.2. A decision maker is risk averse i f and only i f his u t i l i t y function is concave. The proof of Lemma 2.2. can be found in Keeney and Raiffa(1976). Note that the u t i l i t y function in Example 2.1. is concave over the defined range. Therefore, this decision maker is risk averse. And indeed, consistent with Definition 2.7., i t holds that u[E(w'+z)] > E[u(w'+z)]. For example, o o choosing w1 = 0.5 and z = <0.5.-0.5> , we have u[E(w'+z)] = u[E(w')] = o 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 is risk 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 risk aversion, so that by Definition 2.7. we have u[E(w +z)] > E[u(w +z)] . Using (2.1.) we see that U L E ( W + Z ) ] o o o u(w +2), which for an increasing function yields o (2.14.) E(w +z) > w +2 . o o Noting that E(w +z) = w + E(z) , we see that (2.13.) follows directly from - 15 -(2.14.). Second, assuming (2.13.) gives directly (2.14.), so that U[E(WQ+Z)] > u(w +z) = E[u(w +z)], as the u t i l i t y function is increasing. Hence, the o o / decision maker is risk averse. This completes the proof of Lemma 2.3. • Using Lemma 2.3. and Definition 2.2. we easily arrive at the following Lemma 2.4. A decision maker with an increasing u t i l i t y function is risk averse i f and only i f his risk premium is positive for a l l nondegenerate gambles. For the risk 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 is risk 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 is risk neutral i f and only i f his risk premium is zero for a l l nondegenerate gambles. Now assume u(w) to be linear, 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 u[E(w +£)] = E[u(w +Z )L o o o o o tl - 16 -Hence, according to Definition 2.7., the decision maker is risk neutral, which proves the following lemma. Lemma 2.7. A decision maker with a linear u t i l i t y function u(w) = a + $w (8 > 0) is risk neutral. We are now able to investigate whether a decision maker is risk averse or not. What we would like to do next is to state a measure of risk aversion so that an indication is provided when one decision maker is more risk averse than another. The definition we are about to give here stems from Pratt (1964) and Arrow(1965,1971), and i t assumes u(w) is continuous and twice differentiable. Definition 2.8. The absolute risk aversion at w, R (w), of a u t i l i t y ^ — ^ — — — — i ^ — — — — — • • A function u(w), is defined by: (2.16.) R ( W ) . _ « ^ L . A u'(w) The absolute risk aversion R (w) is a function of w, and i t certainly seems A to be a measure with many desirable properties. Note that u'(w) is always positive for monotonically increasing functions and that u"(w) is negative for concave functions, so that R (w) is positive for risk averse decision makers. Similarly, i t can be shown that R.(w) = 0 for risk neutral decision A makers and R^(w) < 0 for risk prone decision makers. Two related lemmas follow: - 17 -Lemma 2.8. Two s t r a t e g i c a l l y equivalent u t i l i t y functions have the same absolute r i s k avers ion R,(w) . A Proof : For u^(w) = a + Bu^Cw) , where B > 0 , we have ui'Cw) Bu"(w) u''(w) (2 .17. ) R A 1 ( w ) = - - ^ = S = - JL = R (w). A > 1 uj(w) Bu'(w) u 2(w) A ' Z Lemma 2.9 . I f two u t i l i t y functions have the same absolute r i s k avers ion R^(w), then they are s t r a t e g i c a l l y equivalent , u"(w) d Proof : — R.(w) = , , . = — [ l o g u' (w)] so that by in teg ra t ion (with C. A u (w) dw 1 being an i n t e g r a t i o n constant) J — R^(w) dw = log u'(w) + , which y i e l d s a f ter exponentiat ing -JR (w)dw C (2.18) e A = e 1 u'(w) . Integra t ion of (2 .18.) gives (with C an in teg ra t ion constant) - J R (w)dw C (2.19.) J e A dw = e 1 u'(w) + C 0 Now, R.(w) determines u(w) up to p o s i t i v e l i n e a r transformations , by not ing that —JR (w)dw (2.20.) u(w) = a + Btje A dw] - 18 --c -c where a = — e and 8 = e > 0 are two constants. Finally, i f R ,(w) = R „(w), then i t follows from (2.20.) and Definition 2.6., that A,l A,2 the corresponding u^(w) and u^(w) are strategically equivalent, i.e. u^(w) ^ u2(w) . • Related to the absolute risk aversion R,(w) are two other measures of A risk aversion. One is the relative risk aversion R (w) as defined by Pratt R (1964) and Arrow(1965,1971); the other is the partial relative risk aversion - 5 Rp(w;wo), as introduced by Menezes and Hanson(1970) . We w i l l now provide the definitions of R„(w) and R (w;w ) . R P o Definition 2.9. The relative risk aversion at w, RR(w), of a u t i l i t y function u(w), is defined by (2.21.) n / N U"(W) R (w) = - w R u'(w) Definition 2.10. The partial relative risk aversion Rp(w;wQ) of a u t i l i t y function u(w), is defined by (2.22.) u"(w + w) o R (w;w ) = — w y ° u'(w + w) where w is some fixed wealth level, o Zeckhauser and Keeler(1970) established a direct relationship between RA(w) R (w), and R (w;w ), which we w i l l present through the following lemma. R P o 5. These a r e not 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 . E s p e c i a l l y i n t h e f i n a n c e l i t e r a t u r e d i f f e r e n t m e a s u r e s o f r i s k a v e r s i o n have been i n t r o d u c e d , d i s c u s s e d , and 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 ( 1 9 7 5 ) , M i 1 1 f r ( 1 9 7 5 ) , K u b i n s t e i n ( 1 9 7 6 ) , and K a l l b e r g and Z i e m b a ( 1 9 7 8 ) . - 19 -Lemma 2.10. R^ Cwjw ) = R_(w + w) — w .R.(w + w) . P o R o 0 A 0 The proof of Lemma 2.10. is obvious. Let us now illustrate the idea of a risk aversion measure with the following example. Example 2.4. Consider two decision makers with two different u t i l i t y functions: u^w) = w - (l/4)w (see Examples 2.1. to 2.3.) and u2(w) = 2 w - (l/3)w , both defined over the range 0 <^  w £ 3/2 . Using Definition 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 is less risk averse than the second decision maker. But what does this actually mean? The answer is that one decision maker is less risk averse than another implies that for a l l nondegenerate gambles within the defined wealth range his risk 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 is less than TT (w",z) « 0.207 . 1 o I o . In fact, we introduced in Example 2.4. some kind of comparative risk aversion. The following lemma has actually been illustrated in the example. The proof of Lemma 2.11. can be found in Pratt(1964). Lemma 2.11. It holds that TT^ W.Z) < TT^ CW.Z) for a l l w and z, i f and v only i f R (w) < R _(w) for a l l w . A,l A>^ - 20 -In order to make effective use of the absolute risk aversion RA(w) we state another important lemma: Lemma 2.12. R (w) is positive (negative) for a l l w i f and only i f the A decision maker is risk averse (risk prone). Proof: We show the validity of the risk averse case. The proof for the risk prone case uses similar arguments and is therefore omitted. Assume now R ^ ( w ) > 0 . Since u'(w) > 0 , because u(w) is increasing, we have u"(w) = - R (w).u'(w) < 0 , so that u(w) is concave. Using Lemma 2.2. i t follows that the decision maker is risk averse. The converse is straight-forward and has been stated earlier in this chapter. | In addition, i t is clear, using Lemma 2.7., that the following lemma holds: Lemma 2.13. If R.(w) =0 , the decision maker is risk 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 is not discussed here. We have seen, that TT(W ,z) is a function of both w and the gamble z . o o Now we would like to discuss what happens to TT(W ,Z) as w varies. It is o o believed that many decision makers w i l l pay a smaller risk premium as their ( i n i t i a l ) wealth increases. The reason being that one can better afford to take a certain risk as one becomes richer. Of course, this is not a general - 21 -rule, i t only provides motivation to investigate whether a decision maker is decreasingly, constantly, or increasingly risk averse. Pratt(1964) provides several definitions and theorems in that direction, the major definition being: , Definition 2.11. A risk averse decision maker is decreasingly (constantly; increasingly) risk averse i f w(w^,z) for any gamble z decreases (is 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 risk premium for a gamble z be TT(W , Z ) and A be a multip l t i p l i c a t i v e factor such that P(w +Az<0) = 0 , then: (2.24.) | — [TT(W , 2 ) ] = 0 dw o < (i) (2.25.) (2.26.) 8 _ 8 A 9 _ 8 A TT(W J A Z ) o TT(AW , A Z ) o ^ 0 < ^ 0 < ( i i ) ( i i i ) i f the corresponding function (i) R.(w ) , ( i i ) A O Rp(wo;wQ) , ( i i i ) R^^) *-s respectively increasing, constant, or decreasing in w . - 22 -A very useful application of the absolute risk aversion R^(w) and the relative risk aversion R (w) is provided by.the following two lemmas. R Lemma 2.15. If the absolute risk aversion is constant, say R^( w) = c> then: (2.27.) u(w) ^  w i f and only i f RA(w) = 0 , A (2.28.) u(w) ^  - e~ C W i f and only i f R,(w) = c > 0 . A u" (w) Proof: Assume f i r s t u(w) ^  w , so that RA(w) = - ,, ( = 0 . Conversely, A u1(w) 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) - ^ , = c . Conversely, assuming R ^ ( w ) = c > 0, (2.20.) gives u(w) = a - 8e C 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. If the relative risk aversion is constant, say Rt)(w) = c , then: (2.29.) u(w) % w 1 _ C i f and only i f 0 < R D(w) = c < 1 , R (2.30.) u(w) ^  log w i f and only i f R D(w) = c = 1 , R 1-c (2.31.) u(w) a- - w i f and only i f R D(w) = c > 1 . R The proof of Lemma 2.16. is similar to the proof of Lemma 2.15. and can be found in Pratt(1964). We w i l l finish this chapter with the following example: - 23 -3 ~ Example 2.5. Following Lemma 2.14. we see that with (2.24.) |_[Tr(wo,2) ]= 0 o i f R.(w ) is constant in w , so that A o o (2.32.) TT(W',2) = TT(W",2) where w1 f w" 0 0 o o i f R.(w ) is constant as w varies. Then, according to Lemma 2.15. we have A o o -that (2.32.) holds i f u(w) ^  - e ° W . The latter is graphically shown in Figure 2.4. below. u(w) o Figure 2.4. A decision maker with u(w) a, - e C W w i l l state the same risk premium for a nondegenerate gamble z at any ( i n i t i a l ) wealth level. - 24 -3 Mathematical Results for R A(w Q) This chapter w i l l derive ^ ( w 0 ^ > b r i e f l y denoted by R^, as a function of: ( i ) the parameters of a nondegenerate gamble z; and ( i i ) the c e r t a i n t y 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. Some of the derived functions are not e x p l i c i t but i m p l i c i t functions of R ^ ( W Q ) i n terms of ( i ) and ( i i ) ; e.g. R^ i s i m p l i c i t l y expressed i n terms of p and the parameters of z. We w i l l f i r s t discuss the idea behind the mathematical r e s u l t s , then introduce a generalized class of u t i l i t y functions, while the remainder of this chapter gives the function R within a subclass of the generalized A class of u t i l i t y functions. Chapter 4 w i l l give an a p p l i c a t i o n of the r e s u l t s by introducing a new method for s e l e c t i n g a u t i l i t y function. The mathematical r e s u l t s are extended i n Chapter 5, for instance by giving r e s u l t s for other u t i l i t y functions. \ - 25 -3.1. The idea behind the mathematical results. Let u(w) be a continuous, twice differentiable, and increasing u t i l i t y function over wealth w. Pratt(1964) then interprets the absolute risk aversion R.(w) by considering a decision maker's risk premium for a A small, fair gamble z. Thus, let z be a gamble with E(z) = 0 and small 2 2 . variance a = O . We w i l l now expand u(w ) around w on both sides of z o o (2.4.), which gives 2 (3.1.) u(w -TT) = u(w ) - TTU'(W ) + J - u"(w) -o o o 2 and ~2 (3.2.) E[u(w +5)] = E[u(w ) + zu'(w ) + u"(w ) + ] o o o 2 o 2 = u(w ) + ~ u"(w ) + o 2 o Equating (3.1.) and (3.2.) and neglecting higher-order terms (as we are actually considering infinitesimal gambles) gives us (3.3.) u(w ) - TTU'(W ) (=> u(w ) + %a2u"(w ) o o o o from which i t follows that 2TT(W ,Z) (3.4.) R sR(w)«4H V-A A ° a2 G~2 i.e. R is twice the risk premium per unit of variance for infinitesimal - 26 -gambles. Similarly, Pratt(1964) shows, that for z being a nonfair gamble, i t holds that (3.5.) R.[w +E(z")] « 2TT(W ,Z) o a- 2 z and for z = <h;-h> we have (3.6.) R s R ( W ) « |(2p - 1) where p = p(w ,h), A A o h o and in both cases z is assumed to be infinitesimal. Pratt's interpretation of the absolute risk aversion is certainly useful, which can be illustrated through the following example. Example 3.1. The decision makers in Example 2.3. were assumed to behave 2 2 according to the u t i l i t y functions u^ = w - (l/4)w and u^ = w - (l/3)w (0 <^  w <^  3/2) respectively. At an i n i t i a l wealth of w^  = 1 i t was shown that R ^w") = 1, R 0(w") = 2, TT (w",z) = 0.118 and TT0(W",5) = 0.207, A , l o A , 2 o l o 2 o where z = <0.5;-0.5>. Using (3.4.) we can estimate R by R where 2TT (w"s2) (3.7.) R, .(w") = 1 ° = 0.944 (R. . = 1) A, 1 o 2 A, 1 z and A ft ^ 6 2TT (w" 5) ( 3 - 8 , ; RA,2(w") = —^—2 = 1.656 (RA „ = 2). a~2 A ' 2 z Especially in the latter 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 on. Definition 3.1. The R -percentage error is defined by A . (3.9.) R % ESTIMATION ERROR = A RA(w ) - RA(w ) A o A o R.(w ) A o .100 An ill u s t r a t i o n is 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 is limited A in the sense that the formula is an approximation and i t requires small-ranged gambles. On the other hand, the formula applies to any u t i l i t y function. The question now arises whether we are able to come up with exact results and not only approximations or estimations. The answer has clearly proven to be 'yes', however at the price of specifying the shape of the u t i l i t y function (not its essential parameter). This has led to the f i r s t R^  as a function of the risk premium and the parameter of a fai r gamble: THEOREM 3.1. (3.10.) For a decision maker with a quadratic u t i l i t y function u(w) ^ w - cw , where c > 0 and w < — , i t holds that 2c 2TT RA 2 2 where TT = Tr(wo,z) and z a fai r gamble with standard deviation a = 0~. - 28 -Proof: Noting that u ( m )(w) = d u ( w ) = 0 for m = 3,4, , (3.1.) , m becomes dw 2 ~ u(w -TT) = u(w ) - TTU'(W ) + %TT u"(w ) and (3.2.) becomes E[u(w +z)] = o o o o o 2 u(w^) + ho~ u"(wo) . Equating the last two equations according to (2.4.) 2 2 gives - im'Cw ) + %TT U"(W ) = %C~ u"(w ), from which the desired result o o z o (3.10.) follows directly. Notice that i f one uses (3.4.) to estimate for the u t i l i t y function u(w) ^  w - cw2, the R % ESTIMATION ERROR becomes A R.% ESTIMATION ERROR = A 2TT 2 2 a - TT 2TT a 2 2TT 2 2 a - TT .100 = ^- (100), where a 2 = a~ 2 . 2 z a Example 3.3. Using the above expression, we are now able to check the ( 11 ft)2 results in Example 3.2.: The R^ ^ % ESTIMATION ERROR = =5.57 and the R 0% ESTIMATION ERROR = ^'^VX = 17.14. The differences are due to A,2 (.1/4; rounding-off errors. Theorem 3.1. has provided the stimulus for a search for similar results with other u t i l i t y functions. The question of which functions can be considered is addressed in the following section. - 29 -3.2. A generalized class of u t i l i t y functions. In this section we w i l l introduce a generalized class of u t i l i t y functions. We w i l l f i r s t state: THEOREM 3.2. If U(w) and u(w) have absolute risk aversion R (w) A,U and R (w) respectively, then i f U'(w) = f(w).u'(w) n j U for some differentiable function f(w) ^  0, i t holds (3.11.) R „ ( w ) = R (w) - ^ T T • A,U A,u f(w) T. f p ( ^  = - U " ( w ) = _ f(w).u"(w) + f'(w).u'(w) _ _ u"(w) _ f'(w £I22E- K A y w ; u'(w) f(w).u'(w) u'(w) f(w - » ( ^  f'(w) = R. (w) —s- . A,u f(w) Now let us define a generalized class of u t i l i t y functions u(w) by (3.12.) u'(w) ^  f(w), r B w a — - + n 6wa with w > 0, B > 0, a > 0, T + 0, + n > 0, and T) = 1 i f x -»• °°. Using Theorem 3.2. i t is easy to verify that the absolute risk aversion R.(w) corresponding to (3.12.) i s : p ( , _ u"(w) _ aBw 3 f ' ( w ) (3.13.) RA(w) - - - ~ ^ + n T - 30 -Choosing f(w) = we define u(w) = u(w;£,a ,T ,B ,T | ) b y (3.14.) u*(w) = u ' C w j ^ . a . T . B . n ) ~ w? 'Bw a - T + n According to (3.13.) we have for the absolute risk aversion corresponding to u(w;£ 5a , T , B 5 r i ) : (3.15.) R(w) A _ aBw' a-l Bwc w + n The relative risk aversion corresponding to (3.14.) is by Definition 2.9, (3.16.) RR(w) !BwC - e. 5W + n S e l e c t i n g E, = 0, we o b t a i n a c l a s s o f u t i l i t y f u n c t i o n s u ( w;0 , a , T,8> n ) a s d e f i n e d b y (3.17.) u' ( w ; 0 , a , T , 8 , r i ) ^ r B w a ^ — + n Multiplying the right hand side of (3.17.) by _ TIT c - gives T and substituting (3.18.) 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 (3.19.) u(w;0,l,T,B,n) ^ B(1-T)| T l - T (T t 1) which is 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 explicitly, for some i t w i l l TIT suffice to substitute c = —^ , so that by (3.15.) we have (3.20.) R ( w) = _ L _ A w + c which directly explains the name of the HARA-class. We w i l l now continue to give cr i t e r i a for (3.14.) to be s t r i c t l y decreasingly risk 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.) is s t r i c t l y decreasingly risk averse i f : (3.21.) (3.22.) w > d, T > 0 and f i n i t e , and 0 <. £ <_ xa, where d = < max 0 - ^ 8 ' ( 3 ( x a - 0 where - 32 -(3.23.) K = i (a-l) + 2? - n 2( ^\2 ^  2M\h a (a-1) +.a {—  T ; 26 T - a Proof: According to Lemma 2.12. u(w; E,, a, T , 3 >n) is risk averse i f R ^ ( w ) is positive, so that with (3.15.) we have a-1 (3.24.) aBw Bw - ^ > 0 w + n which gives (3.25.) w > B(Ta-c-) V. when K ± T a . Furthermore, by Definition 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|) is s t r i c t l y decreasingly risk averse when in addition i t holds that ^-[R,(w)] < 0, or: dw A (3.26.) fBwa 1 a(a-l)(Bw a 2 ) - r o a - 1 i apw (aBwa V • TBw w We w i l l solve (3.26.) for w3 by finding the roots of — [R.(w)] = 0 , i.e. the •> ° dw A left hand side of (3.26.) set equal to zero. We now multiply the LHS of 2 (3.26.) by -|— and set this equal to zero. After reordering the terms, we find a quadratic equation in w , namely - 33 -(3.27.) ( f - a ) | - ( w a ) 2 + [ 1 | + a ( a - l ) ] $ n w a + fy\2 = 0 which gives the following two roots for wc (3.28.) w a(a-l) + T ± n a (a-l) + a .-^ 1% 1 2 23. T i - a T = K. Note that 'the LHS of (3.27.) has a negative limiting value for w ->- 0 0 , as r ft2 (— - a ) — < 0 for 6,T > 0 and 0 < £ < Ta. Assume now n > 0, so that T T = K > K , as — ( - - a) < 0 for 8,T > 0 and & < a. Hence, the LHS of - + T T T (3.27.) is negative for w3 > K and w3 < K in this case, so that w3 > K or ,Va , w > [K_] is one solution of (3.26.). Assuming n < 0 gives K_ £ <_ 0, as [a(a-l) + % 2 • ^f(ax-C)! 2, .»2 2 4^1 V-a (a-l) + a • — T k 2E > a(a-l) + for = T 0 <. £ < Ta, so that for n <_ 0 the condition wa > K_ w i l l always be overruled by w3 > 0 which is included in (3.21.) and (3.22.). Combining now (3.25.), w > [K ] ^ 3 and the i n i t i a l conditions •^W— + n > 0 (or: w > [- ^ ] ^ a ) and T p w > 0, the conditions (3.21.) to (3.23.) have been obtained. • For u(w;0,a,T,6>i~|) i t I s easy to see that (3.21.) to (3.23.) become (3.29.) T > 0 and w > max[0, -c, c(a-l)] V* where again c = . The conditions (3.29.) were earlier derived by Pratt(1964) P for his class of u t i l i t y functions (3.18.) to be s t r i c t l y decreasingly risk averse. - 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 restricted to the function R A I for u(w;0,l ,T , 8,T)), where T = -1, \, 1, 2 and 0 0 respectively, and for u(w;0,0,T,8,ri). The reason behind this selection of u t i l i t y functions is twofold. F i r s t , for these u t i l i t y functions a l l desired functions R could be obtained, so that a very complete presentation results. Second, for the purpose of the new assessment procedure, as discussed in Chapter 4, these functions suffice in explaining and illustrating the method. However, in 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. The derivation of the contents of Table 3.1. can be found in Pratt(1964) and Merton(1971). Table 3.1. U t i l i t y Function: Name: E x p l i c i t Expression with FIT c = -5— (Exponential: c=6) P Absolute r i s k aversion as a function of w : u(w;0,0,T,8,n) li n e a r w constant u(w;0,1,-1,8,n) quadratic - (w + c) increasing u(w;0,l,%,B,n) square root V w + c decreasing u(w;0,l,l,8,n) logarithmic log (w + c) decreasing u(w;0,l,2,8,n) hyperbolic 1 w + c decreasing u(w;0,l,°°,c,l) exponential -cw - e constant - 36 -3.3. Risk aversion as a function of certainty equivalent or  r i s k premium for the gamble z = <h,p;-h>. Consider the gamble z = <h,p;-h>, where E(z) = h(2p-l) : (3.31.) + h Figure 3.1. 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, for u(w;0,1,T,6,n) with T f 1, (3.32.) r e s u l t s i n : rS(w +z) 3(1-T) + n l-T (3.33.) B ( l - x ) 6(w +h) o + n l-T + (l-p) 6(w -h) + n l-T Using (3.20.) and further reducing (3.33.) provides for T t 1 (3.34.) 1 + z R A l l - T = P 1 + hR^l-T + (1-p) 1 -hR.l-T Substitution of z = h(2p-l) - TT provides e a s i l y a r e l a t i o n s h i p containin - 37 -TT = TT(W ,Z). R as a function of z and TT for u(w;0,l ,x , B , n ) with x ^  1 O 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 differently. 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 RA(w ) by R . A o J A 1 . For u(w;0,1,-1,8,11) ^  - (w+c) , with c = - the relationship (3.34.) p becomes: (3.35.) (1-2RJ 2 = p ( l - h R ) 2 + ( l - p ) ( l + h R A ) 2 A A A which gives (3.36.) R - 2[h(2p-l) - 2j _ 2TJ_ A h 2 - 2 2 4p(l-p)h 2 + 2h(2p-l)TT - TT2 and for the fai r f i f t y - f i f t y gamble z = <h;-h> the result becomes quite simple: (3.37.) R = _ 2 2 = _2TT f o r z = <h;-h>. A i 2 ~ 2 v 2 2 h - z h - TT / v ' 2°_. For u(w;0,l ,%,8 ,n) ^ s/w+T , with c = -~, (3.34.) gives <£p (3.38.) y l + 22R = pVl + 2hR + ( l - p ) V l - 2hR A A • A - 38 -so that (3.39.) = 2p(l-p)[h(2p-l) - z] A ,„ 2 „ ,x2. 2 (2p -2p+l) h - 2h(2p-l)z + z .2 2p(l-p)ir . 2,, ,22 2 4p (1-p) h + TT and for the fai r f i f t y - f i f t y gamble z = <h;-h> this gives: (3.40.) R = -22 = 2TT A ,2 x ,.2 2 2 h + 4z h + 4TT for z = <h:-h>. 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(wo+c+z) = p.log(wo+c+h) + (1-p).log(wo+c-h) which gives, after noting that according to (3.20.) Wq + c = [R ] , the following result: * r 2 5- + h(2p-l) - TT = RA (3.42.) + h ^- - h 1-p so that for the fair f i f t y - f i f t y gamble z = <h;-h> we have - 39 -(3.43.) v = ~ 2 G = 2 T T A ,2 „2 .2 2 h + z h + TT for z = <h;-h>, 4^. For u(w;0,1,2,3,11) ^  - w + c > with c = (3.34.) gives (3.44.) 1 = p 1-p 1 + %zR 1 + %hR + 1 - %hR. A A A so that (3.45.) ^ = 2[h(2p-l) - z]  A h 2 - h(2p-l)z 2TT 4p(l-p)h + h(2p-l)TT and for the fair f i f t y - f i f t y gamble z = <h;-h> this gives (3.46.) RA = -2z = 2TT 2 2 ti h for z = <h;-h>. 5^. For u(w;0,l,co j C ,1) ^ - e C W (3.34.) provides (3.47.) lim zR. 1-T 1 + p. lim X-Ko hR. 1-T 1 + + (1-p).lim hR. 1-T 1 -or, using lim X-x» 1 + 1-T _ k . e , this becomes (3.48.) -zR„ -hR, hR, A A , x A i = p.e + (l-p).e - 40 -Hence, -zR -[h(2p-l) - Tr]R4 A _ A e = e (3.49.) hR = 2p.cosh(hR) + (l-2p).e A where the hyperbolic cosine of x is defined by cosh(x) = %(e X + e X) For the fair f i f t y - f i f t y gamble z = <h;-h> the following result then easily obtained: -zR TTR (3.50.) e = e A = cosh(hR) for I = <h;-h>. A 6 . For u(w;0,0,T,$,ri) ^ w we w i l l of course have R = 0, as the A decision maker is risk neutral with a linear u t i l i t y function. A summary of the results in this section is provided through Tables 3.2. and 3.3., which appear after section 3.5. - 41 -3.4. Risk aversion as a function of p r o b a b i l i t y equivalent for  the gamble z = <h,p;-h>. Consider the gamble z = <h,p;-h>, so that by D e f i n i t i o n 2.5. and (2.8.) we have for u(w;0,1,x, 8,n) with x ? 1 : B( I -T) 6w ^ o l-T (3.51.) B(1-T) rB(w +h) o l-T + n + (1-P) B(w -h) l-T + n Using (3.20.) and further reducing (3.51.) provides for x ^ 1 (3.52.) 1 = p hR. l-T 1 + + (1-p) hR. l-T 1 -Hence, for p = p(w ,p,h) i t holds o 1 -(3.53.) hR. l-T ~ 1 -P = hR. l-T 1 + hR. l-T 1 -The desired functions R^ for u(w;0,1 , x , B>n) vith x 1 can be derived from (3.52.) or (3.53.) by specifying x- For x = 1 we w i l l derive a r e s u l t i n a more d i r e c t way. - 42 -1°. For u(w;0,l,-l ,8,ri) * ~ (w + c) , with c = - ^ , (3.52.) gives (3.54.) 1 = p(l - hR A) 2 + ( l - p ) ( l + hR A) 2 which gives (3.55.) R = 2-(2p - 1) A h so that (3.56.) p = khRA + k which can be verified through (3.53.) as well. 2° . For u(w;0,l,%,8,n) ^  v*w + c , with c= jg, (3.52.) gives (3.57.) 1 = pVl + 2hRA + (l-p)vl " 2hRA which gives, after taking both sides to the power two and after some rearranging: (3.58.) (1 - 2p)hRA + p(l-p) = p(l-p)vl - 4h 2R A 2 so that (3.59.) R, - -2^2P-D(P-1) A h(l-2p+2p 2) 2 - 43 -whereas (3.53.) w i l l give an explicit 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. However, using (2.8.), we have (3.60.) 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 ] 1, o A log(l-hR A) ( 3 ' 6 1 - ) P = log(l-hR A) - log(l+hR A) ' £ . For u(w;0,l,2,6,n) ^  - w ]_ c , with c = (3.52.) gives (3.62.) 1 = p[l + %hR ] 1 + ( l - p ) [ l - %hR ] 1 i\ A so that (3.63.) R = 2 ( 2P ~ l ) A h yielding (3.64.) p = %jhR + % A which can be verified through (3.53.) as well. - 44 -31. For u(w;0,l,°°,c,l) ^  - e C w (3.52.) provides (3.65.) 1 = p. 1 im hR. l - T 1 + + (l-p).lim hR. l - T 1 -which gives, using lim 1 + * T l - T = e -k (3.66.) R A = l . l o g _ £ _ 6°. 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 in this section is provided through Table 3.2., which appears after section 3.5. - 45 -3.5. Risk avers ion as a funct ion of gain equivalent for 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 vby (2 .7 . ) we have for u(w;0,1,T,B,n), with T ? 1: B(l-T) Bw 1-T (3 .67.) B(1-T) B(w +h) T rB(w -h) ^ T _ _ £ _ + n + ( 1 _ p ) | _ _ £ _ + n Using (3 .20.) and further reducing (3.67.) provides for T / 1: (3 .68.) 1 = p hR. 1-T 1 •+ + (1-p) 1 -h R A ) 1 _ T / which we w i l l use i n most of the fo l lowing der iva t ions of R. as a funct ion A of h for the u t i l i t y functions l i s t e d i n Table 3 .1 . 1 ° • For u ( w ; 0 ) l , - l , g , n ) - (w + c) , with c = - J , (3 .68.) gives P (3.69.) 1 = p ( l - h R A ) 2 + ( l - p ) ( l + h R A ) 2 so that (3 .70.) R = 2ph - 2 ( l -p )h  A p h 2 + ( l - p ) h 2 - 46 -which becomes for p = \ : (3-71.) R = -P^n for z = <n>h>. A h + h 2J\ For u(w;0 ,l,%,B,n) ^ W + c , w i t h c = -§ , (3.68.) g ives 2p (3.72.) 1 = p [ i + 2hRA]^ + ( l - p ) [ l - 2hR A] % which gives after several calculations: 7 o x „ _ 2p(l-p)[ph+(p-l)h] A tp h + (l-p)^h] For p = \ (3.73.) becomes (3.74.) R . = 4 ( H - H ) A (h + h ) 2 for z = <h;-h>, 3^. For u(w;0,l,l,B,ri) ^ 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.) w + c = [R 1 1 o A (3.76.) . 1 = [1 + h R j P [ l - h R j 1 P . A A - 47 -Formula (3.76.) becomes for p ^  h (3.77.) R. - ^ A hh for z = <h;-h>. 4°. For u(w;0,l,2,B,n) ^  — , with c = ^  , (3.68.) gives = w + c p (3.78.) 1 = p[l + %hR A] _ 1 + ( l - p ) [ l - ^ hRA] 1 so that (3.79.) = 2[ph - (l-p)h] hh which for p = % becomes (3.80.) R. = h - h hh for z = <h;-h> 5°. For u(w;0,l,°°,c,l) % - e ° W (3.68.) provides (3.81.) 1 = p•1im l - T 1 + + (l-p)-lim -r-Ko hR. l - T 1 -which gives, using lim 1 + -l - T -k (3.82.) ~ h RA h R A 1 = p•e + (1-p)-e For p = h (3.82.) becomes - 48 -83.) 1 = 2 —hR hR • A ^ A e + e For u(w;0,0,T,8,n) ^ w we simply have n" = 1-P . P . h -and R. = 0 A A summary of the r e s u l t s i n t h i s section i s provided through es 3.2. and 3.3., which appear on the following pages. Table 3.2. Summary table of the re s u l t s for the gamble z = <h,p;-h>. A b s o l u t e R i s k A v e r s i o n R, as a f u n c t i o n o f : A U t i l i t y F unction: C e r t a i n t y Equivalent 2 Risk Premium IT P r o b a b i l i t y E q u i v a l e n t p Gain Equivalent h Li n e a r : u(w;0,0,T,|3,ri) R A = 0 R A = 0 R A = 0 R A = 0 Quadratic: u(w;0,l,-l ,B,n) 1 „ _ 2[h(2p-l) - 2] RA = 2TT R A - i ( 2 p - 1) „ _ 2ph - 2(l-p)h " h 2 -,2 2 4p(l-p)h 2+2h(2p-l)-Tr 2 A - 2 2 A pli + ( l - p ) h Z Square Root: u(w;0,l ,%,6,ri) RA = 2p ( l - p ) t h ( 2 p - l ) - 2] R _ 2p(l-p)TT JJ _ -2p(2j5-l)(f5-l) JJ _ 2p(l-p)[ph+(p-l)h] (2p 2-2p+l) 2h 2-2h(2p-l)2+2 2 A 4 p 2 ( l - p ) 2 h 2 + IT 2 A h [ l - 2 p + 2 p 2 ] 2 A [p 2h+ ( l - p ) 2 h ] 2 Logarithmic: u(w;0,l,l ,8 ,n ) 1 f l lP f l 1 1 _ P 2 = - ~ + £ +h i - h A '•A J '•A J h(2p-l) - ir = ' 1 f l l p f l l 1 _ p P = l o g ( l - h R A ) l o g ( l - h R A ) - l o g ( l + h R A ) 1= ( l + h R A )P ( l - h R A ) 1 _ P Hyperbolic: u(w;0,l,2,6,n) R _ 2[h(2p-l) - 2] r - 2 7 r R A = 1(2* - 1) „ _ 2[ph - ( l - p ) h ] A h 2 - h(2p-l)2 4p(l-p)h +h(2p-l)TT K _ • * • hh Exponential: u(w;0,l,°°,c,l) -2RA e = hR 2p.cosh(hR A) + (l-2p)e A -[h(2p - D - TT]R A e = hR. 2p.cosh(hR A)+(l-2p)e A -hR A hR A 1 = pe + ( l - p ) e Table 3.3. Summary table of the results for the gamble z = <h;-h>. Absolute Risk Aversion R. as a f u n c t i o n of : A U t i l i t y Function: Name: Certainty Equivalent 2 Risk Premium TT Gain Equivalent h u(w;0,0,T,6,n) Linear R A = 0 R A= 0 RA = 0 u(w;0,l,-l,B,n) Quadratic ' r - ~ 2 2 A .2 a2 h - 2 r - 2 1 T A u 2 2 h - TT _ 2(h-h) A c2 .2 h + h u(w;0,l,%,B,n) Square Root R " " 2 2 A .2 + . .2 h + 4z r - 2 7 T A , 2 . 2 h + 4TT „ _ 4(h-h) A " 2 A (h + h r u(w;0,l,l,B,n) Logarithmic r - ~ 2 z A .2 + .2 h + z r - 2 7 T A . 2 2 h + TT R h - h A ~ ~ hh u(w;0,l,2,B,n) Hyperbolic R = " 2 2 A 2 A h R = H A 2 A h \ - V h A hh u(w;0,l,°°5c,l) Exponential -2RA e = cosh(hR.) A uR A e = cosh(hR.) A , r -hR, hR l 1 , 1 A A 1 = - e + e - 51 -3.6. Risk aversion as a function of certainty equivalent or  r i s k premium for the gamble z = <h,p;0>. -Consider the gamble z = <h,p;0>, where E(z) = ph : (3.84.) Figure 3.2. z = <h,p;0> so that by (2.1.) (3.85.) u(w +z) = p.u(w +h) + (l-p).u(w ) o r o o Hence, for u(w;0,1,T,8,n) with T ± 1, (3.85.) r e s u l t s i n : rB(w +z) x l-T .6(1-T) + n (3.86.) 8(1-T) B(W +h) ^ o — = — + n l-T + (l-p) Bw l-T + n Using (3.20.) and further reducing (3.86.) gives for T ^ 1 (3.87.) l-T = P 1 + hR , A l-T + (1 - p) Substitution of z = ph - TT i n (3.87.) provides 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 as a function of 2 and TT for u(w;0,l , T , $ , r i ) o A with T f 1 can be derived from (3.87.). The case T = 1 w i l l be derived differently. l l . For u(w;0,l,-l,B,n) ~ - (w + c ) 2 , with c = - §, (3.87.) gives P (3.88.) (1-2R ) 2 = p(l-hR ) 2 + ( 1 - p ) A A which gives (3.89.) R. - 2{Vh-*\ = ; r 2I-A ,2 „2 ..2 . . 2 ph - z p(l-p)h + 2phTT - TT and for the f i f t y - f i f t y gamble z = <h;0> (3.90.) R A= = - 2 for S = <h;0>. h - 22 h + 4hfT - 4TT 2^. For u(w;0 Jl )% Jg )r 1) ^  + c , with c = (3.87.) gives (3.91.) Vl + 22R = pVl + 2hR + ( 1 - p ) A A so that (3.92.) R = ~2p(l-p)(z-ph) = 2p(l-p)TT A [2-hp 2] 2 [p(l-p)h - TT] 2 and for the f i f t y - f i f t y gamble z = <h;0> we obtain : ex n o \ rt 4 h - 8 2 8TT , N (3.93.) R = = for z = <h;0>. A (42-hr ( h - 4 T T ) 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.) w + c = [R ] o A the following result: (3.95.) 1 + 2RA = 1 + (ph - TT)RA = [1 + hR ] P A A A or log(l+zR A) log[l + (ph—TT)R ] (3.96.) p= = A_ log(l+hR A) log(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 = ^ f o r 2 = < h ; 0 > , 2 (h - 2TT) For u(w;0,l,2,B,n) ^  " — J — » w i t h c = ^ 5 , (3.87.) gives w + c p - 54 -(3.98.) 1 + %zRA 1 + %hRA + ( 1 - p ) so that (3.99.) R = 2(ph-z) = 2jn_ A (l-p)h2 (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.) RA = 2h - 4z 8TT hz h(h-2Tr) for z = <h;0>. For u(w;0,l,°°,c,l) i> - e C w (3.87.) provides (3.101.) lim zR. 1-T 1 + p. lim hR. 1-T 1 + + ( 1 - p ) which gives, using lim X " K O 1 + -T 1-T = e -k (3.102.) P = -zRA -(ph-Tr)R e - 1 e -hR. -hR. e - 1 e - 1 and for the f i f t y - f i f t y gamble z = <h;0> (3.103.) —zRA -(%h-Tr)R, - h R A ; e + 1 for z = <h 6 . For u(w;0,0,T,6,n) ^  w we w i l l again have R = 0. A summary of the results in this section is provided through Tables 3.4. and 3.5., which appear after section 3.8. - 55 -3.7. Risk aversion R as a function of probability equivalent for  the gamble z = <h,p;0>. Consider the gamble z = <h,p;0>, and note that Definition 2.5. does not apply for this gamble. Definition 2.5. is the definition as known in the literature, hence i t is included under the preliminaries. For the purpose of this section we w i l l need: Definition 3.2. A p r o b a b i l i t y p such that the d e c i s i o n maker i s i n d i f f e r e n t between W Q + %h a n d the gamble z = <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 e q u i v a l e n t and i s denoted by p = pCw^jh) . Hence, p = p(w^,h) is defined by (3.104.) u(w +^ h) = p.u(w +h) + (l-p).u(w ) o o . r o so that for u(w;0,1,T,3,n) with T f 1 we have T B(1-T) rB(w +hh) ^ ° . — - — + n l-T (3.105.) B(I -T) B(W +h) o 1-T + n + (l-p) Bw 1-T + Tl Using (3.20.) and further reducing (3.105.) gives for T ^ 1 - 56 -(3.106.) 1 + 2T l-T hR. l-T 1 + + ( 1 - p ) Hence (3.107.) l-T P = 1 -X L T l-T 1 -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 in Table 3.1. l j \ For u(w;0,l,-l,B,n) * - (w + c) , with c = - j, (3.106.) gives (3.108.) (1.- %hR A) 2 = p(l - hR A) 2 + ( 1 - p ) wh i ch g ive s (3.109.) R = *<1 - 2P> A h(l - 4p) so that (3.110.) 4 - hR. 4(2 - hR ) A which can be verified through (3.107.) as well, V - 57 -2^. For u(w;0,l,%,B,n) ^ Vw + c , with c = jg, (3.106.) gives (3.111.) VI + hR = p V l + 2hR + ( 1 - p ) A A r which gives . - 4 p ( 2 p - l ) ( p - l ) (3.112.) R. = A [2p 2-l] 2h Formula (3.112.) does not permit a way of expressing p as a function of h R , however applying (3.107.) gives 1 - VI + hR (3.113.) p = A • 1 - VI + 2hR A 3^. For u(w;0,l,l,6,n) ^  log (w + c) , with c = (3.106.) and (3.107.) P cannot be applied. However, applying (3.104.) gives us (3.114.) log(w +%h+c) = p.log(w +h+c) + (1-p).log(w +c) o o o which gives, after noting that according to (3.20.) w + £ = [R ] o A log(l + %hR ) (3.115.) p = 2- . log(l + hRA) -1 4^. For u(w;0,l,2,B,n) ^ - £-^r , with c = ^ , (3.106.) gives - 58 -(3.li6.) which gives (3.117.) (1 + %jhR ) 1 = p(l + %hRA) 1 + ( 1 - p ) A A RA = 2(2p - 1) (1 - p)h so that (3.118.) 2 + hR. 4 + hR. which can verified through (3.107.) as,well, 5jL For u(w;0,l,°°,c,l) % - e C W , (3.106.) provides (3.119.) lim 1 + hR *\ A 2T 1-T = p.lim X "K» hR. 1-T 1 + + ( 1 - p ) which gives, using lim X-K» 1 + 1-T = e -k (3.120.) R A = h' l oS 11 " P. so that (3.121.) —%hR 1 - e P = -hR. 1 - e which can be verified through (3.107.) as well. - 59 -6°. For u(w;0,0,T,B,r|) ^  w we simply have R = 0 and p s h. A summary of the results in this section is provided through Table 3.4., which appears after section 3.8. - 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 Definition 2.4. does not apply for this gamble. Definition 2.4. concerning gain equivalents is best known, hence included under the preliminaries. For the purpose of th section we w i l l now state: Definition 3.3. An amount h such that the decision maker is indifferent between the status quo Wq plus p-h, and the gamble z = <fi,p;0> is called the gamble's gain equivalent and is denoted by h = h(w o >p,h). Hence, h = h(w ,p,h) is defined by o (3.122.) u(w + ph) = p-u(w + h) + (l-p)-u(w ) o o o so that for u(w;0,1,T,B,n) with T f 1 we have T B U-T) B(w +ph) ^ o _ — + r, 1-T (3.123.) B(1-T) rB(w +h) i 1-T rBw 1-T V. J + (1-p) Using (3.20.) and furhter reducing (3.123.) gives for T ± 1: - 61 -(3.124.) phR l - T 1 + = P 1 + l - T + (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 listed in Table 3.1. For u(w;0 , l , - l , B,Ti) ^ - (w + c) , with c = ~ g » (3.124.) gives (3.125.) (1 - phR A) 2 = p(l - hR A) 2 + (1- P) which gives (3.126.) v - 2(h - h) so that for p= % (3.126.) becomes (3.127.) R = 4(h - h)  A " 2h 2 - h 2 for z = <h;0>. 2^. For u(w;05l,%,6,n) ^ Vw + c , with c = , (3.124.) gives (3.128.) t l + 2phR = p[l + 2hR + (1-p) which gives after several calculations (3.129.) R = 2(l-p)(h-h)  A " (h - ph) 2 - 62 -Hence, for the f i f t y - f i f t y gamble z=<h;0>, i.e. p = % , we have (3-130.) R = M h ' ^ for I = <h;0>. A (2h - h r 3^. 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(wQ+c+ph) = p-log(wQ+c+h) + (1-p)•log(wQ+c) -1 which gives, after noting that according to (3.20.) w + c = [R ] o A (3.132.) [1 + hR.] P = 1 + phR . A A Hence, for p = h this gives (3.133.) R = 4 ( f i I h ) for z = <h;0>. A h 4i. For u(w;0,l,2,B,n) ^  ]— , with c = , (3.124.) gives " W + C K ° (3-134.) [1 + %PhR ] 1 = p[l + %h R ] 1 + ( i - p ) so that 2(h-h) (3.135.) R = A (l-p)hh - 63 -Hence, for p = % this gives (3.136.) RA = 4(h - h) hh for z = <h:0>. I V For u(w;0,l,°°,c,l) ^  - e C W (3.124.) provide (3.137.) lim -r - X » 1 + phR l - T p • lim •r -X» hR. 1 + l - T + (1-p) which gives, using lim 1 + l - T = e -k (3.138.) -hRA -phRA p-e = e - (1-p) so that for p = % (3.139.) -hR. —%hR A „ A , e = 2e - 1 for z = <h;0>. 6° . For u(w;0,0 , T , 85r)) ^ w we w i l l a g a i n s i m p l y s t a t e h = h a n d R = 0. A summary of the r e s u l t s i n th i s section i s provided through Tables 3.4. and 3.5., which appear on the following pages. Table 3.4. Summary table of the results for the gamble z = <h,p;0>. A b s o l u t e R i s k A v e r s i o n R, as a f u n c t i o n o f : A U t i l i t y Function: C e r t a i n t y Equivalent z Risk Premium ir P r o b a b i l i t y E q u i v a l e n t p Gain Equivalent h Lin e a r : u(w;0,0,T,6,Tl) R A = 0 R A = 0 R A = 0 R A * ° v Quadratic: u(w;0,1,-1,6,n) „ _ 2(ph-2) A u2 7 ph - g R - 2 1 1  A p(l-p)h 2+2phTT-ir 2 ' _ 4(1-20) A h(l-4§) „ _ 2(h - h) A -2 ,2 h - ph Square Root: uU.O.l.if.B.n) _ -2p(l-p)(z-ph) A r- K 2 1 2 Iz-hp J R _ 2p(l- P)T7 A [ p ( l - p ) h - TT ] 2 R _ - 4 p ( 2 p - l ) ( p - l ) A h [ 2 f > 2 - l ] 2 jy _ 2(l- p ) ( h " h ) A - 2 A [ h - p h ] 2 Logarithmic: u(w;0,l,l,i3,n) l o g ( l + zR ) p - A l o g [ l + (ph-TT)R ] n - A l o g ( l + lshR A) A 1 + phR = [1 + hR J P A A r l o g ( l + hR A) P l o g ( l + hR.) A P " l o g ( l + hR A) Hyperbolic: u(w;0,l,2,B,n) _ 2(ph-z) A ( l - p ) h z R - 2 T A (l-p)h(ph-TT) „ - 2(2p-l) KA h ( l - p ) 2(h-h) A * A (l-p)hh Exponential: u(w;0,l, O T,c,l) e = 1 + p e A - l -(ph-TT)R , -hR, i -1+p e -1 -hR "PhR. pe = e A -(1-p) ( Table 3.5. Summary ruble of the results for the gamble 1 • <h;0>. U t i l i t y Function: u(w;0,l,-l,B,n) u ( w;0 , l , % , B , r i ) U(W;0,1,1,B,TI) Name: Quadratic Absolute Risk Aversion R^ as a Certainty ^ | R i g k premium TT Equivalent z f u n c t i o n of : RA . 2 2h - 42 u(w;0 , l,2,6,n) u(w;0,l,°°,c,l) Square Root Logarithmic Hyperbolic Exponential h~ - 2z 2 R. = 8TT A 2 . ± A h +4hTr—4TT R. = 4h - 82 A (42 - hV h - 22 RA = T2 R. = 8TT A (h - 4TTV RA = 2h - 42 h2 ,"2RA . if/™* + x r 2 R. = 8TT A (h - 2irV R. = 8rr A h(h - 2TT) - ( ^ h - T T ) R A Gain Equivalent h - 4(h - h) RA " -2 ,2 A 2h - h 4(h - h) (2h - h ) 2 RA = 4(h - h) RA 4(h - h) hh -hR. —%hR = 2e " - 1 - 66 -4 The R A-method In t h i s chapter we w i l l 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 function 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. In section 4.1. the a l t e r n a t i v e u t i l i t y functions are assumed to be the f i v e r i s k averse u t i l i t y functions l i s t e d i n Table 3.1. The remainder of t h i s chapter w i l l extend the R^-method to include u t i l i t y functions that are not s t r i c t l y r i s k averse over t h e i r e n t i r e region. - 67 -4.1. The R - method for risk averse decision makers. A— 1 • • '• : •— In this section we w i l l assume that the decision maker behaves according to a single risk 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. Also, the mathematical extensions in Chapter 5 w i l l give direction as to how one may expand the set of u t i l i t y functions from which one is chosen. From a notational point of view we w i l l c a l l A the set of risk averse u t i l i t y functions listed in Table 3.1. Hence: Definition 4.1. The class A of risk averse u t i l i t y functions contains u t i l i t y functions (4.1.) u(w;0,l,T,B,n) and x e T, where T = (-l,^,l,2,oo) with c = i f T < 0 0 a n d (4.2.) 6 c = 3 and n = 1 i f 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 closely. The selection procedure starts off by obtaining the decision maker's response to two or more reference f gambles. We suggest taking at least three different gambles, and the result i of the R^-method w i l l only benefit by taking more gambles. The required responses may be chosen from the four indicated in Chapter 3, i.e. the risk premium, the certainty equivalent, the probability equivalent, or the gain equivalent. The question as to how we may be able to arrive at the decision maker's exact response w i l l be deferred un t i l later; for the moment we w i l l - 68 -assume that the responses are assessed without error, and that the decision maker responds consistently according to one (unknown) risk averse u t i l i t y function. Now, let us suppose, for example, that the decision maker provides us with TT^, ^ 2 a n d TT^ as his risk premiums for the gambles z^ = <h^,p;-h^>, 22 = <^ 12' P' _" 12 >' a m * 23 = < h3»P' _^3 > respectively, where ? £ • We are then able to calculate the value of R. for the u t i l i t y functions within A class A, according to the results listed in Table 3.2. (or Table 3.3. for the special case when p = %) . This way we obtained for each u t i l i t y function within A three values for the function R , which may or may not be equal. A If a l l three responses lead to the same value for R^  for 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. is the value of the absolute risk aversion A at the i n i t i a l wealth level w . That i s , R. = R.(w ) has a specific value no o A A o r matter what gamble the decision maker i s faced with. Let us cl a r i f y this point with the following example, since i t i s an essential facet of the R -method. Example 4.1. A decision maker is assumed to behave according to (4.3.) u(w;0,l,2,B,n) ^ - — r — - where c = -2-£ w T c P He states 25 and 16 as the risk premiums for 2^ = <50;-50> and 2^ = <40;-40> respectively. Is this consistent with the function R taking on the same - 69 -value at w^ ? Yes, according to (3.46.) we find that indeed :TT1 h2 More often none of the u t i l i t y functions within A w i l l attain the same values for R^ at Wq from several responses. In this case we cannot directly 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 is "best" is highly subjective and dependent on the choice of an appropriate criterion. 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 relative standard deviation. What we mean A by 'relative' w i l l be explained in the formal description of the R^-method. We w i l l now present this stepwise description of the R -method for risk averse u t i l i t y functions. For the time being we w i l l limit ourselves to the use of risk premiums and the gamble z = <h,p;-h> to aid understanding. 2TT (4.4.) A, 1 o 1 = 2(25) = 2(16) (50) 2 (40) 2 Step 1: Present the decision maker with n.^ 2 different gambles 2 = <h^,p^;-tu> ( i = l,2,...,n) where h_^  > 0. The decision maker's response in terms of risk premiums is = TT^(WO,Z\) ( i = l,2,...,n) respectively, where Wq is the decision maker's (yet unknown) i n i t i a l endowment. It is assumed that the assessment of the risk premiums is completed without errors, and that the decision maker responds consistently according to a single (unknown) u t i l i t y function. Step 2: Check the risk premiums to determine whether the decision maker is risk averse, i.e. TT^  > 0 for a l l i . (If not, then refer to section 4.2. where risk proneness is considered.) Step 3: Calculate the values for the functions according to the results in 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: / 21 22 z n u(w;0,l,-l,B,n) R A(w o,-l,l) R A(w o,-l,2) RA(wo,-l,n) , u(w;0.1,3s,g,n) R A(w o,^,l) RA(wo,JS,n) • • • • u(w;0,l,°°,c,l) R A(w o,~,l) R A(w o,-,2) R A(w o,»,n) i.e. R.(w ,x,i) is the value of the function R, according to the results in A o A Table 3.2. for the gamble z and for 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 relative standard deviation i s - 71 -a(R ,x*) a(R ,x) (4.5.) — = Min - . | U ( R A , T * ) | xeT |p(R A,x)| Select u(w;0,l,x*,8,n) as the u t i l i t y function within A that conforms most closely to the decision maker's behavior. Step 5: Choose as an estimation of R.(w ) : * A o (4.6.) R a(W Q) = u(R A,x*) Step 6: Obtain the decision maker's status quo WQ. Step 7: If x* °°, then we select the parameter c = R A ( W Q ) > a s i t : holds that R. (w ) = c . If x* < 0 0, we easily derive from (3.20.) A o (4.7.) c = |- - w RA ° 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. A decision maker i s presented with the following three gambles: 2 1 = <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 risk premiums. According to Lemma 2.4. his. responses are consistent with risk aversion, as TT_^ > 0 for i = 1,2,3. The calculations in Steps 3 and 4 can now be performed using the results listed in Tables 3.2. and 3.3. The results of the calculations are shown in Table 4.1. on the next page. 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 closely to the decision maker's risk attitude. In accordance with Step 5, we choose R. (w ) = U(R,,T*) = .472 as an estimation of R.(w ). A o A A o 2 Suppose further, that Wq = 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 ' ) U ( W ) - - w - 4~.763 Table 4.1. is 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. We feel that the literature contains sufficient directions for this problem and we refer to the suggested method by Becker, DeGroot, and Marschak(1964). Also, the discussions in Grether and Plott(1979), Lichtenstein and Slovic(1973), and Slovic(1972), who actually a l l employed this method, provide a good reference. The criterion we use in the R -method to determine which u t i l i t y function A is "best", is the relative standard deviation, or the absolute value of the coefficient of variation. As mentioned before, we feel that the variation Table 4.1. The results of calculations referring to Example 4.2. U t i l i t y function RA as a function of A the risk premium TT R (w ,T,1) A O R.Cw ,T,2) A O RA(wo,t,3) V a a l u l u(w;0,l,-l,f3,n) 2TT 4p(l-p)h 2+2h(2p-l)-Tr 2 .234 .667 .800 .567 .242 .43 u(w;O,l,*s,0,n) 2P(1-P)TT 2 2 2 2 4p (1-p) h + TT .415 .250 .160 .275 .106 .39 u(w;0,l,l,B.n) h(2p-l) - TT = - | R A (l l p f l ) 1 _ p + i +h ^ -h |RA J l R A J .462 .400 .308 .390 .063 .16 u(w;0,l,2,8,ri) 2TT 4p(l-p)h 2+h(2p-l)Tr .471 .500 .444 .472 .023 .05 u(w;0,l , o o, c,l) e-[h(2p-l ) -T r]R A _ hR 2p.cosh(hRA)+(l-2p)e .469 .609 .676 .585 .086 .15 - 74 -of the n R -values should be as small as, possible for the selected function, however allowing greater variation for a greater mean u of the R^-yalues. This is exactly what is implied by using the relative standard deviation as our criterion. Note further that in the case where for one u t i l i t y function within A a l l R.(w ,T,i) attain the same value, the relative standard deviation A o in Step 4 reaches i t s minimum at zero. v. Example 4.3. A decision maker with (yet unknown) i n i t i a l wealth w^  indicates .25 and 1 as his risk premiums for the gambles z^ = <1;-1> and z^ = <2;-2> respectively. The results of the calculations in the R^-method are shown in Table 4.2. below. Table 4.2. U t i l i t y function R A(w o,x,l) Ra(WO,T,2) 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 Clearly, the u t i l i t y function u(w;0,l,2,B',n) is selected. Note that in 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). This is because the functional A o A o forms for R as a function of the risk premium in Table 3.3. are different A and cannot be equal i f the risk premiums are not equal. Consider for example two gambles 2^ = <h^ ;-h^ > and z^ = " ^ j l ^ , where h^ ^ h.^ for which the decision maker states TT^ and TT^ as his respective risk .premiums. Suppose Ra(Wq,2,1) = Ra(Wq,2,2), as was the case in Example 4.3. We then have by (3.46.): 2TT1 2TT0 (4.9.) -f- = h l h2 from which i t follows that -n j TT^  as h^ ^ h^. We are now able to show that, for instance, R (w ,1,1) / R A ( W .1,2), since by (3.43. and (4.9.) we have A O A O the following. If TT^ f 0 ^ n^, then 2 2 2 2 1 h l + *1 h1 1 h2 ^ 1 (4.10.) R fw ,1,1) 2TT 2IT, 2 1 2TT„ 2 1 A _ O 1 ± I h2 A 2 4 . 1 2TT2 2"2 Ra(Wq,1,2) as I T . I T „ . Hence, R. (w ,1,1) R. (w ,1,2). If TT. = 0 and ir. 7 0 or vice i I A o A o 1 z versa, then the R -values would be different, i.e. R (w ,2,1) ^ R (w ,2,2). A A O A O Other inequalities can be shown similarly. The functional forms for R^ as a function of the risk premium in Table 3.2. do not directly guarantee similar inequalities. Hence, in theory we may encounter the same R^-values for more than one u t i l i t y function within A, however we feel that in practice this problem w i l l not arise. In circumstances where this problem does occur, we should increase n (= number of reference gambles) in the R^-method. - 76 -The functional forms for 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 in Table 3.3. for R. as a function of A the gain equivalent in 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 probability or gain equivalents, as long as multiple A response modes are employed, as w i l l be discussed in the next paragraph. The R^-method, as introduced in this section, employs the risk premium for the gamble 2 = <h,p;-h> as the only response mode. It should be clear, however, that with our results in Chapter 3 the R -method can employ multiple response modes for both z = <h,p;-h> and z = <h,p;0>. This is because must theoretically attain a single value at Wq, namely R^Cw^), regardless of which type of equivalent or gamble are used. For example, one reference gamble may ask for the certainty equivalent for the gamble z^ = <h^,p^;-h^>, another for the probability equivalent for z^ = ^ ^yp^^» and a third one might ask for the gain equivalent for = <h^ ;0>. The following example illustrates 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(wQ,z^) = 180. He indicates that he is also indifferent between the gamble = <100,fi;0> and winning 50 for sure, when the probability equivalent £ = p(wQ,100) = .55. See Definition 3.2. The results of our calculations are summarized in Table 4.3., where we can easily select u(w;0,l,2,B,n) without even having to use the criterion of relative standard deviation in Step 4 of the RA~method. - 77 -Table 4.3. U t i l i t y function R. aa a function of A h for the gamble R, as a function of A f) for the gamble z^ u(w;0,l,-l,g,n) .00377 .00333 u(w;0,l, 35.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 This exmaple therefore demonstrates that the equality of the two R^-values i n the column for R^ as a function of h does not impose any d i f f i c u l t i e s . The following examples provide i l l u s t r a t i o n s of several mathematical c a l c u l a t i o n s , manipulations, and r e l a t i o n s h i p s , which are a l l based on our r e s u l t s i n Chapter 3. I t may well be used i n the R^-method, for instance as consistency checks for the assessment of the responses to reference gambles. Example 4.5. A decision maker i s assumed to behave according to (4.11.) u(w;0,l,2,B,n) * - ^ where 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, = ^ - r 5 — — for i = 1,2. A h 2 l Therefore, for i = 1 R^ = -2(-25)/(50) 2 = .02, and for i = 2 we have .02 = -2[z^/(20) 2] or = - 4. It is important to note, that we do not have to know the explicit values of the i n i t i a l wealth Wq 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 risk premium for the gamble 2^ = <50,. 75;-50>? Expressing T C \ = IT^(W o,2^) e x p l i c i t l y by (3.45.) results in 4p(l-p)h 2R A (4.13.) TT . = for z. = <h.,p;-h.>. 1 2-h.(2p-l)RA 1 1 1 For p = .75, h 3 = 50 and RA = .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.) 2 Using (3.46.), we easily establish TT^ = [h^/h^J TT^ » where n\ = ^ ( ^ i ^ ) with z\ = < ^ 1 ^ » _ ^ > f° r = 1»2. By the figures of Example 4.5. we then obtain for 2 instance = [20/50] (25) = 4, so that = - 4. In general: rM2 (4.14.) TT. = T T 1 TT. 3 Ih.J i - 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 is assumed to behave according to the u t i l i t y 2 function u(w;0,l,-l,g,n) ^ - (w + c) , with c = - -j}. For the gamble z = p 1 <h,p;-50>, for what value of p w i l l the decision maker state h = 60, given that his risk premium for = <40;0> is equal to 5. According to (3.90.) we have = 8(5)/[(40) 2+4(40)(5)-4(5) 2] = ^fj • Then, by (3.70.), we obtain h2R + 2h (4.15.) p = — A -^—z-2(h+h) - R (h -h ) 2 so that with R = YTK> h = 60, and h = 50, we have p ~ .71. - 80 -4.2. The R -method including risk proneness and risk neutrality. A \ The R - method as discussed in section 4.1. assumed, that the decision A maker conforms to a single risk averse u t i l i t y function over the entire wealth region. Actually, this assumption confounds two assumptions (1) that the decision maker i s risk averse over the entire domain, and (2) that he conforms to just one u t i l i t y function over the whole region. Let us f i r s t consider the case where the decision maker s t i l l behaves according to one u t i l i t y function, however he is not risk averse over the whole domain. With our results and remarks in Chapter 3 i t w i l l be straight-forward to discover risk neutrality, in which case the linear u t i l i t y function u(w;0,0,T,B,ri) w i l l be selected. It is also easy to find out whether the decision maker is risk prone. Namely, i f he indicates negative risk premiums for reference gambles, then the decision maker is risk prone according to a lemma similar to Lemma 2.4. However, Chapter 3 does not provide a tool for selecting the "best" function from among alternative risk prone u t i l i t y functions. Only i f we derive similar mathematical results, can we employ the R^-method for risk prone u t i l i t y functions. Below we use two examples to give direction as to how such results may be obtained. Example 4.9. Consider the risk prone quadratic u t i l i t y function given by 2 (4.16) u(w) ^  (w + c) where c > 0 over the region w > - c. Note that for this u t i l i t y function and the gamble - 81 -z = <h;-h> (2.4.) becomes (4.17.) (w +C-TT) 2 = ^ (w +c+h)2 + 5g(w +c-h)2.% o p 0 u"(wo) Noting that R. = R (w ) = = ; , we can substitute w + c A A o » / \ w + c o u (w ) o o - - i — in (4.17.), so that RA (4.19.) [~ T~ = H[-JT+ h ] 2 + h[~R~- h ] 2 A A A from which we easily derive the following result: (4.19.) RA = - ~ — j for z = <h;-h> h - TT Note that this result is equivalent to the function R, for the risk averse A quadratic u t i l i t y function u(w;0,l,-l ,3 ,n) ^ - (w + c ) 2 , where c = - -JJ-. P However, according to Lemma 2.4., we know that TT w i l l be positive, whereas i t can be proved that for a risk prone u t i l i t y function IT w i l l be negative. In other words, (3.37.) and (4.19.) differ in that (3.37.) w i l l contain a positive and (4.19.) a negative risk premium. Other results for the u t i l i t y function (4.16.) w i l l follow similarly. Example 4.9. Consider the risk prone exponential u t i l i t y function given by (4.20.) u(w) ^ e 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 -TT) c(w +h) c(w -h) (4.21.) e 0 = he ° + he ° < -c, and some reduction, gives \ cosh(hR ) A This result i s equivalent to (3.50.) for the risk averse exponential u t i l i t y —cw ^ function u(w;0,l,°°,c,l) ^ - e , however different in the sense that TT w i l l be positive for the risk averse and negative for the risk prone exponential u t i l i t y 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 risk 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 risk prone u t i l i t y function. Let us now consider the case where the decision maker is risk averse (or risk prone) over the whole region, however he does not conform to a single risk averse (or risk prone) u t i l i t y function. Suppose the decision maker conforms to u,(w) for w < w + b, and to u«(w) for w > w + b, where b is a 1 o z — o constant. We can then employ the R -method for a risk averse (or risk prone) decision maker in order to obtain the u t i l i t y function within class A (or which after noting that R^  TTR (4.22.) e A - 83 -class P) to which he conforms most closely over the region w < w + b. o Similarly, we w i l l use the R -method over the region w >_ w + b in order A O to assess (w). It should be clear that this process can easily be extended to the case where the decision maker conforms to three or more risk averse (or risk 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 arrive at the value of b, however several reference gambles may be sufficient to provide a good estimation for b. Also, we should use a smoothing u t i l i t y function around b in order to connect u^(w) and ^(w). An alternative method for assessing the u t i l i t y function of a decision maker that conforms to a single risk averse u t i l i t y function over the whole region, or to more than one risk averse u t i l i t y function over different 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 in Chapter 3. Let us f i n a l l y consider the case where the decision maker is assumed to be risk averse over part of his wealth domain and risk prone over another part of this domain. We w i l l simplify this case as follows. Suppose the decision maker conforms to the risk prone u t i l i t y function u (w) for w < w + b, and to the risk averse u t i l i t y function u (w) for w > w + b. o J a = o We w i l l follow the literature, e.g. Kahneman and Tversky(1979), by assuming that b w i l l be around zero. That i s , as compared to his i n i t i a l wealth WQ, the decision maker is assumed to be risk averse for gains and risk prone for losses. For assessing the risk averse u t i l i t y function for w >. w^  we can - 84 -employ the R -method using the gamble z = <h ,p ;0>. However, our results J\ J . A. x in Chapter 3 do not provide us with a similar gamble to be used for losses. In other words, we would prefer to have mathematical results for the gamble Iry = <0,p^;-h2> so that the results for this gamble can be used in an R^ -method for assessing the risk prone u t i l i t y function u^(w) for w < WQ. Fortunately, i t is not hard to obtain results for z^. Notice that z^ is equivalent to z^ by using the following transformation: (4.23.) h 2 = - and p 2 = 1 - p^ so that by applying (4.23.) to our results in Tables 3.4. and 3.5. we w i l l directly obtain the desired results. In order to connect the functions u (w) and u (w) we should use a smoothing process. This could be done by a p finding the best u t i l i t y function u(w) using the gamble 2 = <h,p;-h>, where h is taken to be relatively small. We can also think of using the sum-of-exponentials u t i l i t y function (see section 5.5.) for this purpose. With the remarks and results in this section we actually gave direction 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. The underlying u t i l i t y function may be risk averse, risk prone or even both. 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 of our results in Chapter 3, where we expressed R^  = R A ( W Q ) a s a 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 Definition 4.1. The purpose of extending our results is to improve the R^-method as discussed in Chapter 4. The following sections cover a wide variety of possible extensions. In section 5.1. we w i l l give some direction 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 certainly be improved i f the u t i l i t y function can be selected from among a greater number of alternatives. In section 5.2. we w i l l note that we could also establish results using R^  = R^Cw^ instead of R^ . This may particularly be useful when R^  can only be expressed as a function explicitly including the i n i t i a l wealth. Another extension can be found in the choice of the reference gamble. In section 5.3. we w i l l show, that i t is feasible to derive similar results for gambles other than 2 = <h,p;-h> and z = <h,p;0>. To be specific, we w i l l b r i e f l y discuss the gamble z = <h^,p;ti2> where h^ ^ h^, the gamble solely given by i t s mean and variance, the normally distributed gamble, and f i n a l l y the uniformly distributed gamble. Section 5.4. focusses on how to derive our results by means of Taylor - 86 -series expansions on both sides of the equation (2.4.) • We feel that results for 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 in Chapter 3 for the well known and widely used sum-of-exponentials u t i l i t y function. Finally, section 5.6. w i l l conclude this chapter with a brief summary of the thesis. - 8,7 -5.1. Extended results 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 ,r i ) defined by (3.14.). The results of Chapter 3 are confined to six 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 arrive at similar results for those functions. A subclass of (3.14.) is formed by (3.18.), of which Pratt(1964) gave several special cases, e.g. (5.1.) and (5.2.) and (5.3.) By specializing (5.1.) one step further we obtain (5.4.) - 88 -where sinh w^ i s the inverse hyperbolic sine function of w. From (5.2.) we e a s i l y derive the following special case: (5.5.) u(w;0,2,l,8,g) ^ arctan(w) Ac t u a l l y , Pratt(1964) introduced his special cases (5.1.) to (5.3.) i n -cluding a transformation, namely the su b s t i t u t i o n w by w + c. This way, (5.4.) and (5.5.) become respectively: (5.6.) u(w) ^ sinh ^(w + c) and (5.7.) u(w) % arctan(w + c) Other sp e c i a l cases of u(w;£,a,T ,0,ri) include (5.8.) u(w;a-l,a,T,8,r i ) ^ f a - l f 8 sa T . 1~T + n ds = rr, a Bw ( l-T)aBI + c a w + c 1-T where c = jrr B and 2 3n (5.9.) u(w;0,2,-l,B,n) ^ -w(w + c) , where c = ^ - 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 possibilities of the R -method. At this moment we feel that such results are very hard to obtain. However, the following example illustrates 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 2(w +c) (5.13.) tan[arctan(w +c+h) + arctan(w +c-h)] = -1 - [(w +c) 2-h 2] o Equating (5.12.) and (5.13.) according to (5.11.) gives after cross-- 90 -multiplication and some reduction: (5.14.) 7T(W + C ) 2 - ( T T 2 + h 2 ) ( w +C) + T T ( l + h 2 ) = 0 o o which yields (5.15.) w + c = o 2 ,2 AI TT + h + y r 2 .2,2 2 [TT - h J - 4TT 2TT though only (5.16.) w + c = o TT2 + h 2 - ]j[-n2 - h 2 ] 2 - 4TT2 2 7 T is applicable because of the following: For h > 0 and TT 0, RA(w ) should A o reach zero, so that after noting that u"(w ) 2(w +c) (5.17.) R,(w ) = - o _ o A ° u'(w ) 1 + (w +c) 2 o o i t follows that only (5.16.) should be considered. Finally, inserting (5.16.) into (5.17.) gives (5.18.) o r 2 .2 / 2 .2,2 . 2, , \ 2TT[TT +h - VLTT -h J -4TT J A o 4 u4 , 2.2. /r' 2 ^ 2,2 . 2 TT +h — (TT +h )V[TT -h 3 -4TT so that we derived the absolute risk aversion at the i n i t i a l wealth w as a o function of h and the risk 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 like to have a closer look at the absolute risk aversion R as a function of the risk 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 (5.19.) 2TT 2 2 h + a(x) -TT x = - l , h, l , 2, where a(-l) = -1, a(%) = 4, a(l) = 1, and a(2) = 0. The function a(x) is given in Figure 5.1. for the specified values T = -1. %, 1, and 2. - ; i * -O(T) -1 Figure 5.1. The function a(x) in (5.19.) for T = -1, k, I, and 2. From (5.19.) we easily derive the following: (5.20.) a(T) = 2 TTRA T = -1, %, 1, 2, but, of course, i t would be of much more interest to express a d ) solely as a function of the parameter T . In that way we would have achieved a significant generalization of our previous results. - 92 -Note that we can visualize, for instance, a hyperbolic shaped line through the three points in the f i r s t quadrant of Figure 5.1. Also note that the fourth point (r,a) = (-1,-1) is the mirror image of (r,a) = (1,1) with respect to the line a = - T. If indeed i t were possible to derive a(T) solely as a function of T, we would have at once mathematical results for a l l functions included by u(w;0,l,T,g,n)• For example, suppose a(-b) to be known, then we directly establish the function for the u t i l i t y function 3 / u(w;0,l,-%,g,n) ^  - (w + c ) 2 through (5.19.) by plugging in the value of a(-b)- At this point, however, we have not established such extended results. - 93 -5.2. Extended results for the relative risk aversion R ^ ^ ) ' In Chapter 3 we derived R.(w ) as a function of the parameters of a A O gamble z, and a response variable of that gamble (i.e. certainty equivalent, or risk premium, or probability equivalent, or gain equivalent). We are also able to derive similar results for the relative risk aversion R (w ) R o at the status quo w .. By Definition 2.9. we have R s R (w ) = w R.(w ) = o R R o o A o w^ R^ . To be specific now, R^  can be expressed as a function of: (i) the parameters of a nondegenerate gamble z, as a fraction of the i n i t i a l wealth w^  in case of gains and losses; and ( i i ) the certainty equivalent, or the risk premium, or the gain equivalent of that gamble, as a fraction of w^ . An i l l u s t r a t i o n of the preceding is a decision maker having a quadratic u t i l i t y function u(w; 0,1 ,--1, B ,Tl) 'v* - (w + c ) 2 where c = - 5. At his P TT i n i t i a l wealth w he states TT* = — as his relative risk premium for the o w ° h h i f a i r , nondegenerate relative gamble z = <h*;-h*> = <—;- —> . By (3.37.) w w o o we have: (5.21.) R = w R o 2TT , 2 2 h - TT 2TT* (h*) 2 - (TT*) 2 Hence we expressed for this decision maker R^  = ^R^Q^ & S A ^ U N C T ^ O N °^ 1. Note that h* can be s t a t e d l i k e "one f i f t h of the d e c i s i o n Baker's status quo" and his r e s p o n s e •ay be l i k e "ten percent of my status quo". - 94 -h* s —— , which is the standard deviation h of a fair f i f t y - f i f t y gamble w ° TT as a fraction of w , and TT* * — , which is the gamble's risk premium as a o w o fraction of w . The results for the other u t i l i t y functions in Table 1 o follow similarly. 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 is provided^ by the special exponential u t i l i t y function defined by c / (5.22.) u(w) % - e where c > max (0, 2w) Consider the fa i r f i f t y - f i f t y gamble z = <h;-h>, so that by (2.3.) we have for the function (5.22.): w -TT - w +h . w -h / c n \ ° l O , 1 O (5.23.) e = --e + --e u"(w ) 2w + c As R = R (w ) = - w — = — , i t follows that R R o o , ( v u (w ; w o o (5.24.) c = w (R - 2) o R Substitution of (5.24.) into (5.23.) yields y 2 v 2 v 2 (5.25.) e 1 - * - i - e 1 + l 1 * • i . e 1 " " * which gives R as an implicit function of h* and TT*. R / - 95 -So far we have shown, that for some u t i l i t y functions we might be able to express the relative risk aversion = R^CW0) a s a function of (i) and ( i i ) . However, we have not explicitly discussed the purpose of doing this. To understand this point, we should note that our results in Chapter 3 derived the function to be independent from the i n i t i a l wealth W q. This is an important observation, since therefore we do not require the knowledge of the decision maker's status quo W q in order to obtain his response (e.g. risk premium) to a nondegenerate gamble, nor do we need to know W q in order to calculate the value of the function R^ for this gamble. So, i f i t is not feasible to obtain the function R. without w , we may s t i l l be able to arrive A o at a result where R^ is expressed as a function of (i) and ( i i ) . That i s , in this case the knowledge of W q would not be required in order to evaluate 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^ = <h,p;0>. Fir s t , results may be obtained for a generalization of z and z^, namely the gamble z = <h^,p;h2>. Case 1 below w i l l center on this gamble. Second, instead of looking at discrete gambles, we may think of considering con-tinuous gambles. Several continuous gambles are illustrated in the Cases 2, 3 and 4 below. Case 1 : Consider the gamble z = <h^,p;h2> where h^ f h^, and E(z) = ph 1 + (l-p)h 2 = p(h x - h 2) + h 2 : z : Figure 5.2. 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 (5.26.) yields the following relationship: - 97 -rB(wo+z) 1-T BU-T) + n (5.27.) rB(w +h. ) 1-T B(1-T) 2—L- + n + (1-P) rB(wo+h2) 1-T Using (3.20.) and further reducing (5.27.) gives: (5.28.) 2R. 1-T 1 + = p 1 + h l V 1-T (1-p) 1 + h 2 V 1-T Explicit results are given for two cases by specifying T. Setting T - 1 gives by (5.28.) for u(w;0,l,-l,8,r,) ^ - (w + c ) 2 , with c = - j (5.29.) (1 - 2R ) 2 = p(l - h xR A) + (1-P)(1 " h 2R A)' which gives (5.30) 2[2 - ph x - (l-p)h 2l R a ~ 2 2 - ph 2 - (l-p)h 2 which is a generalization of respectively (3.36.) and (3.89.), as can directly be shown by selecting = h and respectively h 2 = - h and h 2 = 0. setting T = 2 in (5.28.) we obtain for u(w;o,l,2,p\n) ^ - w + c > w i t h By c = 2n (5.31.) 1-P 1 + b2R, 1 + 1 + bh 2R A which gives (5.32.) 2[2 - ph x - (l-p)h 2] A h h 2 - ph22 - (1-P)h12 - 98 -which is a generalization of respectively (3.45.) and (3.99.), as can directly be shown by selecting h^ = h and respectively = -h and h^ = 0. Case 2 : Consider a gamble z solely given by its mean and variance. We are thinking of I being continuous, although our statements are equally valid for z having a discrete distribution. As no specification of the distribution of z has been given, we do not expect to arrive at complete results for many different u t i l i t y functions^ However, the example given in this section should serve as a f i r s t step towards other results possibly to be obtained by good approximations. A gamble solely given by i t s mean and variance i s especially interesting from a finance theoretic point of view. For example, the original Markowitz(1952) theory of portfolio selection assumes that the decision maker's (investor's) preferences can be ranked completely in terms of mean and variance of the portfolio return. Also, the discussion on the mean-standard deviation analysis in Tsiang(1972), Bierwag(1974), Borch(1974), Levy(1974), and Tsiang(1974) illustrates the interest in gambles solely given by mean and variance (or standard deviation). We w i l l not further digress 2 on the finance theoretic literature of this topic, but instead emphasize that most of the literature on gambles solely given by their mean and variance mention the special role of the quadratic u t i l i t y function in 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 related references include Biervag(1973), Borch(1969,1973), Feldstein(1969), G l u s t o f f and Nigro(1972), Hanoch and Levy(1970), and Tobin(1958,1969). - 99 -gamble I depends only on the mean u and the variance a of the gamble, and hence not on the distribution of the gamble. Namely: E[u(z)] = E[-(z+c) Z] - E(2 2) - 2cE(z) - c 2 (5.33.) 2 2 2 - (y + o ) - 2cu - c 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.): 2 2 (5.34.) - (w + y - TT + c) = E[- (w + 1 + c) ] o o Developing the right hand side of (5.34.) gives (5.35.) E[-(w +z+c)2]= -(w +c) 2 - 2(w +c)y - (y 2+o 2) o o o Substitution of (5.35.) into (5.34.) yields after some rearranging: 2 2 (5.36.) - 2(w +C)TT = a + 2yrr - TT o According to (3.20.) we have = - — ^ — - , which gives with (5.36.) o - 100 -(5.37.) R - 2 l T A 2 2 a + 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 fai r gamble we have y = 0 and hence (5.37.) results directly in (3.10.). Also, considering the gamble z = <h;-h>, which has y = 0 and 2 2 . a = h , we obtain (3.37.) from (5.37.). Similarly, for the gamble z = 2 2 <h;0> with y = ^ h and c = fch we easily derive (3.90.) from (5.37.) Case 3 : Consider the normally distributed gamble z with mean y and 2 „ 2 variance o , i.e. P(z<z) = / f^(z)dz , where — 0 0 _ ( g - y ) 2 i " 2 ° 2 (5.38.) f„(z) = —i=»e where Tt = 3.14. . N 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 J N —CO 2 2 (5 40 ) i ~ c w -cy+^c 0 1 r_ 2,2 v:>.tu.; ) ^ o +°° -;[z-y+co ] = _ f / e 2o^ d 5 - 101 -1.2 2 -cw -cu+^c o (5.40.) = - e ° which easily gives / c / i -v " c 2 -cy+%c 2a 2 (5.41.) e = e so that, using 2 = u - TT , where ir is the risk premium for the gamble z, and c = R , we have: (5.42.) R = &LZ». = ^ a2 o 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 in a similar way: P (5 43 ) R = 2 ( y ~ g ) = ?! U ' J J KA 2 2 2^ 2 „ 2 a + u - z a + 2yrr - TT Of course, this result also follows directly from (5.37.) in Case 2, Case 4 : Consider the gamble z uniformly distributed on [h ,h ] . For this h l + n2 gamble, where E(z) = , (2.1.) gives (5.44.) u(w +2) = / u(w +z)f (2)d2 o o UN where f T T l, is defined by UN - 102 -(5.45.) h 2 " h l hj < z < h 2 elsewhere Hence, for u(w;0,1,T,8,n) with x ^ 1 (5.44.) results in: rB(w +z) ^ 1-T B(I-T) + n + 0 0 - — r — f u(w + z)dz hrh2 -oo ° (5.46.) ( h 2 - h 1 ) B < £ ( l - T ) ( 2 - T ) B(w +h„) > 2 T rB(w +h,) i 2 T o "2' + n o r + n which gives, using (3.20.), for T M (5.47.) T -- + z RA 1-T ( h 2 - h 1 ) ( 2 - T ) ,2-T — + h RA 1 2-T or (5.48.) 1 + zR.^ A 1-T ( h 2 - h 1 ) ( 2 - T ) 1+ h2 RA 2-T 1+ h l R A 2-T By selecting different values for T we w i l l be able to derive the results - 103 -for this uniformly distributed gamble. Eor example, let us select T = -1, i.e. the quadratic u t i l i t y function. For this function (5.48.) gives: (5.49.) RA(1-2V2 " 3lh7h7 (1-h.R.)3 - (l-h.R A) 3 2 A 1 A from which i t follows that (5.50.) 22 - (h +h ) p = ^ A Jl l r,2 2 - 3[h 2 + h ^ + hj] Of course, this result could have Case 2, by noting that y = E(z) = uniformly distributed gamble z. been obtained directly from (5.37.) in 1 2 1 2 2^h2+hl^ a n d 0 = 12"^ h2~ hl-' f o r t h e \ - 104 -5.4. The results using Taylor series expansions. The idea behind the mathematical results, as pointed out in section 3.1., was to expand both sides of the equation ( 2 . 4 . ) u(w - TT) = E[u(w + z)] o o using Taylor series. Originally we derived our results by expanding (2.4.) explicitly for the functions in Table 3.1., i.e. we explicitly calculated ( 3 . 1 . ) and ( 3 . 2 . ) . For example, consider the hyperbolic u t i l i t y function u(w;0,1,1,B,n) y - — - — , where c = For this function, from now on w + c p simply denoted by u(w), we have tc c i •» (m), \ _ f -1 * ( 5 . 5 1 . ) u (w) = w + c m. so that ( 3 . 1 . ) becomes TT111/ , Nm+1 (m), u(w - TT) = u(w ) - E. — ( - 1 ) -u (w ) o o m=l m. o ( 5 . 5 2 . ) = u(w ) - ? m o m=l , , Nm+1 (w +c) o and ( 3 . 2 . ) can also be explicitly calculated for the fair gamble z = <h;-h>, where again the u t i l i t y function is 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 -• , , m s i o 2 m=l , .... \m+l 2 m-1 (w +c) o 1. ? h m (w +c) o m+1 2m u(w ) -o m=l (w +c) o 2m+l Equating (5.52.) and (5.53.) according to (2.4.) gives: (5.54.) m= 1 w + c j w + c v o 2m u"(w ) 2 so that with R, s R, (w ) = , , °v = —— we obtain A A o u ( w ) w + c o o (5.55.) = S 1 t ( % h R A ) 2 ] m . m=l A m=l A The formula (5.55.) easily yields: (5.56.) 1 - %TTR, - 1 = 1 - (*>hR V A - 1 as w + c o < 1 and (%hR V A w + c o < 1 can be assumed without loss of generality. Then, (5.56.) gives (5.57.) h 2 - 106 -which was d i f f e r e n t l y derived i n (3.46.). Note that t h i s result displays an RA% ESTIMATION ERROR equal to zero. Other examples are s i m i l a r l y derived, e s p e c i a l l y for the quadratic u t i l i t y function. The l a t t e r was shown i n the proof of Theorem 3.1., where i t holds, that u^(w;0,1 ,-1,B,n) = 0 for m = 3, 4, Certainly, the r e s u l t s i n Chapter 3 w i l l not be easier obtained by using Taylor series expansions. However, we believe that the preceding example may serve as a basis for deriving r e s u l t s for u t i l i t y functions other than those l i s t e d i n Table 3.1. or for gambles other than those dealt with i n Chapter 3. i - 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 is the sum-of-exponentials u t i l i t y function, defined by (5.63.) u ( w ) _ e-*w _ b . e_ c w where a,b,c>0 , a*c For this u t i l i t y function we w i l l present in this section some mathematical results similar to those given in Chapter 3. Let TT. = TT.(W ,h.), i = 1, 2, 3, be the risk 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.). It is assumed that f f h^. According to (2.4.) we easily obtain (5.64.) u(w - T T . ) = E[u(w +2.)] i = 1, 2, 3, O 1 o i which yields for (5.63.): -a(w -TT.) -c(w -TT ) e ° 1 + b-e 0 1 (5.65.) 1 2 f-a(w+h.) -c(w +h ) -a(w -h ) -c(w e ° 1 + b-e ° 1 + e ° 1 + b-e so that after rearranging we obtain the following: - 108 -(5.66.) -aw aTT i 1 ah. -ah. e 1 + e 1 -cw + b-e CTT. 1 ch. -eh. I l e + e = 0 Thus (5.67.) If we define -(c-a)w b-e aTT. e - c o s h ( a h . ) I CTT. cosh(ch.) I i = 1, 2, 3. (5.68.) aTT. e - cosh(ah.) H. s 1 CTT. e 1 - cosh(ch.) I i = 1, 2, 3, then (5.67.) i s (5.69.) -(c-a)w b-e i = 1, 2, 3, so that the following results: u"(w ) R. s R (w ) = 2_ A A o' u'(w ) o 9 -aw -cw I o ,2 o a e + be e -aw -cw o , , o ae + bee (5.70.) _ - -(c-a)w 2 2, o a + c be -(c-a)w + cbe 2 2 U a - c H. a - cH. I i = 1, 2, 3. - 109 -Hence, we found 2 2 2 2 2 2 U a - c H. a - c H „ a - c H, (5.71.) 1 = ?. = i a - cH^ a - cH^ a - cH^ which after cross-multiplications reduces to: (5.72.) H l = H2 = H3 One of the above three equations is redundant, while the other two equations (e.g. H^  = H^  and H^  = H^) provide through existing numerical procedures solutions for a and c. If the value w of the decision maker's i n i t i a l o wealth is known, the third parameter b is 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 is known to behave according to this sum-of-exponentials u t i l i t y function. Example: A decision maker with a sum-of-exponentials u t i l i t y function u(w) —aw — cw . . _ - e - b-e is known to state the following risk premiums for the gambles z^ = <1;-1>, z"2 = <2;-2>, and z^ = <3;-3>: (5.73.) TTj = . 50 and TT2 = 1.50 and TT3 = 2.57 The values in (5.73.) are a r t i f i c i a l l y selected inorder to arrive at integer parameter values later on. The equations (5.72.) with defined by (5.68.) become for these values: - 110 -' air a i T 2 air^ e - cosh(a) e - cosh(2a) e - cosh(3a) (5.74.) CTT 1 CTT_ C T T -e - cosh(c) e - cosh(2c) e - cosh(3c) A numerically derived solution of (5.74.) yields: 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 in addition the decision maker's i n i t i a l wealth Wq is known to be wc 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 l l -5.6. Summary. The basis for the R^-method is provided by the mathematical results in Chapter 3. Essential is the idea that for a number of u t i l i t y functions, the absolute risk 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 ( i i ) the decision maker's response to that gamble, in particular: the risk premium, or the certainty equivalent, or the probability equivalent, or the gain equivalent of that gamble. In Chapter 3 we derived the results for five risk 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 five alternative u t i l i t y functions also belong to the well known HARA-class. We obtained results referring to both the gamble 2 = <h,p;-h> and the gamble z = <h,p;0>. The R -method as discussed in section 4.1. employs these mathematical A results to obtain a procedure for selecting among alternative u t i l i t y functions. The basic idea of the R -method i s , that R.(w ) has a specific A A o value, no matter what gamble the decision maker is faced with. The R^-method determines which u t i l i t y function, chosen from the five risk averse u t i l i t y functions, the decision maker conforms to most closely. The function to be selected should have the least relative standard 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 calculating - 112 -the corresponding R values as given in Chapter 3. Section 4.2. extends A the R^-method to incorporate the risk prone and the risk neutral cases. 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