This paper addresses the situation when marketable gambles are restricted to be small. It is easily shown that the necessary conditions for local" Sta

Transcription

1 Basic Risk Aversion Mark Freeman 1 School of Business and Economics, University of Exeter It is demonstrated that small marketable gambles that are unattractive to a Standard Risk Averse investor cannot be made attractive even if certain independent background risks that decrease expected marginal utility are added. 1 Introduction Following Pratt and Zeckhauser (1987), several authors have attempted to answer the question: Can introducing a background gamble make a second, previously unattractive, independent gamble attractive?". This is an important issue as it challenges the intuition that investors will act in a more risk averse manner in an incomplete market than in a complete market. Main papers in the area include Franke, Stapleton and Subrahmanyam (1995) (mean zero background gambles), Gollier and Pratt (1996) (nonpositive mean background gambles Risk Vulnerability), Pratt and Zeckhauser (1987) (unattractive background gambles Proper Risk Aversion) and Kimball (1993) (expected marginal utility increasing background gambles Standard Risk Aversion). In general, this has proved to be an extremely difficult problem to resolve. The only easily tractable necessary and sufficient conditions are attributable to Kimball (1993) who shows that if an investor has decreasing absolute risk aversion (DARA) and decreasing absolute prudence (DAP) across the consumption domain, then no unattractive marketable gamble can be made attractive by introducing any independent background gamble that increases expected marginal utility 2 1 This paper was prepared when the author was visiting Kellogg Graduate School of Management, Northwestern University. He is grateful for the helpful comments on earlier versions of this paper by Christian Gollier, Richard Stapleton, Michael Theobald, Philippe Weil and participants at seminars at Northwestern and Warwick Universities. All errors do, of course, remain my own. 2 Kimball (1993) proves that DAP and DARA are necessary and sufficient for a slightly different condition than this. However, appendix G of the working paper, Kimball (1991), demonstrates that the Kimball (1993) definition of Standard Risk Aversion and the current definition are equivalent. 1

2 This paper addresses the situation when marketable gambles are restricted to be small. It is easily shown that the necessary conditions for local" Standard Risk Aversion are still DAP and DARA. The main result of this paper is that, in this case, a more general statement can be made. It will be demonstrated that, given DAP and DARA, there exists a set of background gambles that do not increase marginal utility but that still will never make small, independent, previously unattractive marketable gambles attractive. This enlarged set of background gambles has interpretation; the elements are those that aggravate (in the sense of Kimball (1993)) small, mean zero independent risks. It is thus demonstrated that the necessary and sufficient conditions for local Standard and local Basic Risk Aversion are identical, even though Basicness is a special case of Standardness. This is, therefore, an important extension to Kimball (1993) and the related literature. 2 Basic Risk Aversion Consider an investor who consumes a single consumption good. The preferences of the investor are described by avon Neumann-Morgenstern utility function U defined over a continuous domain D that contains all possible consumption outcomes for this individual. The notation U (n) is used throughout to denote the n th derivative of the utility function. It will be assumed throughout this paper that U is four times differentiable across the domain. Use the notation A(c) = U (2) (c)=u (1) (c) to denote the Pratt-Arrow coefficient of absolute risk aversion, P(c) = U (3) (c)=u (2) (c) the prudence (Kimball (1990)) and T (c) = U (4) (c)=u (3) (c) the temperance (Kimball (1992), Eeckhoudt, Gollier and Schlesinger (1996)) of the utility function. DAP is equivalent to T (c) P(c) for all c and DARA is equivalent to P(c) A(c) for all c. It will be assumed that, at all points within the domain, the investor has positive marginal utility, positive risk aversion (U (2) (c) < 0) and a positive precautionary savings motive (U (3) (c) > 0). Two equivalent definitions of Basic Risk Aversion follow: Definition 1 An investor is said to be Basic Risk Averse if adding a statistically independent background" gamble that aggravates a small zero-mean gamble makes an undesirable marketable" risk remain undesirable 2

3 Definition A An investor is said to be Basic Risk Averse if, 8c 2 D, for all background" gambles ~x such that E[U (2) (c +~x)]» U (2) (c) (1) and for all marketable gambles" ~y independent of ~x such that E[U(c + ~y)]» U(c), then: E[U(c +~x +~y)]» E[U(c +~x)] (2) That Definition 1 and Definition A are equivalent follows directly from the definition of aggravation. ~x aggravates another gamble, ~z if... an equal chance of one change in the outcome random variable or the other in isolation is preferred to an equal chance of no change or both changes" (Kimball (1993) p.592): E[U(c +~x +~z) U(c +~x) U(c +~z)+u(c)]» 0 (3) This can be compared with the definition of Standard Risk Aversion in Kimball (1993) 3, where the background gamble aggravates a small, certain, reduction in wealth. Alternatively, Standard Risk Aversion is defined by Definition A if equation 1 is replaced by: E[U (1) (c +~x)] U (1) (c) (4) Basic Risk Aversion, which considers background gambles that make small mean preserving increases in variance more painful, is a natural extension of Standard Risk Aversion, which considers background gambles that make a small certain reduction in wealth more painful. The first theorem shows that the set of background gambles considered by Standard Risk Aversion is a subset of the background gambles considered by Basic Risk Aversion. That is, Basic Risk Aversion is at least as strong a condition as Standard Risk Aversion. Theorem 1 Basic Risk Aversion is a special case of Standard Risk Aversion 3 Taking into account the equivalence statement given in Appendix G in Kimball (1991). 3

4 For any c 2 D define ± B (c) f~xje[u (2) (c +~x)]» U (2) (c)g and ± S (c) f~xje[u (1) (c + ~x)] U (1) (c)g. These are the sets of background gambles considered by Basic and Standard Risk Aversion respectively at c. To prove, for a Standard Risk Averse investor, that ± S (c) ± B (c) for all c 2 D establishes the result. From Kimball (1993), it is known that the necessary and sufficient condition for Standard Risk Aversion is T (c) P(c) A(c) for all c 2 D. T (c), P(c) are, respectively, the Pratt-Arrow coefficients of risk aversion for the (pseudo) utility functions U (2), U (1) respectively at c. In other words T (c) P(c) for all c 2 D means that the (pseudo) utility function U (2) is globally more risk averse than the (pseudo) utility function U (1). As ± B (c); ± S (c) are, respectively, the gambles that are unattractive at c under the (pseudo) utility functions U (2) ; U (1), it follows that ± S (c) ± B (c) for all c 2 D. The theorem is proven. This is important because it adds to a set of nesting" results. Standard Risk Aversion is a special case of Proper Risk Aversion, which in turn is a special case in Risk Vulnerability, which is a special case of Decreasing Absolute Risk Aversion. See Gollier and Pratt (1996). Definition 2 Local Basic (Standard) Risk Aversion is defined as Basic (Standard) Risk Aversion with the additional constraint that the marketable gamble ~y is small" in the sense of Pratt (1964). This should be contrasted with the definition of localness in Gollier and Pratt (1996), where the background gamble is restricted to be small but no constraint is placed on the size of the marketable gamble. The main result of this letter can now be presented. Theorem 2 Local Basic Risk Aversion, Local Standard Risk Aversion and Standard Risk Aversion are equivalent conditions. Following Kimball (1993), this result will be demonstrated by showing that T (c) P(c) A(c) for all c 2 D are necessary and sufficient conditions for Local Standard and Local Basic Risk Aversion. As ~y is small and unattractive in the absence of background risk, it is both necessary and sufficient for Local Basic (Standard) Risk Aversion to show that, for all c 2 D, the derived utility function E[U(c +~x)] is locally more risk averse than U(c) for all ~x 2 ± B (c) (± S (c)): 4

5 E[U (2) (c +~x)] E[U (1) (c +~x)] U (2) (c) U (1) (c) To establish the necessity of DAP and DARA for Local Basic (Standard) Risk Aversion, two small background gambles that are in ± B (c) (± S (c)) will be considered. As these background gambles are small and U (3) (c) > 0, equation 5 can be rewritten as: E[~x][P(c) A(c)] 1» 2 ff2 ~x P(c)[T (c) A(c)] (6) The first background gamble to consider is ~x = ffl, a certain small decrease in wealth. As the second (third) derivative of U is negative (positive) at all c 2 D, this is in ± B (c); ± S (c) for all c 2 D. Substituting this ~x into equation 6 establishes that P(c) A(c) is necessary for all c 2 D. The second small background gamble to consider is a ~x such that E[~x] = 1 2 ff2 ~x X (c), where X (c) = P(c) for Standard Risk Aversion and X (c) = T (c) for Basic Risk Aversion. It is clear that this gamble is in ± B (c); ± S (c). By substituting this background gamble into equation 6 it is clear that T (c) P(c) is also necessary 4. The necessity of DAP and DARA have been established. Next, sufficiency is established. Consider some ~x in ± S (c) or ± B (c). In either case, ~x will not be attractive to the (pseudo) utility function U (2) at c. Let the risk premium for ~x under U (2) (c) be given by ffi 0. As T (c) P(c), and thus the (pseudo) utility U (1) is less risk averse than U (2) for all c 2 D, and from Pratt (1964), the risk premium for ~x under U (1) (c) must be lower: say ffi ψ for ψ 0. Therefore: (5) E[U (2) (c +~x)] E[U (1) (c +~x)] = U (2) (c ffi) U (1) (c ffi + ψ) U (2) (c ffi) U (1) (c ffi) U (2) (c) U (1) (c) (7) The first inequality comes from decreasing marginal utility and the second from DARA. The sufficiency, and hence the theorem, has been proven. Theorem 3 Basic Risk Aversion is a strictly stronger condition than Standard Risk Aversion 4 By substituting X (c) = A(c) into equation 6, the local condition for Proper Risk Aversion can also be established: P Aand T =A 2 (A=P). 5

6 That T (c) P(c) A(c) for all c 2 D is not sufficient for Basic Risk Aversion is demonstrated by counterexample. The motivation for the example is as follows. Consider local" Basic Risk Aversion in the sense of Gollier and Pratt (1996) that is, where the marketable gamble can be of any size but the background gamble is small. For some c 2 D take a large marketable gamble that is neither attractive nor unattractive: E[U(c + ~y)] = U(c). Since ~x is small: E[U(c +~x +~y) U(c +~x)] = E[~x]E[U (1) (c +~y) U (1) (c)] ff2 ~x E[U (2) (c +~y) U (2) (c)] (8) Under DAP and DARA, U (2) is more risk averse than U (1) which in turn is more risk averse than U. E[ U (1) (c +~y)] = U (1) (c ß) and E[U (2) (c + ~y)] = U (2) (c ß ρ) for some positive ß; ρ. The necessary and sufficient condition for Basic (Standard) Risk Aversion for small ~x is: E[~x][U (1) (c ß) U (1) (c)] ff2 ~x[u (2) (c ß ρ) U (2) (c)]» 0 (9) Suppose that the small background gamble ~x is neutral under the (pseudo) utility function U (2) : E[~x] = 1 2 ff2 ~x T (c). Substituting this into equation 9 gives: Z c c ß [P( ) T(c)][ U (2) ( )]d Z c ß c ß ρ U (3) (ν)dν (10) The right hand side of equation 10 depends (through ρ) on how much more risk averse U (2) is than U (1) over the range of ~y. The left hand side for Basic Risk Aversion depends (through T (c)) only on the risk aversion of U (2) at c. Therefore, if U (2) has unusually high risk aversion at c compared with other points over the range of ~y, it is not clear that equation 10 need hold. Given this insight, it is easy to demonstrate by counterexample that DAP and DARA are not sufficient for Basic Risk Aversion in the large". For example, set D = [13:3; 15]. Set U(c) = P 5 i=0 a ic i where a 0 = 39=15, a 1 = 246=(15 2 ), a 2 = 464=(15 3 ), a 3 = 441=(15 4 ), a 4 = 210=(15 5 ) and a 5 =40=(15 6 ). It is easily demonstrated that over D, this utility has positive first derivative, positive risk aversion and prudence and exhibits DAP and DARA. Define b 1 =6a 3, b 2 =12a 4 and b 3 =20a 5. Set: 6

7 p 1 = P 5 i=1 a i(14:8 i 13:4 i ) P 5 i=1 a i(14:9 i 13:4 i ) ; p 2 = P 3 i=1 b i(14:8 i 14:7 i ) P 3 i=1 b i(14:9 i 14:7 i ) (11) Let the marketable gamble ~y give +0:1 with probability p 1 and 1:4 with probability 1 p 1. Let the background gamble ~x give +0:1 with probability p 2 and 0:1 with probability 1 p 2. At c =14:8, E[U(14:8+~y)] = U(14:8) and E[U (2) (14:8+~x)] = U (2) (14:8). These gambles are considered by Basic Risk Aversion at c = 14:8. However E[U(14:8 +~x + ~y) U(14:8 +~x)] = 3:8245E 8. It is thus demonstrated that DAP and DARA are not sufficient in the large". This results is reminiscent of Gollier and Pratt's (1996) result that local properness is sufficient for Risk Vulnerability and that global Risk Vulnerability is a strictly stronger condition than local Risk Vulnerability. As a final observation, the definition of Standard Risk Aversion in Kimball (1993) replaces the term makes an undesirable marketable risk remain undesirable" with the term aggravates an undesirable marketable risk". This is equivalent to replacing equation 2 with equation 3 in definition A and setting ~z = ~y. Appendix G of Kimball (1991) shows that these two definitions are equivalent when considering Decreasing Absolute Risk Aversion, Proper Risk Aversion and Standard Risk Aversion 5. This, though, is not true for Basic Risk Aversion. To see this, take ~z = ~y = ffl: a small certain decrease in wealth. In this case, equation 3 is equivalent to E[U (1) (c +~x)] U (1) (c). This constraint will not hold for all background gambles considered by Basic Risk Aversion since the (pseudo) utility function U (2) is more risk averse than U (1). 3 Conclusion This paper has analyzed how a Standard Risk Averse investor evaluates a small, previously unattractive, marketable gamble when a second independent background gamble is introduced. This paper extends the Standardness 5 This also applies to Local Standard Risk Aversion. That DAP and DARA are necessary to ensure that E[U(c +~x +~y)] E[U(c +~x)]» 0 for small ~x 2 ± S (c) means that they are also necessary for the stronger constraint of equation 3 with ~z =~y. The sufficiency of DAP and DARA for global Standardness under the Kimball definition ensures that these conditions are sufficient for local Standardness under the same definition. 7

8 result to show that there is a class of background gambles that do not increase expected marginal utility but still ensure that the marketable gamble is unattractive. The enlarged set of background gambles contains elements that aggravate small, independent, mean zero risks. References Eeckhoudt, L., Gollier, C. and Schlesinger, H.: 1996, Changes in background risk and risk taking behavior, Econometrica 64, Franke, G., Stapleton, R. C. and Subrahmanyam, M. G.: 1995, The size of background risk and the theory of risk bearing. Mimeo: University of Konstanz. Gollier, C. and Pratt, J. W.: 1996, Risk vulnerability and the tempering effect of background risk, Econometrica 64, Kimball, M. S.: 1990, Precautionary saving in the small and the large, Econometrica 58, Kimball, M. S.: 1991, Standard risk aversion. Paper: 99. NBER Technical Working Kimball, M. S.: 1992, Precautionary motives for holding assets, New Palgrave dictionary of money & finance. Kimball, M. S.: 1993, Standard risk aversion, Econometrica 61, Pratt, J. W.: 1964, Risk aversion in the small and the large, Econometrica 32, Pratt, J. W. and Zeckhauser, R. J.: 1987, Proper risk aversion, Econometrica 55,