The Values of Relative Risk Aversion Degrees

Transcription

1 The Values of Relative Risk Aversion Degrees Johanna Etner CERSES CNRS and University of Paris Descartes 45 rue des Saints-Peres, F Paris, France Abstract This article proposes to use lotteries to interpret higher degrees of relative risk aversion (relative prudence, relative temperance,etc.). Following Eeckhoudt and Schlesinger (2008) and Eeckhoudt, Etner and Schroyen (2009), an interpretation of benchmark value for nth order relative risk aversion is provided. Keywords: Relative Risk Aversion, Relative Prudence, Relative Temperance. JEL Classification Numbers: D81 1

2 1 Introduction Eeckhoudt and Schlesinger (2006) have given an interpretation of the signs of successive derivatives of a von Neumann-Morgenstern (vnm) utility function (u(.)) by reference to a preference for harm disaggregation applied to additive lotteries. In this way they could reinterpret the concepts of prudence and temperance outside the decision models, in which these concepts had been defined, e.g., by Kimball (1990, 1993). Recently, Eeckhoudt, Etner and Schroyen (2009) present a comparison of simple lotteries which enables them to elicit in general terms whether relative risk aversion and relative prudence exceeds the standard values of 1 and 2. Nevertheless, it silent about relative temperance and more generally the nth order derivative. Caballé and Pomansky (1996) generalized the Arrow-Pratt index of absolute risk aversion to higher orders. They define the nth order index of absolute risk aversion as A n (w) = u(n+2) (w) u (n+1) (w), for n 0. Similarly, one can define the nth order index of relative risk aversion as R n (w) = w u(n+2) (w), for n 0. u (n+1) (w) In savings decision, Eeckhoudt and Schlesinger (2008) provide necessary and sufficient conditions on preferences for an nth-degree change in risk to increase saving. They show the pertinence of relative degree such as relative prudence or temperance. More precisely, they show that benchmarck value for the nth order relative risk aversion is n. In this paper, we propose to extend Eeckhoudt, Etner and Schroyen (2009) results to any relative degree and provide an interpretation of these successive benchmark values. Our paper is organized as follows. In the next section, after a brief exposition of the preference for harm disaggregation applied to multiplicative lotteries, we illustrate our approach for risk aversion and prudence. The two cases give the intuition for the results presented in section 3 where the benchmark values for any relative degree of risk aversion are formally obtained. We then briefly conclude. 2 Relative Risk Aversion and Relative Prudence We briefly present the two comparaisons of lotteries used in Eeckhoudt, Etner and Schroyen (2009) to interpret the signs 1 and 2-derivatives of u(.). Consider a decision maker endowed with an initial wealth, X, who faces the prospect of two losses expressed as shares of wealth denoted respectively k and r with 0 < k < 1 and 0 < r < 1 occurring each with probability 1 2. The decision maker can apportion the harms in two different ways yielding lotteries A 1 and B 1 defined by Insert Figure 1 2

3 As it shows, for B 1 to be preferred to A 1, risk aversion must be strong enough in the sense that RRA 1. In this case, we will say that individual presents preference for first order harm disaggregation. We denote by C1 the previous condition: Condition 1 (C1) For any wealth X, X u (X) u (X) 1. To elicit the intensity of prudence, replace one of the proportional losses (say r) by a zero mean risky return, ε [ 1, ), which is disliked by a risk averse decision maker. Hence we now have to compare lotteries A 2 and B 2 defined as: Insert Figure 2 A decision maker will prefer B 2 only if prudence is high enough that means RP 2. Following Eeckhoudt and Schlesinger (2006), we define two others lotteries B 2 and A 2 from B 1 and A 1 : Insert Figure 3 Let us define some conditions on utility function. Condition 2 (C2) For any wealth X, u (X) [1 RRA]+X u (X) [2 RP ] 0 Proposition 1 B 2 is preferred to A 2 iff condition C2 is verified. Proof. Notations: B 1 = (x(1 k 0 ), x(1 k 1 )) A 1 = (x(1 k 0 )(1 k 1 ), x) (1 k 2 )B 1 = (x(1 k 0 )(1 k 2 ), x(1 k 1 )(1 k 2 )) Eu[(1 k 2 )B 1 ] = 0, 5u[(x(1 k 0 )(1 k 2 )] + 0.5u[x(1 k 1 )(1 k 2 ))] With our notations, B 2 is preferred to A 2 iff Eu[(1 k 2 )B 1 ]+Eu[A 1 ] Eu[(1 k 2 )A 1 ]+Eu[B 1 ] Eu[(1 k 2 )B 1 ] Eu[(1 k 2 )A 1 ] Eu[B 1 ] Eu[A 1 ]. (1) i) Sufficiency A sufficient condition is : h (k 2 ) Eu[(1 k 2 )B 1 ] Eu[(1 k 2 )A 1 ] increasing in k 2. So that E [ B 1 1 u [(1 k 2 )B 1 ] ] E [ A 1 1 u [(1 k 2 )A 1 ] ] 0 3

4 E [ B 1 1 u [(1 k 2 )B 1 ] ] E [ A 1 1 u [(1 k 2 )A 1 ] ] with t B 1 = t (x(1 k 0 ), x(1 k 1 )) = ( x(1 k0 ) x(1 k 1 ) ( x(1 and t A 1 = t k0 )(1 k (x(1 k 0 )(1 k 1 ), x) = 1 ) x ) ). By noting v (A 1 ) t A 1 u [(1 k 2 )A 1 ] and v (B 1 ) t B 1 u [(1 k 2 )B 1 ], we obtain the following sufficient condition: Ev (B 1 ) Ev (A 1 ). By applying the result that is Eu (B 1 ) Eu (A 1 ) iff 1 + X u (X) u (X) 0 (RRA > 1) to function v (.) v (X) + Xv (X) 0 with v (X) = X u (X). We then obtain: v (X) = u (X) X u (X) v (X) = 2u (X) X u (X) and [u (X) + X u (X)] X [2u (X) + X u (X)] 0 u (X) [1 RRA] + X u (X) [2 RP ] 0 ii) Necessity. Consider the gambles A 2 and B 2 with k 0 = k 1 = k 2 = k, a small positive number. Inequality (1) then becomes 3 4 u[x(1 k)2 ] u[x] 1 4 u[x(1 k)3 ] + 3 u[x(1 k)]. (2) 4 Using a third order Taylor expansion of u[x(1 k)], u[x(1 k) 2 ], u[x(1 k) 3 ] around k = 0, inequality (2) can be written as xk 3 u (x) + 3x 2 k 3 u (x) + x 3 k 3 u (x) + Θ(k 4 ) 0 with lim k 0 Θ(k 4 ) k 3 = 0, we have established that B 2 A 2 for any k (0, 1) implies u (X) [1 RRA] + X u (X) [2 RP ] 0 for any x > 0. Suppose condition C1 holds, a necessary condition for condition C2 is RP > 2. Consequently, this last condition becomes necessary but not sufficient for the comparison between A 2 and B 2. Notice that condition C2 can be written as follows: [ RP 2 1 RRA ] RRA 4

5 Two effects can be distinguished: the one concerning the comparison between lotteries A 1 and B 1 which is done by the relative risk aversion value and the one concerning the preference for second order harm disaggregation which is done by the relative prudence value. If the individual presents preference for first order harm disaggregation (RRA > 1), then lottery B 2 is built with the worst association (multiplicative loss with the worst lottery, A 1 ). Then, preference for second order harm disaggregation has to be relatively strong (large RP ). Else, we retreive the Eeckhoudt, Etner, Schroyen (2009) result (RP 2). More particularly, when the decision maker is indifferent between A 1 and B 1 (RRA = 1), the first effect is nil and only the second effect is pertinent. 3 nth Relative Risk Aversion Let us define recursively B n+1 and A n+1, for any k n+1 (0, 1), Now we compare the two lotteries in an expected utility (EU) framework and we consider the case of n-harm desagragation (B n is preferred to A n ) decision maker. Similarly to the two first cases, we define, for all n 2, condition C n on n derivatives of u: Condition 3 (Cn) for any wealth X > 0, ( 1) n n k=1 an k Xk 1 [ ku (k) + Xu (k+1)] 0 with a n 1 = 1; a n n = 1 n and a n k = an 1 k 1 + kan 1 k k = 2 to n 1. Next result gives some conditions for the comparison between B n and A n. Lemma 1 For an EU decision maker with, if B n A n (condition C n holds), then B n+1 A n+1 under condition C n+1. u: Proof. Suppose that B n A n under the condition C n on n derivatives of ( 1) n n k=1 [ a n kx k 1 ku (k) + Xu (k+1)] 0 with a n 1 = 1; a n n = 1 n and a n k = an 1 k 1 + kan 1 k k = 2 to n 1. Consider now a decision maker, endowed with initial wealth x, faces with the two lotteries A n+1 and B n+1. Then we have that for any k (0, 1), B n+1 A n+1 Eu[(1 k n+1 )B n ] + Eu[A n ] Eu[(1 k n+1 )A n ] + Eu[B n ] Eu[(1 k n+1 )B n ] Eu[(1 k n+1 )A n ] Eu[B n ] Eu[A n ]. (3) 5

6 Define a function h( ) such that h (k n+1 ) def = Eu[(1 k n+1 )B n ] Eu[(1 k n+1 )A n ]. Then a sufficient condition for B n+1 A n+1 is thus that h is increasing in k n+1. That is E [ t B n u [(1 k n+1 )B n ]] E [ t A n u [(1 k n+1 )A n ]]. Now define function v such that v (A n ) t A n u [(1 k n+1 )A n ] and similarly v (B n ) t B n u [(1 k n+1 )B n ], the sufficient condition for having B n+1 A n+1 writes: Ev (B n ) Ev (A n ). By applying condition C n to v (.) with v (X) = X u (X), we obtain an equivalent condition C n+1.with a n+1 k = a n k 1 + kan k k = 2 to n. This result is difficult to interpret and do not ellicit an explicit relative degree. For a useful interpretation of successive n order relative rsk aversion values, we propose to consider that the individual is neutral to all n- harm disaggregations: Proposition 2 For an EU decision maker with, for all n orders, B n A n, B n+1 A n+1 for any k n+1 (0, 1) if X u(n+2) (X) n + 1 for any wealth u (n+1) (X) X > 0. Proof. Suppose that B k A k for all k = 1 to n 1, thus, ( 1) k k j=1 ak j Xj 1 [ ju (j) + Xu (j+1)] = 0. B n A n iff ( 1) n n k=1 [ a n kx k 1 ku (k) + Xu (k+1)] = 0 ( [ ( 1) n a n nx n 1 nu (n) + Xu (n+1)]) = 0. Consequently, if B k A k for all k = 1 to n, a sufficient condition for B n+1 A n+1 is ( 1) n+1 [ (n + 1) u (n+1) + Xu (n+2)] 0 with u (n+1) < 0 (resp. 0) if (n + 1) even (uneven) which is equivalent to X u(n+2) (X) u (n+1) (X) n + 1. Notice the difference between additive and multiplicative risks. For additive risks, Eeckhoudt and Schlesinger (2006) provide an interpretation of the signs of n order derivatives, which characterize particularly prudence, temperance or edginess. Our analysis shows that for multiplicative harms, n th order harm disaggregation conflicts with an opposite effect and dominates if X u(n+1) (X) u (n+) (X) exceeds n. 4 Conclusion The existing literature on savings, insurance and portfolio choices under risk has revealed that quite often comparative statics results depend, among other 6

7 things, upon the values of the coefficients of relative risk aversion, relative prudence or relative temperance. More specifically the benchmark value of nth order relative risk aversion, taken into consideration inside these models, is n. In this paper, we generalize the results obtained in Eeckhoudt - Etner - Schroyen (2009) and give a more fundamental interpretation of the n benchmark values which is independent of the institutional environment in which the choice is made. Finally, the analysis of risk aversion has provided several equivalent concepts that can be applied to the problem of comparing degrees of inequality aversion (Atkinson (1970)). As the same way, our results have direct applications in the area of income distribution. References [1] Atkinson, A. B., 1970, On the Measurement of Inequality. Journal of Economic Theory, 2, [2] Eeckhoudt, L., J. Etner and F. Schroyen, 2009, The values of relative risk aversion and prudence: A context-free interpretation, Mathematical Social Science, 58(1), 1-7. [3] Eeckhoudt, L. and H. Schlesinger, 2006, Putting Risk in its Proper Place, American Economic Review 96, [4] Eeckhoudt, L. and H. Schlesinger, 2008, Changes in risk and the demand for saving, Journal of Monetary Economics, 55, [5] Kimball, M. S., 1990, Precautionary Savings in the Small and in the Large, Econometrica 58, [6] Kimball, M. S., 1993, Standard Risk Aversion, Econometrica 61, Figures 7

8 Figure 1: Lotteries A 1 and B 1 8

9 Figure 2: Lotteries A 2 and B 2 9

10 Figure 3: Lotteries A 2 and B 2 10