October 2011 WORKING PAPER SERIES 2011-ECO-05 Even (mixed) risk lovers are prudent David Crainich CNRS-LEM and IESEG School of Management Louis Eeckhoudt IESEG School of Management (LEM-CNRS) and CORE (Université Catholique de Louvain) Alain Trannoy Aix-Marseille School of Economics, EHESS IESEG School of Management Catholic University of Lille 3, rue de la Digue F-59000 Lille www.ieseg.fr Tel: 33(0)3 20 54 58 92 Fax: 33(0)3 20 57 48 55
Even (mixed) risk lovers are prudent David Crainich CNRS (LEM, UMR 8179) and Iéseg School of Management Louis Eeckhoudt Iéseg School of Management and CORE (Université Catholique de Louvain) Alain Trannoy Aix-Marseille School of Economics, EHESS 1
1. Introduction In many if not all textbooks of microeconomics and finance, at least one chapter is usually devoted to an analysis of risk attitudes. Risk averters and risk lovers are usually described in an expected utility framework respectively by the concavity ( u '' < 0) or the convexity ( u '' > 0 ) of their utility function. From there on however the treatment of the two fundamental risk attitudes diverges. Risk lovers seem to be forgotten and the attention concentrates almost exclusively upon risk averters whose coefficients of absolute and relative risk aversion are discussed in details while specific assumptions are made about their behavior (e.g. decreasing absolute risk aversion or constant relative risk aversion). Besides, again only for risk averters, further properties of the successive derivatives of the utility function start being discussed, giving rise to now well known notions such as prudence ( u ''' > 0 ) or temperance 1 ( u '''' < 0 ). When the alternating pattern of signs of successive derivatives of u is maintained when their number n tends to infinity, one obtains "mixed risk aversion", a term coined by Caballé and Pomansky [1996]. This concept is further described in Eeckhoudt and Schlesinger [2006]. Despite the silence about the features of their utility function, risk lovers exist and their important social role is ambiguously perceived. Sometimes they are seen positively in a perspective of risk sharing because they are willing to accumulate the risks risk averters wish to get rid of. However at other times - as in the recent financial crisis - they are suspected to have induced excess risk taking in financial institutions. The purpose of this note is to pay attention to properties of risk lovers' utility function beyond its convexity. First we show that - contrarily to a priori expectations - risk lovers are prudent and want to accumulate precautionary savings exactly as risk averse decision makers (D.M.'s) do. As a result prudence seems to be a very widespread trait of behaviour since it is shared both by risk averters and by risk lovers. While they agree with mixed risk averse D.M.'s at the third order, mixed risk lovers are intemperant contrarily to mixed risk averters. In fact mixed risk lovers distinguish themselves from mixed risk averters by the signs of even derivatives of their utility function while they agree on the signs of all odd derivatives. 1 The notions of prudence and temperance are also called respectively downside risk aversion (Menezes, Geiss and Tressler [1980]) and outer risk aversion (Menezes and Wang [2005]). 2
Our note is organized as follows. Since risk aversion can be linked to a preference for combining good with bad (see Eeckhoudt and Schlesinger [2006]) and Eeckhoudt, Schlesinger and Tsetlin [2009]) we look in section 2 at the implication of a preference for combining good with good which gives rise to a risk loving behavior. The consequences for the signs of successive derivatives of the utility function are formally proved in section 3. Because prudence is still often associated with precautionary saving we show in section 4 in which sense risk loving and precautionary saving do co-exist. The last section is devoted to implications for empirical validations. 2. Mixed risk lovers and "combining good with good" In two companion papers Eeckhoudt and Schlesinger [2006] and Eeckhoudt, Schlesinger and Tsetlin [2009] show how the preference for combining good with bad not only explains risk aversion ( u '' < 0 in the expected utility model) but also the alternating sign of the successive derivatives of u. As one can expect, the preference for combining good with good gives rise to risk loving. Indeed start from a binary lottery L with equally likely outcomes x and y (x < y so that y is "good" and x is "bad"): L x y Then consider the possible allocation of k (k > 0) to one branch of L so that one faces either A 2 or B 2. A 2 x + k y B 2 x y + k (2.1) If one likes to combine good with good, B 2 should be preferred to A 2 since in B 2, +k (which is good) is attached to the best outcome of L. 3
It is then easily seen that combining good with good gives rise to risk loving since lotteries A 2 and B 2 have the same mean while B 2 has a larger spread. In Rothschild and Stiglitz terminology [1970], B 2 is a mean preserving spread of A 2, which is appreciated by risk lovers. To analyze third order effects start again from L and wonder how to allocate a zero mean risk εɶ to one branch of L, keeping in mind that for a risk lover εɶ is a "good". One generates either lotteries A 3 or B 3 where: A 3 x +ɶ ε y B x y +ɶ ε (2.2) If a decision maker prefers to combine good with good he should state a preference for B 3. Notice that the risk lover who states his preference for B 3 behaves exactly in the same way as a risk averse decision maker. Indeed someone who likes to combine good with bad dislikes a zero mean risk and wishes to attach it to the best outcome of L, i.e. y so that both risk lovers and risk averters agree - for different reasons - to attach the zero mean risk to the best outcome of L. As a result while they diverge at the second order risk averters and risk lovers express the same preference at the third order so that they exhibit prudence (downside risk aversion), i.e. u ''' > 0 as formally shown in the next section.. Notice finally that lotteries A 3 and B 3 have the same mean and variance while B 3 represents a downside risk reduction vis-à-vis A 3 (B 3 is skewed to the right while A 3 is skewed to the left). Hence risk lovers and risk averters are both downside risk averters. To look at the sign of the 4 th derivative of u for a decision maker, start again from lottery L and consider the allocation to each branch of two independent zero mean risk εɶ and ɶ θ which have the same variance and where ɶ θ has less downside risk than εɶ so that both for risk averters and risk lovers ɶ θ ɶ ε (θ ɶ is "good"). One can generate either A 4 or B 4 : A 4 x + ɶ θ B 4 x +ɶ ε (2.3) y +ɶ ε y + ɶ θ 4
If a decision maker likes to "combine good with good" he should prefer B 4 to A 4 and we show in section 4 that this preference is equivalent to u '''' > 0 in the expected utility model. In Kimball's terminology [1992], this decision maker is intemperant. Notice that a decision maker who likes to combine "good with bad" will prefer A 4 and as is well known from Eeckhoudt and Schlesinger [2006], this preference corresponds to u '''' < 0. 2 Our analysis so far illustrates the main difference between risk lovers and risk averters. They disagree about the sign of even derivatives of u ( u '', u '''', ) but agree on the sign of odd ones ( u ', u ''', ). This can easily be confirmed at higher levels by considering two random variables with one of them being an n th degree increase in risk compared to the other one (see Ekern [1980]). For instance in (2.3) εɶ is a 3 rd degree increase in risk vis-à-vis ɶ θ. It is then easily shown (see section 3) that decision makers who systematically like to combine good with good have all successive derivatives of u positive. 3. Formal proofs The proofs essentially follow those adopted for mixed risk averse decision makers with of course the appropriate changes in the direction of some inequalities. We give them here for the sake of completeness. For a decision maker who follows the rules of expected utility, the preference for B 2 in (2.1) implies: or 1 1 1 1 u( x) + u( y + k) > u( x + k) + u( y) (3.1) 2 2 2 2 u( y + k) u( y) > u( x + k) u( x) (3.2) (3.2) holds true for all x, y and k iff: u '( y) > u '( x) > 0 And since y > x, this corresponds to u '' > 0. Hence the preference for combining good with good leads to the convexity of u which is the indication of a risk loving behaviour. 2 In Eeckhoudt and Schlesinger [2006] the proof is done by considering instead the allocation of two independent zero mean risks. It is shown in Eeckhoudt, Schlesinger and Tsetlin [2009] that the two approaches lead to the same result. 5
We now show that the same preference induces prudence (or downside risk aversion, i.e. u ''' > 0 ). From the preference for B 3 in (2.2) we obtain in the expected utility model that: or 1 1 1 1 u( x) + E[ u( y + ɶ ε )] > E[ u( x + ɶ ε )] + u( y ) 2 2 2 2 E[ u( y + ɶ ε )] E[ u( x + ɶ ε )] > u( y) u( x) (3.3) For this to be true for all y and x (with y > x) it is necessary and sufficient that: E[ u '( z + ɶ ε )] > u '( z ) And this is true by Jensen's inequality iff u ' is convex, i.e. iff u ''' > 0. To prove that people who like to combine good with good are intemperant, we return to (2.3). In the expected utility model B 4 > A 4 implies: 1 1 1 1 E[ u( x + ɶ ε )] + E[ u( y + ɶ θ )] > E[ u( x + ɶ θ )] + E[ u( y + ɶ ε )] 2 2 2 2 or 1 1 E[ u( y + ɶ θ )] E[ u( x + ɶ θ )] > E[ u( y + ɶ ε )] E[ u( x + ɶ ε )] (3.4) 2 2 For this to be true for all y and x with y > x it is necessary and sufficient that: E[ u '( z + ɶ θ )] > E[ u '( z + ɶ ε )] (3.5) Since ɶ θ is a downside risk reduction in risk of εɶ (i.e. a third order effect) this can be true iff u ' has a positive fourth order derivative, i.e. if u '''' > 0 (intemperance). To go to higher orders one follows exactly the same pattern which confirms that decision makers who like to combine good with good have all derivatives of u positive. 3. Risk lovers and precautionary savings Following Kimball's seminal contribution (1990), the notion of prudence ( u ''' > 0 ) is still today very much linked with precautionary savings. Since we know from sections 2 and 3 that risk lovers are prudent, we examine now how and why they also develop precautionary savings. Let us start with a very simple two period model in which the interest rate and the impatience rate are zero. In the absence of risk in the second period the decision maker's objective is: 6
Max u( y s) + u( s) (4.1) s where s stands for savings and where y represents total current resources. We assume that borrowing beyond y is not allowed so that 0 s y. For risk lovers, u '' > 0 implies that corner solutions prevail: either s * = 0 or s* = y. Because of the simple structure of the problem, the decision maker is indifferent between the two solutions which yield a total utility equal to u(0) + u( y). Now we introduce risk in the second period so that the objective becomes: Max u( y s) + u( s +ɶ ε ) (4.2) s Again the convexity of u excludes interior solutions. If s * = 0 is selected total utility is: u( y) + E[ u( ɶ ε )] while at s* = y, total utility is: u(0) + E[ u( y +ɶ ε )] The solution s* = y will be optimal for a risk lover iff u(0) + E[ u( y + ɶ ε )] > u( y) + E[ u( ɶ ε )] This is exactly condition (3.3) with x = 0, so that risk lovers who are prudent choose to devote all their current resources to savings. Since in the absence of future risk they may choose not to save at all the existence of the zero mean risk never reduces precautionary savings and in some cases stimulates it. Notice also that if the interest rate and/or the impatience rate are not zero, then under certainty there is no longer indifference between the corner solutions s * = 0 or s* = y. However in this framework, using basically the same reasoning as above, it is possible to show that the presence of a future income risk never reduces the optimal savings level and strictly increases it in some cases. Hence like the decision makers who combine good with bad, risk lovers choose to build up precautionary savings. 7
5. Conclusion When decision makers consistently choose to combine good with good they exhibit risk loving, prudence, intemperance and the successive derivatives of their utility function are positive. On the contrary decision makers who like to combine good with bad are risk averse, prudent, temperant and the successive derivatives of u alternate in sign. We also noticed that both risk averters and risk lovers want to accumulate precautionary savings because they are all prudent ( u ''' > 0 ). Hence our analysis suggests that prudence should be a universal trait of behaviour, almost in the same way as we agree that u ' > 0 is a natural and indisputable assumption. It is worth noticing that this theoretical result is confirmed to a wide extent by the recent experimental literature on higher order risk attitudes. For instance in a paper that uses both a very large sample (N=3457) of the Dutch population and the result of a laboratory experiment with undergraduate student participants (N=109), Noussair, Trautman and van de Kuilen [2011] obtain that "prudence is more prevalent than temperance" (Wilcoxon sign-rank test, p<.01), a result also confirmed in the experiments of Deck and Schlesinger [2010] 3. Besides, table 3 in Noussair et al. also clearly shows that prudence is more widespread than risk aversion and/or temperance, especially in the laboratory experiment group where on average 89% of the answers indicates prudence against 72% for risk aversion and 62% for temperance (see their table 3). Besides in the same group the rank correlation between risk aversion and prudence is slightly negative and insignificant (= -.039) while the rank correlation between risk aversion and temperance is positive and highly significant (=.367). Such observations are in accordance with our theoretical results which predict that risk averse and risk-loving decision-makers should be prudent while only risk averters should be temperant 4. Of course more experiments that jointly examine risk aversion, prudence and temperance are necessary to fully test the prediction of our model. 3 Other very recent experimental papers on higher order risk attitudes are Ebert and Wiesen [2010] and Meier and Ruger [2010]. 4 The results in the general population are less clearcut than in the laboratory experiment but they nevertheless point in the same direction. 8
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