Risk preferences and stochastic dominance Pierre Chaigneau pierre.chaigneau@hec.ca September 5, 2011
Preferences and utility functions The expected utility criterion Future income of an agent: x. Random future income denoted by x [x, x]. A lottery is simply a probability distribution of x in the [x, x] interval. A lottery i is defined by a probability density function, f i, or equivalently by a cumulative distribution function, F i (with F i (k) = k x f i(s)ds = Pr( x k) ). Economic agents have preferences over lotteries: they know how to rank several possible probability distributions of future income. We know that if individual preferences satisfy certain axioms, then lotteries can be compared using the expected utility criterion: an individual with wealth w prefers a lottery A to a lottery B if and only if E[u(w + x A )] > E[u(w + x B )]
The risk-return tradeoff Definition of the certainty equivalent CE: E[u(w + x)] = u ( w + CE ) Definition of the risk premium Π: E[u(w + x)] = u ( w + E[ x] Π ) In the special case where E[ x] = 0, we have CE = Π. For example, with CARA utility (u(w) = exp{ αw}) and a normally distributed variable, the certainty equivalent is linear in the mean and the variance of the payoff. The risk-return tradeoff at the margin: E[u(w + x)] = u(w) + u (w)e[ x] + 0.5u (c)e[ x 2 ] + o(x 2 ) A risk averse agent (u (w) < 0) is willing to bear a small risk x only if E[ x] > 0.
Why not simply use the mean and the variance of returns? Only valid in special cases Lotteries cannot be compared on the basis of their respective means and variances, unless The investor only cares about the first two moments of his future wealth: either he is risk neutral or he has quadratic utility. The probability distribution is fully parameterized by its mean and its variance. In general, higher-order moments matter: skewness, kurtosis. Arrow-Pratt risk premium: proportional to the variance of the risk. But it is an approximation, which only applies to small risks. Fact: daily returns are approximately normal (but fat tails). Compounded i.i.d. returns are lognormally distributed.
Why not simply use the mean and the variance of returns? An example borrowed from Brunnermeier s = 1 s = 2 E[r] σ Investment 1 5% 20% 12.5% 7.5% Investment 2 50% 60% 5.0% 55.0% Investment 3 5% 60% 32.5% 27.5% Investment 1 mean-variance dominates investment 2. However, investment 3 does not mean-variance dominate investment 1...
Why not simply use the mean and the variance of returns? Another example borrowed from Brunnermeier s = 1 (proba = 0.5) s = 2 (proba = 0.5) Investment 4 3% 5% Investment 5 3% 8% Sharpe ratio: commonly used measure to rank portfolios: SR = E[r] r f σ r For r f = 0, investment 4 has a higher Sharpe ratio than investment 5!
Stochastic dominance Given some properties of u, what can we say about the ranking of different lotteries? Put differently, given some characteristics of the agent s preferences, do we know which kind of distribution of future revenues he prefers? Make minimal assumptions about preferences: postulate only risk aversion, say. Fundamental question: are there some conditions under which a lottery A is preferred to a lottery B by ALL economic agents of a certain type? This is the case if E[u( x A )] > E[u( x B )] for all u with certain characteristics (to be defined). Then we say that A dominates B.
Stochastic dominance When can we say that a lottery A is preferred to a lottery B? State-by-state dominance: the payoff of lottery A is higher than the payoff of lottery B in every possible state A dominates B. Several types of stochastic dominance: First order stochastic dominance: for all increasing utility functions (u > 0) Second order stochastic dominance: for all risk averse agents (u > 0 and u < 0) Third order stochastic dominance: for all risk averse and prudent agents (u > 0, u < 0, and u > 0)
First order stochastic dominance Applies to all increasing u, but only to a few lotteries F A first order stochastically dominates F B if and only if, for all k [x, x], F A (k) F B (k) which means that, k [x, x], Pr( x A k) Pr( x B k) Example 1: compare these two lotteries: x 1 N (µ, σ) and x 2 N (µ + h, σ), with h > 0. Example 2: compare these two lotteries: x 1 N (µ, σ) and x 2 N (µ + h, aσ), with h > 0, a > 1.
First order stochastic dominance 1 ) ) 0 [ [
First order stochastic dominance Applies to all increasing u, but only to a few lotteries Intuitive interpretation: shift in probability weights. Criterion which applies to all economic agents who prefer more money to less (whether they are risk averse or not), but which only enables to compare a narrow subset of lotteries... FOSD is not the same as state-by-state dominance. Example: the payoffs are given (say, $1, $2, $3), and different lotteries offer these payoffs with different probabilities we may have FOSD, but not state-by-state dominance.
White noises A pure, zero-mean risk Which changes in risk (i.e., in the probability distribution of the payoff) decrease the expected utility of the lottery for all risk averse agents? Consider two lotteries: Lottery A yields 1 with proba 0.5 and 3 with proba 0.5. Lottery B yields 1 with proba 0.5, 2 with proba 0.25, and 4 with proba 0.25. Same expected payoff (2). Lottery B is a compound lottery: playing lottery B is equivalent to playing lottery A and another lottery with zero-mean. It is equivalent to adding a white noise to lottery A.
White noises Risk averse agents are averse to white noises Adding a white noise (lottery with zero mean) to any lottery reduces the expected utility of all risk averse agents. Another example: E x [ E ɛ [ u( x + ɛ) ]] < E x [ u( x + E ɛ [ ɛ]) ] = E x [ u( x) ] The inequality follows once again from Jensen inequality. Whenever we need to compare two lotteries, can we show that one lottery is equal to the other compounded by a white noise? Two types of white noises (in any case, E[ ɛ x = x] = 0 x): The existence of the white noise ɛ is conditional on the outcome x of the first lottery example on the preceding slide. The existence of the white noise ɛ is unconditional on the outcome x of the first lottery example on this slide.
Mean-preserving spread A change in risk that preserves the expected payoff Mean-preserving spread: spreading the probability distribution in such a way as to leave the expected payoff unchanged. Example: from x N (a, σ 2 ) to x N (a, bσ 2 ). x b is a mean-preserving spread of x a if (i) E[ x b ] = E[ x a ], and (ii) there exists an interval X such that f b (x) f a (x) for all x X and f b (x) f a (x) for all x / X. Whenever we need to compare two lotteries, can we show that one lottery is a mean-preserving spread of the other?
Mean-preserving spread: normal distributions F b (with σ = 2) is a MPS of F a (with σ = 1) sigma=1 sigma=2-3 -2-1 0 1 2 3 sigma=1 sigma=2-3 -2-1 0 1 2 3
Mean-preserving spread and second-order stochastic dominance (1) Assume that F B is a mean-preserving spread (MPS) of F A. This implies, by definition of a MPS: x x x[f B (x) f A (x)]dx = 0 (1) Integrate by parts the LHS, remembering that x [ w ] x x (x)v(x)dx = w(x)v(x) w(x)v (x)dx x x x Here, set w (x) = f B (x) f A (x) and v(x) = x. We get x x x[f B (x) f A (x)]dx = [ ] x [F B (x) F A (x)]x x x x [F B (x) F A (x)]dx (2)
Mean-preserving spread and second-order stochastic dominance (2) The first term on the RHS is equal to zero. Combining (2) with (1), if F B is a MPS of F A, then x Also, if F B is a MPS of F A, then S(x) = x x x [F B (x) F A (x)]dx = 0 [F B (z) F A (z)]dz 0 x (3) This is the notion of second-order stochastic dominance. Definition: F A second order stochastically dominates F B if and only if x x F A (z)dz x x F B (z)dz x
Mean-preserving spread and second-order stochastic dominance (3) Why is it the case? Let us calculate expected utilities: E[u( x i )] = x x u(x)f i (x)dx Integrating by parts (set w (x) = f i (x) and v(x) = u(x)) E[u( x i )] = [ ] x x u(x)f i (x) u (x)f i (x)dx x x Since F A (x) = F B (x) = 0 and F A ( x) = F B ( x) = 1, we get x E[u( x B )] E[u( x A )] = u (x)[f B (x) F A (x)]dx (4) x
Mean-preserving spread and second-order stochastic dominance (4) Integrate by parts equation (4), with w (x) = S (x) = [F B (x) F A (x)] (see (3)) and v(x) = u (x). [ E[u( x B )] E[u( x A )] = u ] x x (x)s(x) + u (x)s(x)dx x x But the first term is zero since S(x) = S( x) = 0. Finally, E[u( x B )] E[u( x A )] = x x u (x)s(x)dx (5)
Mean-preserving spread and second-order stochastic dominance (5) We know that u (x) < 0 if the agent is risk averse, so that F A second order stochastically dominates F B : S(x) > 0 for all x, which is the case if F B is a MPS of F A (see (3)). We have shown that if F B is a MPS of F A, then F A second-order stochastically dominates F B : for all risk averse agents. E[u( x A )] > E[u( x B )] The second order stochastic dominance criterion requires one more assumption (u < 0, not too far-fetched!) but it enables to compare more probability distributions than the first order stochastic dominance criterion.
Second order stochastic dominance Applies to all increasing and concave u If two probability distributions F A and F B have the same mean, then the following four statements are equivalent: All risk averse agents (u > 0, u < 0) prefer lottery A to lottery B: E[u( x A )] > E[u( x B )] F A second order stochastically dominates F B : for any x [x, x], x x F A (z)dz x x F B (z)dz F B is obtained by applying a sequence of MPS to F A. The random variable x B is obtained by adding a white noise to x A :. x B = x A + ɛ where E[ ɛ x A = x] = 0 for any x
Diversification It s good for you If x and ỹ are two i.i.d. random variables, then z 1 2 x + 1 2ỹ is a reduction in risk with respect to x (i.e., x is a MPS of z) See section 2.1.4. of the textbook. All risk averse agents prefer to reduce risks by diversifying. Suppose the CAPM holds. Compare holding the market portfolio (with an arbitrarily large number of assets) to holding only one asset with a β of 1. Is the latter a MPS of the former? Indexing, international diversification, diversification into alternative asset classes. What about risk loving gamblers?
Aversion to downside risk Do you prefer your wealth to be random when you are wealthy or when you are poor? With lottery a, you have 1000 + ɛ with probability 0.5 and 2000 with probability 0.5. With lottery b, you have 1000 with probability 0.5 and 2000 + ɛ with probability 0.5. Experiments: people tend to prefer lottery b: aversion to downside risk.
Aversion to downside risk and prudence Suppose that a lottery pays off z 1 < z 2 < < z n, where each state i occurs with probability 1 n. For a given state i (1 i < n), a second independent lottery with zero-mean payoff ɛ is added to the payoff z i. Then the expected utility is V i 1 n E ɛ[u(z i + ɛ)] + k i 1 n u(z k) Would adding the second lottery to the first in state j > i instead of i raise the agent s expected utility? This is the case if and only if V j > V i. n(v j V i ) = E ɛ [u(z j + ɛ)] E ɛ [u(z i + ɛ)] (u(z j ) u(z i )) n(v j V i ) = zj z i ( E ɛ [u (x + ɛ)] u (x) ) dx
Aversion to downside risk and prudence V j > V i for any collection of {z i } i=1,...n if and only if E ɛ [u (x + ɛ)] > u (x) for any x (since z j z i can be arbitrarily small). E ɛ [u (x + ɛ)] > u (x) for any x if and only u > 0. Definition: An agent is prudent if and only if adding a zero-mean risk to his future wealth increases his optimal level of savings. Proposition: An agent is prudent if and only if u > 0. Proposition: the agent is prudent if and only if he is averse to downside risk. Prudence precautionary motive for saving. Impact of market incompleteness (cannot trade certain risks) on the savings rate, and on the equilibrium risk-free rate. Impact of an increase in the volatility of the economic environment.
Aversion to downside risk and third order stochastic dominance Extra material, optional Consider an agent who is risk averse and prudent (u > 0, u < 0, and u > 0). This agent prefers the probability distribution F a to F b if and only if and x y x x zdf a (z) x x zdf b (z) [ Fb (z) F a (z) ] dz dy 0 x [x, x] x x Third order stochastic dominance.
Downside beta Chen Ang Xing (RFS 2006) Break down a stock s beta into its upside beta, β + (= stock β conditional on the market return being above average), and its downside beta, β (= stock β conditional on the market return being below average). Stocks strongly exposed to downside risk strongly covary with the market when the market falls. Investors more sensitive to downside risk require a higher expected return for holding assets with a large exposure to downside risk. Sort stocks into portfolios according to their β, then compute average returns on these portfolios.
Downside beta: the results Chen Ang Xing (RFS 2006) portfolio return low β 3.5% high β 14.0% low β 2.7% high β 14.5% low β + 5.7% high β + 9.8%
Several degrees of risk increases To summarize... Second-degree risk increase : mean-preserving spread. Implies a higher variance. Third-degree risk increase : increase in downside risk, i.e., dispersion transfer from higher to lower levels of wealth, which leaves mean and variance unchanged. Implies a lower skewness. Fourth-degree risk increase : increase in outer risk, i.e., dispersion transfer from the center of the distribution to its tails, which leaves mean, variance, and skewness unchanged. Implies a higher kurtosis.
Taking multiple risks Suppose that I offer you the following gamble: I will toss a fair coin, you earn $110 if it s heads, you lose $100 if it s tails. Do you accept this bet? Does your attitude change if you bet twice instead (on the same bet)? What about a hundred times?
Taking multiple risks Be careful not to misinterpret the law of large numbers. Risk is reduced if instead of being exposed to one risk of size 1, the agent is exposed to n i.i.d. risks of size 1 n. Risk is not reduced is the agent is exposed to two (or a hundred) sources of risk instead of one. In the example above, the gamble would become more appealing if the size of the bet diminished in proportion to the number of bets: for n bets, the gain (resp. loss) on each bet would be 110 n (resp. 100 n ). In this case with the subdivision of the bet size, in the limit, as the number of bets n tends to infinity, the law of large numbers indeed applies and the average net gain approaches $5.
The tempering effect of background risk Intuitively, being exposed to one risk should lower the willingness of an economic agent to bear another risk. Definition: Preferences are characterized by risk vulnerability if the presence of an exogenous background risk with nonpositive mean (including a pure risk) increases the aversion to other independent risks. Definition: Risk aversion is standard if absolute risk aversion and absolute prudence are decreasing with wealth. Proposition: Standardness is a sufficient condition for risk vulnerability. Proposition: Preferences are risk vulnerable if absolute risk aversion is decreasing and convex. (The Proof follows) This condition means that the risk premium is decreasing with wealth at a decreasing rate. 1 In particular, absolute risk aversion is decreasing and convex with CRRA utility. 1 Notice that absolute risk aversion cannot be positive, decreasing and
Proof of the second Proposition Preferences are risk vulnerable iff the indirect utility function is more concave than u, for any risk x with nonpositive mean: E[u (z + x)] E[u (z + x)] (z) u u (z) for all z E[A(z + x)u (z + x)] A(z)E[u (z + x)] To get this result, first note that DARA implies that cov(a(z + x), u (z + x)) 0 so that for all z E[A(z + x)u (z + x)] E[A(z + x)]e[u (z + x)] Furthermore, if absolute risk aversion is convex, then Jensen inequality implies that E[A(z + x)] A(z + E[ x]) A(z)
The tempering effect of background risk: implications and applications What is the impact on risk taking (and on the market price of risk) of introducing or raising healthcare insurance, unemployment insurance, disability insurance? What about a change in the probability of a deep recession which would significantly lower labor incomes and raise the probability of unemployment? What about a terrorist threat, the possibility of a pandemic, or of a shortage of food or water? The Great Moderation and the market price of risk from 2003 to Q2 2007.
Exercises Homework 2.1 2.5 Exercises on zonecours.
Acknowledgements: Some sources for this series of slides include: The slides of Martin Boyer, for the same course at HEC Montreal. Asset Pricing, by John H. Cochrane. Finance and the Economics of Uncertainty, by Gabrielle Demange and Guy Laroque. The Economics of Risk and Time, by Christian Gollier.