Risk aversion and choice under uncertainty Pierre Chaigneau pierre.chaigneau@hec.ca June 14, 2011
Finance: the economics of risk and uncertainty In financial markets, claims associated with random future payoffs are traded. What should be their price? (asset pricing) In a frictionless market, it depends on the economic fundamentals, and on the time preferences and the risk preferences of economic agents. Price of time, price of risk. Once you understand these time and risk preferences, you can understand changes in asset prices.
A simple example Why intuitions only get you so far... Consider a lottery whose payoffs are $1,000 with probability 0.4, and 0 with probability 0.6. What would you be willing pay to participate in this lottery? $400? $300? Intuitively, it should depend on your preferences: are you adventurous or cautious? It will also depend on your wealth...... to your exposure to other sources of risk...... and to the number of opportunities you have of playing the lottery (casinos usually have rules against playing too much ). Bottom line: even this simple example is too complicated to be solved intuitively. We need a model!
Another simple example The gamble is the following: someone offers you to flip a fair coin. If it s tails you receive $1 and that s it. If it s heads the coin is flipped again. If it s tails you receive $2 and that s it. If it s heads the coin is flipped again. At every round, the amount paid if tails obtains doubles. How much would be willing to pay to play this gamble?
Another simple example A shocking result Suppose that you are risk neutral. Then the price of this gamble is given by its expected payoff. With probability 1 2 you get $1. With probability 1 4 you get $2. With probability 1 8 you get $4. The expected payoff is ( 1 ) i 2 i 1 = 2 i=1 i=1 1 2 =
Another simple example This gamble has an infinite value! Yet with a 93% probability, it pays off $8 or less. The probability to get more than $64 is only 1%... It seems that economic agents either discount these very high payoffs (this is the idea behind the utility function), or underweight these extreme probabilities (this is the idea behind risk neutral probabilities). Solution to this puzzle: risk aversion.
Lotteries We assume that the outcome is unidimensional: money! The set of possible outcomes is X {x s } s=1,...,s. s denotes the state of the world. We denote the random outcome by x. A lottery L is described by a vector a probabilities (p 1,..., p S ). Example 1: Suppose that S = 3, and the possible outcomes are x 1 = 1, x 2 = 10, x 3 = 16. Lottery La is defined by p 1 = 0.1, p 2 = 0.8 (p 3 = 0.1). Lottery L b is defined by p 1 = 0.2, p 2 = 0.3 (p 3 = 0.5). Example 2: S =. The outcomes are x 1 = 0, x 2 = 1, x 3 = 2, etc. Any lottery is defined by the set of probabilities {p 1, p 2, p 3,... }, which must be such that s=1 p s = 1. Given a set of possible outcomes X, the set of all lotteries is L {(p 1,..., p S ) R+ p S 1 + + p S = 1}. A degenerated lottery takes one value x s X with probability 1.
Preferences In consumer theory, preferences are defined over baskets of goods. In financial economics, preferences are defined over lotteries. Axioms on preferences: Completeness: for any two lotteries La and L b, either L a L b or L b L a. Transitivity: If La L b and L b L c then L a L c. Continuity: If La L b L c, there exists a scalar α [0, 1] such that L b αl a + (1 α)l c. NB: a preference relation which is both complete and transitive is rational.
Preferences Theorem: Given a preference relation which satisfies these axioms, there exists a preference functional U which maps any lottery L L into R such that L a L b U(L a ) U(L b ) NB: U is not unique. Consider any increasing function g. Then V ( ) g(u( )) represents the same preferences as U, in the sense that it will yield the same ordering of lotteries (put differently, an individual with preference functional U would always choose the same lotteries as an individual with preference functional V : they would behave in the same way).
Preferences We may want to impose other axioms on preferences: Time consistency: preferences are time consistent if a choice (possibly contingent upon the realization of some random variables) which is optimal at any time t is also optimal at any time T > t. Violation of time inconsistency: hyperbolic discounting, monetary policy, provision of incentives, liquidation of a firm, etc. Anytime there is a conflict between ex-ante and ex-post optimality. Resolution mechanisms: commitment or renegotiation (very important in corporate finance). Independence: for all α [0, 1], L a L b αl a + (1 α)l c αl b + (1 α)l c. If the lottery L a is preferred to the lottery L b, then a mix of L a with a lottery L c is also preferred to the same mix of L b with L c, for any L c.
Choice criterion: expected utility Theorem(von Neumann and Morgenstern): If the preference relation satisfies the axioms above, then it can be represented by a preference functional which is linear in probabilities. This means that, for each outcome x s, u s s.t. S S L a L b ps a u s ps b u s s=1 s=1 Or, with outcomes continuously distributed in [0, ), there exists a function u such that: L a L b 0 u(x s )dφ a (x s ) 0 u(x s )dφ b (x s ) where Φ a is the cumulative distribution function (c.d.f.) of x with the lottery a. In any case, we also write that E a [u( x)] E b [u( x)]. NB: The utility function u is not unique. Any increasing linear transformation of u will give the same ranking of lotteries.
Risk aversion Definition: An agent is risk averse if he dislikes all zero-mean risks at all wealth levels. This implies that, for a risk averse agent, E[u(w + ɛ)] < E[u(w)] = u(w) = u(e[w]) = u(e[w + ɛ]), where E[ ɛ] = 0. Proposition: An agent is risk averse if and only if his utility function u is concave. Jensen inequality: E[u( x)] u(e[ x]) if and only if u is concave What are the implications of concavity for marginal utility? What is the form of the utility function of a risk neutral agent? Risk aversion explains the behavior of economic agents: purchase of insurance, purchase of risky assets. It is a crucial element in GE asset pricing models. Why doesn t it matter in derivatives pricing?
To summarize Some perspective, and expected utility theory in practice The class of models that combine linearity in probabilities with a nonlinear utility function remains the workhorse in much of modern economics. The seminal contributions of Kenneth J. Arrow (1965) and John W. Pratt (1964) provided the foundations for measuring individual risk attitudes using the curvature of the utility function. von Gaudecker et al., AER 2011. Only assume preferences over lotteries. Expected utility theorem. Existence of a utility function. u is usually given: Functional form. Calibration: risk aversion.
Applying the expected utility criterion The possible outcomes are x 1 = 1, x 2 = 10, x 3 = 16. Lottery L a is defined by p 1 = 0.1, p 2 = 0.8 (p 3 = 0.1). Lottery L b is defined by p 1 = 0.2, p 2 = 0.3 (p 3 = 0.5). Consider the utility function such that u (x) = x γ. 1 γ = 0 γ = 0.5 γ = 1 γ = 2 E a [u( x)] 9.7 6.1 2.12 0.2 E b [u( x)] 11.2 6.3 2.08 0.3 1 It can be represented by u(x) = ln(x) for γ = 1, and u(x) = x1 γ 1 γ otherwise.
The risk premium Given the utility function, can we quantify the tradeoff between risk and return? In other words, can we find the maximum amount of money that the agent would be willing to pay to eliminate a pure risk? This amount is called the risk premium, Π. In the case of a risk ɛ which is additive in wealth, it solves: E[u(w + ɛ)] = u(w Π) where E[ ɛ] = 0 This amount Π is positive if the agent is risk averse: E[u(w + ɛ)] < u(w) if u is concave Another similar concept is the certainty equivalent. Given a lottery x (whose expected payoff is not necessarily zero), the certainty equivalent CE is the value of the lottery: E[u(w + x)] = u(w + CE)
The Arrow-Pratt approximation (1) A closed form expression for the tradeoff between risk and return Consider a pure risk ɛ additive in wealth: end-of-period wealth = w + ɛ. Then the absolute risk premium (expressed as the amount of wealth) associated with this risk is Π 1 2 A(w)σ2 where var[ ɛ] = σ 2 and A(w) u (w) u (w) > 0. Important limitation: this approximation is only approximately valid for small risks. The risk premium is increasing in the variance of the risk. The risk premium is increasing in the concavity of the utility function (which is a measure of risk aversion) at w: the coefficient of absolute risk aversion A(w) is a measure of the tradeoff between risk and return at w. The coefficient of absolute risk aversion measures the agent s preferences regarding additive risks (with pure & small risks).
The Arrow-Pratt approximation (2) A closed form expression for the tradeoff between risk and return Consider a pure risk ɛ multiplicative in wealth, so that the end-of-period wealth is w(1 + ɛ). Then the relative risk premium (expressed as a share of wealth) associated with this risk is ˆΠ 1 2 R(w)σ2 where var[ ɛ] = σ 2 and R(w) w u (w) u (w) > 0. Important limitation: this approximation is only approximately valid for small risks. The risk premium is increasing in the variance of the risk. The risk premium is increasing in the concavity of the utility function at w: the coefficient of relative risk aversion R(w) is a measure of the tradeoff between risk and return at w. The coefficient of relative risk aversion measures the agent s preferences regarding pure and small multiplicative risks.
Differences in risk aversion Who is more risk averse? Who is willing to pay more to escape a given risk? Consider a linear transformation of the utility function: ν(x) = a + bu(x). Does that change the risk premium Π? Proposition: the following statements are equivalent: The Arrow-Pratt risk premium is larger for an individual with utility function v than for an individual with utility function u. For all w, A v (w) > A u (w). v is a concave transformation of u.
Wealth and risk aversion Intuitively, a millionaire will not be as averse to winning or losing $100 as a struggling student. Definition: Preferences are characterized by decreasing absolute risk aversion (DARA) if the risk premium associated to any pure additive risk is decreasing in wealth. Preferences are DARA if and only if A(w) is decreasing in w. A necessary condition is u (w) > 0 (this is called prudence), i.e., a convex marginal utility.
Constant absolute risk aversion CARA The coefficient of absolute risk aversion A(w) is a constant function of wealth with the following utility function (also known as negative exponential): As you can check, A(w) = ρ. u(w) = exp{ ρw} With these preferences, an economic agent is as averse to an additive risk if he owns $10 or $1,000,000: constant absolute risk aversion. Unrealistic... but these utility functions are tractable (especially with a normally distributed risk) and are used in stylized models. Increasing relative risk aversion.
Quadratic utility Vintage: popular in the 1950s Simple form: u(w) = aw bw 2 Advantage: preferences are fully described by the mean and the variance of the payoff. Drawbacks: Increasing absolute risk aversion. Only defined on the interval of wealth where marginal utility is increasing. Not prudent: u (w) = 0.
Constant relative risk aversion CRRA, or power utility The coefficient of relative risk aversion A(w) is a constant function of wealth with the following utility function: { w 1 γ u(w) = 1 γ if γ 1 ln(w) if γ = 1 As you can check, R(w) = γ. Constant RRA. With these preferences, an economic agent is as averse to a multiplicative risk whether he owns $10 or $1,000,000. It is also known as the isoelatic utility function: the elasticity of substitution between consumption at any two points in time is equal to 1 γ. Decreasing absolute risk aversion. Widely used in macroeconomics and finance, especially in calibration exercises. Usual assumption: γ [1, 4].
What is your coefficient of relative risk aversion? Which proportion of your wealth would you be willing to pay to escape the risk of your wealth either increasing or decreasing by 10% with equal probabilities? Suppose you own $50,000. If you participate in the lottery, your wealth will be $45,000 with probability 0.5, or $55,000 with probability 0.5. What percentage of $50,000 would you be willing to pay to escape this risk? Answer the same question with 30% instead of 10%. Of course, your answer will also depend on personal circumstances, notably to what extent you are exposed to other sources of risk. More on this later on.
What is your coefficient of relative risk aversion? coef RRA k = 10% k = 30% γ = 0.5 0.3% 2.3% γ = 1 0.5% 4.6% γ = 4 2.0% 16.0% γ = 10 4.4% 24.4% γ = 40 8.4% 28.7% A RRA less than one may be considered a low risk aversion, whereas a RRA higher than four may be considered a high risk aversion.
Caveats of the expected utility criterion The Allais paradox. Your wealth is $6,000. You have the option to pay $2,000 to get $8,000 with probability 0.5 or $0 with probability 0.5. Your wealth is $12,000. You can either choose to lose $8,000 with probability 0.5 or $0 with probability 0.5, or pay $6,000 to escape this lottery. Framing: preferences over lotteries depend on whether they are presented as a gamble or as insurance. Loss aversion.
Caveats of the expected utility criterion Prospect theory (Kahnman and Tversky). Habit formation (Campbell and Cochrane). Recursive utility.
Takeaway If a preference relation satisfies certain axioms, the expected utility criterion can be used to rank lotteries. A risk averse agent has a concave utility function. We can derive the price of a small risk, whether it is additive or multiplicative in wealth. We can use different functional forms for the utility function, with different implications for ARA and RRA as a function of wealth.
Exercises Homework will be solved next week. 1.1 1.3 1.5 Consider an investor with a relative risk aversion which is equal to 4 at any level of wealth. His current wealth is $100. What is his certainty equivalent for an asset whose payoff is equal to $100 with probability 0.5 and $0 otherwise? (think about a bond with a risk of default)
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.