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Chapter 2 — Risk and Risk Aversion

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Chapter 2 — Risk and Risk Aversion Chapter 2 — Risk and Risk Aversion The previous chapter discussed risk and introduced the notion of risk aversion. This chapter examines those concepts in more detail. In particular, it answers questions like, when can we say that one prospect is riskier than another or one agent is more averse to risk than another? How is risk related to statistical measures like variance? Risk aversion is important in finance because it determines how much of a given risk a person is willing to take on. Consider an investor who might wish to commit some funds to a risk project. Suppose that each dollar invested will return x dollars. Ignoring time and any other prospects for investing, the optimal decision for an agent with wealth w is to choose the amount k to invest to maximize [uw (+− kx( 1) )]. The marginal benefit of increasing investment gives the first order condition ∂u 0 = = u′( w +− kx( 1)) ( x − 1) (1) ∂k At k = 0, the marginal benefit isuw′( ) [ x − 1] which is positive for all increasing utility functions provided the risk has a better than fair return, paying back on average more than one dollar for each dollar committed. Therefore, all risk averse agents who prefer more to less should be willing to take on some amount of the risk. How big a position they might take depends on their risk aversion. Someone who is risk neutral, with constant marginal utility, would willingly take an unlimited position, because the right-hand side of (1) remains positive at any chosen level for k. For a risk-averse agent, marginal utility declines as prospects improves so there will be a finite optimum. Conversely, if [x ]< 1, risk-averse and risk-neutral agents would always choose k* = 0.1 We would like to expand upon this with results like: An agent who is more risk averse will invest less in any project than another agent who is less risk averse. The more risk a project has, the smaller is the optimal position for any risk averse agent. The greater the risk, the higher must be the expected outcome to induce the same level of commitment. To verify these claims, we obviously must have precise definitions of riskier and more risk averse. To start, we will consider only single-argument, monotonic increasing, concave utility functions. Also we will ignore time; all risks are resolved immediately, or equivalently all amounts are stated in future value terms. Also ignored are any pre-risk withdrawals from wealth. Finally, it is assumed that expectations exist. A sufficient condition for this to be true is either bounded utility or bounded outcomes. Neither is necessary, however, and many of the examples will violate both conditions. All results should be interpreted to have the additional provision “provided the expectations exist.” Risk conveys one of two related notions. The first is the uncertainty of outcomes. The risky prospect has more dispersed outcomes. This concept is a purely statistical. The other notion is risk is something that risk-averse agents do not like. However, not everything that is disliked by a risk averter is risk. Because utility is increasing, decreasing any outcome will reduce expected utility. However, decreasing very good outcomes may well reduce dispersion. Our definition of risk must take this into consideration. The solution to this dilemma is the one already adopted. The common practice, and the one adopted in the previous chapter is to define a risk premium relative to the expected payoff. The basic economic justification for using the 1 If it is possible to select k < 0, that is, to short the investment, then all risk-neutral or risk averse investors would wish to do that. This would not be possible in equilibrium where investments are in zero or positive supply. Risk and Risk Aversion — © Jonathan Ingersoll 1 version: June 25, 2019 expectation is that it is the only criterion for a risk-neutral agent with linear utility, and linear functions provides the closure on the set of concave, or risk averse, functions. So when comparing among prospects with the same expectation, we will define risk as that which is disliked by risk averters. Whether or not something is disliked, however, depends upon the particular utility function used in the evaluation. So risk is a property that is defined for a class of utility functions. Naturally we would like a definition which is as broadly applicable as possible. Risk: The Basics Rothschild and Stiglitz defined risk and developed its properties for the class of concave utilities. That is, they asked and answered the question: Under what conditions do all risk averse agents weakly prefer x̃ to ỹ; i.e., when is it true that [()]ux≥ [()] uy for all concave u. Asking this question for all concave utilities rather than for all increasing, concave utilities automatically equates the expectations because both u(x) = x and u(x) = −x are weakly concave functions, and the first always prefers a higher mean while the second always prefers a lower mean. This is where we will begin our examination, though we will then look at increasing, concave utilities. To reiterate, x is said to be less risky than y if [()]ux≥ [()] uy (2) for all concave (but not necessarily increasing) u. The random variable x̃ is strictly less risky than ỹ if the inequality in (2) is strict for all strictly concave utility functions. Unlike the preference relation introduced in the previous chapter, this riskiness ordering is not complete. It is possible to find outcomes xyand and utility functions u1 and u2 such that [ux1 ( )] > [uy1 ( )] while [ux22 ( )]< [ uy ( )] so neither may be said to be riskier, yet they are not equivalent either. For example, consider the random variables xyand with 1133 Pr{xx= 0} =44 = Pr{ = 4} = , Pr{ y = 1} = 4 , Pr{ y = 9} = 4 (3) Both variables have an expected value of 3. For the utility function u(x) = x1/2, the expected utilities of x and y are both 1.5. However, for the utility function wα/α so x͂ is strictly preferred when ½ < α < 1 and y͂ is strictly preferred for α < ½. In fact as α → 0, [u(x͂ )] → −∞. So these two prospects cannot be ranked on riskiness in the class of all concave utility functions. Riskiness is a complete ordering only when the class of utilities or permitted random variables or both are restricted. The set of quadratic utility functions composes such a restricted 2 class. Expected quadratic utility is [u ( x )] =−− x bx bvar[ x ]. So for two random outcomes with the same expectation, the one with the smaller variance is preferred under quadratic utility. This ordering is obviously complete because risk can be measured by the variance, a single the real number, and the greater than ordering is complete over the real numbers. If the random variables are restricted to be normally distributed, then variance again provides a complete ranking of riskiness. For a normally distributed variable with mean µ and standard deviation σ, expected utility is ∞∞ [()]u x = u ( µ+σ e )() φ e de = [( u µ+σ e ) + u ( µ−σ e )]() φ e de . (4) ∫∫−∞ 0 where ẽ is a standard normal variable with mean 0 and variance 1. Increasing variance decreases expected utility Risk and Risk Aversion — © Jonathan Ingersoll 2 version: June 25, 2019 ∂ [()]ux ∞ =[u′′ ( µ+σ e ) − u ( µ−σ e )] e φ ( e ) de ≤ 0 . (5) ∂σ ∫0 Because marginal utility is weakly decreasing, the final integrand is always negative or zero so an increase in variance always weakly reduces expected utility.2 Variance is a valid measure of riskiness in many models used in Finance, but even when there is a complete ordering for riskiness based on some statistic, variance may not play that role. Consider the class of utility functions uxc( ; )=−− x c | x x0 | for c <1. (6) These utility functions are piece-wise linear with a kink at x = x0 where the slope decreases from 1 + c to 1 − c. For this class, the expected utility for any gamble is [(;)]uxc=−− []x c [| x x0 |] (7) The expected utility for gambles with the same expectation is smaller whenever the absolute deviation of x͂ from x0 is larger. This statistic provides a complete ordering but is not based on variance. One random variable can be riskier than another even if it has a smaller variance. Similar examples can be constructed for other measures of central tendency. While no single parameter like variance is sufficient to determine the risk of a random outcome for all concave utilities, it is true that random variables that are riskier must have a higher variance. This is obvious because variance does measure riskiness for some members of the class, namely quadratic utilities While variance is not the correct measure of riskiness in all cases, the idea that increased dispersion creates more risk is valid when dispersion is defined properly. It is natural to think that for ỹ = x̃ + ε, that ε with []ε= 0is the added risk. Some obvious ways that might describe additional risk are (i) ε is uncorrelated with x͂ ,3 (ii) ε is a fair game with respect to x͂ , or (iii) ε is independent of x͂ . This list forms a natural hierarchy as (iii) ⇒ (ii) ⇒ (i). A lack of correlation between x and ε means var[xx+ε ] ≥ var[ ], but that is insufficient to guarantee an increase in risk as the following example illustrates Pr xxε +ε [x ]=⋅11 10 +⋅ 30 +⋅ 1 50 = 20 1 10− 5 5 236 2 [ε= ]1 ⋅− ( 5) + 11 ⋅ 10 + ⋅− ( 5) = 0 (8) 1 20 10 30 2 36 3 ε=1 ⋅ ⋅− + 11 ⋅ ⋅ + ⋅ − = 1 [x ]2 10 ( 5) 36 30 10 50( 5) 0 .
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