CARA utility and risk aversion
August 26, 2021
This is a brief note on the CRRA (constant relative risk aversion) utility func- tion:
ln(c) with γ = 1.
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1. If 0 < γ < 1, the function is always positive for positive c values (it is
bounded from below and unbounded from above). See Figure 1, Panel A.
2. If γ = 1, the function is the logarithmic function, i.e., it can be negative. It is unbounded from below and from above. See Figure 1, Panel B.
3. If γ > 1, the function is always negative. It is unbounded from below, but bounded from above. See Figure 1, Panel C.
4. Notice that the negative utility (for some levels of γ) is not an issue. Consumption/investment choices depend on the derivative of the utility function (recall our conditions for utility maximization). Hence, the neg- ativity of the utility function does not affect optimal choices.
In all cases, the CRRA utility is concave. The definition of a concave function is the following:
u(w×c1 +(1−w)×c2)≥w×u(c1)+(1−w)×u(c2), (1)
for all w ∈ [0, 1].
You can see visually from Figure 2 that the CRRA utility function satisfies the concavity property in Eq. (1). In words, what we are saying is that “the utility of a weighted average of outcomes should be larger than (or equal to) the weighted average of the utilities”.
Now, think about the outcomes as a lottery. You get c1 with probability w and you get c2 with probability 1 − w. If you have a CRRA utility, because u(w×c1 +(1−w)×c2) = u(E(c)) ≥ w×u(c1)+(1−w)×u(c2) = E(u(c)), the utility associated with the expected outcome (i.e., u(w × c1 + (1 − w) × c2) = u(E(c))) is larger than the expected utility associated with the two outcomes (i.e., w × u(c1) + (1 − w) × u(c2) = E(u(c))). Said differently, if I were to give you the expected outcome E(c) = w×c1 +(1−w)×c2, you would take it
1 c1−γ withγ>0 u(c) = 1−γ
and not play the lottery. This is, however, precisely what you would do if you were risk averse. You would take the expected outcome rather than playing and having some chance (i.e., 1 − w) of getting more than the expected outcome (i.e., u(c2) > u(E(c))).
Conclusion:
1. Concave utility functions represents risk-averse behavior. Because the CRRA utility function is always concave (i.e., it is concave for all 0 < γ < ∞), it always represents risk-averse behavior.
2. Linear utility functions would, of course, represent risk-neutral behavior.
3. Convex utility functions would, instead, represent risk-loving behavior.
Some observations:
1. Notice that the concavity and, therefore, the risk-aversion of the CRRA utility function increases with γ. This is intuitive. As γ increases, the function goes from unbounded from above (the case 0 < γ ≤ 1) to bounded from above (the case γ > 1). Hence, the function is becoming more “concave,” thereby yielding an increase in risk aversion.
2. Coefficient of Absolute Risk Aversion. In light of the previous point, it seems natural to define risk aversion using the second derivative of the utility function (u′′(.)). The second derivative, in fact, measures curvature and, therefore, concavity. This would, however, be problematic: let’s see why. Because optimal choices depend on the first derivative of the utility function, a utility function u(.) leads to the same optimal choice as a linear (or affine) transformation u(.) = a + bu(.) of the same utility, where a and b are numbers. For our C-CAPM problem, in fact, given u(.), we derived
ptu (ct) = βEt(u (ct+1)(pt+1 + dt+1)), If we had used u(.) = a + bu(.), we would have obtained
ptbu (ct) = βEt(bu (ct+1)(pt+1 + dt+1)), which is the same as
ptu (ct) = βEt(u (ct+1)(pt+1 + dt+1)),
once you delete the constant b. It is, therefore, interesting to select a measure of risk aversion which is related to the second derivative of the utility function (u′′(.)) but is immune to affine transformations. The coefficient of Absolute Risk Aversion, achieves this goal. It is
ARA = −u′′(.), u′ (.)
i.e., the negative value of the second derivative of the utility function divided by the first derivative of the utility function. Notice that (1) the measure is always positive (the second derivative – by concavity – is negative and the first derivative – by non-satiation – is positive) and (2) it is the same for all affine transformations u(.) = a + bu(.). In fact,
−u′′(.) = −bu′′(.) = −u′′(.). u′ (.) bu′ (.) u′ (.)
3. Absolute risk aversion is related to the notion of risk premium. An agent is risk-averse if he/she dislikes every lottery with an expected payoff of zero (i.e., if he/she dislikes uncertainty):
U(c) ≥ E[U(c + ε)], (2)
where ε is a random variable with mean 0 and variance σε2. Notice that Eq. (2) is the same as the definition of concavity in Eq. (1), just written in a different way. In order to take the lottery, the agent would have to be given a risk premium π such that
U(c) = E[U(c + ε + π)] (3) Assume that ε and π are small. The first is random (mean 0 and variance
σε2) and the latter is not. We have, by Taylor expansion,
U(c) = E[U(c + ε + π)]
≃ E[U(c) + (ε + π)U′(c) + (ε + π)2U′′(c)]
≃U(c)+(E(ε)+π)U′(c)+(E(ε2)+ π2 +2E(ε)π)U′′(c)
=0 σε2 ≃0 =0 ≃ U(c) + πU′(c) + σε2U′′(c),
which implies π = −σ2 U′′(c) for the left-hand side in Eq. (3) to be the ε U′(c)
same as the right-hand side. The result is intuitive. What the agent wants
to be paid to take a zero mean lottery is proportional to risk aversion
(captured here by the coefficient of absolute risk aversion ARA = − u′′ (c) ) u′ (c)
and the variance of the outcomes of the lottery (which is a proxy for the extent of uncertainty).
4. ARA measures attitude towards lotteries which are not proportional to consumption (or wealth). Relative risk aversion (RRA) measures, in- stead, attitude towards lotteries that are proportional to consumption (or wealth). In this case, the risk premium would be such that
U(c) = E[U((1 + ε + π)c]
≃ E[U(c) + (ε + π)cU′(c) + (ε + π)2c2U′′(c)]
≃U(c)+(E(ε)c+πc)U′(c)+(E(ε2)c2 + π2 c2 +2E(ε)πc2)U′′(c)
=0 σε2 ≃0 =0 ≃ U(c) + πcU′(c) + σε2c2U′′(c),
which implies, now, π = −σ2 cU ′′ (c) . ε U′(c)
5. Finally, we note that the RRA of the CRRA utility function is constant and equal to γ, hence the name of the function. Write u′(c) = (1 − γ)c(1−γ−1)/(1 − γ) = c−γ and u′′(c) = −γc−γ−1. Thus, the RRA of the
CRRA utility is −cU′′(c) = −−γc−γ−1c = γ. So, the assumed utility is U ′ (c) c−γ
such that agents’ risk aversion with respect to lotteries defined in absolute dollar amounts (as captured by ARA) is decreasing in wealth but agents’ risk aversion with respect to lotteries defined as a percentage of wealth (as captured by RRA) is constant. One implication of this is that, for investors with CRRA utilities, the selection of portfolios (which are lotteries defined as a percentage of wealth – i.e., x% in one asset and (1 − x)% in another asset, say) does not depend on the amount of wealth.
u(c)= 1 c1−γ withγ=0.7 1−γ
u(c) = log(c) c
u(c)= 1 c1−γ withγ=2
Figure 1: The figure represents the CRRA utility function
valuesofγ>0. Thefirstpanelisfor0<γ<1(Iusedγ=0.7). Thesecond
panel is for γ = 1 (notice that 1 c1−γ = log(c) in this case). The third panel
1−γ is for 1 < γ < ∞ (I used γ = 3).
1 c1−γ for various 1−γ
u(c2) u(c) =
u(w×c1 +(1−w)×c2)
w × u(c1) + (1 − w) × u(c2)
1 c1−γ with γ = 0.7 1−γ
0 c1 w × c1 + (1 − w) × c2 c2 Figure 2: The CRRA utility function is concave.
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