CRRA Utility: How Risk Aversion Changes with Gamma and Wealth
Summary
The document explains how the parameter gamma in constant relative risk aversion utility relates to both relative and absolute risk aversion. For a concave utility function, an investor is risk-averse; relative risk aversion measures the willingness to pay as a fraction of wealth to avoid a proportional gamble, while absolute risk aversion measures the amount paid to avoid a fixed-size gamble. Under CRRA utility, relative risk aversion equals gamma, so increasing gamma raises aversion to proportional risk.
Absolute risk aversion is relative risk aversion divided by wealth. Thus, at a fixed wealth level, a larger gamma also raises absolute risk aversion, while greater wealth lowers the absolute amount an investor would pay to avoid a fixed-size gamble. The discussion notes that zero gamma corresponds to linear, risk-neutral utility and that the limiting case at gamma equal to one is logarithmic utility. These distinctions resolve the apparent conflict between interpreting the utility curve and comparing risk measures; the answer gives definitions and relationships, but no empirical evidence or investment prescription.
Key ideas
- CRRA utility has constant relative risk aversion equal to gamma.
- Increasing gamma raises the fraction of wealth an investor would pay to avoid proportional risk.
- Absolute risk aversion equals relative risk aversion divided by wealth.
- At a fixed gamma, absolute risk aversion falls as wealth rises.
- Zero gamma represents risk neutrality, while the limiting case at one is logarithmic utility.
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Full text
# CRRA Ultility， simple question
# CRRA Ultility， simple question
for CRRA, does increasing gamma leads to increase in risk-aversion?
Looking at the curve, I think increasing gamma leads to less in risk-aversion (since the risk preimum is less). But in terms of absolute risk aversion, CRRA = $\gamma /X$. Looks like increasing $\gamma $ leads to high risk-aversion. Which is right?
## Answer by RRL (score 2, accepted)
https://quant.stackexchange.com/a/54187
If the utility function $W \mapsto U(W)$ (where $W$ is wealth) is concave, then the individual is risk-averse and unwilling to accept any actuariually fair gamble.
We can distinguish between absolute risk aversion ($ARA$) and relative risk aversion ($RRA$)
$$ARA(W) = - \frac{U''(W)}{U'(W)},\quad RRA(W) = - W\frac{U''(W)}{U'(W)}$$
Here, $ARA(W)$ determines the absolute amount the individual is willing to pay to avoid a gamble of a given absolute size. Similarly, $RRA(W)$ determines the relative amount, i.e., fraction of wealth, the individual is willing to pay to avoid a gamble of a given size relative to wealth. A derivation for $ARA$ is given here and is easily modified for $RRA$ by replacing $\epsilon$ and $\delta$ with $\epsilon/W$ and $\delta/W$, respectively.
As you would expect, a $CRRA$ utility function has constant relative risk aversion $\gamma$,
$$RRA(W) = -W \frac{U''(W)}{U'(W)} = - W \frac{d}{dW} \log U'(W) = \gamma$$
Without loss of generality in terms of constants, we can solve for $U$ as
$$U(W) = \frac{W^{1-\gamma}-1}{1-\gamma}$$
To ensure concavity (risk aversion) we must have $\gamma > 0$. The case where $\gamma = 0$ corresponds to a linear utility function (risk neutrality) and in the limit as $\gamma \to 1$ we have , by L'Hopital's rule,
$$\lim_{\gamma \to 1}U(W) = \log W$$
With $\gamma$ fixed the fraction of wealth the individual pays to avoid a gamble is, of course, independent of wealth since this is $CRRA$. Nevertheless the fraction of wealth paid would increase as $\gamma$ increases.
However, since $ARA(W) = RRA(W)/W$, the absolute amount paid decreases with increasing wealth.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.