# What is the significance of Relative Risk Aversion

I know that the relative risk aversion is defined as $$R(c) = cA(c)=\frac{-cu''(c)}{u'(c)}$$ where $u(c)$ denotes the utility curve as a function of wealth $c$.

But I do not understand the intuition for it. Can you explain the intuition for relative risk aversion?

• I tried to make it clear but I'm not sure I succeeded, if you have any further questions don't hesitate to ask. – Bob Jansen Aug 4 '13 at 16:48

In utility theory the basic assumption is that $u(c)$ is strictly monotonically increasing in wealth: people prefer more over less. Hence, $\forall c, u'(c) > 0$. The second assumption is that the amount of utility added, as $c$ increases, diminishes, so $\forall c, u''(c) < 0$. Combining these two observations we have that $$\forall c, A(c) = \frac{-u''(c)}{u'(c)} > 0.$$

This can be interpreted as follows, if for a particular $c$ $u'(c)$ is large $A(c)$ will be small. Thus if utility curve is sensitive to increases in wealth the risk aversion is low. For $u''(c)$ the reverse holds: if $u''(c)$ for a particular value of $c$ risk aversion will be low. $A(c)$ captures both sensitivities and also produces some kind of a trade-off between them.

The quantity $R(c)$ is just $A(c)$ scaled by the wealth. This scaling has the advantage that this quantity is not sensitive to a change in numéraire of $c$.

Consider a two period model. At time $t = 0$, an asset's price is $1$. At time $t = 1$, the asset's price is equal to $1 + \epsilon$ with probability $\frac{1}{2}$, and equal to $\frac{1}{1 + \epsilon}$ with probability $\frac{1}{2}$. An agent has initial wealth $c$ and utility function $U(\cdot)$. We wish to determine $\alpha^*$, the proportion of the agent's wealth which should be invested to maximize expected utilty. I claim that $\alpha^*$ is proportional to $\frac{1}{R(c)}$.
Suppose that the agent invests $\alpha \%$ of his wealth. Then at time $t = 1$, his expected utility is $$\frac{1}{2} U \left(c - c\alpha + c\alpha(1 + \epsilon) \right) + \frac{1}{2}U \left( c - c\alpha + c\frac{\alpha}{1 + \epsilon} \right).$$ This can be rewritten as $$\frac{1}{2}U \left(c + c \alpha \epsilon \right) + \frac{1}{2}U \left(c - \frac{c \alpha \epsilon}{1 + \epsilon} \right).$$
We are assuming that $\epsilon$ is very small (recall the infinitesimal bit I mentioned earlier), so we will approximate this expression with a second order Taylor expansion. It becomes $$U(c) + \frac{1}{2}c \alpha \epsilon U'(c) + \frac{1}{4} c^2\alpha^2 \epsilon^2 U''(c) - \frac{1}{2}\frac{c \alpha \epsilon}{1 + \epsilon}U'(c) + \frac{1}{4}\frac{c^2 \alpha^2 \epsilon^2}{(1 + \epsilon)^2}U''(c).$$ We're maximizing utility here, so we want to choose the best $\alpha$. Ignoring the first term, which doesn't depend on $\alpha$, and combining terms, it is equivalent to maximize $$c \alpha \epsilon U'(c) \frac{\epsilon}{1 + \epsilon} + \frac{1}{2}c^2 \alpha^2 \epsilon^2 U''(c) \left( 1 + \frac{1}{(1 + \epsilon)^2} \right).$$ Divide through by constants (everything but $\alpha$) to get $$\alpha U'(c) \frac{1}{1 + \epsilon} + \frac{1}{2} c \alpha^2 U''(c) \left( 1 + \frac{1}{(1 + \epsilon)^2} \right).$$ Letting $\epsilon$ tend to zero, this becomes $$\alpha U'(c) + \frac{1}{2} c \alpha^2 U''(c).$$ Take derivatives with respect to $\alpha$ and set equal to zero. This gives $$U'(c) + c \alpha U''(c) = 0,$$ or $\alpha^* = -\frac{U'(c)}{c U''(c)}$.
One last note. There is a similar mechanism going on in the expected utility problem for power utility in a Black-Scholes market, studied by Merton a long time ago. For example, when the market dynamics follow $$dX_t = \mu X_t dt + \frac{1}{2} \sigma X_t dW_t,$$ and the agent has power utility $U(x) = \frac{1}{\alpha}x^\alpha$, $0 < \alpha < 1$, it is optimal for the agent to always hold $\frac{\mu}{\sigma}\alpha \%$ of his wealth in the risky asset. You may calculate that the relative risk aversion of such an agent is $\frac{1}{\alpha}$, and the constant of proportionality, $\frac{\mu}{\sigma}$, is the asset's sharpe ratio.