Relative Risk Aversion and Optimal Risky Investment

# What is the significance of Relative Risk Aversion


# 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?

## Answer by Bob Jansen (score 5, accepted)

https://quant.stackexchange.com/a/8645

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 $c$.

By the way, the Wikipedia page is excellent.

## Answer by quasi (score 3)

https://quant.stackexchange.com/a/8650

Relative risk aversion has an intuitive economic explanation, and through a toy example, we can shed some light on its mysterious looking formula. Consider an agent with constant relative risk aversion (i.e. power or log utility) and some asset with a fixed "attractiveness" (essentially sharpe ratio, more on this later). For the agent to invest optimally, he wants to invest a proportion of his wealth into the asset, and this proportion is inversely related to his relative risk aversion. For agents with a more general utility function, relative risk aversion is no longer constant, and so this reasoning doesn't hold true globally. It is still true, however, in an infinitesimal sense. Let's go into this in more detail:

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.