risk aversion and convexity of indifference curve This is a question from the CFA exam. With respect to utility theory, the most risk-averse investor will have an indifference curve with : (a) greatest slope coefficient (b) most convexity The answer is A but I think B is also correct. The utility function is described by $U=E(r)-\frac{1}{2}A\sigma^2$ where A is the measure of risk aversion. I want to understand this mathematically. How should we measure convexity? Is this just the second derivative or the curvature? And why is B incorrect?
It would seem that both options are correct given the specific mean-variance utility function.
Use $\mu$ to denote the expected value. In the $(\sigma,\mu)$-plain, an indifference curve representing a particular utility level $\overline U$ is given by
\begin{equation}
\overline U=\mu-\frac12 A\sigma^2.
\end{equation}
Applying implicit differentiation, we can obtain the slope of the indifference curve to be
\begin{equation}
\frac{\partial \mu}{\partial \sigma}=A\sigma,
\end{equation}
which is increasing in $A$, the parameter of absolute risk aversion. This shows that option (a) is correct.
Moreover, if we measure the degree of convexity of the indifference curve by the value of its second derivative, we'd get
\begin{equation}
\frac{\partial^2\mu}{\partial \sigma^2}=A,
\end{equation}
which is again increasing in $A$. This suggests that option (b) is also correct.
This conclusion seems consistent with the graph you show: within risk aversion, the indifference curves with higher curvature are also the ones with steeper slopes.