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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? enter image description here

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  • $\begingroup$ Is this question verbatim from the exam? It makes little sense to talk about the indifference "curve" of a mean-variance utility, which is just one single point. $\endgroup$ – Herr K. Apr 12 at 16:09
  • $\begingroup$ @HerrK. This is from utility theory from portfolio management. Yes, it is verbatim. They describe the utility function as a parabola. I have added a figure for better illustration. $\endgroup$ – Daniel Apr 12 at 16:20
  • $\begingroup$ I see. So the indifference is between expected return and risk. $\endgroup$ – Herr K. Apr 12 at 16:33
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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.

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  • $\begingroup$ Yeah, but I am suspecting convexity is not measured by the second derivative here. That's the only thing where it could go wrong. $\endgroup$ – Daniel Apr 12 at 20:30
  • $\begingroup$ @Daniel: The second derivative is the most natural measure, and I'm not aware of another way to measure the degree of a function's convexity. $\endgroup$ – Herr K. Apr 13 at 0:18
  • $\begingroup$ What about curvature as in multivariable calculus? Or the convexity approximation used in bond duration? $\endgroup$ – Daniel Apr 13 at 1:14
  • $\begingroup$ @Daniel: The multivariate notion does not apply here, since an indifference curve is essentially a real-valued function from $\mathbb R$ to $\mathbb R$. As measuring the degree of a function's convexity is more of a math problem, you may want to ask that on MSE. $\endgroup$ – Herr K. Apr 13 at 1:56
  • $\begingroup$ we may need to first parametrize the function and then take the curvature $\kappa$ $\endgroup$ – Daniel Apr 13 at 2:59
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This seems to be specific to the CFA exam and is a badly formulated question. First, an indifference curve for some fixed utility level can be viewed as a function mapping $\sigma$ to $\mu$. At any value of $\sigma$, this function has a slope (which is given by $\sigma A$). No economist would call this a "slope coefficient" in this context, as this term is only used in regression analysis. Second, convexity is a property of a function, not a quantity, so it makes little sense to use the term "most convexity".

If the second derivative (which is $A$) is used as a measure of "convexity", then both answers A and B are correct, which makes no sense in the context of a single choice exam question.

If the curvature of the function is used as a measure of "convexity", then this curvature is first increasing and then decreasing in $A$, so for a given value of $\sigma$ you can indeed have two investors with the first having a higher "slope coefficient" but a lower "convexity" than the second. With this interpretation, only answer A is correct. However, I doubt that it was this what the exam authors had in mind. I think it is more likely that they just posed a stupid question.

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