<p>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 <span class="math-container">$\sigma$</span> to <span class="math-container">$\mu$</span>. At any value of <span class="math-container">$\sigma$</span>, this function has a <em>slope</em> (which is given by <span class="math-container">$\sigma A$</span>). No economist would call this a &quot;<em>slope coefficient</em>&quot; in this context, as this term is only used in regression analysis. Second, convexity is a <em>property</em> of a function, not a quantity, so it makes little sense to use the term &quot;<em>most convexity</em>&quot;.</p>
<p>If the second derivative (which is <span class="math-container">$A$</span>) is used as a measure of &quot;convexity&quot;, then both answers A and B are correct, which makes no sense in the context of a single choice exam question.</p>
<p>If the <a href="https://en.wikipedia.org/wiki/Curvature" rel="nofollow noreferrer">curvature</a> of the function is used as a measure of &quot;convexity&quot;, then this curvature is first increasing and then decreasing in <span class="math-container">$A$</span>, so for a given value of <span class="math-container">$\sigma$</span> you can indeed have two investors with the first having a higher &quot;slope coefficient&quot; but a lower &quot;convexity&quot; 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.</p>

CC BY-SA 4.0
2021-04-13T11:36:17.617Z
2021-04-13T11:36:17.617Z
VARulle
<p>It would seem that both options are correct given the specific mean-variance utility function.</p>
<p>Use <span class="math-container">$\mu$</span> to denote the expected value. In the <span class="math-container">$(\sigma,\mu)$</span>-plain, an indifference curve representing a particular utility level <span class="math-container">$\overline U$</span> is given by
<span class="math-container">\begin{equation}
\overline U=\mu-\frac12 A\sigma^2.
\end{equation}</span>
Applying implicit differentiation, we can obtain the slope of the indifference curve to be
<span class="math-container">\begin{equation}
\frac{\partial \mu}{\partial \sigma}=A\sigma,
\end{equation}</span>
which is increasing in <span class="math-container">$A$</span>, the parameter of absolute risk aversion. This shows that option (a) is correct.</p>
<p>Moreover, if we measure the degree of convexity of the indifference curve by the value of its second derivative, we'd get
<span class="math-container">\begin{equation}
\frac{\partial^2\mu}{\partial \sigma^2}=A,
\end{equation}</span>
which is again increasing in <span class="math-container">$A$</span>. This suggests that option (b) is also correct.</p>
<p>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.</p>

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2021-04-12T20:30:41.357Z
Yeah, but I am suspecting convexity is not measured by the second derivative here. That's the only thing where it could go wrong.
aff0gato
CC BY-SA 4.0
2021-04-13T00:18:44.533Z
@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.
Herr K.
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2021-04-13T01:14:53.513Z
What about curvature as in multivariable calculus? Or the convexity approximation used in bond duration?
aff0gato
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2021-04-13T01:56:04.733Z
@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](https://math.stackexchange.com/).
Herr K.
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2021-04-13T02:59:43.063Z
we may need to first parametrize the function and then take the curvature $\kappa$
aff0gato
CC BY-SA 4.0
2021-04-12T18:50:33.317Z
2021-04-12T18:50:33.317Z
Herr K.
<p>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 <span class="math-container">$U=E(r)-\frac{1}{2}A\sigma^2$</span> 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?
<a href="https://i.sstatic.net/l4J7O.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/l4J7O.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2021-04-12T16:09:14.190Z
Is this question verbatim from the exam? It makes little sense to talk about the indifference "curve" of a [mean-variance utility](https://www.tau.ac.il/~spiegel/teaching/corpfin/mean-variance.pdf), which is just one single point.
Herr K.
CC BY-SA 4.0
2021-04-12T16:20:05.217Z
@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.
aff0gato
CC BY-SA 4.0
2021-04-12T16:33:09.027Z
I see. So the indifference is between expected return and risk.
Herr K.
<finance><financial-economics><portfolio-theory>
risk aversion and convexity of indifference curve
aff0gato
# risk aversion and convexity of indifference curve

(Asked by: username_0 on 2021-04-12)

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](https://i.sstatic.net/l4J7O.png)](https://i.sstatic.net/l4J7O.png)

**Tags:** finance, financial-economics, portfolio-theory

> **Comments:**
> **username_1**: Is this question verbatim from the exam? It makes little sense to talk about the indifference "curve" of a [mean-variance utility](https://www.tau.ac.il/~spiegel/teaching/corpfin/mean-variance.pdf), which is just one single point.  
> **username_0**: @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.  
> **username_1**: I see. So the indifference is between expected return and risk.  

---

## 2 answers

### Answer by username_2 on 2021-04-13. Score: 2

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](https://en.wikipedia.org/wiki/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.


---

### Answer by username_1 on 2021-04-12. Score: 2

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.

> **Comments:**
> **username_0**: Yeah, but I am suspecting convexity is not measured by the second derivative here. That's the only thing where it could go wrong.  
> **username_1**: @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.  
> **username_0**: What about curvature as in multivariable calculus? Or the convexity approximation used in bond duration?  
> **username_1**: @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](https://math.stackexchange.com/).  
> **username_0**: we may need to first parametrize the function and then take the curvature $\kappa$  

---

