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Given utility function $U(w) = -e^{-w}$ of an investor (where $w$ denotes wealth) and two assets - risky and safe, will the investor's amount of investment into the risky asset be indifferent of his wealth?

The graph of $U(w)$ is strictly concave, so I feel like the investor will put in more money into the risky asset as his wealth increases and gain more utility from it. On the other hand, when I test using the Arrow-Pratt measure of risk aversion, I find that $r(w) = -\frac{u''(w)}{u'(w)} = 1$ which tells that the investor will be indifferent.

This is contradictory, so where am I going wrong?

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Risk aversion is given by the sign of $r(w)$ according to these cases:

  • $r(w) > 0 \implies$ the individual’s utility function is concave and they are risk averse.
  • $r(w) < 0 \implies$ the individual’s utility function is convex and they are risk loving.
  • $r(w) = 0 \implies$ the individual’s utility function is linear and they are risk neutral.

We have

$U(w) = - e^{-w}$

$U’(w) = e^{-w}$

$U’’(w) = - e^{-w}$

Therefore,

$r(w) = - \frac{u’’(w)}{u’(w)} = - \frac{ - e^{-w}}{e^{-w}} = 1$, so your $r$ value is correct.

Since $r(w) = 1 > 0$, the individual in fact has a concave utility function as they are risk averse, so there is no contradiction.

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  • $\begingroup$ Appreciated, thank you! Can't upvote you without signing up fully, so sorry about that. $\endgroup$
    – riskyasset
    Apr 9, 2023 at 13:38

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