Portfolio choice and risk aversion

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?

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.

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