# Investor's optimization problem with risk aversion

Consider an investor with initial wealth $$w$$ and has to decide how to invest it. There is a riskless asset with rate of return $$r$$. The risky asset has return $$x_i$$ with probability $$\pi_i$$ for $$i=1,2,3,...,n$$. Denote by $$\alpha$$ the fraction of wealth that investor puts into the risky asset, so that $$1- \alpha$$ is the fraction that he puts in the riskless asset. Write down the investor's optimization problem.

This is a question from my homework test. I want to specifically confirm my answer of part a), I wrote $$max \sum_{i=1}^{n} \pi _{i}[U((1-\alpha )w(1+r)-\alpha w+\sum_{i=1}^{n}x_i]$$

This, we have to maximise w.r.t. $$\alpha$$ Can anyone confirm?

• Here $x_i$ means the interest rate/rate of return, right? – superhulk Sep 26 '18 at 17:47
• No, I think it's the whole amount which you'll get with probability $$\pi$$. Or I'm confused should the optimisation roblem be : – Henam Sep 27 '18 at 4:56
• $$U[\alpha w+\alpha w\sum_{i=1}^{n}x_{i}\pi _{i}+(1-\alpha )w(1+r)]$$ – Henam Sep 27 '18 at 5:06
• Did you get your exam back? It doesn't make sense to me for $x_i$ to be anything other than a rate of return, comparable with $r$. Why do you think it's a coupon amount? – Kenneth Rios Sep 30 '18 at 4:40
• It may be, I don't know. – Henam Sep 30 '18 at 9:12

You didn't specify a utility function over wealth so I'm using $$u(\cdot)$$. Assuming that $$x_i$$ is indeed a rate of return, the investor's optimization problem would be
$$\DeclareMathOperator*{\argmax}{arg\,max} \argmax_{\alpha \in \ [0,1]} \sum_{i=1}^{n} \pi_iu(w(1-\alpha)r + w\alpha x_i);$$
that is, the investor wishes to find the optimal $$\alpha \in [0,1]$$ that maximizes expected utility over returns.
In words: with probability $$\sum_{i=1}^{n} \pi_i = 1$$ the investor earns the return $$w(1-\alpha)r$$ from the riskless asset, but with probability $$\pi_i$$ the investor earns the return $$w\alpha x_i$$ from the risky asset.
• We've to do this by assuming $U(.)$ only. Can you help out in part b also? – Henam Oct 2 '18 at 4:51
• Hey, I've a confusion, should we add $w$ in the utility function, since that's the principal amount and it should be added in the utility function. The utility function you've written does only include the interest amount. – Henam Oct 26 '18 at 17:25