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?

  • $\begingroup$ Here $x_i$ means the interest rate/rate of return, right? $\endgroup$ – superhulk Sep 26 '18 at 17:47
  • $\begingroup$ No, I think it's the whole amount which you'll get with probability $$\pi $$. Or I'm confused should the optimisation roblem be : $\endgroup$ – Henam Sep 27 '18 at 4:56
  • $\begingroup$ $$U[\alpha w+\alpha w\sum_{i=1}^{n}x_{i}\pi _{i}+(1-\alpha )w(1+r)]$$ $\endgroup$ – Henam Sep 27 '18 at 5:06
  • $\begingroup$ 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? $\endgroup$ – Kenneth Rios Sep 30 '18 at 4:40
  • $\begingroup$ It may be, I don't know. $\endgroup$ – 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.

  • $\begingroup$ We've to do this by assuming $U(.)$ only. Can you help out in part b also? $\endgroup$ – Henam Oct 2 '18 at 4:51
  • $\begingroup$ If you are still interested in a response to part b, I suggest you open a new question that is clearly written in MathJax (no images please) and show all the work you were able to do and where you got stuck. I may be able to get to it -- no promises though. But someone else could give it a look as well. $\endgroup$ – Kenneth Rios Oct 2 '18 at 15:03
  • $\begingroup$ 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. $\endgroup$ – Henam Oct 26 '18 at 17:25
  • $\begingroup$ No, because the investor wants to maximize returns by optimally distributing wealth over the assets. It doesn't matter what the initial wealth is, just the ratio between investments. $\endgroup$ – Kenneth Rios Oct 26 '18 at 17:57

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