<p>You didn't specify a utility function over wealth so I'm using <span class="math-container">$u(\cdot)$</span>. Assuming that <span class="math-container">$x_i$</span> is indeed a rate of return, the investor's optimization problem would be</p>

<p><span class="math-container">$$
\DeclareMathOperator*{\argmax}{arg\,max}
\argmax_{\alpha \in \ [0,1]} \sum_{i=1}^{n} \pi_iu(w(1-\alpha)r + w\alpha x_i);
$$</span></p>

<p>that is, the investor wishes to find the optimal <span class="math-container">$\alpha \in [0,1]$</span> that maximizes expected utility over returns. </p>

<p>In words: with probability <span class="math-container">$\sum_{i=1}^{n} \pi_i = 1$</span> the investor earns the return <span class="math-container">$w(1-\alpha)r$</span> from the riskless asset, but with probability <span class="math-container">$\pi_i$</span> the investor earns the return <span class="math-container">$w\alpha x_i$</span> from the risky asset.</p>

CC BY-SA 4.0
2018-10-02T04:51:24.890Z
We've to do this by assuming $U(.)$ only. Can you help out in part b also?
Henam
CC BY-SA 4.0
2018-10-02T15:03:02.850Z
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.
Kenneth Rios
CC BY-SA 4.0
2018-10-26T17:25:56.870Z
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
CC BY-SA 4.0
2018-10-26T17:57:46.783Z
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.
Kenneth Rios
CC BY-SA 4.0
2018-10-01T22:18:43.537Z
2018-10-01T22:26:23.833Z
2018-10-01T22:26:23.833Z
Kenneth Rios
<p>Consider an investor with initial wealth <span class="math-container">$w$</span> and has to decide how to invest it. There is a riskless asset with rate of return <span class="math-container">$r$</span>. The risky asset has return <span class="math-container">$x_i$</span> with probability <span class="math-container">$\pi_i$</span> for <span class="math-container">$i=1,2,3,...,n$</span>. Denote by <span class="math-container">$\alpha$</span> the fraction of wealth that investor puts into the risky asset, so that <span class="math-container">$1- \alpha$</span> is the fraction that he puts in the riskless asset. 
Write down the investor's optimization problem.</p>

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

<p>This, we have to maximise w.r.t. <span class="math-container">$\alpha$</span>
Can anyone confirm?</p>

CC BY-SA 4.0
2018-09-26T17:47:42.510Z
Here $x_i$ means the interest rate/rate of return, right?
superhulk
CC BY-SA 4.0
2018-09-27T04:56:48.190Z
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
CC BY-SA 4.0
2018-09-27T05:06:12.703Z
$$U[\alpha w+\alpha w\sum_{i=1}^{n}x_{i}\pi _{i}+(1-\alpha )w(1+r)]$$
Henam
CC BY-SA 4.0
2018-09-30T04:40:29.743Z
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
CC BY-SA 4.0
2018-09-30T09:12:54.093Z
It may be, I don't know.
Henam
CC BY-SA 4.0
2018-09-30T09:15:14.133Z
Nope, not yet! Can you help out even if $x_i$ is a rate of return. I'm unable to get a clue even ? Thank you so much!
Henam
CC BY-SA 4.0
2018-10-01T00:34:31.627Z
To make the question more readable, consider typing it in instead of posting dim photos of it.
Sander Heinsalu
CC BY-SA 4.0
2018-10-01T17:43:38.180Z
Done, sorry @SanderHeinsalu. Won't happen again.
Henam
<investment><optimization><risk-aversion>
Investor's optimization problem with risk aversion
Henam
# Investor's optimization problem with risk aversion

(Asked by: username_0 on 2018-09-26)

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?

**Tags:** investment, optimization, risk-aversion

> **Comments:**
> **username_1**: Here $x_i$ means the interest rate/rate of return, right?  
> **username_0**: No, I think it's the whole amount which you'll get with probability $$\pi $$.                    Or I'm confused should the optimisation roblem be :  
> **username_0**: $$U[\alpha w+\alpha w\sum_{i=1}^{n}x_{i}\pi _{i}+(1-\alpha )w(1+r)]$$  
> **username_2**: 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?  
> **username_0**: It may be, I don't know.  
> **username_0**: Nope, not yet! Can you help out even if $x_i$ is a rate of return. I'm unable to get a clue even ? Thank you so much!  
> **username_3**: To make the question more readable, consider typing it in instead of posting dim photos of it.  
> **username_0**: Done, sorry @username_3. Won't happen again.  

---

## 1 answers

### Answer by username_2 on 2018-10-01. Score: 0 | Accepted answer

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.

> **Comments:**
> **username_0**: We've to do this by assuming $U(.)$ only. Can you help out in part b also?  
> **username_2**: 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.  
> **username_0**: 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.  
> **username_2**: 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.  

---

