<p>Suppose a risk-averse investor with differentiable Bernoulli utility <span class="math-container">$u$</span> and wealth <span class="math-container">$w$</span> that can be allocated between asset <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>. Both <span class="math-container">$X,Y$</span> have positive expectations. If the investor invests a fraction <span class="math-container">$a$</span> of their wealth in <span class="math-container">$X$</span> and <span class="math-container">$1-a$</span> in <span class="math-container">$Y$</span>, their wealth will be <span class="math-container">$aXw+(1-a)Yw$</span> in the next period.</p>
<p>How do I argue that <span class="math-container">$a=0.5$</span> is the optimal? I think I need to compare the random variable <span class="math-container">$Z_a=aX+(1-a)Y$</span> to <span class="math-container">$Z_{0.5}=0.5X+0.5Y$</span>. How do I proceed?</p>

CC BY-SA 4.0
2023-10-28T06:26:22.573Z
I do not think this is true. Assume that $X$ pays an amount 100 with probability 0.99 and $0$ with probability 0.01, and $Y$ pays an amount $100$ with probability 0.01 and $0$ with probability .99. I'm pretty sure the risk-averse inester will invest more in $X$ than in $Y$.
tdm
CC BY-SA 4.0
2023-10-28T23:52:47.980Z
It is true under three conditions: X and Y have equal expected returns, equal variances and their covariance = 0. In this case, you can just minimize the portfolio variance without worrying about the returns and that leads to $a = 0.5$.
mark leeds
<risk-aversion>
How do I show that it's optimal to invest in equal amount of 2 risky assets via mean-preserving spread?
Ludwig Gershwin
# How do I show that it's optimal to invest in equal amount of 2 risky assets via mean-preserving spread?

(Asked by: username_0 on 2023-10-27)

Suppose a risk-averse investor with differentiable Bernoulli utility $u$ and wealth $w$ that can be allocated between asset $X$ and $Y$. Both $X,Y$ have positive expectations. If the investor invests a fraction $a$ of their wealth in $X$ and $1-a$ in $Y$, their wealth will be $aXw+(1-a)Yw$ in the next period.

How do I argue that $a=0.5$ is the optimal? I think I need to compare the random variable $Z\_a=aX+(1-a)Y$ to $Z\_{0.5}=0.5X+0.5Y$. How do I proceed?

**Tags:** risk-aversion

> **Comments:**
> **username_1**: I do not think this is true. Assume that $X$ pays an amount 100 with probability 0.99 and $0$ with probability 0.01, and $Y$ pays an amount $100$ with probability 0.01 and $0$ with probability .99. I'm pretty sure the risk-averse inester will invest more in $X$ than in $Y$.  
> **username_2**: It is true under three conditions: X and Y have equal expected returns, equal variances and their covariance = 0. In this case, you can just minimize the portfolio variance without worrying about the returns and that leads to $a = 0.5$.  

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