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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?

Ludwig Gershwin
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  • 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 Oct 28 '23 at 06:26
  • 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 Oct 28 '23 at 23:52

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