We've got given annual results of two stock companies described with following values:

Company X: expected value $\mu_X=0.05$, standard deviation $\sigma_X=0.02$

Company Y: expected value $\mu_Y=0.07$, standard deviation $\sigma_Y=0.03$

We also know that a cross correlation between results of company X and Y is $\rho=-0.5, \rho=\frac{cov(X,Y)}{\sqrt{var(X)var(Y)}}$. How do we should divide z dollars between these companies to obtain:

a) maximum revenue (expected return on a portfolio)?

b) minimum risk (variance of a portfolio)?

I don't even expect a full solution, just a brief explanation which would help me to understand a practical meaning of above values.

  • $\begingroup$ Wondering if there's a bit of context missing. Sounds like a question after the introduction of an algorithm which takes those inputs as values and computes how to optimize revenue or risk..? $\endgroup$ – user979 May 11 '14 at 22:44
  • $\begingroup$ So you have to allocate a proportion $p$ of $z$ (i.e. $pz$) dollars to X and $(1-p)z$ dollars to Y. (hint: make $z=1$ to start with - if you allocate $p$ to X and $1-p$ to Y, what is the mean and variance of revenue?) $\endgroup$ – Glen_b May 11 '14 at 23:09
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    $\begingroup$ $E(revenue)=p\mu_X+(1-p)\mu_Y \quad Var(revenue)=p\rho_X^2+(1-p)\rho_Y^2+2cov(X,Y)$ Do I get it right? $\endgroup$ – kozooh May 11 '14 at 23:39
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    $\begingroup$ I think I missed sth and it should be $2p(1-p)cov(X,Y)$. Then I need to find $max E$ and $min Var$? $\endgroup$ – kozooh May 11 '14 at 23:49
  • $\begingroup$ Kozooh, this question is not well defined: because the returns for the two assets are random, the revenue cannot be optimized. However, some properties of its distribution can be optimized, such as expected revenue, or chance that the revenue exceeds a predetermined threshold. Please edit your post to address this point. As far as optimizing the risk (usually defined as the negative of the expected loss) goes, you have enough information to write it as a function of the proportion of the $z$ dollars allocated to asset $X$. Use Calculus (or algebra) to minimize that function. $\endgroup$ – whuber May 12 '14 at 13:39

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