I have a weighted sum,

weighted sum = w1*mu1 + (1-w1)*mu2


variance weighted sum = (w1^2)*var1 + ((1-w1)^2)*var2 + 2*w1*(1-w1)*cov

in which

mu1 = mean 1; mu2 = mean 2; var1 = variance for mean 1; var2 = variance for mean 2; cov = covariance; w1 = weight (ranging from 0 to 1)

Even though I can compute easily the minimum variance, I am now interested in finding the w1 that gives the largest ratio (weighted sum)/sqrt(variance_weighted sum).

Do you have any ideas on how one could do that? Any references?

Thanks in advance!



Many thanks, Robert! Does your solution take into account that the denominator is the square root of var_weighted_sum? and that the ratio may be either positive or negative? Actually, I am looking for the largest absolute ratio.

The formulation seems to provide weights that do not give the largest ratio. Example:

mu1 = 0.8125358 mu2 = 0.1312268

var1 = 0.123922 var2 = 0.010128

cov = 0.0021274

I know that for this example, the weight (w1) that gives the highest ratio R


R = (w1*mu1 + (1-w1)*mu2)/sqrt((w1^2)*var1 + ((1-w1)^2)*var2 + 2*w1*(1-w1)*cov)

is w1 = 0.354, giving a R = 2.5867

According to your solution, the w1 = 0.0795, giving a R = 1.885

Since my background is in Biosciences, I will be really grateful if you could comment on that. Am I doing something wrong?


It's easy enough to take the derivative and solve for that = 0. The result I get is $w_1 = \frac{\mu_2 cov - \mu_1 var_2}{(\mu_1+\mu_2) cov - \mu_1 var_1 - \mu_2 var_2}$. This critical point is not necessarily in the interval $[0,1]$ and might be a minimum or inflection rather than a maximum, so the maximum might be here or at 0 or 1. It's also possible that there is no maximum because the variance weighted sum is 0 at some point.

  • $\begingroup$ Oops: typo. That should be $w_1 = \frac{\mu_2 cov - \mu_1 var_2}{(\mu_1 + \mu_2) cov - \mu_1 var_2 - \mu_2 var_1}$ $\endgroup$ – Robert Israel Apr 22 '11 at 19:52
  • $\begingroup$ Perfect! thank you so much, Robert! $\endgroup$ – Tiago Apr 23 '11 at 11:45

You should look at the bazillion references on "Markowitz portfolio optimization" -- this is covered, in particular, in Rob Vanderbei's book on convex optimization.


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