Weighting an average to minimize variance

Summary

Suppose you have $100 to invest in two independent assets, A and B, and you want to minimize volatility. Suppose A is more volatile than B. Then putting all your money on A would be the worst thing to do, but putting all your money on B would not be the best thing to do. The optimal allocation would be some mix of A and B, with more (but not all) going to B. We will formalize this problem and determine the optimal allocation, then generalize the problem to more assets. Two variables Let X and Y be two independent random variables with finite variance and assume at least one of X and Y is not constant. We want to find t that minimizes subject to the constraint 0 ≤ t ≤ 1. Because X and Y are independent, Taking the derivative with respect to t and setting it to zero shows that So the smaller the variance on Y, the less we allocate to X. If Y is constant, we allocate nothing to X and go all in on Y. If X and Y have equal variance, we allocate an equal amount to each. If X has twice the variance of Y, we allocate 1/3 to X and 2/3 to Y. Multiple variables Now suppose we have n independent random variables Xi for i running from 1 to n, and at least one of the variables is not constant. Then we want to minimize subject to the constraint and all ti non-negative. We can solve this optimization problem with Lagrange multipliers and find that for all 1 ≤ i, j ≤ n. These (n − 1) equations along with the constraint that all the ti sum to 1 give us a system of equations whose solution is Incidentally, the denominator has a name: the (n − 1)st elementary symmetric polynomial in n variables. More on this in the next post. Related posts