Weighting a mean to reduce variance

Suppose you will have $100 to spend money on two impartial belongings, A and B, and also you need to reduce volatility. Suppose A is extra unstable than B. Then placing all of your cash on A could be the worst factor to do, however placing all of your cash on B wouldn’t be the perfect factor to do.

The optimum allocation could be some mixture of A and B, with extra (however not all) going to B. We are going to formalize this drawback and decide the optimum allocation, then generalize the issue to extra belongings.

Two variables

Let X and Y be two impartial random variables with finite variance and assume a minimum of certainly one of X and Y isn’t fixed. We need to discover t that minimizes

topic to the constraint 0 ≤ t ≤ 1. As a result of X and Y are impartial,

Taking the by-product with respect to t and setting it to zero reveals that

So the smaller the variance on Y, the much less we allocate to X. If Y is fixed, we allocate nothing to X and go all in on Y.  If X and Y have equal variance, we allocate an equal quantity to every. If X has twice the variance of Y, we allocate 1/3 to X and a couple of/3 to Y.

A number of variables

Now suppose we now have n impartial random variables X_i for i working from 1 to n, and a minimum of one of many variables isn’t fixed. Then we need to reduce

topic to the constraint

and all t_i non-negative. We will resolve this optimization drawback with Lagrange multipliers and discover that

for all 1 ≤ i, j ≤ n. These (n − 1) equations together with the constraint that each one the t_i sum to 1 give us a system of equations whose resolution is

By the way, the denominator has a reputation: the (n − 1)st elementary symmetric polynomial in n variables. Extra on this within the subsequent submit.

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