Weighting an average to minimize variance

Suppose you have $100 to invest in two independent assets, A andB, and you want to minimize volatility. SupposeA is more volatile thanB. Then putting all your money onA would be the worst thing to do, but putting all your money onB would not be the best thing to do.

The optimal allocation would be some mix ofA andB, with more (but not all) going toB. We will formalize this problem and determine the optimal allocation, then generalize the problem to more assets.

Two variables

LetX andY 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. BecauseX andY are independent,

Taking the derivative with respect tot and setting it to zero shows that

So the smaller the variance onY, the less we allocate toX. IfY is constant, we allocate nothing toX and go all in onY. 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 haven independent random variables X_i 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 t_i 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 t_i 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

• Symmetric funcions and U-statistics
• Regular solids and Monte Carlo integration
• Lagrange multiplier set up. Now what?

The post Weighting an average to minimize variance first appeared on John D. Cook.