https://www.johndcook.com/blog/2025/11/12/minimum-variance/

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Weighting an average to minimize variance

Posted on 12 November 2025 by John

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

\text{Var}[tX + (1-t)Y]

subject to the constraint 0 <= t <= 1. Because X and Y are independent,

\text{Var}[tX + (1-t)Y] = t^2 \text{Var}[X] + (1-t)^2 \text{Var}[Y]

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

t = \frac{\text{Var}[Y]}{\text{Var}[X] + \text{Var}[Y]}

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 X[i] for i running
from 1 to n, and at least one of the variables is not constant. Then
we want to minimize

\text{Var}\left[ \sum_{i=1}^n t_i X_i \right] = \sum_{i=1}^n t_i^2 \
text{Var}[X_i]

subject to the constraint

\sum_{i=1}^n t_i = 1

and all t[i] non-negative. We can solve this optimization problem
with Lagrange multipliers and find that

t_i \text{Var}[X_i] = t_j \text{Var}[X_j]

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

t_i = \frac{\prod_{j \ne i} \text{Var}[X_j]}{\sum_{i = 1}^n \prod_{j
\ne i} \text{Var}[X_j]}

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?

Categories : Statistics Uncategorized
Tags : Probability and Statistics
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John D. Cook

John D. Cook, PhD

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