Let’s say you have $100 to invest in two independent assets, A And bAnd you want to reduce volatility. suppose A is more unstable than bthen invest all your money A This would be the worst thing to do, but invest all your money b Wouldn’t be the best thing to do.
The optimal allocation will be some mixture A And bwith more (but not all) bWe will formalize this problem and determine the optimal allocation, then generalize the problem to more assets,
two variables
let x And Y Let there be two independent random variables with finite variance and at least one of them take the value x And Y Is not stable. we want to find Tea he does the minimum
![\text{var}[tX + (1-t)Y]](https://www.johndcook.com/minvar1.svg "Weighting an average to minimize variance 2"){kind=small}
subject to the constraint 0 Tea ≤ 1. because x And Y are free,
![\text{var}[tX + (1-t)Y] = t^2 \text{Var}[X] + (1-t)^2 \text{Var}[Y]](https://www.johndcook.com/minvar2.svg "Weighting an average to minimize variance 3"){kind=small}
taking the derivative with respect to Tea and setting it to zero indicates that
![t = \frac{\text{Var}[Y]}{\text{Var}[X] + \text{Var}[Y],](https://www.johndcook.com/minvar3.svg "Weighting an average to minimize variance 4"){kind=small}
Therefore the variance will be smaller Ythe less we allocate xIf Y is static, we don’t allocate anything x and go all in YIf x And Y Having equal variance, we allocate equal amount to each. If x has twice the variance of YWe allocate 1/3 x and by 2/3 Y,
multiple variables
Now suppose we have n independent random variable xI For I running since 1 nAnd at least one variable 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]](https://www.johndcook.com/minvar4.svg "Weighting an average to minimize variance 5")
subject to obstruction
and all TeaI 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]](https://www.johndcook.com/minvar6.svg "Weighting an average to minimize variance 7"){kind=small}
for all 1 I, J nin (n − 1) Equations with constraints that all TeaI The addition of 1 gives 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],](https://www.johndcook.com/minvar7x.svg "Weighting an average to minimize variance 8")
Incidentally, each has a name: (n − 1) st initial symmetric polynomial n Variable. More information on this in the next post.