``` 最小化方差的加权平均 ```

``` 最小化方差的加权平均 ```
Weighting an average to minimize variance

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

本文讨论了优化投资配置以最小化投资组合波动率的方法。在选择两个独立的资产（A和B）时，如果A比B更具波动性，最佳策略不是将所有资金投入其中任何一个。相反，混合配置是最佳的，并且应偏向波动性较低的资产（B）。最佳配置通过数学确定，取决于每个资产的方差。资产Y的方差越低，对其的配置就越大。如果Y没有方差（是常数），则完全投资于Y。方差相等意味着配置相等。这个原则可以扩展到*n*个资产。每个资产（Xi）的最佳配置与其方差成反比。具体来说，分配给每个资产的权重是其反方差除以所有反方差之和。这确保了给予低风险资产更高的权重，从而最小化整体投资组合的波动率。

## 最小化方差的加权平均 - 摘要最近的 Hacker News 讨论围绕一篇博客文章展开，该文章详细介绍了一种加权平均以最小化方差的方法。该技术适用于结合来自不同精度设备的测量结果，建议根据其方差的反比例对每个测量结果进行加权。讨论强调了它在物理科学（结合测量结果以达成共识标准）和数值算法评估等领域的相关性。许多评论员指出，该方法与贝叶斯方法和卡尔曼滤波器等概念相符，尤其是在假设高斯分布时。一个关键点是，分配*一些*投资于更具波动性的资产，由于分散投资带来的好处，实际上可以*降低*整体投资组合的方差，这一观点颇具反直觉。然而，人们对依赖准确的方差估计表示担忧，尤其是在动态系统中，以及在不考虑实际因素的情况下应用纯统计方法的局限性。讨论还涉及更清晰的数学符号的需求以及平台上的数学渲染的潜在好处。

原文

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.