Introduction

In this chapter, we will delve into the concept of computing the variance of a weighted sum of two random variables. Understanding how to calculate the variance of such a combination is crucial in various statistical and probabilistic analyses. We will explore the mathematical principles underlying this computation and provide practical examples to illustrate its application.

Variance of a Weighted Sum

Consider two random variables, $\mathrm{X}$ and $\mathrm{Y}$, with corresponding variances $\operatorname{Var}(\mathrm{X})$ and $\operatorname{Var}(\mathrm{Y})$, and let $a$ and $b$ be constants. The weighted sum $Z=a X+b Y$ is a linear combination of $X$ and $\mathrm{Y}$. The variance of $\mathrm{Z}$, denoted as $\operatorname{Var}(\mathrm{Z})$, can be calculated as follows:

$$\operatorname{Var}(Z)=a^2 \cdot \operatorname{Var}(X)+b^2 \cdot \operatorname{Var}(Y)+2 a b \cdot \operatorname{Cov}(X, Y)$$

Here, $\operatorname{Cov}(X, Y)$ represents the covariance between $\mathrm{X}$ and $\mathrm{Y}$.

Example: Let’s illustrate this with an example. Suppose we have two investment portfolios, Portfolio A and Portfolio $B$, with returns $X$ and $Y$, respectively. We want to calculate the variance of a combined portfolio $\mathrm{Z}=0.6 \mathrm{X}+0.4 \mathrm{Y}$, where the weights ( $\mathrm{a}$ and $\mathrm{b}$ ) are 0.6 and 0.4, respectively.

Solution: Given that $\operatorname{Var}(X)=0.02, \operatorname{Var}(Y)=0.03$, and $\operatorname{Cov}(X, Y)=0.015$, we can plug in these values into the formula:

$$\operatorname{Var}(Z)=(0.6)^2 \cdot 0.02+(0.4)^2 \cdot 0.03+2 \cdot 0.6 \cdot 0.4 \cdot 0.015$$

Calculating this gives us $\operatorname{Var}(\mathrm{Z}) \approx 0.0255$.

Significance and Interpretation

The formula for calculating the variance of a weighted sum highlights how the weights of the random variables influence the combined variance. Additionally, the covariance between the random variables affects the total variance. A positive covariance contributes to the overall variance, while a negative covariance may partially offset the variance caused by the individual random variables.