Expected Return and Variance for a Two Asset Portfolio

Expected Return for a Two Asset Portfolio

The expected return of a portfolio is equal to the weighted average of the returns on individual assets in the portfolio.

$$R_p = w_1R_1 + w_2R_2$$

• $R_p$ = expected return for the portfolio
• $w_1$ = proportion of the portfolio invested in asset 1
• $R_1$ = expected return of asset 1

Expected Variance for a Two Asset Portfolio

The variance of the portfolio is calculated as follows:

$$\sigma_{p}^2 = w_{1}^2\sigma_{1}^2 + w_{2}^2\sigma_{2}^2 + 2w_{1}w_{2}Cov_{1,2}$$

• $Cov_{1,2}$ = covariance between assets 1 and 2
• $\text{Cov}_{1,2} = \rho_{1,2} \cdot \sigma_{1} \cdot \sigma_{2}$; where ρ = correlation between assets 1 and 2

The above equation can be rewritten as:

$$\sigma_{p}^2 = w_{1}^2\sigma_{1}^2 + w_{2}^2\sigma_{2}^2 + 2w_{1}w_{2} \rho_{1,2} \sigma_{1} \sigma_{2}$$

Keep in mind that this is the calculation for portfolio variance. If a test question asks for the standard deviation then you will need to take the square root of the variance calculation. Percentage values can be used in this formula for the variances, instead of decimals.

Example 

The following information about a two stock portfolio is available:

The weights for the two assets are:

$$\begin{align*} w_A &= \frac{20,000}{50,000} = 40\% \\\ w_B &= \frac{30,000}{50,000} = 60\% \end{align*}$$ $$\textbf{Expected Returns} = 0.40 \times 0.12 + 0.60 \times 0.20 = 16.8\%$$ $$\textbf{Variance} = (0.40)^2(0.20)^2 + (0.60)^2(0.30)^2 + 2(0.40)(0.60)(0.25)(0.20)(0.30) \\\ = 0.046$$ $$\textbf{Standard deviation} = \sqrt{0.046} = 0.2145 \text{ or } 21.45\%$$

Expected Variance for a Three Asset Portfolio

$$\sigma_p^2 = w_1^2 \sigma_{1}^2 + w_2^2 \sigma_{2}^2 + w_3^2 \sigma_{3}^2 + 2w_1w_2 \text{Cov}_{1,2} + 2w_1w_3 \text{Cov}_{1,3} + 2w_2w_3 \text{Cov}_{2,3}$$