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Expected Return and Variance for a Two Asset Portfolio

Securities & MarketsπŸ“ŠInvestment ManagementApril 15, 2012 Β· 2 min read

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.

Rp=w1R1+w2R2R_p = w_1R_1 + w_2R_2
  • RpR_p = expected return for the portfolio
  • w1w_1 = proportion of the portfolio invested in asset 1
  • R1R_1 = expected return of asset 1

Expected Variance for a Two Asset Portfolio

The variance of the portfolio is calculated as follows:

Οƒp2=w12Οƒ12+w22Οƒ22+2w1w2Cov1,2\sigma_{p}^2 = w_{1}^2\sigma_{1}^2 + w_{2}^2\sigma_{2}^2 + 2w_{1}w_{2}Cov_{1,2}
  • Cov1,2Cov_{1,2} = covariance between assets 1 and 2
  • Cov1,2=ρ1,2β‹…Οƒ1β‹…Οƒ2\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:

Οƒp2=w12Οƒ12+w22Οƒ22+2w1w2ρ1,2Οƒ1Οƒ2\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.

ExampleΒ 

The following information about a two stock portfolio is available:

Β Stock AStock B
Amount20,00030,000
Expected Returns12%20%
Standard Deviation20%30%
Correlation0.25

The weights for the two assets are:

wA=20,00050,000=40%Β wB=30,00050,000=60%\begin{align*} w_A &= \frac{20,000}{50,000} = 40\% \\\ w_B &= \frac{30,000}{50,000} = 60\% \end{align*} ExpectedΒ Returns=0.40Γ—0.12+0.60Γ—0.20=16.8%\textbf{Expected Returns} = 0.40 \times 0.12 + 0.60 \times 0.20 = 16.8\% Variance=(0.40)2(0.20)2+(0.60)2(0.30)2+2(0.40)(0.60)(0.25)(0.20)(0.30)Β =0.046\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 StandardΒ deviation=0.046=0.2145Β orΒ 21.45%\textbf{Standard deviation} = \sqrt{0.046} = 0.2145 \text{ or } 21.45\%

Expected Variance for a Three Asset Portfolio

Οƒp2=w12Οƒ12+w22Οƒ22+w32Οƒ32+2w1w2Cov1,2+2w1w3Cov1,3+2w2w3Cov2,3\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}

Check your understanding

6 questions

    1. What does the expected return formula $R_p = w_1R_1 + w_2R_2$ assume about the weights $w_1$ and $w_2$?
    1. A portfolio has 40% in Asset 1 ($R_1 = 8%$, $\sigma_1 = 10%$) and 60% in Asset 2 ($R_2 = 14%$, $\sigma_2 = 20%$), with a correlation of 0.30. What is the portfolio variance?
    1. Why is the portfolio variance formula $\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\text{Cov}_{1,2}$ generally less than the weighted average of individual variances?
    1. Using the example in the lesson (Stock A: $w=40%$, $\sigma=20%$; Stock B: $w=60%$, $\sigma=30%$; $\rho=0.25$), what would happen to portfolio variance if the correlation increased to 1.0?
    1. How many unique covariance terms appear in the variance formula for a three-asset portfolio?
    1. An analyst calculates a two-asset portfolio variance of 0.0529. What is the portfolio standard deviation, and what does this figure represent?
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