Asset A has an expected return of 18%, Asset B has an expected return of 10%, and the rate ofreturn on T-bills is 7%. The covariance matrix for the two risky assets is as follows:Asset A BA 1600 90B 90 400a) Calculate the expected return and standard deviation of a portfolio consisting of 20%T-bills, 35% Asset A, and 45% Asset B.
Question
Asset A has an expected return of 18%, Asset B has an expected return of 10%, and the rate ofreturn on T-bills is 7%. The covariance matrix for the two risky assets is as follows:Asset A BA 1600 90B 90 400a) Calculate the expected return and standard deviation of a portfolio consisting of 20%T-bills, 35% Asset A, and 45% Asset B.
Solution
a) To calculate the expected return and standard deviation of a portfolio, we need to use the following formulas:
Expected Return (E[Rp]) = w1R1 + w2R2 + w3R3 Standard Deviation (σp) = sqrt[(w1^2σ1^2) + (w2^2σ2^2) + 2w1w2Cov(R1,R2)]
where:
- w1, w2, and w3 are the weights of the assets in the portfolio (in this case, 20% T-bills, 35% Asset A, and 45% Asset B)
- R1, R2, and R3 are the expected returns of the assets (in this case, 7% for T-bills, 18% for Asset A, and 10% for Asset B)
- σ1 and σ2 are the standard deviations of the assets (in this case, the square root of the variances given in the covariance matrix, so sqrt(1600) = 40 for Asset A and sqrt(400) = 20 for Asset B)
- Cov(R1,R2) is the covariance between the returns of the two assets (in this case, 90)
Step 1: Calculate the expected return of the portfolio:
E[Rp] = (0.200.07) + (0.350.18) + (0.45*0.10) = 0.014 + 0.063 + 0.045 = 0.122 or 12.2%
Step 2: Calculate the standard deviation of the portfolio:
σp = sqrt[(0.35^240^2) + (0.45^220^2) + 20.350.4590] = sqrt[(0.12251600) + (0.2025*400) + 56.7] = sqrt[196 + 81 + 56.7] = sqrt[333.7] = 18.27%
So, the expected return of the portfolio is 12.2% and the standard deviation is 18.27%.
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