Question

You manage a risky portfolio with expected rate of return of 12% and a standard deviation of 24%.  The T-bill rate is 3%.

1. Suppose that your client decides to invest in your portfolio a proportion, y, of the total investment budget so that the overall portfolio will have an expected rate of return of 8.40%.  
   1. What is the proportion y?
   2. What is the standard deviation of the rate of return on your client’s portfolio at this new level y?

Answer 1

Expected Return of the risky portfolio = R₁ = 12%

Standard deviation of the risky portfolio = σ₁ = 24%

T-Bills are risk-free assets

Return on risk-free asset = R₂ = 3%, standard deviation of risk-free asset = σ₂ = 0

weight invested in risky portfolio = w₁ = y

weight invested in risk-free asset = w₂ =1-y

a. Expected return of the overall portfolio of the client = E[R_P] = 8.4%

Expected return of the portfolio is calculated using the formula: E[R_P] =w₁*R₁ + w₂*R₂

8.4% = y*12% + (1-y)*3%

8.4% = y*12% + 3% - y*3%

8.4% - 3% = y*12% - y*3%

5.4% = y*9%

y = 5.4%/9% = 0.6

y = 60%

Answer -> y = 60% or 0.6

b. Variance on the overall portfolio is calculated using the formula:

σ_P² = w₁²*σ₁² + w₂*σ₂² + 2*w₁*w₂*ρ*σ₁*σ₂

where ρ is correlation coefficient between risky portfolio and risk-free asset

Since σ₂ = 0

We get, σ_P² = w₁²*σ₁² + 0 + 0 = w₁²*σ₁²

Standard deviation is the square-root of variance

So, standard deviation of the overall portfolio = σ_P = [w₁²*σ₁²]^1/2 = w₁*σ₁

y = 60%, σ₁ = 24%

Standard deviation of the portfolio = σ_P = w₁*σ₁ = 60%*24% = 14.4%

Answer -> Standard deviation of the client's overall portfolio = 14.4%