Question

A portfolio consists of the following two funds:
A: E(r) = 13%, Stdev = 16%, amount invested = $16,000
B: E(r) = 9%, Stdev = 12%, amount invested = $8,000
Corr(A,B) = 0.6, Risk-free rate = 4%.
What is the Sharpe ratio of the portfolio? (Hint: you need to find portfolio return and portfolio standard deviation).

Answer 1

Expected Return on A = E(r_A) = 13%, Standard deviation of A = σ_A = 16%

Expected Return on B = E(r_B) = 9%, Standard deviation of B = σ_B = 12%

Amount invested in A = 16000, Amount invested in B = $8000

Total amount invested = 16000 + 8000 = 24000

Weight of A in the portfolio = W_A = 16000/24000 = 0.666666666666667

Weight of B in the portfolio = W_B = 8000/24000 = 0.333333333333333

Expected Return of the portfolio

Expected Return of the portfolio is calculated using the formula:

Expected return of portfolio = E(r_P) = W_A*E(r_A) + W_B*E(r_B) = 0.666666666666667*13% + 0.333333333333333*9% = 0.1166666667 = 11.66666667 %

Standard Deviation of portfolio

Correlation between A and B = Corr(A,B) = ρ = 0.6

Variance of the portfolio is calculated using the formula:

Variance of the portfolio = σ_P² = W_A²*σ_A² + W_B²*σ_B² + 2*W_A*W_B*ρ*σ_A*σ_B = (0.666666666666667)²*(16%)² + (0.333333333333333)²*(12%)² + 2*(0.666666666666667)*(0.333333333333333)*0.6*16%*12% = 0.0113777777777778 + 0.0016 + 0.00512 = 0.0180977777777778

Standard Deviation is square-root of variance

Standard Deviation of the portfolio = σ_P = 0.0180977777777778^1/2 = 0.134527981393381 = 13.4527981393381 %

Sharpe's Ratio

Expected Return on Portfolio = E(r_P) = 11.66666667 %

Standard Deviation of portfolio = σ_P = 13.4527981393381 %

Risk-free rate = r_F = 4%

Sharpe's ratio is calculated using the formula:

Sharpe's ratio = [E(r_P) - r_F]/σ_P = (11.66666667% - 4%)/13.4527981393381% = 0.0766666667/13.4527981393381% = 0.569893831052254

Answer -> Sharpe's Ratio = 0.569893831052254