Question

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.7%. The probability distributions of the two risky funds are:

  

	Expected Return	Standard Deviation
Stock fund (S)	18%	47%
Bond fund (B)	7%	41%

The correlation between the two fund returns is .0317.

  

What is the standard deviation for the minimum-variance portfolio of the two risky funds? (Do not round intermediate calculations. Enter your answer as a decimal number rounded to 4 decimal places.)

Answer 1

Stock Fund: Expected Return or R1 = 18% | Standard Deviation or Sigma1 = 47%

Bond Fund: Expected Return or R2 = 7% | Standard Deviation or Sigma2  = 41%

Correlation between two funds = 0.0317

To find the Standard Deviation of Minimum variance portfolio, we first need to find the weight allocation for each fund in minimum variance portfolio using the below Minimum Variance Portfolio, .

W1 = (Sigma2² - Correlation (1,2) * Sigma1 * Sigma2) / (Sigma1² + Sigma2² - 2*Correlation(1,2)*Sigma1*Sigma2)

Weight of Stock Fund = (41%² - 0.0317 * 47% * 41%) / (47%² + 41%² - 2*0.0317*47%*41%)

Solving the above equation, we will get the weight allocation for Stock fund in the Minimum Variance Portfolio

Weight of Stock Fund = 57.01%

Weight of Bond Fund = 1 - Weight of Stock Fund = 1 - 57.01% = 42.99%

Now we will calculate the Standard deviation of the Minimum variance portfolio using the weights and each asset's standard deviation & correlation.

Standard Dev of portfolio = (Variance1 * W1² + Variance2 * W2² + 2*W1*W2*Sigma1*Sigma2*Correlation(1,2))^1/2

SD of Minimum Variance Portfolio = ((47%*57.01%)² + (41%*42.99%)² + 2*57.01%*42.99%*47%*41%*0.0317)^1/2

Solving the above equation, we will get the Standard Deviation of Minimum variance portfolio.

Standard Deviation of Minimum variance portfolio = (0.105854)^1/2

Standard Deviation of Minimum variance portfolio = 32.5351% or 32.54%