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 rate of 5.7%. The probability distribution of the risky funds is as follows:

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

The correlation between the fund returns is 0.17.

Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio. (Do not round intermediate calculations and round your final answers to 2 decimal places. Omit the "%" sign in your response.)

Portfolio invested in the stock	%
Portfolio invested in the bond	%
Expected return	%
Standard deviation	%

Answer 1

We need to calculate the weights for the stocks and the bonds, so the formula for the weights calculation would be;

[Expected Return of the stock – Risk free rate]* Variance of bond – [Expected Return of the bond – Risk Free rate]* Covariance of the stock & bond/ [Expected Return of the stock – Risk free rate]* Variance of bond + [Expected Return of the bond – Risk Free rate]* Variance of the stock – [Expected Return of the stock – Risk free rate + Expected Return of the bond – Risk Free rate]* Covariance of the stock & bond

Covariance of the (bond, stock) = Correlation of stock, bond* SD of stock* SD of bond

Correlation of stock, bond = 0.17

SD of Stock = 47%

SD of Bond = 41%

Covariance of stock, bond = 0.17* 47* 41 = 327.59

Weights of the stock = [0.18-0.057]*1681 – [0.07-0.057]*327.59/ [0.18-0.057]*1681 + [0.07 – 0.057]* 2209 – [0.18- 0.057 + 0.07 – 0.057] * 327.59

Weights of the stock = 202.50/ 280.03 = 72.30% (Approx Weight)

Weight of the bond = 1- weight of the stock = 1-0.7230 = 27.70%

Expected Return of the Portfolio = Weight of the stock* Expected Return of the stock + Weight of the bond* Expected Return of the bond

Or, Expected return of the portfolio = 0.7230*18% + 0.2770*7% = 14.953%

Standard Deviation of the portfolio = {(Weight of the stock) ^ 2 * (SD of Stock) ^ 2 + (Weight of the bond) ^ 2 * (SD of bond) ^ 2 + (2* weight of the stock * weight of the bond * SD of stock * SD of bond * correlation)} ^ (1/2)

SD of the portfolio = {(0.7230) ^ 2* (0.47) ^2 + (0.2770) ^2 * (0.41) ^2 + (2* 0.7230* 0.2770 * 0.47* 0.41* 0.17)} ^ (0.50)

SD of portfolio = {(0.11547) + (0.012898) + (0.01312)} ^ (0.50) = 37.62%

Portfolio invested in the stock	72.30 %
Portfolio invested in the bond	27.70 %
Expected return	14.953 %
Standard deviation	37.62 %