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 3.0%. The probability distributions of the risky funds are Expected Return 12% Stock fund (S) Bond fund (B) Standard Deviation 41% 30% 5% The correlation between the fund returns is .0667. Suppose now that your portfolio must yield an expected return of 9% and be efficient, that is, on the best feasible CAL. a. What is the standard deviation of your portfolio? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Standard deviationſ %

Answer 1

First we need to calculate the weight of stock in the optimal portfolio

(rstock – Trisk free) o Bond - (r Bond – Frisk free)po Stock X O Bond) W Stock = 7 (rstock – Trisk free) o Bond + (r Bond - Prisk free) ostock - plostock X Bond) (( Bond – Trisk free) + (rStock – Trisk free)) W = weight T= return o = standard deviation (0.12 - 0.03)0.32 – (0.05 -0.03) x 0.0667 x (0.41 x 0.3) (0.12 -0.03)0.32 + (0.05 – 0.03)0.412 – 0.0667 x (0.41 x 0.3)((0.05 – 0.03) + (0.12 - 0.03)) W Stock = 0.75

Weight of the bond is 1-weight of stock = 1-0.75 = 0.25

Portfolio Expected return is calcualted by solving the following equation:

Portfolio standard deviation is calculated by solving the following equation:

Expected return risky portfolio = Weightstockr Stock + WeightBondr Bond Expected retur risky portfolio = 0.75 X 12 +0.25 x 5 Expected return risky portfolio = 10.25%

Orisky portfolio = V(Weight stock stock)+(Weight Bond Bond) +2Weight stock stock Weight Bond Bond X correlartion stock, Bond

Orisky portfolio = V(0.75x41)2+(0.25x 30)2+(x0)2+(0x0)2+(0x0)2+(0x0)2+2(0.75 x 41x0.25x30x0.0667)

Orisky portfolio = 32.13%

This is the calculation only of the risky portfolio.

Now we need to calculate the proportion of a portfolio comprising the risky portfolio and the T bill to get a return of 9% Once we know the weights, we can calculate the standard deviation of this portfolio

Portfolio Expected return is calcualted by solving the following equation:

Expected retur n portfolio = Weight Riskyr Risky + WeightTbilllThill 9% = Weight Risky 10.25% + (1 - Weight Risky)3% Weight Risky = 82.76% Weightīball = 100% – 82.76% = 17.24%

As tbill is not risky it has a stadard deviation of 0 so the portfolio stadard deviation is as follows:

O portfolio = Weight Risky X O Risky Oportfolio = 82.76% x 32.13% portfolio = 26.59%

So the answer to the first question is 26.59%

Proportions of bond and stock in this portfolio are as follows:

Asset	Weight	Portfolio weight	Final weight
Stock	75.00%	82.76%	0.8276 x 75 = 62.07%
Bond	25.00%	82.76%	0.8276 x 25 = 20.69%