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 4.9%.

The probability distributions of the risky funds are:

Stock Fund (S): Expected Return= 10% Standard Deviation= 39%

Bond Fund (B): Expected Return= 5% Standard Deviation= 33%

The correlation between the fund returns is .0030. Suppose now that your portfolio must yield an expected return of 8% 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.)

b-1. What is the proportion invested in the T-bill fund? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

b-2. What is the proportion invested in each of the two risky funds? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Answer 1

The Optimal Risky portfolio weights for a two security (A and B) portfolio is given by

and W_B= 1 - W_A

These weights correspond to the point on the best feasible Capital allocation line which is tangent from the Risk free Asset point (0,RFR)

Let A be the stock fund and B be the bond fund

Here, E(R_A) = 10%, stdev(A) = 39%

E(R_B) = 5%, stdev(B) = 33%

R_f= 4.9% , Correlation coefficient = 0.003

So, W_A = [(0.10-0.049)*0.33^2 - (0.05-0.049)*0.39*0.33*0.003] / [(0.10-0.049)*0.33^2 +(0.05-0.049)*0.39^2 - (0.10-0.049+ 0.05-0.049)*0.39*0.33*0.003]

   =0.005554/0.005686

=0.9767 =97.67%

and W_B = 1-0.9767=0.0233 =2.33%

This is the optimal portfolio weights i.e. 0.9767 invested in Stock fund and 0.0233 invested in Bond fund

The return of a portfolio is the weighted return of the two stocks

So Return of this portfolio = 0.9767 * 10% +0.0233 *5% = 9.88%

The standard deviation of a portfolio is given by

Where Wi is the weight of the security i,

is the standard deviation of returns of security i.

and is the correlation coefficient between returns of security i and security j

So, standard deviation of portfolio =sqrt (0.9767^2*0.39^2+0.0233^2*0.33^2+2*0.9767*0.0233*0.39*0.33*0.003)

=sqrt(0.145175)

=0.381018 =38.10%

To achieve expected return of 8%, let w be the proportion invested in the optimal risky portfolio determined above and (1-w) be the proportion invested in riskfree asset

w*9.88%+(1-w)*4.9% =8%

=> w =0.622045

So, the portfolio required to achieve 8% return has 62.20% invested in optimal risky portfolio and 37.80% in Risk free asset

a) Standard Deviation of the portfolio = 0.622045* 38.10% = 23.70%

b-1) proportion invested in the T bill fund = 37.80% (as calculated above)

b-2)

Proportion invested in Stock fund  = 0.9767* 62.20% = 0.607559 or 60.76%

Proportion invested in Bond fund  = 0.0233* 62.20% = 0.014486 or 1.45%