Chapter 7, Problem 5PS

Summary Introduction

To calculate: The investment opportunity set of the two risky funds is to be tabulated along with the graph.

Introduction: The portfolio risk is defined as the combination of assets which carries its own risk with each investment.

The standard deviation is used to determine that in which manner the values from a data set vary from its mean value. This is calculated by the square root of the variance.

  $standarddeviation=\sqrt{variance}$

The expected return is defined as the return which is obtained on the risky asset that is expected in future.

Answer to Problem 5PS

Expected return for the portfolio = 13.39%

Standard deviation for the portfolio = ${\sigma }_p=0.1392$

The graph is represented as −

  

Explanation of Solution

The following formula will be used to calculate the expected return of the portfolio for different proportions of stock and bond-

  $E(r_p)=w_s\times E(r_s)+w_B\times E(r_B)$.................... Equ (1)

Where,

  $w_s$ = weight of stock in percentage

  $E(r_s)$ = expected return for stocks

  $w_B$ = weight of bonds in percentage

  $E(r_B)$ = expected return for bonds

The following formula will be used to calculate the standard deviation of the portfolio for different proportions of stock and bond-

  ${\sigma }_p=\sqrt{w_s^2\times {\sigma }_s^2+w_B^2\times {\sigma }_B^2+2w_s{w}_B\times {\sigma }_s\times {\sigma }_B\times {\rho }_{sB}}$

The three mutual funds are −

1. Stock fund
2. Long term government and corporate bond fund
3. T-bill money market fund with yield 8%

The probability distribution of the risk fund is given as −

	Expected return	Standard deviation
Stock fund (S)	20%	30%
Bond fund (B)	12	15

The correlation between fund return = 0.10

The minimum variance portfolio is calculated as-

  $Cov(r_s,r_B)=\rho \times {\sigma }_s\times {\sigma }_B$

  $Cov(r_s,r_B)=0.1\times 30\%\times 15\%=0.0045$

The weight of the stock fund in minimum variance portfolio is calculated as-

  $W_{MIN}(S)=\frac{{\sigma }_B^2-Cov(r_s,r_B)}{{\sigma }_s^2+{\sigma }_B^2-2\times Cov(r_s,r_B)}$

  $W_{MIN}(S)=\frac{0.0225-0.0045}{0.09+0.0225-2\times (0.0045)}=0.1739$

For bonds funds:

  $W_{MIN}(B)=1-W_{MIN}(S)$

  $W_{MIN}(B)=1-0.1739=0.8261$

Put the calculated values in Equ (1) for expected return −

  $E(r_p)=w_s\times E(r_s)+w_B\times E(r_B)$

  $E(r_p)=0.1739\times 20\%+0.8261\times 12\%$

  $E(r_p)=0.1339=13.39\%$

Put the calculated values in Equ (2) for the standard deviation-

  ${\sigma }_p=\sqrt{w_s^2\times {\sigma }_s^2+w_B^2\times {\sigma }_B^2+2w_s{w}_B\times {\sigma }_s\times {\sigma }_B\times {\rho }_{sB}}$

  ${\sigma }_p=\sqrt{{0.1739}^2\times {0.30}^2+{0.8261}^2\times {0.15}^2+2\times 0.1739\times 0.8261\times 0.30\times 0.15\times 0.10}$

  ${\sigma }_p=0.1392$

The graph between expected return and standard deviation is called as investment opportunity set which is represented as −