Chapter 7, Problem 9PS

A

Summary Introduction

To compute: The standard deviation of the portfolio is to be determined.

Introduction: The portfolio risk is refers to 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 computed 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 9PS

The standard deviation of the portfolio is 16.5%.

Explanation of Solution

The following equation will be used to compute the standard deviation and the expected return-

  $E(r_c)=r_f+\left[E(r_p)-r_f\right]\frac{{\sigma }_c}{{\sigma }_p}$

Where,

$E(r_c)$ = expected return

$r_f$ = rate of return of risk

$E(r_p)$ = portfolio expected return

${\sigma }_c$ = standard deviation for investor portfolio

${\sigma }_p$ = standard deviation for optimal portfolio

When E(r_p) and ${\sigma }_p$ are the expected return and the standard deviation for the optimal portfolio then is it is best feasible CAL.

Given that −

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

Risk free rate = 8%

The expected return on the portfolio can be computed by using the following formula −

  $E(r_p)={\displaystyle \sum _{i=1}^n{w}_i{r}_i}$

Where

w = weight on the asset

r = rate on the asset

The weight for the stock fund on optimal portfolio can be computed as −

  $w_s=\frac{R_s{\sigma }_B^2-R_B\mathrm{cov}(r_s,r_B)}{R_s{\sigma }_B^2+R_B{\sigma }_s^2-(R_s+R_B)\mathrm{cov}(r_s,r_B)}$

Or,

  $w_s=\frac{R_s{\sigma }_B^2-R_B{\sigma }_s{\sigma }_{B}corr(r_s,r_B)}{R_s{\sigma }_B^2+R_B{\sigma }_s^2-(R_s+R_B){\sigma }_s{\sigma }_{B}corr(r_s,r_B)}$

Given that −

  $R_s=(0.20-0.08)$

  $R_s=(0.12-0.08)$

  ${\sigma }_s=0.30$

  ${\sigma }_B=0.15$

  $corr(r_s,r_B)=0.10$

Put the given values in above equation

  $w_s=\frac{\left(0.20-0.08\right){(0.15)}^2-\left(0.12-0.08\right)(0.30)(0.15)(0.10)}{\left(0.20-0.08\right){(0.15)}^2+\left(0.12-0.08\right){(0.30)}^2-\left(\left(0.20-0.08\right)+\left(0.12-0.08\right)\right)(0.30)(0.15)(0.10)}$

  $w_s=\frac{0.00252}{0.00558}=0.452$

Weight for stock fund = 0.452

For the bond fund −

  $w_B=1-w_s$

  $w_B=1-0.452=0.548$

Weight for bond fund = 0.548

The expected return is −

  $E(r_p)={\displaystyle \sum _{i=1}^n{w}_i{r}_i}$

  $E(r_p)=w_s{r}_s+w_B{r}_B$

  $E(r_p)=0.452\times 0.20+0.548\times 0.12=0.156$

Expected return = 15.6%

The standard deviation of the portfolio is computed as −

  ${\sigma }_p=\sqrt{{\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=1}^2{w}_i{w}_j\mathrm{cov}(r_i,r_j)}}}$

  ${\sigma }_p=\sqrt{w_s^2{\sigma }_s^2+w_B^2{\sigma }_B^2+2w_s{w}_B{\sigma }_s{\sigma }_{B}corr(r_s,r_B)}$

  ${\sigma }_p=\sqrt{{(0.452)}^2{(0.30)}^2+{(0.548)}^2{(0.15)}^2+2(0.452)(0.548)(0.30)(0.15)(0.10)}=0.165$

The standard deviation of portfolio = 16.5%

B

Summary Introduction

To compute: The proportion invested in T-bill fund every year and each of the two risky funds is to be determined.

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 computed 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 9PS

The tabular form −

Fund	Proportion
T-bill	21.2%
Stock fund	35.6%
Bond fund	43.2%

Explanation of Solution

The following formula will be used to compute the risky portfolio and risk free asset-

  $E(r_c)=w_{p}E(r_p)+w_f{r}_f$

Consider, in risky asset ‘w’ is the proportion of the portfolio and given as −

  $E(r_c)=w\times E(r_p)+(1-w)r_f$

  $w=\frac{E(r_c)-r_f}{E(r_p)-r_f}$

Put the values in above Equation −

  $w=\frac{0.14-0.08}{0.156-0.08}=0.789$

In risky portfolio, portfolio of investor = 0.789 or 78.9%

The risk free asset = 21.1%

The tabular form is given as −

Fund	Proportion
T-bill	21.2%
Stock fund	$78.9\%\times 0.452=35.6\%$
Bond fund	$78.9\%\times 0.548=43.2\%$