Chapter 8, Problem 17PS

a.

Summary Introduction

To determine: Calculate the excepted excess returns, alpha values and the residual variances for the stocks

Introduction: The Capital Asset Pricing Model explains the bond in between the systematic risk of an asset and the return that are expected.

Answer to Problem 17PS

The excepted excess returns, alpha values and the residual variances for the stocks are shown in table.

Explanation of Solution

Given Information:

Forecast returns, standard deviations and the beta values are given.

The capital asset pricing model describes the expected return on beta based security. This model is used for determine the expected return on asset, which is based on systematic risk.

  $Expected\mathrm{Re}turn=RiskFreeRate+Beta[ExpectedMarketRisk-RiskFreeRate]$

  $=r_f+\beta [E(r_M)-r_f]$

  $E_r$ , the expected return

  $r_f$ , the risk free rate of interest

  $\beta$ , the systematic risk

  $r_M$ , return on market index

As, the value of micro and macro forecasts of each stocks are given, substitute these values in the equations to determine the value of alpha,

	ALPHA ( $a$ )	EXPECTED EXCESS RETURN $E(r_i)-r_f$
$a_A$	1.60%	12%
$a_B$	-4.40%	10%
$a_C$	3.40%	9%
$a_D$	-4.00%	4%

From the above table it shows that in stock A, C have positive alpha values and for B, D the alpha values are negative.

STOCK	RESIDUAL VALUE
A	${\sigma }^2(e_A)={58}^2=3,364$
B	${\sigma }^2(e_B)={71}^2=5,041$
C	${\sigma }^2(e_C)={60}^2=3,600$
D	${\sigma }^2(e_D)={55}^2=3,025$

b.

Summary Introduction

To determine: The optimal risky portfolio

Introduction: The optimal asset allocation is the best attainable capital allocation in which a risk free asset is added. The optimal asset allocation depends upon the degree of risk return.

Answer to Problem 17PS

1.0486 is the optimal risky portfolio

Explanation of Solution

Given Information:

Forecast returns, standard deviations and the beta values are given.

The capital asset pricing model describes the expected return on beta based security. This model is used for determine the expected return on asset, which is based on systematic risk.

First to determine optimal active portfolio,

	$\frac{a}{{\sigma }^2(e)}$	$\frac{\frac{a}{{\sigma }^2}(e)}{\frac{S_a}{{\sigma }^2}(e)}$
A	0.00048	-0.6142
B	-0.00088	1.1265
C	0.00094	-1.2181
D	-0.00132	0.7058
Total	-0.00078	1

The alpha of active portfolios,

  $\begin{array}{l}a=[-0.6142\times 1.6]+[1.1265\times (4.4)]-[1.2181\times 3.4]+[1.7058\times (-4.0)]\\ =-16.90\%\end{array}$

The beta of active portfolio

  $\begin{array}{l}\beta =[-0.6142\times 1.3]+[1.1265\times (1.8)]-[1.2181\times 0.7]+[1.7058\times 1]\\ =2.08\%\end{array}$

The portfolio is higher than the individual stock beta. So, the standard deviation of portfolio is,

  $\begin{array}{l}$ {\sigma }^2(e)=[{(-0.6142)}^2\times {58}^2]+[{1.1265}^2\times {(71)}^2]+[{(-1.2181)}^2\times {60}^2]+[{1.7058}^2\times {55}^2]$=21,809.6\end{array}$

  $\sigma (e)=\sqrt{21,809.6}=147.68\%$

It results high residual standard deviation

The optimal risky portfolio, $W_0=-0.05124$

The adjustment of beta, $W^{*}=-0.0486$

As the value is positive the position in stock is positive alpha and the negative position in stocks with negative alphas.

In index portfolio, the position is,

  $\begin{array}{l}1-(-0.0486)\\ =1.0486\end{array}$

c.

Summary Introduction

To determine: The Sharpe ratio for the optimal portfolio

Introduction: The Sharpe ratio is used to measure the accumulated performance of an aggregate investment portfolio or an individual stock. It evaluates the performance of equity investment to the rate of return.

Answer to Problem 17PS

0.3662 is the Sharpe ratio for the optimal portfolio

Explanation of Solution

Given Information:

Forecast returns, standard deviations and the beta values are given.

Sharpe ratio is mostly used to measure the risk return. For this, first compute the expected return on investment or individual stock and then subtract it from the risk free rate of return. Generally, when the ratio is greater than 1, it is considered as acceptable by the investors.

First to calculate the information ratio for the active portfolio,

  $\begin{array}{l}A=\frac{a}{\sigma (e)}\\ =\frac{-16.90}{147.68}=-0.1144\end{array}$

  $A^2=0.0131$

The Sharpe ratio for the optimal portfolio,

  $\begin{array}{l}{S}^2=S_M{}^2+A^2\\ ={\left(\frac{8}{23}\right)}^2+0.0131=0.1341\end{array}$

  $\Rightarrow S=\sqrt{0.1341}=0.3662$

d.

Summary Introduction

To determine: The position in the active portfolio improve Sharpe ratio with a purely passive index strategy.

Introduction: The Sharpe ratio is used to measure the accumulated performance of an aggregate investment portfolio or an individual stock. It evaluates the performance of equity investment to the rate of return.

Answer to Problem 17PS

The active portfolio improve Sharpe ratio with a purely passive index strategy.

Explanation of Solution

Given Information:

Forecast returns, standard deviations and the beta values are given.

Sharpe ratio is mostly used to measure the risk return. For this, first compute the expected return on investment or individual stock and then subtract it from the risk free rate of return. Generally, when the ratio is greater than 1, it is considered as acceptable by the investors.

Calculation of Beta,

  $\begin{array}{l}{\beta }_P=W_M+(W_A\times {\beta }_A)\\ =1.0486+[(-0.0486)\times 2.08]\\ =0.95\end{array}$

  $\begin{array}{l}TheExcess\mathrm{Re}turnValue,(R_P)=a_P+{\beta }_{P}E(R_M)\\ =[(-0.0486)\times (-16.90\%)]+(0.95\times 8\%)]\\ =8.42\%\end{array}$

The variance,

  ${\sigma }_P{}^2=\$528.94$

As, A = 2.8, the optimal position is calculated as,

  $y=\frac{8.42}{0.01\times 2.8\times 528.94}=0.5685$

If use the passive strategy,

  $y=\frac{8}{0.01\times 2.8\times {23}^2}=0.5401$

The difference in Sharpe ratio between the risky portfolio and the market,

  $=0.5685-0.5401=0.0284$

The performance has been developed.

e.

Summary Introduction

To determine: The complete portfolio with a coefficient of risk of 2.8

Introduction: The Capital Asset Pricing Model explains the relationship in between the systematic risk of an asset and the return that are expected.

Answer to Problem 17PS

The complete portfolio is shown in the table

Explanation of Solution

Given Information:

Forecast returns, standard deviations and the beta values are given.

The capital asset pricing model describes the expected return on beta based security. This model is used for determine the expected return on asset, which is based on systematic risk.

Calculation of Beta,

  $\begin{array}{l}{\beta }_P=W_M+(W_A\times {\beta }_A)\\ =1.0486+[(-0.0486)\times 2.08]\\ =0.95\end{array}$

  $\begin{array}{l}TheExcess\mathrm{Re}turnValue,(R_P)=a_P+{\beta }_{P}E(R_M)\\ =[(-0.0486)\times (-16.90\%)]+(0.95\times 8\%)]\\ =8.42\%\end{array}$

The variance,

  ${\sigma }_P{}^2=\$528.94$

As, A = 2.8, the optimal position is calculated as,

  $y=\frac{8.42}{0.01\times 2.8\times 528.94}=0.5685$

If use the passive strategy,

  $y=\frac{8}{0.01\times 2.8\times {23}^2}=0.5401$

The difference in Sharpe ratio between the risky portfolio and the market,

  $=0.5685-0.5401=0.0284$

BILLS	1-0.5685	43.15%
M	0.5685x0.0486	59.61%
A	0.5685x(-0.0486)x(-0.6142)	1.70%
B	0.5685x(-0.0486)x1.1265	-3.11%
C	0.5685x(-0.0486)x(-1.21181)	3.37%
D	0.5685x(-0.0486)x1.7058	-4.71%
		100.00%