Chapter 18, Problem 4CP

Summary Introduction

(A)

To compute:

Calculate expected excess returns, alpha values, and residual variances for these stocks.

Introduction:

Calculate the Asset, expected return, beta values and residual variances using the table and use the formula to determine the final values.

Explanation of Solution

The table summarizing the micro forecasts for different stocks is as follows:

Asset	Expected return	Beta	Residual standard deviation
Stock A	20%	1.3	58%
Stock B	18%	1.8	71%
Stock C	17%	0.7	60%
Stock D	12%	1.0	55%

The table summarizing the macro forecasts is as follows:

Asset	Expected return	Standard deviation
T-bill	8%	0%
Passive equity portfolio	16%	23%

Formula for alpha value is as follows:

${\alpha }_i=R_i-\left[R_F+{\beta }_i\left(R_M-R_F\right)\right]$

Here,

$\begin{array}{l}{\text{α}}_{\text{i}}\text{=Alpha}\text{value}\\ {\text{R}}_{\text{i}}\text{=Expected}\text{return}\text{of}\text{Stock}\\ {\text{R}}_{\text{F}}\text{=Expected}\text{return}\text{of}\text{T-bill}\\ {\text{β}}_{\text{i}}\text{=Beta}\text{of}\text{stock}\\ {\text{R}}_{\text{M}}\text{=Expected}\text{return}\text{of}\text{passive}\text{portfolio}\end{array}$

Formula for expected excess return is as follows:

$\text{Expected}\text{excess}{\text{return=R}}_{\text{i}}{\text{-R}}_{\text{F}}$

Here,

$\begin{array}{l}{\text{R}}_{\text{i}}\text{=Expected}\text{return}\text{of}\text{Stock}\\ {\text{R}}_{\text{F}}\text{=Expected}\text{return}\text{of}\text{T-bill}\end{array}$

Calculate the expected returns, alpha value and residual values.

The table shows the calculation of expected excess returns, alpha of different stocks as follows:

Stock	Alpha$\left(\alpha \right){\alpha }_i=R_i-\left[R_F+{\beta }_i\left(R_M-R_F\right)\right]$	Expected excess return$R_i-R_F$
A	$\begin{array}{l}{\alpha }_A=20\%-\left[8\%+1.3\left(16\%-8\%\right)\right]\\ =20\%-\left[8\%+10.4\%\right]\\ =20\%-18.4\%\\ =1.6\%\end{array}$	$20\%-8\%=12\%$
B	$\begin{array}{l}{\alpha }_B=18\%-\left[8\%+1.8\left(16\%-8\%\right)\right]\\ =18\%-\left[8\%+14.4\%\right]\\ =18\%-22.4\%\\ =-4.4\%\end{array}$	$18\%-8\%=10\%$
C	$\begin{array}{l}{\alpha }_C=17\%-\left[8\%+0.7\left(16\%-8\%\right)\right]\\ =17\%-\left[8\%+5.6\%\right]\\ =17\%-13.6\%\\ =3.4\%\end{array}$	$17\%-8\%=9\%$
D	$\begin{array}{l}{\alpha }_D=12\%-\left[8\%+1.0\left(16\%-8\%\right)\right]\\ =12\%-\left[8\%+8\%\right]\\ =12\%-16\%\\ =-4.0\%\end{array}$	$12\%-8\%=4\%$

From the calculation, it can be seen that stocks C and A have positive alphas and the other two stocks, that is, B and D have negative alphas.

The variances of the stocks can be calculated by squaring the standard deviation

The table summarizing the calculation of variances of the stocks is as follows:

Stocks	Residual Variance   $Variance={\sigma }^2$
A	$\begin{array}{l}{\sigma }_A^2={\left(58\right)}^2\\ =3,364\end{array}$
B	$\begin{array}{l}{\sigma }_B^2={\left(71\right)}^2\\ =5,041\end{array}$
C	$\begin{array}{l}{\sigma }_C^2={\left(60\right)}^2\\ =3,600\end{array}$
D	$\begin{array}{l}{\sigma }_D^2={\left(55\right)}^2\\ =3,024\end{array}$

Conclusion

Thus, the values are found above in the table.

Summary Introduction

(B)

To compute:

Construct the optimal risky portfolio.

Introduction:

To construct the optimal risky portfolio, we need to determine optimal active portfolio and substituting the values to the table to get the standard deviation value.

Explanation of Solution

To build an optimal portfolio, firstly optimal active portfolio needs to be determined.

Treynor-Black technique weight of each stock formula is as follows:

$\text{Weight}\text{of}\text{each}\text{stock=}\frac{\frac{\text{α}}{{\text{σ}}^{\text{2}}}}{{\displaystyle \sum \frac{\text{α}}{{\text{σ}}^{\text{2}}}}}$

Here,

$\alpha$ is alpha value

${\alpha }^2$ is variance

The following table shows the construction of active portfolio using the Treynor-Black technique:

Stocks	$\frac{\alpha }{{\sigma }^2}$	$Weightofeachstock=\frac{\frac{\alpha }{{\sigma }^2}}{{\displaystyle \sum \frac{\alpha }{{\sigma }^2}}}$
A	$\begin{array}{l}\frac{\alpha }{{\sigma }^2}=\frac{0.016}{33.64}\\ =0.000476\end{array}$	$\begin{array}{l}=\frac{0.000476}{-0.000775}\\ =-0.6142\end{array}$
B	$\begin{array}{l}\frac{\alpha }{{\sigma }^2}=\frac{-0.044}{50.41}\\ =-0.000873\end{array}$	$\begin{array}{l}=\frac{-0.000873}{-0.000775}\\ =1.1265\end{array}$
C	$\begin{array}{l}\frac{\alpha }{{\sigma }^2}=\frac{0.034}{36.00}\\ =0.000944\end{array}$	$\begin{array}{l}=\frac{0.000944}{-0.000775}\\ =-1.2181\end{array}$
D	$\begin{array}{l}\frac{\alpha }{{\sigma }^2}=\frac{-0.040}{30.25}\\ =-0.001322\end{array}$	$\begin{array}{l}=\frac{-0.001322}{-0.000775}\\ =1.7058\end{array}$
Total	-0.000775	1.0000

From the weights calculated in the above table, the forecast of the active portfolio can be made and the following formula is used to calculate the alpha of this portfolio:

${\alpha }_P=w_A\times {\alpha }_A+w_B\times {\alpha }_C+w_D\times {\alpha }_D$

Here,

$w_A,w_B,w_{C}and{w}_D$ are the weights of the stocks A, B, C and D respectively and${\alpha }_A,{\alpha }_B,{\alpha }_{C}and{\alpha }_D$ are the alphas of the stocks, A, B, C and D respectively.

Substitute the required values from the tables above as follows:

$\begin{array}{l}{\alpha }_P=w_A\times {\alpha }_A+w_B\times {\alpha }_C+w_D\times {\alpha }_D\\ =\left[-0.6142\times 1.6\right]+\left[1.1265\times \left(-4.4\right)\right]+\left[-1.2181\times 3.4\right]+\left[1.7058\times \left(-4.0\right)\right]\\ =\left[-0.98272\right]+\left[-4.95660\right]+\left[-4.14154\right]+\left[-6.8232\right]\\ =-16.90\%\end{array}$

Thus, the alpha of this active portfolio is$-16.90\%$ .

The beta of this active portfolio can be calculated using the following formula:

$\beta =w_A\times {\beta }_A+w_B\times {\beta }_C+w_D\times {\beta }_D$

Substitute the required values from the tables above as follows:

$\begin{array}{l}\beta =w_A\times {\beta }_A+w_B\times {\beta }_C+w_D\times {\beta }_D\\ =\left[-0.6142\times 1.3\right]+\left[1.1265\times 1.8\right]+\left[-1.2181\times 0.70\right]+\left[1.7058\times 1\right]\\ =\left[-0.79846\right]+\left[2.02770\right]+\left[-0.85267\right]+\left[1.70580\right]\\ =2.08\end{array}$

Thus, the beta of this active portfolio is$2.08$ .

The variance and standard deviation of the portfolio can be calculated using the following formula:

$Variance\left({\sigma }^2\right)=w_A^2\times {\sigma }_A^2+w_B^2\times {\sigma }_B^2+w_C^2\times {\sigma }_C^2+w_D^2\times {\sigma }_D^2$

Substitute the required values from the tables above as follows:

$\begin{array}{l}Variance\left({\sigma }^2\right)=w_A^2\times {\sigma }_A^2+w_B^2\times {\sigma }_B^2+w_C^2\times {\sigma }_C^2+w_D^2\times {\sigma }_D^2\\ =\left[{\left(-0.6142\right)}^2\times 3,364\right]+\left[{1.1265}^2\times 5,041\right]+\\ \left[{\left(-1.2181\right)}^2\times 3,600\right]+\left[{1.7058}^2\times 3,025\right]\\ =1,269.04+6,397.04+5,341.56+8,802.00\\ =21,809.64\end{array}$

Thus, the variance of the active portfolio is$21,809.64$ .

The standard deviation is the square root of the variance and is thus calculate as follows:

Standard Deviation$\begin{array}{l}=\sqrt{Variance}\\ =\sqrt{21809.6}\\ =147.68\%\end{array}$

Thus, the standard deviation of this portfolio is$147.68\%$ .

Conclusion

The standard deviation of this portfolio is$147.68\%$ .

Summary Introduction

(C)

To compute:

What is Sharpe's measure for the optimal portfolio and how much of it is contributed by the active portfolio? What is the M2?Introduction:

Explanation of Solution

To evaluate the Sharpe measure of this active portfolio, the appraisal ratio is calculated and the Sharpe measure for the portfolio of market.

Active portfolio's appraisal ratio is calculated using the following equation where alpha of the portfolio is-16.90% and standard deviation is 147.68%.

$\begin{array}{l}{A}^2={\left(\frac{\alpha }{\sigma }\right)}^2\\ ={\left(\frac{-16.90}{147.65}\right)}^2\\ ={\left(-0.1144\right)}^2\\ =0.0131\end{array}$

The square of the Sharpe measure of the optimized portfolio is as follows:

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

The Sharpe measure of the optimized portfolio is as follows:

$\begin{array}{l}{S}_P=\sqrt{S^2}\\ =\sqrt{0.1341}\\ =0.3662\end{array}$

The Sharpe measure of the market is as follows:

  $\begin{array}{l}{S}_M=\frac{ExpectedreturnofT-bills}{Standarddeviationofpassiveequityportfolio}\\ =\frac{8}{23}\\ =0.3478\end{array}$

There is difference between the Sharpe measure of optimized portfolio and Sharpe measure of market as follows:

$\begin{array}{l}Difference=Sharpemeasureoftheoptimizedportfolio-\\ Sharpemeasureofthemarket\\ =0.3662-0.3478\\ =0.0184\end{array}$

Modigliani-squared measure, that is,$M^2$ can be calculated using the following formula:

$M^2=E\left(r_P^{•}\right)-E\left(r_M\right)$

Here,

$\begin{array}{l}E\left(r_P^{•}\right)=r_f+S_P{S}_M\\ r_f=EcpectedreturnofT-bills\\ S_P=\text{}Sharpemeasureoftheoptimizedportfolio\\ S_M=\text{}StandardDeviationofpassiveequityportfolio\\ E\left(r_M\right)=Expectedreturnofpassiveequityportfolio\end{array}$

Calculate$E\left(r_P^{•}\right)$ as follows:

$\begin{array}{l}E\left(r_P^{•}\right)=r_f+S_P{S}_M\\ =8\%+\left(0.3662\times 23\%\right)\\ =8\%+8.4226\%\\ =16.423\%\end{array}$

Modigliani-squared measure, that is,$M^2$ can be calculated as follows:

$\begin{array}{l}{M}^2=E\left(r_P^{•}\right)-E\left(r_M\right)\\ =16.423\%-16\%\\ =0.423\%\end{array}$

Hence, the value of$M^2$ is$0.423\%$ .

Conclusion

Hence, the amount contributed by the active portfolio is$16.423\%$ and the value of$M^2$ is$0.423\%$