ESSENTIALS OF INVESTMENTS SELECT CHAPT
ESSENTIALS OF INVESTMENTS SELECT CHAPT
17th Edition
ISBN: 9781307126228
Author: Bodie
Publisher: MCG/CREATE
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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.

Expert Solution
Check Mark

Explanation of Solution

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

    AssetExpected returnBetaResidual 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:

    AssetExpected returnStandard deviation
    T-bill
    8%
    0%
    Passive equity portfolio
    16%
    23%

Formula for alpha value is as follows:

αi=RiRF+βiRMRF

Here,

αi=AlphavalueRi=ExpectedreturnofStockRF=ExpectedreturnofT-billβi=BetaofstockRM=Expectedreturnofpassiveportfolio

Formula for expected excess return is as follows:

Expectedexcessreturn=Ri-RF

Here,

Ri=ExpectedreturnofStockRF=ExpectedreturnofT-bill

Calculate the expected returns, alpha value and residual values.

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

    StockAlphaααi=RiRF+βiRMRFExpected excess returnRiRF

    A

    αA=20%8%+1.316%8%=20%8%+10.4%=20%18.4%=1.6%20%8%=12%

    B

    αB=18%8%+1.816%8%=18%8%+14.4%=18%22.4%=4.4%18%8%=10%

    C

    αC=17%8%+0.716%8%=17%8%+5.6%=17%13.6%=3.4%17%8%=9%

    D

    αD=12%8%+1.016%8%=12%8%+8%=12%16%=4.0%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:

    StocksResidual Variance

      Variance=σ2

    A

    σA2=582=3,364

    B

    σB2=712=5,041

    C

    σC2=602=3,600

    D

    σD2=552=3,024
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.

Expert Solution
Check Mark

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:

Weightofeachstock=ασ2ασ2

Here,

α is alpha value

α2 is variance

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

    Stocksασ2Weightofeachstock=ασ2ασ2
    A
    ασ2=0.01633.64=0.000476

      =0.0004760.000775=0.6142

    B

      ασ2=0.04450.41=0.000873

      =0.0008730.000775=1.1265

    C
    ασ2=0.03436.00=0.000944

      =0.0009440.000775=1.2181

    D
    ασ2=0.04030.25=0.001322

      =0.0013220.000775=1.7058

    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:

αP=wA×αA+wB×αC+wD×αD

Here,

wA,wB,wCandwD are the weights of the stocks A, B, C and D respectively andαA,αB,αCandαD are the alphas of the stocks, A, B, C and D respectively.

Substitute the required values from the tables above as follows:

αP=wA×αA+wB×αC+wD×αD=0.6142×1.6+1.1265×4.4+1.2181×3.4+1.7058×4.0=0.98272+4.95660+4.14154+6.8232=16.90%

Thus, the alpha of this active portfolio is16.90% .

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

β=wA×βA+wB×βC+wD×βD

Substitute the required values from the tables above as follows:

β=wA×βA+wB×βC+wD×βD=0.6142×1.3+1.1265×1.8+1.2181×0.70+1.7058×1=0.79846+2.02770+0.85267+1.70580=2.08

Thus, the beta of this active portfolio is2.08 .

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

Varianceσ2=wA2×σA2+wB2×σB2+wC2×σC2+wD2×σD2

Substitute the required values from the tables above as follows:

Varianceσ2=wA2×σA2+wB2×σB2+wC2×σC2+wD2×σD2=0.61422×3,364+1.12652×5,041+1.21812×3,600+1.70582×3,025=1,269.04+6,397.04+5,341.56+8,802.00=21,809.64

Thus, the variance of the active portfolio is21,809.64 .

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

Standard Deviation=Variance=21809.6=147.68%

Thus, the standard deviation of this portfolio is147.68% .

Conclusion

The standard deviation of this portfolio is147.68% .

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:

Expert Solution
Check Mark

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%.

A2=ασ2=16.90147.652=0.11442=0.0131

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

S2=SM2+A2=ExpectedreturnofTbillsStandarddeviationofpassiveequityportfolio2+A2=8232+0.0131=0.1341

The Sharpe measure of the optimized portfolio is as follows:

SP=S2=0.1341=0.3662

The Sharpe measure of the market is as follows:

  SM=ExpectedreturnofTbillsStandarddeviationofpassiveequityportfolio=823=0.3478

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

Difference=SharpemeasureoftheoptimizedportfolioSharpemeasureofthemarket=0.36620.3478=0.0184

Modigliani-squared measure, that is,M2 can be calculated using the following formula:

M2=ErPErM

Here,

ErP=rf+SPSMrf=EcpectedreturnofTbillsSP=SharpemeasureoftheoptimizedportfolioSM=Standard DeviationofpassiveequityportfolioErM=Expectedreturnofpassiveequityportfolio

CalculateErP as follows:

ErP=rf+SPSM=8%+0.3662×23%=8%+8.4226%=16.423%

Modigliani-squared measure, that is,M2 can be calculated as follows:

M2=ErPErM=16.423%16%=0.423%

Hence, the value ofM2 is0.423% .

Conclusion

Hence, the amount contributed by the active portfolio is16.423% and the value ofM2 is0.423%

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