SOLUTION NOTES FOR EXERCISES ON FUNCTIONS
4. Code for Standard Deviation of Cash Flows
8.2 ACTIVE–PASSIVE MANAGEMENT
In practice, many investment managers assume that most shares are fairly valued, but that some are overpriced and some are underpriced. In the light of the CAPM, the extent to which a share is mispriced is measured by its alpha value.
Treynor and Black developed a model for portfolio managers who focus on a portfolio of mispriced shares, thereby assuming that share markets are not entirely efficient. First, they showed how the optimal active portfolio can be blended with the passive market portfolio to create the optimal risky portfolio. Second, they detailed how the optimal risky portfolio can then be combined with the risk-free asset to give the optimal portfolio.
These two steps correspond to combining two risky assets and then a risk-free and a risky asset as outlined in our treatment of the generic portfolio problems (Problem Two and Problem One in sections 6.8 and 6.7 respectively). The Treynor–Black approach uses the Appraisal ratio of performance measurement and underlies the ‘core-satellite’ approach to investment management that is making a comeback.
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A B C D E F G H I
Active-Passive Exercise (Treynor-Black)
Risk-free rate 7.0%
Equity risk premium 8.0%
Equity market risk 20.0%
RRA value (A) 3.0
A: Find Optimal Active Portfolio
Exp Rew Beta Spec Risk Rew / Var Weight
Share 1 1.0% 1.13 13.0% 0.592 98.2%
Share 2 1.5% 1.13 16.0% 0.586 97.2%
Share 3 -2.0% 1.04 25.0% -0.320 -53.1%
Share 4 -0.5% 0.91 14.0% -0.255 -42.3%
Optimal Active Portfolio 3.7% 1.27 24.8% 0.603
Figure 8.2 Treynor– Black’s model in the AP sheet
The example shown in Figure 8.2 contains details of four shares that the portfolio manager thinks are mispriced. For example, share 1 is expected to have a 1.0% excess return in the next year (over and above the return expected under the CAPM). The first task is to create the optimal active portfolio, given the expected rewards, beta and specific risk estimates for the four mispriced shares. Treynor–Black argue that we should maximise the Appraisal ratio for the active portfolio. They show that this is equivalent to having holdings for the individual shares that are proportional to their individual expected reward divided by specific variance. These ratios are in cells G12:G15, their total sum is in cell G17 and hence the weights for the four shares in the optimal active portfolio are calculated in cells I12:I15. For example, share 1 has a weight of 98.2% (DG12/G17) in the optimal active portfolio. Combining the four constituents with these weights, the optimal active portfolio has an expected excess return of 3.7%, a beta of 1.27 and specific risk of 24.8% (assuming that the specific risks of the chosen shares are independent).
The second task is to combine the optimal active portfolio with the passive market portfolio (the latter having an excess return equal to the equity risk premium in cell B5, and total risk equal to the equity market risk in cell B6). Here we can apply the solution to generic portfolio Problem Two (described in section 6.8). This focused on combining two risky assets, but the methodology can be applied equally to combining portfolios. Thus the weight for the active portfolio displayed in cell I23 of Figure 8.3 uses the formula:
=Prob2OptimalRiskyWeight1(B24,E24,B4,B29,E29,B31,B7)
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A B C D E F G H I J
B: Find Optimal Risky Portfolio
Active Passive Solve Prob2 : optimal risky weights
Return above CAPM 3.7% Return above Rf 8.0% w active 32.8%
Return 20.9% 15.0% w passive 67.2%
Beta 1.27 1.00 Optimal risky portfolio
Xs return 9.9%
Spec risk 24.8% 0.0% Return 16.9%
Total risk 35.5% 20.0% Alpha 1.2%
Beta 1.09
corr(a,p) 0.72 Spec risk 8.1%
Total risk 23.3%
Sharpe ratio 0.40 Sharpe ratio 0.43
Figure 8.3 Finding the optimal risky portfolio in the AP sheet
The function has as inputs the return and risk for the two portfolios (B24, E24, B29, E29), the correlation between them (B31), the risk-free rate (B4) and the risk aversion coefficient (B7). It shows that 32.8% of the investment in risky assets should be in the active portfolio. In passing, note that the optimal risky portfolio has a Sharpe ratio greater than that for the passive portfolio, that is it provides more reward for risk.
The final task is to split the total investment between the optimal risky portfolio and the risk-free asset. Again the solution to this problem has been met before, this time generic portfolio Problem One (in section 6.7). The weight for the optimal risky portfolio in Figure 8.4, cell B41, uses the formula:
=Prob1OptimalRiskyWeight(I28,B4,I32,B7)
Performance Measurement and Attribution 143 where the function inputs are the return and risk on the optimal risky portfolio (I28, I32), the risk-free rate (B4) and the investor’s risk aversion coefficient (B7). Here the proportion invested in risky assets is under two-thirds (61.2%).
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A B C D E F G H I J
C: Find Optimal Portfolio
Solve Prob1 : split between risk-free and risky
Risk-free 38.8%
Risky 61.2%
Optimal portfolio
Risk-free 38.8% Xs return 6.1%
Share 1 19.7% Return 13.1%
Share 2 19.5% Alpha 0.7%
Share 3 -10.7% Beta 0.67
Share 4 -8.5% Spec risk 5.0%
Passive 41.1% Total risk 14.2%
Figure 8.4 Deciding the risky – risk-free proportions in the Treynor– Black model, AP sheet
The active and passive portfolios are illustrated on the risk–return chart in Figure 8.5.
Blending the active and passive portfolios produces the portfolios on the frontier between them. The optimal risky portfolio occurs where the straight line from the risk-free asset (7% return) is tangential to this frontier. The risk aversion coefficient influences the position on the line of the final optimal portfolio, which in our example includes 38.8%
of risk-free investment.
Active−Passive Exercise
Active
Passive
Risk-Free
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0%
Portfolio risk
Portfolio return
Problem2 Problem1
Figure 8.5 Chart sheet for Treynor– Black’s model in the EQUITY3 workbook