Investments, tenth edition

 The Theory of Active 

Portfolio Management 

 CHAPTER TWENTY-SEVEN 

     27.1 

Optimal Portfolios and Alpha Values 

  In Chapter 8 we showed how to form an optimal risky portfolio with a single-index model. 

 Table 27.1  summarizes the steps in this optimization, commonly known as the Treynor-

Black model.  

1

   The outlined procedure uses the index model that ignores nonzero cova-

riance values across residuals. This is sometimes called the  diagonal model,  because it 

assumes that the covariance matrix of residuals has nonzero entries only on the diagonals. 

Moreover, we saw that despite significant correlation between some pairs of residuals in 

the portfolio construction example we used in Chapter 8, for example, between Shell and 

BP, the efficient frontiers formed from the index model and the Markowitz model were 

barely distinguishable (see Figure 8.5 of Chapter 8).   

1

 We know from Chapter 10 that a multiple-index model such as that of Fama and French may better describe 

security returns. In that case, the passive market-index portfolio will be augmented with positions in the addi-

tional factor portfolios (for example, the size and value portfolios in the FF model). However, the rest of the 

Treynor-Black procedure will remain unchanged. 

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  Applied Portfolio Management

 For illustration, in this chapter we continue with the example employed in Chapter 8. 

 Spreadsheet 27.1  recaps the data and results of this exercise. Table D in the spreadsheet 

shows the improvement in the Sharpe ratio over the    passive  market-index  portfolio  

offered by adding the    active  portfolio    to the mix. To better appreciate this improvement 

we have included the  M -square measure of performance.  M -square is the incremental 

expected return of the optimized portfolio compared to the passive alternative once the 

active portfolio is mixed with bills to provide the same total volatility as the index portfolio 

(for a review, see Chapter 24).