Investments, tenth edition

  The Optimal Risky Portfolio   

Panel 5 of  Spreadsheet 8.1  displays calculations for 

the optimal risky portfolio. They follow the summary procedure of Section 8.4 (you should 

try to replicate these calculations in your own spreadsheet). In this example we allow short 

sales. Notice that the weight of each security in the active portfolio (see row 52) has the 

same sign as the alpha value. Allowing short sales, the positions in the active portfolio are 

quite large (e.g., the position in BP is .7349); this is an aggressive portfolio. As a result, 

the alpha of the active portfolio (2.22%) is larger than that of any of the individual alpha 

forecasts. However, this aggressive stance also results in a large residual variance (.0404, 

which corresponds to a residual standard deviation of 20%). Therefore, the position in the 

active portfolio is scaled down (see Equation 8.20) and ends up quite modest (.1718; cell 

C57), reinforcing the notion that diversification considerations are paramount in the opti-

mal risky portfolio. 

 The optimal risky portfolio has a risk premium of 6.48%, standard deviation of 14.22%, 

and a Sharpe ratio of .46 (cells J58–J61). By comparison, the Sharpe ratio of the index 

portfolio is .06/.1358  5  .44 (cell B61), which is quite close to that of the optimal risky 

portfolio. The small improvement is a result of the modest alpha forecasts that we used. 

In Chapter 11 on market efficiency and Chapter 24 on performance evaluation we demon-

strate that such results are common in the mutual fund industry. Of course, a few portfolio 

managers can and do produce portfolios with better performance. 

 The interesting question here is the extent to which the index model produces results 

that are inferior to that of the full-covariance (Markowitz) model.  Figure  8.5  shows the 

efficient frontiers from the two models with the example data. We find that the difference is 

in fact small.  Table 8.2  compares the compositions and expected performance of the global 

minimum variance ( G ) and the optimal risky portfolios derived from the two models.   

 The standard deviations of efficient portfolios produced from the Markowitz model and 

the index model are calculated from the covariance matrixes used in each model. As dis-

cussed earlier, we cannot be sure that the covariance estimates from the full covariance 

model are more accurate than those from 

the more restrictive single-index model. 

However, by assuming the full covariance 

model to be more accurate, we get an idea 

of how far off the two models can be. 

 

 

Figure  8.5 

 shows that for conservative 

portfolios (closer to the minimum-variance 

portfolio  

G), 

 the index model underesti-

mates the volatility and hence overestimates 

performance. The reverse happens with 

portfolios that are riskier than the index, 

which also include the region near the opti-

mal portfolio. Despite these differences, 

what stands out from this comparison is 

that the outputs of the two models are in 

fact extremely similar, with the index model 

perhaps calling for a more conservative 

position. This is where we would like to be 

with a model relying on approximations.