Investments, tenth edition

  

  S&P 500  

  Active Pf A  

  

  HP  

  Dell  

  WMT  

  Target  

  BP  

  Shell  

  s  

2

 ( e ) 

 0.0705  

0.0572  

0.0309  

0.0392  

0.0297  

0.0317 

  

  

 25.7562 

  a / s  

2

 ( e ) 

 2.0855  

3.0641  

6.2544  

7.1701  

6.0566  

1.1255 

  

  

  1.0000 

 

 w  

0

 ( i )  

0.0810  

0.1190  

0.2428  

0.2784  

0.2352  

0.0437 

  

  

  

 [ w  

0

 ( i )] 

2

  

 0.0066  

0.0142  

0.0590  

0.0775  

0.0553  

0.0019 

  a  

 A 

  

  

  0.2018 

 

 

  

  

  

  

  

  

  s  

2

 ( e  

 A 

 ) 

  

  0.0078 

 

 

  

  

  

  

  

  

  w  

0

  

  

  7.9116 

 

 

  

  

  

  

  

  

  w  *  

 0.5661  

 0.4339 

 

 

 0.0351  

0.0516  

0.1054  

0.1208  

0.1020  

0.0190 

  

  

  Overall 

Portfolio  

  

  

  

  

  

  

 Beta  

1  

 0.9538 

 

0.9800  

0.0351  

0.0516  

0.1054  

0.1208  

0.1020  

0.0190 

 Risk premium 

 0.06  

 0.2590 

 

0.1464  

0.0750  

0.1121  

0.0689  

0.0447  

0.0880  

0.0305 

 Standard 

deviation 

 0.1358  

 0.1568 

 

0.1385  

0.3817  

0.2901  

0.1935  

0.2611  

0.1822  

0.1988 

 Sharpe ratio 

 0.44  

 1.65 

 

1.0569  

 

  

  

  

  

  

  M -square  

0  

 0.1642 

 

0.0835  

 

  

  

  

  

  

 Benchmark risk 

  

  

 0.0385  

 

  

  

  

  

  

 Table 27.5 

 The optimal risky portfolio with the analysts’ new forecasts (benchmark risk constrained to 3.85%) 

3

 In application to debates about social issues, you might define a fanatic as one who enters the debate with a prior 

that is so tight that no argument will influence his posterior, making the debate altogether a waste of time. 

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  C H A P T E R  

2 7

  The Theory of Active Portfolio Management 

959

   Adjusting Forecasts for the Precision of Alpha 

 Imagine that you have just downloaded from Yahoo! Finance the analysts’ forecasts we 

used in the previous section, implying that Target’s alpha is 28.1%. Should you conclude 

that the optimal position in Target, before adjusting for beta, is  a / s  

2

 ( e )  5  .281/.198 

2

   5  

7.17  (717%)? Naturally, before committing to such an extreme position, any reasonable 

manager would first ask: “How accurate is this forecast?” and “How should I adjust my 

position to take account of forecast imprecision?” 

 Treynor and Black  

4

   asked this question and supplied an answer. The logic of the answer 

is quite straightforward; you must quantify the uncertainty about this forecast, just as you 

would the risk of the underlying asset or portfolio. A Web surfer may not have a way to 

assess the precision of a downloaded forecast, but the employer of the analyst who issued 

the forecast does. How? By examining the    forecasting  record    of previous forecasts issued 

by the same forecaster.  

 

Suppose that a security analyst provides the portfolio manager with forecasts of 

alpha at regular intervals, say, the beginning of each month. The investor portfolio is 

updated using the forecast and held until the update of next month’s forecast. At the end 

of each month,  T,  the realized abnormal return of Target’s stock is the sum of alpha plus 

a residual:

 

   u(T)

5 R

TGT

(T)

2 b

TGT

R

M

(T)

5 a(T) 1 e(T) 

 (27.4)   

 where beta is estimated from Target’s security characteristic line (SCL) using data for peri-

ods prior to  T, 

 

   SCL:        R

TGT

(t)

5 a 1 b

TGT

R

M

(t)

1 e(t),        t , T 

 (27.5)   

 The 1-month, forward-looking forecast  a  

 f 

 ( T  ) issued by the analyst at the beginning of 

month  T  is aimed at the abnormal return,  u ( T  ), in Equation 27.4. In order to decide how to 

use the forecast for month  T,  the portfolio manager uses the analyst’s forecasting record. 

The analyst’s record is the paired time series of all past forecasts,  a  

 f 

 ( t ), and realizations, 

 u ( t ). To assess forecast accuracy, that is, the relationship between forecast and realized 

alphas, the manager uses this record to estimate the regression:

 

   u(t)

5 a

0

1 a

1

a

f

(t)

1 e(t) 

 (27.6)   

 Our goal is to adjust alpha forecasts to properly account for their imprecision. We will 

form an    adjusted  alpha     forecast   a ( T  ) for the coming month by using the original fore-

casts  a  

 f 

 ( T  ) and applying the estimates from the regression Equation 27.6, that is,

 

   a(T)

5 a

0

1 a

1

a

f

(T) 

 (27.7)   

 The properties of the regression estimates assure us that the adjusted forecast is the 

“best linear unbiased estimator” of the abnormal return on Target in the coming month,  T.  

 “Best” in this context means it has the lowest possible variance among unbiased fore-

casts that are linear functions of the original forecast. We show in Appendix A that the 

value we should use for  a  

1

  in Equation 27.7 is the  R -square of the regression Equation 

27.6. Because  R -square is less than 1, this implies that we “shrink” the forecast toward 

zero. The lower the precision of the original forecast (the lower its  R -square), the more 

we shrink the adjusted alpha back toward zero. The coefficient  a  

0

  adjusts the forecast 

upward if the forecaster has been consistently pessimistic, and downward for consistent 

optimism.  

  

4

 Jack Treynor and Fischer Black, “How to Use Security Analysis to Improve Portfolio Selection,”  Journal of 

Business,  January 1973. 

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960

P A R T   V I I

  Applied Portfolio Management