Investments, tenth edition

  Stock  

  HP  

  Dell  

  WMT  

  Target  

  BP  

  Shell  

 Current price 

 32.15  

25.39  

48.14  

49.01  

70.8  

68.7 

 Target price 

 36.88  

29.84  

57.44  

62.8  

83.52  

71.15 

 Implied alpha 

  0.1471   

 0.1753   

 0.1932   

 0.2814   

 0.1797   

 0.0357 

 Table 27.2 

 Stock prices and analysts’ 

target prices 

 Figure 27.1 

Rates of return on the S&P 500 (GSPC) and the six stocks  

+60

HP

BP

RDS-B

GSPC

WMT

TGT

DELL

+40

+20

−20

−40

−60

July

September

Rate of Return (%)

November

January

March

May

0

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  C H A P T E R  

2 7

  The Theory of Active Portfolio Management

955

  

  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  *  

 0.0000  

 1.0000 

 

 

 0.0810  

0.1190  

0.2428  

0.2784  

0.2352  

0.0437 

  

  

  

  Overall 

Portfolio  

  

  

  

  

  

  

 Beta  

1  

 0.9538 

 

0.9538  

0.0810  

0.1190  

0.2428  

0.2784  

0.2352  

0.0437 

 Risk premium 

 0.06  

 0.2590 

 

0.2590  

0.2692  

0.2492  

0.2304  

0.3574  

0.2077  

0.0761 

 SD  

0.1358  

 0.1568 

 

0.1568  

0.3817  

0.2901  

0.1935  

0.2611  

0.1822  

0.1988 

 Sharpe ratio 

 0.44  

 1.65 

 

1.6515  

 

  

  

  

  

  

  M -square  

0  

 0.1642 

 

0.1642  

 

  

  

  

  

  

 Benchmark risk 

  

  

 0.0887  

 

  

  

  

  

  

 Table 27.4 

 The optimal risky portfolio with constraint on the active portfolio ( w  

 A 

  # 1) 

 Table 27.3 

 The optimal risky portfolio with the analysts’ new forecasts 

  

  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  *  

  2 4.7937    5.7937  

 

 0.4691163  

0.6892459  

1.4069035  

1.6128803  

1.3624061  

0.2531855 

  

  

  

  Overall 

Portfolio  

  

  

  

  

  

  

 Beta  

 1 

 

 0.9538   

0.7323  

 

 

0.4691  

 

 

0.6892  

 

 

1.4069  

 

 

1.6129  

 

 

1.3624  

 

 

0.2532 

 Risk premium 

  0.06 

 

 0.2590   

1.2132  

 

 

0.2692  

 

 

0.2492  

 

 

0.2304  

 

 

0.3574  

 

 

0.2077  

 

 

0.0761 

 SD  

 0.1358   

 0.1568   

0.5224  

 

 

0.3817  

 

 

0.2901  

 

 

0.1935  

 

 

0.2611  

 

 

0.1822  

 

 

0.1988 

 Sharpe ratio 

  0.44 

 

 1.65 

  

2.3223  

 

  

  

  

  

  

  M -square  

 0 

 

 0.1642   

0.2553  

 

  

  

  

  

  

 Benchmark risk    

  

   0.5146  

 

  

  

  

  

  

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956 

P A R T   V I I

  Applied Portfolio Management

 Is this a satisfactory solution? This would depend on the organization. For hedge funds, 

this may be a dream portfolio. For most mutual funds, however, the lack of diversifica-

tion would rule it out. Notice the positions in the six stocks: Walmart, Target, and British 

Petroleum alone account for 76% of the portfolio. 

 Here we have to acknowledge the limitations of our example. Surely, when the invest-

ment company covers more securities, the problem of lack of diversification would largely 

vanish. But it turns out that the problem with extreme long/short positions typically per-

sists even when we consider a larger number of firms, and this can gut the practical value 

of the optimization model. Consider this conclusion from an important article by Black 

and Litterman  

2

   (whose model we will present in Section 27.3):  

  the mean-variance optimization used in standard asset allocation models is extremely 

sensitive to expected return assumptions the investor must provide . . . The optimal port-

folio, given its sensitivity to the expected returns, often appears to bear little or no rela-

tion to the views the investor wishes to express. In practice, therefore, despite obvious 

conceptual attractions of a quantitative approach, few global investment managers regu-

larly allow quantitative models to play a major role in their asset allocation decisions.  

 This statement is more complex than it reads at first blush, and we will analyze it in 

depth in Section 27.3. We bring it up in this section, however, to point out the general con-

clusion that “few global investment managers regularly allow quantitative models to play 

a major role in their asset allocation decisions.” In fact, this statement also applies to many 

portfolio managers who avoid the mean-variance optimization process altogether for other 

reasons. We return to this issue in Section 27.4.  

  Restriction of Benchmark Risk 

 Black and Litterman point out a related important practical issue. Many investment man-

agers are judged against the performance of a    benchmark,    and a benchmark index is pro-

vided in the mutual fund prospectus. Implied in our analysis so far is that the passive 

portfolio, the S&P 500, is that benchmark. Such commitment raises the importance of 

   tracking  error.    Tracking error is estimated from the time series of differences between 

the returns on the overall risky portfolio and the benchmark return, that is,  T  

 E 

   5   R  

 P 

   2   R  

 M 

 . 

The portfolio manager must be mindful of benchmark risk, that is, the standard deviation 

of the tracking error. 

 The tracking error of the optimized risky portfolio can be expressed in terms of the beta 

of the portfolio and thus reveals the benchmark risk:

   Tracking error

5 T

E

5 R

P

2 R

M

R

P

5 w

*

A

a

A

1 31 2 w

*

A

(1

2 b

A

)

4R

M

1 w

*

A

e

A

T

E

5 w

*

A

a

A

2 w

*

A

(1

2 b

A

)R

M

1 w

*

A

e

A

Var (T

E

)

5 3w

*

A

(1

2 b

A

)

4

2

 Var (R

M

)

1 Var (w

*

A

e

A

)

5 3w

*

A

(1

2 b

A

)

4

2

s

M

2

1 3w

*

A

s(e

A

)

4

2

Benchmark risk

5 s(T

E

)

5 w

*

A

"(1 2 b

A

)

2

s

M

2

1 3s(e

A

)

4

2

  

(27.1)

   

 Equation 27.1 shows us how to calculate the volatility of tracking error and how to set the 

position in the active portfolio,    w

*

A

,  to restrict tracking risk to any desired level. For a unit 

investment in the active portfolio, that is, for    w

*

A

5 1,  benchmark risk is

 

   s(T

E

; w

*

A

5 1) 5 "(1 2 b

A

)

2

s

M

2

1 3s(e

A

)

4

2

  

(27.2)   

  

2

 Fischer Black and Robert Litterman, “Global Portfolio Optimization,”  Financial Analysts Journal,   September/

October 1992. © 1992, CFA Institute. Reprinted with permission from the CFA Institute. 

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  C H A P T E R  

2 7

  The Theory of Active Portfolio Management

957

 For a desired benchmark risk of  s

0

  ( T

E

 ) we would restrict the weight of the active port-

folio to

w

A

(T

E

)

5

s

0

(T

E

)

s (T

E

; w

*

A

5 1)

 

 (27.3)   

 Obviously, introducing a constraint on tracking risk entails a cost. We must shift weight 

from the active to the passive portfolio.  Figure 27.2  illustrates the cost. The portfolio opti-

mization would lead us to portfolio  T,  the tangency of the capital allocation line (CAL), 

which is the ray from the risk-free rate to the efficient frontier formed from  A  and  M.

Reducing risk by shifting weight from  T  to  M  takes us down the efficient frontier, instead 

of along the CAL, to a lower risk position, reducing the Sharpe ratio and  M -square of the 

constrained portfolio.  

 Notice that the standard deviation of tracking error using the “meager” alpha fore-

casts in  Spreadsheet 27.1  is only 3.46% because the weight in the active portfolio is only 

17%. Using the larger alphas based on analysts’ forecasts with no restriction on portfolio 

weights, the standard deviation of tracking error is 51.46% (see  Table  27.3 ), more than 

any real-life manager who is evaluated against a benchmark would be willing to bear. 

However, with weight of 1.0 on the active portfolio, the benchmark risk falls to 8.87% 

( Table 27.4 ). 

 Finally, suppose a manager wishes to restrict benchmark risk to the same level as it was 

using the original forecasts, that is, to 3.46%. Equations 27.2 and 27.3 instruct us to invest 

W

A

5  .43 in the active portfolio. We obtain the results in  Table 27.5 . This portfolio is mod-

erate, yet superior in performance: (1) its standard deviation is only slightly higher than 

that of the passive portfolio, 13.85%; (2) its beta is .98; (3) the standard deviation of track-

ing error that we specified is extremely low, 3.85%; (4) given that we have only six securi-

ties, the largest position of 12% (in Target) is quite low and would be lower still if more 

securities were covered; yet (5) the Sharpe ratio is a whopping 1.06, and the  M -square  is 

an impressive 8.35%. Thus, by controlling benchmark risk we can avoid the flaws of the 

unconstrained portfolio and still maintain superior performance.     

 Figure 27.2 

Reduced efficiency when benchmark risk is lowered  

M

CAL

T

A

5

6

7

8

9

10

11

12

13

12

14

16

18

20

22

24

26

Pf Standard Deviation

Portfolio Mean

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958

P A R T   V I I

  Applied Portfolio Management

    27.2 

The Treynor-Black Model and Forecast Precision 

  Suppose the risky portfolio of your 401(k) retirement fund is currently in an S&P 500 index 

fund, and you are pondering whether you should take some extra risk and allocate some 

funds to Target’s stock, the high-performing discounter. You know that, absent research 

analysis, you should assume the alpha of any stock is zero. Hence, the mean of your    prior 

distribution    of Target’s alpha is zero. Downloading return data for Target and the S&P 500 

reveals a residual standard deviation of 19.8%. Given this volatility, the prior mean of zero, 

and an assumption of normality, you now have the entire prior distribution of Target’s alpha. 

 One can make a decision using a prior distribution, or refine that distribution by expend-

ing effort to obtain additional data. In jargon, this effort is called  the experiment.   The 

experiment as a stand-alone venture would yield a probability distribution of possible out-

comes. The optimal statistical procedure is to combine one’s prior distribution for alpha 

with the information derived from the experiment to form a    posterior  distribution     that 

reflects both. This posterior distribution is then used for decision making. 

 A “tight” prior, that is, a distribution with a small standard deviation, implies a high 

degree of confidence in the likely range of possible alpha values even before looking at 

the data. In this case, the experiment may not be sufficiently convincing to affect your 

beliefs, meaning that the posterior will be little changed from the prior.  

3

   In the context of 

the present discussion, an active forecast of alpha and its precision provides the experiment 

that may induce you to update your prior beliefs about its value. The role of the portfolio 

manager is to form a posterior distribution of alpha that serves portfolio construction.