Investments, tenth edition

 Spreadsheet 27.2

Sensitivity of the Black-Litterman portfolio to confidence in views          

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Table 1: Bordered Covariance Matrix from Historical Excess Returns

and Market-Value Weights and Calculation of Baseline Forecasts

Bonds

Stocks

Weights

0.25

0.75

Bonds

Stocks

sumproduct

11.65

170.21

Market portfolio variance V(M) = sum(c11,d11) = 

Coefficient of risk aversion of representative investor =

Baseline market portfolio risk premium = 0.01A*V(M) =

Covariance with R

M

Baseline risk premiums

Proportion of covariance attributed to expected returns

0.01

1.247920819

0.256237542

Covariance matrix of expected returns

Bonds

Stocks

Bonds

Stocks

Table 2: Views, Confidence, and Revised (Posterior) Expectations

View: Difference between returns on bonds and stocks, Q =

View embedded in baseline forecasts Q

E

 =

Variance of Q

E

 = Var(R

B

 − R

S

)

Var[E(R

B

)] − Cov[E(R

B

),E(R

S

)] =

Cov[E(RB),E(RS)] − Var[E(RB)]  =

Difference between view and baseline data, D =

Confidence measured by standard deviation of view Q

Possible SD

0

1

1.73

3.00

6.00

Variance

0

1.5

3

9

36

Baseline

E(R

B

|P)

1.90

1.72

1.64

1.52

1.43

1.40

E(R

S

|P)

1.40

3.33

4.24

5.56

6.43

6.81

A

C

D

E

F

G

H

I

B

0.25

64

40.8

0.75

40.8

289

46.6

226.95

1.40

6.81

0.64

0.408

0.408

2.89

0.5

−5.41

2.71

0.23

−2.48

5.91

181.86

3

5.46

e

X

c e l

Please visit us at 

www.mhhe.com/bkm

 Figure 27.5 

Sensitivity of Black-Litterman portfolio performance to  confidence 

level  

−3.0

−2.0

−1.0

0.0

1.0

2.0

3.0

0

1

2

3

4

5

6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

M-square–View is False

M-square–View is Correct

Weight in Bonds

Confidence (SD)

SD of View = 1.73%

Risk Adjusted Performance (%)

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P A R T   V I I

  Applied Portfolio Management

and incorrect. The weight in bonds declines as the confidence in the view falls (the SD of 

the view increases). With no confidence in the view (SD very large), the weight in bonds 

falls to 0.3, determined by the baseline forecast. At this point, the portfolio is passive; its 

 M -square  is  zero.   

 Notice that the  M -square profile is asymmetric. With great confidence in the view and the 

resultant large position in bonds, the gain in  M -square when the view is correct is smaller 

than the loss in  M -square when the view is incorrect. With less confidence and therefore 

a smaller position in bonds, the “game” becomes more symmetric between a correct and 

incorrect view. Since determination of the SD of a view is quite abstract, the graph tells us 

that to err on the side of skepticism may well be the prudent choice.    

    27.4 

Treynor-Black versus Black-Litterman: 

Complements, Not Substitutes 

  Treynor, Black, and Litterman have earned a place among the important innovators of the 

investments industry. Wide implementation of their models could contribute much to the 

industry. The comparative analysis of their models presented here is not aimed at elevating 

one at the expense of the other—in any case, we find them complementary—but rather to 

clarify the relative merits of each. 

 First and foremost, once you reach the optimization stage, the models are identical. Put 

differently, if users of either model arrive at identical input lists, they will choose identical 

portfolios and realize identical performance measures. In Section 27.6, we show that these 

levels of performance should be far superior to passive strategies, as well as to active strat-

egies that do not take advantage of the quantitative techniques of these models. The models 

differ primarily in the way they arrive at the input list, and analysis of these differences 

shows that the models are true complements and are best used in tandem.