Investments, tenth edition

  Step 4: Revised (Posterior) Expectations 

 The baseline forecasts of expected returns derived from market values and their covari-

ance matrix comprise the prior distribution of the rates of return on bonds and stocks. The 

manager’s view, together with its confidence measure, provides the probability distribution 

arising from the “experiment,” that is, the additional information that must be optimally 

integrated with the prior distribution. The result is the posterior: a new set of expected 

returns, conditioned on the manager’s views. 

 To acquire intuition about the solution, consider what the baseline expected returns 

imply about the view. The expectations derived from market data were that the expected 

return on bonds is 1.40% and on stocks 6.81%. Therefore, the baseline view is that  E ( R  

 B 

 )  2  

 E ( R  

 S 

 )  5   2 5.41%. In contrast, the manager thinks this difference is  Q   5   R  

 B 

   2   R  

 S 

   5  .5%. 

Using the BL linear-equation notation for market expectations:

    Q

E

5 PR

E

T

 P

5 (1, 21)

 R

E

5 3E(R

B

), E(R

S

)

4 5 (1.40%, 6.81%)

 

 Q

E

5 1.40 2 6.81 5 25.41%

  

(27.12)

   

  

8

 A simpler view that bonds will return 3% is also legitimate. In that case  P   5  (1, 0) and the view is really like an 

alpha forecast in the Treynor-Black model. If all views were like this simple one, there would be no difference 

between the TB and BL models. 

  

9

 Absent specific information shedding light on the SD of the view, for example, the track record of the source of 

the view, the SD calculated from the covariance matrix of the baseline forecasts is commonly used. In that case, 

the SD would be that of  Q  

 E 

  in Equation 27.13:    s(Q

E

)

5 ".0002714 5 .0165 (1.65%).  

  

10

 Notice that the view is expressed as a row vector with as many elements as there are risky assets (here, two) 

applied to the row vector of returns. The manager’s view ( Q ) is obtained from the vector,  P,  marking the assets 

included in the view, times their actual returns. We denote the return row vector,  R,  with a superscript “ T ”  (for 

transpose—turning a row vector into a column), and therefore compute the “sumproduct” of the two vectors. 

 More generally, any view that is a linear combination of the relevant excess returns 

can be presented as an array (in Excel, an array would be a column of numbers) that 

 multiplies another array (column) of excess returns. In this case, the array of weights is 

 P   5  (1,  2 1) and the array of excess returns is ( R  

 B 

 ,  R  

 S 

 ). The value of this linear combina-

tion, denoted  Q,  reflects the manager’s view. In this case,  Q   5  .5% must be taken into 

account in optimizing the portfolio.  

8

    

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966 

P A R T   V I I

  Applied Portfolio Management

 Thus, the baseline “view” is  2 5.41% (i.e., stocks will outperform bonds), which is vastly 

different from the manager’s view. The difference,  D,  and its variance are

   

 D

5 Q 2 Q

E

5 .005 2 (2.0541) 5 .0591

 s

2

(D)

5 s

2

(

e) 1 s

2

(Q

E

)

5 .0003 1 s

2

(Q

E

)

 s

2

(Q

E

)

5 Var3E(R

B

)

2 E(R

S

)

4 5 s

E(R

B

)

2

1 s

E(R

S

)

2

2 2Cov3E(R

B

), E(R

S

)

4

 

5 .000064 1 .000289 2 2 3 .0000408 5 .0002714

 

 s

2

(D)

5 .0003 1 .0002714 5 .0005714

  

(27.13)

   

 Given the large difference between the manager’s and the baseline views, we would expect 

a significant change in the conditional expected returns from those of the baseline and, as 

a result, a very different optimal portfolio. 

 The expected returns conditional on the view is a function of four elements: the baseline 

expectations,  E ( R ); the difference,  D,  between the manager’s view and the baseline view 

(see Equation 27.13); the contribution of the asset return to the variance of  D;  and the total 

variance of  D.  The result is:

    E(R

0view) 5 R 1 D

 

Contribution of asset to s

D

2

s

D

2

 E(R

B 

0

 

P)

5 E(R

B

)

1

D

5s

E(R

B

)

2

2 Cov3E(R

B

), E(R

S

)

46

s

D

2

 

5 .0140 1

.0591(.000064

2 .0000408)

.0005714

5 .0140 1 .0024 5 .0164

 

 E(R

S 

0

 

P)

5 E(R

S

)

1

D

5Cov3E(R

B

), E(R

S

)

4 2 s

E(R

S

)

2

6

s

D

2

 

(27.14)

 

5 .0681 1

.0591(.0000408

2 .000289)

.0005714

5 .0681 2 .0257 5 .0424    

 We see that the manager increases his expected returns on bonds by .24% to 1.64%, and 

reduces his expected return on stocks by 2.57% to 4.24%. The difference between the 

expected returns on stocks and bonds is reduced from 5.41% to 2.60%. While this is a very 

large change, we also realize that the manager’s private view that  Q  5 .5% has been greatly 

tempered by the prior distribution to a value roughly halfway between his private view and 

the baseline view. In general, the degree of compromise between views will depend on the 

precision assigned to them. 

 The example we have described contains only two assets and one view. It can easily be 

generalized to any number of assets with any number of views about future returns. The 

views can be more complex than a simple difference between a pair of returns. Views can 

assign a value to  any  linear combination of the assets in the universe, and the confidence 

level (the covariance matrix of the set of á values of the views) can allow for dependence 

across views. This flexibility gives the model great potential by quantifying a rich set of 

information that is unique to a portfolio manager. Appendix B to the chapter presents the 

general BL model.