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21 A step-by-step guide to the Black–Litterman model

Table 2.2 Recommended portfolio weights

Asset class Weight based on

Weight based on Weight based on

Market

historical w

Hist

CAPM GSMI implied equilibrium

capitalization

w

GSMI

return vector

weight w

mkt

US Bonds 114432% 2133% 1934% 1934%

Int’l Bonds −10459% 519% 2613% 2613%

US Large Growth 5499% 1080% 1209% 1209%

US Large Value −529% 1082% 1209% 1209%

US Small Growth −6052% 373% 134% 134%

US Small Value 8147% −049% 134% 134%

Int’l Dev. Equity −10436% 1710% 2418% 2418%

Int’l Emerg. Equity 1459% 214% 349% 349%

High 114432% 2133% 2613% 2613%

Low −10459% −049% 134%

134%

In Table 2.2, equation (2.2) is used to find the optimum weights for three portfo-

lios based on the return vectors from Table 2.1. The market capitalization weights are

presented in the final column of Table 2.2.

Not surprisingly, the historical return vector produces an extreme portfolio. Those

not familiar with mean-variance optimization might expect two highly correlated return

vectors to lead to similarly correlated vectors of portfolio holdings. Nevertheless, despite

the similarity between the CAPM GSMI return vector and the implied equilibrium return

vector

, the return vectors produce two rather distinct weight vectors (the correlation

coefficient is 66%). Most of the weights of the CAPM GSMI-based portfolio are

signifi-

cantly different than the benchmark market capitalization-weighted portfolio, especially

the allocation to international bonds. As would be expected (since the process of

extract-

ing the implied equilibrium returns using the market capitalization weights is reversed),

the implied equilibrium return vector

leads back to the market capitalization-weighted

portfolio. In the absence of views that differ from the implied equilibrium return, investors

should hold the market portfolio. The implied equilibrium return vector

is the market-

neutral starting point for the Black–Litterman model.

2.3 The Black–Litterman model

2.3.1 The Black–Litterman formula

Prior to advancing, it is important to introduce the Black–Litterman formula and provide

a brief description of each of its elements. Throughout this article,

K is used to represent

the number of views and

N is used to express the number of assets in the formula. The

formula for the new combined return vector

ER is

−1

ER =

−1

+P

−1

P

−1

+P

−1

Q (2.3)

where ER is the new (posterior) combined return vector (N ×1 column vector), is a

scalar,

is the covariance matrix of excess returns (N ×N matrix), P is a matrix that

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