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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
historical w
CAPM GSMI implied equilibrium
return vector
weight w
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%
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
, 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
cantly different than the benchmark market capitalization-weighted portfolio, especially
the allocation to international bonds. As would be expected (since the process of
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
ER is
ER =
Q (2.3)
where ER is the new (posterior) combined return vector (N ×1 column vector), is a
is the covariance matrix of excess returns (N ×N matrix), P is a matrix that

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