An Example Using Black-Litterman

To illuminate concepts but minimize the size of the problem and associated math, consider a portfolio of stocks and bonds with returns data beginning monthly in June of 1978. Stocks are U.S. equity with a custom benchmark (R3000, S&P 500) and the benchmark for fixed income is the Lehman Aggregate. Let's ignore the benchmarks for now and concentrate on the program returns. Suppose that the forward-looking returns assumptions are that, in equilibrium, these will earn 7.10 and 5.30 percent, respectively. Thus Π = (7.10, 5.30)′. Using the historical returns, we estimate the covariance matrix to be:

Volatilities are the square roots of the diagonal elements and indicate that stocks and bonds have volatilities equal to 16.57 percent and 8.27 percent, respectively, since 1978. Consider now a meeting in which two views are proposed. View one is that stocks will outperform bonds by 300 basis points. View two is that bonds will earn 6.0 percent. (Thus, view one implies that the stocks will earn 9.0 percent). Returns are thereby expected to change to:

The view matrix is now

Simply adopting these views would suggest we have 100 percent confidence in each. Suppose we do not. Instead, we relate confidence in a view to the volatilities contained in V by computing the diagonal matrix:

We link views P to underlying volatilities V by setting:

and where p₁ is the first row of P. The second row operation produces Ω₂ = 68. Replacing these values into Ω, we get:

Recall that whereas V models uncertainty with respect to the prior Π, Ω models uncertainty with respect to views. Note that the view and prior uncertainty associated with the bond return are identical at 68. This reflects the confidence in the bond return view being tied to the historical volatility in V but which is unrelated to volatility in stocks. The confidence in the view on stocks, however, does involve the view on the bonds and, because of their positive covariation, the confidence in the view on stocks suffers somewhat (Ω₁ = 286 > V₁ = 275).

Solving equation μ∗ for the posterior expected returns yields:

with posterior covariance matrix:

V^∗ has no zero or negative eigen values and its condition number is 5.42, implying that the posterior covariance matrix is positive semi-definite and invertible. (Actually, as long as V and Ω are each positive semi-definite, then the sum of two positive semi-definite matrices is itself positive semi-definite.)

These posterior parameter estimates can then be substituted into the (unconstrained) mean-variance optimization problem:

from which the view portfolio w is solved. Using μ∗ and V∗ (and assuming λ = 2.25), the view portfolio consists of a (0.28, 0.72) mix of the stocks and bonds. Compare this allocation with the prior allocation at Π = (7.1, 5.3)′ and V, which requires a mix of (0.21, 0.79), respectively. The view, therefore, shifts the portfolio in the direction of stocks. Most of this shift is due to the 300 bps expected spread between stock and bond returns but which is attenuated by the (lack of) confidence in the view itself.

Had we simply substituted the views Q for μ earlier and retained the prior covariances in V, the allocation would not have shifted far enough (it would now be 0.24, 0.76, respectively) because the covariance matrix V does not account for how our confidence in the views affect returns covariances. This is a very important point. A view implicitly, but fundamentally, alters the return stream and its covariances with the remaining assets. While it is tempting to insert one's view on returns, it is wrong to blithely assume that risks don't change to reflect those views. In essence, V alone is misspecified.