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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 ...