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Black-Litterman with eight asset classes and three views: implied returns, posterior returns and weights (Idzorek's example)

Tool	Black-Litterman
Source	Idzorek, T. M. (2005). A Step-by-Step Guide to the Black-Litterman Model. Ibbotson Associates. Market weights of Tables 2 and 6, covariance matrix of Table 5, the three views of section 2.2, results of Tables 4, 6 and 7.
Level	Graduate

Problem

Eight asset classes have the market weights 19.34, 26.13, 12.09, 12.09, 1.34, 1.34, 24.18 and 3.49 % and a known covariance matrix. The market premium is 3 %, so the risk aversion is 3 % divided by the market variance. An investor holds three views: international developed equity will return 5.25 %; international bonds will outperform US bonds by 0.25 %; US growth (large and small, 0.9 / 0.1) will outperform US value by 2 %. With tau = 0.025 and each view as uncertain as the market-implied return of its own combination, find the implied equilibrium returns, the posterior returns and the unconstrained weights.

Inputs to Enter

Market weights and volatilities of the eight classes; the correlation matrix (the tool's default, computed from the paper's covariance matrix)
Risk aversion 3.0658; risk-free rate 0 (returns above the risk-free rate); tau 0.025
Views: Intl Dev Equity 5.25 %; Intl Bonds - US Bonds 0.25 %; 0.9 US Large Growth + 0.1 US Small Growth - 0.9 US Large Value - 0.1 US Small Value 2 %

Open the Black-Litterman tool (loaded with this example)

Answers

Market variance 97.85 %², volatility 9.89 %; premium 3.0658 · 97.85 / 100 = 3.00 %; Sharpe ratio 0.303
US Bonds: covariance with the market 2.7022 %², implied return 3.0658 · 2.7022 / 100 = 0.08 %; all eight: 0.08, 0.67, 6.42, 4.08, 7.43, 3.70, 4.80, 6.60 %
View 1 alone: market-implied 4.80 %, variance 283.55 %², uncertainty 0.025 · 283.55 = 7.089, weight 0.5, posterior 4.801 + 0.5 · (5.25 − 4.801) = 5.03 %
Posterior returns with the three views: 0.07, 0.50, 6.50, 4.32, 7.59, 3.94, 4.93, 6.84 % (the paper's Table 6)
Weights: 29.89, 15.58, 9.36, 14.82, 1.04, 1.64, 27.81, 3.49 %, adding to 103.63 % (Table 6 prints the same within 0.01)
Positions in the three views: +3.63, −10.55 and −3.04 %; the class named in no view keeps its 3.49 %
Every view at 100 % confidence: 43.8, 1.6, 3.8, 20.4, 0.4, 2.3, 35.2, 3.5 % (Table 7)

Notes

The tool's default inputs are this case; press Submit. The form takes volatilities and correlations, so the paper's covariance matrix was converted (3 and 4 decimals); from the printed matrix itself 27 of 32 cells of the paper's Tables 1 and 6 are reproduced to the printed digit and the rest within 0.01 (test_black_litterman.php). The implied returns restate the market weights and covariances; they are not forecasts, and the weights describe the views entered. A calculator for education and analysis, not investment advice.