Financial
One-asset Black-Litterman example: blend a prior and one absolute view
Explore a scalar normal-prior update with one absolute return view; distinguish uncertainty in the estimated mean from asset return variance.
State the scalar model assumptions
This is the one-asset, one-absolute-view special case with pick coefficient P=1. It is not the full multi-asset Black-Litterman optimizer: correlated assets and relative views need covariance matrices and a view matrix.
Pi and Q are supplied estimates on the same excess-return basis and time horizon. The prior variance of the unknown mean is tau*sigma²; the independent Gaussian view-error variance is omega. They must be positive. Higher stated precision weights a view more heavily, but does not prove that the view is accurate.
The posterior estimation variance M is uncertainty about the mean. It is not the asset return variance or a promise of lower investment risk. The common posterior predictive covariance convention adds the original asset covariance to M. This worksheet neither derives equilibrium Pi nor calculates portfolio weights.
| Black-Litterman Input | Cell / Formula | Equal Confidence (50/50) | High Conviction (omega = 0.0005) |
|---|---|---|---|
| Equilibrium Return Pi | B2 = 6.0% (0.06) | tau=0.05 (B3), Vol=20% (B4) | Prior Var = 0.05*0.04 = 0.0020 |
| PM View Return Q & omega | B5 = 10.0% (0.10) | omega B6 = 0.0020 | Prior Prec=500 | View Prec=2000 |
| Posterior Expected E[R]_BL | Precision-weighted blend | 8.00% (0.0800 midpoint) | 9.20% (0.0920 -> 80% view!) |
| Posterior Variance Var_BL | 1 / (Prec_prior + Prec_view) | 1 / 1000 = 0.0010 | 1 / 2500 = 0.0004 |
Black-Litterman Posterior Expected Return E[R]_BL
Weights equilibrium prior B2 by 1 / (B3 * B4^2) and subjective view B5 by 1 / B6.
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B3<=0,B4<=0,B6<=0),"Invalid inputs",ROUND(((B2 / (B3 * (B4 ^ 2))) + (B5 / B6)) / ((1 / (B3 * (B4 ^ 2))) + (1 / B6)), 4)))Posterior Estimation Variance Var_BL
Before rounding, M is below both positive input estimation variances under the assumed independent Gaussian information model. Four-decimal display rounding can erase small values or strict inequalities; it does not make uncertainty vanish.
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B3<=0,B4<=0,B6<=0),"Invalid inputs",ROUND(1 / ((1 / (B3 * (B4 ^ 2))) + (1 / B6)), 4)))Try the example
Use the original values or change them to test your understanding.