Black-Litterman Views

Black-Litterman Model

The Black-Litterman model (1990) combines a market equilibrium prior with investor views to produce a posterior estimate of expected returns.

The Framework

• Prior: Market equilibrium returns π (implied by market cap weights and a risk model)
• Views: An investor expresses a view that a linear combination of assets will return Q, with uncertainty Ω
• Posterior: The BL formula blends the prior and views using Bayes' theorem

Simplified Formula

For a single view expressed as a vector P (portfolio weights for the view) and target return Q:

$\mu_{BL} = \pi + \tau \Sigma P^T \left(P \tau \Sigma P^T + \Omega\right)^{-1} (Q - P \pi)$

In our simplified version, we use the identity matrix for Σ (unit covariance), so:

$\mu_{BL,i} = \pi_i + \frac{\tau P_i}{\tau \sum_j P_j^2 + \Omega} (Q - P \cdot \pi)$

where τ (tau) is a scalar controlling prior uncertainty (default = 0.05).

Your Task

Implement bl_posterior_return(pi, omega, P, Q, tau=0.05) where:

• pi — list of prior expected returns
• omega — scalar view uncertainty
• P — list of view portfolio weights
• Q — scalar view expected return

Return a list of posterior expected returns, each rounded to 4 decimal places.