What is Bayesian asset allocation and how does it differ from mean-variance?

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AcadiFi · Accepted Answer

Bayesian asset allocation combines prior beliefs (equilibrium or manager views) with observed market data using Bayes' theorem, producing posterior expected returns that are more stable and less extreme than classical mean-variance optimization (MVO). The dominant implementation is Black-Litterman (BL), which blends CAPM equilibrium returns with investor views.

Classical MVO problems:
- Estimation error in expected returns dominates optimal weights.
- Small changes in inputs cause huge allocation shifts ('error maximization').
- Optimal portfolios often have extreme weights (100% one asset, short others).

Bayesian/BL solution:
1. **Prior** — reverse-optimize market-cap weights to derive equilibrium returns π. These are treated as the consensus 'prior'.
2. **Views** — investor specifies views (absolute or relative), each with confidence expressed as a covariance matrix Ω.
3. **Posterior** — combine via BL formula: E[R] = [(τΣ)⁻¹ + P'Ω⁻¹P]⁻¹·[(τΣ)⁻¹π + P'Ω⁻¹Q], where τ scales prior uncertainty, Σ is asset covariance, P is view matrix, Q is view vector.

Example: Braddick Pension has equilibrium implied returns for global equity 7%, global bonds 3.5%. Manager view: EM equity will outperform developed by 2% (confidence 50%). BL posterior tilts EM weight up modestly rather than dramatically, reflecting limited conviction. Compare to plugging +2% EM return into naive MVO: portfolio shifts 40% into EM — unreasonable.

```mermaid
flowchart LR
    A[Market Prior π] --> C[Bayes Update]
    B[Investor Views Q] --> C
    C --> D[Posterior E[R]]
    D --> E[MVO with posterior]
    E --> F[Stable weights]
```

Benefits: weights are diversified by default (equilibrium), deviations proportional to view confidence, intuitive specification. CFA candidates should memorize the BL formula structure and understand τ (typically 0.025-0.05) controls how much views move posterior from prior. Firms like Quentwood Asset Management use BL extensively for multi-asset portfolios where classical MVO produces unworkable allocations.

What is Bayesian asset allocation and how does it differ from mean-variance?

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