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Latent Variable Estimation in Bayesian Black-Litterman Models
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Reformulates the Bayesian Black-Litterman portfolio model as a Bayesian network that treats investor views (q, Omega) as latent variables learned from feature data, yielding view-free portfolio weights.
Problem
Classical Black-Litterman requires subjective investor views (forecast vector q and uncertainty Omega), which are heuristic, biased, and when estimated by external models cause incoherent learning and error propagation in portfolio optimization.
Approach
Treat views and their uncertainty as latent variables in a single Bayesian network with asset features, using a shared-latent parametrization (SLP-BL) for closed-form posterior over returns, plus a feature-influenced views model (FIV-BL) that yields a Student-t/mixture posterior; both recover classical BL and Markowitz as special cases.
Key results
- On SPDR sector ETFs, BL (80d) reaches Sharpe 0.69 versus 0.50 for Markowitz (80d) and 0.59 for S&P500.
- On Dow Jones data, BL (120d) reaches Sharpe 0.87 versus 0.57 for Markowitz (120d) and 0.52 for DJIA, with max drawdown 39.81% vs 46.73%.
- The paper reports roughly 49.8% mean Sharpe improvement and 55.1% turnover reduction over Markowitz and indices.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace traditional Markowitz with feature-based SLP-BL (BL in the results table) | Portfolio optimization with 50d rolling window · SPDR Sector ETFs | Sharpe Ratio: 0.35 -> 0.66 | Helped |
| Replace traditional Markowitz with feature-based SLP-BL (BL in the results table) | Portfolio optimization with 80d rolling window · SPDR Sector ETFs | Sharpe Ratio: 0.50 -> 0.69 | Helped |
| Replace traditional Markowitz with feature-based SLP-BL (BL in the results table) | Portfolio optimization with 100d rolling window · SPDR Sector ETFs | Sharpe Ratio: 0.57 -> 0.70 | Helped |
| Replace traditional Markowitz with feature-based SLP-BL (BL in the results table) | Portfolio optimization with 120d rolling window · SPDR Sector ETFs | Sharpe Ratio: 0.57 -> 0.70 | Helped |
| Replace traditional Markowitz with feature-based SLP-BL (BL in the results table) | Portfolio optimization with 150d rolling window · SPDR Sector ETFs | Sharpe Ratio: 0.45 -> 0.68 | Helped |
| Replace traditional Markowitz with feature-based SLP-BL (BL in the results table) | Portfolio optimization with 50d rolling window · Dow Jones Index | Sharpe Ratio: 0.45 -> 0.78 | Helped |
| Replace traditional Markowitz with feature-based SLP-BL (BL in the results table) | Portfolio optimization with 80d rolling window · Dow Jones Index | Sharpe Ratio: 0.51 -> 0.84 | Helped |
| Replace traditional Markowitz with feature-based SLP-BL (BL in the results table) | Portfolio optimization with 100d rolling window · Dow Jones Index | Sharpe Ratio: 0.55 -> 0.85 | Helped |
| Replace traditional Markowitz with feature-based SLP-BL (BL in the results table) | Portfolio optimization with 120d rolling window · Dow Jones Index | Sharpe Ratio: 0.57 -> 0.87 | Helped |
| Replace traditional Markowitz with feature-based SLP-BL (BL in the results table) | Portfolio optimization with 150d rolling window · Dow Jones Index | Sharpe Ratio: 0.62 -> 0.80 | Helped |
Limitations
- Only the SLP-BL model is empirically evaluated; the FIV-BL model requires approximation (IW prior on posterior covariance, constant mean) and is not tested.
- Backtests use a single feature set of nine generic indicators and fixed hyperparameter rules of thumb.
Experimental setup
Datasets: SPDR Sector ETFs prices and volume Dow Jones constituent stock prices and volume
Benchmarks: SPDR Sector ETFs backtest Dow Jones Index backtest
Metrics: cumulative return CAGR Sharpe ratio maximum drawdown volatility
Baselines: Markowitz (MV) equal-weighted portfolio S&P 500 DJIA
Builds on: Black-Litterman model Black-Litterman-Bayes Markowitz mean-variance optimization
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