Multi-asset portfolio optimization and out-of-sample performance: an evaluation of Black–Litterman, mean-variance, and naïve diversification approaches

Abstract

The Black–Litterman model aims to enhance asset allocation decisions by overcoming the problems of mean-variance portfolio optimization. We propose a sample-based version of the Black–Litterman model and implement it on a multi-asset portfolio consisting of global stocks, bonds, and commodity indices, covering the period from January 1993 to December 2011. We test its out-of-sample performance relative to other asset allocation models and find that Black–Litterman optimized portfolios significantly outperform naïve-diversified portfolios (1/N rule and strategic weights), and consistently perform better than mean-variance, Bayes–Stein, and minimum-variance strategies in terms of out-of-sample Sharpe ratios, even after controlling for different levels of risk aversion, investment constraints, and transaction costs. The BL model generates portfolios with lower risk, less extreme asset allocations, and higher diversification across asset classes. Sensitivity analyses indicate that these advantages are due to more stable mixed return estimates that incorporate the reliability of return predictions, smaller estimation errors, and lower turnover.

Keywords:

• portfolio optimization
• Black–Litterman
• mean-variance
• minimum-variance
• Bayes–Stein
• naïve diversification
• 1/N
• Markowitz

JEL Classification:

• C61
• G11

Acknowledgements

We are grateful to Chris Adcock (editor) and two anonymous referees for helpful advice and comments that significantly improved the quality of this research. The authors also thank Lawrence Kryzanowski, Paul Söderlind, Michael Frömmel, as well as participants at the European Financial Management Symposium 2012, the 19th Annual Meeting of the German Finance Association 2012, the International Annual Conference of the German OR Society 2012, and the Verein fuer Socialpolitik Annual Congress 2012 for helpful comments and suggestions.

Notes

1. Value-weighted portfolios weight constituents proportional to their relative market weights and price-weighted portfolios allocate the fraction of each constituent proportional to its actual market price.

2. Moreover, benchmark portfolios with rebalancing are more common in institutional asset management and we expect the performance of the portfolios with rebalancing to be less sensitive to the evaluation period compared to the buy-and-hold portfolio because the portfolio composition of the buy-and-hold portfolio varies over time based on the relative performance of the assets during the evaluation period.

3 See Opdyke (2007) for a detailed description of the Sharpe ratio test.

4. The results for the other investor types are qualitatively similar so that we only report the performance measures for the moderate investor. We use the same continuous sample to compute optimized portfolios and divide the resulting return time series into four sub-samples, thereby avoiding the problems of rolling sample estimations on a discontinuous sample.

5. It seems surprising that restricting asset weights results in higher Sharpe ratios for conservative investors, but to an inferior performance for moderate and aggressive investors. The explanation is that the restriction of the asset weights for the aggressive (moderate) investor results in a maximum portfolio weight of government bonds below 22.5% (50%). Therefore, in periods of stock market downturns, the asset allocation algorithm cannot fully reallocate the portfolio into the safe asset, thereby loosing performance relative to the unrestricted case.

6. In the BS approach the sample covariance matrix is inflated by the factor , where T is the sample size and N is the number of assets. While for larger observation windows the covariance matrix inflation in the BS approach plays only a minor role, it gets pronounced for a shorter estimation window of only 12 months as the inflation factor increases to 2.2 (T = 12; N = 5). In this case, the expected portfolio volatility increases by the factor . Because the volatility constraint requires the expected portfolio volatility to be below 10%, inflating the covariance matrix by a factor of 2.2 is equal to setting the volatility constraint to a level of 6.74% instead of 10%.

7. For the stock-only case, we do not compute strategically weighted portfolios as strategic weights are only used to determine the fraction of stocks relative to bonds and commodities and we do not have any reason to set different strategic weights for different industries or countries on an ex ante basis.