Using Mean-Variance Optimization in the Real World: Black-Litterman vs. Resampling
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The document compares two techniques for making mean-variance optimization (MVO) more usable in real-world asset allocation: Black-Litterman and resampling. Both techniques help address limitations of MVO, such as unintuitive portfolios caused by estimation error. The experiment found that while both techniques led to more diversified portfolios than historical inputs alone, Black-Litterman created the most intuitive portfolios for practical use.
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Using Mean-Variance Optimization in the Real World: Black-Litterman vs. Resampling
- 1. Using Mean-Variance Optimization in the Real World: Black-Litterman vs. Resampling Jill Adrogue Zephyr Associates, Inc. September 15, 2005
- 2. Making Mean-Variance Optimization Usable • Mean-Variance Optimization (MVO) has been little used in practice. • Both Black-Litterman and Resampling, when combined with MVO, create more diversified portfolios. • Only Black-Litterman creates intuitive portfolios that are usable in the real world. • Portfolios on the resampled frontier include active risk caused by the forecasts and averaging process.
- 3. MVO and the Asset Allocation Process • Mean-Variance Optimization leads to unintuitive, undiversified portfolios. • Until recently, MVO has mostly been used as window dressing. MVO, though a powerful algorithm, has not found its place in practical asset allocation.
- 4. The Power of MVO • Mean-Variance Optimization was developed by Nobel Laureate Harry Markowitz in 1952. • Markowitz discovered that an investor can reduce the volatility of a portfolio and increase its return at the same time. • Diversification: The risk of a portfolio can be decreased by combining assets whose returns move in different directions under certain market conditions.
- 5. MVO in Two Stages 1. Calculate the forecasts. – Calculate forecasts for returns, standard deviations and correlations for the set of assets in which you can invest. – This is often done using historical data. 2. Calculate the Efficient Frontier. – The efficient frontier is the set of portfolios that minimizes risk at the possible levels of return. – A portfolio can be selected from the frontier based on risk, utility maximization, maximum Sharpe Ratio, etc.
- 6. The Mechanics 1. Create or calculate Forecasts for Return, Risk and Correlations for a set of assets. These parameters describe a multivariate return distribution. 2. Calculate the Efficient Frontier. – Assume that all portfolios have positive weights (no short-selling) and add to 100. – Calculate the minimum variance portfolios and maximum return portfolio using the forecasts. – Calculate the portfolio that minimizes risk for each of 98 portfolios between the minimum variance and maximum return portfolios. This set of 100 portfolios is the efficient frontier.
- 7. The Efficient Frontier 16% Maximum Return Portfolio 14% 12% Annualized Return 10% 8% Minimum Variance Portfolio 6% 4% 2% 0% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% Annualized Risk (Standard Deviation)
- 8. Limitations of MVO • Returns are very difficult to forecast. – MVO requires forecasts on ALL assets. – Historical returns are very poor forecasts. • Input Sensitivity--MVO is highly sensitive to the return forecasts. – Small changes in return assumptions often lead to large changes in the optimal allocations. Estimation Error is built into forecasting and magnified by MVO
- 9. Estimation Error Leads to Unusable Portfolios • Portfolios are very concentrated (no diversification). • Portfolios are unintuitive. Both of these issues must be solved to make MVO a practical real-world tool.
- 10. Two Approaches to Creating Diversified Portfolios with MVO • Black-Litterman – Technique developed by Fischer Black and Robert Litterman of Goldman Sachs to create better return estimates. • Resampling – Technique developed by Richard Michaud to average over the statistical equivalence region and create a new efficient frontier.
- 11. An Experiment to Compare the Two Techniques • Select a set of assets. • Calculate an efficient frontier using Historical Inputs, Resampling and Black- Litterman Inputs. • Compare the resulting portfolios.
- 12. The Assets Return Std. Dev. US Bonds 7.4% 4.2% Int'l Bonds 8.4% 9.4% Large Growth 11.8% 18.3% Large Value 12.8% 14.2% Small Growth 10.4% 24.0% Small Value 14.0% 16.3% Int'l Equity 8.4% 16.7% Emerging Markets 12.5% 22.7% Historical Data January 1987-July 2005
- 13. Are the Portfolios Diversified? • First, let’s look at the diversification of the portfolios resulting from the three techniques.
- 14. Using Historical Forecasts in MVO Leads to Highly Concentrated Portfolios 100% 90% 80% 70% 60% Allocation 50% 40% 30% 20% 10% 0% 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 Portfolio US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets
- 15. Black-Litterman Implied Returns • Black-Litterman Implied Returns are consistent with MPT and CAPM. • Black-Litterman Implied Returns are the returns that put the market in equilibrium. • Black-Litterman Implied Returns are calculated using Reverse Optimization. The inputs are the market capitalizations and covariance matrix of the assets, and the risk premium for the set of assets.
- 16. Black-Litterman Returns as Forecasts • Black-Litterman Implied Returns make excellent forecasts for use with MVO. The result is diversified, intuitive portfolios.
- 17. Black-Litterman Implied Returns Lead to Diversified Portfolios 100% 90% 80% 70% 60% Allocation 50% 40% 30% 20% 10% 0% 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 Portfolio US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets
- 18. Resampling 1. Estimate returns, standard deviations and correlations for a set of assets. Michaud does Stage this using historical data. 1 of MVO 2. Run a Monte Carlo simulation, creating a new data set. Calculate the return, standard deviation and correlations of the new data set. Stage 3. Create an efficient frontier using the new 2 of inputs. MVO 4. Repeat steps 2 and 3 500 times. Add’l 5. Calculate the average allocations to the Step assets for a set of predetermined return intervals. This is the new efficient frontier. This procedure has U.S. Patent #6,003,018 by Michaud et al., December 12, 1999
- 19. The Resampled Frontier 1.40% 1.20% Small Value Emerging Markets Large Value 1.00% Large Growth Small Growth Monthly Return 0.80% Int'l Equity Int'l Bonds US Bonds 0.60% Historical Frontier 0.40% Resampled Frontier 0.20% 0.00% 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% Monthly Risk (Standard Deviation)
- 20. Resampling also Leads to Diversified 100% Portfolios 90% 80% 70% 60% Allocation 50% 40% 30% 20% 10% 0% 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 Portfolio US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets
- 21. A Closer Look at the Resampled Frontier
- 22. Where is the Frontier? 1.40% 1.20% Small Value Large Value Emerging Markets 1.00% Large Growth Small Growth Monthly Return 0.80% Int'l Equity Int'l Bonds US Bonds 0.60% 0.40% 0.20% 0.00% 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% Monthly Risk (Standard Deviation) Need to select one set of portfolios, but there is no theoretical motivation for Michaud’s averaging
- 23. Portfolio #50 1.40% 1.20% Small Value Large Value Emerging Markets 1.00% Large Growth Port 50 Historical Port 50 Resampled Small Growth Monthly Return 0.80% Int'l Equity Int'l Bonds US Bonds 0.60% Portfolios of rank 50 Resampled Frontier 0.40% Historical Frontier Portfolio 50 Historical Port 50 Resampled 0.20% 0.00% 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% Monthly Risk (Standard Deviation)
- 24. Consequences of Averaging to Create the Resampled Frontier • Frontier is Suboptimal. • Outliers tilt the allocations. • Very small allocations to assets throughout frontier. • It is possible to get an upward sloping frontier.
- 25. The Resampled Frontier Is Suboptimal 1.40% 1.20% Small Value Emerging Markets Large Value 1.00% Large Growth Small Growth Monthly Return 0.80% Int'l Equity Int'l Bonds US Bonds 0.60% Historical Frontier 0.40% Resampled Frontier 0.20% 0.00% 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% Monthly Risk (Standard Deviation)
- 26. Frequencies and Averaged Weights Distribution of Weights to Large Value for Portfolio 84 Resampled Weight is 21% 300 250 200 Frequency Allocation in 150 Resampled Frontier is 100 21% 50 0 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95%100% Allocation
- 27. Allocations to Every Asset in Every Portfolio Allocations to Int'l Equity in Resampled Frontier 5.0% 4.5% 4.0% 3.5% 3.0% Allocation 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 Portfolio
- 28. Are the Portfolios Intuitive? • Next, let’s look at the allocations of the portfolios. Specifically, consider two questions: – Do the allocations make sense for real- world investment? – What kind of active risk would I be taking relative to a neutral asset allocation?
- 29. Historical Data Sharpe Return Risk Ratio US Bonds 7.44% 4.16% 0.967 Int’l Bonds 8.40% 9.42% 0.529 Large Growth 11.76% 18.26% 0.457 Large Value 12.84% 14.24% 0.662 Small Growth 10.44% 24.04% 0.292 Small Value 14.04% 16.32% 0.651 Int’l Equity 8.40% 16.73% 0.298 Emerging Markets 12.48% 22.72% 0.399 January 1987-July 2005
- 30. Historical Portfolios 100% Emerging Markets 90% 80% 70% Small Value 60% Allocation 50% 40% 30% Large Value US Bonds Global Bonds 20% 10% 0% 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 Portfolio US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets
- 31. Forecasts and the Resampled Frontier • The Portfolios from the Resampled Frontier are heavily influenced by the original forecasts. • Remember, making forecasts is hard.
- 32. Resampled Portfolios 100% 90% Emerging Markets 80% 70% Small Value 60% Allocation 50% 40% Large Value 30% US Bonds Global Bonds 20% Large 10% Growth 0% 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 Portfolio US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets
- 33. Do the Resampled Portfolios Make Sense? Resampled Portfolio #25 Resampled Portfolio #50 Emerging Emerging Markets Markets 7% 11% US Bonds Small Value 10% 26% Small Value Large Value 19% 6% Large Growth US Bonds 1% 59% Large Value Int'l Bonds Int'l Bonds 14% Large 17% 27% Growth 3% Resampled Portfolio #75 Emerging US Bonds Markets 6% 17% Int'l Bonds 23% Small Value Large 28% Growth 6% Large Value 20%
- 34. The Market Portfolio: A Neutral Portfolio Weight US Bonds 21% Int'l Bonds 14% Large Growth 15% Large Value 15% Small Growth 1% Small Value 1% Int'l Equity 29% Emerging Markets 3%
- 35. Using Resampling Means Taking an Unintentional Active Risk 80% 70% 60% Market Portfolio 50% Resampled Max Sharpe Ratio Portfolio Allocation 40% 30% 20% 10% 0% US Bonds Int'l Bonds Large Large Value Small Small Value Int'l Equity Emerging Growth Growth Markets
- 36. Resampling results in taking active risk—why take bets without a reason?
- 37. The Black-Litterman Model: A Better Way to Take Active Risk • Black-Litterman starts with the Implied Returns, which come from the market portfolio and are a neutral starting point. • If you want to take a bet away from the market portfolio, Black-Litterman allows you to incorporate Views. • The Black-Litterman mixed estimation technique incorporates views so that the active risk you take makes sense and reflects your views.
- 38. Implied Returns as Forecasts • The Implied Returns make excellent forecasts for MVO in the absence of views. • Using the Implied Returns with MVO results in intuitive portfolios.
- 39. Portfolios Created Using the Implied Returns Make Sense Implied Returns Portfolio #25 Implied Returns Portfolio #50 Emerging Emerging Markets 5% Markets 3% US Bonds Int'l Equity 24% 14% Int'l Equity Small Value 28% 3% Large Value US Bonds 5% 56% Large Small Value Int'l Bonds Growth 2% 14% 7% Small Int'l Bonds Growth 1% Large Value Large 10% 14% Growth 14% Implied Returns Portfolio #75 Emerging Int'l Bonds Markets 1% 8% Large Growth 22% Int'l Equity 47% Large Value Small 20% Growth 2%
- 40. Portfolio # 25 Resampled Portfolio #25 Implied Returns Portfolio #25 Emerging Emerging Markets 7% Markets 5% Small Value Int'l Equity 10% 14% Small Value Large Value 3% 6% Large Value US Bonds Large 5% US Bonds 56% Growth 59% Large 1% Growth 7% Int'l Bonds Int'l Bonds 17% 10%
- 41. Portfolio #50 Resampled Portfolio #50 Implied Returns Portfolio #50 Emerging Markets Emerging 11% Markets 3% US Bonds US Bonds 26% 24% Int'l Equity Small Value 28% 19% Small Value Int'l Bonds 2% 14% Small Large Value Int'l Bonds Growth 1% 14% Large Large Large Value 27% Growth 3% 14% Growth 14%
- 42. Portfolio #75 Resampled Portfolio #75 Implied Returns Portfolio #75 Emerging US Bonds Emerging Int'l Bonds Markets Markets 1% 6% 8% 17% Large Int'l Bonds Growth 23% 22% Int'l Equity 47% Small Value Large 28% Growth 6% Large Value Large Value Small 20% 20% Growth 2%
- 43. The Implied Returns are a Neutral Starting Point 35% 30% Market Portfolio 25% Implied Returns Max Sharpe Ratio Portfolio 20% Allocation 15% 10% 5% 0% US Bonds Int'l Bonds Large Large Value Small Small Value Int'l Equity Emerging Growth Growth Markets
- 44. Views Allow You to Take Intentional Active Risk • Views allow you to take an active risk away from the market portfolio. • Views only have to be expressed for those assets about which you have special knowledge or strong opinions.
- 45. The Implied Returns are Combined with Your Views to Create New Black- Litterman Forecasts Implied Returns Views Black-Litterman Forecast Returns
- 46. Risk Aversion Covariance Market Capitalization Uncertainty of Coefficient Matrix Weights Views Views λ = (E (r ) − r f ) σ 2 (Σ ) ( wmkt ) (Q ) (Ω ) Implied Equilibrium Return Vector Π = λΣwmkt Prior Equilibrium Distribution View Distribution N ~ (Π, τΣ ) N ~ (Q, Ω ) New Combined Return Distribution ( [ −1 ( N ~ E[ R ], (τΣ ) + P ' Ω −1 P )] −1 )
- 47. Sample View • Sample View: Large Growth will have an annualized return of 14% (Implied Return is 12.2%).
- 48. An Active Bet Toward Large Growth 35% IR/Market Portfolio 30% Portfolio with View 25% 20% 15% 10% 5% 0% US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets
- 49. Conclusion • Both Black-Litterman and Resampling result in diversified portfolios. • Black-Litterman also provides intuitive portfolios. • Black-Litterman allows you to take purposeful active risk with the use of Views.
- 50. Sources • Black, Fischer, and Robert Litterman. “Global Portfolio Optimization.” Financial Analysts Journal, September/October 1992, pp. 28-43. • Grinold Richard C. and Ronald N. Kahn. Active Portfolio Management. 2nd ed. New York: McGraw- Hill, 1999. • Harvey, Campbell. “Estimation Error and Portfolio Optimization.” Available http://faculty.fuqua.duke.edu/~charvey/Teaching/CDROM_BA453_2003/Estimation_error_and.ppt. • He, Guangliang, and Robert Litterman. “The Intuition Behind Black-Litterman Model Portfolios.” Investment Management Research, Goldman, Sachs & Company, December 1999. • Idzorek, Tom. “A Step by Step Guide to the Black-Litterman Model. Available http://faculty.fuqua.duke.edu/~charvey/Teaching/BA453_2005/Idzorek_onBL.pdf • Litterman, Robert, and the Quantitative Resources Group, Goldman Sachs Asset Management. Modern Investment Management: An Equilibrium Approach. New Jersey: John Wiley & Sons, 2003. • Markowitz, Harry M. "Portfolio Selection." Journal of Finance 7, no. 1 (March 1952), pp 77-91. • Michaud, Richard. Efficient Asset Management. Boston, MA: Harvard Business School Press. 1998. • Scherer, Bernd. “Portfolio Resampling: Review and Critique.” Financial Analysts Journal. November/December 2002, pp98-109.