Using Mean-Variance Optimization in the Real World: Black-Litterman vs. Resampling

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  • 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.