using mean-variance optimization in the real world: black-litterman vs. resampling
DESCRIPTION
The presentation by Jill Adrogue.TRANSCRIPT
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Using Mean-Variance Optimization in the Real World: Black-Litterman vs. Resampling
Jill AdrogueZephyr Associates, Inc.
September 15, 2005
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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.
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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.
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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.
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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.
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The Mechanics1. 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.
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The Efficient FrontierMaximum Return
Portfolio
Minimum VariancePortfolio
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2%
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16%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18%Annualized Risk (Standard Deviation)
Annu
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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
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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.
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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.
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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.
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The Assets
Historical Data January 1987-July 2005
22.7%12.5%Emerging Markets16.7%8.4%Int'l Equity16.3%14.0%Small Value24.0%10.4%Small Growth14.2%12.8%Large Value18.3%11.8%Large Growth
9.4%8.4%Int'l Bonds4.2%7.4%US Bonds
Std. Dev.Return
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Are the Portfolios Diversified?
• First, let’s look at the diversification of the portfolios resulting from the three techniques.
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Using Historical Forecasts in MVO Leads to Highly Concentrated Portfolios
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Portfolio
Allo
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US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets
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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.
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Black-Litterman Returns as Forecasts
• Black-Litterman Implied Returns make excellent forecasts for use with MVO. The result is diversified, intuitive portfolios.
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Black-Litterman Implied Returns Lead to Diversified Portfolios
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Portfolio
Allo
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US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets
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Resampling1. Estimate returns, standard deviations and
correlations for a set of assets. Michaud does this using historical data.
2. Run a Monte Carlo simulation, creating a new data set. Calculate the return, standard deviation and correlations of the new data set.
3. Create an efficient frontier using the new inputs.
4. Repeat steps 2 and 3 500 times.
5. Calculate the average allocations to the 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
Stage 1 of
MVO
Stage 2 of
MVO
Add’lStep
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US Bonds
Small Growth
Small Value
Int'l Equity
Emerging MarketsLarge Value
Int'l Bonds
Large Growth
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
1.20%
1.40%
0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00%
Monthly Risk (Standard Deviation)
Mon
thly
Ret
urn
Historical Frontier
Resampled Frontier
The Resampled Frontier
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Resampling also Leads to Diversified Portfolios
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US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets
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A Closer Look at the Resampled Frontier
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Where is the Frontier?
Need to select one set of portfolios, but there is no theoretical motivation for Michaud’s averaging
US Bonds
Small Growth
Small Value
Int'l Equity
Emerging MarketsLarge Value
Int'l Bonds
Large Growth
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
1.20%
1.40%
0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00%
Monthly Risk (Standard Deviation)
Mon
thly
Ret
urn
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Portfolio #50
US Bonds
Small Growth
Small Value
Int'l Equity
Emerging Markets
Port 50 Resampled
Large Value
Int'l Bonds
Large Growth
Port 50 Historical
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0.80%
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Monthly Risk (Standard Deviation)
Mon
thly
Ret
urn
Portfolios of rank 50
Resampled Frontier
Historical Frontier
Portfolio 50 Historical
Port 50 Resampled
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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.
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The Resampled Frontier Is Suboptimal
US Bonds
Small Growth
Small Value
Int'l Equity
Emerging MarketsLarge Value
Int'l Bonds
Large Growth
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
1.20%
1.40%
0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00%
Monthly Risk (Standard Deviation)
Mon
thly
Ret
urn
Historical Frontier
Resampled Frontier
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Distribution of Weights to Large Value for Portfolio 84Resampled Weight is 21%
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0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95%100%
Allocation
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yFrequencies and Averaged Weights
Allocation in Resampled Frontier is
21%
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Allocations to Every Asset in Every Portfolio
Allocations to Int'l Equity in Resampled Frontier
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
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4.5%
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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?
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Historical Data
0.39922.72%12.48%Emerging Markets0.29816.73%8.40%Int’l Equity0.65116.32%14.04%Small Value0.29224.04%10.44%Small Growth0.66214.24%12.84%Large Value0.45718.26%11.76%Large Growth0.5299.42%8.40%Int’l Bonds0.9674.16%7.44%US Bonds
SharpeRatioRiskReturn
January 1987-July 2005
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Historical Portfolios
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Portfolio
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US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets
Emerging Markets
US Bonds Global BondsLarge Value
Small Value
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Forecasts and the Resampled Frontier
• The Portfolios from the Resampled Frontier are heavily influenced by the original forecasts.
• Remember, making forecasts is hard.
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Resampled Portfolios
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US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets
US Bonds Global Bonds
Large Value
Small Value
Emerging Markets
Large Growth
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Do the Resampled Portfolios Make Sense?
Resampled Portfolio #25
Large Growth
1%
Large Value6%
Small Value10%
Int'l Bonds17%
Emerging Markets 7%
US Bonds59%
Resampled Portfolio #50
US Bonds26%
Int'l Bonds27%
Large Value14%
Small Value19%
Emerging Markets
11%
Large Growth 3%
Resampled Portfolio #75
US Bonds6%
Int'l Bonds23%
Large Growth
6%
Small Value28%
Emerging Markets
17%
Large Value 20%
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The Market Portfolio: A Neutral Portfolio
3%Emerging Markets29%Int'l Equity1%Small Value1%Small Growth
15%Large Value15%Large Growth14%Int'l Bonds21%US Bonds
Weight
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Using Resampling Means Taking an Unintentional Active Risk
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US Bonds Int'l Bonds LargeGrowth
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Small Value Int'l Equity EmergingMarkets
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Market PortfolioResampled Max Sharpe Ratio Portfolio
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Resampling results in taking active risk—why take bets
without a reason?
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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.
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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.
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Portfolios Created Using the Implied Returns Make Sense
Implied Returns Portfolio #25
US Bonds56%
Large Growth
7%
Large Value5%
Small Value3%
Emerging Markets 5%
Int'l Equity14%
Int'l Bonds10%
Implied Returns Portfolio #50
US Bonds24%
Int'l Bonds14%
Int'l Equity28%
Large Value14%
Large Growth 14%
Small Growth 1%
Small Value2%
Emerging Markets 3%
Implied Returns Portfolio #75
Large Growth
22%
Large Value20%
Int'l Equity47%
Small Growth 2%
Int'l Bonds8%
Emerging Markets 1%
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Portfolio # 25
Resampled Portfolio #25
Large Growth
1%
Large Value6%
Small Value10%
Int'l Bonds17%
Emerging Markets 7%
US Bonds59%
Implied Returns Portfolio #25
US Bonds56%
Large Growth
7%
Large Value5%
Small Value3%
Emerging Markets 5%
Int'l Equity14%
Int'l Bonds10%
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Portfolio #50
Resampled Portfolio #50
US Bonds26%
Int'l Bonds27%
Large Value14%
Small Value19%
Emerging Markets
11%
Large Growth 3%
Implied Returns Portfolio #50
US Bonds24%
Int'l Bonds14%
Int'l Equity28%
Large Value14%
Large Growth 14%
Small Growth 1%
Small Value2%
Emerging Markets 3%
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Portfolio #75
Resampled Portfolio #75
US Bonds6%
Int'l Bonds23%
Large Growth
6%
Small Value28%
Emerging Markets
17%
Large Value 20%
Implied Returns Portfolio #75
Large Growth
22%
Large Value20%
Int'l Equity47%
Small Growth 2%
Int'l Bonds8%
Emerging Markets 1%
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The Implied Returns are a Neutral Starting Point
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US Bonds Int'l Bonds LargeGrowth
Large Value SmallGrowth
Small Value Int'l Equity EmergingMarkets
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Market PortfolioImplied Returns Max Sharpe Ratio Portfolio
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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.
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The Implied Returns are Combined with Your Views to Create New Black-
Litterman Forecasts
Implied Returns Views
Black-Litterman Forecast Returns
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View DistributionPrior Equilibrium Distribution
Risk AversionCoefficient
CovarianceMatrix
Market CapitalizationWeights
( )
Implied Equilibrium Return Vector
New Combined Return Distribution
ViewsUncertainty of
Views
( )Σ mktw( ) 2)( σλ frrE −=
mktwΣ=Π λ
( )Q ( )Ω
( )Ω,~ QN
( ) ( )[ ]( )111 '],[~−−− Ω+Σ PPREN τ
( )ΣΠ τ,~N
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Sample View
• Sample View: Large Growth will have an annualized return of 14% (Implied Return is 12.2%).
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An Active Bet Toward Large Growth
0%
5%
10%
15%
20%
25%
30%
35%
US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity EmergingMarkets
IR/Market PortfolioPortfolio with View
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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.
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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.
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