using mean-variance optimization in the real world: black-litterman vs. resampling

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

Jill AdrogueZephyr Associates, Inc.

September 15, 2005

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.

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.

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.

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.

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.

The Efficient FrontierMaximum Return

Portfolio

Minimum VariancePortfolio

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16%

0% 2% 4% 6% 8% 10% 12% 14% 16% 18%Annualized Risk (Standard Deviation)

Annu

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ed R

etur

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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

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.

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.

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.

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

Are the Portfolios Diversified?

• First, let’s look at the diversification of the portfolios resulting from the three techniques.

Using Historical Forecasts in MVO Leads to Highly Concentrated Portfolios

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US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity Emerging Markets

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.

Black-Litterman Returns as Forecasts

• Black-Litterman Implied Returns make excellent forecasts for use with MVO. The result is diversified, intuitive portfolios.

Black-Litterman Implied Returns Lead to Diversified Portfolios

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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

US Bonds

Small Growth

Small Value

Int'l Equity

Emerging MarketsLarge Value

Int'l Bonds

Large Growth

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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)

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thly

Ret

urn

Historical Frontier

Resampled Frontier

The Resampled Frontier

Resampling also Leads to Diversified Portfolios

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A Closer Look at the Resampled Frontier

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


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%


Mon

thly

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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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0.20%

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%


Mon

thly

Ret

urn

Portfolios of rank 50

Resampled Frontier

Historical Frontier

Portfolio 50 Historical

Port 50 Resampled

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.

The Resampled Frontier Is Suboptimal

US Bonds

Small Growth

Small Value

Int'l Equity


Int'l Bonds

Large Growth

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0.60%

0.80%

1.00%

1.20%

1.40%

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Mon

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Historical Frontier

Resampled Frontier

Distribution of Weights to Large Value for Portfolio 84Resampled Weight is 21%

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Allocation

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yFrequencies and Averaged Weights

Allocation in Resampled Frontier is

21%

Allocations to Every Asset in Every Portfolio

Allocations to Int'l Equity in Resampled Frontier

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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?

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

Historical Portfolios

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Emerging Markets

US Bonds Global BondsLarge Value

Small Value

Forecasts and the Resampled Frontier

• The Portfolios from the Resampled Frontier are heavily influenced by the original forecasts.

• Remember, making forecasts is hard.

Resampled Portfolios

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US Bonds Global Bonds

Large Value

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Large Growth

Do the Resampled Portfolios Make Sense?

Resampled Portfolio #25

Large Growth

1%

Large Value6%

Small Value10%

Int'l Bonds17%

Emerging Markets 7%

US Bonds59%


US Bonds26%

Int'l Bonds27%

Large Value14%

Small Value19%

Emerging Markets

11%

Large Growth 3%


US Bonds6%

Int'l Bonds23%

Large Growth

6%

Small Value28%

Emerging Markets

17%

Large Value 20%

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

Using Resampling Means Taking an Unintentional Active Risk

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US Bonds Int'l Bonds LargeGrowth

Large Value SmallGrowth

Small Value Int'l Equity EmergingMarkets

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Market PortfolioResampled Max Sharpe Ratio Portfolio

Resampling results in taking active risk—why take bets

without a reason?

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.

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.

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%


US Bonds24%

Int'l Bonds14%

Int'l Equity28%

Large Value14%

Large Growth 14%

Small Growth 1%

Small Value2%

Emerging Markets 3%


Large Growth

22%

Large Value20%

Int'l Equity47%

Small Growth 2%

Int'l Bonds8%

Emerging Markets 1%

Portfolio # 25


Large Growth

1%

Large Value6%

Small Value10%

Int'l Bonds17%

Emerging Markets 7%

US Bonds59%


US Bonds56%

Large Growth

7%

Large Value5%

Small Value3%

Emerging Markets 5%

Int'l Equity14%

Int'l Bonds10%

Portfolio #50


US Bonds26%

Int'l Bonds27%

Large Value14%

Small Value19%

Emerging Markets

11%

Large Growth 3%


US Bonds24%

Int'l Bonds14%

Int'l Equity28%

Large Value14%

Large Growth 14%

Small Growth 1%

Small Value2%

Emerging Markets 3%

Portfolio #75


US Bonds6%

Int'l Bonds23%

Large Growth

6%

Small Value28%

Emerging Markets

17%

Large Value 20%


Large Growth

22%

Large Value20%

Int'l Equity47%

Small Growth 2%

Int'l Bonds8%

Emerging Markets 1%

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

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.

The Implied Returns are Combined with Your Views to Create New Black-

Litterman Forecasts

Implied Returns Views

Black-Litterman Forecast Returns

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

Sample View

• Sample View: Large Growth will have an annualized return of 14% (Implied Return is 12.2%).

An Active Bet Toward Large Growth

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US Bonds Int'l Bonds Large Growth Large Value Small Growth Small Value Int'l Equity EmergingMarkets

IR/Market PortfolioPortfolio with View

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.

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.

using mean-variance optimization in the real world: black-litterman vs. resampling

Education

set of portfolios

mvo blacklitterman technique

blacklitterman inputs

limitations of mvo returns

maximumreturn portfolios

return forecasts

resulting portfolios

minimum variance portfolios