Optimal Diversification 102727757

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Optimal Diversification A Unified Framework

May 3, 2011

Portfolio Modeling | IPRS

2011 PORTFOLIO MANAGEMENT CONFERENCE

PLEASE SEE ANALYST CERTIFICATION(S) AND IMPORTANT DISCLOSURES STARTING AFTER PAGE 23


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What Portfolios Are Diversified?

Diversified portfolio US Fixed Income and Equity (Barclays Capital US Aggregate, S&P 500) Some solutions: equal weights, market, weights = 1/vol, minimum volatility

Portfolios are very different Not clear what they have in common and how they differ Common feature: No Expected Return Forecast

Barclays Capital

US Agg

S&P

500

Equal Weights 32% 95%

Market Weights 22% 96%

Equal Vol 73% 71%

Minimum Vol 94% 17%

Correlation betweenPortfolios and Assets

Volatility

Max

Drawdown

Equal Weights 2.30 -27.1

Market Weights 2.79 -34.0

Equal Vol 1.32 -9.4

Minimum Vol 1.07 -5.5

Stats of Portfolio Returns%/mo 19912010

Source: Barclays Capital Source: Barclays Capital


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Objective

A unified understanding of popular diversification solutions Each solution = a result of assumptions

Consequences Quality of assumptions quality of solution Different assumptions different solution No assumption is universally right no universal best solution


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Agenda

1. Portfolio Construction General Framework Optimal Risk Return Tradeoff

2. From Portfolio Construction to Diversification Diversification: Expected returns assumed a function of risk

3. Diversification assumptions about expected returns Construct portfolios Examples


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1. Portfolio Construction Framework

Portfolio construction goal: best risk-return tradeoff Definitions forexpected returns, risk, and best tradeoff

Pick one from each column

Portfolio

Risk Measure

Asset

Risk Measure

Expected

Returns Measure

Portfolio

Scaling

VolatilityMarginal Contribution to

portfolio riskMany possibilities

Target portfolio

risk level

VaRTotal Contribution to

portfolio risk

Target portfolio

expected return

Expected ShortfallTarget portfolio

leverage

Downside DeviationOther measure

1. Portfolio Construction Framework 2. From PC to Diversification 3. Exp Returns Assumptions


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1. Portfolio Construction Framework (cont)

Tradeoff: Expected Return/Risk E.g. Sharpe ratio

Tradeoff can be at portfolio or asset level Portfolio level: best tradeoff Maximum tradeoff Asset level: best tradeoff Equal tradeoff across all assets Same result only if Asset Risk Measure = Marginal Contributions



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1. Portfolio Construction Framework Example

Pick one from each column

Choices Delivering Basic Risk-Parity Setup

PortfolioRisk Measure AssetRisk Measure ExpectedReturns Measure PortfolioScaling BestTradeoff

Volatility

Marginal

Contribution to

portfolio risk

The same for all

assets

Target portfolio

risk level

At portfolio level: Max ER/Risk

VaRTotal

Contribution to

portfolio risk

Target portfolioexpected return

At asset level: ER/Risk thesame

Expected ShortfallTarget portfolio

leverage

DownsideDeviation



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2. From Portfolio Construction to Diversification

Portfolio Construction Main goal: best risk-return tradeoff

Diversification Construct portfolios without explicit expected returns (ER) A special case of portfolio construction: no ER

Diversification framework Assume ER depend on risk Each assumption about ER A different diversified portfolio Simple and intuitive assumptions Common diversified portfolios



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2. Diversification Framework

Special case of general portfolio construction framework

PortfolioRisk Measure

AssetRisk Measure

ExpectedReturns Measure

PortfolioScaling

BestTradeoff

Volatility

Marginal

Contribution to

portfolio risk

A function of risk

special todiversification

Target portfolio

risk level


VaRTotal Contribution

to portfolio risk

Target portfolio

expected returns At asset level: ER/Risk the sameExpected Shortfall

Target portfolio

leverage = 0

DownsideDeviation



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3. Diversification Examples

What we will use for our examples: Mean-Variance Optimal (MVO) portfolios Various assumptions about expected returns

PortfolioRisk Measure

AssetRisk Measure

ExpectedReturns Measure

PortfolioScaling

BestTradeoff

Volatility

Marginal

Contribution to

portfolio risk

A function of risk

special todiversification

Target portfolio

risk level


VaRTotal Contribution

to portfolio risk

Target portfolio

expected returns At asset level: ER/Risk the sameExpected Shortfall

Target portfolio

leverage = 0

DownsideDeviation

1. Portfolio Construction Framework 2. From PC to Diversification 3. Diversification Examples

1 P tf li C t ti F k 2 F PC t Di ifi ti 3 Di ifi ti E l


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Expected Return Assumptions

Typical Expected Returns Assumptions and Resulting Diversified Portfolios

Assumption about ERAdditional

Assumptions

Mean-Var OptimumPortfolios

A1.1 Equal across assets Global Minimum Volatility

A1.2 Equal across assetsVolatilities and correlations

are constantEqual Weight

A2.1 Proportional to Volatility Global Minimum Volatilityon correlations

A2.2 Proportional to Volatility Correlations are constantWeights proportional to

1/Vol (Equal-vol)

A3.1 Equal to a linear combination factor betas Portfolio of factors

A3.2 Equal to a linear combination factor betas Factor = market CAPM (market portfolio)

1. Portfolio Construction Framework 2. From PC to Diversification 3. Diversification Examples

1 P tf li C t ti F k 2 F PC t Di ifi ti 3 1 E R t E l


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A1: Expected Returns Equal across Assets

Portfolio Risk Asset Risk Measure Expected Returns Portfolio Scaling Best Tradeoff

VolatilityMarginal Contribution

to portfolio riskEqual across assets

Target portfolio

leverage = 0


Optimal Portfolio Global Minimum Volatility (GMV) Equal weight portfolio (if no views on corrs and vols)

If assets on different scales Wrong assumption No diversification

US Tsy S&P 500 Comm

GMV 86% 19% 18%

Equal Weights 9% 64% 84%

40/40/20 14% 81% 66%

Correlation between Various

Portfolios and Assets, 19902010Example 1:

Barclays Capital US TreasuryS&P 500

S&P Commodities

1. Portfolio Construction Framework 2. From PC to Diversification 3.1 Exp Returns Equal

Source: Barclays Capital

1 Portfolio Construction Framework 2 From PC to Diversification 3 2 Exp Returns Proportional to Vol


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A2: Expected Returns/Vol Are the Same



to portfolio riskProportional to Vol

Target portfolio

leverage = 0


Optimal Portfolio Global Minimum Volatility portfolio using correlation matrix (MVC) Volatility-weighted portfolio (if no views on correlations); Equal Vol

Do analysis in terms of vol-stabilized assets They have the same ER apply results A1

Weights depend only on correlations No correlation views

Beta of on a factorf=

iii rr /~=

constfi *,ii constw /=

iw~

ir~

1. Portfolio Construction Framework 2. From PC to Diversification 3.2 Exp Returns Proportional to Vol

Nwi /1~=

)(nCorrelatio)~(Covariance rr =



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Example 1 (contd): Asset Class Portfolio

US Tsy, Equity, and Commodities: Barclays Capital US Tsy, S&P 500, S&P GS Comm Four diversification methods: Min Vol on Corr (MVC), Equal Vol, Equal Wgt, 40/40/20

Correlation BetweenOptimal Portfolio and Assets

Stats of Portfolio Returns %/mo 19832010

Risk optimization lowers portfolio vol and drawdown, staying fully invested

Min Vol

(MVC)

Equal

Vol

Equal

Weight 40/40/20

Mean 0.34 0.38 0.40 0.41

Volatility 1.54 1.57 2.57 2.29

Drawdown -11.1 -16.2 -40.2 -33.8

US Tsy S&P 500 Comm

Min Vol (MVC) 55% 54% 54%

Equal Vol 52% 57% 62%

Equal Weight 9% 64% 84%

40/40/20 14% 81% 66%

Difference

due to vols

Difference

due to corrs





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Example 1 (contd): Asset Class Portfolio

How Min Vol on Corr (MVC) works Less weight (vol adjusted) on more correlated asset; Still all assets have the same correlation with final portfolio

Optimum Min Vol on Corr Portfolio and Underlying Assets

US Tsy S&P 500 Comm

Avg correlation w/other assets 3% 9% 0%

Avg weight (vol-adjusted) 34% 30% 36%

Avg weight (original assets) 71% 12% 17%

Correlation w/realized portfolio,monthly 19832010

55% 54% 54%



1 Portfolio Construction Framework 2 From PC to Diversification 3.2 Exp Returns Proportional to Vol


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Example 2: S&P 500 Sector Portfolio

22 industry sectors of SP500 Homogenous large universe; correlations high; correlations and vols similar What to expect from theory

Min Vol on Corr (MVC) Equal Vol because correlations are similar Equal Vol Equal Weight because vols are similar Min Vol Market because correlations are high

Stats of Correlations between

Optimal Portfolios and AssetsStats of Portfolio Returns %/mo 19932010

Empirical results match expectationsFor homogenous and correlated universe all these assumptions are reasonable

Min Vol

(MVC)

Equal

Vol

Equal

Weight

Market Weight

Mean 0.33 0.53 0.56 0.47

Volatility 4.09 4.11 4.39 4.43

Drawdown -58.1 -50.4 -53.1 -54.8

Min

Corr

Max

Corr

Avg

Corr

Min Vol (MVC) 51% 79% 64%

Equal Vol 50% 88% 70%

Equal Weight 47% 88% 70%

Market Weight 44% 86% 67%

Difference

due to vols

Difference

due to corrs





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Example 2 (contd): S&P 500 Sector Portfolio

Why correlations may fail: we cannot estimate them well and weights aresmall Errors in corr forecast create 1:1 errors in weights

Behavior in two correlation regimes of the utilities sector w/other sectors Jul-00 Jun-02: realized vs. forecasted corrs are different Jan-03 Dec-04: realized vs. forecasted corrs are similar

Portfolio Volatilities in Two Correlation Regimes

In large universes, imprecise/unstable correlations create issues

Unstable

Correlations

Stable

Correlations

Period Jul-00Jun-02 Jan-03Dec-04

Vol of MVC Portfolio 4.3 2.8

Vol of EqVol Portfolio 3.7 3.1

Vol MVC Vol EqVol 0.6 -0.3

1. Portfolio Construction Framework 2. From PC to Diversification 3 p etu s opo t o a to o




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Example 3: US Treasuries Duration Buckets

6 duration buckets of BarCap US Treasuries Semi-Homogenous small universe; corrs high; corrs and vols differ

A better way may be to optimize directly over risk factors

Min Vol on Corr (MVC) portfolio 50% 13y, 50% 2030y Takes mostly level and slope curve risk If one wants to also take convexity risk move to a portfolio of risk factors

Stats of Portfolio Returns %/mo 19942010

Min Vol

(MVC)

Equal

Vol

Equal

Weight

Market Weight

Mean 0.14 0.19 0.24 0.20

Volatility 0.78 1.17 1.70 1.36

Drawdown -4.6 -7.2 -9.8 -8.4

Difference

due to vols

Difference

due to corrs

Vols and corrs contribute separately to lower portfolio risk, staying fully invested

p p

Min

Corr

Max

Corr

Avg Corr

Min Vol 87.7% 96.0% 93.1%

Equal Vol 85.4% 97.7% 93.5%

Equal Wgt 79.0% 96.6% 91.7%

Mkt Wgt 82.6% 97.2% 93.0%

Stats of correlations between

optimal portfolios and assets


1. Portfolio Construction Framework 2. From PC to Diversification 3.3 Exp Returns = Combin of Beta


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Multi-factor case: ER are a linear combination of factor betas If linear combination of betas = Optimum portfolio of factors then:

To get optimum factor portfolio use previous assumptions We need factorcovariance and assumptions about factorER

A3: Expected Returns = Beta_factor*Constant



to portfolio risk

Linear Combination

of factor betas

Target portfolio

leverage = 0


Investors demand compensation to carry certain sources of risk (factors) Higher beta to these risk factors higher Expected Returns

Optimum portfolio should contain only risks that carry Expected Returns Simple case: one risk factor, factor = market CAPM

Optimum portfolio of assets Optimum portfolio of factors

Optimum portfolio contains only factors



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Example 4: Re-Do Cross-Assets Using Sector Details

US Treasuries, S&P 500, S&P GS Commodities Use sector details: 6 Tsy duration sectors, 22 equity industry sectors, and 5

commodity sectors

Assume: one risk factor per asset class Final portfolio contains only three asset class factors

Asset

Market-Weight Factors Optimum-Weight Factors

US Tsy SP500 Comm US Tsy SP500 Comm

Mean 0.20 0.47 0.35 0.14 0.31 0.08

Volatility 1.36 4.53 6.52 0.78 4.16 3.07

Drawdown -8.4 -55.8 -67.9 -4.6 -58.1 -44.9

Stats of Factors Constructed with Two Definitions

%/mo 19942010

Factor definitions Market weights (asset class indices) Optimum portfolios within the asset class (see Examples 23)




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Example 4 (contd): Cross-Assets Using Sector Details

Diversification methods we will use Use all 33 assets directly, no factors Use market-weight factors (indices); optimize over factors (same as Example 1) Use optimum asset-class factors; optimize over factors Use a mixture of market-weight and optimum factors; optimize over factors

Optimize = Min Vol on Corr

No Factors

Market

Factors

Optimum

Factors Mixed

Mean 0.02 0.24 0.09 0.11

Volatility 1.25 1.45 0.88 0.85

Drawdown -15.8 -11.1 -9.1 -7.1

Stats of Optimum Portfolio Returns for Various Factor Definitions

(%/mo 19962010)

Using entire universe (no factors) gives an unstable matrix large drawdown Optimum factors portfolio has lower risk than market factors portfolio; their correlation

is 85%

Can use factors with different definitions mix and match

For large homogeneous universes (like equity), market factor makes more sense



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Conclusions

Unified framework on diversification Solutions assumptions

Diversification = Portfolio construction ER = F(Risk)

Quality of solutions = quality of assumptions Validated for various settings

Framework customizable based on Assumptions you feel comfortable with Information you have about risk Your preferences


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