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Financial Econometrics In Action: Analyzing Hedge Funds and

Systemic Risk

© 2010 by Andrew W. LoAll Rights Reserved

Andrew W. Lo, MIT and NBER Econometric Methods Lecture

July 16, 2010

NBEROutline of Presentation

Basic Facts About Hedge Funds Hedge-Fund Returns Hedge-Fund Risks and Liquidity Hedge Funds and Systemic Risk August 1998, August 2007,

May 2010,…

© 2010 by Andrew W. Lo, All Rights Reserved Page 27/16/2010

Basic Facts About Hedge Funds

References

Fung, W. and D. Hsieh, 1999, “A Primer on Hedge Funds”, Journal of Empirical Finance 6, 309–331.

Lo, A., 2002, “The Statistics of Sharpe Ratios”, Financial Analysts Journal 58, 36–52. Lo, A., 2008, Hedge Funds: An Analytic Perspective, Chapters 1–2. Princeton, NJ:

Princeton University Press.

© 2010 by Andrew W. Lo, All Rights Reserved Slide 37/16/2010

NBERWhat Are Hedge Funds?Unregulated Investment Companies For “qualified” (sophisticated) investors Need not satisfy regulatory requirements (few investment constraints) High fees, high performance (historically), and high attrition

Alfred Winslow Jones First “hedge fund” in 1949 (market exposure vs. stock selection):

Magnify stock selection (leverage). reduce market exposure (short positions) Hence the term “hedge” Charged 20% incentive fee Eventually included several managers (fund of funds)


NBERWhat Are Hedge Funds?An Example Suppose a mutual fund has $10MM of capital

– It can purchase $10MM of IBM– If IBM goes up by 10%, mutual fund makes $1MM– If IBM goes down by 10%, mutual fund loses $1MM

Suppose a hedge fund has $10MM– It can purchase $30MM of IBM (like a margin account)– It can simultaneously sell $30MM of GE– If IBM goes up by 10%, hedge fund makes $3MM– If GE goes down by 10%, hedge fund makes $3MM– What if IBM goes down and GE goes up by 10%?


NBERWhat Are Hedge Funds?Today: Over 10,000 hedge funds worldwide, controlling $1.7 trillion in assets Typical fee structure: 2% fixed, 20% incentive, high-water mark Most hedge funds are Registered Investment Advisors Most common legal form: Limited Liability Corporation (LLC) Most common corporate structure:

– Management company (the business entity)– Fund (the assets), master-feeder structure

Master Fund (Delaware L.P.)

Domestic Feeder(Delaware L.P.)

Offshore Feeder(Cayman Ltd.)

Brokerage Account


NBERWhat Are Hedge Funds?The Hedge-Fund Industry Is Heterogeneous: Closer to 20–30 cottage industries CSFB/Tremont Index Categories:

Does not include private equity, venture capital, real estate, natural resources, insurance

Event-Driven Multi-StrategyFixed Income ArbitrageGlobal MacroLong/Short EquityManaged FuturesMulti-StrategyRisk Arbitrage

Hedge FundsConvertible ArbitrageDedicated ShortsellerDistressedEmerging MarketsEquity Market NeutralEvent-Driven


NBERWhat Are Hedge Funds?Why Should We Care About Hedge Funds? Hedge funds play a key role in the financial industry

– During normal times, hedge funds are the “tip of the spear”– During bad times, hedge funds are the “canary in the cage”

As unregulated entities, hedge funds innovate rapidly Due to leverage, hedge funds have disproportionate impact on markets Investors in hedge funds include:

– Private investors– Fund of funds– Central banks and sovereign wealth funds– Insurance companies– Pension funds


NBER

Estimated assets

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

Asse

ts (U

Market positions 1,887 1,877 1,968 2,620 2,674 4,247 5,231

539 625 820 972 1,105 1,464 1,910

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

38 58 95 167 167 185 256 367 374 456 490

95 145 238 418 418 463 640 918 935 1,140 1,225

Sources: through Q4 2009 – HFR industry report. 2010 projections – based on CS analysis

2,730

1,440

3,670

1,600

2009 2010

4,200

1,770

6.0

5.0

4.0

3.0

2.0

1.0

$ trillions

Assets In The Hedge Fund Industry, 1990–2010

What Are Hedge Funds?


NBERWhat Are Hedge Funds?


NBERWhat Are Hedge Funds?

Hedge Funds Are The “Galapagos Islands” of Finance Relatively low barriers to entry and exit High levels of compensation Competition and adaptation are extreme New “species” are coming and going constantly Strategies wax and wane over time:

– Risk arbitrage waxing– Statistical arbitrage waning

Supports Adaptive Markets Hypothesis (Lo, 2004, 2005)


NBERBasic Empirical FactsAggregate and Individual Data: CSFB/Tremont hedge-fund indexes

– 14 indexes (13 style categories)– Jan 1994 to present

TASS individual hedge-fund returns– Feb 1977 to present– Monthly returns, AUM, style, fund info

Other sources for individual hedge-fund data– HFR, CISDM, MSCI, Eureka– Some overlap, but many unique funds in each dataset– See Fung and Hsieh (2007)


NBERBasic Empirical Facts

Source: Fung and Hsieh (2007)

Intersections of Hedge-Fund Databases as of 2005




Number of Funds in TASS, HFR, and CISDM Databases




Change in Style Composition of TASS Hedge Funds(Percentage of Total AUM)


NBERBasic Empirical FactsSummary Statistics, Jan 1994 to Feb 2009



Sharpe Ratio Measures Reward/Risk Trade-Off For monthly returns:

Sometimes, Rf is assumed to be 0 SR measures reward per unit risk Typical SR for long-only equities is 0.25 to 0.50 Typical SR for liquid hedge funds is 0.25 to 1.00 Typical SR for illiquid hedge funds is 0.50 to 1.50


NBERBasic Empirical FactsHigh Sharpe Ratios May Be Due To: High-frequency trading (market-making) Illiquid securities (low volatility from smooth returns) Good information (too good?) Fraud?

Investors Are Drawn ToHigh Sharpe Ratios


NBERBasic Empirical FactsConsider A Simple Example of Market-Making: Each day, you can make or lose 0.5% Probability of winning is 58%, probability of losing is 42% Expected return is 8 basis points a day (0.08%) Suppose risk-free rate is 1% per year Assume 250 days of independent bets Sharpe ratio is:


NBERBasic Empirical FactsAn Aside On Estimating Sharpe Ratios (Lo, 2002): For IID returns with finite fourth moments:


NBERBasic Empirical FactsAn Aside On Estimating Sharpe Ratios (Lo, 2002): For stationary and ergodic returns, use GMM:


NBERBasic Empirical FactsAn Aside On Estimating Sharpe Ratios (Lo, 2002): Time aggregation:


NBERBasic Empirical FactsSummary Statistics for Individual Mutual Funds and Hedge Funds

FundStart Date End Date

Sample Size Mean SD ρ1 ρ2 ρ3 p(Q 11)

(%) (%) (%) (%) (%) (%)

Vanguard 500 Index Oct-76 Jun-00 286 1.30 4.27 -4.0 -6.6 -4.9 64.5 Fidelity Magellan Jan-67 Jun-00 402 1.73 6.23 12.4 -2.3 -0.4 28.6 Investment Company of America Jan-63 Jun-00 450 1.17 4.01 1.8 -3.2 -4.5 80.2 Janus Mar-70 Jun-00 364 1.52 4.75 10.5 0.0 -3.7 58.1 Fidelity Contrafund May-67 Jun-00 397 1.29 4.97 7.4 -2.5 -6.8 58.2 Washington Mutual Investors Jan-63 Jun-00 450 1.13 4.09 -0.1 -7.2 -2.6 22.8 Janus Worldwide Jan-92 Jun-00 102 1.81 4.36 11.4 3.4 -3.8 13.2 Fidelity Growth and Income Jan-86 Jun-00 174 1.54 4.13 5.1 -1.6 -8.2 60.9 American Century Ultra Dec-81 Jun-00 223 1.72 7.11 2.3 3.4 1.4 54.5 Growth Fund of America Jul-64 Jun-00 431 1.18 5.35 8.5 -2.7 -4.1 45.4

Convertible/Option Arbitrage May-92 Dec-00 104 1.63 0.97 42.7 29.0 21.4 0.0 Relative Value Dec-92 Dec-00 97 0.66 0.21 25.9 19.2 -2.1 4.5 Mortgage-Backed Securities Jan-93 Dec-00 96 1.33 0.79 42.0 22.1 16.7 0.1 High Yield Debt Jun-94 Dec-00 79 1.30 0.87 33.7 21.8 13.1 5.2 Risk Arbitrage A Jul-93 Dec-00 90 1.06 0.69 -4.9 -10.8 6.9 30.6 Long/Short Equities Jul-89 Dec-00 138 1.18 0.83 -20.2 24.6 8.7 0.1 Multi-Strategy A Jan-95 Dec-00 72 1.08 0.75 48.9 23.4 3.3 0.3 Risk Arbitrage B Nov-94 Dec-00 74 0.90 0.77 -4.9 2.5 -8.3 96.1 Convertible Arbitrage A Sep-92 Dec-00 100 1.38 1.60 33.8 30.8 7.9 0.8 Convertible Arbitrage B Jul-94 Dec-00 78 0.78 0.62 32.4 9.7 -4.5 23.4 Multi-Strategy B Jun-89 Dec-00 139 1.34 1.63 49.0 24.6 10.6 0.0 Fund of Funds Oct-94 Dec-00 75 1.68 2.29 29.7 21.1 0.9 23.4

Mutual Funds

Hedge Funds


NBERBasic Empirical FactsCorrelations Between CS/Tremont Indexes and Other Indexes

January 1994 to February 2009






Hedge-Fund Returns

References

Ackermann, C., McEnally, R. and D. Ravenscraft, 1999, “The Performance of Hedge Funds: Risk, Return, and Incentives'', Journal of Finance 54, 833–874.

Lo, A., 2008, “Where Do Alphas Come From?: A New Measure of the Value of Active Investment Management”, Journal of Investment Management 6, 1–29.


NBERMotivationTotal Return Comparison, Jan 1994 to Feb 2009


NBER

What Do We Mean By “Active” Management? Excess expected return above some benchmark “Active” bets means departures from the benchmark Ft

Benchmarks are considered “passive”– Easy to achieve, very little skill involved, low fees

Traditional measure of active management—IR:

But what if there are other “systematic” factors? Such factors can yield positive IR, but require no skill

Motivation


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For example:

In this case, excess returns may be passively achieved One natural implication: “hedge-fund beta” products So what do we really mean by “active management” in a multi-

factor world?

⇒ Where Do Alphas Come From?

Motivation

αi ??


NBERA New MeasureA New Definition of Active Management: Active management consists of time-varying portfolio weights

based on information Passive management consists of either non-stochastic

portfolios, or where the time-variation is not information-based, e.g., value-weighted portfolios, random portfolios

To measure the value of active management, ask whether time-variation in weights increases expected return E[Rt]

We don’t need to know Xt-1 to estimate the value of active management


NBERA New MeasureA New Measure of the Value of Active Management:

Recall the definition of covariance:

True for sample estimators as well:


NBERA New Measure

A New Measure of the Value of Active Management:

Active Passive



Two sources of expected returns:1. Average exposures to risk premia µi

2. Correlation between weights and returns (asset-timing ability)

Note: weights are determined prior to returns (otherwise, arbitrage is possible: buy low/sell high), hence they capture predictability of current asset returns by past information



Active component can be estimated without covariances Active component can be estimated without weights ωi

In fact, only average weights and returns are needed Ideal for institutional investors of active managers and hedge

funds (position transparency unnecessary)


NBERA New MeasureImplicit Assumptions: Returns are stationary and parameters are known Xt is information (need not be private)

Properties of Determine Active Value-Added If weights are non-stochastic, covariances are 0 If weights are uncorrelated with returns, covariances are 0 Passive portfolio earns risk premia on average Unlike IR, θ is a dynamic measure of predictability


NBERExample 1

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Constant weights (no timing) No active management Active component: 0% Passive component: 100%


NBERExample 1

Time-varying weights Adds 0.13% per month Active component: 9.7% Passive component: 90.3%

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NBERExample 1

Time-varying weights Subtracts 0.13% per month Active component: -12.05% Passive component: 112.05%

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NBERExample 2Market Timing Equities vs. cash, where equity returns follow AR(1):

Consider 0/1 investment policy based on lagged return


NBERExample 2Market Timing Then expected returns can be computed explicitly:


NBERExample 2


NBERThe Sampling IntervalConsider Warren Buffett vs. Steven Cohen Buffett makes a few decisions per year Cohen makes a few decisions per minute One symptom: very different Sharpe ratios! How frequently do we need to sample their weights?

Do these terms capture the variation in the manager’s weights, i.e., decisions?


NBERThe Sampling IntervalCase 1: Decision Interval Coincides with Sampling Interval


NBERThe Sampling IntervalCase 2: Decision Interval Longer Than Sampling Interval





Recall the definition of covariance:

True for sample estimators as well:


NBERExample 1

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Constant weights (no timing) No active management Active component: 0% Passive component: 100%


NBERExample 1

Time-varying weights Adds 0.13% per month Active component: 9.7% Passive component: 90.3%

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NBERExample 1

Time-varying weights Subtracts 0.13% per month Active component: -12.05% Passive component: 112.05%

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NBERExample 2Market Timing Equities vs. cash, where equity returns follow AR(1):

Consider 0/1 investment policy based on lagged return


NBERExample 2Market Timing Then expected returns can be computed explicitly:


NBERExample 2


NBERThe Sampling IntervalConsider Warren Buffett vs. Steven Cohen Buffett makes a few decisions per year Cohen makes a few decisions per minute One symptom: very different Sharpe ratios! How frequently do we need to sample their weights?

Do these terms capture the variation in the manager’s weights, i.e., decisions?


NBERThe Sampling IntervalCase 1: Decision Interval Coincides with Sampling Interval




NBEREmpirical AnalysisConsider Contrarian Trading Strategy: Lo and MacKinlay (1990)


NBEREmpirical Analysis

Profitability can be quantified:


NBEREmpirical AnalysisApply This To NASDAQ Size-Decile Returns January 2, 1990 to December 29, 1995 Use only 5 smallest deciles to maximize returns May not be realistic, but is illustrative

R1t+1 R2t+1 R3t+1 R4t+1 R5t+1

R1t 10.0% 21.5% 15.8% 18.1% 16.7% R2t 23.4% 15.4% 20.2% 19.7% 15.8% R3t 26.2% 25.0% 15.2% 23.9% 21.6% R4t 25.4% 27.0% 24.3% 18.2% 18.7% R5t 25.4% 26.6% 26.5% 26.2% 19.4%

Lead/Lag Effect Among NASDAQ Five Smallest Size-Decile Portfolios


NBEREmpirical AnalysisCumulative Return of Contrarian Strategy on NASDAQ Deciles 1–5

January 2, 1990 to December 29, 1995


NBEREmpirical Analysis

Summary Statistics for NASDAQ Decilesand Contrarian Strategy, Daily

January 2, 1990 to December 29, 1995

Statistic Decile 1 Decile 2 Decile 3 Decile 4 Decile 5 Rpt

Mean*250 27.4% 17.5% 14.0% 13.7% 12.8% 31.4% SD*sqrt(250) 12.2% 9.8% 8.9% 9.1% 9.5% 7.9% SR*sqrt(250) 2.25 1.78 1.58 1.50 1.35 3.95 Min -2.9% -2.7% -2.7% -3.3% -3.5% -2.2% Median 0.1% 0.1% 0.1% 0.1% 0.1% 0.1% Max 6.7% 3.6% 2.0% 2.1% 2.3% 2.4% Skew 0.6 0.0 -0.5 -0.7 -0.9 -0.1XSKurt 5.1 2.4 2.1 3.1 3.9 1.7ρ1 10.0% 15.4% 15.2% 18.2% 19.4% 4.7% ρ2 10.3% 7.7% 10.1% 13.9% 10.5% 0.9% ρ3 5.7% 4.2% 7.5% 9.2% 11.0% 7.5%


NBEREmpirical AnalysisMeasure of Active Management (Daily):

Estimate SE t-stat

Portfolio Mean * 250 31.4% 0.3% 91.00 Risk Premia * 250 -0.6% 3.5% -0.17 Active Component * 250 32.0% 3.5% 9.24 Active Ratio 101.9% 0.3% 354.40

More than 100% of returns are active! Why is passive component negative? Mean reversion!

– On average, holding losers and shorting winners is costly


NBEREmpirical AnalysisNow Consider Monthly Transparency Manager provides month-end weights and monthly returns

(summed) for all positions How informative is this information for the strategy? Note:


NBEREmpirical AnalysisSummary Statistics for NASDAQ Deciles

and Contrarian Strategy, MonthlyJanuary 1990 to December 1995

Statistic Decile 1 Decile 2 Decile 3 Decile 4 Decile 5 Rpt

Mean*12 27.5% 17.4% 13.9% 13.7% 12.8% -4.0% SD*sqrt(12) 20.6% 17.7% 15.6% 15.0% 15.9% 8.8% SR*sqrt(12) 1.34 0.98 0.89 0.91 0.80 -0.45 Min -8.0% -11.3% -9.0% -9.7% -11.4% -6.6% Median 1.6% 1.0% 1.2% 1.0% 1.5% -0.5% Max 26.4% 21.5% 18.1% 16.8% 16.2% 9.7% Skew 1.2 0.9 0.7 0.5 0.1 1.2XSKurt 3.3 3.0 2.1 2.0 1.2 4.7ρ 1 36.4% 43.7% 43.2% 41.4% 45.0% 8.2% ρ 2 17.3% 16.6% 18.9% 10.1% 13.7% 15.7% ρ 3 -5.6% -2.7% -3.1% -7.8% -7.0% -3.2%


NBEREmpirical AnalysisMeasure of Active Management (Monthly):

Estimate SE t-statPortfolio Mean * 12 -4.0% 1.0% -3.98 Risk Premia * 12 0.1% 4.0% 0.03 Active Component * 12 -4.1% 4.1% -1.01 Active Ratio 102.6% 11.8% 8.66

Total return is negative based on month-end weights! Active component is responsible for this negative return Results bear no relation to the daily strategy!


NBERExtension To Multi-Factor ModelsConsider Multi-Factor Model for Individual Securities:

S&P 500 Lehman Bond US Dollar GSCI VIX Credit

MSCI U.S. Growth/Value U.S. Large/Small Cap Foreign Growth/Value Foreign Large/Small Cap Liquidity

Option-Based Strategies Property & Casualty Real Estate Timber Hurricane Earthquakes

Possible Factors:


NBER

Now Consider An Active Manager:

Extension To Multi-Factor Models

Security Selection

Factor Timing

Factor Exposures


NBER

Expected Returns Are the Sum of Three Components:1. Risk Premia (Average Betas)2. Security Selection (Alpha)3. Factor Timing (Time-Varying Portfolio Betas)

Active Passive



NBER

Expected Returns Are the Sum of Three Components:1. Risk Premia (Average Betas)2. Security Selection (Alpha)3. Factor Timing (Time-Varying Portfolio Betas)



NBERSummary“Active” Management Involves Changing Weights Measure of active management must involve weights Correlation between weights and returns is key Good managers have large positive correlations Implies new definitions for “active” and “passive”

Constant- or undirected-weight component is “passive” “Passive” is not always simple, but is often easier to implement Investors should pay less for passive management Investors should pay more for active management

Implies new methods for performance attribution Requires new data: average weights, average returns Sampling interval must be at least as fine as decision interval Factor models provide additional transparency


Hedge-Fund Risks and Liquidity

References

Fung, W. and D. Hsieh, 2001, “The Risk in Hedge Fund Strategies: Theory and Evidence from Trend Followers'', Review of Financial Studies 14, 313–341.

Getmansky, M., Lo, A. and I. Makarov, 2004, “An Econometric Analysis of Serial Correlation and Illiquidity in Hedge-Fund Returns”, Journal of Financial Economics74, 529–609.

Lo, A., 2001, “Risk Management For Hedge Funds: Introduction and Overview”, Financial Analysts Journal 57, 16–33.


NBER

© 2010 by Andrew W. Lo, All Rights Reserved7/16/2010 Page 72

Risk Models

Hedge Fund Risk Exposures are Different: Heterogeneity of Hedge-Fund Styles

Phase-Locking Regime Shifts Nonlinearities

Long/Short Equity Fund– Market Beta– Industry/Sector Exposure– Value/Growth– Stock-Loan Constraints– Execution Costs

Fixed-Income Arb Fund– Yield Curve Model– Credit Exposure– Liquidity Exposure– Leverage Constraints– Macroeconomic Factors

NBER


Risk ModelsCorrelation Analysis Can Be Misleading: Correlation Is A Linear Measure Hedge Fund Returns May Be Nonlinear One Example: “Phase-Locking”

NBER

When no event occurs (It = 0):

When an event occurs (It = 1):


Risk Models

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Risk Models

Unconditional Correlation:

Let p = 0.001, σz = 10σε :

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Two-State Markov Regime-Switching Model:

State 1 State 0

Risk Models

NBER

Regime-Switching Probability Estimates forCSFB/Tremont Hedge-Fund Indexes

January 1994 to September 2006


Risk Models

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Regime-Switching Probability Estimates for CSFB/TremontFixed-Income Arbitrage Index


Risk Models

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Regime-Switching Probability Estimates for CSFB/TremontConvertible Arbitrage Index


Risk Models

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Regime-Switching Probability Estimates for CSFB/TremontManaged Futures IndexJanuary 1994 to September 2006

Risk Models

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Risk Models

Aggregate Hedge-Fund Indexes: Estimate Parameters Via MLE Estimate Unconditional State Probabilities Aggregate Probabilities Across Indexes

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Risk Models

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Liquidity

Definition of Liquidity: Price, Time, and Size Many Hedge Funds Are Not “Liquid” Liquidity Risk Has Multiple Dimensions

– Valuation, Correlation, Timing Nothing Wrong With Illiquidity, If Fully Disclosed Problems When Illiquidity Is Not Recognized Simple Indicator of Illiquidity:

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