Risk Management For Hedge Funds: Introduction and Overview

Transcription

1 Risk Management For Hedge Funds: Introduction and Overview Andrew W. Lo Harris & Harris Group Professor Director, Laboratory for Financial Engineering Massachusetts Institute of Technology March 2004 Sloan Innovation Period Seminar c 2004 by A. Lo

2 Outline 1. Industry Overview 2. Why Risk Management? 3. Current Risk Management Practices 4. Data Issues and Survivorship Bias 5. Dynamic Risk Management 6. Correlation and Risk Adjustments 7. Risk and Performance Attribution 8. Estimating Smoothing and Illiquidity Profiles 9. Conclusions 10. Additional References c 2004 by A. Lo

3 Industry Overview Hedge Funds: Unregulated Investment Companies For Qualified (Sophisticated) Investors Need Not Satisfy Regulatory Requirements Few Constraints on Investment Strategies High Fees, High Performance, High Attrition Alfred W. Jones: Established First Hedge Fund in 1949 Market Exposure vs. Stock Selection Market Exposure = Long Position Short Position Capital Magnify Stock Selection (Leverage) Reduce Market Exposure (Short Positions) Hence The Term Hedge Charged 20% Incentive Fee Eventually Included Several Managers (Fund of Funds) MIT c 2004 by A. Lo Page 1/53

4 Today: Over 7,000 Hedge Funds Over $800B In Assets (More If Other Institutions Included) Typical Fee Structure: 1% Fixed, 20% Incentive High Water Mark Various Types MIT c 2004 by A. Lo Page 2/53

5 MIT c 2004 by A. Lo Page 3/53 Summary Statistics For Hedge Fund Index Returns Strategy Monthly Data, January 1996 to November 1999 Number of Mean S.D. Sharpe Max ˆρ 1 ˆρ 2 ˆρ 3 ˆρ 4 Funds (%) (%) Ratio D.D. (%) (%) (%) (%) Currency Trading ED-Distress ED-Merger-Arb EM-Equity Emerging Market EM-Fixed Income Event Driven Fund of Funds Futures Trading Growth High Yield Macro Opportunistic Other Relative Value RV-Convertible RV-EQWLS RV-Option-Arb RV-Other-Stat-Arb ShortSelling Value EDCorp Restructure* Market Timing* RV-Mortgage-Backed* RV-CreditSpread* Equal-Weighted Index* Funds-Weighted Index* SP Source: AlphaSimplex Group

6 Motivation For Hedge Fund Investments: Alpha Fewer Regulatory Constraints Alpha Quicker To Exploit Market Opportunities Alpha Incentive Fees Attract Unique Talent Pool Alpha Exciting, Sexy, Stimulating Alpha Typical Hedge Fund Manager: Manager Knows Best (Requires Broad Discretion) Trading Strategies Highly Proprietary (No Transparency) Total Return Is Ultimate (And Often Only) Objective Risk Management Not Important (A Necessary Evil) Regulatory Constraints Should Be Avoided Little Intellectual Property Manager Is The Fund Typical Institutional Investor: Fiduciaries Must Understand The Investment Process Manager s Investment Policies May Need To Be Constrained Performance Is Multi-Faceted (Risk, Tracking Error) Risk Management and Risk Transparency Are Essential Institutions Are Highly Regulated (ERISA, etc.) Process Is As Important As People Gap Between Managers and Institutional Investors MIT c 2004 by A. Lo Page 4/53

7 Why Risk Management? Isn t Performance More Important Than Risk Control? Risk Control Can Be A Source Of Alpha! E[R ] and SD[R ], R Max[R, 10%] SD[R] E[R] 5% 0% 5% 10% 15% 20% 5% 10% 25% 50% 75% 100% 4.6% 0.0% 5.0% 10.0% 15.0% 20.0% 4.4% 4.9% 5.0% 5.0% 5.0% 5.0% 3.1% 0.7% 5.2% 10.0% 15.0% 20.0% 7.8% 8.9% 9.6% 9.9% 10.0% 10.0% 2.2% 5.1% 8.5% 12.3% 16.4% 20.8% 18.3% 19.8% 21.1% 22.2% 23.1% 23.8% 10.7% 13.2% 15.9% 18.9% 22.2% 25.7% 38.7% 39.9% 41.0% 42.2% 43.3% 44.4% 17.7% 20.2% 22.7% 25.5% 28.4% 31.5% 61.5% 62.3% 63.2% 64.1% 65.0% 66.0% 23.5% 25.9% 28.5% 31.2% 34.0% 37.0% 85.7% 86.2% 86.8% 87.5% 88.2% 88.9% Expected values E[R ] (first rows) and standard deviations SD[R ] (second rows) of R Max[R, 10%] for lognormally distributed return R with expectation E[R] and standard deviation SD[R]. Source: AlphaSimplex Group, LLC. MIT c 2004 by A. Lo Page 5/53

8 E[W T ] = $10,000 (1 + E[R ]) T R Max[R, 10%], T = 20 SD[R] E[R] 5% 0% 5% 10% 15% 20% E[W T ]: $3,585 $10,000 $26,533 $67,275 $163,665 $383,376 10% $5,298 $11,513 $27,581 $67,798 $163,842 $383,414 25% $10,890 $19,994 $39,698 $83,540 $182,072 $402,061 50% $21,932 $37,182 $66,670 $125,482 $245,455 $493,399 75% $41,992 $67,805 $114,255 $200,149 $362,635 $675, % $75,951 $118,721 $191,951 $320,299 $549,922 $967,825 Expected values of end-of-period wealth E[WT ] = W 0(1 + E[R ]) T for truncated return R Max[R, κ] for lognormally distributed return R with expectation E[R], standard deviation SD[R], truncation point κ = 10%, over T periods for initial wealth W 0 = $10,000. Rows labelled E[W T ] report the expected values of end-of-period wealth, W 0 (1 + E[R]) T, for the return R. Source: AlphaSimplex Group, LLC. MIT c 2004 by A. Lo Page 6/53

9 Current Risk Management Practices Origins of Current Risk Management Practices: Derivatives Trading Desks Complex OTC Securities Specialized Portfolios Dynamic Hedging Risks Main Tool VAR Value at Risk is an estimate, with a predefined confidence interval, of how much one can lose from holding a position over a set horizon. Potential horizons may be one day for typical trading activities or a month or longer for portfolio management. The methods described in our documentation use historical returns to forecast volatilities and correlations that are then used to estimate the market risk. These statistics can be applied across a set of asset classes covering products used by financial institutions, corporations, and institutional investors. JP Morgan, Introduction to RiskMetrics (1995) Typical Risk Management Protocol: 1. Identify Risk Exposures 2. Evaluate Value-At-Risk (VAR) 3. Target Risks To Hedge 4. Select Hedging Vehicles 5. Evaluate Post-Hedge VAR MIT c 2004 by A. Lo Page 7/53

10 Several Important Shortcomings: VAR Is Generic VAR Is Static VAR Is Parametric VAR Is Focused on Outliers VAR Is Purely Statistical Risk Measurement Vs. Risk Management Different Hedge Funds Have Different Risks: Different Securities and Contract Terms Different Markets Different Horizons Different Portfolio Construction Methods Different Trading Styles Different Liquidity Different Objectives For Example, Compare Risk Exposures Of: Equity Hedge Fund Fixed-Income Hedge Fund MIT c 2004 by A. Lo Page 8/53

11 Equity Hedge Fund: Investment Style: Value, Growth, Quantitative Factor-Model Specification, Estimation, Updates Portfolio Construction and Optimization Stock Loan Considerations Execution Costs and Other Implementation Issues Performance Evaluation and Attribution Fixed-Income Hedge Fund: Yield Curve Model Specification, Estimation, Updates Prepayment Model Specification, Estimation, Updates Optionality (Callable, Convertible, Puttable, etc.) Credit Risk Inflationary Pressures, Central Banking Policy Macroeconomic Factors and Events Current Risk Management Practices Are Inadequate: Mark-to-Market vs. Mark-to-Model Stress Testing Scenarios Capital Adequacy Requirements MIT c 2004 by A. Lo Page 9/53

12 Five Unique Aspects of Hedge Fund Investments: 1. Data Issues and Survivorship Bias 2. Dynamic Risk Management 3. Correlation and Risk Adjustments 4. Risk and Performance Attribution 5. Psychology of Risk Preferences MIT c 2004 by A. Lo Page 10/53

13 Data Issues and Survivorship Bias What You Cannot Measure, You Cannot Control : Many Hedge Fund Databases Exist Complete Database of Hedge Funds Does Not Exist Mostly Monthly NAV s For Illiquid Funds, NAV s Are Ambiguous Heterogeneous Terms (Lock-Ups, Incentives, HWM s) AUM Not Always Reported Compare With Mutual Fund Industry Survivorship Bias: Survivors Will Have Higher Average Returns Survivors Will Have Lower Volatility This Is True Even If Managers Have No Alpha! Past Performance Future Returns MIT c 2004 by A. Lo Page 11/53

14 Consider n Funds, IID: ˆα 1 σ 1, ˆα 2 σ 2,..., ˆα n σ n F (x) (1) Assume All n Funds Have α j = 0, j = 1,..., n X j ˆα j /σ j Called Sharpe Ratio Risk-Adjusted Performance Index F (x) Contains All Knowable Information About X j Suppose We Select The Best Fund: X (n) Arg Max [ X 1,..., X n ] (2) Pr ( X (n) < x ) = Pr ( Max[X 1,..., X n ] < x ) = Pr ( X 1 < x,..., X n < x ) = Pr ( X 1 < x ) Pr ( X n < x ) Pr ( X (n) < x ) = F n (x) (3) E [ X (n) ] = Var [ X (n) ] = xdf (x) (4) x 2 df (x) E [ X (n) ] 2 (5) δ = F n (C δ ) C δ = F 1 (δ 1/n ) (6) MIT c 2004 by A. Lo Page 12/53

15 Mean, SD, and Quantiles of X (n) For Selected n For IID {X j } N (0, 1) n E [X (n)] SD [X (n)] C C n = 1 n = 10 n = 20 n = Density MIT c 2004 by A. Lo Page 13/53

16 What If X j s Are Correlated? Selection Bias Less Severe Difficult To Obtain Analytic Results Consider Special Case (Equi-Correlated X j s): Corr[ X i, X j ] = ρ, i j (7) Assume Normality Distribution of X (n) Computable N=1 N=10 N=20 N= (a) ρ = N=1 N=10 N=20 N= (b) ρ = 0.50 MIT c 2004 by A. Lo Page 14/53

17 Mean, SD, and Quantiles of X (n) For Selected n For Equi-Correlated {X j } N (0, 1) n E [X (n)] SD [X (n)] C C ρ = 0% ρ = 10% ρ = 25% ρ = 50% MIT c 2004 by A. Lo Page 15/53

18 Dynamic Risk Management Three Types of Investment Strategies: Passive Active Hyperactive Hedge Fund Investments Are Dynamic: Rarely Buy-and-Hold Fees Reflect Active Management Information-Driven Trading Sensitive To Economic and Market Conditions Effects Are Generally Nonlinear Static Risk Analytics Are Not Appropriate Example: Capital Decimation Partners, L.P. Market Neutral Equity Hedge Fund Established January 1992 Initial Assets: $10M 8-Year Track Record (96 Months) MIT c 2004 by A. Lo Page 16/53

19 Capital Decimation Partners, L.P. Performance Summary January 1992 to December 1999 Statistic SPX CDP Monthly Mean 1.4% 3.7% Monthly Std. Dev. 3.6% 5.8% Min Month 8.9% 18.3% Max Month 14.0% 27.0% Annual Sharpe Ratio # Negative Months 36/96 6/96 Correlation 100.0% 59.9% Total Return 367.1% % MIT c 2004 by A. Lo Page 17/53

20 MIT c 2004 by A. Lo Page 18/53 Month Capital Decimation Partners, L.P. Monthly Performance History SPX CDP SPX CDP SPX CDP SPX CDP SPX CDP SPX CDP SPX CDP SPX CDP Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year

21 CDP vs. S&P 500 Returns, Frequency More Monthly Return CDP SP500 MIT c 2004 by A. Lo Page 19/53

22 MIT c 2004 by A. Lo Page 20/53 Total Return $30 $25 $20 $15 $10 $5 S&P 500 $0 Jan- 92 Jul- 92 CDP Jan- 93 Jul- 93 Jan- 94 S&P 500 vs. CDP Total Return Jul- 94 Jan- 95 Jul- 95 Jan- 96 Date Jul- 96 Jan- 97 Jul- 97 Jan- 98 Jul- 98 Jan- 99 Jul- 99

23 Capital Decimation Partners II, L.P. Week P t Position Value Financing t ($) (Shares) ($) ($) , , , , , , , , , ,013 1,204,981 1,240, ,128 1,000,356 1,024, ,510 1,150,101 1,185, , , , , , , , , , , , , , , , , ,711 44, , ,205 76, , , , , , , , , , , , , , , , , , , , ,777 95, , ,526 87, , ,832 71, , ,408 83, , ,986 59, ,445 97,782 45, ,140 85,870 33,445 Positions In XYZ Stock Over Six Months: Initial Price: $40 Expected Return: 7% Standard Deviation: 13% MIT c 2004 by A. Lo Page 21/53

24 Trading Strategy Is A Synthetic Option! Delta-Hedging Strategy (Black-Scholes Formula) Replicates Short European Put On: 10,000,000 Shares of XYZ Strike: $25 Time To Maturity: 2 Years Borrowing Rate: 5% Put Options Worth Only $14,693 Profitable Most of the Time How Can You Tell If A Manager Is Doing This? New Analytics Are Needed Risk Attribution Performance Attribution MIT c 2004 by A. Lo Page 22/53

25 Correlation and Risk Adjustments Hedge Funds As Diversifying Investment: Low Correlation To Market Indexes Market Neutral Why Neutral? How Neutral? Why Neutral? Does Diversification Make Sense For You? Too Much Diversification May Mean Lower E[R] R p = 1 n [ E[R 1 ] + E[R 2 ] + + E[R n ] ] + [ ] 1 ɛ 1 + ɛ ɛ n n }{{} 0 (8) Depends on Quality of Manager Selection Process Complete Diversification T-Bill Returns MIT c 2004 by A. Lo Page 23/53

26 MIT c 2004 by A. Lo Page 24/53 Correlation Matrix For Hedge Fund Index Returns Monthly Data, January 1996 to November Currencies ED-Distress ED-Merger-Arb EM-Equity Emerging Market EM-Fixed Income Event Driven Fund of Funds Futures Trading Growth High Yield Macro Opportunistic Other Relative Value RV-Convertible RV-EWLS RV-Option-Arb RV-Other-Stat-Arb ShortSelling Value SP Source: AlphaSimplex Group, LLC

27 How Neutral? Where Does Alpha Come From? E[R i ] = β 1i E[Λ 1 ] + + β Ki E[Λ K ] + β (K+1)i E[Λ K+1 ] + + β Mi E[Λ M ] (9) }{{} α i Cannot Be Completely Neutral If E[R i ] > R f Do You Know Your Risk Exposures? Are You Comfortable With Your Risk Exposures? Risk Attribution Analysis Correlation and Portfolios of Hedge Funds: Risk Attribution Essential, But Not Enough Correlation Is A Linear Measure of Association Hedge Fund Returns Are Nonlinear Phase-Locking Behavior Is Just One Example R it = α i + β i Λ t + I t Z t + ɛ it (10) Z t = Crisis Component (σ 2 z /σ2 λ 1) (11) I t = { 0 with probability 1 p = with probability p = (12) Corr [ R it, R jt I t = 0 ] = β i β j σ 2 λ β 2 i σ2 λ + σ2 ɛ i β 2 j σ 2 λ + σ2 ɛ j (13) MIT c 2004 by A. Lo Page 25/53

28 Corr [ R it, R jt I t = 1 ] = β i β j σ 2 λ + σ2 z β 2 i σ2 λ + σ2 z + σ2 ɛ i β 2 j σ 2 λ + σ2 z + σ2 ɛ j (14) 1 For Large σ 2 z But Consider Usual Correlation Coefficient: Corr[R it, R jt ] = β i β j σλ 2 + pσ2 z βi 2σ2 λ + pσ2 z + σ2 ɛ,i βj 2σ2 λ + pσ2 z + σ2 ɛ,j p p + σ ɛ,i 2 /σ2 z p + σ ɛ,j 2 /σ2 z For β i, β j 0 For p = And σ 2 ɛ,j /σ2 z = 0.100: Corr[R it, R jt ] = (15) Risk Model For Hedge Funds: Must Accommodate Phase-Locking Behavior Other Nonlinearities Are Important Time Series Properties Are Critical Mixed Discrete/Continuous Phenomena MIT c 2004 by A. Lo Page 26/53

29 For Example, Consider Asymmetric Market Betas: R it = α i + β + i Λ+ t + β i Λ t + ɛ it (16) Λ + t Λ t { Λt if Λ t > 0 0 otherwise { Λt if Λ t 0 0 otherwise (17) (18) Λ t Λ + t + Λ t (19) Special Case, β + i = β i = β i (One-Factor Risk Model): R it = α i + β i Λ + t + β i Λ t + ɛ it (20) = α i + β i Λ t + ɛ it (21) For Event-Driven Distress Index: R it = Λ + t Λ t + ɛ it, R 2 = 0.36 Beta Changes Sign Depending on Up/Down Market! Nonlinear Risk Exposure Should This Be Considered Market Neutral? MIT c 2004 by A. Lo Page 27/53

30 Regression Analysis For Hedge Fund Index Returns Monthly Data, January 1996 to November 1999 Strategy ˆα t(ˆα) ˆβ+ t( ˆβ + ) ˆβ t( ˆβ ) R 2 Currency Trading ED-Distress ED-Merger-Arb EM-Equity Emerging Market EM-Fixed Income Event Driven Fund of Funds Futures Trading Growth High Yield Macro Opportunistic Other Relative Value RV-Convertible RV-EQWLS RV-Option-Arb RV-Other-Stat-Arb ShortSelling Value Source: AlphaSimplex Group MIT c 2004 by A. Lo Page 28/53

31 Risk and Performance Attribution Sharpe Ratio Central To Finance: µ E[R t ], σ 2 Var[R t ] (22) Mean-Variance Efficiency Performance Attribution Risk Management SR µ R f σ (23) But Sharpe Ratios Must Be Estimated: ˆµ = 1 T ˆσ 2 = 1 T T R t (24) t=1 T (R t ˆµ) 2 (25) t=1 ŜR = ˆµ R f ˆσ (26) How Accurate Are Such Estimators? MIT c 2004 by A. Lo Page 29/53

32 The IID Case: T (ˆµ µ) a N ( T (ˆσ 2 σ 2 ) a N ( 0, σ 2 ) (27) 0, 2σ 4 ) (28) T (ŜR SR) a N ( 0, V IID ) V IID = ( ) g 2 σ 2 + µ ( ) g 2 σ 2 2σ 4 (29) V IID = 1 + (µ R f) 2 2σ 2 = SR2 (30) SE(ŜR) = ( SR2 )/T (31) MIT c 2004 by A. Lo Page 30/53

33 Standard Errors For Sharpe Ratio: IID Case Asymptotic standard errors V IID /T for the Sharpe ratio estimator ŜR under the assumption of independently and identically distributed returns, for various values of the sample size T and the true Sharpe ratio SR, where V IID = SR2. SR Sample Size T MIT c 2004 by A. Lo Page 31/53

34 The Non-IID Case: Assume That Returns Are Stationary and Ergodic: Use GMM To Estimate ˆµ and ˆσ 2 Apply Delta Method T (ŜR SR) a N ( 0, V GMM ) (32) V GMM = g θ Σ g SE[ŜR] a = θ (33) V GMM /T (34) Time Aggregation: Multiperiod Returns IID Case: R t (q) R t + R t R t q+1 (35) SR(q) = E[R t(q)] R f (q) Var[Rt (q)] T (ŜR(q) q SR) = q(µ R f) q σ = q SR (36) a ( N 0, V IID (q) ) (37) V IID (q) = q V IID = q ( SR2 ) (38) MIT c 2004 by A. Lo Page 32/53

35 Non-IID Case: Var[R t (q)] = q 1 q 1 i=0 j=0 Cov[R t i, R t j ] (39) = qσ 2 + 2σ 2 q 1 (q k)ρ k (40) k=1 SR(q) = η(q) SR (41) η(q) q q + 2 q 1 k=1 (q k)ρ k (42) MIT c 2004 by A. Lo Page 33/53

36 Scale Factors η(q) for AR(1) Returns Scale factors η(q) for time-aggregated Sharpe ratios SR(q) = η(q)sr, when returns follow an AR(1) process with autoregressive coefficient ρ, for various aggregation values q and autocorrelation coefficients. ρ Aggregation Value q % % % % % % % % % % % % % % % % % % % MIT c 2004 by A. Lo Page 34/53

37 MIT c 2004 by A. Lo Page 35/53 Autocorrelation-Adjusted Sharpe Ratios Monthly and annual Sharpe ratio estimates for a sample of mutual funds and hedge funds, based on monthly total returns from various start dates through June 2000 for the mutual-fund sample and various start dates through December 2000 for the hedge-fund sample. ˆρ k denotes the k-th autocorrelation coefficient, and Q 11 denotes the Ljung-Box (1978) Q-statistic T (T +2) 11 k=1 ρ2 k /(T k) which is asymptotically χ 2 11 under the null hypothesis of no serial correlation. ŜR denotes the usual Sharpe ratio estimator (ˆµ R f )/ˆσ based on monthly data where R f is assumed to be 5.0%/12 per month, and ŜR(12) denotes the annual Sharpe ratio estimator which takes into account serial correlation in monthly returns. All standard errors are based on GMM estimators using Newey and West s (1987) procedure with truncation lag m=3 for entries in the SE 3 and SE 3 (12) columns, and m=6 for entries in the SE 6 (12) column. Fund Mutual Funds: Monthly Annual Start ˆµ ˆσ ˆρ T 1 ˆρ 2 ˆρ 3 p-value of Date (%) (%) (%) (%) (%) Q 11 (%) ŜR SE 3 12 ŜR ŜR(12) SE 3 (12) SE 6 (12) Vanguard 500 Index Fidelity Magellan Investment Company of America Janus Fidelity Contrafund Washington Mutual Investors Janus Worldwide Fidelity Growth and Income American Century Ultra Growth Fund of America Hedge Funds: Convertible/Option Arbitrage Relative Value Mortgage-Backed Securities High Yield Debt Risk Arbitrage A Long/Short Equities Multi-Strategy A Risk Arbitrage B Convertible Arbitrage A Convertible Arbitrage B Multi-Strategy B Fund of Funds

38 Role of Serial Correlation in Alternative Investments: Basic Property of the Data Risk and Performance Attribution Time Diversification Liquidity Performance Smoothing Let R it Denote True (Economic) Returns: R it = µ i + β i Λ t + ɛ it, E[Λ t ] = E[ɛ it ] = 0 (43a) ɛ it, Λ t IID (43b) Var[R it ] σ 2 i (43c) Let R o it Denote Observed Returns: R o it = θ 0 R it + θ 1 R it θ k R it k (44a) θ j [0, 1], j = 1,..., k (44b) 1 = θ 0 + θ θ k (44c) Proposition 1: Under (44c), the statistical properties of observed returns are characterized by: E[Rit o ] = c µ µ i = µ i (45) Var[Rit o ] = c2 σ σ2 i σi 2 (46) SR o i = c s SR i SR i (47) βi o = c β β i β i (48) MIT c 2004 by A. Lo Page 36/53

39 Cov[R o it, Ro it m ] = Corr[R o it, Ro it m ] = where: ( k m ) j=0 θ jθ j+m σi 2 if 0 m k 0 if m > k k m j=0 θ j θ j+m kj=0 θ 2 j if 0 m k 0 if m > k (49) (50) (51) c µ θ 0 + θ θ k (52a) c 2 σ θ θ θ2 k (52b) c s 1 θ θ2 k (52c) c β θ 0 (52d) But There Must Be A Limit To Smoothing: Cannot Maintain Differences Indefinitely Even Illiquid Securities Are Eventually Transacted Periodic External Auditing Agents MIT c 2004 by A. Lo Page 37/53

40 Proposition 2: If θ j [0, 1] and 1 = θ θ k, then (T ) ( R o i1 + Ro i2 + + Ro it ) (R i1 + R i2 + + R it ) = k 1 (R j R T j )(1 j=0 j θ i ) i=0 E[ (T )] = 0 Var[ (T )] = 2σ 2 i ξ k 1 j=0 k 1 j=0 1 1 j l=0 j l=0 θ l 2 θ l 2 = 2σ 2 i ξ k Consider Various Smoothing Profiles: θ j = θ j = 1 (Straightline) k + 1 k + 1 j (Sum-of-Years) (k + 1)(k + 2)/2 (53a) (53b) θ j = δj (1 δ), δ (0, 1) (Geometric) (53c) 1 δk+1 MIT c 2004 by A. Lo Page 38/53

41 Impact on Variance of (T ): ξ = k(2k + 1) 6(k + 1) (54a) ξ = k(3k2 + 6k + 1) 15(k + 1)(k + 2) (54b) ξ = δ2 ( 1 + δ k (2 + 2δ + δ k ( 1 2δ + k(δ 2 1)))) (δ 2 1)(δ k+1 1) 2 (54c) MIT c 2004 by A. Lo Page 39/53

42 MIT c 2004 by A. Lo Page 40/53 k Implications of Various Smoothing Profiles θ 0 θ 1 θ 2 θ 3 θ 4 θ 5 c β c σ c s ρ o 1 ρ o 2 ρ o 3 ρ o 4 ρ o 5 ξ (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Straightline Smoothing Sum-of-Years Smoothing Geometric Smoothing (δ = 0.25) Implications of three different smoothing profiles for observed betas, standard deviations, Sharpe ratios, and serial correlation coefficients of a fund with IID true returns. Straightline smoothing is given by θ j = 1/(k+1); sum-of-years smoothing is given by θ j = (k+1 j)/[(k+1)(k+2)/2]; geometric smooothing is given by θ j = δ j (1 δ)/(1 δ k+1 ). c β, c σ, and c s denote multipliers associated with the beta, standard deviation, and Sharpe ratio of observed returns, and ρ o j denotes the j-th autocorrelation coefficient of observed returns. Source: AlphaSimplex Group, LLC.

43 Estimating Smoothing and Illiquidity Profiles Set k = 2 And Estimate MA(2) Via Maximum Likelihood: With Constraint: Rit o = θ 0 R it + θ 1 R it 1 + θ 2 R it 2 (55a) R it IID N (µ i, σi 2 ) (55b) 1 = θ 0 + θ 1 + θ 2 (56) As A Specification Check, Impose Different Normalization: σ 2 i ˆσ 2 nw (57) TASS Dataset: November 1977 to January 2001 Live (1,512) and Graveyard (927) Funds Graveyard Database Started In Funds Deleted (Quarterly Returns, Gross Fees) Monthly Returns, Net of Fees Funds With At Least 5 Years of History 909 Funds (651 Live, 258 Dead) MIT c 2004 by A. Lo Page 41/53

44 MIT c 2004 by A. Lo Page 42/53 TASS Hedge Funds with 25 Smallest Estimated Smoothing Parameters ˆθ 0 Code Category Period T Status ˆθ0 SE(ˆθ 0 ) ˆθ1 SE(ˆθ 1 ) ˆθ2 SE(ˆθ 2 ) ˆξ 1201 Non-Directional/Relative Value Event Driven Not Categorized Non-Directional/Relative Value Fund of Funds Non-Directional/Relative Value Non-Directional/Relative Value Fund of Funds Fixed Income Directional Non-Directional/Relative Value Non-Directional/Relative Value Fund of Funds Non-Directional/Relative Value Non-Directional/Relative Value Non-Directional/Relative Value Non-Directional/Relative Value Non-Directional/Relative Value Event Driven Fund of Funds Not Categorized Fixed Income Directional Fund of Funds Event Driven Event Driven Non-Directional/Relative Value First 25 funds of ranked list of 908 hedge funds in the TASS Hedge Fund Combined (Live and Graveyard) database with at least five years of returns history during the period from November 1977 to January 2001, ranked in increasing order of the estimated smoothing parameter ˆθ 0 of the MA(2) smoothing process Rt o = θ 0R t + θ 1 R t 1 + θ 2 R t 2, subject to the normalization 1 = θ 0 + θ 1 + θ 2, and estimated via maximum likelihood.

45 Estimated Smoothing Coefficients ˆθ θ Fund Category Estimated Smoothing Index ˆξ ξ Fund Category MIT c 2004 by A. Lo Page 43/53

46 MIT c 2004 by A. Lo Page 44/53 Summary of MA(2) Parameter Estimates For TASS Hedge Funds Category Monthly Data, November 1977 to January 2001 N MA(2) with Constrained Sum ˆθ 0 ˆθ1 ˆθ2 ˆξ Mean SD Mean SD Mean SD Mean SD Not Categorized US Equity Hedge European Equity Hedge Asian Equity Hedge Global Equity Hedge Dedicated Shortseller Fixed-Income Directional Convertible Fund (Long Only) Event Driven Non-Directional/Relative Value Global Macro Global Opportunity Natural Resources Pure Leveraged Currency Pure Managed Futures Pure Emerging Market Pure Property Fund of Funds All Means and standard deviations of maximum likelihood estimates of MA(2) smoothing process Rt o = θ 0R t + θ 1 R t 1 + θ 2 R t 2, ξ θ0 2 + θ2 1 + θ2 2, for 908 hedge funds in the TASS combined database with at least five years of returns history during the period from November 1977 to January 2001.

47 MA(2) Parameter Estimates For Indexes and Mutual Funds Monthly Data, Various Sample Periods Series Period T Mean SD ˆρ 1 ˆρ 2 ˆρ 3 R ˆβ 2 (%) (%) (%) (%) (%) (%) Ibbotson Small Company Ibbotson Long-Term Government Bonds Ibbotson Long-Term Corporate Bonds Ibbotson Large Company AXP Extra Income Fund (INEAX) Vanguard 500 Index Trust (VFINX) CSFB/Tremont Indices: Aggregate Hedge Fund Index Convertible Arbitrage Dedicated Short Bias Emerging Markets Equity Market Neutral Event Driven Fixed Income Arbitrage Global Macro Long/Short Managed Futures Series Period T ˆθ0 SE(ˆθ 0 ) ˆθ1 SE(ˆθ 1 ) ˆθ2 SE(ˆθ 2 ) ˆξ Ibbotson Small Company Ibbotson Long-Term Government Bonds Ibbotson Long-Term Corporate Bonds Ibbotson Large Company AXP Extra Income Fund (INEAX) Vanguard 500 Index Trust (VFINX) CSFB/Tremont Indices: Aggregate Hedge Fund Index Convertible Arbitrage Dedicated Short Bias Emerging Markets Equity Market Neutral Event Driven Fixed Income Arbitrage Global Macro Long/Short Managed Futures MIT c 2004 by A. Lo Page 45/53

48 Classical and Adjusted Sharpe Ratios for TASS Hedge Funds Monthly Data, November 1977 to January 2001 Sharpe Ratios For Combined Sample Category N SR SR SR Mean SD Mean SD Mean SD Not Categorized US Equity Hedge European Equity Hedge Asian Equity Hedge Global Equity Hedge Dedicated Shortseller Fixed-Income Directional Convertible Fund (Long Only) Event Driven Non-Directional/Relative Value Global Macro Global Opportunity Natural Resources Pure Leveraged Currency Pure Managed Futures Pure Emerging Market Pure Property Fund of Funds All Means and standard deviations of Sharpe ratios of 908 hedge funds in the TASS Hedge Fund Combined (Live and Graveyard) database with at least five years of returns history during the period from November 1977 to January SR is the standard Sharpe ratio, SR is the smoothing-adjusted Sharpe ratio of Lo (2002), and SR is the smoothing-adjusted Sharpe ratio using ˆσ NW. All Sharpe ratios are computed with respect to a 0 benchmark. Caveats: Illiquidity and Performance Smoothing Are Related Cannot Distinguish With Current Data Need Not Imply Foul Play! MIT c 2004 by A. Lo Page 46/53

49 Conclusions Risk Transparency Is Essential: Individual Investors Desire Transparency Institutional Investors Demand Transparency Requires Unique Set of Risk Analytics Risk Transparency vs. Position Transparency: Position Transparency Risk Transparency Most Managers Do Not Want To Reveal Positions Most Investors Cannot Interpret Positions Compromise Risk Transparency! Question: Can Strategies Be Reverse Engineered? Answer: Usually Not Five Unique Aspects of Hedge Fund Investments: 1. Data Issues and Survivorship Bias 2. Dynamic Risk Management 3. Correlation and Risk Adjustments 4. Risk and Performance Attribution 5. Psychology of Risk Preferences MIT c 2004 by A. Lo Page 47/53

50 A theory should be made as simple as possible, but not simpler.. Albert Einstein MIT c 2004 by A. Lo Page 48/53

51 Additional References Ackermann, C., McEnally, R. and D. Ravenscraft, 1999, The Performance of Hedge Funds: Risk, Return, and Incentives, Journal of Finance 54, Agarwal, V. and N. Naik, 2000a, Performance Evaluation of Hedge Funds with Buy-and- Hold and Option-Based Strategies, Hedge Fund Centre Working Paper No. HF 003, London Business School. Agarwal, V. and N. Naik, 2000b, On Taking the Alternative Route: The Risks, Rewards, and Performance Persistence of Hedge Funds, Journal of Alternative Investments 2, Agarwal, V. and N. Naik, 2000c, Multi-Period Performance Persistence Analysis of Hedge Funds Source, Journal of Financial and Quantitative Analysis 35, Agarwal, V. and N. Naik, 2002, Risks and Portfolio Decisions Involving Hedge Funds, Institute of Finance and Accounting Working Paper No. 364, London Business School. Asness, C., Krail, R. and J. Liew, 2001, Do Hedge Funds Hedge?, The Journal of Portfolio Management 28, Baquero, G., Horst, J. and M. Verbeek, 2002, Survival, Look-Ahead Bias and the Performance of Hedge Funds, Erasmus University Rotterdam Working Paper. Bernstein, P., 1992, Capital Ideas. New York: Free Press. Bernstein, P., 1996, Against the Gods. New York: John Wiley & Sons. Bhattacharya, S. and P. Pfleiderer, 1985, Delegated Portfolio Management, Journal of Economic Theory 36, Brown, S. and W. Goetzmann, 2001, Hedge Funds With Style, NBER Working Paper No Brown, S., Goetzmann, W., Ibbotson, R. and S. Ross, 1992, Survivorship Bias in Performance Studies, Review of Financial Studies 5, Brown, S., Goetzmann, W. and R. Ibbotson, 1999, Offshore Hedge Funds: Survival and Performance , Journal of Business 72, Brown, S., Goetzmann, W. and B. Liang, 2002, Fees on Fees in Funds of Funds, Yale ICF Work Paper No MIT c 2004 by A. Lo Page 49/53

52 Brown, S., Goetzmann, W. and J. Park, 1997, Conditions for Survival: Changing Risk and the Performance of Hedge Fund Managers and CTAs, Yale School of Management Work Paper No. F 59. Brown, S., Goetzmann, W. and J. Park, 2000, Hedge Funds and the Asian Currency Crisis, Journal of Portfolio Management 26, Brown, S., Goetzmann, W. and J. Park, 2001, Careers and Survival: Competition and Risks in the Hedge Fund and CTA Industry, Journal of Finance 56, Campbell, J., Lo, A., and C. MacKinlay, 1997, The Econometrics of Financial Markets. Princeton, NJ: Princeton University Press. Edwards, F., and M. Caglayan, 2001, Hedge Fund and Commodity Fund Investments in Bull and Bear Markets, The Journal of Portfolio Management 27, Elton, E. and M. Gruber, 2002, Incentive Fees and Mutual Funds, to appear in Journal of Finance. Farmer, D. and A. Lo, 1999, Frontiers of Finance: Evolution and Efficient Markets, Proceedings of the National Academy of Sciences 96, Fung, W. and D. Hsieh, 1997a, Empirical Characteristics of Dynamic Trading Strategies: The Case of Hedge Funds, Review of Financial Studies 10, Fung, W. and D. Hsieh, 1997b, Investment Style and Survivorship Bias in the Returns of CTAs: The Information Content of Track Records, Journal of Portfolio Management 24, Fung, W. and D. Hsieh, 1999, A Primer on Hedge Funds, Journal of Empirical Finance 6, Fung, W. and D. Hsieh, 2000, Performance Characteristics of Hedge Funds and Commodity Funds: Natural versus Spurious Biases, Journal of Financial and Quantitative Analysis 35, Fung, W. and D. Hsieh, 2001, The Risk in Hedge Fund Strategies: Theory and Evidence from Trend Followers, Review of Financial Studies 14, Getmansky, M., Lo, A. and I. Makarov, 2004, An Econometric Model of Serial Correlation and Illiquidity In Hedge Fund Returns, to appear in Journal of Financial Economics. Goetzmann, W., Ingersoll, J. and S. Ross, 1997, High Water Marks and Hedge Fund Management Contracts, to appear in Journal of Finance. MIT c 2004 by A. Lo Page 50/53

53 Goetzmann, W., Ingersoll, J., Spiegel, M. and I. Welch, 2002, Sharpening Sharpe Ratios, National Bureau of Economic Research Working Paper No. W9116. Hendricks, D., Patel, J. and R. Zeckhauser, 1997, The J-Shape of Performance Persistence Given Survivorship Bias, Review of Economics and Statistics 79, Horst, J., Nijman, T. and M. Verbeek, 2001, Eliminating Look-Ahead Bias in Evaluating Persistence in Mutual Fund Performance, Journal of Empirical Finance 8, Jobson, J. and R. Korkie, 1981, Performance Hypothesis Testing with the Sharpe and Treynor Measures, Journal of Finance 36, Kao, D., 2002, Battle for Alphas: Hedge Funds versus Long-Only Portfolios, Financial Analysts Journal 58, Leroy, S., 1973, Risk Aversion and the Martingale Property of Stock Returns, International Economic Review 14, Liang, B., 1999, On the Performance of Hedge Funds, Financial Analysts Journal 55, Liang, B., 2000, Hedge Funds: The Living and the Dead, Journal of Financial and Quantitative Analysis 35, Liang, B., 2001, Hedge Fund Performance: , Financial Analysts Journal 57, Lo, A., 1994, Data-Snooping Biases in Financial Analysis, in H. Russell Fogler, ed.: Blending Quantitative and Traditional Equity Analysis, Charlottesville, VA: Association for Investment Management and Research. Lo, A., ed., 1997, Market Efficiency: Stock Market Behaviour In Theory and Practice, Volumes I and II, Cheltenham, UK: Edward Elgar Publishing Company. Lo, A., 1999, The Three P s of Total Risk Management, Financial Analysts Journal 55, Lo, A., 2001, Risk Management For Hedge Funds: Introduction and Overview, to appear in Financial Analysts Journal 57, Lo, A., 2002, The Statistics of Sharpe Ratios, Financial Analysts Journal 58, Lo, A. and C. MacKinlay, 1990, Data-Snooping Biases in Tests of Financial Asset Pricing Models, Review of Financial Studies 3, MIT c 2004 by A. Lo Page 51/53

54 Lo, A. and C. MacKinlay, 1999, A Non-Random Walk Down Wall Street. Princeton, NJ: Princeton University Press. Lochoff, R., 2002, Hedge Funds and Hope, The Journal of Portfolio Management 28, Lucas, R., 1978, Asset Prices in an Exchange Economy, Econometrica 46, Samuelson, P., 1965, Proof that Properly Anticipated Prices Fluctuate Randomly, Industrial Management Review 6, Schneeweis, T. and R. Spurgin, 1996, Survivor Bias in Commodity Trading Advisor Performance, Journal of Futures Markets 16, Sharpe, W., 1994, The Sharpe Ratio, Journal of Portfolio Management 21, Spurgin, R., 2001, How to Game Your Sharpe Ratio, The Journal of Alternative Investments 4, Weisman, A., 2002, Informationless Investing and Hedge Fund Performance Measurement Bias, The Journal of Portfolio Management 28, MIT c 2004 by A. Lo Page 52/53