Risk Management For Hedge Funds: Introduction and Overviewweb.mit.edu/katykam/Public/slides2004.pdf · 2004. 3. 23. · Credit Risk In ationary Pressures, Central Banking Policy Macroeconomic

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

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

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

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


Summary Statistics For Hedge Fund Index Returns

Monthly Data, January 1996 to November 1999

StrategyNumber of Mean S.D. Sharpe Max ρ1 ρ2 ρ3 ρ4

Funds (%) (%) Ratio D.D. (%) (%) (%) (%)

Currency Trading 14 0.9 2.7 0.65 −10.1 17.8 −8.5 −5.8 −17.3ED-Distress 16 1.2 1.8 1.57 −11.4 32.8 11.5 −7.1 2.6ED-Merger-Arb 28 1.2 1.1 2.47 −4.5 14.6 −5.9 −17.5 −12.1EM-Equity 29 2.9 9.4 0.90 −46.0 25.2 5.1 −3.5 −10.6Emerging Market 76 2.1 5.3 1.09 −34.3 33.8 5.7 0.7 −4.9EM-Fixed Income 20 1.6 2.9 1.35 −12.5 19.1 2.1 2.6 −0.8Event Driven 92 1.2 1.3 2.18 −7.0 38.7 5.8 −12.4 −11.0Fund of Funds 192 1.0 1.1 1.96 −2.9 23.9 9.5 −2.8 −5.3Futures Trading 46 1.0 2.9 0.66 −8.7 3.4 −9.6 −10.5 −9.0Growth 83 2.2 3.8 1.64 −14.0 2.9 −6.3 0.2 −10.8High Yield 13 0.8 0.9 1.46 −5.3 31.4 19.5 2.1 12.1Macro 25 1.2 3.3 0.86 −23.3 9.7 6.8 9.4 −3.0Opportunistic 29 1.6 2.4 1.73 −18.3 26.9 6.1 9.8 −6.4Other 55 1.3 2.2 1.44 −9.5 −4.0 1.3 1.5 −24.5Relative Value 133 1.1 0.7 3.63 −2.3 32.4 16.6 6.6 6.2RV-Convertible 28 1.1 0.9 2.50 −3.0 52.3 27.1 16.3 7.5RV-EQWLS 46 1.0 1.0 1.92 −2.7 16.4 −8.6 −13.4 −2.2RV-Option-Arb 13 2.5 6.1 1.16 −16.7 −2.1 4.5 6.1 14.7RV-Other-Stat-Arb 7 1.3 1.8 1.62 −5.4 23.7 −13.3 −7.1 5.0ShortSelling 12 −0.4 4.7 −0.58 −43.5 2.0 −11.3 −4.0 −12.6Value 56 1.4 2.5 1.37 −17.1 27.0 3.1 −12.3 −20.7

EDCorp Restructure* 5 1.9 3.2 1.62 −10.7 23.7 −5.8 −28.5 −20.9Market Timing* 18 2.1 2.4 2.39 −4.9 −31.5 9.9 9.9 −10.5RV-Mortgage-Backed* 12 1.2 1.2 2.46 −4.5 61.3 43.9 31.0 24.7RV-CreditSpread* 2 1.3 2.0 1.55 −7.5 26.1 5.7 −7.6 −1.1

Equal-Weighted Index* — 1.3 1.4 2.23 −4.3 22.6 8.0 −1.5 −8.7Funds-Weighted Index* — 1.4 1.8 1.94 −7.2 18.7 2.0 −2.8 −12.4SP500 — 1.4 3.6 0.92 −15.6 −11.1 −0.9 0.5 −14.3

Source: AlphaSimplex Group

MIT

c©2004

by

A.Lo

Page

3/53

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


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%−4.6% 0.0% 5.0% 10.0% 15.0% 20.0%

4.4% 4.9% 5.0% 5.0% 5.0% 5.0%

10%−3.1% 0.7% 5.2% 10.0% 15.0% 20.0%

7.8% 8.9% 9.6% 9.9% 10.0% 10.0%

25%2.2% 5.1% 8.5% 12.3% 16.4% 20.8%

18.3% 19.8% 21.1% 22.2% 23.1% 23.8%

50%10.7% 13.2% 15.9% 18.9% 22.2% 25.7%38.7% 39.9% 41.0% 42.2% 43.3% 44.4%

75%17.7% 20.2% 22.7% 25.5% 28.4% 31.5%61.5% 62.3% 63.2% 64.1% 65.0% 66.0%

100%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 deviationSD[R]. Source: AlphaSimplex Group, LLC.


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[WT ]: $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 $402,06150% $21,932 $37,182 $66,670 $125,482 $245,455 $493,399

75% $41,992 $67,805 $114,255 $200,149 $362,635 $675,371100% $75,951 $118,721 $191,951 $320,299 $549,922 $967,825

Expected values of end-of-period wealth E[W ∗T ] = W0(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 W0 = $10,000. Rows labelled ‘E[WT ]’report the expected values of end-of-period wealth, W0(1 + E[R])T , for the return R. Source:AlphaSimplex Group, LLC.


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 confidenceinterval, of how much one can lose from holding a positionover a set horizon. Potential horizons may be one day fortypical trading activities or a month or longer for portfoliomanagement. The methods described in our documentation usehistorical returns to forecast volatilities and correlations that arethen used to estimate the market risk. These statistics can beapplied across a set of asset classes covering products used byfinancial 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


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


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


Five Unique Aspects of Hedge Fund Investments:





5. Psychology of Risk Preferences


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 6⇒ Future Returns


Consider n Funds, IID:

α1

σ1,

α2

σ2, . . . ,

αn

σn∼ F (x) (1)

• Assume All n Funds Have αj = 0, j = 1, . . . , n

• Xj ≡ αj/σj Called “Sharpe Ratio”

• Risk-Adjusted Performance Index

• F (x) Contains All Knowable Information About Xj

• Suppose We Select The “Best” Fund:

X∗(n) ≡ Arg Max [ X1, . . . , Xn ] (2)

Pr ( X∗(n) < x ) = Pr ( Max[X1, . . . , Xn] < x )

= Pr ( X1 < x, . . . , Xn < x )

= Pr ( X1 < x ) · · ·Pr ( Xn < x )

Pr ( X∗(n) < x ) = F n(x) (3)

E [ X∗(n) ] =

∫xdF (x) (4)

Var [ X∗(n) ] =

∫x2dF (x) − E [ X∗(n) ]2 (5)

δ = Fn(Cδ) ⇒ Cδ = F

−1(δ

1/n) (6)


Mean, SD, and Quantiles of X∗(n) For Selected n

For IID {Xj} ∼ N (0, 1)

n E [X∗(n)] SD [X∗(n)] C0.025 C0.975

1 0.00000 1.00000 −1.960 1.960

5 1.16296 0.66898 −0.055 2.572

10 1.53875 0.58681 0.506 2.803

15 1.73591 0.54867 0.779 2.932

20 1.86748 0.52507 0.960 3.020

25 1.96531 0.50844 1.093 3.087

30 2.04276 0.49582 1.197 3.140

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

-5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00

Den

sity

n = 1n = 10n = 20n = 50


What If Xj’s Are Correlated?

• Selection Bias Less Severe

• Difficult To Obtain Analytic Results

• Consider Special Case (Equi-Correlated Xj’s):

Corr[ Xi , Xj ] = ρ , i 6= j (7)

• Assume Normality

• Distribution of X∗(n) Computable

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1N=1 N=10N=20N=50

(a) ρ = 0.00

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1N=1 N=10N=20N=50

(b) ρ = 0.50


Mean, SD, and Quantiles of X∗(n) For Selected n

For Equi-Correlated {Xj} ∼ N (0, 1)

n E [X∗(n)] SD [X∗(n)] C0.025 C0.975

ρ = 0%

10 1.54 0.59 0.500 2.80320 1.87 0.53 0.960 3.02050 2.25 0.46 1.467 3.287

100 2.51 0.43 1.796 3.477500 3.04 0.37 2.440 3.888

ρ = 10%

10 1.46 0.64 0.283 2.79820 1.77 0.59 0.694 3.01450 2.13 0.54 1.142 3.280

100 2.38 0.52 1.440 3.469500 2.88 0.47 2.016 3.877

ρ = 25%

10 1.33 0.71 −0.018 2.78220 1.62 0.68 0.338 2.99250 1.95 0.64 0.731 3.251

100 2.17 0.62 0.988 3.435500 2.63 0.59 1.496 3.828

1000 2.81 0.59 1.687 3.986

ρ = 50%

10 1.09 0.82 −0.499 2.71620 1.32 0.80 −0.227 2.90550 1.59 0.78 0.078 3.135

100 1.77 0.77 0.278 3.297500 2.15 0.75 0.680 3.637


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)


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 0.98 1.94

# Negative Months 36/96 6/96

Correlation 100.0% 59.9%

Total Return 367.1% 2721.3%


Capital Decimation Partners, L.P.

Monthly Performance History

1992 1993 1994 1995 1996 1997 1998 1999Month

SPX CDP SPX CDP SPX CDP SPX CDP SPX CDP SPX CDP SPX CDP SPX CDP

Jan 8.2 8.1 −1.2 1.8 1.8 2.3 1.3 3.7 −0.7 1.0 3.6 4.4 1.6 15.3 5.5 10.1Feb −1.8 9.3 −0.4 1.0 −1.5 0.7 3.9 0.7 5.9 1.2 3.3 6.0 7.6 11.7 −0.3 16.6Mar 0.0 4.9 3.7 3.6 0.7 2.2 2.7 1.9 −1.0 0.6 −2.2 3.0 6.3 6.7 4.8 10.0Apr 1.2 3.2 −0.3 1.6 −5.3 −0.1 2.6 2.4 0.6 3.0 −2.3 2.8 2.1 3.5 1.5 7.2May −1.4 1.3 −0.7 1.3 2.0 5.5 2.1 1.6 3.7 4.0 8.3 5.7 −1.2 5.8 0.9 7.2Jun −1.6 0.6 −0.5 1.7 0.8 1.5 5.0 1.8 −0.3 2.0 8.3 4.9 −0.7 3.9 0.9 8.6Jul 3.0 1.9 0.5 1.9 −0.9 0.4 1.5 1.6 −4.2 0.3 1.8 5.5 7.8 7.5 5.7 6.1Aug −0.2 1.7 2.3 1.4 2.1 2.9 1.0 1.2 4.1 3.2 −1.6 2.6 −8.9 −18.3 −5.8 −3.1Sep 1.9 2.0 0.6 0.8 1.6 0.8 4.3 1.3 3.3 3.4 5.5 11.5 −5.7 −16.2 −0.1 8.3Oct −2.6 −2.8 2.3 3.0 −1.3 0.9 0.3 1.1 3.5 2.2 −0.7 5.6 3.6 27.0 −6.6 −10.7Nov 3.6 8.5 −1.5 0.6 −0.7 2.7 2.6 1.4 3.8 3.0 2.0 4.6 10.1 22.8 14.0 14.5Dec 3.4 1.2 0.8 2.9 −0.6 10.0 2.7 1.5 1.5 2.0 −1.7 6.7 1.3 4.3 −0.1 2.4

Year 14.0 46.9 5.7 23.7 −1.6 33.6 34.3 22.1 21.5 28.9 26.4 84.8 24.5 87.3 20.6 105.7

MIT

c©2004

by

A.Lo

Page

18/53

015304560

Frequency

-0.2

0

-0.1

0

0.00

0.10

0.20

Mor

e

Monthly Return

CDP vs. S&P 500 Returns, 1992-1999

CDP SP500

MIT

c©2004

by

A.Lo

Page

19/53

S&P 500 vs. CDP Total Return

$0

$5

$10

$15

$20

$25

$30

Jan-92

Jul-92

Jan-93

Jul-93

Jan-94

Jul-94

Jan-95

Jul-95

Jan-96

Jul-96

Jan-97

Jul-97

Jan-98

Jul-98

Jan-99

Jul-99

Date

Tota

l Ret

urn

S&P 500 CDP

MIT

c©2004

by

A.Lo

Page

20/53

Capital Decimation Partners II, L.P.

Week Pt Position Value Financing

t ($) (Shares) ($) ($)

0 40.000 7,057 282,281 −296,9741 39.875 7,240 288,712 −304,5852 40.250 5,850 235,456 −248,9183 36.500 33,013 1,204,981 −1,240,6294 36.875 27,128 1,000,356 −1,024,8655 36.500 31,510 1,150,101 −1,185,8096 37.000 24,320 899,841 −920,9817 39.875 5,843 232,970 −185,1118 39.875 5,621 224,153 −176,4799 40.125 4,762 191,062 −142,159

10 39.500 6,280 248,065 −202,28011 41.250 2,441 100,711 −44,13812 40.625 3,230 131,205 −76,20213 39.875 4,572 182,300 −129,79614 39.375 5,690 224,035 −173,94715 39.625 4,774 189,170 −137,83416 39.750 4,267 169,609 −117,81417 39.250 5,333 209,312 −159,76818 39.500 4,447 175,657 −124,94019 39.750 3,692 146,777 −95,07320 39.750 3,510 139,526 −87,91721 39.875 3,106 123,832 −71,87222 39.625 3,392 134,408 −83,29623 39.875 2,783 110,986 −59,10924 40.000 2,445 97,782 −45,61725 40.125 2,140 85,870 −33,445

Positions In XYZ Stock Over Six Months:

• Initial Price: $40

• Expected Return: 7%

• Standard Deviation: 13%


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


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]

Rp =1

n

[E[R1] + E[R2] + · · · + E[Rn]

]+

1

n

[ε1 + ε2 + · · · + εn

]

︸︷︷︸≈ 0

(8)

• Depends on Quality of Manager Selection Process

• Complete Diversification ⇒ T-Bill Returns


Correlation Matrix For Hedge Fund Index Returns


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

1. Currencies 100.02. ED-Distress 4.7 100.03. ED-Merger-Arb −11.1 54.4 100.04. EM-Equity −21.0 42.8 47.2 100.05. Emerging Market −11.1 45.6 50.2 86.3 100.06. EM-Fixed Income −5.8 28.7 34.2 43.8 76.3 100.07. Event Driven −5.7 85.8 80.8 54.6 58.8 40.1 100.08. Fund of Funds 7.9 59.2 52.4 44.3 56.8 49.8 70.8 100.09. Futures Trading 45.3 3.0 −7.7 −6.1 5.5 15.3 0.5 30.1 100.010. Growth −3.7 49.1 47.6 34.2 42.7 29.2 60.5 63.2 6.9 100.011. High Yield 8.0 54.2 16.7 25.2 33.5 35.4 48.2 35.8 7.2 12.9 100.012. Macro 28.5 10.9 8.7 5.5 16.9 30.1 17.1 44.1 51.5 14.5 17.1 100.013. Opportunistic 9.7 49.3 40.5 29.0 43.8 43.4 60.3 74.7 19.4 68.0 24.4 45.0 100.014. Other 8.7 53.9 51.6 37.6 52.7 48.0 64.3 68.5 27.3 76.9 20.3 29.6 73.6 100.015. Relative Value 12.0 48.9 36.9 37.5 39.4 26.0 53.9 46.1 13.5 19.1 51.0 18.5 34.2 31.6 100.016. RV-Convertible 8.1 52.2 36.3 28.6 41.6 37.3 54.4 45.4 6.9 25.9 49.6 22.7 47.1 33.6 56.9 100.017. RV-EWLS 6.5 30.5 43.5 33.5 26.2 12.6 42.5 34.0 4.2 34.3 17.1 9.0 26.3 41.4 50.8 13.0 100.018. RV-Option-Arb −0.3 7.1 1.5 14.6 10.6 3.4 8.9 3.0 −2.1 −20.3 12.2 6.5 4.7 −14.0 47.3 2.0 −4.0 100.019. RV-Other-Stat-Arb 10.2 19.9 −0.2 17.2 12.4 −2.4 13.6 19.4 0.6 22.4 8.1 −12.8 10.2 14.4 30.2 8.1 7.0 5.3 100.020. ShortSelling 15.1 −48.1 −53.8 −35.2 −43.4 −30.1 −61.8 −49.9 8.6 −85.7 −12.3 −4.3 −59.1 −67.9 −9.6 −28.6 −29.5 21.6 −11.0 100.021. Value −10.6 65.7 59.3 47.9 62.0 43.5 78.4 69.4 8.7 74.3 26.0 15.2 67.0 77.7 36.6 45.5 34.3 −4.3 20.3 −73.8 100.022. SP500 11.1 40.7 46.6 28.6 43.1 35.4 51.0 54.6 19.3 78.0 18.1 20.5 60.4 71.3 23.4 28.3 41.5 −16.3 7.6 −70.1 63.9 100.0

Source: AlphaSimplex Group, LLC

MIT

c©2004

by

A.Lo

Page

24/53

How Neutral?

• Where Does Alpha Come From?

E[Ri] = β1iE[Λ1] + · · · + βKiE[ΛK] +

β(K+1)iE[ΛK+1] + · · · + βMiE[ΛM ]︸︷︷︸

αi

(9)

• Cannot Be Completely Neutral If E[Ri] > Rf

• 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

Rit = αi + βiΛt + ItZt + εit (10)

Zt = Crisis Component (σ2z/σ2

λ � 1) (11)

It =

{0 with probability 1−p = 0.9991 with probability p = 0.001

(12)

Corr[Rit, Rjt|It = 0

]=

βiβjσ2λ√

β2i σ2

λ+ σ2

εi

√β2

j σ2λ

+ σ2εj

(13)


Corr[Rit, Rjt|It = 1

]=

βiβjσ2λ + σ2

z√β2

i σ2λ

+ σ2z + σ2

εi

√β2

j σ2λ

+ σ2z + σ2

εj

(14)

≈ 1 For Large σ2z

But Consider Usual Correlation Coefficient:

Corr[Rit, Rjt] =βiβjσ

2λ + pσ2

z√β2

i σ2λ

+ pσ2z + σ2

ε,i

√β2

j σ2λ

+ pσ2z + σ2

ε,j

≈ p√p + σ2

ε,i/σ2z

√p + σ2

ε,j/σ2z

For βi, βj ∼ 0

For p = 0.001 And σ2ε,j/σ2

z = 0.100:

Corr[Rit, Rjt] ≈ 0.0010

0.1010= 0.0099 . (15)

Risk Model For Hedge Funds:

• Must Accommodate Phase-Locking Behavior

• Other Nonlinearities Are Important

• Time Series Properties Are Critical

• Mixed Discrete/Continuous Phenomena


For Example, Consider Asymmetric Market Betas:

Rit = αi + β+i Λ+

t + β−i Λ−

t + εit (16)

Λ+t ≡

{Λt if Λt > 00 otherwise

(17)

Λ−t ≡

{Λt if Λt ≤ 00 otherwise

(18)

⇒ Λt ≡ Λ+t + Λ−

t (19)

Special Case, β+i = β−

i = βi (One-Factor Risk Model):

Rit = αi + βiΛ+t + βiΛ

−t + εit (20)

= αi + βiΛt + εit (21)

For Event-Driven Distress Index:

Rit = 1.95 − 0.11Λ+t + 0.58Λ−

t + εit , R2 = 0.36

• Beta Changes Sign Depending on Up/Down Market!

• Nonlinear Risk Exposure

• Should This Be Considered “Market Neutral”?


Regression Analysis For Hedge Fund Index Returns


Strategy α t(α) β+ t(β+) β− t(β−) R2

Currency Trading 0.93 1.97 0.05 0.34 0.13 0.81 0.01ED-Distress 1.95 7.84 −0.11 −1.50 0.58 6.95 0.36ED-Merger-Arb 1.35 7.99 0.04 0.91 0.27 4.78 0.27EM-Equity 3.78 2.41 0.16 0.34 1.49 2.84 0.11Emerging Market 2.64 3.20 0.21 0.88 1.18 4.27 0.23EM-Fixed Income 1.88 3.99 0.07 0.49 0.56 3.56 0.16Event Driven 1.61 9.35 −0.01 −0.26 0.43 7.37 0.41Fund of Funds 1.07 6.89 0.08 1.84 0.27 5.13 0.33Futures Trading 0.69 1.35 0.18 1.23 0.13 0.76 0.04Growth 1.49 3.65 0.69 5.80 0.98 7.13 0.62High Yield 1.11 8.05 −0.08 −1.92 0.19 4.10 0.15Macro 0.61 1.09 0.30 1.84 0.05 0.28 0.05Opportunistic 1.35 3.95 0.33 3.31 0.52 4.53 0.37Other 1.41 5.58 0.23 3.05 0.69 8.19 0.57Relative Value 1.36 12.22 −0.04 −1.27 0.15 4.02 0.15RV-Convertible 1.25 8.44 −0.01 −0.31 0.18 3.55 0.14RV-EQWLS 0.87 5.64 0.09 2.04 0.14 2.65 0.17RV-Option-Arb 4.48 4.29 −0.78 −2.56 0.33 0.95 0.07RV-Other-Stat-Arb 1.40 4.38 −0.02 −0.18 0.11 0.99 0.01ShortSelling 0.04 0.07 −0.67 −3.94 −1.25 −6.41 0.51Value 1.46 4.49 0.24 2.54 0.69 6.41 0.45

Source: AlphaSimplex Group


Risk and Performance Attribution

Sharpe Ratio Central To Finance:

µ ≡ E[Rt] , σ2 ≡ Var[Rt] (22)

SR ≡µ − Rf

σ(23)

• Mean-Variance Efficiency

• Performance Attribution

• Risk Management

But Sharpe Ratios Must Be Estimated:

µ =1

T

T∑

t=1

Rt (24)

σ2

=1

T

T∑

t=1

(Rt − µ)2

(25)

SR =µ − Rf

σ(26)

How Accurate Are Such Estimators?


The IID Case:√

T (µ − µ)a∼ N

(0 , σ

2)

(27)

√T (σ

2 − σ2)

a∼ N(

0 , 2σ4

)(28)

√T (SR − SR)

a∼ N ( 0 , VIID )

VIID =

(∂g

∂µ

)2

σ2 +

(∂g

∂σ2

)2

2σ4(29)

VIID = 1 +(µ − Rf)2

2σ2= 1 + 1

2SR2

(30)

SE(SR) =√

(1 + 12SR2)/T (31)


Standard Errors For Sharpe Ratio: IID Case

Asymptotic standard errors√

VIID/T for the Sharpe ratio estimator SRunder the assumption of independently and identically distributed returns,

for various values of the sample size T and the true Sharpe ratio SR, where

VIID=1+12SR2.

SRSample Size T

12 24 36 48 60 125 250 500

0.50 0.306 0.217 0.177 0.153 0.137 0.095 0.067 0.047

0.75 0.327 0.231 0.189 0.163 0.146 0.101 0.072 0.0511.00 0.354 0.250 0.204 0.177 0.158 0.110 0.077 0.055

1.25 0.385 0.272 0.222 0.193 0.172 0.119 0.084 0.0601.50 0.421 0.298 0.243 0.210 0.188 0.130 0.092 0.065

1.75 0.459 0.325 0.265 0.230 0.205 0.142 0.101 0.0712.00 0.500 0.354 0.289 0.250 0.224 0.155 0.110 0.077

2.25 0.542 0.384 0.313 0.271 0.243 0.168 0.119 0.0842.50 0.586 0.415 0.339 0.293 0.262 0.182 0.128 0.091

2.75 0.631 0.446 0.364 0.316 0.282 0.196 0.138 0.0983.00 0.677 0.479 0.391 0.339 0.303 0.210 0.148 0.105


The Non-IID Case:

• Assume That Returns Are Stationary and Ergodic:

• Use GMM To Estimate µ and σ2

• Apply Delta Method

√T (SR − SR)

a∼ N ( 0 , VGMM ) (32)

VGMM =∂g

∂θΣ

∂g

∂θ′ (33)

SE[SR]a=

√VGMM/T (34)

Time Aggregation:

• Multiperiod Returns

Rt(q) ≡ Rt + Rt−1 + · · · + Rt−q+1 (35)

• IID Case:

SR(q) =E[Rt(q)] − Rf(q)

√Var[Rt(q)]

=q(µ − Rf)

√q σ

=√

q SR (36)

√T (SR(q) − √

q SR)a∼ N

(0 , VIID(q)

)(37)

VIID(q) = q VIID = q (1 + 12SR

2) (38)


• Non-IID Case:

Var[Rt(q)] =

q−1∑

i=0

q−1∑

j=0

Cov[Rt−i, Rt−j] (39)

= qσ2 + 2σ2q−1∑

k=1

(q−k)ρk (40)

SR(q) = η(q) SR (41)

η(q) ≡ q√q + 2

∑q−1k=1

(q−k)ρk

(42)


Scale Factors η(q) for AR(1) Returns

Scale factors η(q) for time-aggregated Sharpe ratios SR(q) = η(q)SR, whenreturns follow an AR(1) process with autoregressive coefficient ρ, for various

aggregation values q and autocorrelation coefficients.

ρAggregation Value q

2 3 4 6 12 24 36 48 125 250

90% 1.03 1.05 1.07 1.10 1.21 1.41 1.60 1.77 2.67 3.70

80% 1.05 1.10 1.14 1.21 1.43 1.81 2.14 2.42 3.79 5.32

70% 1.08 1.15 1.21 1.33 1.65 2.19 2.62 3.00 4.75 6.6860% 1.12 1.21 1.30 1.46 1.89 2.55 3.08 3.53 5.63 7.94

50% 1.15 1.28 1.39 1.60 2.12 2.91 3.53 4.06 6.49 9.1540% 1.20 1.35 1.49 1.75 2.36 3.27 3.98 4.58 7.35 10.37

30% 1.24 1.43 1.60 1.91 2.61 3.65 4.44 5.12 8.23 11.6220% 1.29 1.52 1.73 2.07 2.88 4.04 4.93 5.68 9.14 12.92

10% 1.35 1.62 1.86 2.25 3.16 4.45 5.44 6.28 10.12 14.31

0% 1.41 1.73 2.00 2.45 3.46 4.90 6.00 6.93 11.18 15.81

−10% 1.49 1.85 2.16 2.66 3.80 5.39 6.61 7.64 12.35 17.47

−20% 1.58 1.99 2.33 2.90 4.17 5.95 7.31 8.45 13.67 19.35−30% 1.69 2.13 2.53 3.17 4.60 6.59 8.10 9.38 15.20 21.52

−40% 1.83 2.29 2.75 3.48 5.09 7.34 9.05 10.48 17.01 24.11−50% 2.00 2.45 3.02 3.84 5.69 8.26 10.21 11.84 19.26 27.31

−60% 2.24 2.61 3.37 4.30 6.44 9.44 11.70 13.59 22.19 31.50

−70% 2.58 2.76 3.86 4.92 7.45 11.05 13.77 16.04 26.33 37.43−80% 3.16 2.89 4.66 5.91 8.96 13.50 16.98 19.88 32.96 47.02

−90% 4.47 2.97 6.47 8.09 12.06 18.29 23.32 27.61 46.99 67.65


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 variousstart dates through June 2000 for the mutual-fund sample and various start dates through December 2000 for the hedge-fund sample.ρk denotes the k-th autocorrelation coefficient, and Q11 denotes the Ljung-Box (1978) Q-statistic T (T +2)

∑11k=1 ρ2

k/(T −k) which is

asymptotically χ211 under the null hypothesis of no serial correlation. SR denotes the usual Sharpe ratio estimator (µ−Rf )/σ based

on monthly data where Rf is assumed to be 5.0%/12 per month, and SR(12) denotes the annual Sharpe ratio estimator which takesinto 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 “SE3” and “SE3(12)” columns, and m=6 for entries in the “SE6(12)” column.

FundStart

Tµ σ ρ1 ρ2 ρ3 p-value of

Monthly Annual

Date (%) (%) (%) (%) (%) Q11 (%)SR SE3

√

12 SR SR(12) SE3(12) SE6(12)

Mutual Funds:

Vanguard 500 Index 76.10 286 1.30 4.27 −4.0 −6.6 −4.9 64.5 0.21 0.06 0.72 0.85 0.26 0.25Fidelity Magellan 67.01 402 1.73 6.23 12.4 −2.3 −0.4 28.6 0.21 0.06 0.73 0.66 0.20 0.21Investment Company of America 63.01 450 1.17 4.01 1.8 −3.2 −4.5 80.2 0.19 0.05 0.65 0.71 0.22 0.22Janus 70.03 364 1.52 4.75 10.5 −0.0 −3.7 58.1 0.23 0.06 0.81 0.80 0.17 0.17Fidelity Contrafund 67.05 397 1.29 4.97 7.4 −2.5 −6.8 58.2 0.18 0.05 0.61 0.67 0.23 0.23Washington Mutual Investors 63.01 450 1.13 4.09 −0.1 −7.2 −2.6 22.8 0.17 0.05 0.60 0.65 0.20 0.20Janus Worldwide 92.01 102 1.81 4.36 11.4 3.4 −3.8 13.2 0.32 0.11 1.12 1.29 0.46 0.37Fidelity Growth and Income 86.01 174 1.54 4.13 5.1 −1.6 −8.2 60.9 0.27 0.09 0.95 1.18 0.47 0.40American Century Ultra 81.12 223 1.72 7.11 2.3 3.4 1.4 54.5 0.18 0.07 0.64 0.71 0.27 0.25Growth Fund of America 64.07 431 1.18 5.35 8.5 −2.7 −4.1 45.4 0.14 0.05 0.50 0.49 0.19 0.20

Hedge Funds:

Convertible/Option Arbitrage 92.05 104 1.63 0.97 42.6 29.0 21.4 0.0 1.26 0.28 4.35 2.99 1.04 1.11Relative Value 92.12 97 0.66 0.21 25.9 19.2 −2.1 4.5 1.17 0.17 4.06 3.38 1.16 1.07Mortgage-Backed Securities 93.01 96 1.33 0.79 42.0 22.1 16.7 0.1 1.16 0.24 4.03 2.44 0.53 0.54High Yield Debt 94.06 79 1.30 0.87 33.7 21.8 13.1 5.2 1.02 0.27 3.54 2.25 0.74 0.72Risk Arbitrage A 93.07 90 1.06 0.69 −4.9 −10.8 6.9 30.6 0.94 0.20 3.25 3.83 0.87 0.85Long/Short Equities 89.07 138 1.18 0.83 −20.2 24.6 8.7 0.1 0.92 0.06 3.19 2.32 0.35 0.37Multi-Strategy A 95.01 72 1.08 0.75 48.9 23.4 3.3 0.3 0.89 0.40 3.09 2.18 1.14 1.19Risk Arbitrage B 94.11 74 0.90 0.77 −4.9 2.5 −8.3 96.1 0.63 0.14 2.17 2.47 0.79 0.77Convertible Arbitrage A 92.09 100 1.38 1.60 33.8 30.8 7.9 0.8 0.60 0.18 2.08 1.43 0.44 0.45Convertible Arbitrage B 94.07 78 0.78 0.62 32.4 9.7 −4.5 23.4 0.60 0.18 2.06 1.67 0.68 0.62Multi-Strategy B 89.06 139 1.34 1.63 49.0 24.6 10.6 0.0 0.57 0.16 1.96 1.17 0.25 0.25Fund of Funds 94.10 75 1.68 2.29 29.7 21.1 0.9 23.4 0.56 0.19 1.93 1.39 0.67 0.70

MIT

c©2004

by

A.Lo

Page

35/53

Role of Serial Correlation in Alternative Investments:

• Basic Property of the Data

• Risk and Performance Attribution

• Time Diversification

• Liquidity

• Performance Smoothing

Let Rit Denote True (Economic) Returns:

Rit = µi + βiΛt + εit , E[Λt] = E[εit] = 0 (43a)

εit , Λt ∼ IID (43b)

Var[Rit] ≡ σ2i (43c)

Let Roit Denote Observed Returns:

Roit = θ0 Rit + θ1 Rit−1 + · · · + θk Rit−k (44a)

θj ∈ [0, 1] , j = 1, . . . , k (44b)

1 = θ0 + θ1 + · · · + θk (44c)

Proposition 1: Under (44c), the statistical properties ofobserved returns are characterized by:

E[Roit] = cµ µi = µi (45)

Var[Roit] = c2

σ σ2i ≤ σ2

i (46)

SRoi = cs SRi ≥ SRi (47)

βoi = cβ βi ≤ βi (48)


Cov[Roit, Ro

it−m] =

(∑k−mj=0 θjθj+m

)σ2

i if 0 ≤ m ≤ k

0 if m > k

(49)

Corr[Roit, R

oit−m] =

∑k−mj=0

θjθj+m∑k

j=0 θ2j

if 0 ≤ m ≤ k

0 if m > k

(50)

(51)

where:

cµ ≡ θ0 + θ1 + · · · + θk (52a)

c2σ ≡ θ

20 + θ

21 + · · · + θ

2k (52b)

cs ≡ 1√θ20 + · · · + θ2

k

(52c)

cβ ≡ θ0 (52d)

But There Must Be A Limit To Smoothing:

• Cannot Maintain Differences Indefinitely

• Even Illiquid Securities Are Eventually Transacted

• Periodic External Auditing Agents


Proposition 2: If θj ∈ [0, 1] and 1 = θ0 + · · · + θk, then

∆(T ) ≡(Ro

i1 + Roi2 + · · · + Ro

iT

)−

(Ri1 + Ri2 + · · · + RiT )

=

k−1∑

j=0

(R−j − RT−j)(1 −j∑

i=0

θi)

E[∆(T )] = 0

Var[∆(T )] = 2σ2i

k−1∑

j=0

1 −

j∑

l=0

θl

2

= 2σ2i ξ

ξ ≡k−1∑

j=0

1 −

j∑

l=0

θl

2

≤ k

Consider Various Smoothing Profiles:

θj =1

k + 1(Straightline) (53a)

θj =k + 1 − j

(k + 1)(k + 2)/2(Sum-of-Years) (53b)

θj =δj(1 − δ)

1 − δk+1, δ ∈ (0, 1) (Geometric) (53c)


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 + δk(2 + 2δ + δk(−1 − 2δ + k(δ2 − 1))))

(δ2 − 1)(δk+1 − 1)2(54c)


Implications of Various Smoothing Profiles

kθ0 θ1 θ2 θ3 θ4 θ5 cβ cσ cs

ρo1 ρo

2 ρo3 ρo

4 ρo5 ξ

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

Straightline Smoothing

0 100.0 — — — — — 1.00 1.00 1.00 0.0 0.0 0.0 0.0 0.0 —1 50.0 50.0 — — — — 0.50 0.71 1.41 50.0 0.0 0.0 0.0 0.0 25.02 33.3 33.3 33.3 — — — 0.33 0.58 1.73 66.7 33.3 0.0 0.0 0.0 55.63 25.0 25.0 25.0 25.0 — — 0.25 0.50 2.00 75.0 50.0 25.0 0.0 0.0 87.54 20.0 20.0 20.0 20.0 20.0 — 0.20 0.45 2.24 80.0 60.0 40.0 20.0 0.0 120.05 16.7 16.7 16.7 16.7 16.7 16.7 0.17 0.41 2.45 83.3 66.7 50.0 33.3 16.7 152.8

Sum-of-Years Smoothing

0 100.0 — — — — — 1.00 1.00 1.00 0.0 0.0 0.0 0.0 0.0 —1 66.7 33.3 — — — — 0.67 0.75 1.34 40.0 0.0 0.0 0.0 0.0 11.12 50.0 33.3 16.7 — — — 0.50 0.62 1.60 57.1 21.4 0.0 0.0 0.0 27.83 40.0 30.0 20.0 10.0 — — 0.40 0.55 1.83 66.7 36.7 13.3 0.0 0.0 46.04 33.3 26.7 20.0 13.3 6.7 — 0.33 0.49 2.02 72.7 47.3 25.5 9.1 0.0 64.95 28.6 23.8 19.0 14.3 9.5 4.8 0.29 0.45 2.20 76.9 54.9 35.2 18.7 6.6 84.1

Geometric Smoothing (δ = 0.25)

0 100.0 — — — — — 1.00 1.00 1.00 0.0 0.0 0.0 0.0 0.0 —1 80.0 20.0 — — — — 0.80 0.82 1.21 23.5 0.0 0.0 0.0 0.0 4.02 76.2 19.0 4.8 — — — 0.76 0.79 1.27 24.9 5.9 0.0 0.0 0.0 5.93 75.3 18.8 4.7 1.2 — — 0.75 0.78 1.29 25.0 6.2 1.5 0.0 0.0 6.54 75.1 18.8 4.7 1.2 0.3 — 0.75 0.78 1.29 25.0 6.2 1.6 0.4 0.0 6.65 75.0 18.8 4.7 1.2 0.3 0.1 0.75 0.77 1.29 25.0 6.2 1.6 0.4 0.1 6.7

Implications of three different smoothing profiles for observed betas, standard deviations, Sharpe ratios, and serialcorrelation coefficients of a fund with IID true returns. Straightline smoothing is given by θj = 1/(k+1); sum-of-yearssmoothing is given by θj = (k+1−j)/[(k+1)(k+2)/2]; geometric smooothing is given by θj = δj(1−δ)/(1−δk+1).cβ , cσ , and cs denote multipliers associated with the beta, standard deviation, and Sharpe ratio of observed returns,and ρo

j denotes the j-th autocorrelation coefficient of observed returns. Source: AlphaSimplex Group, LLC.

MIT

c©2004

by

A.Lo

Page

40/53

Estimating Smoothing andIlliquidity Profiles

Set k=2 And Estimate MA(2) Via Maximum Likelihood:

Roit = θ0 Rit + θ1 Rit−1 + θ2 Rit−2 (55a)

Rit ∼ IID N (µi, σ2i ) (55b)

With Constraint:

1 = θ0 + θ1 + θ2 (56)

As A Specification Check, Impose Different Normalization:

σ2i ≡ σ

2nw (57)

TASS Dataset:

• November 1977 to January 2001

• Live (1,512) and Graveyard (927) Funds

• Graveyard Database Started In 1994

• 30 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)


TASS Hedge Funds with 25 Smallest Estimated Smoothing Parameters θ0

Code Category Period T Status θ0 SE(θ0) θ1 SE(θ1) θ2 SE(θ2) ξ

1201 Non-Directional/Relative Value 199409–200101 77 1 0.464 0.040 0.365 0.031 0.171 0.041 0.3784346 Event Driven 199412–200011 71 1 0.471 0.041 0.335 0.034 0.195 0.041 0.371180 Not Categorized 198906–199608 87 0 0.485 0.041 0.342 0.032 0.173 0.040 0.382

1204 Non-Directional/Relative Value 199510–200101 64 1 0.492 0.049 0.339 0.038 0.169 0.048 0.3861584 Fund of Funds 199601–200101 61 1 0.504 0.046 0.245 0.042 0.251 0.042 0.377518 Non-Directional/Relative Value 199312–200005 78 0 0.514 0.034 0.142 0.037 0.343 0.028 0.403971 Non-Directional/Relative Value 199409–200012 76 1 0.515 0.038 0.176 0.039 0.309 0.033 0.392

1234 Fund of Funds 199410–200012 75 1 0.527 0.061 0.446 0.034 0.027 0.059 0.4772185 Fixed Income Directional 199108–200101 114 1 0.532 0.040 0.265 0.034 0.202 0.037 0.395

26 Non-Directional/Relative Value 199303–200101 95 1 0.533 0.047 0.305 0.036 0.162 0.044 0.403171 Non-Directional/Relative Value 199304–200101 94 1 0.536 0.040 0.193 0.039 0.271 0.035 0.398

1696 Fund of Funds 199501–200001 61 0 0.536 0.067 0.406 0.041 0.058 0.064 0.456120 Non-Directional/Relative Value 198207–199810 196 0 0.546 0.032 0.238 0.027 0.215 0.028 0.402

1396 Non-Directional/Relative Value 199507–200101 67 1 0.548 0.056 0.255 0.047 0.197 0.050 0.4042774 Non-Directional/Relative Value 199501–200006 66 0 0.553 0.058 0.262 0.047 0.185 0.052 0.4091397 Non-Directional/Relative Value 199410–200101 76 1 0.556 0.055 0.260 0.044 0.184 0.049 0.410

57 Non-Directional/Relative Value 199210–200101 100 1 0.561 0.052 0.311 0.038 0.127 0.048 0.4284158 Event Driven 199510–200101 64 1 0.565 0.062 0.261 0.050 0.174 0.055 0.4171773 Fund of Funds 199506–200101 68 1 0.569 0.056 0.194 0.051 0.238 0.048 0.417415 Not Categorized 198807–199608 98 0 0.570 0.054 0.306 0.039 0.124 0.049 0.434

1713 Fixed Income Directional 199601–200101 61 1 0.573 0.067 0.269 0.052 0.158 0.059 0.4261576 Fund of Funds 199504–200010 67 1 0.575 0.066 0.293 0.048 0.132 0.059 0.4341633 Event Driven 199304–199901 70 0 0.576 0.066 0.309 0.047 0.115 0.060 0.4401883 Event Driven 199306–200101 92 1 0.578 0.053 0.236 0.044 0.187 0.046 0.4241779 Non-Directional/Relative Value 199512–200101 62 1 0.584 0.067 0.241 0.054 0.175 0.058 0.429

First 25 funds of ranked list of 908 hedge funds in the TASS Hedge Fund Combined (Live and Graveyard) database withat least five years of returns history during the period from November 1977 to January 2001, ranked in increasing orderof the estimated smoothing parameter θ0 of the MA(2) smoothing process Ro

t = θ0Rt + θ1Rt−1 + θ2Rt−2, subject to thenormalization 1 = θ0 + θ1 + θ2, and estimated via maximum likelihood.

MIT

c©2004

by

A.Lo

Page

42/53

Estimated Smoothing Coefficients θ0

0.40

0.60

0.80

1.00

1.20

1.40

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Fund�Category

θθθθ0

Estimated Smoothing Index ξ

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Fund�Category

ξξξξ


Summary of MA(2) Parameter Estimates For TASS Hedge Funds

Monthly Data, November 1977 to January 2001

Category N

MA(2) with Constrained Sum

θ0 θ1 θ2 ξ

Mean SD Mean SD Mean SD Mean SD

Not Categorized 111 1.04 0.33 0.01 0.18 −0.05 0.23 1.27 1.06US Equity Hedge 162 0.95 0.21 0.06 0.15 −0.01 0.15 1.00 0.54European Equity Hedge 22 0.82 0.15 0.08 0.09 0.11 0.11 0.72 0.28Asian Equity Hedge 5 0.94 0.30 0.05 0.19 0.01 0.11 1.00 0.71Global Equity Hedge 27 0.91 0.14 0.11 0.10 −0.02 0.09 0.88 0.27Dedicated Shortseller 7 1.03 0.20 0.03 0.12 −0.06 0.13 1.12 0.48Fixed-Income Directional 13 0.76 0.17 0.15 0.10 0.08 0.12 0.67 0.27Convertible Fund (Long Only) 15 0.84 0.35 0.18 0.08 −0.02 0.35 0.98 1.36Event Driven 109 0.81 0.17 0.15 0.13 0.04 0.11 0.74 0.28Non-Directional/Relative Value 85 0.82 0.24 0.09 0.25 0.09 0.14 0.82 0.62Global Macro 24 0.95 0.18 0.08 0.11 −0.03 0.15 0.99 0.40Global Opportunity 1 0.74 — 0.16 — 0.10 — 0.58 —Natural Resources 3 0.91 0.19 0.02 0.11 0.07 0.09 0.87 0.34Pure Leveraged Currency 26 1.10 0.34 0.03 0.13 −0.13 0.25 1.41 1.38Pure Managed Futures 93 1.13 0.32 −0.05 0.22 −0.08 0.18 1.47 1.19Pure Emerging Market 72 0.83 0.16 0.15 0.09 0.02 0.13 0.76 0.31Pure Property 1 1.23 — −0.31 — 0.07 — 1.62 —Fund of Funds 132 0.85 0.21 0.13 0.13 0.03 0.13 0.81 0.50

All 908 0.92 0.26 0.08 0.17 0.00 0.17 0.98 0.76

Means and standard deviations of maximum likelihood estimates of MA(2) smoothing process Rot = θ0Rt +

θ1Rt−1 + θ2Rt−2, ξ ≡ θ20 + θ2

1 + θ22, 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.

MIT

c©2004

by

A.Lo

Page

44/53

MA(2) Parameter Estimates For Indexes and Mutual Funds

Monthly Data, Various Sample Periods

SeriesPeriod

TMean SD ρ1 ρ2 ρ3 β

R2

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

Ibbotson Small Company 192601–200112 912 1.35 8.63 15.6 1.7 −10.6 1.27 66.9Ibbotson Long-Term Government Bonds 192601–200112 912 0.46 2.22 6.7 0.3 −8.3 0.07 2.8Ibbotson Long-Term Corporate Bonds 192601–200112 912 0.49 1.96 15.6 0.3 −6.0 0.08 5.2Ibbotson Large Company 192601–200112 912 1.03 5.57 9.8 −3.2 −10.7 1.00 100.0AXP Extra Income Fund (INEAX) 198401–200112 216 0.67 2.04 35.4 13.1 2.5 0.21 20.7Vanguard 500 Index Trust (VFINX) 197609–200112 304 1.16 4.36 −2.3 −6.8 −3.2 1.00 100.0

CSFB/Tremont Indices:Aggregate Hedge Fund Index 199401–200210 106 0.87 2.58 11.2 4.1 −0.4 0.31 24.9Convertible Arbitrage 199401–200210 106 0.81 1.40 56.6 42.6 15.6 0.03 1.1Dedicated Short Bias 199401–200210 106 0.22 5.29 7.8 −6.3 −5.0 −0.94 58.6Emerging Markets 199401–200210 106 0.54 5.38 29.4 1.2 −2.1 0.62 24.0Equity Market Neutral 199401–200210 106 0.89 0.92 29.4 18.1 8.4 0.10 21.1Event Driven 199401–200210 106 0.83 1.81 34.8 14.7 3.8 0.23 30.2Fixed Income Arbitrage 199401–200210 106 0.55 1.18 39.6 10.8 5.4 0.02 0.7Global Macro 199401–200210 106 1.17 3.69 5.6 4.6 8.3 0.24 7.5Long/Short 199401–200210 106 0.98 3.34 15.9 5.9 −4.6 0.48 36.7Managed Futures 199401–200210 106 0.55 3.44 3.2 −6.3 0.7 −0.12 2.5

SeriesPeriod

T θ0 SE(θ0) θ1 SE(θ1) θ2 SE(θ2) ξ

Ibbotson Small Company 192601–200112 912 0.82 0.03 0.13 0.02 0.04 0.03 0.69Ibbotson Long-Term Government Bonds 192601–200112 912 0.92 0.05 0.06 0.03 0.01 0.03 0.86Ibbotson Long-Term Corporate Bonds 192601–200112 912 0.84 0.04 0.14 0.02 0.02 0.03 0.73Ibbotson Large Company 192601–200112 912 0.92 0.05 0.09 0.03 −0.01 0.03 0.85AXP Extra Income Fund (INEAX) 198401–200112 216 0.67 0.03 0.24 0.03 0.09 0.04 0.51Vanguard 500 Index Trust (VFINX) 197609–200112 304 1.12 0.17 −0.03 0.07 −0.09 0.07 1.26

CSFB/Tremont Indices:Aggregate Hedge Fund Index 199401–200210 106 0.86 0.12 0.09 0.08 0.04 0.08 0.76Convertible Arbitrage 199401–200210 106 0.49 0.01 0.26 0.03 0.25 0.03 0.37Dedicated Short Bias 199401–200210 106 0.99 0.20 0.08 0.09 −0.07 0.10 0.99Emerging Markets 199401–200210 106 0.75 0.08 0.24 0.05 0.01 0.07 0.62Equity Market Neutral 199401–200210 106 0.71 0.06 0.18 0.05 0.12 0.06 0.54Event Driven 199401–200210 106 0.68 0.05 0.23 0.05 0.09 0.06 0.52Fixed Income Arbitrage 199401–200210 106 0.63 0.04 0.28 0.04 0.08 0.05 0.49Global Macro 199401–200210 106 0.91 0.14 0.04 0.08 0.05 0.08 0.84Long/Short 199401–200210 106 0.82 0.10 0.13 0.07 0.06 0.07 0.68Managed Futures 199401–200210 106 1.04 0.23 0.04 0.10 −0.08 0.11 1.08


Classical and Adjusted Sharpe Ratios for TASS Hedge Funds

Monthly Data, November 1977 to January 2001

Category

Sharpe Ratios For Combined Sample

NSR SR∗ SR∗∗

Mean SD Mean SD Mean SD

Not Categorized 111 1.12 1.09 1.06 0.87 1.00 0.82US Equity Hedge 162 1.26 0.75 1.31 0.75 1.23 0.69European Equity Hedge 22 1.43 0.74 1.43 0.80 1.33 0.74Asian Equity Hedge 5 0.50 0.39 0.52 0.39 0.49 0.37Global Equity Hedge 27 0.90 0.61 0.95 0.66 0.89 0.61Dedicated Shortseller 7 0.28 0.59 0.32 0.64 0.30 0.61Fixed-Income Directional 13 2.02 2.35 1.80 2.23 1.68 2.06Convertible Fund (Long Only) 15 1.83 1.20 1.66 0.85 1.55 0.80Event Driven 109 2.36 1.45 2.21 1.57 2.08 1.47Non-Directional/Relative Value 85 2.20 1.86 2.03 2.39 1.89 2.22Global Macro 24 1.08 0.67 1.14 0.73 1.07 0.70Global Opportunity 1 −0.56 — −0.39 — −0.37 —Natural Resources 3 0.60 0.25 0.56 0.23 0.52 0.21Pure Leveraged Currency 26 0.63 0.49 0.65 0.50 0.61 0.47Pure Managed Futures 93 0.54 0.55 0.63 0.60 0.60 0.56Pure Emerging Market 72 0.39 0.45 0.36 0.44 0.34 0.40Pure Property 1 0.42 — 0.45 — 0.41 —Fund of Funds 132 1.44 1.01 1.30 0.88 1.22 0.82

All 908 1.32 1.24 1.27 1.27 1.19 1.18

Means and standard deviations of Sharpe ratios of 908 hedge funds in the TASS Hedge FundCombined (Live and Graveyard) database with at least five years of returns history duringthe period from November 1977 to January 2001. SR is the standard Sharpe ratio, SR∗

is the smoothing-adjusted Sharpe ratio of Lo (2002), and SR∗∗ is the smoothing-adjustedSharpe ratio using σNW. 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!


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 6= 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:





5. Psychology of Risk Preferences


A theory should be made as simple as possible, but not

simpler..

– Albert Einstein


Additional References

Ackermann, C., McEnally, R. and D. Ravenscraft, 1999, “The Performance of Hedge

Funds: Risk, Return, and Incentives”, Journal of Finance 54, 833–874.

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, 6–23.

Agarwal, V. and N. Naik, 2000c, “Multi-Period Performance Persistence Analysis of

Hedge Funds Source”, Journal of Financial and Quantitative Analysis 35, 327–342.

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, 6–19.

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, 1–25.

Brown, S. and W. Goetzmann, 2001, “Hedge Funds With Style”, NBER Working PaperNo. 8173.

Brown, S., Goetzmann, W., Ibbotson, R. and S. Ross, 1992, “Survivorship Bias inPerformance Studies”, Review of Financial Studies 5, 553–580.

Brown, S., Goetzmann, W. and R. Ibbotson, 1999, “Offshore Hedge Funds: Survival andPerformance 1989–1995”, Journal of Business 72, 91–118.

Brown, S., Goetzmann, W. and B. Liang, 2002, “Fees on Fees in Funds of Funds”, Yale

ICF Work Paper No. 02–33.


Brown, S., Goetzmann, W. and J. Park, 1997, “Conditions for Survival: Changing

Risk and the Performance of Hedge Fund Managers and CTAs”, Yale School ofManagement Work Paper No. F–59.

Brown, S., Goetzmann, W. and J. Park, 2000, “Hedge Funds and the Asian CurrencyCrisis”, Journal of Portfolio Management 26, 95–101.

Brown, S., Goetzmann, W. and J. Park, 2001, “Careers and Survival: Competition andRisks in the Hedge Fund and CTA Industry”, Journal of Finance 56, 1869–1886.

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 Investmentsin Bull and Bear Markets”, The Journal of Portfolio Management 27, 97–108.

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, 9991–9992.

Fung, W. and D. Hsieh, 1997a, “Empirical Characteristics of Dynamic TradingStrategies: The Case of Hedge Funds”, Review of Financial Studies 10, 275–302.

Fung, W. and D. Hsieh, 1997b, “Investment Style and Survivorship Bias in the Returns ofCTAs: The Information Content of Track Records”, Journal of Portfolio Management

24, 30–41.

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

6, 309–31.

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, 291–307.

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

Getmansky, M., Lo, A. and I. Makarov, 2004, “ An Econometric Model of SerialCorrelation 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.


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 PerformancePersistence Given Survivorship Bias”, Review of Economics and Statistics 79, 161–

170.

Horst, J., Nijman, T. and M. Verbeek, 2001, “Eliminating Look-Ahead Bias in EvaluatingPersistence in Mutual Fund Performance”, Journal of Empirical Finance 8, 345–373.

Jobson, J. and R. Korkie, 1981, “Performance Hypothesis Testing with the Sharpe and

Treynor Measures”, Journal of Finance 36, 889–908.

Kao, D., 2002, “Battle for Alphas: Hedge Funds versus Long-Only Portfolios”, Financial

Analysts Journal 58, 16–36.

Leroy, S., 1973, “Risk Aversion and the Martingale Property of Stock Returns”,

International Economic Review 14, 436–446.

Liang, B., 1999, “On the Performance of Hedge Funds”, Financial Analysts Journal 55,72–85.

Liang, B., 2000, “Hedge Funds: The Living and the Dead”, Journal of Financial and

Quantitative Analysis 35, 309–326.

Liang, B., 2001, “Hedge Fund Performance: 1990–1999”, Financial Analysts Journal 57,

11–18.

Lo, A., 1994, “Data-Snooping Biases in Financial Analysis”, in H. Russell Fogler, ed.:Blending Quantitative and Traditional Equity Analysis, 1994. 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, 13–26.

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

Lo, A., 2002, “The Statistics of Sharpe Ratios”, Financial Analysts Journal 58, 36–50.

Lo, A. and C. MacKinlay, 1990, “Data-Snooping Biases in Tests of Financial Asset

Pricing Models”, Review of Financial Studies 3, 431–469.


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,

92–99.

Lucas, R., 1978, “Asset Prices in an Exchange Economy”, Econometrica 46, 1429–1446.

Samuelson, P., 1965, “Proof that Properly Anticipated Prices Fluctuate Randomly”,Industrial Management Review 6, 41–49.

Schneeweis, T. and R. Spurgin, 1996, “Survivor Bias in Commodity Trading AdvisorPerformance”, Journal of Futures Markets 16, 757–772.

Sharpe, W., 1994, “The Sharpe Ratio”, Journal of Portfolio Management 21, 49–58.

Spurgin, R., 2001, “How to Game Your Sharpe Ratio”, The Journal of Alternative

Investments 4, 38–46.

Weisman, A., 2002, “Informationless Investing and Hedge Fund PerformanceMeasurement Bias”, The Journal of Portfolio Management 28, 80–91.


Documents

Risk Management For Hedge Funds: Introduction and Overviewweb.mit.edu/katykam/Public/slides2004.pdf · 2004. 3. 23. · Credit Risk In ationary Pressures, Central Banking Policy Macroeconomic