The nonlinearities of hedge fund returns
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The Nonlinearities of Hedge Fund Returns. Joint work with E. Derman. [email protected] www.riskdata.com +33 1 44 54 35 00. Raphaël Douady Research Director, Riskdata ®. Traditional Equity Fund. FUND realized returns. y = 1,1051x - 0,0071. R. 2. = 0,8909.

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The Nonlinearities of Hedge Fund Returns

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The nonlinearities of hedge fund returns

The Nonlinearities of Hedge Fund Returns

Joint work with E. Derman

  • [email protected]

  • www.riskdata.com

  • +33 1 44 54 35 00

Raphaël Douady

Research Director, Riskdata®


Non linear modeling

Traditional Equity Fund

FUND realized returns

y = 1,1051x - 0,0071

R

2

= 0,8909

SP500 observed Returns

Non Linear Modeling?

  • Do Hedge Funds support Linear Modeling?

Fund Return = a + bx Index Return + e


Non linear modeling1

y = 0,4398x - 0,0215

R

2

= 0,0754

FUND realized returns

SP500 observed Returns

Non Linear Modeling?

  • Nonlinear Modeling

    • Missing nonlinearity  Erroneous a and b

    • Correlation is more than meaningless: misleading

Fund Return = F(Index Return) + e


Testing nonlinearity

Testing Nonlinearity

  • 1000 Hedge Funds

    • Distribution across Strategies similar to overall HF population

    • Including Dead Funds and their last return

    • Monthly Returns

  • Analysis Period

    • Jan 95 to Jun 05 (restricted to Fund existence)

  • Methodology

    • Select, among investable factors, the most explanatory

    • F-test of Quadratic and Cubic regression vs. Linear regression

    • Identify Funds that reject Linear model with Confidence 95%


Hedge fund database

Hedge Fund Database


Nonlinearity test

Nonlinearity Test

  • Only ¼ of Hedge Funds support Linear Modeling


Nonlinearity test by strategy

100%

90%

80%

70%

60%

50%

40%

30%

20%

10%

0%

VALUE

MACRO

MISCELANEOUS

LONG ONLY

SHORT BIAS

MORTGAGES

COMMODITY

DISTRESSED

CURRENCIES

MARKET TIMER

REGULATION D

EVENT DRIVEN

OPPORTUNISTIC

MULTI-STRATEGY

FINANCE SECTOR

OTHER ARBITRAGE

SMALL/MICRO CAP

COUNTRY SPECIFIC

LONG/SHORT EQUITY

EMERGING MARKETS

SHORT:TERM TRADING

SPECIAL SITUATIONS

HEALTHCARE SECTOR

OPTIONS STRATEGIES

TECHNOLOGY SECTOR

CTA/MANAGED FUTURES

STATISTICAL ARBITRAGE

MERGER/RISK ARBITRAGE

MARKET NEUTRAL EQUITY

CONVERTIBLE ARBITRAGE

FIXED INCOME ARBITRAGE

CAPITAL STRUCTURE ARBITRAGE

FIXED INCOME (NON-ARBITRAGE)

Nonlinearity Test by Strategy


Linear and nonlinear strategies

Linear and Nonlinear Strategies

  • Nonlinear Strategies

    • Short Bias, Market Timer, Currencies, Tech Sector, Stat Arb., Short Term Trading, CTA, L/S Equity

      • Dynamic Trading imply Optional Profile (Black-Scholes-Merton)

    • Convert. Arb., Option Strategies

      • Nonlinear Instruments

    • Merger/Risk Arb., Mortgages, Fixed Income Arb.

      • Correlation break under Liquidity Stress

  • Linear Strategies

    • Country specific, Emerging Markets, Commodities, Small Caps

      • Directional portfolio

      • Low turnover


Nonlinear profiles

3%

2%

Period

30/06/2001

30/06/2005

1%

0%

-3%

-2%

-1%

0%

1%

2%

3%

-1%

Fund Return

-2%

2

y = 25.75x

+ 0.01x - 0.00

y = 0.07x + 0.00

-3%

2

2

R

= 0.14

R

= 0.01

-4%

Sector Finance USA

Nonlinear Profiles

  • Long/Short Equity with Quadratic Shape

Strategy: “to achieve capital appreciation through the application of analytically and statistically based trading strategies, the beta neutral strategy is designed to return 12-14% annually with low drawdowns and low correlation to S&P 500”


Simulation trend follower

Simulation: Trend Follower

  • Quadratic shape

  • Negative outliers

Strategy: Buy when Index Spot > 1M moving average

Sell when Index Spot < 1M moving average


Alternative factors

Alternative Factors?

  • Create “nonlinear” factors

    • Returns of a Trend Following strategy

    • Trend Followers and Mean Reverters are now Linear with respect to this “Alternative Factor”

  • Literature

    • Naik-Agarval:Factor = returns of an Option Roll-over

    • Fung-Hsieh:Factor = returns of a Barrier Option

  • Industry practice: Hedge Fund Indices

    • Average of HF returns of a given strategy

    • HFR, EDHEC, etc.


Cubic profile

20%

3

2

y = 66.81x

+ 2.86x

+ 0.03x - 0.00

2

R

= 0.35

15%

Period

31/03/2001

31/07/2005

10%

5%

y = 0.50x + 0.00

2

R

= 0.25

0%

Fund Return

-15%

-10%

-5%

0%

5%

10%

15%

-5%

-10%

-15%

COMEX GOLD INDEX

Cubic Profile

  • CTA

Strategy:“The primary objective of the Advisor is the capital appreciation of the Company’s assets through the speculation in commodity futures contracts and cash currencies (FX). The Advisor will attempt to meet the objective of capital appreciation by making trading decisions based upon a proprietary trading method. (…)It believes that future price movements in all markets may be more accurately anticipated by historical price movements within a quantitative or technical analysis than by fundamental economic analysis. Since non-directional and limited price directional trading strategies are employed, major long-term price movements are not necessarily needed for the program to be successful. Rather, diverse models that have yielded good risk/reward characteristics in the past are combined with other uncorrelated models to form a robust trading program that is less dependent on any one particular market characteristic”


Specific hedge fund index

20%

15%

Period

31/03/2001

31/07/2005

10%

5%

0%

Fund Return

-15%

-10%

-5%

0%

5%

10%

15%

-5%

2

y = 9.17x

- 0.13x - 0.02

-10%

2

R

= 0.35

-15%

TREND STRATEGY ON COMEX GOLD INDEX

Specific Hedge Fund Index

  • CTA with respect to Alternative Factor

  • Reverse engineer CTA signal: Trend follower on Trend Following Strategy

  • Quadratic on Quadratic  Cubic shape


Modeling with lags

Modeling with Lags

  • Do Hedge Fund support Modeling without Lags?

    • Null Hypothesis: The past has no influence on Fund returns

      Fund Return = F(Index Return) + e

    • Alternative Hypothesis: Fund returns are sensitive to

      • Past Fund returns

      • Past Index returns

        Fund Rtn = F(Idx Rtn) + G(Past Idx Rtn) + H(Past Fund Rtn) + e

      • Function H  Serial correlation

      • Function G  Delayed market impact


Test for lags

Test for Lags

  • Only ¼ of Hedge Funds support Modeling without Lags


Average serial correlation

Average Serial Correlation


Serial correlation

Serial Correlation

  • Less liquid strategies display higher serial correlation

  • Literature

    • Andy Lo: Many reasons for Serial Correlation

      • Illiquid Securities are serially correlated

      • Delayed influence of Markets on Illiquid Securities

      • Time mismatch between Trades and Fund Reporting

      • Return smoothing and “Hard to price” securities


Need for nonlinear dynamic model

Need for Nonlinear Dynamic Model

  • Hypothesis Testing for the need of Nonlinear functional form and Lagged Factor Values

  • Models:

    • LS : Linear Static (without lags)

    • LD : Linear Dynamic (with lags)

    • NLS : Nonlinear Static (without lags)

    • NLD : Nonlinear Dynamic (with lags)


Conclusions

CONCLUSIONS

  • For 89% of Hedge Funds, the Linear Model without Lags is Rejected with 95% Confidence

  • Nonlinear Modeling is even more important than Lag Modeling

    • Nonlinear Model without Lags rejected for ~1/2 of Hedge Funds

    • Linear Model with Lags rejected for ~ 3/4 of Hedge Funds

  • Nonlinear Returns ≠ Nonlinear instruments

    • Not Captured by Holdings analysis

    • Due to Dynamic Trading and Liquidity Stress

  • Lagged influence ≠ Serial Correlation≠ Return smoothing

    • ~1/2 of Hedge Funds are influenced by Past Factor Returns

    • Smoothing is only one of the explanations for Serial Correlation


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