# Presentation: 359813

## Dynamic Investment Opportunities and the Cross-Section of Hedge Fund Returns: Implications of Higher-Moment Risks for Performance

- Vikas Agarwal
- Georgia State University
- Gurdip Bakshi
- University of Maryland
- Joop Huij
- Erasmus University

- _18__th__ Annual Derivative Securities and__ Risk Management Conference_

- April 11, 2008

## Overview and Main Themes

- Investors require premiums for bearing higher-moment risks: hate volatility, negative skewness and large moves
- 	Rubinstein (1973); Merton ICAPM (1973); Kraus and Litzenberger (1976); Vanden (2006); among others
- Higher moment risks are priced in option returns – vast literature EQ[VOL]>EP[VOL] for S&P 500 index

## Overview and Main Themes (contd.)

- Hedge funds have the capability and flexibility to extract risk premiums
- Hedge funds have nonlinear option-like return structure
- Hedge funds trade dynamically and take state-contingent bets in response to changes in macroeconomic conditions and arbitrage opportunities [Fung and Hsieh (2001); Mitchell and Pulvino (2001); Agarwal and Naik (2004)]
- Hedge funds should exhibit exposures to higher moments of market returns

## Research Questions

- To what extent are hedge funds exposed to higher-moment market risks?
- How much is the premium for the higher moments and how much of the hedge fund performance may be attributable to the higher moment risks?
- What are the implications for excluding higher moment risk factors while evaluating hedge fund performance?
- What about funds of hedge funds, which are less vulnerable to biases? (other issues: diversification and compensation)
- How large are the exposures in case of mutual funds?

## Hedge fund and mutual fund data

- Monthly net-of-fee returns of 3,771 hedge funds and 1,062 funds of hedge funds from Lipper (previously TASS) database from Jan 1994 to Dec 2004
- Exclude funds that do not report on monthly basis and those that do not have 12 consecutive returns over the sample period
- Survivorship-bias free due to inclusion of funds that have disappeared from the database
- Monthly net-of-fee returns of about 9,769 mutual funds from CRSP mutual fund database from Jan 1994 to Dec 2004

## Measures of higher-moments of market risk

- Precise estimation of higher moments require long time-series data
- Need forward-looking measures
- Estimate the intrinsic values of the second, third, and fourth moments of equity returns from S&P 100 index options traded on CBOE by spanning and valuing the relevant payoffs as shown in Bakshi, Kapadia, and Madan (RFS 2003)
- Higher moments of other markets (fixed income, commodities, etc.) may also be relevant but matching high-quality option data unavailable

## Time-Series of the Value of Claims to Higher Moments

## Time-Series of the Value of Claims to Higher Moments (contd.)

## Time-Series of the Value of Claims to Higher Moments (contd.)

## Measures of risk-adjusted performance

- Fung and Hsieh (2004) seven-factor model –market, size, bond, credit, and three option factors on bonds, currencies, and commodities – FH-7 MODEL
- Carhart (1997) four-factor model – market, size, book-to-market, and momentum – CARHART-4 MODEL

## Methodology (I)

- Standard asset pricing tests using pooled time-series cross-sectional data
- First construct a set of base assets that have significant dispersion in the sensitivity to higher-moment risks
- Form decile portfolios of hedge funds sorted based on their exposures to innovations in higher moments - VOL, SKEW, and KURT

## Methodology (II)

- Rolling window regressions over the past 12 months
- Short estimation window – compromise between estimating coefficients with precision and allowing for time-varying factor loadings (Bollen and Whaley 2007) – for robustness, perform Bayesian analysis
- Keep number of factors minimum in constructing the portfolios but control for all possible factors in the post-formation period (similar to Ang et al. (JF 2006))
- Wait for three months for the post-formation period to allow for lockups and redemption period

## Rank Portfolios

- Jan
- ‘94

- Jan
- ‘95

- Feb
- ‘95

- Mar
- ‘95

- Portfolio formation period (I)

- Out-of-sample equally-weighted returns for sorted portfolios (I)

- Feb
- ‘94

- Portfolio formation period (II)

- Out-of-sample equally-weighted returns for sorted portfolios (II)

- Ranking Period

- Investment Period

- Dec
- ‘04

- Apr
- ‘95

- Three-month wait (I)

- May
- ‘95

- Three-month wait (II)

- Wait

## Independent sorts on higher moments

- LARGE
- AND
- SIGNIFICANT
- SPREADS
- IN ALPHAS

## Three-way sorts on higher moments

- SPREADS
- IN ALPHAS
- CONTINUE
- TO BE
- LARGE AND
- SIGNIFICANT

## Bootstrapping Results

- HISTOGRAM SHOWS HOW BIG A SPREAD IN ALPHAS IS OBTAINED BY CHANCE  IF ZERO ALPHA IS IMPOSED IN THE FH-7 MODEL SPECIFICATION

- BOOTSTRAPPED ALPHAS BETWEEN EXTREME PORTFOLIOS AT THE 95% CONFIDENCE LEVEL ARE BETWEEN -8.5% AND 8.5% P.A.

## Higher moment risk factors based on hedge fund returns

- LARGE AND SIGNIFICANT PREMIUMS

- EXISTING
- MODELS
- CAN’T
- EXPLAIN

## Post-Ranking Regressions and Risk Story

- STRONG PATTERNS IN
- POST-RANKING LOADINGS RISK-BASED EXPLANATION

- IMPROVEMENT
- IN EXPLANATORY
- POWER

- SPREADS
- IN ALPHAS
- BECOME
- SMALL

## Augmented FH-7 with higher moment factors

- BARS ON THE DIAGONAL INDICATE THE % OF FUNDS THAT ARE RANKED IN THE SAME DECILE USING THE TWO MODELS. OFF-DIAGONAL BARS REPRESENT INCONSISTENT DECILE RANKINGS.

- NEW MODEL
- CAN CONSIDERABLY ALTER HEDGE FUNDS’ ALPHA-BASED RANKINGS

## Robustness tests

- Liquidity
- Fung and Hsieh 9-factor model
- Out-of-the-money put option factor
- Bayesian estimates
- Backfilling bias

## Hedge Fund Strategies with extreme exposures

## Higher moment exposures for FOFs

- FOFs popular among institutional investors; less susceptible to data biases
- Three possibilities regarding higher moment exposures
- Effective risk management policy would imply neutralizing of higher moment risks
- Improving performance to increase compensation would suggest taking on higher moment risks
- Unintentional diversification due to investment in disparate strategies

## Substantial Reduction in Exposure for FOFs

- (1) MAGNITUDE OF EXPOSURES ARE 40% OF THOSE FOR THE COMBINED SAMPLE
- (2) REDUCTION IN THE X-SECTIONAL DISPERSION OF BETAS

## Summary of results for FOFs

- Sensitivities of FOFs to higher moment risks ameliorated across the board
- FOFs may be natural vehicles to offset higher moment exposures
- Hitherto unidentified additional benefit accruing to FOF investors

## Higher moments not priced in mutual funds

- No discernible pattern in alphas
- Given the higher dispersion in exposures for hedge funds, our test design on hedge funds is crucial for recovering the risk premiums

- LOWER DISPERSION
- IN BETAS COMPARED
- TO HEDGE FUNDS

- LOWER SPREAD
- IN ALPHAS

## Summary

- Striking patterns in alphas when hedge funds are sorted on exposures to higher moment risks
- Significant and large risk premiums for the three higher moments
- 	Volatility :   ─6.55%
- 	Skewness: 3.40%
- 	Kurtosis:    ─ 2.39%
- Far lower dispersion for funds of hedge funds

## Summary (contd.)

- No patterns in mutual funds — differences in trading strategies compared to hedge funds
- Including higher moment risk factors flattens alphas and improves explanatory power
- Hedge funds earn a substantial portion of their returns for bearing higher moment risks