Papers

Value at risk, GARCH modelling and the forecasting

of hedge fund return volatility

Roland Fu

, Dieter G. Kaiser and Zeno Adams

Department of Empirical Research and Econometrics, University of Freiburg,

Platz der Alten Synagoge, Freiburg im Breisgau, D-79085, Germany.

Tel: þ49 0 761 203 2341, Fax: þ49 0 761 203 2340,

E-mail: [email protected]

Received (in revised form): 29th August, 2006

Dr Roland Fu

ss is Lecturer at the Department of Empirical Research and Econometrics and Assistant Professor at the

Department of Finance and Banking at the University of Freiburg, Germany. He holds an MBA from the University of

Applied Science in Lo

rrach, M.Ec. and PhD degree in Economics from the University of Freiburg. His research

interests are in the field of applied econometrics, alternative investments as well as international and real estate

finance. He has (co-) authored several articles in finance journals and book chapters. Further, he is a member of the

Verein fu

r Socialpolitik and of the German Finance Association.

Dieter G. Kaiser is responsible for the institutional research at Benchmark Alternative Strategies in Frankfurt,

Germany. He has worked for Dresdner Kleinwort Wasserstein and Cre

dit Agricole Asset Management in Frankfurt

where he was responsible for the fund-of-hedge-funds Marketing Support. He has written several articles on the

subject of alternative investments that have been published in professional as well as academic journals. He is the

co-author and co-editor of five books on alternative investments published by John Wiley & Sons, Risk Books and

Gabler. He holds a Diploma in Business Administration and a Master of Arts in Banking & Finance from the HfB -

Business School of Finance and Management in Frankfurt, where he has also lectured on alternative investments

since 2003.

Zeno Adams is Research Assistant at the Department of Empirical Research and Econometrics, University of

Freiburg, Platz der Alten Synagoge, D-79085 Freiburg im Breisgau, Germany.

Practical applications

The Value at Risk (VaR) approach is widely used within the asset and risk management of traditional and

alternative investments. From the investor’s point of view, an adequate VaR model should

indicate how the positions of the hedge fund portfolio should be sized for the best protection against

downside risk. Due to the skewness and excess kurtosis of daily financial return distributions, the normal

VaR has its drawbacks particularly when it is applied to hedge funds. In addition to the Cornish–Fischer

VaR, which explicitly accounts for non-normally distributed returns, we examine the conditional

volatility characteristics of daily hedge fund management style returns. The inclusion of time-varying

conditional volatility into the VaR measurement enables us to trace the actual return process more

effectively. We show that such a GARCH-type VaR is a superior measure of downside risk, especially for

trading strategies that exhibit negative skewness and excess kurtosis and offer daily pricing.

2 Journal of Derivatives & Hedge Funds Volume 13 Number 1 2007

www.palgrave-journals.com/dutr

Journal of Derivatives

& Hedge Funds,

Vol. 13 No. 1, 2007,

pp. 2–25

r 2007 Palgrave

Macmillan Ltd

1753-9641 $30.00

Abstract

This paper examines the conditional volatility

characteristics of daily management style returns and

compares the out-of-sample forecasts of different Value

at Risk (VaR) approaches, namely, the normal,

Cornish–Fisher (CF), and the so-called GARCH-

type VaR. The examination of the conditional

volatility of hedge fund styles and composite returns

shows important differences concerning persistence,

mean reversion and asymmetry in the period under

consideration. Hedge fund returns exhibit significant

negative skewness and excess kurtosis, which cannot be

captured in the normal VaR whereas the CF-VaR

results in a systematic downward shift of the

conventional VaR. The GARCH-type VaR,

however, includes the time-varying conditional

volatility and is able to trace the actual return process

more effectively. Since the forecast performance cannot

detect which of the three VaR types can match the

time-varying risk adequately, an adjusted hit ratio

takes the size of the hits as well as the average VaR

into account. According to this, the GARCH-type

VaR outperforms the other VaRs for most of the hedge

fund style indices.

Journal of Derivatives & Hedge Funds (2007) 13,

2–25. doi:10.1057/palgrave.dutr.1850048

Keywords: hedge funds; Value at Risk;

GARCH models; forecasting

INTRODUCTION

Since the breakdown of Long Term Capital

Management (LTCM), risk management and the

transparency of hedge funds have become

dramatically important and evolved into

outstanding fields for practitioners and academic

researchers. Value at Risk (VaR) is one of the

most important concepts widely used for risk

management by banks and financial institutions.

Since VaR can be easily computed by capturing

risk in only one figure, it has gained increasing

popularity in the past. Although there are several

forms of financial risk, we focus on market risk

in this paper, that is, the unexpected changes in

stock returns.

The literature on VaR has become quite

expansive (eg Hendricks

, Beder

, Marshall and

Siegel

). However, the conventional VaR

assumes that returns follow a normal or

conditional normal distribution. Particularly, in

the case of skewed and fat-tailed returns, the

assumption of normality leads to substantial bias

in the VaR estimation and results in an

underestimation of volatility.

In contrast to mutual funds, different trading

instruments, such as arbitrage, leverage and short

selling, characterise hedge funds. These trading

instruments are highly dynamic and often

exhibit low systematic risk (Fung and Hsieh

Since hedge funds use options or option-like

trading strategies or strategies that lose money

during down-market phases, they may generate

non-normal payoffs. In addition, Liang

emphasises the higher Sharpe ratios and lower

market risk as well as the higher abnormal

returns of hedge funds investments. Moreover, it

has been well documented that monthly return

distributions of most hedge fund indices show

extremely high negative skewness, positive

excess kurtosis and, significantly, positive first-

order serial correlation. In the context of the

frequently used mean–variance approach, these

return properties inevitably result in an

underestimation of the true volatility.

Favre and Galeano

suggest a modified

method of VaR by implementing a Cornish–

Fisher (CF) expansion, which is used to control

for skewness and kurtosis.

Agarwal and Naik

introduce a mean-conditional VaR (CVaR)

Value at risk, GARCH modelling and the forecasting of hedge fund return volatility 3

framework that explicitly accounts for negative

tail risk.

As the conventional VaR refers only to

the frequency of extreme events, the CVaR

focuses on both frequency and size of losses in

the case of extreme events. Kellezi and Gilli

introduce a risk capital measure based on the

Extreme Value Theory (EVT). The EVT focuses

only on extreme values, that is, the tail of the

distribution, rather than the whole distribution.

However, Danielsson and de Vries

show that

the EVT can be accurately used only for very

extreme events and often does not provide good

results at more conventional 5 per cent VaR

levels. Furthermore, EVT assumes an identically

and independently distributed (iid) framework

that is not consistent with most financial data.

In this paper, we use a GARCH-type VaR by

modelling and forecasting conditional volatility,

using GARCH and EGARCH, and then

implementing the time-varying volatility in the

VaR. In doing so, we also control for skewness

and kurtosis. Volatility forecasting is important

not only in risk management and market timing

for single hedge funds, but also in the context of

portfolio diversification including hedge funds.

The knowledge of future volatilities allows

portfolio managers to control the risk

temporally, for example, sell an asset or portfolio

before a dramatic increase in volatility takes place

(Engle and Patton

). Furthermore, by means of

information on the volatility process in general,

and the development of volatility in particular,

the risk pricing of the market can be

determined.

To our knowledge, there are no empirical

studies that introduce GARCH-type forecasts

into the conventional VaR framework to

simultaneously account for time-varying

volatility, serial correlation, skewness and

kurtosis in hedge fund returns.

This paper is organised as follows: Firstly, the

next section describes different hedge funds

strategies from the data provider, Standard

&Poor’s, according to the different management

styles. The concepts of conventional VaR and

CF expansion are then briefly introduced.

Following this, the stylised facts of volatility

and the two conditional volatility models,

GARCH(p,q) and EGARCH(p,q), that should

capture these features are discussed.

Subsequently the conditional variances for the

hedge funds styles under consideration are

estimated using alternative model specifications,

and their volatility characteristics are analysed.

The GARCH-type models are then applied to

estimate the daily VaR of the different hedge

fund styles. The accuracy of one-step-ahead VaR

forecasts is evaluated by different ratios that

measure the distance between the observed and

forecasted VaR values. Some concluding

remarks are offered in the final section.

HEDGE FUNDS STYLES AND THEIR

STRATEGIES

Investment strategies used by hedge funds tend to

be quite different from those of traditional mutual

funds. Basically, every hedge fund embarks on its

own preferred strategy, which leads to a very

heterogeneous asset class ‘hedge funds’. However,

hedge funds may be classified into a number of

different strategy groups depending on the main

type of strategy followed (Kat and Lu

). In

referring to market exposure as classification

criteria, Purcell and Crowley

distinguish

between three different styles of hedge funds.

These broader categories encompass the relative

value, the event-driven and the opportunistic

strategy. Based on market exposure to traditional

asset classes, Agarwal and Naik

also classify

4 Fu

ss, Kaiser and Adams

hedge fund strategies into categories: directional

and non-directional. Ineichen

conducts a

subdivision on the hedge funds styles: arbitrage,

event-driven and directional (see Figure 1). In

this study, we adapt the classification proposed by

the latter to match the system of the hedge funds

index provider S&P, as we use their indices later

in the empirical section of this paper.

Arbitrage management style

The ‘arbitrage’ strategies, also known as ‘relative

value’ or ‘non-directional’ strategies, try to take

advantage of temporarily wrong valuations

between different financial instruments and

attempt to offer their investors very low market

exposure. Independent of the market situation,

the strategy aims at achieving absolute returns by

avoiding or reducing market (beta) risk, industry

risk, interest rate risk (duration), etc. Thus, this

so-called ‘equity market neutral’ strategy takes

advantage of short-term pricing inefficiencies

between stocks or groups of stocks, which usually

behave identically. In applying this mean-reverting

strategy, the directional market risk is eliminated

or neutralised as far as possible so that the beta of

the overall situation is ideally zero. Capocci’s

analysis comes to the conclusion that most of the

market neutral funds are not significantly exposed

to the equity market, but tend to be more

exposed during bear market than during bull

market without being negatively impacted.

The strategy ‘convertible arbitrage’

exploits the differences in the value between

a convertible bond and the underlying stock.

It does this by buying (selling) the underrated

(overrated) convertible bond and, at the

same time, also selling (buying) the stock.

Agarwal et al.

show that the key risk factors in

convertible arbitrage strategies are: equity (and

volatility) risk, credit risk and interest rate risk.

They also demonstrate that the risk-adjusted

returns of convertible arbitrage hedge funds

are affected by mismatches between supply of

and demand for convertible bonds. They also

show that convertible arbitrageurs escape some

of the losses experienced by long-only

convertible bond mutual funds by changing

their risk exposures in response to the Long-

Term Capital Management (LTCM) crisis.

Hedge Fund Investment Techniques

Arbitrage

Equity Market Neutral

Fixed Income Arbitrage

Convertible Arbitrage

Event-Driven

Merger Arbitrage

Distressed

Special Situations

Directional/Tactical

Equity Long/Short

Managed Futures

Global Macro

Systematic Market

Exposure

high

low

anagement

Styles

Strategies

Figure 1: Hedge fund styles and strategies

Source: Adapted from Ineichen

Value at risk, GARCH modelling and the forecasting of hedge fund return volatility 5

‘Fixed income arbitrage’ funds search for false

valuations or anomalies of bonds in order to

achieve arbitrage profits in securities of different

maturities, credit ratings and volatilities (with

high leverage factors). The overall situation

should have a duration of zero, that is, the

interest rate risk should be neutralised

(immunisation). The advantage of this strategy

is the attainment of liquidity and credit risk

premiums. Leverage is also applied to increase

absolute returns.

Fung and Hsieh,

as well as

Jaeger and Wagner

show that the fixed income

arbitrage strategy bears a risk profile similar to a

short option, with a risk of significant losses but

otherwise steady returns. They also

demonstrated that the heaviest losses of fixed

income arbitrage occur in ‘flight to quality’

scenarios, when credit spreads suddenly widen,

liquidity evaporates and emerging markets fall

sharply. In their analysis of the risk and return

characteristics of fixed income arbitrage

strategies, Duarte et al.

find that these tend to

produce significant alphas after controlling for

traditional bond and equity market risk factors.

Furthermore, even after taking into account the

typical hedge fund fees, these alphas remain

significant, and some fixed income arbitrage

strategies actually produce positively skewed

returns.

Event-driven management style

The goal of the ‘event-driven’ strategy is to take

advantage of price anomalies triggered by

pending or upcoming firm transactions such as

mergers, restructurings, liquidations and

insolvencies. The success of this strategy comes

from a false judgement of the situation and the

uncertainty of other investors in the case of take-

overs, reorganisations or management buyouts.

‘Merger arbitrage’ invests simultaneously in long

and short positions by purchasing the stocks of

the company being taken over and at the same

time, selling those of the take-over company.

Generally, stocks that are the object of a take-

over gain in value while stocks of the take-over

company fall in value. According to Mitchell and

Pulvino,

merger arbitrage strategies display

rather high correlations to the equity markets

when the latter declines and, in turn, display low

correlations when stocks trade up or sideways. In

other words, the payout profile of merger

arbitrage strategies corresponds directly to a

short position in a put option on announced

merger deals.

The sub-strategy ‘distressed securities’ invests

in assets of companies that have financial or

operational problems, are bankrupt or are

expecting such an economic situation. Such a

situation means reorganisation, insolvencies,

liquidations and/or other restructurings of

companies. A typical strategy is to buy the stocks

with a realised backwardation that results from

the tight situation. The stocks are then kept until

the process of restructuring is completed and the

value of the company has appreciated.

Depending on the style of the fund manager,

investments are made in bank or corporate loans,

pecuniary claims, equity shares, preference shares

and warrants. The risk factors of distressed

securities strategies come with a simple set of

exposures to credit, equity (particularly small cap

equity) and liquidity risks. Jaeger and Wagner

show that with an alpha between 3 and 4 per

cent per annum, distressed securities funds and

its peers in the event-driven discipline offer the

highest level of alpha in the hedge fund industry.

The success of the ‘special situations’ strategy

results from the ability to correctly judge and

take advantage of a broad range of corporate

6 Fu

ss, Kaiser and Adams

events such as spin-offs, index reshufflings,

insolvency reconstruction and share buy-backs.

This strategy also includes closed-end fund

arbitrage, which involves the purchase and

hedging of closed-end funds that may be trading

at significantly different levels of their net asset

values (NAVs).

Directional/tactical management style

The directional management styles, which are

characterised by significant market exposure

(long and short), contain the strategy indices

equity hedge, commodity trading advisors

(CTAs)/managed futures and global macro.

‘Equity hedge’ or ‘long/short’ equity funds are

basically comprised of long positions in stocks

that are hedged at any time through tactical short

selling and/or stock index options. Most equity

hedge funds have a long bias (Kat and Lu

Besides stocks, equity hedge funds can also be

invested in other assets on a limited scale.

According to Fung and Hsieh

as well as Jaeger

and Wagner,

most of the equity hedge

managers have exposure to both the broad

equity market and, in particular, to small cap

stocks. They argue that it may be easier for

equity hedge managers to find opportunities in a

rising market, and to go short in large cap stocks

and buy small ones instead.

‘Managed futures’ funds take long and short

positions in liquid and listed futures contracts,

for example, foreign exchange, interest rates,

stock market indices and commodities. Most of

the CTAs pursue a trend following strategy in

which a trend is replicated to where one can

earn from increasing as well as decreasing prices.

The results of Schneeweis and Spurgin

indicate

that the technical trading rule and market

momentum variables explain a significant part of

the sources of managed futures funds returns.

According to Jaeger and Wagner,

‘managed

futures’ hedge fund strategy is the only one that

displays a negative alpha. Fung and Hsieh

show

that the returns of trend following funds during

extreme market shifts can be explained by a

combination of primitive trend following

strategies on currencies, commodities, three-

month interest rates and US bonds, but not by

strategies on stock indices. Kat

shows that

when investors account for ‘managed futures’ in

conventional portfolios, this allows them to

achieve a very substantial degree of overall risk

reduction at limited costs. The author concludes

that adding managed futures to a portfolio of

stocks and bonds reduces a portfolio’s standard

deviation more efficiently than other hedge

funds strategies do, without the undesirable side-

effects of skewness and kurtosis. In relevant

literature, for example, Schneeweis and

Spurgin,

Schneeweis,

Ineichen,

Kat,

Lamm,

considered managed futures or CTAs

are as a separate alternative investment class

rather than as a hedge fund strategy. However,

as some index providers, for example, Barclay,

CS/Tremont, Eurekahedge or S&P, also publish

a managed futures strategy index, this strategy is

also considered in our analysis as well.

The strategy of ‘global macro’ is based on

macroeconomic top-down analysis, where fund

managers try to profit from major economic

trends and events in the global economy.

Typically, large shifts in interest rates, currency,

bonds and equity prices are quickly utilised by

making extensive use of leverage and derivatives

(Kat and Lu

). Jaeger and Wagner

show that

global macro managers do better in strong bond

markets, have exposure to the risk characteristic

of managed futures and also have some non-

linear exposure to the broad equity market.

Value at risk, GARCH modelling and the forecasting of hedge fund return volatility 7

CONVENTIONAL VAR AND THE CF

EXPANSION

The VaR is a downside risk measurement used

widely by financial institutions for internal and

external purposes, with the attractive

characteristics of expressing the risk in only one

figure. It is the estimated loss of an asset that

within a given period, normally 1 or 10 days,

will only be exceeded by a certain small

probability y (mainly 1 or 5 per cent). Thus,

with a probability of 95 per cent, the one day 5

per cent VaR shows the negative return that will

not be exceeded within this day:

prob½return

o VaR

¼y ð1Þ

where O

denotes the information set available at

time t.

Statistically speaking, the 5 per cent quantile

of the probability density function of the asset

returns is considered. Assuming that the returns

are normally distributed under a foregoing

observation period of one year, the VaR can be

calculated as the deviation of the value from the

distribution function Z times the standard

deviation s from its mean m:

VaR ¼ðZ  s mÞð2Þ

In general, a higher probability value and a

longer period increase the VaR. Hence, in this

paper, a period of one day and a probability of 95

per cent are assumed.

The VaR is based on the assumption of

normally distributed returns. However, financial

asset and particularly hedge fund returns often

feature excess kurtosis and negative skewness,

which make the probability of extreme returns

more likely than under a normal distribution.

Thus, applying the VaR under the assumption of

normality can lead to a systematic underestimation

of the actual risk. Favre and Galeano

apply a CF

expansion by adjusting the critical value according

to the distribution function Z:

¼ Z þ

ðZ

 1ÞS þ

ðZ

 3ZÞ

K 

ð2Z

 5ZÞS

ð3Þ

with S and K astheskewnessandtheexcesskurtosis

of the empirical distribution, respectively. If the

return distribution is normal, that is, S and K are

equal to zero, then Z

¼Z according to equation

(2). In contrast, for non-normally distributed hedge

fund returns, it can be expected that the VaR

thresholdobtainedfromCFexpansionwillbemore

accurate in comparison with the estimated threshold

from conventional VaR.

MODELS OF CONDITIONAL PRICE

VOLATILITY

Generally speaking, a volatility model has to be

able to forecast the variance of a time series of

returns preferably well. Thus, to forecast the

quantiles for VaR applications, the predictability

of volatility is required. Engle and Patton

mention common stylised facts of asset price

volatility process that should also hold for hedge

fund prices. Many of these stylised facts can be

viewed as properties of the volatility forecasts,

and can be taken as a starting point when

examining its consistency with the time series

data.

If current returns or current volatility shocks

have an influence on the expected variance

several periods in the future, then the volatility

process of returns is called persistent. This

property is based on the observation of volatility

clustering in the price process whereupon

clusters of high and low price changes occur

over time. Mandelbrot

and Fama

empirically

conclude that large stock price changes are often

8 Fu

ss, Kaiser and Adams

followed by large price variation, and there are

periods in which one can observe consecutive

small changes.

Volatility is referred to as mean reverting (or

stationary) if, after a period of high or low

volatility, a reversion to a normal level of risk

eventually occurs over time. Mean reversion of

volatility is thus regarded as the normal level of

volatility that is always reached after short-run

deviations. In this way, the conditional variance

fluctuates around the unconditional volatility

depending on positive and negative differences.

Many volatility models assume that the

variance of asset prices is influenced

asymmetrically by positive and negative

innovations. Hence, it is very unlikely that

positive and negative shocks have the same effect

on the conditional volatility of assets or hedge

fund returns. The influence of the sign of the

price innovation on volatility, and the negative

correlation between asset returns and changes in

volatility, is called leverage effect or risk

premium effect. The leverage effect describes

the circumstance when decreasing stock prices

increase the debt to equity ratio which, in turn,

results in a higher volatility of returns for

shareholders. The risk premium effect suggests

that increased volatility caused by news reduces

the demand for the stock because of risk-averse

market participants (Engle and Patton

). Besides

purely endogenous effects, which are normally

considered in univariate volatility models by

only using information from the history of a

time series, other exogenous factors (like interest

rates, exchange rates, etc.) and/or deterministic

events (like announcements by companies,

economic news, time-of-day effects, etc.) can be

relevant for the volatility of a time series

(Bollerslev and Melvin,

Engle et al.,

Engle

and Mezrich

GARCH(p,q) model

The conditional mean m

and the conditional

variance h

of a time series of returns R

are

defined as the mean and the variance of a

random variable with their expectation

influenced by the knowledge of another random

variable:

¼ E½r

t1

and

¼ E ðr

 m

t1



ð4Þ

with m

being the mean of r

given the

information from the past period t1, O

t1

This means that R

is generated by the following

process:

¼ m

ﬃﬃﬃﬃ

;

with:

E½e

t1

¼0 ð5Þ

and

Var½e

t1

¼1

A GARCH(p,q) model assumes that the returns

in the mean equation is composed of the

conditional mean m and the price innovation

¼ m þ e

;

with

t1

 N ð0; h

Þð6Þ

However, to account for the serial dependence

structure in the hedge fund returns, which are

often caused by infrequent trading, the mean

equation is modelled by an ARMA process:

¼ m

k¼1

tk

l¼1

tl

þ e

ð7Þ

The conditional variance h

in a GARCH(p,q)

model is defined as a function of the past squared

error terms e

tj

and the conditional variance of

Value at risk, GARCH modelling and the forecasting of hedge fund return volatility 9

past periods h

ti

¼ o þ

j¼1

tj

i¼1

ti

;

with : e

¼ R

 m

ð8Þ

The parameters in equation (8) have to meet the

following restrictions: o>0, a

, and b

>0 for

i ¼1, y, p and j ¼1, y, q. The introduction of

this non-negativity condition is necessary in

order to exclude a negative variance. In addition,

to provide stationarity the following condition

must hold:

j¼1

i¼1

o1: ð9Þ

If the sum of a

and b

have values close to one,

the volatility is highly persistent, that is, volatility

has a very long ‘memory’. The condition of

stationarity also includes the mean reverting

property if the sum of the ARCH and GARCH

term is significantly less than one. Thus, it

appears that although the return volatility has

quite a long memory, the volatility process

nevertheless returns to its mean. One can test for

mean reversion by comparing the average

(stationary) unconditional mean of the

GARCH(p,q) process with the sample estimate

of the unconditional variance. The uncondi-

tional mean of the GARCH(p,q)

process is calculated as the ratio between o and

(1a

b

Previous tests have shown that a lag structure

of p ¼1 and q ¼1 is already sufficient to attain

adequate estimation results. Thus, the choice

of parameters in the empirical analyses in the

next section are restricted to a GARCH(1,1)

model.

For estimating the parameter vector

Y ¼(o, a

, b

), the maximum likelihood

technique is applied under the assumption that

the residuals e

follow a conditional Gaussian

(normal) distribution.

Asymmetric conditional volatility

models

As already mentioned, not only the magnitude

but also the sign of an innovation can influence

volatility. Hence, a relationship between the

volatility of returns and the returns themselves is

assumed, to have a negative sign, that is,

decreasing asset returns lead to an increasing

volatility and vice versa (Engle and Ng

In relevant literature, several approaches are

used to model the leverage effect. There are the

QGARCH(p,q), Quadratic GARCH and the

NGARCH(p,q) model and the Non-linear

Asymmetric GARCH, among other GARCH

modifications (Sentana,

Engle and Ng

). The

EGARCH(p,q) model, developed by Nelson,

assumes normally distributed residuals and can

be expressed as:

logðh

Þ¼o þ

i¼1

logðh

ti

j¼1

tj

ﬃﬃﬃﬃﬃﬃﬃ

tj



ﬃﬃﬃ



þ g

tj

ﬃﬃﬃﬃﬃﬃﬃ

tj

ð10Þ

Unlike the linear GARCH(p,q) model in

equation (8), the conditional variance in the

EGARCH model is formulated in logarithms.

In doing so, all restrictions become irrelevant,

particularly the non-negativity condition for the

parameters of the conditional variance.

On the

one hand, the unconditional standard deviation

is considered by the parameter a

, that includes

the effects of price innovations j periods in the

past, and on the other by the term g

that

considers the effect of the sign. In a model

comparison where a number of different

ARCH, GARCH, EGARCH and other more

complex semi-parametric and non-parametric

models were compared, Pagan and Schwert

10 Fu

ss, Kaiser and Adams

could provide empirical evidence that a simple

EGARCH(1,1) model is already a sufficient

parameterisation to adequately model the

dynamics of price innovations. As a result of

the modelling of volatility clustering, positive

parameter values should result for a

and, in

order to take the leverage effect into account,

negative values result for g

GARCH-type VaR

Although the CF expansion includes the

deviations from a normal distribution, it ignores

the volatility clustering of the returns. However,

as ARCH effects establish the fact that one bad

day with highly negative returns makes a

consecutive bad day more likely than without

ARCH effects, the unconditional standard

deviation s is replaced by the estimated

conditional standard deviation of the GARCH

process:

VaR

¼ðZ 

ﬃﬃﬃﬃ

 m

Þð11Þ

EMPIRICAL ANALYSIS OF STOCHASTIC

VOLATILITY

Data

For the empirical analysis of the stochastic

volatilities, daily data from the S&P hedge fund

index series (SPHG) are utilised, as this is the

hedge fund database with the longest daily track

record, starting in September 2002. However,

the use of daily indices has several drawbacks.

Firstly, all available daily hedge fund indices are

based on managed account platforms. In their

empirical analysis on hedge funds based on

managed accounts, Haberfelner et al.

show that

the managed account composite with 0.37

exhibits a significant lower Sharpe ratio than the

Eurekahedge composite (1.08). Moreover,

investors obtain a premium of 8.19 per cent per

annum for not going the managed account way.

Thus, one can conclude that the SPHG

benchmark for the hedge fund universe suffers

from selection bias as managers of account

platform have strict requirements in terms of

transparency, liquidity and investability.

Secondly, from the investor’s perspective,

‘traditional’ hedge fund indices also suffer from

survivorship, selection and self-selection biases.

However, it is important to mention that these

are non-investable hedge fund indices. Fung and

Hsieh

argue that fund of hedge fund data are

less prone to these data biases than non-

investable hedge fund indices. Drawing on

conclusions from Fung and Hsieh

48,

we are

of the opinion that the investable hedge fund

indices used in this study are good proxies for

the advancement of diversified hedge fund

portfolios.

Descriptive statistics

The observation period ranges from 30th

September, 2002 until 31st May 2006. During

this period, 926 return observations resulted,

whereby the last month serves as an out-of-

sample forecast period. By using daily data, a

sufficiently large sample is available that makes

the existence of ARCH effects, and the

occurrence of volatility clustering, in particular,

a requirement for estimating GARCH models.

Denoting P

as the price or the NAV of a

particular hedge fund style at the time t, the

continuous return for the period t1 until

t is calculated as follows:

¼ ln

t1



 100 ð12Þ

Value at risk, GARCH modelling and the forecasting of hedge fund return volatility 11