Value at risk, GARCH modelling and the forecasting of hedge fund return volatility
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DOI: 10.1057/palgrave.jdhf.1850048
- Cite this article as:
- Füss, R., Kaiser, D. & Adams, Z. J Deriv Hedge Funds (2007) 13: 2. doi:10.1057/palgrave.jdhf.1850048
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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.
Keywords
hedge fundsValue at RiskGARCH modelsforecastingINTRODUCTION
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 Hendricks1, Beder2, Marshall and Siegel3). 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 Hsieh4). 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, Liang5 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 Galeano6 suggest a modified method of VaR by implementing a Cornish–Fisher (CF) expansion, which is used to control for skewness and kurtosis.7 Agarwal and Naik8 introduce a mean-conditional VaR (CVaR) framework that explicitly accounts for negative tail risk.9 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 Gilli10 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 Vries11 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 Patton12). 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
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's17 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.18 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.
‘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.19 Fung and Hsieh,20 as well as Jaeger and Wagner21 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.22 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,23 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 Wagner21 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 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 Lu13). Besides stocks, equity hedge funds can also be invested in other assets on a limited scale. According to Fung and Hsieh24 as well as Jaeger and Wagner,21most 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 Spurgin25 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,21 ‘managed futures’ hedge fund strategy is the only one that displays a negative alpha. Fung and Hsieh26 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. Kat27 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,28 Schneeweis,29 Ineichen,16 Kat,27 Lamm,30 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 Lu13). Jaeger and Wagner21 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.
CONVENTIONAL VAR AND THE CF EXPANSION
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.
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 Patton12 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. Mandelbrot31 and Fama32 empirically conclude that large stock price changes are often 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 Patton12). 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,33 Engle et al.,34 Engle and Mezrich35).
GARCH(p,q) model
If the sum of α_{j} and β_{i} 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 unconditional mean of the GARCH(p,q) process is calculated as the ratio between ω and (1−α_{j}−β_{i}).
Previous tests have shown that a lag structure of p=1 and q=1 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.39 For estimating the parameter vector Θ=(ω, α_{j}, β_{i}), the maximum likelihood technique is applied under the assumption that the residuals ɛ_{t} follow a conditional Gaussian (normal) distribution.40
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 Ng41).
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.44 On the one hand, the unconditional standard deviation is considered by the parameter α_{j}, that includes the effects of price innovations j periods in the past, and on the other by the term γ_{j}α_{j} 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 Schwert45 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 α_{1} and, in order to take the leverage effect into account, negative values result for γ_{1}.
GARCH-type VaR
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.46 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 Hsieh47 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 Hsieh48, 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.
Descriptive statistics, 09/30/2002–04/28/2006
Asset | Indices | Meanr̄_{ann.} (in %) | Volatility σ¯_{ann.} (in %) | Skewness | Kurtosis | Jarque–Bera test |
---|---|---|---|---|---|---|
Hedge fund strategies | SPHG composite | 6.396 | 2.076 | −0.262 | 3.965 | 45.456^{**} |
SPHG arbitrage | 2.791 | 3.272 | 0.109 | 3.836 | 28.101^{***} | |
SPHG event driven | 9.204 | 2.090 | 0.086 | 5.402 | 218.39^{***} | |
SPHG directional | 7.013 | 4.767 | −0.242 | 4.097 | 54.146^{***} | |
SPHG managed futures | 4.733 | 16.193 | −0.380 | 4.258 | 75.538^{***} | |
Financial assets | S&P 500 | 14.906 | 14.207 | 0.332 | 5.284 | 213.10^{***} |
NASDAQ composite | 16.904 | 20.789 | 0.239 | 4.800 | 130.66^{***} | |
Wilshire growth | 12.799 | 14.062 | 0.269 | 4.299 | 74.456^{***} | |
Wilshire value | 19.092 | 14.946 | 0.236 | 5.785 | 300.59^{***} | |
Wilshire small cap | 23.077 | 16.688 | −0.104 | 3.198 | 3.115^{*} | |
Wilshire large cap | 15.405 | 13.941 | 0.312 | 5.178 | 193.43^{***} | |
J.P. Morgan bond index | 2.030 | 4.810 | −0.276 | 4.550 | 105.384^{***} | |
GS commodity index | 18.084 | 23.661 | 0.081 | 3.403 | 7.087^{**} |
In reference to the second moment, the hedge fund composite and the three style indices exhibit the lowest volatilities, whereas managed futures simultaneously display extraordinary variability and low annualised returns. However, using only volatility as a risk measurement can only be reasonable if it can be assumed that the observed returns are normally distributed. According to the test statistics of the Jarque–Bera normality test, the null hypothesis of normally distributed returns can only be accepted for the index of small capitalised companies. In contrast, the evidence of leptokurtosis or positive excess kurtosis of hedge fund returns supports the existence of ARCH effects in these time series. Most of the other financial assets are also leptokurtically distributed, which is a typical characteristic of daily equity returns. Furthermore, the composite, directional and managed futures indices are negatively skewed.
Autocorrelations
Strategy indices | Lag(1) | Lag(2) | Lag(3) | Lag(4) | Lag(5) | Lag(10) | Lag(15) | Lag(20) |
---|---|---|---|---|---|---|---|---|
SPHG composite | ||||||||
ACF | 0.071^{**} | 0.049 | 0.041 | 0.034 | 0.051 | 0.018 | 0.024 | −0.030 |
PACF | 0.071^{**} | 0.044 | 0.035 | 0.027 | 0.044 | 0.015 | 0.002 | −0.050 |
SPHG arbitrage | ||||||||
ACF | −0.189^{**} | −0.107^{**} | 0.010 | −0.001 | 0.046 | 0.026 | 0.047 | 0.020 |
PACF | −0.189^{**} | −0.148^{**} | −0.044 | −0.025 | 0.039 | 0.026 | 0.065 | 0.020 |
SPHG event driven | ||||||||
ACF | 0.056 | 0.016 | 0.058 | 0.027 | 0.043 | 0.098 | 0.098^{**} | 0.039 |
PACF | 0.056 | 0.013 | 0.057 | 0.021 | 0.039 | 0.042 | 0.089^{**} | 0.022 |
SPHG directional | ||||||||
ACF | 0.084^{**} | −0.021 | −0.011 | 0.000 | −0.006 | −0.013 | −0.043 | −0.028 |
PACF | 0.084^{**} | −0.028 | −0.007 | 0.001 | −0.007 | −0.006 | −0.061^{**} | −0.025 |
SPHG managed fut. | ||||||||
ACF | −0.023 | −0.036 | −0.003 | 0.064 | −0.057 | 0.005 | −0.024 | −0.026 |
PACF | −0.023 | −0.037 | −0.005 | 0.063 | −0.055 | −0.003 | −0.024 | −0.033 |
ARMA-modelling
Indices | SPHG composite | SPHG arbitrage | SPHG event driven | SPHG directional | SPHG managed fut |
---|---|---|---|---|---|
Model | ARMA (1,1) | ARMA (1,2) | — | ARMA (1,2) | — |
μ̂ | 0.028^{***} | 0.012^{**} | 0.037^{***} | 0.030^{***} | 0.019 |
(0.007) | (0.005) | (0.004) | (0.010) | (0.035) | |
φ̂_{1} | 0.947^{***} | −0.222^{***} | — | 0.879^{***} | — |
(0.034) | (0.033) | (0.153) | |||
θ̂_{1} | −0.913^{***} | — | — | −0.794^{***} | — |
(0.044) | (0.156) | ||||
θ̂_{2} | — | −0.155^{***} | — | −0.082^{**} | — |
(0.034) | (0.035) | ||||
AIC | −1.233 | −0.378 | — | 0.438 | — |
SIC | −1.217 | −0.359 | — | 0.460 | — |
Q(5) | 0.598 | 3.269 | 8.480 | 0.270 | 7.828 |
Q^{2}(5) | 4.803 | 46.940^{***} | 14.278^{**} | 4.080 | 1.241 |
LM-test | 1.068 | 9.461^{***} | 3.393^{*} | 2.161 | 0.033 |
Test for ARCH effects — squared returns
Indices | ρ̂_{1} | ρ̂_{2} | ρ̂_{3} | ρ̂_{4} | ρ̂_{5} | ρ̂_{10} | ρ̂_{15} | LB(5) | LB(10) | LB(15) |
---|---|---|---|---|---|---|---|---|---|---|
SPHG composite | 0.007 | 0.039 | 0.007 | −0.010 | 0.030 | 0.038 | 0.042 | 2.371 | 7.692 | 19.255 |
SPHG arbitrage | 0.174 | 0.077 | 0.072 | 0.067 | 0.157 | 0.113 | 0.100 | 64.22^{***} | 93.61^{***} | 108.9^{***} |
SPHG event-driven | 0.045 | 0.080 | 0.009 | 0.050 | 0.007 | 0.019 | 0.045 | 10.09^{*} | 20.80^{**} | 23.62^{**} |
SPHG directional | 0.046 | −0.017 | 0.049 | 0.010 | 0.001 | 0.034 | 0.049 | 4.509 | 16.55^{*} | 31.74^{***} |
SPHG managed fut. | 0.004 | 0.003 | 0.029 | 0.026 | 0.007 | 0.030 | 0.006 | 1.331 | 3.845 | 11.72 |
Parameter estimation
Since volatility clustering is a specific characteristic of speculative prices, the conditional second moment is modelled using a variation of the GARCH process originally suggested by Bollerslev37. By estimating these conditional volatility models for the particular hedge fund style indices, it is possible, in the second step, to check the consistency of certain properties of volatility that are again important for the forecasting of volatility.
ARMA-GARCH (p,q) model for hedge fund indices
Parameter | SPHG composite | SPHG arbitrage | SPHG event-driven | SPHG directional | SPHG_{2} managed fut_{3}. |
---|---|---|---|---|---|
Mean equation | |||||
ĉ | 0.030^{***} | 0.009^{**} | 0.038^{***} | 0.032^{***} | 0.023^{**} |
(0.008) | (0.004) | (0.004) | (0.011) | (0.009) | |
φ̂_{1} | 0.951^{***} | −0.180^{***} | — | 0.901^{***} | — |
(0.026) | (0.035) | (0.076) | |||
θ̂_{1} | −0.911^{***} | — | — | −0.835^{***} | — |
(0.035) | (0.086) | ||||
θ̂_{2} | — | −0.168^{***} | — | −0.060^{*} | — |
(0.037) | (0.035) | ||||
Variance equation | |||||
ω̂ | 0.001 | 0.001 | 0.0001 | 0.001^{***} | 0.001^{*} |
(0.001) | (0.001) | (0.000) | (0.001) | (0.001) | |
α̂ | 0.040^{**} | 0.057^{***} | 0.043^{**} | 0.019^{*} | 0.023^{**} |
(0.016) | (0.018) | (0.019) | (0.010) | (0.011) | |
β̂ | 0.920^{***} | 0.918^{***} | 0.929^{***} | 0.968^{***} | 0.963^{***} |
(0.039) | (0.029) | (0.035) | (0.012) | (0.016) | |
AIC | −1.247 | −0.433 | −1.237 | 0.409 | 0.418 |
SIC | −1.215 | −0.401 | −1.215 | 0.446 | 0.439 |
LogL | 568.94 | 201.41 | 562.88 | −177.50 | −184.89 |
Q(5) | 0.518 | 4.174 | 11.276^{**} | 0.427 | 4.844 |
Q^{2}(5) | 1.486 | 1.146 | 2.974 | 2.182 | 2.507 |
ARCH-LM test | 0.000 | 0.001 | 0.023 | 0.273 | 0.210 |
J.B. | 39.691^{***} | 12.808^{***} | 188.41^{***} | 23.065^{***} | 27.333^{***} |
α̂+β̂ | 0.960 | 0.975 | 0.973 | 0.987 | 0.985 |
HLP | 17.151 | 27.618 | 25.163 | 51.368 | 47.370 |
σ¯_{ann.} (in %) | 2.08 | 3.13 | 2.12 | 4.92 | 4.91 |
However, since the sum of the GARCH terms are still smaller than one, all strategies show a mean-reverting behaviour. The property of mean reversion exists if the unconditional variance from the GARCH(1,1) process is close to the estimated unconditional sample variance. The unconditional variance from the GARCH models is computed as the ratio of the constant ω̂ and the expression (1−α̂−β̂). For instance, for the arbitrage style an unconditional variance of 0.0391 results which corresponds to an annualised volatility of σ¯_{ann.}=3.128 per cent that is thus approximately the same as the estimated sample volatility (3.272) from Table 1. Taken together, all unconditional volatilities acquired from the GARCH processes are close to their sample estimators except for the managed futures index. The Ljung–Box test for testing on dependencies of the residuals Q(5) and squared residuals Q^{2}(5) as well as the ARCH–LM(1) test, confirm that no linear or non-linear dependencies exist, and that the models take the heteroscedasticity and the changing unconditional and conditional variance in the return-time series into account. Finally, the Jarque–Bera statistic suggests that skewness and kurtosis in the standardised residuals are reduced but not completely eliminated.
ARMA-EGARCH(p,q) model for hedge fund indices
Parameter | SPHG composite | SPHG arbitrage | SPHG event-driven | SPHG directional | SPHG managed fut_{2}. |
---|---|---|---|---|---|
Mean equation | |||||
ĉ | 0.031^{**} | 0.008^{*} | 0.038^{***} | 0.034^{**} | 0.032^{***} |
(0.013) | (0.004) | (0.004) | (0.013) | (0.010) | |
φ̂_{1} | 0.976^{***} | −0.187^{***} | — | 0.947^{***} | — |
(0.015) | (0.035) | (0.044) | |||
θ̂_{1} | −0.934^{***} | — | — | −0.873^{***} | — |
(0.021) | (0.056) | ||||
θ̂_{2} | — | −0.156^{***} | — | −0.062^{*} | — |
(0.037) | (0.035) | ||||
Variance equation | |||||
ω̂ | −0.797^{***} | −0.204^{***} | −0.631^{**} | −1.366^{***} | −1.647^{***} |
(0.271) | (0.077) | (0.294) | (0.415) | (0.538) | |
α̂ | 0.142^{***} | 0.134^{***} | 0.171^{***} | 0.140^{*} | 0.131^{*} |
(0.052) | (0.040) | (0.060) | (0.074) | (0.078) | |
γ̂ | −0.110^{***} | −0.009 | −0.058 | −0.172^{***} | −0.175^{***} |
(0.038) | (0.022) | (0.040) | (0.055) | (0.056) | |
β̂ | 0.833^{***} | 0.970^{***} | 0.876^{***} | 0.483^{***} | 0.362^{*} |
(0.061) | (0.017) | (0.065) | (0.165) | (0.217) | |
AIC | −1.255 | −0.434 | −1.241 | 0.423 | 0.426 |
SIC | −1.218 | −0.397 | −1.214 | 0.465 | 0.452 |
LogL | 573.71 | 203.15 | 565.84 | −182.80 | −187.46 |
Q(5) | 0.502 | 4.314 | 8.832 | 1.471 | 5.313 |
Q^{2}(5) | 1.112 | 0.833 | 3.363 | 2.761 | 3.393 |
ARCH-LM test | 0.082 | 0.009 | 0.158 | 0.001 | 0.008 |
J.B. | 20.441^{***} | 11.439^{***} | 179.70^{***} | 24.586^{***} | 24.775^{***} |
Conditional variance forecasts
Conditional volatility forecasts
T+1 (%) | T+2 (%) | T+3 (%) | T+4 (%) | T+5 (%) | T+6 (%) | TIC | |
---|---|---|---|---|---|---|---|
SPHG composite | 2.399 | 2.350 | 2.309 | 2.277 | 2.249 | 2.227 | 0.8315 |
SPHG arbitrage | 3.025 | 3.027 | 3.030 | 3.032 | 3.035 | 3.037 | 0.9274 |
SPHG event-driven | 2.181 | 2.179 | 2.178 | 2.176 | 2.175 | 2.174 | 0.6924 |
SPHG directional | 4.800 | 4.841 | 4.819 | 4.808 | 4.803 | 4.800 | 0.8979 |
SPHG managed futures | 16.46 | 16.43 | 16.41 | 16.38 | 16.36 | 16.34 | 0.9844 |
Furthermore, the values of the TIC suggest that the forecasts for the event-driven and the composite indices can be considered clearly superior to the random walk, whereas the forecasts for arbitrage, managed futures and directional indices are quite close to what a random walk would predict.
Comparison of VaR forecasts
The forecasted conditional variance of the GARCH(1,1) and EGARCH(1,1) models can be implemented in the VaR equation in order to obtain forecasts for the GARCH-type VaR. This requires that the standard deviation σ_{t} from equation (2) be replaced by the square root of the forecasted conditional GARCH-Variance h_{t} from equations (7) and (9):
One-month value at risk forecasts
Indices | TIC | HMAE | HRMSE | ||||||
---|---|---|---|---|---|---|---|---|---|
VaR | CF-VaR | GARCH- type VaR | VaR | CF-VaR | GARCH- type VaR | VaR | CF-VaR | GARCH- type VaR | |
SPHG composite | 0.0152 | 0.0382 | 0.1532 | 0.0797 | 0.0657 | 0.2859 | 0.0085 | 0.0065 | 0.1342 |
SPHG arbitrage | 0.0206 | 0.0196 | 0.0478 | 0.0403 | 0.0380 | 0.0728 | 0.0424 | 0.0404 | 0.0975 |
SPHG event-driven | 0.0236 | 0.0236 | 0.1093 | 0.0360 | 0.0387 | 0.1236 | 0.0496 | 0.0498 | 0.2350 |
SPHG directional | 0.0429 | 0.0433 | 0.1228 | 0.0699 | 0.0719 | 0.2722 | 0.0899 | 0.0910 | 0.2809 |
SPHG managed futures | 0.0262 | 0.0296 | 0.0485 | 0.0393 | 0.0459 | 0.0782 | 0.0539 | 0.0611 | 0.0101 |
Adjusted Hit ratios of the alternative value at risk approaches
× 10^{−3} | VaR | CF-VaR | GARCH-type VaR |
---|---|---|---|
SPHG Composite | 0.1447 | 0.1337 | 0.1134 |
SPHG Arbitrage | 0.1269 | 0.0958 | 0.1491 |
SPHG Event-driven | 0.1216 | 0.1142 | 0.0945 |
SPHG Directional | 2.5586 | 2.2395 | 2.1815 |
SPHG Managed futures | 85.992 | 58.960 | 86.824 |
CONCLUSIONS
In this paper, alternative volatility models such as GARCH(p,q) and EGARCH(p,q) for particular hedge fund strategies were estimated and compared with each other. The examination of the conditional volatility of hedge fund strategy returns shows important differences concerning persistence, mean reversion and asymmetric effects among the strategies considered. The knowledge of the conditional volatility of hedge fund returns can be used to enhance the VaR estimation and forecast. While the normal VaR is generally based on the assumption of normally distributed returns, the CF expansion tends to result in a systematic downward shift in the VaR estimates. Furthermore, both VaR measurements are very inertial, meaning that they do not react immediately to small or large price changes. In contrast, the GARCH-type VaR is able to trace the return process. However, skewness and kurtosis are not completely eliminated by the GARCH modelling and, thus, some bias remains in the VaR estimations and forecasts. Performance ratios indicate an inferior forecast ability compared to the normal and CF-VaR. This is due to the fact that the GARCH-type VaR is more sensitive to changes in the return process. However, by introducing an adjusted hit ratio, one can demonstrate that the GARCH-type VaR adjusts much better to the time-varying risk. Hence, by allowing such a temporal risk control, GARCH-type VaR offers an enhanced protection against downside risk in a portfolio including hedge funds.