The VAR Implementation Handbook (PROFESSIONAL FINANCE & INVESTM) - Hardcover

Gregoriou, Greg N.

Hardcover

ISBN 10: 007161513X ISBN 13: 9780071615136

Verlag: McGraw Hill, 2009

Inhaltsangabe

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For investors, risk is about the odds of losing money, and Value at Risk (VaR) is grounded in that common-sense fact. VAR modeling answers, “What is my worst-case scenario?” and “How much could I lose in a really bad month?”

However, there has not been an effective guidebook available to help investors and financial managers make their own VaR calculations--until now.

The VaR Implementation Handbook is a hands-on road map for professionals who have a solid background in VaR but need the critical strategies, models, and insights to apply their knowledge in the real world.

Heralded as “the new science of risk management,” VaR has emerged as the dominant methodology used by financial institutions and corporate treasuries worldwide for estimating precisely how much money is at risk each day in the financial markets. The VaR Implementation Handbook picks up where other books on the subject leave off and demonstrates how, with proper implementation, VaR can be a valuable tool for assessing risk in a variety of areas-from equity to structured and operational products.

This complete guide thoroughly covers the three major areas of VaR implementation--measuring, modeling risk, and managing--in three convenient sections. Savvy professionals will keep this handbook at their fingertips for its:

• Reliable advice from 40 recognized experts working in universities and financial institutions around the world
• Effective methods and measures to ensure that implemented VaR models maintain optimal performance
• Up-to-date coverage on newly exposed areas of volatility, including derivatives

Real-world prosperity requires making informed financial decisions. The VaR Implementation Handbook is a step-by-step playbook to getting the most out of VaR modeling so you can successfully manage financial risk.

Über diesen Titel

Über die Autorin bzw. den Autor

Greg N. Gregoriou is professor of finance in the School of Business and Economics at State University of New York (Plattsburgh). He has published twenty-five books and is coeditor for the peer-reviewed Journal of Derivatives and Hedge Funds and editorial board member for the Journal of Wealth Management, Journal of Risk Management in Financial Institutions, and Brazilian Business Review.

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THE VaR IMPLEMENTATION HANDBOOK

The McGraw-Hill Companies, Inc.

Copyright © 2009 The McGraw-Hill Companies
All right reserved.
ISBN: 978-0-07-161513-6

Contents

Chapter One

Calculating VaR for Hedge Funds

M. Billio, M. Getmansky, and L. Pelizzon

ABSTRACT

It is well known that hedge funds implement dynamic strategies; therefore, the exposure of hedge funds to various risk factors is nonlinear. In this chapter, we propose to analyze hedge fund tail event behavior conditional on nonlinearity in factor loadings. In particular, we calculate VaR for different hedge fund strategies conditional on different states of the market risk factor. Specifically, we are concentrating on dynamic risk factors that are switching from a market regime or state that we call normal to two other regimes that could be identified as "crisis" and "bubble" and that are usually characterized, respectively, by (1) largely low returns and high volatility and (2) high returns. We are proposing a factor model that allows for regime switching in expected returns and volatilities and compare the VaR determined with this methodology with the other VaR approaches like GARCH(1,1), IGARCH(1,1), and Cornish Fisher.

INTRODUCTION

In recent years, the flow of funds into alternative investments for pension funds, endowments, and foundations has experienced a dramatic increase. Unfortunately, the very fact that hedge funds and commodity trading advisors (managed futures funds) have only lately come into prominence during the last decade, has meant that they generally have only recently been considered as substitutes or as additions to other more "traditional" private-equity-based alternative investment vehicles.

Hedge funds are considered by some to be the epitome of active management. They are lightly unregulated investment vehicles with great trading flexibility, and they often pursue highly sophisticated investment strategies. Hedge funds promise "absolute returns" to their investors, leading to a belief that they hold factor-neutral portfolios. They have grown in size noticeably over the past decade and have been receiving increasing portfolio allocations from institutional investors. According to press reports, a number of hedge fund managers have been enjoying compensation that is well in excess of US$10 million per annum.

It is well known that hedge funds implement dynamic strategies; therefore, the exposure of hedge funds to various risk factors is nonlinear. In this chapter, we propose to analyze hedge fund value at risk (VaR) conditional on nonlinearity in factor loadings. In the current VaR literature there are some papers arguing in favor or against of certain VaR models for hedge funds [see Liang and Park (2007), Bali et al. (2007), and Gupta and Liang (2005), for example]. We add to the literature by proposing a model that takes into consideration the dynamic exposure of hedge funds to market and other risk factors. Moreover, it is important to perform a consistent comparison of major VaR models in order to determine the model with the best performance. The main objective of our work is thus, to propose a model of VaR based on regime switching of hedge fund returns, and to provide a consistent comparison of the VaR estimation based on regime switching and three other major VaR models: GARCH(1,1), IGARCH(1,1), and Cornish–Fisher.

The structure of the chapter is as follows. The first section provides an overview of hedge fund literature and hedge fund strategies. The following section presents models used to calculate VaR and to perform backtesting analyses. The final two sections describe hedge funds datasets and their properties and present results of our analysis. Finally, some concluding remarks are provided.

HEDGE FUNDS

The tremendous increase in the number of hedge funds and availability of hedge fund data has attracted a lot of attention in the academic literature that has been concentrated on analyzing hedge funds styles (Fung and Hsieh, 2001; Mitchell and Pulvino, 2001), performance and risk exposure (Bali et al., 2005; Gupta and Liang, 2005; Agarwal and Naik, 2004; Brealey and Kaplanis, 2001; Edwards and Caglayan, 2001; Schneeweis et al., 2002; and Fung and Hsieh, 1997), liquidity, systemic risk, and contagion issues (Billio et al., 2008; Boyson et al., 2007; Chan et al., 2005; Getmansky et al., 2004). All of the above studies find that risk–return characteristics of hedge fund strategies are nonlinear, that hedge funds implement dynamic strategies and exhibit nonlinear and nonnormal payoffs.

Hedge fund strategies greatly differ from each other and have different risk exposures. Fung and Hsieh (2001) analyzed a trend following strategy and Mitchell and Pulvino (2001) studied a risk arbitrage strategy. Both studies find the risk–return characteristics of the hedge fund strategies to be nonlinear and stress the importance of taking into account optionlike features while analyzing hedge funds. Moreover, Agarwal and Naik (2004) show that the nonlinear optionlike payoffs, also called asset-based style factors [ABS-factors introduced by Fung and Hsieh (2002)], are not restricted just to these two strategies but are an integral part of payoffs of various hedge fund strategies.

Hedge funds may exhibit nonnormal payoffs for various reasons such as their use of options, or more generally dynamic trading strategies. Unlike most mutual funds (Koski and Pontiff, 1999), hedge funds frequently trade in derivatives. Furthermore, hedge funds are known for their opportunistic nature of trading and a significant part of their returns arise from taking state contingent bets. For this reason the inclusion of dynamic risk factor exposures is extremely relevant in the VaR calculation. In this work we show the relevance of this issue by introducing a model based on dynamic and state-contingent factor loadings that is able to capture VaR hedge fund risk exposures and compare this model with other models that do not allow for dynamic risk exposures.

VALUE AT RISK

Theoretical Definition

The VaR is considered as a measure of downside risk. It is a measure of the left tail risk of a financial series. Value at risk is the maximum amount of loss that can happen over a given horizon at a certain confidence level. It usually appears in statements like

The maximum loss over one day is about $47 million at the 95 percent confidence level.

If we assume that F is the Cumulative Distribution function of return process we can define VaR as

F(VaR) = α (1.1)

where α is the corresponding probability for the specified confidence level. For instance, for 99 percent confidence level α = 0.01. Using a density function f of returns, VaR can also be equivalently defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

The main question pertinent to the VaR analysis is thus how to forecast the return distribution f over the specified horizon. This study uses the Markov regime-switching approach to forecast the return distribution and thus to compute the VaR. The results are compared with estimates of VaR obtained by other methods like GARCH(1,1), IGARCH(1,1), and Cornish Fisher. In the next section, a brief discussion of these methods is provided along with relative estimation methods.

Empirical Issues

Estimation Methods

When it comes to applying the theoretical formulae to compute the VaR for a specific data set, there are a number of problems which force us to make strong assumptions. The first problem we are going to face is that it is impossible to know the true probability distribution (density) of returns. A number of parametric and nonparametric methods are suggested to overcome this problem.

The traditional nonparametric way is to take the appropriate per-centile of a historical return distribution. This is achieved by sorting the returns in ascending order and taking the (αN)th element of the sorted series, where α is the corresponding probability. If αN is not an integer value, VaR is interpolated. That is, for instance, if αN = 4.6, the linear interpolation is obtained as follows:

VaR = 4th obs + 0.6 (5th obs - 4th obs) (1.3)

The simplest case, in parametric models, is where the return series is assumed to be normally distributed with the mean and variance estimated form the available sample data. This assumption of normality of returns seems to be in conflict with the empirical properties of most financial time series. More specifically, hedge fund returns, due to the dynamic nature of the trading strategies, are known to highly deviate from normality. In the next four subsections we will have a brief discussion of other methods that try to relax the strong assumption of normality. The parametric and semiparamet-ric methods discussed below assume a specific distribution of returns. They use estimation methods like maximum likelihood estimation (MLE) to estimate the relevant parameters of the distribution, which are then used to forecast the future return distribution and thus VaR. In all these models, we denote the return of hedge funds at the end of each period t as r_t.

GARCH(1,1) Model

The GARCH(1,1) approach allows the conditional distribution of the return series to have a time-varying variance. In this study we assume that the returns are conditionally normally distributed with conditional mean μ_t and conditional variance h_t. That is,

r_t ~ N (μ_t, σ²_t

with

where I_t-1 is all the available information at time t-1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

This gives the following model:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

where ε_t ~ N (0, 1).

Regarding the estimation of the conditional mean, given the relevance of factor loadings we consider a factor model where the return dynamics could be represented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

with F_i being certain market risk factors.

The conditional variance h_t is modeled as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

where all the parameters ω, θ, and β should be nonnegative to guarantee a positive conditional variance at all times.

The parameters of all the GARCH models are estimated by MLE. Once we have the estimates of the parameters, we can get the forecast of the conditional mean and variance of the return for the next period. This enables us to compute the one-period VaR using the forecasted normal distribution of returns. Therefore, the VaR is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

where [bar.μ_t] and [bar.h_t] are the forecasted conditional mean and variance, respectively, and η is the appropriate quantile in the N (0, 1) distribution.

RiskMetrics IGARCH(1,1) Model

The JP Morgan RiskMetrics approach uses the GARCH(1,1) framework with specific constraints on the parameters. The conditional mean and the constant term in the conditional variance equation are assumed to be zero and θ + β = 1. Thus, we have the model:

r_t ~ N(0, h_t) (1.10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

This gives the return as

r_t = ε_t √h_t (1.12)

where ε_t ~ N (0, 1). The conditional variance h_t is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

After estimating the parameter θ by MLE method, we get the VaR as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)

Cornish–Fisher Method

The traditional and perhaps the most naïve approach to estimate VaR is to assume the returns are normally distributed with mean μ and variance σ². The VaR is thus given by VaR = (μ - ησ), with η as the appropriate quantile point in the N (0,1) distribution. Considering the fact that hedge funds use dynamic trading strategies and various empirical studies showed that hedge fund return series strictly deviate from the Gaussian distribution [Bali et al. (2007), Gupta and Liang (2005), Agarwal and Naik (2004), Schneeweis et al. (2002), Brealey and Kaplanis (2001), Edwards and Caglayan (2001), and Fung and Hsieh (1997)), this is a very strong assumption to make.

The Cornish–Fisher (CF) approach tries to adjust for the nonnormality of the returns by correcting the critical value η for excess kurtosis and skewness of the historical time series of returns. This is done by taking

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.15)

where η is the critical value for the standard Gaussian distribution and S and K are the empirical skewness and kurtosis of the return series, respectively. The CF VaR is thus given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16)

Markov Regime-Switching Models

Following Billio and Pelizzon (2000), this study considers two different types of regime- switching models.

A. Simple Regime-Switching Models

In simple regime-switching models (SRSM), we consider that the volatility of the return process has different regimes or states. For instance, in a case of two regimes, we can consider high volatility and low volatility regimes. This assumption is consistent with the volatility clustering we observe in financial data series. We can also let returns to have different expectations in different regimes. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.17)

where μ_t and σ_t are the mean and standard deviation of the return r_t, respectively, ε_t is a unit variance and zero mean white noise, and S_t is a Markov chain variable with n states and a transition probability matrix Γ. Theoretically, it is possible to consider as many states as possible. In practice, however, it is usually sufficient to consider two or three states. Considering only two states, 0 and 1, results in the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.18)

Since ε_t ~ IIN (0, 1), we can write the distribution of r_t as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.19)

where P_ij is the probability of switching from regime i to j and P_ii is the probability that the Markov chain does not switch and stays in the same regime. Since P_ij + P_ii = 1, we can write the transition matrix as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.20)

As for the GARCH model, we also consider a factor specification, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.21)

The VaR for the next period is analytically defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.22)

where α is the corresponding probability for the desired confidence level, h is the horizon, and P(S_t+h| It) is the forecasted probability of the regime S at the end of the horizon, which can be obtained from the Hamilton filter [see Hamilton (1994)].

B. Regime-Switching Beta Models

In regime-switching beta models (RSBM) we capture hedge fund exposure to market and other risk factors based on the state of the market, S_mt. In the simplest case, where market (β) and other risk factor (θ) exposures are based on switching of only one market risk factor (r_mt), we obtain the following model:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.23)

where i indicates a hedge fund strategy, m is the market risk factor, ε_t and ε_it are both IIN (0, 1) distributed, and S_mt and Sit are independent Markov chains related to the market risk factor and the hedge fund strategy, respectively.

The VaR in this case is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.24)

Evaluation

As discussed in Kupiec (1995) and Lopez (1997) a variety of tests are available to test the null hypothesis that the observed probability of occurrence over a reporting period equals to α. In other words, α measures the number of exceptions observed in the data, i.e., the number of times the observed returns are lower than VaR. In our work two evaluation methods are used in evaluating VaR model accuracy: the proportion of failure (PF) test (Kupiec, 1995) and the time until first failure (TUFF) test (Kupiec, 1995).

(Continues...)

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Details zur Ausgabe

• Verlag: McGraw Hill
• Erscheinungsdatum: 2009
• Sprache: Englisch
• ISBN 10: 007161513X
• ISBN 13: 9780071615136
• Einband: Tapa dura
• Anzahl der Seiten: 560
• Kontakt zum Hersteller: Nicht verfügbar