Carol Alexander is a Professor of Risk Management at the ICMA Centre, University of Reading, and Chair of the Academic Advisory Council of the Professional Risk Manager’s International Association (PRMIA). She is the author of Market Models: A Guide to Financial Data Analysis (John Wiley & Sons Ltd, 2001) and has been editor and contributor of a very large number of books in finance and mathematics, including the multi-volume Professional Risk Manager’s Handbook (McGraw-Hill, 2008 and PRMIA Publications). Carol has published nearly 100 academic journal articles, book chapters and books, the majority of which focus on financial risk management and mathematical finance.

Professor Alexander is one of the world’s leading authorities on market risk analysis. For further details, see www.carolalexander.org

Market Risk Analysis

Volume IV: Value at Risk ModelsBy Carol Alexander

John Wiley & Sons

Copyright © 2008 Carol Alexander
All right reserved.
ISBN: 978-0-470-99788-8

Chapter One

IV.1 Value at Risk and Other Risk Metrics

IV.1.1 INTRODUCTION

A market risk metric is a measure of the uncertainty in the future value of a portfolio, i.e. a measure of uncertainty in the portfolio’s return or profit and loss (P&L). Its fundamental purpose is to summarize the potential for deviations from a target or expected value. To determine the dispersion of a portfolio’s return or P&L we need to know about the potential for individual asset prices to vary and about the dependency between movements of different asset prices. Volatility and correlation are portfolio risk metrics but they are only sufficient (in the sense that these metrics alone define the shape of a portfolio’s return or P&L distribution) when asset or risk factor returns have a multivariate normal distribution. When these returns are not multivariate normal (or multivariate Student t) it is inappropriate and misleading to use volatility and correlation to summarize uncertainty in the future value of a portfolio. Statistical models of volatility and correlation, and more general models of statistical dependency called copulas, are thoroughly discussed in Volume II of Market Risk Analysis. The purpose of the present introductory chapter is to introduce other types of risk metric that are commonly used by banks, corporate treasuries, portfolio management firms and other financial practitioners.

Following the lead from both regulators and large international banks during the mid-1990s, almost all financial institutions now use some form of value at risk (VaR) as a risk metric. This almost universal adoption of VaR has sparked a rigorous debate. Many quants and academics argue against the metric because it is not necessarily sub-additive, which contradicts the principal of diversification and hence also the foundations of modern portfolio theory. Moreover, there is a closely associated risk metric, the conditional VaR, or what I prefer to call the expected tail loss (ETL) because the terminology is more descriptive, that is sub-additive. And it is very simple to estimate ETL once the firm has developed a VaR model, so why not use ETL instead of VaR? Readers are recommended the book by Szeg (2004) to learn more about this debate.

The attractive features of VaR as a risk metric are as follows:

It corresponds to an amount that could be lost with some chosen probability. It measures the risk of the risk factors as well as the risk factor sensitivities. It can be compared across different markets and different exposures. It is a universal metric that applies to all activities and to all types of risk. It can be measured at any level, from an individual trade or portfolio, up to a single enterprise-wide VaR measure covering all the risks in the firm as a whole. When aggregated (to find the total VaR of larger and larger portfolios) or disaggregated (to isolate component risks corresponding to different types of risk factor) it takes account of dependencies between the constituent assets or portfolios.

The purpose of this chapter is to introduce VaR in the context of other ‘traditional’ risk metrics that have been commonly used in the finance industry. The assessment of VaR is usually more complex than the assessment of these traditional risk metrics, because it depends on the multivariate risk factor return distribution and on the dynamics of this distribution, as well as on the risk factor mapping of the portfolio. We term the mathematical models that are used to derive the risk metric, the risk model and the mathematical technique that is applied to estimate the risk metrics from this model (e.g. using some type of simulation procedure) the resolution method.

Although VaR and its related measures such as ETL and benchmark VaR have recently been embraced almost universally, the evolution of risk assessment in the finance industry has drawn on various traditional risk metrics that continue to be used alongside VaR. Broadly speaking, some traditional risk metrics only measure sensitivity to a risk factor, ignoring the risk of the factor itself. For instance, the beta of a stock portfolio or the delta and gamma of an option portfolio are examples of price sensitivities. Other traditional risk metrics measure the risk relative to a benchmark, and we shall be introducing some of these metrics here, including the omega and kappa indices that are currently favoured by many fund managers.

The outline of the chapter is as follows. Section IV.1.2 explains how and why risk assessment in banking has evolved separately from risk assessment in portfolio management. Section IV.1.3 introduces a number of downside risk metrics that are commonly used in portfolio management. These are so called because they focus only on the risk of underperforming a benchmark, ignoring the ‘risk’ of outperforming the benchmark.

The reminder of the chapter focuses on VaR and its associated risk metrics. We use the whole of Section IV.1.4 to provide a thorough definition of market VaR. For instance, when VaR is used to assess risks over a long horizon, as it often is in portfolio management, we should adjust the risk metric for any difference between the expected return and the risk free or benchmark return. However, a non-zero expected excess return has negligible effect when the risk horizon for the VaR estimate is only a few days, as it usually is for banks, and so some texts simply ignore this effect.

Section IV.1.5 lays some essential foundations for the rest of this book by stating some of the basic principles of VaR measurement. These principles are illustrated with simple numerical examples where the only aim is to measure the VaR

at the portfolio level, and where

the portfolio returns are independent and identically distributed (i.i.d.).

Section IV.1.6 begins by stressing the importance of measuring VaR at the risk factor level: without this we could not quantify the main sources of risk. This section also includes two simple examples of measuring the systematic VaR, i.e. the VaR that is captured by the entire risk factor mapping. We consider two examples: an equity portfolio that has been mapped to a broad market index and a cash-flow portfolio that has been mapped to zero-coupon interest rates at standard maturities.

Section IV.1.7 discusses the aggregation and disaggregation of VaR. One of the many advantages of VaR is that is can be aggregated to measure the total VaR of larger and larger portfolios, taking into account diversification effects arising from the imperfect dependency between movements in different risk factors. Or, starting with total risk factor VaR, i.e. systematic VaR, we can disaggregate this into stand-alone VaR components, each representing the risk arising from some specific risk factors. Since we take account of risk factor dependence when we aggregate VaR, the total VaR is often less than the sum of the stand-alone VaRs. That is, VaR is often sub-additive. But it does not have to be so, and this is one of the main objections to using VaR as a risk metric. We conclude the section by introducing marginal VaR (a component VaR that is adjusted for diversification, so that the sum of the marginal VaRs is approximately equal to the total risk factor VaR) and incremental VaR (which is the VaR associated with a single new trade).

Section IV.1.8 introduces risk metrics that are associated with VaR, including the conditional VaR risk metric or expected tail loss. This is the average of the losses that exceed the VaR. Whilst VaR represents the loss that we are fairly confident will not be exceeded, ETL tells us how much we would expect to lose given that the VaR has been exceeded. We also introduce benchmark VaR and its associated conditional metric, expected shortfall (ES). The section concludes with a discussion on the properties of a coherent risk metric. ETL and ES are coherent risk metrics, but when VaR and benchmark VaR are estimated using simulation they are not coherent because they are not sub-additive.

Section IV.1.9 introduces the three fundamental types of resolution method that may be used to estimate VaR, applying each method in only its most basic form, and to only a very simple portfolio. After a brief overview of these approaches, which we call the normal linear VaR, historical VaR and normal Monte Carlo VaR models, we present a case study on measuring VaR for a simple position of $1000 per point on an equity index. Our purpose here is to illustrate the fundamental differences between the models and the reasons why our estimates of VaR can differ so much depending on the model used. Section IV.1.10 summarizes and concludes.

Volume IV of the Market Risk Analysis series builds on the three previous volumes, and even for this first chapter readers first require an understanding of: quantiles and other basic concepts in statistics (Section I.3.2); the normal distribution family and the standard normal transformation (Section I.3.3.4); stochastic processes in discrete time (Section I.3.7.1); portfolio returns and log returns (Section I.1.4); aggregation of log returns and scaling of volatility under the i.i.d. assumption (Section II.3.2.1); the matrix representation of the expectation and variance of returns on a linear portfolio (Section I.2.4); univariate normal Monte Carlo simulation and how it is performed in Excel (Section I.5.7). risk factor mappings for portfolios of equities, bonds and options, i.e. the expression of the portfolio P&L or return as a function of market factors that are common to many portfolios (e.g. stock index returns, or changes in LIBOR rates) and which are called the risk factors of the portfolio (Section III.5).

There is a fundamental distinction between linear and non-linear portfolios. A linear portfolio is one whose return or P&L may be expressed as a linear function of the returns or P&L on its constituent assets or risk factors. All portfolios except those with options or option-like structures fall into the category of linear portfolios.

It is worth repeating here my usual message about the spreadsheets on the CD-ROM. Each chapter has a folder which contains the data, figures, case studies and examples given in the text. All the included data are freely downloadable from websites, to which references for updating are given in the text. The vast majority of examples are set up in an interactive fashion, so that the reader or tutor can change any parameter of the problem, shown in red, and then view the output in blue. If the Excel data analysis tools or Solver are required, then instructions are given in the text or the spreadsheet.

IV.1.2 AN OVERVIEW OF MARKET RISK ASSESSMENT

In general, the choice of risk metric, the relevant time horizon and the level of accuracy required by the analyst depend very much on the application:

A typical trader requires a detailed modelling of short-term risks with a high level of accuracy. A risk manager working in a large organization will apply a risk factor mapping that allows total portfolio risk to be decomposed into components that are meaningful to senior management. Risk managers often require less detail in their risk models than traders do. On the other hand, risk managers often want a very high level of confidence in their results. This is particularly true when they want to demonstrate to a rating agency that the company deserves a good credit rating. Senior managers that report to the board are primarily concerned with the efficient allocation of capital on a global scale, so they will be looking at long-horizon risks, taking a broad-brush approach to encompass only the most important risks.

The metrics used to assess market risks have evolved quite separately in banking, portfolio management and large corporations. Since these professions have adopted different approaches to market risk assessment we shall divide our discussion into these three broad categories.

IV.1.2.1 Risk Measurement in Banks

The main business of banks is to accept risks (because they know, or should know, how to manage them) in return for a premium paid by the client. For retail and commercial banks and for many functions in an investment bank, this is, traditionally, their main source of profit. For instance, banks write options to make money on the premium and, when market making, to make profits from the bid-ask spread. It is not their business, at least not their core business, to seek profits through enhanced returns on investments: this is the role of portfolio management. The asset management business within a large investment bank seeks superior returns on investments, but the primary concern of banks is to manage their risks.

A very important decision about risk management for banks is whether to keep the risk or to hedge at least part of it. To inform this decision the risk manager must first be able to measure the risk. Often market risks are measured over the very short term, over which banks could hedge their risks if they chose to, and over a short horizon is it standard to assume the expected return on a financial asset is the risk free rate of return. So modelling the expected return does not come into the picture at all. Rather, the risk is associated with the unexpected return – a phrase which here means the deviation of the return about its expected value – and the expected rate of return is usually assumed to be the risk free rate.

Rather than fully hedging all their risks, traders are usually required to manage their positions so that their total risk stays within a limit. This limit can vary over time. Setting appropriate risk limits for traders is an important aspect of risk control. When a market has been highly volatile the risk limits in that market should be raised. For instance, in equity markets rapid price falls would lead to high volatility and equity betas could become closer to 1 if the stock’s market correlation increased. If a proprietary trader believes the market will now start to rise he may want to buy into that market so his risk limits, based on either volatility or portfolio beta, should be raised. Traditionally risk factor exposures were controlled by limiting risk factor sensitivities. For instance, equity traders were limited by portfolio beta, options traders operated under limits determined by the net value Greeks of their portfolio, and bond traders assessed and managed risk using duration or convexity. However, two significant problems with this traditional approach have been recognized for some time.

The first problem is the inability to compare different types of risks. One of the reasons why sensitivities are usually represented in value terms is that value sensitivities can be summed across similar types of positions. For instance, a value delta for one option portfolio can be added to a value delta for another option portfolio; likewise the value duration for one bond portfolio can be added to the value duration for another bond portfolio. But we cannot mix two different types of sensitivities. The sum of a value beta, a value gamma and a value convexity is some amount of money, but it does not correspond to anything meaningful. The risk factors for equities, options and bonds are different, so we cannot add their sensitivities. Thus, whilst value sensitivities allow risks to be aggregated within a given type of trading activity, they do not aggregate across different trading units. The traditional sensitivity-based approach to risk management is designed to work only within a single asset class.

(Continues…)

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