Quantitative Risk Management: Concepts, Techniques And . PDF

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Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.1Risk in PerspectiveIn this chapter we provide a non-mathematical discussion of various issues that formthe background to the rest of the book. In Section 1.1 we begin with the nature ofrisk itself and discuss how risk relates to randomness; in the financial context (whichincludes insurance) we summarize the main kinds of risks encountered and explainwhat it means to measure and manage such risks.A brief history of financial risk management and the development of financialregulation is given in Section 1.2, while Section 1.3 contains a summary of theregulatory framework in the financial and insurance industries.In Section 1.4 we take a step back and attempt to address the fundamental questionof why we might want to measure and manage risk at all. Finally, in Section 1.5 weturn to quantitative risk management (QRM) explicitly and set out our own viewsconcerning the nature of this discipline and the challenge it poses. This section inparticular should give more insight into our choice of methodological topics in therest of the book.1.1RiskThe Concise Oxford English Dictionary defines risk as “hazard, a chance of badconsequences, loss or exposure to mischance”. In a discussion with students takinga course on financial risk management, ingredients that are typically discussed areevents, decisions, consequences and uncertainty. It is mostly only the downside ofrisk that is mentioned, rarely a possible upside, i.e. the potential for a gain. Whilefor many people risk has largely negative connotations, it may also represent anopportunity. Much of the financial industry would not exist were it not for thepresence of financial risk and the opportunities afforded to companies that are ableto create products and services that offer more financial certainty to their clients.For financial risks no single one-sentence definition of risk is entirely satisfactory.Depending on context, one might arrive at notions such as “any event or action thatmay adversely affect an organization’s ability to achieve its objectives and execute itsstrategies” or, alternatively, “the quantifiable likelihood of loss or less-than-expectedreturns”.1.1.1Risk and RandomnessRegardless of context, risk strongly relates to uncertainty, and hence to the notion ofrandomness. Randomness has eluded a clear, workable definition for many centuries;For general queries, contact [email protected]

Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.41. Risk in Perspectiveit was not until 1933 that the Russian mathematician A. N. Kolmogorov gave anaxiomatic definition of randomness and probability (see Kolmogorov 1933). Thisdefinition and its accompanying theory provide the language for the majority of theliterature on risk, including this book.Our reliance on probability may seem unsatisfactorily narrow to some. It bypassesseveral of the current debates on risk and uncertainty (Frank Knight), the writings onprobabilistic thinking within economics (John Maynard Keynes), the unpredictability of unprecedented financial shocks, often referred to as Black Swans (NassimTaleb), or even the more political expression of the known, the unknown and theunknowable (Donald Rumsfeld); see the Notes and Comments section for moreexplanation. Although these debates are interesting and important, at some pointclear definitions and arguments are called for and this is where mathematics as a language enters. The formalism of Kolmogorov, while not the only possible approach,is a tried-and-tested framework for mathematical reasoning about risk.In Kolmogorov’s language a probabilistic model is described by a triplet(Ω, F , P ). An element ω of Ω represents a realization of an experiment, in economics often referred to as a state of nature. The statement “the probability thatan event A occurs” is denoted (and in Kolmogorov’s axiomatic system defined)as P (A), where A is an element of F , the set of all events. P denotes the probability measure. For the less mathematically trained reader it suffices to acceptthat Kolmogorov’s system translates our intuition about randomness into a concise,axiomatic language and clear rules.Consider the following examples: an investor who holds stock in a particularcompany; an insurance company that has sold an insurance policy; an individualwho decides to convert a fixed-rate mortgage into a variable one. All of these situations have something important in common: the investor holds today an assetwith an uncertain future value. This is very clear in the case of the stock. For theinsurance company, the policy sold may or may not be triggered by the underlying event covered. In the case of a mortgage, our decision today to enter into thisrefinancing agreement will change (for better or for worse) the future repayments.So randomness plays a crucial role in the valuation of current products held by theinvestor, the insurance company and the home owner.To model these situations a mathematician would now define the value of a riskyposition X to be a function on the probability space (Ω, F , P ); this function is calleda random variable. We leave for the moment the range of X (i.e. its possible values)unspecified. Most of the modelling of a risky position X concerns its distributionfunction FX (x) P (X x): the probability that by the end of the period underconsideration the value of the risk X is less than or equal to a given number x.Several risky positions would then be denoted by a random vector (X1 , . . . , Xd ),also written in bold face as X; time can be introduced, leading to the notion ofrandom (or so-called stochastic) processes, usually written (Xt ). Throughout thisbook we will encounter many such processes, which serve as essential buildingblocks in the mathematical description of risk.For general queries, contact [email protected]

Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.1.1. Risk5We therefore expect the reader to be at ease with basic notation, terminology andresults from elementary probability and statistics, the branch of mathematics dealingwith stochastic models and their application to the real world. The word “stochastic”is derived from the Greek “stochazesthai”, the art of guessing, or “stochastikos”,meaning skilled at aiming (“stochos” being a target). In discussing stochastic methods for risk management we hope to emphasize the skill aspect rather than theguesswork.1.1.2Financial RiskIn this book we discuss risk in the context of finance and insurance (although manyof the tools introduced are applicable well beyond this context). We start by givinga brief overview of the main risk types encountered in the financial industry.The best-known type of risk is probably market risk: the risk of a change inthe value of a financial position or portfolio due to changes in the value of theunderlying components on which that portfolio depends, such as stock and bondprices, exchange rates, commodity prices, etc. The next important category is creditrisk: the risk of not receiving promised repayments on outstanding investments suchas loans and bonds, because of the “default” of the borrower. A further risk categoryis operational risk: the risk of losses resulting from inadequate or failed internalprocesses, people and systems, or from external events.The three risk categories of market, credit and operational risk are the main oneswe study in this book, but they do not form an exhaustive list of the full rangeof possible risks affecting a financial institution, nor are their boundaries alwaysclearly defined. For example, when a corporate bond falls in value this is marketrisk, but the fall in value is often associated with a deterioration in the credit qualityof the issuer, which is related to credit risk. The ideal way forward for a successfulhandling of financial risk is a holistic approach, i.e. an integrated approach takingall types of risk and their interactions into account.Other important notions of risk are model risk and liquidity risk. The former isthe risk associated with using a misspecified (inappropriate) model for measuringrisk. Think, for instance, of using the Black–Scholes model for pricing an exoticoption in circumstances where the basic Black–Scholes model assumptions on theunderlying securities (such as the assumption of normally distributed returns) areviolated. It may be argued that model risk is always present to some degree.When we talk about liquidity risk we are generally referring to price or marketliquidity risk, which can be broadly defined as the risk stemming from the lackof marketability of an investment that cannot be bought or sold quickly enough toprevent or minimize a loss. Liquidity can be thought of as “oxygen for a healthymarket”; a market requires it to function properly but most of the time we are notaware of its presence. Its absence, however, is recognized immediately, with oftendisastrous consequences.In banking, there is also the concept of funding liquidity risk, which refers tothe ease with which institutions can raise funding to make payments and meetwithdrawals as they arise. The management of funding liquidity risk tends to beFor general queries, contact [email protected]

Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.61. Risk in Perspectivea specialist activity of bank treasuries (see, for example, Choudhry 2012) ratherthan trading-desk risk managers and is not a subject of this book. However, fundingliquidity and market liquidity can interact profoundly in periods of financial stress.Firms that have problems obtaining funding may sell assets in fire sales to raise cash,and this in turn can contribute to market illiquidity, depressing prices, distorting thevaluation of assets on balance sheets and, in turn, making funding even more difficultto obtain; this phenomenon has been described as a liquidity spiral (Brunnermeierand Pedersen 2009).In insurance, a further risk category is underwriting risk: the risk inherent ininsurance policies sold. Examples of risk factors that play a role here are changingpatterns of natural catastrophes, changes in demographic tables underlying (longdated) life products, political or legal interventions, or customer behaviour (such aslapsation).1.1.3Measurement and ManagementMuch of this book is concerned with techniques for the statistical measurement ofrisk, an activity which is part of the process of managing risk, as we attempt toclarify in this section.Risk measurement. Suppose we hold a portfolio consisting of d underlying investments with respective weights w1 , . . . , wd , so that the change in value of the portfolioover a given holding period (the so-called profit and loss, or P&L) can be written as X di 1 wi Xi , where Xi denotes the change in value of the ith investment. Measuring the risk of this portfolio essentially consists of determining its distributionfunction FX (x) P (X x), or functionals describing this distribution functionsuch as its mean, variance or 99th percentile.In order to achieve this, we need a properly calibrated joint model for the underlying random vector of investments (X1 , . . . , Xd ), so statistical methodology hasan important role to play in risk measurement; based on historical observations andgiven a specific model, a statistical estimate of the distribution of the change invalue of a position, or one of its functionals, is calculated. In Chapter 2 we developa detailed framework framework for risk measurement. As we shall see—and thisis indeed a main theme throughout the book—this is by no means an easy task witha unique solution.It should be clear from the outset that good risk measurement is essential. Increasingly, the clients of financial institutions demand objective and detailed informationon the products that they buy, and firms can face legal action when this informationis found wanting. For any product sold, a proper quantification of the underlyingrisks needs to be explicitly made, allowing the client to decide whether or not theproduct on offer corresponds to his or her risk appetite; the 2007–9 crisis saw numerous violations of this basic principle. For more discussion of the importance of thequantitative approach to risk, see Section 1.5.Risk management. In a very general answer to the question of what risk management is about, Kloman (1990) writes:For general queries, contact [email protected]

Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.1.1. Risk7To many analysts, politicians, and academics it is the management ofenvironmental and nuclear risks, those technology-generated macrorisks that appear to threaten our existence. To bankers and financialofficers it is the sophisticated use of such techniques as currency hedgingand interest-rate swaps. To insurance buyers or sellers it is coordinationof insurable risks and the reduction of insurance costs. To hospitaladministrators it may mean “quality assurance”. To safety professionalsit is reducing accidents and injuries. In summary, risk management isa discipline for living with the possibility that future events may causeadverse effects.The last phrase in particular (the emphasis is ours) captures the general essence ofrisk management: it is about ensuring resilience to future events. For a financialinstitution one can perhaps go further. A financial firm’s attitude to risk is not passive and defensive; a bank or insurer actively and willingly takes on risk, because itseeks a return and this does not come without risk. Indeed, risk management can beseen as the core competence of an insurance company or a bank. By using its expertise, market position and capital structure, a financial institution can manage risksby repackaging or bundling them and transferring them to markets in customizedways.The management of risk at financial institutions involves a range of tasks. Tobegin with, an enterprise needs to determine the capital it should hold to absorblosses, both for regulatory and economic capital purposes. It also needs to managethe risk on its books. This involves ensuring that portfolios are well diversified andoptimizing portfolios according to risk–return considerations. The risk profile ofthe portfolio can be altered

turn to quantitative risk management (QRM) explicitly and set out our own views concerning the nature of this discipline and the challenge it poses. This section in particular should give more insight into our choice of methodological topics in the rest of the book. 1.1 Risk The Concise Oxford English Dictionary defines risk as “hazard, a ...