Mathematical Modeling and Statistical Methods for Risk ...

1 Mathematical Modeling and Statistical Methodsfor Risk ManagementLecture Notesc Henrik Hult and Filip Lindskog2007 Contents1 Some background to financial risk A preliminary example .. Why risk management? .. Regulators and supervisors .. Why the government cares about the buffer capital .. Types of risk .. Financial derivatives .. 42 Loss operators and financial Portfolios and the loss operator .. The general case .. 73 Risk Elementary measures of risk .. Risk measures based on the loss distribution .. 134 Methods for computing VaR and Empirical VaR and ES .. Confidence intervals .. Exact confidence intervals for Value-at-Risk .. Using the bootstrap to obtain confidence intervals .. Historical simulation .. Variance Covariance method.

2 Monte-Carlo Methods .. 265 Extreme value theory for random variables with heavy Quantile-quantile plots .. Regular variation .. 306 Hill Selecting the number of upper order statistics .. 367 The Peaks Over Threshold (POT) How to choose a high threshold.. Mean-excess plot .. Parameter estimation .. Estimation of Value-at-Risk and Expected shortfall .. 438 Multivariate distributions and Basic properties of random vectors .. Joint log return distributions .. Comonotonicity and countermonotonicity .. Covariance and linear correlation .. Rank correlation .. Tail dependence .. 52i9 Multivariate elliptical The multivariate normal distribution .. Normal mixtures .. Spherical distributions .. Elliptical distributions .. Properties of elliptical distributions.

3 Elliptical distributions and risk management .. 6010 Basic properties .. Dependence measures .. Elliptical copulas .. Simulation from Gaussian and t-copulas .. Archimedean copulas .. Simulation from Gumbel and Clayton copulas .. Fitting copulas to data .. Gaussian and t-copulas .. 8111 Portfolio credit risk A simple model .. Latent variable models .. Mixture models .. One-factor Bernoulli mixture models .. Probit normal mixture models .. Beta mixture models .. 9012 Popular portfolio credit risk The KMV model .. CreditRisk+ a Poisson mixture model .. 97A A few probability Convergence concepts .. Limit theorems and inequalities .. 105B Conditional Definition and properties .. An expression in terms the density of (X,Z) .. Orthogonality and projections in Hilbert spaces.

4 108iiPrefaceThese lecture notes aim at giving an introduction to Quantitative Risk Man-agement. We will introduce Statistical techniques used forderiving the profit-and-loss distribution for a portfolio of financial instruments and to compute riskmeasures associated with this distribution. The focus lieson the mathemati-cal/ Statistical Modeling of market- and credit risk. Operational risks and theuse of financial time series for risk Modeling are not treatedin these lecturenotes. Financial institutions typically hold portfolios consisting on large num-ber of financial instruments. A careful Modeling of the dependence betweenthese instruments is crucial for good risk management in these situations. Alarge part of these lecture notes is therefore devoted to theissue of reader is assumed to have a Mathematical / Statistical knowledge correspond-ing to basic courses in linear algebra, analysis, statistics and an intermediatecourse in probability.

5 The lecture notes are written with the aim of presentingthe material in a fairly rigorous way without any use of measure chapters 1-4 in these lecture notes are based on the book [12]which we strongly recommend. More material on the topics pre-sented in remaining chapters can be found in [8] (chapters 5-7), [12](chapters 8-12) and articles found in the list of referencesat the endof these lecture Hult and Filip Lindskog, 2007iii1 Some background to financial risk manage-mentWe will now give a brief introduction to the topic of risk management andexplain why this may be of importance for a bank or financial institution. Wewill start with a preliminary example illustrating in a simple way some of theissues encountered when dealing with risks and risk A preliminary exampleA player (investor/speculator) is entering a casino with aninitial capital ofV0= 1 million Swedish Kroner.

6 All initial capital is used to place bets accordingto a predetermined gambling strategy. After the game the capital isV1. Wedenote the profit(loss) by a random variableX=V1 V0. The distributionofXis called theprofit-and-loss distribution(P&L) and the distribution ofL= X=V0 V1is simply called theloss distribution. As the loss may bepositive this is a risky position, there is a risk of losing some of the a game is constructed so that it gives million Swedish Kronerwith probabilitypand million Swedish Kroner with probability 1 p. Hence,X= with probabilityp, with probability 1 p.( )The fair price for this game, corresponding to E(X) = 0, isp= However,even ifp > the player might choose not to participate in the game with theview that not participating is more attractive than playinga game with a smallexpected profit together with a risk of loosing million Swedish Kroner.

7 Thisattitude is , the choice of whether to participate or not dependson the P&Ldistribution. However, in most cases (think of investing ininstruments on thefinancial market) the P&L distribution is not known. Then youneed to evaluatesome aspects of the distribution to decide whether to play ornot. For thispurpose it is natural to use arisk measure. A risk measure is a mappingfrom the random variables to the real numbers; to every loss random variableLthere is a real number (L) representing the riskiness ofL. To evaluate theloss distribution in terms of a single number is of course a huge simplificationof the world but the hope is that it can give us sufficient indication whether toplay the game or the game ( ) described above and suppose that themean E(L) = ( a positive expected profit) and standard deviation std(L) = of thelossLis known.

8 In this case the game had only two known possible outcomes sothe information about the mean and standard deviation uniquely specifies theP&L distribution, yieldingp= However, the possible outcomes of a typicalreal-world game are typically not known and mean and standard deviation do1not specify the P&L distribution. A simple example is the following:X= with probability , with probability ( )Here we also have E(L) = and std(L) = However, most risk-averseplayers would agree that the game ( ) is riskier than the game ( ) withp= Using an appropriate quantile of the lossLas a risk measure wouldclassify the game ( ) as riskier than the game ( ) withp= However,evaluating a single risk measure such as a quantile will in general not providea lot of information about the loss distribution, although it can provide somerelevant information.

9 A key to a sound risk management is to look for riskmeasures that give as much relevant information about the loss distribution risk manager at a financial institution with responsibility for a portfolioconsisting of a few up to hundreds or thousands of financial assets and contractsfaces a similar problem as the player above entering the casino. Management orinvestors have also imposed risk preferences that the risk manager is trying tomeet. To evaluate the position the risk manager tries to assess the loss distribu-tion to make sure that the current positions is in accordancewith imposed riskpreferences. If it is not, then the risk manager must rebalance the portfolio untila desirable loss distribution is obtained. We may view a financial investor as aplayer participating in the game at the financial market and the loss distributionmust be evaluated in order to know which game the investor is participating Why risk management?

10 The trading volumes on the financial markets have increased tremendously overthe last decades. In 1970 the average daily trading volume atthe New YorkStock Exchange was million shares. In 2002 it was billion shares. Inthe last few years we have seen a significant increase in the derivatives are a huge number of actors on the financial markets taking risky positionsContracts1995 1998 2002 FOREX131818 Interest rate2650 102 Total4780 142 Table 1: Global market in OTC derivatives (nominal value) intrillion US dollars(1 trillion = 1012).and to evaluate their positions properly they need quantitative tools from riskmanagement. Recent history also shows several examples where large losses onthe financial market are mainly due to the absence of proper risk (Orange County)On December 6 1994, Orange County, aprosperous district in California, declared bankruptcy after suffering losses of2around $ billion from a wrong-way bet on interest rates in one of its principalinvestment pools.