Modelling Market Risk

1. 1 Modelling Market Risk Value at Risk and Linear Factor Risk Modelling Laurence Wormald

2. Slide 2

3. Slide 3 Historical Perspective What is Risk? A problem for Conventional Asset Managers A problem for Banks And of course a problem for Hedge Funds Is a certain loss a risk? Is risk absolute or relative? Is risk a sensitivity measure? Can we apply our risk measure to Equities, Bonds, Currencies? Funds, Options, Structured Products?

4. Slide 4 Approaches to Risk Banks Risk as a hedging problem Control losses Look for �Portfolio Insurance� Conventional Asset Managers Risk as a diversification problem When you are as diversified as your benchmark, you are done Control volatility �Risk is good�

5. Slide 5 Expected Return Distribution

6. Slide 6 Parameters Moments of a distribution 1st moment = mean (expected return) 2nd moment = variance (standard deviation, volatility, tracking error) 3rd moment = skewness (measure of asymmetry, = 0 for normal dist) 4th moment = excess kurtosis (measure of fat tails, = 0 for normal dist) Fractiles of a distribution Median = 50% fractile 1st quartile = 25% fractile 99% VaR = 1% fractile

7. Slide 7 Risk Modelling Banks Estimate of probability of loss Left-hand tail of expected return distribution Parametric or non-parametric methodology Conventional Asset Managers Estimate of dispersion (inverse measure of diversification) � variance, volatility or tracking error Statistic of entire expected return distribution Parametric methodology Hedge Funds Variability of P&L/Maximum Drawdown Essentially equivalent to VaR measure

8. Slide 8 Real Market Distributionssource: Pan and Duffie

9. Slide 9 Deviations from Normality The following graphs show how real market returns are nothing like normal If returns were normal and stationary: All the symbols would be within the dotted boxes Daily and monthly measures would be similar, so scatter would be symmetric about diagonal Left and right tail graphs would be very similar

10. Slide 10 Left-tail Behavioursource: Pan and Duffie

11. Slide 11 Right-tail Behavioursource: Pan and Duffie

12. Slide 12 Estimation of Risk Models Ideally � model the entire Expected Distribution of Returns Historically � need to simplify Covariance Matrix-based models (Markowitz) Based on assumed M-V form of investor utility Reduces the portfolio risk to a linear algebraic problem s2 = P�V P Historically a tremendous advantage Ignores effects of all higher moments Originally used with historical covariances Rests on firm theoretical grounds if either: Investors really do exhibit quadratic (symmetric) utility Returns are really multivariate normal

13. Slide 13 Estimation of LFM V = X�FX + S Types of Linear Factor Models X-sectional (micro- or fundamental factors) Time Series (macro-factors) Statistical (principal component factors) Hybrid (any combination of factors) Order in which factors are taken out is vital eg: Northfield (X-sectional/macro/blind hybrid)

14. Slide 14 Factor Risk Models Linear Factor Models as models of Variance Intuitively appealing to market professionals We believe in �driving forces� in most securities markets We would like to know how to �neutralise� certain factor risks LFM allow construction of factor-mimicking portfolios (factor portfolio of trades - FPT) FPT may be constructed by constrained optimisation In Practise All assumptions are routinely violated Factors are different for each asset class �Transient� Factors are invoked to explain residuals Ever more elaborate estimation methods are proposed (Stroyny, diBartolomeo et al) Not suitable for derivatives

15. Slide 15 Value at Risk Conventional VaR Simplicity of expression appeals to non-mathematicians (business types, regulators, marketeers) A fractile of the EDR rather than a dispersion measure Consistent interpretation regardless of shape of EDR May be applied across asset classes and for all new types of securities Easy to calculate if entire EDR is available Calculation usually by a simulation method which generates a sufficiently large set of possible outcomes Not easily related to supposed portfolio risk factors Cannot be optimised (not sub-additive or convex)

16. Slide 16 Other VaR measures Conditional VaR (Expected Shortfall) Combination fractile and dispersion measure One of a class of lower partial moment measures Reveals the nature of the tail More suitable than VaR for stress testing Improvement on VaR in that CVaR is subadditive and coherent (Artzner, Acerbi) When optimising, CVaR frontiers are properly convex This is a more robust downside measure than VaR

17. Slide 17 VaR Models Parametric If fitted to historical data, entail sample selection bias Linear or quadratic (delta-gamma) VaR measures Other modelling assumptions require Monte Carlo methods (repeated sampling from parametric or historical distributions) All subject to model risk Historical Simulation Entails sample selection bias Can avoid most other assumptions Requires a library of pricing functions Extreme Value Theory Special Parametric form of Tail for Stress Testing

18. Slide 18 Risk Decomposition Performance Attribution has provided great insights into investment Can we do the same for risk? Marginal Risk Measures the rate of change of portfolio risk for a small trade in a given security. May be represented as a vector for a given set of securities Can be estimated in TE or VaR terms Trade may be financed from cash, from the rest of the portfolio, or from the benchmark Simply derived algebraically for LFM May be calculated by brute-force or semi-analytic methods for VaR

19. Slide 19 Risk Decomposition Component Risk Measures the fraction of the portfolio risk which can be attributed to the current holding of a particular security We would like this to be additive, so as to aggregate CR to any sub-portfolio Simply derived algebraically for LFM in terms of the Marginal Risk vector May be calculated for VaR (Garman, Hallerbach) - expressed in terms of the Marginal VaR vector Attribution to factors is heavily dependent on the estimation method (Scowcroft et al) Does not provide all that performance attribution does, but vital information for the manager

20. Slide 20 New Investment Asset Classes Structured Products vs Hedge Funds Both now available to certain asset managers and pension funds Both present problems to LFM-based risk management Structured Products � the banker�s approach SP allow the buyer to take a view on a specific scenario while limiting downside May be index-based, capturing systematic risk only Credit Derivatives: now appearing in many �boring� fixed income portfolios Mortgage-Backed Securities: hard to price Explicit optionality and variable leverage Hedge Funds � the asset manager�s approach HF allow the manager much more freedom to pursue alpha For the investor, they exhibit Unknown leverage Implicit optionality

21. Slide 21 Regulatory Issues New emphasis on regulations based on quantitative risk measures Supplement to traditional allocation limits Regulators have focused on downside and ruin Regulators� mission is to avoid mis-selling of funds Product regulations on funds containing derivatives (FDI) via VaR-based approach, inspired by capital adequacy provisions of Basel II UCITS III � in force Jan 2007 Sophisticated UCITS = UCITS which may use FDI for investment purposes, particularly UCITS which employ leverage in their use of FDI and/or use OTC derivatives. Daily VaR reporting plus Stress Testing and Model Testing There is some freedom in the interpretation of what constitutes a sophisticated UCITS

22. Slide 22 Bridging the Gap AMs like to use LFM for portfolio construction Ubiquitous �factor alpha� models Ability to use quadratic optimisation tools Style-based investing is explicitly driven by linearised factors � �factor betas� or �factor tilts� Some factors are purely statistical Investors (especially Banks) and regulators are interested in the impact on VaR Best VaR models are based on Monte Carlo simulations and complex pricing functions LFM betas are NOT represented within VaR model Would like to quantify VaR impact of increasing any individual factor tilt within a portfolio

23. Slide 23 Marginal Factor VaR How do factor risks relate to VaR? � Marginal Factor VaR From the LFM, generate the factor portfolio of trades associated with the factor of interest FPT may be constructed using quadratic optimiser FPT will have unit exposure to Fi, neutral to all other Fj, and be risk-minimised Take the scalar product of this FPT with Marginal VaR vector to obtain the Marginal Factor VaR for this factor Allows VaR-based comparison of factor effects on portfolio

24. Slide 24 Example: Factor VaR Select factor from LFM provided by BITA Risk Statistical factor1 is a global equity factor within a hybrid model Select a universe of large-cap European stocks DJ Stoxx50E � 50 names Create Factor Portfolio of Trades (FPT) using optimiser Constrained optimisation for trade portfolio of zero market value Minimise risk, subject to constraints Estimate Marginal VaR for core portfolio plus FPT using bank VaR model Core portfolio is broadly neutral to SF1 Modified portfolio will have same net market value, but unit exposure to SF1

25. Slide 25 Statistical Factor1 Select factor from LFM provided by BITA Risk BITA Risk model is a hybrid model incorporating Currency Region Industry Statistical factors, all estimated using stepwise regression on weekly historical data The SF1 factor is the first (most significant) statistical factor within the hybrid model For our example, we select universe of large-cap European stocks DJ Stoxx50E: 50 names within Euro market Next slide shows prices and weights as of 29 Aug 2007

27. Slide 27 SF1 FPT Create Factor Portfolio of Trades (FPT) using optimiser Constrained optimisation to create a trade portfolio with zero market value Minimise risk, subject to constraints: Equality constraint: Net Market Value = 0 (balanced long/short FPT) Notional market value of long side of FPT as appropriate (eg 10MEUR for a typical 500MEUR portfolio) Equality constraint: SF1 exposure = 1 Equality constraints: All other factor exposures = 0 Next slide shows weights of SF1 Factor Portfolio of Trades for Stoxx50E universe

29. Slide 29 MVAR on SF1 FPT Estimate marginal VaR for core portfolio plus FPT using bank VaR model Core portfolio is broadly neutral to the factor SF1 Modified portfolio will have same the net market value, but unit exposure to SF1 Hence VaR could be lower or higher after the trade In this example VAR is lowered, suggesting that SF1 is a diversifying or hedging factor for the portfolio This will reveal the Marginal Factor VaR associated with the SF1 factor Next slide shows the CVARs which comprise the Marginal Factor VAR for the SF1 factor applied to the core portfolio The Marginal Factor VAR value is calculated as -81,535 EUR

31. Slide 31 Conclusions Each historically-popular class of risk measure brings its own problems for the modeller The first issue in Risk Model estimation is: parametric/non parametric? If parametric, linear or non-linear? Risk models are typically �fit for purpose�, but not for more than one purpose Portfolio construction Hedging (what risk measure do we want to control?) Pure risk reporting (compliance) Regulators are taking a strong interest in the downside and in stress testing, and demanding VaR reporting The Marginal Factor VaR approach is a new approach which helps to improve portfolio construction within the regulatory VaR framework

32. Slide 32 References Acerbi, C, 2002, �Spectral Measures of Risk: a coherent representation of subjective risk aversion�, Journal of Banking and Finance, 26, 1505-1518 Artzner, P, F Delbaen, J-M Eber and D Heath, 1999, �Coherent Measures of Risk�, Mathematical Finance, 9, 203-228 Asgharian, H, 2004, �A Comparative Analysis of Ability of Mimicking Portfolios in Representing the Background Factors�, Working Papers 2004:10, Lund University Carroll, R B, T Perry, H Yang and A Ho, 2001, �A new approach to component VaR�, Journal of Risk, Volume 3 / Number 3, Spring DiBartolomeo, D, and S Warrick, 2005, �Making covariance-based portfolio risk models sensistive to the rate at which markets reflect new information� in Linear Factor Models in Finance, Elsevier Finance. Giacometti, R, and S O Lozza, 2004,�Risk Measures for Asset Allocation Models� in Risk Measures for the 21st Century, Wiley Finance Garman, M B, 1997, "Taking VaR to pieces", Risk Vol 10, No 10. Grinold, R, and R Kahn, 1999, Active Portfolio Management, 2nd Edition, (New York: McGraw Hill) Hallerbach, W G, 1999 & 2003, "Decomposing Portfolio Value at Risk: A General Analysis", Discussion paper & Journal of Risk, Vol 5, No 2. Pan, J and Duffie, D, 2001, �An Overview of Value at Risk� in Options Markets, edited by G. Constantinides and A. G. Malliaris, London: Edward Elgar Satchell, S E, and L Shi, 2005, "Further Results on Tracking Error, concerning Stochastic Weights and Higher Moments", Working paper. Scherer, B, 2004, Portfolio Construction and Risk Budgeting, Risk Books. Scowcroft, A, and J Sefton, 2005, in Linear Factor Models in Finance, Elsevier Finance. Stroyny, A L, 2005, "Estimating a combined linear factor model" in Linear Factor Models in Finance, Elsevier Finance.