Modelling Market Risk

1
Modelling Market Risk

• Value at Risk and Linear Factor Risk Modelling
• Laurence Wormald

2
Overview

• Fundamentals of Risk
• The Historical Perspective
• Differing Approaches to Risk
• Variance Models vs VaR
• Decomposition of Risk
• New Investment Asset Classes
• Regulatory Issues
• Bridging the Gap between VaR and Factor Models

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
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
Expected Return Distribution
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
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 PL/Maximum Drawdown
• Essentially equivalent to VaR measure

8
Real Market Distributionssource Pan and Duffie
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
Left-tail Behavioursource Pan and Duffie
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Right-tail Behavioursource Pan and Duffie
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 PV 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
Estimation of LFM

• V XFX 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
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
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
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
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
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
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
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 bankers 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 managers approach
• HF allow the manager much more freedom to pursue
  alpha
• For the investor, they exhibit
• Unknown leverage
• Implicit optionality

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
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
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
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
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

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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

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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

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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
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 200410,
  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
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• Scherer, B, 2004, Portfolio Construction and Risk
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• Scowcroft, A, and J Sefton, 2005, in Linear
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• Stroyny, A L, 2005, "Estimating a combined linear
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