MATLAB® Products for Financial Risk Management & Modeling

Use of COPULAS

ANURAG JAIN

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Case Study Topic: Copulas in Risk Management

Quantitative Risk Modeling gaining more attention and exposure after recent crisis

Function that links (couples) univariate margins distributions to create full multivariate distribution (MVD)

Joint distribution function of d standard uniform random variables

Demo : Equity Portfolio Risk Management using Copulas

• Chief Risk Officer needs methods for AGGREGATION OF RISKS: Enterprise Level Mgt.

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Needs, Uses and Target Users

Returns in real world are not normal, simple Pearson correlations don’t always work especially in tails

Fat Tails and Tail Dependence need to be modeled separately

Internal models for credit, market & operational risks (for Bank Capital Allocation based on Basel II): Problem: modeling of joint distributions of different risks

Equity Portfolios: Estimation of covariances alone not sufficient to capture the real extreme movements among individual equities: portfolio risk manager has to optimize allocation

Demo shown later3/29/2010

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Credit Portfolio: Individual default risk of an obligor can be better handled but not the dependence among default risks for several obligors

Need better estimation of credit risk of a portfolio and corresponding VaR, Expected Shortfall

Identify particular sector exposure and dependencies

Energy & Commodities Trading

Spread relationships dominate physical markets and asset hedging activities

Need dependence among various spreads: refinery crack, spark, storage time, geographical (shipping and pipelines)

A commodity or energy trader/quant would need to model these

Needs, Uses and Target Users

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Pricing of Credit Derivatives, Structured Products

First-to-default swap credit-linked products, CDOs, other exotic options etc.

Li made Gaussian Copula famous but needed to look beyond normality assumption {Recall : Formula that killed Wall Street}

Actuarial: Pricing of Life Insurance Products

Relationship between individuals' incidence of disease

Joint survival time distributions of multiple dependent life times

Reinsurance: e.g. Pricing of Sovereign Risk Products

Assess the risk of a large political risk reinsurance portfolio based on historical country risk ratings, sovereign ceilings, default rates and severity assumptions

Needs, Uses and Target Users

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AdvantagesUnderstanding of dependence at a deeper levelHighlight the fallacies and dangers of dependence

only on correlation

Copulas are easily simulated: allow Monte Carlo studies of risk

Express dependence on quantile scale: useful for describing dependence of extreme outcomes

VaR and Expected Shortfall express risk in terms of

quantiles of loss distributions

Allow fitting to MV risk factor data ; separate problem into 2 stepsFinding marginal models for individual risk factors Copula models for their dependence structure

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Most Common Marginal DistributionsMarket portfolio returns

Generalized hyperbolic (GH), or special cases such as:Normal inverse Gaussian (NIG)Student t and Gaussian

Credit portfolio returns

BetaWeibull

Insurance portfolio returns and operational risk

ParetoLog-normalGamma

All (and more) can be handled by “Statistics Toolbox” Model return time series with “Econometrics Toolbox”: ARMA

& GARCH3/29/2010

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Copula Functions in MATLABMATLAB’s “Statistics Toolbox” has many copula related

functions and capabilities

Probability density functions (copulapdf) and the cumulative distribution functions (copulacdf)

Rank correlations from linear correlations (copulastat) and vice versa (copulaparam)

Random vectors (copularnd)

Parameters for copulas fit to data (copulafit)

Available Copulas: Gaussian, Student -t and 3 bivariate Archimedean: One parameter families defined directly in terms of their cdfs: Clayton , Frank, Gumbel

Combined with related toolboxes (Econometrics, Optimization, Financial etc.) MATLAB provides comprehensive, unique platform for risk modeling

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Demo Case Study: Joint Extreme EventsFor 1/1/1996 to 12/31/2000 daily (1262) logarithmic returns

Xt = (Xt1, …Xt5) for 5

stocks (MSFT, GE, INTC, AAPL, IBM), interested in probability: P[X1 ≤ qa(F1),……, X5 ≤ q a (F5)] for a = 0.05

using four different models

MV normal distribution N5( m, S ) calibrated via sample

mean vector and covariance matrix

Gaussian copula CPGa calibrated by estimating P via rank

correlation

Student-t copula Cnt calibrated via covariance matrix and

degrees of freedom

Clayton copula CqCl calibrated by MLE for 5 dimensional

Clayton Copula3/29/2010

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Demo Case Study Results (Probabilities) Model 1): CR

Ga (a,....,a) = 0.035%

Model 2): CPGa (a,….,a) = [0.062%; 0.066%] (for Kendall's or

Spearman's method used to estimate P)

Model 3): Cnt (a,….,a) = 0.162% for n = 4

Model 4): CqCl (a,….,a) = 0.25% for q = 0.465

Comparing these with historical frequency of event in 1996 - 2000 period

qi is a-quantile under empirical distribution of Xi (for a = 0.05 and n = 1262 the qi is the 64th smallest of observations X1i, ..…, Xni), phist = 0.158%

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Demo Case Study SummaryEstimating probability by simple MV normal

distribution underestimates by factor of 4.5

Improvement using Gaussian copula via Spearman or Kendall’s tau Rank correlation

Student – t copula gives the closest match with empirical probability

Clayton copula : Best copula for modeling lower tail dependence

MATLAB function for MLE parameter estimation for Clayton Copula would be a good addition

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Possible Extension & Improvements in Functionality: Future opportunities for demos by AEDependencies among any kind of asset returns can be

modeled using Copulas: Enable risk modeling and estimation, portfolio optimization and allocation, pricing Equities, Indices, Options

Fixed Income, Credit products, Structured products

Hedge funds and other alternatives

Commodities, Electricity

Insurance

Macroeconomic Relationships

Some examples in literature already exist

A separate “Risk Management” toolbox or visible added related functionalities in Financial or Econometrics toolbox

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Alternatives

R is the only other package that has most of the Copula functionalities

Limited copula capabilities in Mathematica

Can work in others: C++, Java etc. but need to build all functions/routines from scratch

MATLAB has many advantages as discussed in following slides

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MATLAB vs. RPower/friendliness of user interfaces and documentation

of MATLAB and R is light years apart

MATLAB has a really mature GUI, help and documentation well laid out and browsable, for R need to search through multiple pages for simple task

MATLAB, unlike R, has a working debugger, tool to find syntax errors and suggest improvements, a file dependency checker

No Standards and Control for R: CRAN package repository features 2274 available packages

Study by D. Knowles, U. Cambridge using (MATLAB R2008b &

R 2.8.0 with Intel Core 2- 1.86 GHz processor, 4 GB RAM) showed MATLAB at par or better than R in speed

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MATLAB allows to develop complete models & applications with other toolboxes

Example: Marginal distributions of an asset may require GARCH modelingImage taken from one of the recorded webinar at MathWorks site

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Deployment with multiple platforms possible

Image taken from one of the recorded webinar at MathWorks site

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Selected References Bouy’e et al., Copulas for Finance, A Reading Guide and Some

Applications, July 2000 Claudio Romano, Applying Copula Functions to Risk Management, part of

PhD Thesis Trivedi & Zimmer, Copula Modeling: An Introduction for Practitioners,

Foundations & Trends in Econometrics, 1(1) (2005) 1–111 Schuermann, Integrated Risk Management in a Financial Conglomerate,

http://nyfedeconomists.org/schuermann/

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Backup: In Mathematical TermsFor an m-variate function F, copula associated with F is

distribution function C : [0, 1]m → [0, 1] that satisfies F(y1, . . . , ym) = C(F1(y1),...,Fm(ym);θ)where θ is a parameter of the copula called the dependence

parameter, which measures dependence between the marginals.

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