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Use of Copulas for risk management and modeling via MATLAB

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  • 1. MATLABProducts forFinancial Risk Management & ModelingUse of COPULAS
    ANURAG JAIN

2. Case Study Topic: Copulas in Risk Management
Demo: Equity Portfolio Risk Management using Copulas

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

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
3/29/2010
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3. Needs, Uses and Target Users
Returns in real world are not normal, simple Pearson correlations dont 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 later
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4. Needs, Uses and Target Users
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
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5. Needs, Uses and Target Users
Pricingof 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
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6. Advantages
Understanding of dependence at a deeper level
Highlight 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 steps
Finding marginal models for individual risk factors
Copula models for their dependence structure
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7. Most Common Marginal Distributions
Market portfolio returns
Generalized hyperbolic (GH), or special cases such as:
Normal inverse Gaussian (NIG)
Student t and Gaussian
Credit portfolio returns
Beta
Weibull
Insurance portfolio returns and operational risk
Pareto
Log-normal
Gamma
All(and more) can be handled by Statistics Toolbox
Model return time series with Econometrics Toolbox: ARMA & GARCH
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8. Copula Functions in MATLAB
MATLABs 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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9. Demo Case Study: Joint Extreme Events
For 1/1/1996 to 12/31/2000 daily (1262) logarithmic returnsXt = (Xt1, Xt5) for 5 stocks (MSFT, GE, INTC, AAPL, IBM),interested in probability: P[X1 qa(F1),, X5 qa (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 CPGacalibrated by estimating P via rank correlation
Student-t copula Cntcalibrated via covariance matrix and degrees of freedom
Clayton copula CqClcalibrated by MLE for 5 dimensional Clayton Copula
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10. Demo Case Study Results (Probabilities)
Model 1): CRGa(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 (fora = 0.05 and n = 1262 the qi is the 64th smallest of observations X1i, .., Xni),
phist = 0.158%
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11. Demo Case Study Summary
Estimating probability by simple MV normal distribution underestimates by factor of 4.5
Improvement using Gaussian copula via Spearman or Kendalls 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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12. Possible Extension & Improvements in Functionality:Future opportunities for demos by AE
Dependencies 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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13. 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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14. MATLAB vs. R
Power/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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15. MATLAB allows to develop complete models & applications with other toolboxes
Example: Marginal distributions of an asset may require GARCH modeling
Image taken from one of the recorded webinar at MathWorks site
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16. Deployment with multiple platforms possible
Image taken from one of the recorded webinar at MathWorks site
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17. Selected References
Bouye 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 & TrendsinEconometrics, 1(1) (2005) 1111
Schuermann, Integrated Risk Management in a Financial Conglomerate, http://nyfedeconomists.org/schuermann/
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18. Backup: In Mathematical Terms
For 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);)
where is a parameter of the copula called the dependence parameter, which measures dependence between the marginals.
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