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Aj Copulas V4


Use of Copulas for risk management and modeling via MATLAB

Use of Copulas for risk management and modeling via MATLAB

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  • 1. MATLAB® Products for Financial Risk Management & ModelingUse of COPULAS
  • 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. 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 later
  • 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
  • 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
  • 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
  • 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
    Insurance portfolio returns and operational risk
    All (and more) can be handled by “Statistics Toolbox”
    Model return time series with “Econometrics Toolbox”: ARMA & GARCH
  • 8. Copula Functions in MATLAB
    MATLAB’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
  • 9. Demo Case Study: Joint Extreme Events
    For 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 ≤ 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 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 Copula
  • 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 (for a = 0.05 and n = 1262 the qi is the 64th smallest of observations X1i, ..…, Xni),
    phist = 0.158%
  • 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 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
  • 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
    Macroeconomic Relationships
    Some examples in literature already exist
    A separate “Risk Management” toolbox or visible added related functionalities in Financial or Econometrics toolbox
  • 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
  • 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
  • 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
  • 16. Deployment with multiple platforms possible
    Image taken from one of the recorded webinar at MathWorks site
  • 17. 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 & TrendsinEconometrics, 1(1) (2005) 1–111
    Schuermann, Integrated Risk Management in a Financial Conglomerate,
  • 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);θ)
    where θ is a parameter of the copula called the dependence parameter, which measures dependence between the marginals.