risk aggregation: copula approach ken seng tan, ph.d., asa, cera canada research chair in...

Risk Aggregation: Copula Approach

Ken Seng Tan, Ph.D., ASA, CERA

Canada Research Chair in

Quantitative Risk ManagementApril 18-19, 2009 [email protected]

[email protected] SOA CERA - EPP 2

Introduction The goal of integrated risk management in a

financial institution is to both measure and manage risk and capital across a diverse range of activities in the banking, securities, and insurance sectors

This requires an approach for aggregating different risk types, and hence risk distributions a problem found in many applications in finance

including risk management and portfolio choice.


Risk Aggregation

Risk Driver


Topics

1. Measures of Association

2. Copulas

3. Which Copula to Use?

4. Applications

5. Concluding Remarks


Topic I Measures of Association

Comovements (or dependence) between variables Pearson correlation

Its potential pitfalls Comonotonic risks Rank correlations Tail dependence

Copulas Which Copula to Use? Applications Concluding Remarks


Pearson Correlation of Coefficient: ρ(X,Y)

Most common measure of dependence Definition:

Properties: -1 ≤ ρ(X,Y) ≤ 1 If X & Y are independent, then ρ(X,Y) = 0. If |ρ(X,Y)| = 1, then X and Y are said to be perfectly linearly

dependent X = aY + b, for nonzero a

linear correlation coefficient Invariant under strictly increasing linear transformations:

( , ) ( ) ( ) ( )( , )

( ) ( ) ( ) ( )

Cov X Y E XY E X E YX Y

Var X Var Y Var X Var Y

, , , 0, 0aX b cY d X Y a c


Is Pearson Correlation a Good Measure of Dependence?

Var(X) and Var(Y) must be finite Problems with heavy-tailed distributions

Possible values of correlation depend on the marginal (and joint) distribution of X and Y All values between -1 and 1 are not necessary attainable

Perfectly positively (negatively) dependent risks do not necessarily have a Pearson correlation of 1 (-1)

Correlation is not invariant under non-linear transformations of risks


Example: Attainable Correlations Suppose X ~ N(0,1), Y ~ N(0,σ2) For a given ρ(X ,Y), what can you say about ρ(eX , eY)?

nonlinear transformation eX and eY are lognormally distributed

Now assume σ = 4: If ρ(X ,Y) = 1

ρ(eX , eY) = 0.01372 = ρ(eZ , eσZ) for Z ~ N(0,1),

If ρ(X ,Y) = -1 ρ(eX , eY) = ρ(eZ , e-σZ) = -0.00025

This implies for -1≤ ρ(X ,Y) ≤ 1 -0.00025 ≤ ρ (eX , eY) ≤ 0.01372

Implications?


What can we conclude from the last example?

Pearson correlation is an effective way to represent comovements between variables if they are linked by linear relationships, but it may be severely flawed in the presence of non-linear links

Need better measures of dependence!


Comonotonic risks

Comonotonicity is an extension of the concept of perfect correlation to random variables with non-linear relations.

Two risks X and Y are comonotonic if there exists a r.v. Z and

increasing functions u and v such that X = u(Z) and Y = v(Z) X and Y are countermonotonic if u increasing and v decreasing, or

vice versa.

Example I: Last example with X ~ N(0,1) and Y ~ N(0,σ2) eX & eY are comonotoic when ρ(X,Y) = 1 eX & eY are countermonotoic when ρ(X,Y) = -1

Example II: ceding company’s risk and reinsurer’s risk Not perfectly (linearly) dependent but they are comonotonic


Rank Correlations

Non-parametric (or distribution-free) measures of association by looking at the ranks of the data Only need to know the ordering (or ranks) of the

sample for each variable and not its actual numerical value

Does not depend on marginal distributions Invariant under strictly monotone transforms

Two variants of rank correlation: Kendall’s Tau (ρτ) Spearman’s Rho (ρS)


Properties of Rank Correlations

Rank correlation measures the degree of monotonic dependence between X and Y, whereas linear correlation measures the degree of linear dependence rank correlations are alternatives to the linear correlation coefficient

as a measure of dependence for nonelliptical distributions

1 , 1 and 1 ( , ) 1SX Y X Y

, ,

If & are comonotonic 1 1

If & are countermonotonic 1 1

If & are independent 0 0

SX Y X Y

X Y

X Y

X Y


Coefficients of Tail Dependence

In risk management, we are often concerned with extreme values, particularly their dependence in the tails

The concept of tail dependence relates to the amount of dependence in the upper-right-quadrant or lower-left-quadrant tail of a bivariate distribution Provide measures of extremal dependence A measure of joint downside risk or joint upside potential A bivariate distribution can have either

upper tail dependence, or lower tail dependence, or both, or none (i.e. tail independence)


Simulated Samples of Some Bivariate Distributions


Topic II

Measures of Association

Copulas What is a copula?

Key results Some examples of copulas Other properties of copulas

Which Copula to Use? Applications Concluding Remarks


Basic Copula Primer Copulas provide important theoretical insights and practical

applications in multivariate modeling The key idea of the copula approach is that a joint distribution

can be factored into the marginals and a dependence function called a copula. The dependence structure is entirely determined by the copula

Using a copula, marginal risks that are initially estimated separately can then be combined in a joint risk (or aggregate) distribution that preserves the original characteristics of the marginals. facilitate a bottom-up approach to multivariate model building; Given marginal distributions, the joint distribution is completely

determined by its copula.


Basic Copula Primer (cont’d) This implies that the multivariate modeling can be

decomposed into two steps: Define the appropriate marginals and Choose the appropriate copula

The separation of marginal and dependence is also useful from a practical (or calibration) point of view;

Copulas express dependence on a quantile scale allow us to define a number of useful alternative

dependence measures useful for describing the dependence of extreme

outcomes the concept of quantile is also natural in risk

management (e.g. VaR)


What is a Copula? A copula C is a

multivariate uniform distribution function (d.f.) with standard uniform marginals

We focus on bivariate case

Bivariate copula:

C(u,v) = Pr( U ≤ u, V ≤ v )

where U, V ~ Uniform(0,1)

Connection between (bivariate) d.f., marginals and copula function:

, ,

Pr ,

Pr ,

Pr ,

,

X Y

X X Y Y

X Y

X Y

F x y

X x Y y

F X F x F Y F y

U F x V F y

C F x F y

, , ,X Y X YF x y C F x F y


Does such a copula always exist?


Mathematical Foundation: Sklar’s Theorem (1959)

Suppose X and Y are r.v. with continuous d.f. FX & FY. If C is any copula, then

is a joint d.f. with marginals FX & FY.

Conversely, if FX,Y(x,y) is a joint d.f. with marginals FX & FY , then there exists a unique copula C such that

Key result: Decomposition of multivariate d.f.

Marginal information is embedded in FX & FY and the dependence structure is captured by the copula C(·,·)

, X YC F x F y

F

X ,Yx, y C FX x , FY y


Examples of Copula:

,

1 12

Independence copula:

, , ,

Gaussian copula (or Normal copula) with correlation :

( , ) ( ( ), ( ); )

where

X Y X Y X Y

Ga

C u v u v C F x F y F x F y F x y

C C u v u v

2

1

( , , ) is the std bivariate normal d.f. with correlation

and is the inverse of the standard normal d.f.

Student copula with degree of freedom and correlation :

t

1 1, ,

1

, ,

where : inverse of the univariate Student d.f. with degrees of freedom

Gumbel copula, Clayton copula, Frank copula, etc ...

t tv v v v

v

C C u v C t u t v

t t v



Copulas: Gaussian vs Student t


Other Properties of Copulas

1. Flexibility Useful when “off-the-shelf” multivariate distributions

inadequately characterize the joint risk distribution

2. Easy to simulate3. Invariant property4. Dependence measures

Offers important insights to modeling dependence via

a) rank correlations b) tail dependence


(1) Flexibility:

The power of copula lies in its flexibility in creating multivariate d.f. via arbitrary marginals Useful when “off-the-shelf” multivariate distributions inadequately

characterize the joint risk distribution

see Jouanin, Riboulet and Roncalli (2004)

Example: In credit risk modeling, the

default time may be modeled as X1 ~ Inverse Gaussian

The recovery rate may be modeled as X2 ~ Beta

Interested in the joint distribution of X1 and X2



2) Easy to simulate

Offers Monte Carlo risk studies risk measures economic capital stress testing …

Simulated samples: Gaussian copula

ρ = 0.7 Gumbel copula:

θ = 2.0 Clayton copula:

θ = 2.2 t-copula:

v = 4 ρ =0.71


3) Invariant Property:

Invariant under strictly increasing transformations of the marginals Let C be a copula for X & Y, If g(.) and h(.) are strictly increasing functions Then C is also the copula for g(X) and h(Y)

This is due to the fact that copula relates the quantiles of the two distributions rather than the original variables

Example: Consider two standard normals X & Y and let their dependence

be represented by the Gaussian copula. Under increasing transforms, eX & eY still have the Gaussian

copula Useful with confidentiality of banks’ or insurers’ data.

Copulas can be estimated even if data is transformed appropriately.



4a) Kendall’s Tau and Spearman’s Rho via Copula

1 1

0 0

1 1

0 0

Both rank correlations depend only on the (unique) copula:

( , ) 4 ( , ) ( , ) 1

( , ) 12 ( , )

Invariant under monotonic transformation

Ga

S

X Y C u v dC u v

X Y C u v uv dudv

ussian copula:

2 6 , arcsin & , arcsin

22

copula: , arcsin (independent of the d.f.)

Useful for fitting copulas to data

SX Y X Y

t X Y


4b) Tail Dependence via Copula

Recall that tail dependence relates to the magnitude of dependence in the upper-right-quadrant or lower-left-quadrant tail of a bivariate distribution the joint exceedance (tail) probabilities at high (and low) quantiles examine tail dependence either for a fixed quantile or asymptotically.

1 1

Joint Exceedance Probability (for Upper Tail Dependence)

1 2 , Pr for quantile close to 1

1Y X

CY F X F

1 1

Joint Exceedance Probability (for Lower Tail Dependence)

, Pr for quantile close to 0Y X

CY F X F


Comparison of Tail dependence: Gaussian vs t copulas (std normal marginals)

copula parameters: =0.7, =3

quantiles lines (vertical and horizontal): 0.5% and 99.5%


Joint Exceedance Probabilities at High Quantitles

Joint exceedance probabilities are given for Normal copula For t-copula, we report the ratio of the joint exceedance

probabilities of t-copula to normal-copula From Table 5.2 of McNeil, Frey and Embrechts (2005)


Joint 99% (or equivalently 1%) Exceedance Probabilities in High Dimensions

Consider daily returns on five stocks with constant ρ = 0.5. Impact on the choice of copula?

Prob. on any day all returns are below 1% quantile

How often does such an event happen on average?

Gaussian 7.48 x 10-5 once every 53.1 years

t (4 d.f.) (7.48 x 10-5) x 7.68 once every 6.9 years


Asymptotic Tail Dependence

Limiting probability Asymptotic upper tail dependence is obtained by taking the limit

α-quantile 1 Asymptotic lower tail dependence is obtained by taking the limit

α-quantile 0 limiting probability > 0 implies tail dependence

Gaussian Asymptotic tail independence (ρ < 1)

t Asymptotic tail dependence (ρ > -1)

Gumbel Asymptotic upper tail dependence (θ > 1)

Clayton Asymptotic lower tail dependence (θ > 0)


Simulated Copulas with Standard Normal Marginals

In all cases, linear correlation is around 0.7

Gumbel copula: θ = 2.0

Clayton copula: θ = 2.2

t copula: v = 4 ρ =0.71


Topic III

Measures of Association Copulas

Which Copula to Use?

Applications Concluding Remarks


Which Copula to Use?

Parameter estimation One-step approach Two-step approach

Model validation Goodness-of-fit test

Kolmogorov-Smirnov test Anderson-Darling test …

Examine tail dependence

Model selection Principle of parsimony Akaike’s Information Criterion (AIC) Schwartz Bayesian Criterion (SBC)

Klugman, Panjer and Willmot (2008) Loss Models: From Data to Decisions.

Venter (2002) “Tails of Copulas” Genest, Remillard and Beaudoinc (in

press): “Goodness-of-fit tests for copulas: A review and a power study”

Given observed data set: { (x1,y1), …, (xT,yT) } how do we select a copula that reflects the underlying characteristics of the data?


Parameter Estimation: One-Step Approach

Direct Maximum Likelihood (ML) method Estimate jointly the marginals and the copula function

using the method of ML nC + nX + nY dimensions optimization problem

, ( , ) ,

# of parameters:

X Y X Y

C X Y

F x y C F x F y

n n n


Parameter Estimation: Two-Step Approach

Inference-functions for Margins (IFM) method Step 1:

for each risk factor, independently determine parametric form of marginal, say, using method of ML

nX parameters for 1st factor and nY parameters for 2nd factor Step 2:

given marginals, determine copula using method of ML nC dimensions optimization problem

Pseudo-likelihood method/Semi-parametric Approach Similar to IFM except that the marginals are the empirical cdf

Rank-correlation-based Method of Moments Calibrating copula by matching to the empirical rank correlations,

independent of marginals


Topic IV

Measures of Association Copulas Choosing the Right Copula

Applications

Concluding Remarks


Frees, Carriere, and Valdez (1996): “Annuity Valuation with Dependent Mortality”

Gompertz marginals (for both males and females) and Frank's copula are calibrated to the joint lives data from a large Canadian insurer.

The estimation results show strong positive dependence between joint lives with real economic significance.

The study shows a reduction of approximately 5% in annuity values when dependent mortality models are used, compared to the standard models that assume independence.


Klugman and Parsa (1999): “Fitting Bivariate

Loss Distributions with Copulas”

Calibrate Frank’s copula to the joint distribution of loss and allocated loss adjustment expense (ALAE) for a liability line using 1,500 claims supplied by Insurance Services Office.

Marginals: Examine a number of severity distributions Loss data: 2-parameter inverse paralogistic distribution ALAE: 3-parameter inverse Burr distribution

Discuss ML inference for copulas and bivariate goodness-of-fit tests

Frees and Valdez (1997) “Understanding relationships using copulas” Using similar data, they adopt Pareto marginals for both

distributions and consider Frank’s copula and Gumbel copula


Kole, Koedijk and Verbeek (2007): “Selecting Copulas for Risk Management”

They show the importance of selecting an accurate copula for risk management.

They extend standard goodness-of-fit tests to copulas. Using a portfolio consisting of stocks, bonds and real estate,

these tests provide clear evidence in favor of the Student's t copula, and reject both Gaussian copula and Gumbel copula. Gaussian copula underestimates the probability of joint extreme

downward movements, while the Gumbel copula overestimates this risk.

Gaussian copula is too optimistic on diversification benefits, while the Gumbel copula is too pessimistic.

These differences are significant. They also conclude that both dependence in the center and

dependence in the tails are important


Rosenberg and Schuermann (2006): “A general approach to integrated risk management with skewed, fat-tailed risks”

A comprehensive study of banks’ returns driven by credit , market, and operational risks

They propose a copula-based methodology to integrate a bank’s distributions of credit, market, and operational risk-driven returns.

Their empirical analysis uses information from regulatory reports, market data, and vendor data most of them are publicly available, industry-wide data

They examine the sensitivity of risk estimates to business mix, dependence structure, risk measure, and estimation method


Fig 2 of Rosenberg

and Schuermann (2006)


Rosenberg and Schuermann (2006) (cont’d)

Their findings: Given a risk type, total risk is more sensitive to

differences in business mix or risk weights than to differences in inter-risk correlations

The choice of copula (normal versus t ) has a modest effect on total risk

Assuming perfect correlation overestimates risk by more than 40%.

Assuming joint normality of the risks, underestimates risk by a similar amount


Concluding Remarks In this presentation,

we discussed various dependence measures, highlighted pitfalls with the commonly used linear correlation;

we introduced copula, particularly its role in modeling dependence and joint risk distributions;

we reviewed various ways of calibrating copula to empirical data;

we also examined some of its applications in insurance, finance, and risk management,


Concluding Remarks (cont’d) A quote from Embrechts (2008) “Copulas: A personal view” :

“… the question “which copula to use?” has no obvious answer. There definitely are many problems out there for which copula modeling is very useful. … Copula theory does not yield a magic trick to pull the model out of a hat.”

Nevertheless copula has some obvious advantages: the separation of marginals and dependence modeling is appealing, particularly

for problems with a large number of risk drivers it can still be a powerful tool, providing a simple way of coupling marginal d.f.

while inducing dependence Tail dependence is important, especially for risk management

“One of my probability friends, at the height of the copula craze to credit risk pricing, told me that “The Gauss–copula is the worst invention ever for credit risk management.” ” Embrechts (2008)

Numerous studies have supported the use of the t-copula, as opposed to the Gaussian copula

“All models are wrong but some are useful” George E.P. Box


References

P. Embrechts (2008) “Copulas: A personal view” www.math.ethz.ch/~embrechts/ A. McNeil, R. Frey, P. Embrechts (2005) “Quantitative Risk Management” Princeton

University Press. J. Yan (2007) “Enjoy the Joy of Copulas: With a Package copula”. Journal of Statistical

Software vol. 21 issue #4. Copula R package (freeware) cran.r-project.org C. Genest, B. Remillard, and D. Beaudoin “Goodness-of-fit tests for copulas: A review and a

power study” forthcoming in Insurance, Mathematics and Economics. E.W. Frees and E.A. Valdez (1997) “Understanding relationships using copulas” North

American Actuarial Journal 2(1):1-25 E.W. Frees, J. Carriere, and E.A. Valdez (1996) “Annuity Valuation with Dependent

Mortality.” Journal of Risk and Insurance, 63(2):229-261. J-F Jouanin, G. Riboulet and T. Roncalli (2004) “Financial Applications of Copula Functions”

in Risk Measures for the 21st Century editor G. Szego. E. Kole, K. Koedijk, and M. Verbeek (2007) “Selecting copulas for risk management”, J of

Banking & Finance 31:2405-2423. S.A. Klugman, H.H. Panjer, and G.E. Willmot (2008) Loss Models: From Data to Decisions.

3rd edition. Wiley. S.A. Klugman and R. Parsa (1999) “Fitting bivariate loss distributions with copulas”

Insurance: Mathematics and Economics 24:139-148. J.V. Rosenberg and T. Schuermann (2006) “A general approach to integrated risk

management with skewed, fat-tailed risks”, J of Financial Economics 79:569-614 G. Venter (2002) “Tails of Copulas”. Proceedings of the Casualty Actuarial Society, LXXXIX

2:68–113.

risk aggregation: copula approach ken seng tan, ph.d., asa, cera canada research chair in...

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