On the modelling of default correlation using copula

functions

Econophysics Final Work. Master in Computational Physics, UB-UPC 2011.

Oleguer Sagarra PascualJune 2011

Quick Review of ContentsIntroduction: The Risk-Credit based Trading

Securizing the Default Risk:

CDS

CDO’s

Modelling the Default Risk:

Assumptions

Individual Default: Credit Curves

Correlated Default: Copula Approach

Pricing the Risk: Li’s Model

Simulating the model: Results

Criticism

Risk-Credit Based Trading IBefore... (Traditional Banking)

Investor puts Money on Bank

Borrower ask money to the Bank

Bank evaluates the Borrower, lends money (takes a risk, or not!) and charges him a penalty, that is returned to the investors.

Key Point: Good Credit Risk assessment. If Borrower defaults (fails to pay), Bank loses money.

Irruption of Derivatives: We can trade with everything!

Why not trade with risk? Securization

Now the Bank sells the risk from the Borrower to an Insurer.

Borrower defaults : Insurer pays a penalty

Borrower pays: Insurer gets payed a periodic premium for assuming the risk.

Advantages:

SPV: Outside the books. No taxes. Capital freed. Allows more Leverage.

Macro-Economic mainstream: “Good: It diffuses risk on the system” (?¿!)

Magic: Bank is risk safe ? No, because the it is doubly exposed to default: By Insurer and/or by Borrower!

Please Remember: More Risk = More Premium = More Business! (Or at least until something goes wrong...). And the banks no longer care about risk... they are “insured”!

Risk-Credit Based Trading II

Individual assets subject to Credit Default events:

Mortgages, Student Debts, Credit Card...

CDS (Credit Default Swaps)

Securizing the Default Risk I

Key Point: Probability S(t) of an asset to survive to time t.

Please note: One can generate many CDS contracts from the same asset! = More volume

As in all derivatives: Cheaper than assets!

Some figures...

Starts in the 90’s: 100 billion* $ by the end of 1998.

Booms on the new millennia**: 1 trillion $ in 2000, 60 trillion $ by 2008.

Securizing the Default Risk II

* 1 American Billion=1000 Million1 American Trillion= 10 000 Million **Li’s first paper appears on 1999

One step beyond: Collateralized Debt Obligations (CDO’s)

Take N default-susceptible assets and pool them together in a portfolio.

Tranche the pool and sell the risk:

Senior: (Low risk: 80%) AAA

Mezzanine: (Med Risk: 15%) BBB

Equity: (High Risk: 5 %) Unrated

Securizing the Default Risk III

Rating becomes independent of the subjacent assets

Securizing the Default Risk IV

Key Point: Joint Probability S(t1,t2,t3...) of survival to k-th default of correlated assets.

Assumptions:

Market is fair : The prices are “correct”.

Market is efficient: Information is accessible to determine evolution of market.

Procedure:

Model individual default probabilities (Marginals)

Model joint default probabilities

Problem: Solution is not unique, if the assets are correlated!

Modelling the Default Risk I

Individual Default Modelling:

3 approaches:

Rating agencies + Historical data

Merton approach (stochastic random walk)

Current Market Data approach

Definitions:

S(t)= 1- F(t) : Survival Function to time t.

h(t) : Hazard Rate Function. Proba of defaulting in the interval [t,t+dt].

Modelling the Default Risk II

We can easily solve this using B.C: (S(0)=1, S(inf)=0)

Modelling the Default Risk III

Assuming h(t) piecewise constant function*,

And the problem is solved (assuming we are able to construct h(t)).

*h(t): Stochastic nature. But in Li’s model is piecewise constant

Joint Default Modelling:

Copula Approach : Characterise correlation of variables with the copula (independently of marginals)

Modelling the Default Risk IV

Problem not unique: Many families of copulas exist

Important feature: Tail Coefficient (extreme events*)

Modelling the Default Risk V

Two Examples: Gaussian (Li’s Model), T-Student

* Such as crisis

Modelling the Default Risk IV

Suppose we have a set of hazard rate functions {hi(t)}...

We generate a set of correlated {Ui=Ti(Ti)} using a copula.

We obtain joint default times via the transform {Ti=F-1

(Ui)}.

Once we have that, it is simple to derive the fair price of the CDO/CDS contract using no-arbitrage arguments.

Pricing the Default Risk I

Li’s procedure:

Infer h(t) piecewise constant from the market for each price, based on the price of the CDS contracts at different maturities T (expiring times).

Determine 1-Factor ρ from market data using ML methods.

Use 1-Factor Gaussian Copula* to generate default times via MC simulation and obtain prices for CDO averaging.

Pricing the Default Risk II

* Extreme additional assumption: Pairwise correlation is constant between assets.

Weaknesses:

Unrealistic assumption for h(t), ρ.

Bad characterisation of extreme events

Massive presence of Bias: Relied on data from CDS, priced from other CDS!

Strengths:

Simple, computationally easy. Few parameters to estimate.

So... (almost) everybody used it!

Pricing the Default Risk III

We apply two tests to both the Student and Gaussian Copula:

Error spread: We apply 5% random errors to both ρ and h(t).

We simulate a crisis, with a h(t) non piece-wise function.

Simulating the model: Results and Criticism I

Relevant Magnitudes:

Mean default time

Mean survival rate

Extreme events: Probability of k-assets defaulting

Times to k-th default

Simulating the model: Results and Criticism II

h(t) functions used:

Piece-wise constant function

Simulating the model: Results and Criticism III

Continuos function with random normal noise

Error check:Simulating the model: Results and Criticism IV

Good convergence as N grows.

Small differences between two copulas.

ρ key factor on convergence.

Number of k-th defaults

Simulating the model: Results and Criticism VI

Increasing ρ clusters events

Small differences between two copulas.

k-th time default

Simulating the model: Results and Criticism VI

h(t) effect is more important than the copula.

In fact, correlation might be included twice in the model.

Mean k-th defaults are more clustered in the Student copula.

Criticism:Theory strongly dependent on h(t).

Is it possible to estimate h(t) from market data?

Reduces correlation to a single factor.

Modelling: Inability to do stress testing.

Inadequate usage of Mathematical/Econophysics formulas.

Very quantitative results. Inconclusive results.

Feedback: Bubble Effect.

Complete fail to reproduce fat tails (extreme events)

A question arises: Could all this have been avoided ?

“All Models are Wrong but some are useful” (George P. Box 1987)