Criticism on Li's Copula Approach
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On the modelling of default correlation using copula
functions
Econophysics Final Work. Master in Computational Physics, UB-UPC 2011.
Oleguer Sagarra PascualJune 2011
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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
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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.
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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
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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.
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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
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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
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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.
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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
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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
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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
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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
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Important feature: Tail Coefficient (extreme events*)
Modelling the Default Risk V
Two Examples: Gaussian (Li’s Model), T-Student
* Such as crisis
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Modelling the Default Risk IV
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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
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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.
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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
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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
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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
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h(t) functions used:
Piece-wise constant function
Simulating the model: Results and Criticism III
Continuos function with random normal noise
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Error check:Simulating the model: Results and Criticism IV
Good convergence as N grows.
Small differences between two copulas.
ρ key factor on convergence.
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Number of k-th defaults
Simulating the model: Results and Criticism VI
Increasing ρ clusters events
Small differences between two copulas.
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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.
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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)
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A question arises: Could all this have been avoided ?
“All Models are Wrong but some are useful” (George P. Box 1987)