THE DEVIL IS IN THE TAILS: ACTUARIAL MATHEMATICS AND THE SUBPRIME MORTGAGE CRISIS

Outline

Root of Subprime mortgage crisis Securitization Gaussian copula approach to CDO pricing Drawbacks of copula-based model in credit

risk Alternative approach to value CDO

Root of Subprime mortgage crisis

Root of crisis: The transfer of mortgage default risk from mortgage lenders to banks, insurance companies, and hedge funds.

This transfer was effected by a process called “securitization”

What is securitization?

Pool of assets

- Bank bundles different kinds of financial assets: mortgages and auto loans

Special-Purpose Vehicle

“SPV”- Mortgage repayments

were transferred to SPV

- SPV is bankruptcy-remote from the bank (default by the bank does not result in a default by the SPV)

- SPV uses repayments to pay coupons on CDO

Senior tranche- Highest priority to

receive coupon payment

- Last to bear losses

Mezzanine tranche- Lower priority to

receive coupon payment

- Second to bear losses

Equity tranche- Lowest priority to

receive coupon payment

- First to bear losses

Collateralized Debt Obligation “CDO”

CDO pricing

Key to value CDOs is modeling the defaults in the underlying portfolios Because coupon payments received by the

holders of CDO tranches depend directly on the defaults occurring in the underlying assets

If we can determine the distribution of the joint default time in the underlying portfolio, then we have a way to value CDO

Copula-based approach is used to model the defaults in underlying portfolios

“ Li Model” - Gaussian Copula approach to CDO pricing

Using copulas allows us to separate individual behavior of marginal distributions from their joint dependency on each other

Let C be a copula and F1,…, Fd be univariate distribution functions H(x1 ,…,xd) := C(F1(x1),…, Fd(xd)), (x1,…, xd) є Rd

The function H is a joint distribution function with margins F1,…, Fd

Let default times (Ti) of d bonds in the underlying portfolio of some CDO. Using Fi denote distribution function of default time Ti for i = 1,…,d

The Li copula approach is to define the joint default time as:

Pr[T1 ≤ t1 ,…, Td ≤ td ] : C (F1(t1),…, Fd(td)),

(t1,…, td) є [0,∞)d

where C is a copula function

In practice, the Li model is used within one-factor or multi-factor framework.

Suppose the d bonds in the underlying portfolio of the CDO have been issued by d companies

Under the one-factor framework, it is assumed that

i , for i = 1,…,d

where and , 1,… d are independent standard normally distributed random variables

Zi - asset value of company i

Z - random variable represents a market factor which is common to all companies

i - random variable represents the factor specific to company i

– correlation between asset values of each pair of companies

The idea is that default by company i occurs if the asset value falls below some threshold value

The default time Ti is related to the one-factor structure by the relationship:

-1 (Fi(Ti ))

With above relationship, the joint df of default times is:

Pr[ T1 ≤ t1 ,…, Td ≤ td ] : C (F1(t1),…, Fd (td))

Once choosing the marginal dfs (Fi), the one-factor Li model is fully specified

Advantages of copula-based model in credit risk

Enable fast computations and easy to calibrate since only pairwise correlation needs to be estimated

Relies on assuming all assets in the underlying portfolio have the same pairwise correlation

However, simplicity and ease of use typically comes at a price – three main drawbacks

1. Inadequate modeling of default clustering

Does not adequately model the occurrence of defaults in the underlying portfolio

During crisis, corporate defaults occur in clusters – if one company defaults then it is likely that other companies also default within a short time period

However, under Gaussian copula model, company defaults become independent as their size of default increase

Asymptotic independence of extreme events for Gaussian carries over to the asymptotic independence for default times – this is not desirable to model defaults which cluster together

Model is based on normal distribution which is easy to understand and resulting in fast computation. But it fails to model the occurrence of extreme events in the tail

2. Inconsistent correlation in tranches

An implied correlation is calculated for each CDO tranche – this is the correlation which makes tranche market price agree with the Gaussian copula model

Using implied correlation to find delta for each tranche Delta measures sensitivity of tranches to

uniform changes in the spreads in underlying portfolio

We expect implied correlation should be the same for each tranche, since it is a property of underlying portfolio. But, Gaussian model gives different implied correlation for each tranche

Also, implied correlations do not move uniformly together – equity tranche can increase more than mezzanine tranche

3. Ability to do stress-testing

Use of copula reduces ability to test systemic economic factors (insufficient macroeconomic modeling)

It does not model economic reality. It is a mathematical structure which fits historical data

When extreme market conditions reign, time dependent model is not powerful

Copula technology is highly useful for stress-testing for portfolios where marginal loss information is available (e.g.: multi-line none-life insurance)

But fail to capture dynamic events in fast-changing markets because there is no natural definition for stochastic process.

Alternative approaches to value CDOs

Two main classes of models for credit risk modeling Structural models

Firm-value approach – models default via dynamics of firm value

Model default via relationship of firm’s assets value to its liabilities value

Default occurs if assets value is less than liabilities value

e.g: One-factor Gaussian copula Hazard rate models

Model the infinitesimal chance of default In these models, the default is some exogenous process

which does not depend on the firm e.g: CreditRisk+