The devil is in the tails: Actuarial mathematics and the subprime mortgage crisis

1. The devil is in the tails: Actuarial mathematics and the subprime mortgage crisis

2. 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

3. 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”

4. What is securitization? Collateralized Debt Obligation “CDO” Pool of assets - Bank bundles different kinds of financial assets: mortgages and auto loans Special-PurposeVehicle “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

5. 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

6. “ 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,…,Fdbe univariatedistribution 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

7. Let default times (Ti) of d bonds in the underlying portfolio of some CDO. Using Fidenote 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

8. 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

9. 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 companyi 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

10. The idea is that default by company i occurs if the asset value falls below some threshold value • The default time Tiis 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

11. 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

12. 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

13. 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

14. 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.

15. 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+