THE DEVIL IS IN THE TAILS:
ACTUARIAL MATHEMATICS
AND THE SUBPRIME
MORTGAGE CRISIS
Outline
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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
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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?
Collateralized Debt
Obligation “CDO”
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 bankruptcyremote 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
CDO pricing
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Key to value CDOs is modeling the defaults in the
underlying portfolios
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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
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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
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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
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In practice, the Li model is used within one-factor or multifactor framework.
Suppose the d bonds in the underlying portfolio of the
CDO have been issued by d companies
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Under the one-factor framework, it is assumed that
Zi = 𝜌 𝑍 + 1 − 𝜌 єi ,
for i = 1,…,d
where 𝜌 ∈ (0,1) and 𝑍, є1,… єd 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
єi - random variable represents the factor specific to
company i
𝜌 – correlation between asset values of each pair of
companies
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The idea is that default by company i occurs if the asset
value Zi falls below some threshold value
The default time Ti is related to the one-factor structure
by the relationship:
Zi = Φ-1 (Fi(Ti ))
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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
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Enable fast computations and easy to calibrate since only
pairwise correlation 𝜌 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
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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
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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
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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
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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
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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+
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