Correlation and Credit Risk
presentation to CAS / SOA ERM Symposium July 30, 2003
John A. Dodson / American Express Financial Advisors / john.a.dodson@aexp.com
Preliminary & Confidential – For discussion purpose only
Portfolio Credit Risk

Generally, asset portfolio gains / losses related to credit
rating migration and impairment through default over time

Focus here on distribution of cumulative aggregate default
losses over a single fixed horizon (E.g. 99 % upper
confidence over 1 year)

Expected Loss (EL) and Unexpected Loss (UL)

Unexpected Loss can be a basis for Economic Capital
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Note: “Coherent” Measures of Risk

Paradox of Default Value-at-Risk (CVaR): diversification
may increase CVaR.

VaR is not “coherent” (Artzner): monotonic, translationinvariant, positive-homogenous, sub-additive

Expected Shortfall (ES) over a single horizon is coherent –
multi-horizon coherent measure remains an open
problem…
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Unexpected Loss Measures

Value-at-Risk

Expected Shortfall
  Prt t ,t t  VaRt1,t 
ES
1
t , t

1
t , t
 E t   t ,t  t  t ,t  t  VaR

N.B.: For a normal distribution, these converge for high confidence
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Credit Default Models

Intensity Models: default rates processes are continuous
– Duffie-Singleton 1996 (including RMG CreditMetrics™)
– Defaults are “inaccessible”
– Correlations are based on empirical factor analysis

Structural Models: asset/liab. processes are continuous
– Merton 1974 (including KMV & RMG CreditGrades™)
– Default are “accessible”
– Correlations come through equity prices
N.B.: There has been recent work to unify these
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Intensity Models for Default


Hazard rates for each obligor are positive, continuous
stochastic processes
Default occurs at a stopping time defined by


exp    hti dt   ~ U 0,1
 0

Default correlations in this context are interpreted to mean
correlations amongst intensity innovations
i

[exercise: prove these correlations carry over to the default indicator processes.]
N.B.: Note that this model is amenable in the actuarial context to multi-factor CIR
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Default Intensity Correlations
K
i

K i
i
j
dh   dt    l K i dWt   l j dWt  : h0i
j 1


i
t



i
i
Drift and Volatility are deterministic functions
K (imperfectly) correlated, systematic factors (contagions)
l’s are entries of Cholesky matrix L, s.t.
 K K  0 

  L  LT
I
 0

Elements of Rho are factor correlations
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Special Cases and Extensions

Note that if K=0 (no correlations), it can be proven that the
expectations of each default probability are sufficient
statistics for the aggregate cumulative loss distribution
over a single horizon; that is, UL from default is
independent of UL from migration (as driven by the
stochastic intensities)

Also note that the loss distribution is guaranteed to be
multi-normal for deterministic volatilities. Stochastic
volatility is the gateway to the general copula approach
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Effective Diversity

Consider the case of a portfolio of N equal holdings of
independent defaultable assets with equal expected default
frequencies and equal default severities.

The single horizon cumulative aggregate loss is simply
proportional to a Binomial(N, p) random variable

The UL in this case (by whatever measure) as a proportion
of the portfolio value serves as a benchmark indicating
effective diversity N
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Example

Consider a portfolio with individual loss distributions
independent but not all equal; in particular consider a
geometric scheme:
UL i   i 1  UL1

In this case, in the limit as the number of obligors goes to
infinity (Central Limit Thm), one can show that the
effective diversity is
1 
N
1 
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Obligor-Specific Risk

Consider the case of K=1 (one systematic factor)

dhti   i dt   i  1  i2 dWt1i  i dWt1


:
h0i
The correlation between the intensities of two obligors is
simply the product of the two systematic correlations
Hypothesis: systematic correlation should be an increasing
function of firm size (E.g., systemic risks)
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Effective Diversity & Specific Risk


In the event that the systematic correlations are taken to be
identical for all obligors, the general effect this correlation
has is to reduce the effective diversity
For the benchmark case this effect is
N
1
N
 2
2
1  N  1  


So for K independent systematic factors,
N
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K
2
12
Heuristic Observations
N  K 2

Consider again the final heuristic result:

The effective value of K is driven to some extent by the
correlations amongst the systematic factors; but the
obligor-specific risk (as determined by the systematic
correlation here) is the dominant factor
We know from simulation experiments that this is
especially true at high confidence
Unfortunately, determination of appropriate models of
obligor-specific risk is largely speculation


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Empirical Observations

If we assume that the factors driving default intensity are
stationary, we can impute effective diversity from the
variability of historical default rates

Using Moody’s data (mostly U. S.) from 1970, these are
– N ≈ 494 ± 147 for high grade issuers
– N ≈ 109 ± 16 for all issuers
– N ≈ 43 ± 5 for high yield issuers

But how can effective diversity depend on quality?
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Conclusions

How can effective diversity depend on credit quality?

The modeling choices are stark: We can either allow factor
loadings to depend explicitly on intensity levels, rendering
the problem intractable…
…or we can acknowledge stochastic factor volatilities and
enter the brave new world of copulas.
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Other Topics…

Valuation and Risk Measurement of Structured Credit
Securities using copulas (multi-period analysis)

Panjer’s Recursion and CSFB CreditRisk+ analytic model

Glasserman’s variance reduction techniques for Monte Carlo
simulation of correlated defaults
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