correlation and credit risk presentation to cas / soa erm symposium july 30, 2003
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Correlation and Credit Risk presentation to CAS / SOA ERM Symposium July 30, 2003. John A. Dodson / American Express Financial Advisors / [email protected]. Portfolio Credit Risk. - PowerPoint PPT PresentationTRANSCRIPT
Preliminary & Confidential – For discussion purpose only
Correlation and Credit Riskpresentation to CAS / SOA ERM Symposium July 30, 2003
John A. Dodson / American Express Financial Advisors / [email protected]
04/19/23 2
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, translation-invariant, 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
N.B.: For a normal distribution, these converge for high confidence
1
,,Pr tttttt VaR
1
,,,1, E ttttttttttt VaRES
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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
Default correlations in this context are interpreted to mean correlations amongst intensity innovations[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
1,0~exp0
Utdhi
it
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Default Intensity Correlations
Drift and Volatility are deterministic functions K (imperfectly) correlated, systematic factors (contagions) l’s are entries of Cholesky matrix L, s.t.
Elements of Rho are factor correlations
iK
j
jt
iiKt
iiiit hdWldWldtdh
jiK 01
:
TKK LLI0
0
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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:
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
11 ULUL i
i
1
1N
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Obligor-Specific Risk
Consider the case of K=1 (one systematic factor)
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)
iti
iti
iiit hdWdWdtdh 0
112 :1
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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
So for K independent systematic factors,
22
1
11
N
NN
2K
N
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Heuristic Observations
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
2KN
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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