CIR 2 - Modeling Correlated Defaults Flashcards
Two main Credit Default models analyzed in Blum 2
- Bernoulli Model (L): Moody’s KMV / Risk Metrics / Most Internal Bank Models
- Poisson Model (L’ ): CreditRisk+
Total defaults for Bernoulli loss statistic L
Loss Percentage for Bernoulli loss statistic L
Properties of Bernoulli Loss Statistic
Meaning of Uniform default probability p in (bernoulli loss statistic)
Means the probability of default of each counterparty is the same, pi = p for all i.
Overview of General Bernoulli Mixture Model
- Specifies explicit dependencies b/w counterparties L’s
- Loss probabilites are random variables w/ distribution F
- Conditional on P, L’s are independent
General Bernoulli Mixture Model distribution and moments
Covariance derivation under the General Bernoulli Mixture Model
Uniform Default Probability and Uniform Correlation
- called uniform portfolios
- Works best for portfolios with all exposures are of approximately the same size and type of risk
Correlation formula for uniform Bernoulli portfolio
Correlation cases for uniform Bernoulli portfolio
- A correlation of 0 happens if and only if there is no randomness at all regarding P. In this case, there is a binomial distribution with default probability p bar
- A correlation of 1 happens with the “rigid” behavior that either:
- All counterparties default
- Or all counterparties survive simultaneously
- Most realistic scenarios woud have a correlation strictly between 0 and 1
Stability of Poisson Models
The sum of independent poisson models is poisson where you just add the underlying parameter values
Overview of General Poisson Mixture Model
Correlation formula for uniform Poisson portfolio
Dispersion ratio of a random variable
- ratio of the variance devided by the mean
*
Overview of common credit models
Overview of Log-return model in Moody’s KMV and RiskMetrics
One-year default probability of firm i conditional on composite factor (Moody’s KMV and RiskMetrics)
Overview of Composite Factor in Log-return model (Moody’s KMV and RiskMetrics’)
- Composite factor captures the level of general market risk, or nonidiosyncratic
risk - Broken into indices (e.g. industry and country indices)
- Weighted average of inices
Overview of CreditRisk+
- analyzes risk factors by sector indexed by the variable s (in contrast with Moody’s which is a factor model)
- each sector has a default intensity following gamma distribution
- Sectors’ default intensities are independent
- two firms are correlated if and only if there is at least one sector where
both firms have a positive sector weight- two firms admit a common source of systematic default risk
- Default risk of firm i is modeled with a mixed Poisson L’i with random intensity lamdai following a gamma distribution
Default intensity formula of Firm i in CreditRisk+ model
- s is the sector index
- ms is the total number of sectors
- wis is the weight in firm i and sector s
- lamdai is the mean default intensity of firm i
- lamda(s) is the mean default intensity parameter for sector s
- Lamdai is the default intensity of Firm i
- Lamda(s) is the default intensity for sector s (random realization)
Overview of Credit Portfolio View (CPV)
- Rating-based approach to modeling dependence of default and migration probabilities on the economic cyle
- Often focuses on migrantion matrices
- Obeserved migration matrix in a given year is conditional
- Can average conditional to get unconditional migration matrices
- Can construct a migration matrix using a 3 Step Shift Algorithm
Step Shift Algorithm to Construct a Migration Matrix in CPV
m¯is is the probability of moving from credit state i to j by the end of the year
Overview of CPV Macro
- Focuses on time series of empirical data using macroeconomic regression
- Macroeconomic variables drive the distribution of default probabilities and migration matrices for each risk segment
- Examples of macroeconomic data:
- GDP
- Unemployement Rate
- Interest Rates
- Currency Exchange Rates
Overview of CPV Direct
- draws segment-specific conditional default probabilities ps from a multivariate gamma distribution
- disadvantage of the CPV Direct model - the CPV Direct model can output a default probability greater than 1
- support of the multivariate gamma distribution is over all positive real numbers
- output of the CPV model is a default probability
- In practice, such scenarios are not very likely and are typically “thrown away” and redrawn
Overview of Dynamic Intensity Models
- theory underlying intensity models has much in common with interest rate term structure models
- basic assumption: every firm admits a default time such that default happens in a time interval [0; T] if and only if the default time of the considered firm appears to be smaller than the planning horizon T
- Default times are driven by an intensity process, a so-called Affine process
- Model the default times as independent Poisson arrivals with intensities following a stochastic differential equation w/ a jump term
Overview of One-Factor Models
- In the context of KMV and RiskMetrics, one-factor models assume one single factor common to all counterparties
- assumes that asset correlation b/w obligors is uniform
- Composite factor in (KMV/RiskMetrics) of all obligors are equal to one single factor
- and the R-squared is replaced by rho, a uniform asset correlation b/w asset value log-returns ri
Components of One-Factor Model
splits the log-return ri into the sum of two pieces:
- A general market component (Y term)
- A firm-specific component (Zi term)
Formula for the conditional default probability of firm i given composite factor Y (One-Factor model)
Join Default Probability (One-Factor Model)
- Consider m loss statistics Li - Binom(1; pi(Y)) with uniform asset correlation
Describe almost sure convergence in words in the lens of the portfolio loss variable
- the percentage loss on the portfolio converges to its expectation on an almost sure basis as m goes to infinity
- observed loss percentage converges to expected loss percentage as we crank up sample size (exposures)
- almost sure convergence in a probabilistic sense # not convergence in a deterministic sense
CreditRisk+ model with only one sector
Under the CreditRisk+ model with only one sector and in the limit as m goes to infinity, the portfolio loss distribution converges to a negative binomial distribution:
Sklar’s Theorem in words
- every multivariate distribution with continuous marginals has a unique copula
representation
Types of Copulas
- The independent copula assumes that X1 and X2 are independent random variables
- The independent copula has the simplest form: C(u1, . . . , um) = u1u2 . . . um
- The Gaussian copula has dependent variables with light tails
- The Student-t copula has dependent variables with heavier tails (and approaches the Gaussian copula as n, degrees of freedom goes to infinity)