CIR 2 - Modeling Correlated Defaults

Question 1

Two main Credit Default models analyzed in Blum 2

Answer 1

• Bernoulli Model (L): Moody’s KMV / Risk Metrics / Most Internal Bank Models
• Poisson Model (L^’ ): CreditRisk+

Question 2

Total defaults for Bernoulli loss statistic L

Answer 2

Question 3

Loss Percentage for Bernoulli loss statistic L

Answer 3

Question 4

Properties of Bernoulli Loss Statistic

Answer 4

Question 5

Meaning of Uniform default probability p in (bernoulli loss statistic)

Answer 5

Means the probability of default of each counterparty is the same, p_i = p for all i.

Question 6

Overview of General Bernoulli Mixture Model

Answer 6

• Specifies explicit dependencies b/w counterparties L’s
• Loss probabilites are random variables w/ distribution F
• Conditional on P, L’s are independent

Question 7

General Bernoulli Mixture Model distribution and moments

Answer 7

Question 8

Covariance derivation under the General Bernoulli Mixture Model

Answer 8

Question 9

Uniform Default Probability and Uniform Correlation

Answer 9

• called uniform portfolios
• Works best for portfolios with all exposures are of approximately the same size and type of risk

Question 10

Correlation formula for uniform Bernoulli portfolio

Answer 10

Question 11

Correlation cases for uniform Bernoulli portfolio

Answer 11

• 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

Question 12

Stability of Poisson Models

Answer 12

The sum of independent poisson models is poisson where you just add the underlying parameter values

Question 13

Overview of General Poisson Mixture Model

Answer 13

Question 14

Correlation formula for uniform Poisson portfolio

Answer 14

Question 15

Dispersion ratio of a random variable

Answer 15

• ratio of the variance devided by the mean
  *

Question 16

Overview of common credit models

Answer 16

Question 17

Overview of Log-return model in Moody’s KMV and RiskMetrics

Answer 17

Question 18

One-year default probability of firm i conditional on composite factor (Moody’s KMV and RiskMetrics)

Answer 18

Question 19

Overview of Composite Factor in Log-return model (Moody’s KMV and RiskMetrics’)

Answer 19

• 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

Question 20

Overview of CreditRisk+

Answer 20

• 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 lamda_i following a gamma distribution

Question 21

Default intensity formula of Firm i in CreditRisk+ model

Answer 21

• s is the sector index
• m_s is the total number of sectors
• w_is is the weight in firm i and sector s
• lamda_i is the mean default intensity of firm i
• lamda_(s) is the mean default intensity parameter for sector s
• Lamda_i is the default intensity of Firm i
• Lamda^(s) is the default intensity for sector s (random realization)

Question 22

Overview of Credit Portfolio View (CPV)

Answer 22

• 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

Question 23

Step Shift Algorithm to Construct a Migration Matrix in CPV

Answer 23

m¯_is is the probability of moving from credit state i to j by the end of the year

Question 24

Overview of CPV Macro

Answer 24

• 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

Question 25

Overview of CPV Direct

Answer 25

* 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

Question 26

Overview of Dynamic Intensity Models

Answer 26

* 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

Question 27

Overview of One-Factor Models

Answer 27

* 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 r_i

Question 28

Components of One-Factor Model

Answer 28

splits the log-return r_i into the sum of two pieces: 1. A general market component (Y term) 2. A firm-specific component (Z_i term)

Question 29

Formula for the conditional default probability of firm i given composite factor Y (One-Factor model)

Answer 29

Question 30

Join Default Probability (One-Factor Model)

Answer 30

* Consider m loss statistics L_i - Binom(1; p_i(Y)) with **uniform asset correlation**

Question 31

Describe almost sure convergence in words in the lens of the portfolio loss variable

Answer 31

* 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

Question 32

CreditRisk+ model with only one sector

Answer 32

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:

Question 33

Sklar’s Theorem in words

Answer 33

* every multivariate distribution with continuous marginals has a unique copula representation

Question 34

Types of Copulas

Answer 34

* 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)