Deck 32: Modeling Correlated Default

Question 1

Consider two firms with one-year probabilities of default of $$p _ { 1 } = 0.10$$ and $$p _ { 2 } = 0.05$$ ,respectively.The correlation of default of the two firms is $$\rho = 0.5$$ .What is the conditional probability of default $$\operatorname { Pr } \left[ D _ { 2 } \mid D _ { 1 } \right]$$ ?

A)0.38
B)0.42
C)0.46
D)0.50

Answer 1

0.38

Question 2

Consider two firms with one-year probabilities of default of $$p _ { 1 } = 0.10$$ and $$p _ { 2 } = 0.05$$ ,respectively.The correlation of default of these two firms is $$\rho = 0.30$$ .What is the probability that both firms default in one year?

A)1.0%
B)1.5%
C)2.0$
D)2.5%

Answer 2

2.5%

Question 3

Consider two firms,each of which has a distance-to-default of 2.The correlation of default of the two firms is $$\rho = 0.5$$ .Assuming bivariate normality,what is the value of a $100 notional first-to-default basket option on these two firms,if the discount rate is zero?

A)$1
B)$2
C)$3
D)$4

Answer 3

$4

Question 4

Which of the following isnot an important benefit of using copula functions for correlated default?

A)Copula distribution functions are computationally more tractable than multivariate normal distributions.
B)Copulas allow separation of the marginal distribution functions and the correlation function.
C)Copulas allow coupling of multivariate distributions where the marginals may be chosen from varied probability distributions.
D)Copulas allow flexible modeling of the tails of the joint distribution.

Question 5

Consider two firms with hazard rates $$\lambda _ { 1 } = 0.10$$ and $$\lambda _ { 2 } = 0.05$$ .The correlation of their default times is zero.What is the approximate probability that both firms default within five years?

A)5%
B)9%
C)13%
D)17%

Question 6

Consider two firms with one-year probabilities of default of $$p _ { 1 } = 0.10$$ and $$p _ { 2 } = 0.05$$ ,respectively.The correlation of default of the two firms is $$\rho = 0.5$$ .What is the conditional probability of default $$\operatorname { Pr } \left[ D _ { 1 } \mid D _ { 2 } \right]$$ ?

A)0.55
B)0.65
C)0.75
D)0.85

Question 7

The value of a CDO (collateralized debt obligation)unambiguously increases when

A)The correlation of default amongst the names in the CDO decreases.
B)The correlation of default amongst the names in the CDO increases.
C)The intensity of default of names in the portfolio decreases.
D)The number of names in the CDO decreases.

Question 8

Consider two firms,each of which has a distance-to-default of 1.The correlation of default of the two firms is $$\rho = 0.5$$ .Assuming bivariate normality,what is the probability that both firms default?

A)4.50%
B)6.25%
C)8.75%
D)10.25%

Question 9

Consider two firms with one-year probabilities of default of $$p _ { 1 } = 0.10$$ and $$p _ { 2 } = 0.05$$ ,respectively.The correlation of default of these two firms is $$\rho = 0.30$$ .What is the price of a $100 notional one-year maturity first-to-default basket option on these two firms? (Assume the discount rate is zero. )

A)$10.00
B)$12.50
C)$15.00
D)$17.50

Question 10

If a firm has a distance-to-default of 2,and we assume a normal distribution,then the probability of a firm defaulting is

A)1.1%
B)1.5%
C)2.1%
D)2.3%

Question 11

You are assessing a credit portfolio with 100 issuers where the hazard rate of default of each name is 0.05.The default correlation of all firms (pairwise)is zero.What is the average time it will take for 10% of the portfolio to default?

A)1/5 year
B)1/2 year
C)1 year
D)2 years

Question 12

A CDO has three tranches,a senior tranche,mezzanine tranche,and equity tranche.Keeping the probabilities of default in the CDO collateral fixed,the value of the equity tranche increases if

A)The correlation of default across all names in the CDO increases.
B)The correlation of default across all names in the CDO decreases.
C)The credit quality of names in the CDO collateral declines.
D)The volatility of credit spreads on names in the CDO collateral increases.

Question 13

Consider two firms with one-year probabilities of default of $$p _ { 1 } = 0.10$$ and $$p _ { 2 } = 0.05$$ ,respectively.The conditional probability of default in one year is $$\operatorname { Pr } \left[ D _ { 1 } \mid D _ { 2 } \right] = 0.7$$ .What is the probability of a first-to-default basket option that pays $100 if any one firm defaults within a year? (Assume zero discount rates. )

A)$10.00
B)$11.50
C)$14.25
D)$17.35

Question 14

A second-to-default (STD)basket option pays off when any one of the companies in a credit basket defaults.The price of the STD basket increases when

A)The credit quality of any issuer in the basket improves.
B)The correlation of default across names in the basket increases.
C)The correlation of default across names in the basket decreases.
D)The volatility of credit spreads for names in the basket decreases.

Question 15

Consider two firms with one-year probabilities of default of $$p _ { 1 } = 0.10$$ and $$p _ { 2 } = 0.05$$ ,respectively.The correlation of default of these two firms is $$\rho = 0.30$$ .What is the price of a $100 notional one-year maturity second-to-default basket option on these two firms? (Assume the discount rate is zero. )

A)$2.50
B)$5.50
C)$11.25
D)$17.75

Question 16

Which of the following isnot a reason to favor the top-down approach to modeling correlated default versus the bottom-up approach?

A)The loss distribution in simple cases is easy to compute in closed-form.
B)The computational complexity of the top-down approach is far less than that of the bottom-up one.
C)It is easier to model CDOs on CDOs.
D)It enables breaking down the portfolio loss name by name.

Question 17

Consider two firms with one-year probabilities of default of $$p _ { 1 } = 0.10$$ and $$p _ { 2 } = 0.05$$ ,respectively.The conditional probability of default in one year is $$\operatorname { Pr } \left[ D _ { 1 } \mid D _ { 2 } \right] = 0.7$$ .What is the probability of a second-to-default basket option that pays $100 if any both firms default within a year? (Assume zero discount rates. )

A)$0.50
B)$1.50
C)$2.50
D)$3.50

Question 18

If you expect default correlations to increase in the future,what trade might you engage in to profit from this view?

A)Buy the senior tranche of a CDO.
B)Buy a second-to-default basket option and sell a first-to-default basket.
C)Sell a second-to-default basket option and buy a first-to-default basket.
D)Short the equity tranche of a CDO.

Question 19

Two firms that have zero default correlation and expected losses conditional on default of $2 million and $3 million,respectively.The probability of loss of the two firms in one year is 0.10 and 0.05,respectively.What is the mean loss of this portfolio?

A)$250,000
B)$350,000
C)$400,000
D)$500,000

Question 20

Consider two firms with one-year probabilities of default of $$p _ { 1 } = 0.10$$ and $$p _ { 2 } = 0.05$$ ,respectively.The conditional probability of default in one year is $$\operatorname { Pr } \left[ D _ { 1 } \mid D _ { 2 } \right] = 0.7$$ .What is the correlation of default of these two firms closest to?

A)0.25
B)0.35
C)0.45
D)0.55

Question 21

A self-exciting model for defaults is one where

A)There are non-random exogenous changes in the rate at which defaults arrive.
B)The volatility of the intensity increases with the level of losses.
C)The rate at which defaults arrive increases with the level of losses.
D)Excitement from defaults is spontaneous!

Question 22

Which of the following is an Archimedean copula over two distribution functions

A) $$C \left( F _ { 1 } , F _ { 2 } \right) = \ln \left\{ - \left[ \left( \ln \frac { 1 } { F _ { 1 } } \right) ^ { a } + \left( \ln \frac { 1 } { F _ { 2 } } \right) ^ { a } \right] ^ { 1 / a } \right\}$$
B) $$C \left( F _ { 1 } , F _ { 2 } \right) = \exp \left\{ - \left[ \left( \ln \frac { 1 } { F _ { 1 } } \right) ^ { a } + \left( \ln \frac { 1 } { F _ { 2 } } \right) ^ { a } \right] ^ { a } \right\}$$
C) $$C \left( F _ { 1 } , F _ { 2 } \right) = \exp \left\{ - \left[ \left( \ln \frac { 1 } { F _ { 1 } } \right) ^ { a } + \left( \ln \frac { 1 } { F _ { 2 } } \right) ^ { a } \right] ^ { 1 / a } \right\}$$
D)None of the above.

Question 23

The difference between implied correlation and base correlation in CDOs is that

A)Base correlation is the smallest correlation implied across all tranches of a CDO.
B)Base correlation is the correlation of cumulative sets of tranches starting with the equity tranche,whereas implied correlation is for single tranches only.
C)All base correlations are implied,but not all implied correlations are base correlations.
D)Base correlation is historical and implied correlation is computed using tranche prices at a point in time.

Question 24

In the Longstaff and Rajan top-down correlated default model,assume that losses $$L _ { t }$$ in a credit portfolio are given by the following dynamic process in a one-factor setting: $$\frac { d L _ { t } } { 1 - L _ { t } } = \gamma d N ( \lambda )$$ where $$\gamma$$ is a fractional loss (of the current portfolio value)that occurs every time there is a default,assumed to be generated by a Poisson process $$N$$ with loss arrival rate $$\lambda$$ (a constant).What is the expected loss of a $100 portfolio in a year if $$\gamma = 0.01$$ and $$\lambda = 2$$ ?

A)$1.5
B)$2.0
C)$2.5
D)$3.0