Deck 31: Reduced-Form Models of Default Risk

Question 1

Suppose we have a zero-coupon bond that pays $100 after one year if the issuing firm is not in default.If the firm is in default the recovery rate is 50%.The simple risk free interest rate for one year is 3% and the risk-neutral probability that the firm defaults is 5%.What is today's fair price for this bond?

A)$87.55
B)$89.23
C)$92.59
D)$94.66

Answer 1

D 11EA6C3E_0C4C_BFF4_9875_F30D400DB45A .

Question 2

The hazard rate for a firm evolves as follows: $$\lambda ( t ) = 0.2 + 0.5 t$$ .The probability of the firm defaulting in the next year is:

A)0.24
B)0.36
C)0.48
D)0.60

Answer 2

0.36

Question 3

There are two ratings in a very simple world: non-default (ND)and default (D).The real-world rating transition matrix per year is given by: $$P = \left[ \begin{array} { c c } 0.95 & 0.05 \\0 & 1\end{array} \right]$$ i.e. ,the probability of defaulting when the current state is non-default is 0.05,and a defaulted bond never leaves that state and has zero recovery.The two-year zero-coupon risk-free rate is 4% (continuously-compounded).The price of a default-risk-bearing two-year $100 face value zero-coupon bond is $88.If the off-diagonal one-period transition probabilities in the real-world transition matrix are multiplied by a premium adjustment $$\pi$$ to get the risk-neutral transition matrix (as in the Jarrow-Lando-Turnbull model),then given the price of the two-year bond,what is the value of $$\pi$$ ?

A)1.97
B)2.00
C)2.03
D)2.10

Answer 3

2.03

Question 4

Suppose the default probability of a firm,conditional on it not having defaulted so far,is 0.10 per year.What is the 5-year survival probability of the firm?

A)49%
B)51%
C)59%
D)61%

Question 5

Empirically,recessions witness a rise in default rates.Which of the following scenarios also accompanies the rise in default rates?

A)Recovery rates rise.
B)Loss given default (LGD)falls
C)LGD rises
D)LGD becomes highly volatile.

Question 6

The current one-year and two-year zero-coupon rates are 6% and 7%,respectively.The one-year and two-year credit spreads are 1% and 2%,respectively.If the recovery rates on this class of bonds is 40% of face value,which of the following numbers most closely approximates the forward probability of default in year 2? Assume that interest rates and yields are in continuously-compounded and annualized terms.Assume also that if default occurs in any year,the recovered amount is received at the end of that year.

A)1%
B)2%
C)5%
D)9%

Question 7

The probability of a firm defaulting each year,given that it has not defaulted in prior years is 10%.What is the probability that it will have defaulted at some time in the first 10 years? Approximately:

A)50%
B)65%
C)80%
D)100%

Question 8

If the rate of defaults per year in a set of companies is given by $$\lambda = 5$$ ,what is the probability of four or more defaults in half a year?

A)0.06
B)0.12
C)0.18
D)0.24

Question 9

There are two ratings in a very simple world: non-default (ND)and default (D).The risk-neutral rating transition matrix per year is given by: $$Q = \left[ \begin{array} { c c } 0.90 & 0.10 \\0 & 1\end{array} \right]$$ i.e. ,the probability of defaulting when the current state is non-default is 0.10,and a defaulted bond never leaves that state and has zero recovery.The three-year zero-coupon risk-free rate is 4% (continuously-compounded).The price of a default-risk-bearing three-year unit face value zero-coupon bond is:

A)0.55
B)0.60
C)0.65
D)0.70

Question 10

There are different recovery conventions.Two common ones are RMV (recovery of market value)and RT (recovery of Treasury value).For a given dollar value recovered on a default bond,it is generally the case that

A)RMV <
RT)
B)RMV $$>$$
RT)
C)Which one is greater depends on the level of interest rates.
D)None of the above.

Question 11

Consider a two-year,annual pay CDS contract,where premiums are paid at the end of the year and if default occurs,it is also assumed to happen at the end of the year (but immediately after the premium payment).Each year there is a 5% risk-neutral probability of the firm defaulting.In default,recovery is 50% (recovery of par,RP).Assume that interest rates are zero.The fair price of this CDS is a spread of

A)200 bps
B)225 bps
C)250 bps
D)275 bps

Question 12

A zero coupon bond with a maturity of one-year pays $1,000 if the issuing firm is not in default.If the firm is in default,the recovery rate is 35%.The risk-free interest rate for one year is 5% (in simple terms with annual compounding)and the risk-neutral probability that the firm defaults is 20%.What is todays price for this bond?

A)$781.28
B)$828.57
C)$885.71
D)$912.92

Question 13

Consider a one-year zero-coupon defaultable bond.Let $$r$$ and $$S$$ denote,respectively,the risk-free interest rate and the spread on the bond,where both are expressed in simple terms with annual compounding.Suppose the risk-neutral probability of default $$\lambda$$ and the recovery rate of the bond in default $$\phi$$ remain fixed.Then,an increase in the risk-free rate must be accompanied by

A)An increase in the spread.
B)A decrease in the spread.
C)No change in the spread.
D)A change in the spread that can be positive,negative,or zero.

Question 14

ABC Inc.has a risk-neutral probability of default of 5% over every half-year period.The loss-given-default (LGD)is 75% of the face value of the debt in ABC Inc.If the risk-free interest rate for one year is 10% on a semiannual compounding basis,find the fair spread for a one-year maturity,semiannual pay CDS contract.Assume that the spread is paid at the beginning of each half-year,while default,if it occurs,occurs at the end of each semiannual period.

A)228 bps
B)357 bps
C)428 bps
D)551 bps

Question 15

Suppose the default intensity of a firm is 0.10.What is the five-year survival probability of the firm closest to?

A)45%
B)50%
C)55%
D)60%

Question 16

Suppose we have a zero-coupon bond that pays $1 after one year if the issuing firm is not in default.If the firm is in default the recovery rate is 40%.The one-year risk free interest rate in simple terms is 5% and the risk-neutral probability that the firm defaults is 10%.What is today's fair price for this bond?

A)$0.875
B)$0.895
C)$0.915
D)$0.935

Question 17

A zero coupon bond with a maturity of one-year pays $1,000 if the issuing firm is not in default.If the firm is in default,the recovery rate is 35%.The risk-free interest rate for one year is 5% and the risk-neutral probability that the firm defaults is 20%.What is the credit spread (over the risk-free rate)on the bond? All yields are in simple terms with annual compounding.

A)7.90%
B)12.90%
C)15.69%
D)20.69%

Question 18

Suppose we have a zero coupon bond that pays $1 after one year if the issuing firm is not in default.If the firm is in default,the recovery rate is 42%.The risk free interest rate for one year is 5%.If the credit spread on the bond is 2.5%,what is the risk-neutral probability of default of the bond? Assume all yields are stated in simple terms with annual compounding.

A)0.01
B)0.02
C)0.03
D)0.04

Question 19

Suppose we have a zero-coupon bond that pays $1 after one year if the issuing firm is not in default.If the firm is in default the recovery rate is 40%.The one-year risk free interest rate in simple terms is 5% and the risk-neutral probability that the firm defaults is 10%.What is the fair credit spread on the bond (again,in simple terms)?

A)5.0%
B)6.7%
C)10.5%
D)11.7%

Question 20

Suppose we have a zero-coupon bond that pays $100 after one year if the issuing firm is not in default.If the firm is in default the recovery rate is 50%.The simple risk-free interest rate for one year is 5% and the risk-neutral probability that the firm defaults is 10%.What is todayÕs fair credit spread for this bond?

A)5.5%
B)6.5%
C)7.5%
D)8.5%

Question 21

The average default rate in the economy is 1.5% of the face value of outstanding debt defaults per year.What is the average time between defaults if there are 1000 firms alive on average?

A)20 days
B)24 days
C)38 days
D)41 days

Question 22

The average default rate in the economy is 1.5% of the face value of outstanding debt defaults per year.How many years will it be on average before half the firms are no longer in existence if no new firms enter the economy?

A)44 years
B)45 years
C)46 years
D)47 years

Question 23

If the hazard rate is $$\lambda = 0.2$$ ,the risk-free rate is zero,then if the price of a one-year $100 face value discount bond is $85,then what is the expected recovery rate $$\phi$$ on default of the bond?

A)17%
B)24%
C)40%
D)55%