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Exam 31: Reduced-Form Models of Default Risk

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Question 1

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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%

E) None of the above
F) All of the above

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C

Question 2

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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.

E) A) and B)
F) B) and C)

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Question 3

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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

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 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.

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?

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?

There are two ratings in a very simple world: non-default (ND) and defaultd. 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 $\pi$ ?

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?

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

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 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?

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:

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?

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.

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) ?

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?

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.

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?

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