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 5% and the risk-neutral probability that the firm defaults is 10%. What is todayÕs fair credit spread for this bond?

Accepted Answer

The fair credit spread is the difference between the yield to maturity of the risky bond and the risk-free rate. The yield to maturity of the risky bond is the rate that equates the present value of the bond's cash flows to its current price. The present value of the bond's cash flows is:PV = $100 / (1 + r)where r is the yield to maturity. The current price of the bond is:Price = $100 * (1 - 0.1) / (1 + r)where 0.1 is the risk-neutral probability of default. Substituting this into the equation for the yield to maturity, we get:$100 / (1 + r) = $100 * (1 - 0.1) / (1 + r)Solving for r, we get:r = 5.5%The fair credit spread is therefore:Fair credit spread = 5.5% - 5% = 0.5%

Question 2

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

Accepted Answer

D

Question 3

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?

Accepted Answer

To calculate the price of the bond, we use the formula that accounts for the expected payoff, which includes both the default and non-default scenarios, discounted at the risk-free rate. The expected payoff is calculated as follows:- Non-default scenario: The bond pays $1,000 with a probability of (1 - 0.20) = 0.80.- Default scenario: The bond pays $1,000 * 35% = $350 with a probability of 0.20.Expected payoff = (0.80 * $1,000) + (0.20 * $350) = $800 + $70 = $870.To find the present value, we discount this expected payoff at the risk-free rate of 5%:Price = $870 / (1 + 0.05) = $870 / 1.05 = $828.57.

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?

Accepted Answer

The 5-year survival probability is calculated as the probability of not defaulting each year, raised to the power of 5. Since the default probability is 0.10, the survival probability each year is 1 - 0.10 = 0.90. Therefore, over 5 years, the survival probability is $0.90^5 = 0.59049$ or approximately 59%.

Question 5

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?

Accepted Answer

The fair price of the bond can be calculated using the expected payoff under the risk-neutral measure, discounted at the risk-free rate. The expected payoff is the probability of no default times the payoff in that case plus the probability of default times the recovery value. Mathematically, this is (1 - 0.05) * $100 + 0.05 * ($100 * 0.5) = $95 + $2.5 = $97.5. Discounting this at the risk-free rate of 3% gives $97.5 / (1 + 0.03) = $94.66.

Question 6

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.

Accepted Answer

The price of the bond can be calculated using the formula: $ \text{Price} = \frac{\text{Expected Payoff}}{1 + \text{Risk-Free Rate} + \text{Credit Spread}} $.The expected payoff is $ (1 - \text{Probability of Default}) \times 1 + \text{Probability of Default} \times 0.42 $.Given that the risk-free rate is 5% (or 0.05) and the credit spread is 2.5% (or 0.025), the denominator becomes $ 1 + 0.05 + 0.025 = 1.075 $.Setting the price equal to the expected payoff divided by 1.075 and solving for the probability of default gives us the risk-neutral probability of default. Rearranging the formula and solving for the probability of default, we find it to be approximately 4%, or 0.04, under the given conditions.

Question 7

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:

Accepted Answer

The answer of The hazard rate for a firm evolves...

Question 8

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?

Accepted Answer

The fair price of the bond can be calculated by taking the expected payoff under the risk-neutral measure and discounting it back at the risk-free rate. The expected payoff is $(1 - 0.1) * $1 + 0.1 * $0.40 = $0.94. Discounting this at the 5% risk-free rate gives $0.94 / (1 + 0.05) = $0.895.

Question 9

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)?

Accepted Answer

To calculate the fair credit spread on the bond, we need to first calculate the expected loss in case of default:
Expected loss = Probability of default * (Face value - Recovery value)
= 0.1 * (1 - 0.4)
= 0.06

Next, we need to calculate the risk-neutral present value of the bond:
PV = Face value / (1 + risk-free rate)
= 1 / (1 + 0.05)
= 0.9524

Finally, the fair credit spread is:
Credit spread = Expected loss / PV
= 0.06 / 0.9524
= 0.063 or 6.3%

Therefore, the closest option is B) 6.7%.

Question 10

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.

Accepted Answer

The forward probability of default in year 2 is the probability that the bond will default in year 2 given that it has not defaulted in year 1. This can be calculated using the following formula:```Forward probability of default in year 2 = (Credit spread in year 2 - Credit spread in year 1) / (1 - Recovery rate)```Plugging in the given values, we get:```Forward probability of default in year 2 = (2% - 1%) / (1 - 40%) = 5%```Therefore, the answer is C) 5%.

Question 11

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 $\pi$ ?

Accepted Answer

The price of a two-year zero-coupon bond in a risk-neutral world can be calculated using the risk-neutral transition matrix and the given bond price. The risk-neutral transition matrix adjusts the real-world probabilities to reflect the market's pricing of risk. The adjustment factor $\pi$ scales the off-diagonal elements of the transition matrix, which represent the transition from non-default to default. Given the bond's price of $88 and the risk-free rate of 4% (continuously compounded), we can calculate the expected payoff under the risk-neutral measure and solve for $\pi$ to match the bond's price. The correct adjustment factor that aligns with these conditions is 2.03, indicating how much the real-world probability of default must be scaled to match the market pricing under risk-neutral valuation.

Question 12

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.

Accepted Answer

The fair spread can be calculated using the formula:
Fair Spread = (Probability of Default * LGD) / (1 - Probability of Default) * Risk-free rate
Probability of Default = 5%
LGD = 75%
Risk-free rate = 10%
Therefore, Fair Spread = (5% * 75%) / (1 - 5%) * 10% = 357 bps, which is closest to option B.

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

Accepted Answer

An increase in the risk-free rate $r$ increases the discount rate for the bond's cash flows, reducing its present value. To maintain the same risk-neutral probability of default $\lambda$ and recovery rate $\phi$, the spread $S$ must increase to compensate for the higher discount rate, keeping the bond's price consistent with its risk profile.

Question 14

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

Accepted Answer

To find the price of a default-risk-bearing three-year unit face value zero-coupon bond, we first calculate the probability of the bond not defaulting over the three years. Using the given transition matrix $Q$, the probability of staying non-default after one year is 0.90. Since the matrix shows that once defaulted, the bond stays in default (with a transition probability of 1 to stay in default), we can calculate the probability of not defaulting over three years as $0.90^3 = 0.729$.Given the risk-free rate is 4% (continuously compounded), the risk-free price of a three-year zero-coupon bond with a face value of 1 is calculated using the formula $P = e^{-rt}$, where $r$ is the risk-free rate and $t$ is the time in years. Substituting $r = 0.04$ and $t = 3$, we get $P = e^{-0.04*3} = e^{-0.12} \approx 0.8869$.The price of the default-risk-bearing bond is then the product of the risk-free price and the probability of not defaulting, which is $0.8869 * 0.729 \approx 0.646$, which rounds to 0.65, making option C the correct answer.

Question 15

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

Accepted Answer

RMV (recovery of market value) typically results in a higher recovery value compared to RT (recovery of Treasury value) because RMV is based on the market value of the bond, which can include factors such as the bond's creditworthiness and market demand, potentially leading to a higher valuation than the more conservative Treasury-based recovery.

Question 16

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

Accepted Answer

The fair price of the CDS can be calculated considering the expected loss and the premium payments. Given a 5% default probability each year and a 50% recovery rate, the expected loss in case of default is 50% of the notional. Since default can occur at the end of the first or second year, we calculate the expected loss for each year and sum them up, considering the time value of money is zero (interest rates are zero).For the first year, the expected loss is 5% (probability of default) * 50% (loss given default) = 2.5% of the notional.For the second year, the default can only occur if it hasn't occurred in the first year. So, the probability of reaching the second year is 95% (100% - 5% probability of default in the first year), and then the probability of default in the second year is again 5%. Thus, the expected loss for the second year is 95% * 5% * 50% = 2.375% of the notional.The total expected loss over the two years is 2.5% + 2.375% = 4.875% of the notional.Since the CDS spread compensates for the expected loss and there are two premium payments (one at the end of each year), the fair CDS spread is the total expected loss divided by the number of payments, which is 4.875% / 2 = 2.4375%. This is approximately 243.75 bps, but since this option is not available, the closest available option is 250 bps.

Question 17

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?

Accepted Answer

The number of defaults can be modeled using a Poisson distribution, where $\lambda$ is the average rate of occurrence. For half a year, the rate $\lambda = 5/2 = 2.5$ defaults. The probability of having $k$ defaults is given by $P(k) = \frac{e^{-\lambda} \lambda^k}{k!}$. To find the probability of four or more defaults, we calculate the probability of 0, 1, 2, and 3 defaults and subtract this sum from 1.$$P(0) = \frac{e^{-2.5} 2.5^0}{0!} = e^{-2.5}$$$$P(1) = \frac{e^{-2.5} 2.5^1}{1!} = 2.5e^{-2.5}$$$$P(2) = \frac{e^{-2.5} 2.5^2}{2!} = \frac{6.25e^{-2.5}}{2}$$$$P(3) = \frac{e^{-2.5} 2.5^3}{3!} = \frac{15.625e^{-2.5}}{6}$$Summing these probabilities gives the probability of having fewer than four defaults. Subtracting this sum from 1 gives the probability of having four or more defaults:$$1 - (P(0) + P(1) + P(2) + P(3)) = 1 - e^{-2.5}(1 + 2.5 + \frac{6.25}{2} + \frac{15.625}{6})$$Calculating this expression gives a value close to 0.24, making option D the correct answer.

Question 18

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:

Accepted Answer

The probability that a firm does not default in a year is 90%. Therefore, the probability that it does not default in 10 years is (0.9)^10 = 0.3487. The complement of this is the probability that it will have defaulted at some time in the first 10 years, which is 1 - 0.3487 = 0.6513 or approximately 65%. Therefore, the best choice is B.

Question 19

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.

Accepted Answer

The credit spread is the difference between the yield on the risky bond and the yield on the risk-free bond. The yield on the risky bond is the expected return on the bond, which is equal to the risk-free rate plus the credit spread. The expected return on the bond is equal to the face value of the bond times the probability of default times the recovery rate plus the face value of the bond times the probability of no default. The probability of default is 20%, the recovery rate is 35%, and the face value of the bond is $1,000. Therefore, the expected return on the bond is $1,000 * 0.20 * 0.35 + $1,000 * 0.80 = $870. The yield on the risky bond is equal to the expected return on the bond divided by the price of the bond. The price of the bond is equal to the face value of the bond times the present value of the expected return on the bond. The present value of the expected return on the bond is equal to the expected return on the bond divided by the risk-free rate. Therefore, the price of the bond is equal to $1,000 * $870 / 1.05 = $828.57. The yield on the risky bond is equal to $870 / $828.57 = 10.50%. The credit spread is equal to the yield on the risky bond minus the yield on the risk-free bond. The yield on the risk-free bond is 5%. Therefore, the credit spread is 10.50% - 5% = 5.50%.

Question 20

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

Accepted Answer

During recessions, the value of collateral assets is likely to fall, making it harder for borrowers to repay their loans. Thus, the Loss given default (LGD), which measures the loss incurred by the lender in case a borrower defaults, is likely to rise.

Quiz 31: Reduced-Form Models of Default Risk

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

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?

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:

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

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?

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:

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

Quiz 31: Reduced-Form Models of Default Risk

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

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?

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

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

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

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:

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

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:

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?