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Options, Futures, and Other Derivatives

John C. Hull

Chapter 25

Credit derivatives - all with Video Answers

Educators

Chapter Questions

Problem 1

Explain the difference between a regular credit default swap and a binary credit default swap.

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Problem 2

A 5-year credit default swap requires a quarterly payment at the rate of 60 basis points per year. The principal is $$\$ 300$$ million and the credit default swap is settled in cash. A default occurs after 4 years and 2 months, and the price of the cheapest deliverable bond is estimated as $40 \%$ of its face value shortly after the default. List the cash flows and their timing for the seller of the credit default swap.

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00:52

Problem 3

Explain the difference between a cash CDO and a synthetic CDO.

Nicole Smina
Nicole Smina
Numerade Educator
00:31

Problem 4

Explain the term "single-tranche trading."

Amrita Bhasin
Amrita Bhasin
Numerade Educator

Problem 5

What is a first-to-default credit default swap? Does its value increase or decrease as the default correlation between the companies in the basket increases? Explain your answer.

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Problem 6

Explain the difference between risk-neutral and real-world default probabilities. Which should be used for valuing CDSs?

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Problem 7

Explain why a total return swap can be useful as a financing tool.

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Problem 8

Suppose that the risk-free zero curve is flat at $7 \%$ per annum with continuous compounding and that defaults can occur halfway through each year in a new 5 -year credit default swap. Suppose that the recovery rate is $30 \%$ and the hazard rate is $3 \%$. Estimate the credit default swap spread. Assume payments are made annually.

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Problem 9

What is the value of the swap in Problem 25.8 per dollar of notional principal to the protection buyer if the credit default swap spread is 150 basis points?

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Problem 10

What is the credit default swap spread in Problem 25.8 if it is a binary CDS?

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Problem 11

How does a 5 -year $n$ th-to-default credit default swap work? Consider a basket of 100 reference entities where each reference entity has a probability of defaulting in each year of $1 \%$. As the default correlation between the reference entities increases what would you expect to happen to the value of the swap when (a) $n=1$ and (b) $n=25$. Explain your answer.

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02:11

Problem 12

What is the formula relating the payoff on a CDS to the notional principal and the recovery rate?

Nick Johnson
Nick Johnson
Numerade Educator
01:26

Problem 13

Show that the spread for a new plain vanilla CDS should be $(1-R)$ times the spread for a similar new binary CDS, where $R$ is the recovery rate.

Jake Zanazzi
Jake Zanazzi
Numerade Educator

Problem 14

Verify that, if the CDS spread for the example in Tables 25.1 to 25.4 is 100 basis points, the hazard rate must be $1.63 \%$ per year. How does the hazard rate change when the recovery rate is $20 \%$ instead of $40 \%$ ? Verify that your answer is consistent with the implied hazard rate being approximately proportional to $1 /(1-R)$, where $R$ is the recovery rate.

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03:06

Problem 15

A company enters into a total return swap where it receives the return on a corporate bond paying a coupon of $5 \%$ and pays LIBOR. Explain the difference between this and a regular swap where $5 \%$ is exchanged for LIBOR.

Jennifer Stoner
Jennifer Stoner
Numerade Educator

Problem 16

Explain how forward contracts and options on credit default swaps are structured.

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Problem 17

"The position of a buyer of a credit default swap is similar to the position of someone who is long a risk-free bond and short a corporate bond." Explain this statement.

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02:03

Problem 18

Why is there a potential asymmetric information problem in credit default swaps?

Pragya Ahuja
Pragya Ahuja
Numerade Educator

Problem 19

Does valuing a CDS using real-world default probabilities rather than risk-neutral default probabilities overstate or understate its value? Explain your answer.

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Problem 20

What is the difference between a total return swap and an asset swap?

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01:45

Problem 21

Suppose that in a one-factor Gaussian copula model the 5 -year probability of default for each of 125 names is $3 \%$ and the pairwise copula correlation is 0.2 . Calculate, for factor values of $-2,-1,0,1$, and 2 : (a) the default probability conditional on the factor value and (b) the probability of more than 10 defaults conditional on the factor value.

Michelle Z.
Michelle Z.
Numerade Educator
00:55

Problem 22

Explain the difference between base correlation and compound correlation.

Trent Speier
Trent Speier
Numerade Educator
00:24

Problem 23

In Example 25.2, what is the tranche spread for the $9 \%$ to $12 \%$ tranche assuming a tranche correlation of 0.15 ?

Bryan Meares
Bryan Meares
Numerade Educator

Problem 24

Suppose that the risk-free zero curve is flat at $6 \%$ per annum with continuous compounding and that defaults can occur at times 0.25 years, 0.75 years, 1.25 years, and 1.75 years in a 2 -year plain vanilla credit default swap with semiannual payments. Suppose that the recovery rate is $20 \%$ and the unconditional probabilities of default (as seen at time zero) are $1 \%$ at times 0.25 years and 0.75 years, and $1.5 \%$ at times 1.25 years and 1.75 years. What is the credit default swap spread? What would the credit default spread be if the instrument were a binary credit default swap?

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Problem 25

Assume that the hazard rate for a company is $\lambda$ and the recovery rate is $R$. The risk-free interest rate is $5 \%$ per annum. Default always occurs halfway through a year. The spread for a 5-year plain vanilla CDS where payments are made annually is 120 basis points and the spread for a 5 -year binary CDS where payments are made annually is 160 basis points. Estimate $R$ and $\lambda$.

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02:06

Problem 26

Explain how you would expect the returns offered on the various tranches in a synthetic CDO to change when the correlation between the bonds in the portfolio increases.

Jennifer Stoner
Jennifer Stoner
Numerade Educator

Problem 27

Suppose that:
(a) The yield on a 5 -year risk-free bond is $7 \%$.
(b) The yield on a 5-year corporate bond issued by company $\mathrm{X}$ is $9.5 \%$.
(c) A 5-year credit default swap providing insurance against company $\mathrm{X}$ defaulting costs 150 basis points per year.
What arbitrage opportunity is there in this situation? What arbitrage opportunity would there be if the credit default spread were 300 basis points instead of 150 basis points?

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03:46

Problem 28

In Example 25.3, what is the spread for (a) a first-to-default CDS and (b) a second-todefault CDS?

Michael Wang
Michael Wang
Numerade Educator

Problem 29

In Example 25.2, what is the tranche spread for the $6 \%$ to $9 \%$ tranche assuming a tranche correlation of 0.15 ?

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Problem 30

The 1-, 2-, 3-, 4-, and 5-year CDS spreads are $100,120,135,145$, and 152 basis points, respectively. The risk-free rate is $3 \%$ for all maturities, the recovery rate is $35 \%$, and payments are quarterly. Use DerivaGem to calculate the hazard rate each year. What is the probability of default in year 1 ? What is the probability of default in year 2 ?

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Problem 31

Table 25.6 shows the 5 -year iTraxx index was 77 basis points on January $31,2008$. Assume the risk-free rate is $5 \%$ for all maturities, the recovery rate is $40 \%$, and payments are quarterly. Assume also that the spread of 77 basis points applies to all maturities. Use the DerivaGem CDS worksheet to calculate a hazard rate consistent with the spread. Use this in the CDO worksheet with 10 integration points to imply base correlations for each tranche from the quotes for January $31,2008$.

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