Homework SourseCan you please explain step by step on how to do this questi
**Can you please explain step by step on how to do this question*** and please show formulas used so I can understand how to do it on my own. thank you.
(4.1) Suppose that the risk-free zero curve is flat at 7% per annum with continuous compounding and that defaults can occur half way through each year in a new five-year credit default swap. Suppose that the recovery rate is 30% and the default probabilities each year conditional on no earlier default is 3% Estimate the credit default swap spread? Assume payments are made annually.
What is the value of the swap in Problem 4.1 per dollar of notional principal to the protection buyer if the credit default swap spread is 150 basis points?
What is the credit default swap spread in Problem 4.1 if it is a binary CDS?
Solution
Below are the steps to calculate the solution along with the formulas
First we need to prepare a table 5 years giving each default probability and survival probability for each year
| Years | Hazard Rate | Default Probability | Survival probability | ||
| 0 | 1 | ||||
| 1 | 0.03 | 0.03*1 | 1-(0.03*1) | ||
| 2 | 0.03 | 0.03*Survival probability of 1st year | (Previous year survival probability - current year default probability) | ||
| 3 | 0.03 | 0.03*Survival probability of 2nd year | (Previous year survival probability - current year default probability) | ||
| 4 | 0.03 | 0.03*Survival probability of 3rd year | (Previous year survival probability - current year default probability) | ||
| 5 | 0.03 | 0.03*Survival probability of 4th year | (Previous year survival probability - current year default probability) | ||
| Now we would calculate the present value of expected regular payment we would assume the expected payment rate is s | |||||
| A | B | C | D | ||
| Years | Probaility of survival (to be taken as calculated above) | Expected payment(Probability of survival*s) | Discount factor | PV of expected payment | |
| 1 | ColumnA*s | e^-(0.07*1) | Column B* Column C | ||
| 2 | ColumnA*s | e^-(0.07*2) | Column B* Column C | ||
| 3 | ColumnA*s | e^-(0.07*3) | Column B* Column C | ||
| 4 | ColumnA*s | e^-(0.07*4) | Column B* Column C | ||
| 5 | ColumnA*s | e^-(0.07*5) | Column B* Column C | ||
| Total | |||||
| The next step would be to calculate the present value of the expected payoff | |||||
| Assuming the notional principal is $ 1 | |||||
| Years | Probability of default( to be taken from above) | Recovery rate | Expected payoff | Discount factor | |
| 0.5 | 0.3 | Notinal principal*(1-recovery rate)*probability of default | e^-(0.035*0.50) | ||
| 1.5 | 0.3 | Notinal principal*(1-recovery rate)*probability of default | e^-(0.035*1.50) | ||
| 2.5 | 0.3 | Notinal principal*(1-recovery rate)*probability of default | e^-(0.035*2.50) | ||
| 3.5 | 0.3 | Notinal principal*(1-recovery rate)*probability of default | e^-(0.035*3.50) | ||
| 4.5 | 0.3 | Notinal principal*(1-recovery rate)*probability of default | e^-(0.035*4.50) | ||
| Total | |||||
| Now we would calculate the present value of expected accrual payments | |||||
| Years | Probability of default( to be taken from above) | Expected accrual payment | Discount factor | PV of expected Accrual payment(Expected accrual payment*discount factor) | |
| 0.5 | Probability of default*0.50*s | e^-(0.035*0.50) | |||
| 1.5 | Probability of default*0.50*s | e^-(0.035*1.50) | |||
| 2.5 | Probability of default*0.50*s | e^-(0.035*2.50) | |||
| 3.5 | Probability of default*0.50*s | e^-(0.035*3.50) | |||
| 4.5 | Probability of default*0.50*s | e^-(0.035*4.50) | |||
| Total | |||||
| The credit default swap spread s would be given by | |||||
| Present value of the expected payment + PV of expected accrual payment = Present value of expected payoff | |||||
| When we put the total of the value received from the above table we would get the value of s which is the credit default swap spread | |||||
| 2) | If the credit default swap is 150 basis point, the value of the swap to buyer would be | ||||
| Present value of expected payoff - (Present value of the expected payment+present value of the expected accrual payment)*credit default swap | |||||
| 3) | If the swap is a binary CDS, the present value of the expected payoffs would be | ||||
| Present value of the expected payment + PV of expected accrual payment = Present value of expected payoff of binary cds | |||||
