Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
ISE 563 Homework 3
Due April 8 2024
Exercise 2.1
Given the following:
• Stock equals 100
• Stock volatility of 40%
• Debt maturity of 5 years
• Debt Face value of 150
• Risk-free rate of 3%
Use Merton’s model to find the asset value and asset volatility?
What is the risk-neutral probability of default over the debt’s maturity and the annualized default probability?
What is the market spread for the debt? What is the implied Recovery Rate?
Exercise 2.2
Given CDS spreads with the following tenors.
|
Tenor |
CDS Spread |
RN Default Prob |
|
1 |
1.0% |
|
|
2 |
1.2% |
|
|
3 |
1.4% |
|
|
4 |
1.6% |
|
|
5 |
1.7% |
|
Risk free rate of 5% flat continuously compounded and a recovery rate of 40%.
What are the risk neutral default probabilities for each year?
What is the value of an existing 5-year CDS with a 200bp spread? What is the par 5-year Binary CDS spread?
Exercise 2.3
Given the following transition matrix and credit spreads per rating category. Assume you initially have a 10-year zero-coupon BBB bond. Over the next ten years, the bonds can change ratings including to default. If default occurs, the LGD is 70%. The 1-year Base transition matrix is given below.
|
|
XB1 Final: Rescaled and Modified Base 1-Year Transition Matrix |
Credit Spread |
||||||||
|
Rating |
AAA |
AA |
A |
BBB |
BB |
B |
CCC |
CC |
D |
|
|
AAA |
0.9501 |
0.0437 |
0.0061 |
0.0000 |
0.0000 |
0.0000 |
0.0000 |
0.0000 |
0.0000 |
0.3% |
|
AA |
0.0052 |
0.9457 |
0.0434 |
0.0056 |
0.0000 |
0.0000 |
0.0000 |
0.0000 |
0.0001 |
0.5% |
|
A |
0.0000 |
0.0175 |
0.9220 |
0.0558 |
0.0030 |
0.0010 |
0.0000 |
0.0000 |
0.0006 |
1.0% |
|
BBB |
0.0000 |
0.0012 |
0.0355 |
0.9200 |
0.0361 |
0.0050 |
0.0006 |
0.0000 |
0.0016 |
1.8% |
|
BB |
0.0000 |
0.0000 |
0.0014 |
0.0523 |
0.8505 |
0.0832 |
0.0052 |
0.0007 |
0.0066 |
3.0% |
|
B |
0.0000 |
0.0000 |
0.0009 |
0.0018 |
0.0505 |
0.8577 |
0.0524 |
0.0030 |
0.0336 |
5.0% |
|
CCC |
0.0000 |
0.0000 |
0.0000 |
0.0028 |
0.0060 |
0.1678 |
0.5430 |
0.0149 |
0.2654 |
9.0% |
|
CC |
0.0000 |
0.0000 |
0.0000 |
0.0000 |
0.0146 |
0.0801 |
0.1166 |
0.1239 |
0.6648 |
12.0% |
|
D |
0.0000 |
0.0000 |
0.0000 |
0.0000 |
0.0000 |
0.0000 |
0.0000 |
0.0000 |
1.0000 |
|
A) What is the distribution of ratings for that bond at year 1, year 5,and year 10?
B) What is the expected value of the bond at 1, 5, and 10 years taking the rating migrations and default into account?
Assume a Great Financial Crisis (GFC) type event occurs in year 1 with the transition matrix given below. For years 2 through 10 use the transition matrix above.
How do your answers to A) and B) in the first part change under the GFC stress?
Exercise 2.4
Price a basket option with the following features.
• Basket with 20 credits
• Each has an annual RN default probability of 1%.
• 30% correlation between assets
• Simulate with annual steps with default occurring at end of year. Collect premiums for entire year (full accrual)
o Do as many simulations as possible.
• Risk-free rate of 5%.
• 100 notional and LGD of 100%
• Price the following three tranches?
o Tranche A loses principle starting with the 13th default and has coupon of 4%.
o Tranche B loses principle starting at 5th default and all principle is gone at 12th default and has coupon of 6%.
o Tranche C equity tranche taking first loss with all principle gone at 4th default and has coupon of 12%