Fixed Income Securities and Risk Management
Assignment
Question 1: This question has two related parts, (a) and (b).
(a) Use the daily yields in the table below to compute a daily standard deviation of yields. Next annualize the daily standard deviation just calculated first using 365 calendar days, then using 250 trading days in a year. Finally, construct a 4 -day moving average equal weight volatility forecast for each day in the period from Day 5 to Day 24, assuming that the expected value of the daily change in yield is zero. Please show your calculations, including formulas used. Please note: for this question 1) if you use Excel please attach your Excel worksheet to your assignment and 2) keep at least 6 decimal places in both your calculations and your final answers.
|
Day
|
Yield (%)
|
|
0
|
4.380
|
|
1
|
4.393
|
|
2
|
4.430
|
|
3
|
4.428
|
|
4
|
4.522
|
|
5
|
4.648
|
|
6
|
4.656
|
|
7
|
4.595
|
|
8
|
4.562
|
|
9
|
5.208
|
|
10
|
4.454
|
|
11
|
4.404
|
|
12
|
4.659
|
|
13
|
4.904
|
|
14
|
4.820
|
|
15
|
3.933
|
|
16
|
4.907
|
|
17
|
4.650
|
|
18
|
4.814
|
|
19
|
4.766
|
|
20
|
4.692
|
|
21
|
4.687
|
|
22
|
4.521
|
|
23
|
4.435
|
(b) Plot in the same graph for each day between Day 5 and Day 24 (inclusive) the computed daily standard deviation of yields and volatility forecast obtained in part (a) above. Please briefly comment on the volatility forecasting method in light of the graph.
Question 2 (Binomial model)
This question has two independent parts, (a) and (b).
(a) Use the binomial tree in the following table to price a putable/callable bond with the following characteristics: a par value of $100, 6% coupon rate payable annually, 4 years of maturity, callable at $102.5, $101.5, and $100 in Year 1, Year 2, and Year 3, respectively, and putable at par starting in Year 1 and thereafter. What is the value of the embedded put option? What is the value of embedded call option? Please show your calculations, including formulas used.
|
Interest rate
|
(in %)
|
Interest rate
|
(in %)
|
|
r0
|
3.5000
|
r2,LL
|
3.7492
|
|
r1,H
|
5.9196
|
r3,HHH
|
12.0003
|
|
r1,L
|
3.9680
|
r3,HHL
|
8.0441
|
|
r2,HH
|
8.3440
|
r3,HLL
|
5.3921
|
|
r2,HL
|
5.5931
|
r3,LLL
|
3.6144
|
(b) Let us assume that the underlying stock price trades at $50.00 with a 30% annual volatility. A convertible bond with a 9 months maturity has a conversion ratio of 20. The convertible bond has a $1,000.00 face value, a 4% annual coupon. Further assume that the risk-free rate is a (continuously compounded) 10%, while the yield to maturity on straight bonds issued by the same company is a (continuously compounded) 15%. We also assume that the call price is $1,100.00. Use a 3 periods binomial model (t/n=3 months, or ¼ year) to value this convertible bond.
Question 3 (Mortgage-backed securities)
This question has two independent parts, (a) and (b).
(a) Complete the following table (in thousands of dollars) assuming a prepayment speed of 165 PSA. The original mortgage balance is $100,000,000, the annual passthrough rate is 9% monthly compounded, and the weighted average maturity (WAM) is 306 months. Assume American mortgages of original maturity of 360 months. WAC is also 9% monthly compounded. Please show your calculations, including formulas used.
|
Month
|
Outstanding balance
|
SMM
|
Interest
|
Scheduled payment
|
Prepayment
|
Total Principal
|
Total Cash flow
|
|
1
|
100.000
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
|
Principal Cash flow
(b) Consider a $25,000,000 passthrough with a WAC of 6.5% and a WAM of 354 months. Given the prepayment speeds assumed in the following table, compute, for each month in the table: SMM, CPR, outstanding balance, mortgage payment, interest, scheduled principal repayment, prepayment, total principal, and total cash flow. Please show your calculations and do not use Excel.
Months Prepayment
|
Months from now
|
Prepayment speed
|
|
1
|
120 PSA
|
|
3
|
100 PSA
|
|
5
|
132 PSA
|
Question 4 (Bullet portfolio, barbell portfolio, and CDO) This question has two independent parts, (a) and (b).
Use the following information to answer part (a).
(a) You are given the following Treasury zero coupon rates (on a BEY basis): 3.9% at six months, 4.9% at three years, and 5.6% at five years, respectively. Suppose you have $100,000 to invest.
(i) What par amount would you invest in each zero coupon bond if you decide to hold a 3-year bullet portfolio?
(ii) What par amount would you invest in each zero coupon bond if you decide to hold a barbell portfolio with equal par amounts in the 6-month and 5-year zeros?
(iii) How does the value of your portfolios in part (4ai) and (4aii) above change if the 6- month zero coupon rate decreases by 10 basis points (bps), the 3-year zero coupon rate increases by 5 bps, whereas the 5-year zero coupon rate remains unchanged? How does your answer differ if you use both duration and convexity or duration only to approximate the changes in the portfolio value?
(b) A CDO of $100 million notional value with five year maturity has the following structure:
|
Tranche
|
Size (million)
|
Spread (bps)
|
O/C target
|
I/C target
|
|
A
|
60
|
50
|
1.35
|
1.6
|
|
B
|
10
|
200
|
1.25
|
1.4
|
|
C
|
10
|
500
|
1.13
|
1.2
|
|
Equity
|
20
|
|
|
|
The CDO collateral has an average annual coupon rate of 10% payable semiannually. The CDO manager charges a deal structuring fee of 100 bps of the notional amount amortized over the life span of the CDO at 5% per annum. The yearly manager fee is 40 bps for the senior tranche and 20 bps for the subordinated tranches. The current risk free interest rate is 6% on a BEY basis.
(i) If there is no default, find the semiannual cash flow distributions to deal restructuring fees, asset manager fees, payments to tranches A, B, and C, and payment to equity tranche. Calculate the O/C and I/C ratios for tranches A, B, and C. Are these ratios above their target levels?
(ii) Assume the collateral portfolio loses $10 million before the first coupon payment date. Redo all of the calculations in part (4bi) above. Are the O/C and I/C ratios for tranches A, B, and C above their target levels? If not, how should we adjust?