Fixed Income Securities and Risk Management

Fixed Income Securities and Risk Management Course Number: AP/ADMS 4504 A & B

Term: Fall 2016 Assignment #2

The due date for this assignment is Nov 29 (Tuesday) for Section A and Dec 5 (Monday day) for Section B. 1.This assignment is to be done individually. You must sign and submit the standard cover page. 2.This assignment is to be either handwritten or printed out. Work that is too difficult to read due to messiness and poor handwriting will receive zero credit. You must show your work to receive full credit. 3.This assignment has five questions and carries a total mark of 100 points. 4.Late assignments will not be accepted whether for technical or any other reason. 5.Decimal places: please keep at least 4 in your calculations and at least 6 for interest rates and at least 2 in your final answers.

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. (10 marks)

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 24 4.424

(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.

(5 marks)

Question 2 (Binomial model) (25 marks) 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. (10 marks)

Interest rate (in %) Interest rate

(in %)

r 0 3.5000 r

2,LL 3.7492

r 1,H

5.9196 r 3,HHH

12.0003

r 1,L

3.9680 r 3,HHL

8.0441

r 2,HH

8.3440 r 3,HLL

5.3921

r 2,HL

5.5931 r 3,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. (15 marks)

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. (10

marks)

Month Outstanding SMM Interest Scheduled Prepay ment

Total Total balance payment Principal Cash flow

1 100,000 2

(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. (10 marks)

Months Prepayment from now 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. (20 marks) (i) What par amount would you invest in each zero coupon bond if you decide to hold a

3-year bullet portfolio? ( 7 marks)

(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? (7 marks)

(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? (6 marks)

(b) A CDO of $100 million notional value with five year maturity has the following

structure: (20 marks)

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? (10 marks)

(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? (10

marks)