A. Determine the future periodic spot interest rates, at one period from now, on a onestep Ho-Lee model, using current periodic spot interest rate of A1 percent, current spot

rate over two periods of A2 percent, and an annual interest rate volatility of .005 (.5%).

Then consider the current spot rate over three periods of A3 percent to find the future

periodic interest rates in a two-step Ho-Lee model. The period is A4 months. Find

the following.

1. The trend (lambda1) in interest rate in the first period.

2. The trend (lambda2) in interest rate in the second period.

3. The future periodic rate on the upside at the end of one period.

4. The future periodic rate on the downside at the end of one period.

5. The future periodic rate at the up-most branch at the end of two periods.

6. The future periodic rate at the middle branch at the end of two periods.

7. The future periodic rate at the down-most branch at the end of two periods.

8. The future periodic rate at the down-most branch at the end of two periods

B. The cash futures price of a 3-month zero coupon bond with a face value of $100 for

delivery in B1 months from now is B2 dollars. Suppose that the current spot interest

rate for a term of B3 months is B4 per cent per annum. Assume continuous

compounding to answer the following:

9. The forward rate of interest for period B1 months to B1+3 months.

10. The spot rate of interest for period 0 to B1 months.

11. The current fair value of a B1-month zero coupon bond with $100 face.

C. Consider C1-month spot interest rates evolving in the following two-step binomial

tree over C2 months, i.e., with C1 months in each of the next two steps. The current

C2-month spot interest rate is 5.15% and the C3-month spot interest rate is 5.3%. Find

the following by assuming monthly compounding.

12. The risk neutral probability for the up move in first step.

13. The risk neutral probability for the up move in second step.

14. The current fair value of a C1-month European call option with a strike price of

$974 written on a C2-month zero coupon bond with face value $1000.

15. The current fair value of a C2-month European put option with a strike price of

$994 written on a C3-month zero coupon bond with face value $1000.

D. Consider D1-month spot interest rates evolving in the following three-step binomial

tree over D4 months, i.e., with D1 months in each of the next three steps. The current

D3-month spot interest rate is 5.15%, the current D2-month spot interest rate is 5.3%,

and the current D4-month spot interest rate is 5.45%. This three-step problem involves

calculation of just the third step, because it is an extension of the two-step Problem C

5.30%

4.30%

5.10%

6.20%

4 50 5.10%

4.00%

3

and, so, you should simply take the risk-neutral probabilities obtained in Problem C to

answer the following by assuming monthly compounding.

16. The risk neutral probability for the up move in the last (third) step.

17. The current fair value of a D3-month European call option with a strike price

of $974 written on a D4-month zero coupon bond with face value $1000.

18. The current fair value of a D2-month European put option with a strike price of

$994 written on a D4-month zero coupon bond with face value $1000.

5

5

E. Suppose that you are long in an original portfolio of 20-year bonds with a total

face value of E1 million dollars and that you want to hedge the portfolio value against

fluctuating interest rates by trading on a 10-year bond and a 30-year bond. You have

estimated (given): DV0110 = E2, DV0120 = E3, and DV0130 = E4 per $100 face value

of 10-year, 20-year and 30-year bonds, respectively. You have also used yield data on

these bonds to estimate the coefficients, b=1.3 and c = 1.6, in the following regression:

20 10 30 . t t tt ∆ = +∆ +∆ + y a by cy ε

Using the given data, determine your optimal trading strategy for the 10-year and 30-

year bonds to hedge 20-year bonds. Report answers for the following:

19. Face value in dollars of 10 year bonds needed for hedging.

20. Face value in dollars of 30 year bonds needed for hedging.

5.1%

5.3%

4.3%

5.1%

5.3%

4.3%

3.7%

4.0%

7.1%

6.2%

Page 4 of 6 4

After hedging, suppose that you observe that the 20-year yield moves by E5 basis

points when the 10-year yield moves by 1 basis point and that the 20-year yield moves

by E6 basis points when the 30-year yield moves 1 basis point. Then:

21. How much will be your profit or loss corresponding to 30-year yield increasing

by 1 basis point?

F. A collared floater is like a variable rate bond, but with an upper limit and a lower

limit on the rate of coupon payment. Consider, for example, annual rates coupon

payment and one-year spot interest rates. In the following table, the current one-year

spot interest rate is denoted by r in column (1). Depending on the current spot interest

rate, the collared floater with a face value of F1 dollars will pay the holder of this

floater a sum a year later, as given in column (2) of the table. The binomial tree below

gives the evolution of one-year spot interest rates over a four year period.

22. What is the current fair value of the four-year collared floater if the risk-neutral

probabilities in the binomial tree are 0.5 and 0.5 for the up and down moves,

respectively?

Current Rate, r

(1)

Collared Floater with

face value F1 pays at

the end of the year

(2)

r < 5% F2

5% ≤ r ≤ 8% F1 x r%

r > 8% F3

4.5%

6.2%

5.2%

6.00%

9.50%

6.25%

4.75%

5.50%

10.50%

8.75%

Page 5 of 6 5

G. A four-year participating cap pays an interest based on a nominal principal of G1,

at the end of the year if the current annual spot interest rate, r, is higher than 5%. If

the current annual spot rate, r, is less than 5%, the participating cap obligates its holder

to lose 30% of (5%-r%) on the nominal principal at the end of the year. The payments

are given in column (2) of the following table, given the rates in column (1). The

binomial tree below gives the evolution of one-year spot interest rates over a four year

period.

23. What is the current fair value of the participating cap if the risk-neutral

probabilities in the binomial tree are 0.5 and 0.5 for up and down moves,

respectively?

Current rate, r

(1)

Buyer gets at end of year

(2)

r > 5% (r% – 5%)xG1

r ≤ 5% -.30(5% – r%)xG1

H. What is the risk premium (spread) of a H1 percent annual coupon, 3.35 year

maturity, and 100-face value bond? The bond pays the annual coupon in two equal

4.0%

5.3%

3.7%

4.8%

6.2%

4.3%

3.1%

3.3%

8.7%

6.8%

Page 6 of 6 6

installments and trades now at a quoted spot price of H2. Assume continuous

compounding and use the term structure of discount factors for riskless bonds, d(t) =

1 – .03t – .0075t2 + .0015t3

. Report answer for the following (Hint: rate at which to

discount future cash flows of a risky bond to find the current value is (as per CAPM)

the riskless rate which changes with the term of the cash flow plus risk premium which

is fixed across all terms; the easier way to solve for risk spread by the method of trial

and error, as done an equation solver):

24. Risk Spread (premium) of the bond

The price is based on these factors:

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