Exercises

lr (Amortization) A debt of \$25,000 is to be amortized over 7 years at 1% interest What value of monthly payments will achieve this?

2. (Cycles and annual worth o) Given a cash flow stream X — (a'0, .v,, .v2, , .v„), a new stream of infinite length is made by successively repeating the corresponding finite stream The interest rate is / Let P and A be the present value and the annual worth, respectively, of stream X Finally, let P^ be the present value of stream Find A in terms of and conclude that A can be used as well as Pco for evaluation purposes.

3. (Uncertain annuity o) Gavin's grandfather, Mr Jones, has just turned 90 years old and is applying for a lifetime annuity that will pay \$10,000 per year, starting 1 year from now, until he dies He asks Gavin lo analyze it for him Gavin finds that according to statistical summaries, the chance (probability) that Mr Jones wilt die at any particular age is as follows:

 age 90 91 92 93 94 95 96 97 98 99 100 101 probability 07 08 09 10 0.1 10 .10 .10 10 07 05 04

Then Gavin {and you) answer the following questions: {a) What is the life expectancy of Mr Jones?

(b) What is the present value of an annuity at 8% interest that has a lifetime equal to Mr Jones's life expectancy? (For an annuity of a nonintegral number of years, use an averaging method.)

(c) What is the expected present value of the annuity?

4. (APR) For the mortgage listed second in Table 3 1 wiiat are the total fees?

5. (Callable bond) The Z Corporation issues a 10%, 20-year bond at a time when yields are 10% The bond has a call provision that allows the corporation to force a bond holder to redeem his or her bond at face value plus 5%, After 5 years the corporation finds that exercise of this call provision is advantageous What can you deduce about the yield at that time? {Assume one coupon payment per year)

6. (The biweekly mortgage®) Here is a proposal that has been advanced as a way for homeowners to save thousands of dollars on mortgage payments: pay biweekly instead of monthly Specifically, if monthly payments are a\ it is suggested that one instead pay .v/2 every two weeks (for a total of 26 payments per year) This will pay down the mortgage faster, saving interest The savings are surprisingly dramatic for this seemingly minor modification—often cutting the total interest payment by over one-third. Assume a loan amount of \$100,000 for 30 years at 10% interest, compounded monthly

{a) Under a monthly payment program, what are the monthly payments and the total interest paid over the course of the 30 years? (b) Using the biweekly program, when will the loan be completely repaid, and what are the savings in total interest paid over the monthly program? (You may assume biweekly compounding for this part)

7. (Annua! worth) One advantage of the annual worth method is that it simplifies the comparison of investment projects that are repetitive but have different cycle times Consider the automobile purchase problem of Example 2 7. Find the annual worths of the two (single-cycle) options, and determine directly which is preferable

8. (Variable-rate mortgage©) The Smith family just took out a variable-rate mortgage on their new home. The mortgage value is \$100,000, the term is 30 years, and initially the interest rate is 8% The interest rate is guaranteed for 5 years, after which time the rate will be adjusted according to prevailing fates The new rate can be applied to their loan either by changing the payment amount or by changing the length of the mortgage

(a) What is tiie original yearly mortgage payment? (Assume payments are yearly )

(b) What will be the mortgage balance after 5 years?

(c) If the interest rate on the mortgage changes to 9% after 5 years, what will be the new yearly payment that keeps the termination time the same?

(cl) Under the interest change in (t), what will be the new term if the payments remain the same?

9. (Bond price) An 8% bond with IB years to maturity has a yield of 9% What is the price of this bond?

10. (Duration) Find the price and duration of a 10-year, 8% bond that is trading at a yield of 10%

11. (Annuity duration o) Find the duration D and the modified duration D^ of a perpetual annuity that pays an amount A at the beginning of each year, with the first such payment being I year from now Assume a constant interest rate r compounded yearly [Hint- It is not necessary to evaluate any new summations ]

12. (Bond selection) Consider the four bonds having annual payments as shown in Table 3 9 They are traded to produce a 15% yield

{a) Determine the price of each bond

(b) Determine the duration of each bond (not the modified duration)

(c) Which bond is most sensitive to a change in yield?

(d) Suppose you owe \$2,000 at the end of 2 years Concern about interest rate risk suggests that a portfolio consisting of the bonds and the obligation should he immunized If VA, Vu, Vc, and VD are the total values of bonds purchased of types A, B, C, and D, respectively, what are the necessary constraints to implement the immunization? [Hint: There are two equations (Do not solve.)]

 End of year payments Bond A Bond B Bond C Bond D Year 1 100 50 0 0 + 1000 Year 2 100 50 0 0 Year 3 100 + 1000 50 + 1000 0 + 1000 0

{e) In order to immunize the portfolio, you decide to use bond C and one other bond Which other bond should you choose? Find the amounts (in total value) of each of these to purchase

(/) You decided in (e) to use bond C in the immunization Would other choices, including perhaps a combination of bonds, lead to lower total cost?

13. (Continuous compounding o) Under continuous compounding die Macaulay duration becomes

Find dP/dk in terms of D and P

14. (Duration limit) Show that the limiting value of duration as maturity is increased to infinity is

For the bonds in Table 3 6 (where A, = .05 and m ™ 2) we obtain D —> 20.5 Note that for large k this limiting value approaches 1/m, and hence the duration for large yields tends to be relatively short

15. (Convexity value) Find the convexity of a zero-coupon bond maturing at time T under continuous compounding (that is, when m -> oo).

16, (Convexity theorem o) Suppose that an obligation occurring at a single time period is immunized against interest rate changes with bonds that have only nonnegative cash flows (as in the X Corporation example) Let P{k) be the value of the resulting portfolio, including the obligation, when the interest rate is / + k and / is the current interest rate By construction P(0) = 0 and P'(0) = 0 In this exercise we show that P(0) is a local minimum; that is, P"(0) > 0 (This property is exhibited by Example 3 10)

Assume a yearly compounding convention The discount factor for time t is d,(k) ~ (1 -J- r 4- A.)"""'. Let d, = d,(0). For convenience assume that the obligation has magnitude I and is due at time 1 The conditions for immunization are then

(a) Show that for all values of a and fi there holds

P"(0)(I +/')2 = ]TV + at + P)c,d, - (r -fa/ -f- fi)d-t i

(b) Show that a and fi can be selected so that the function t1 + at + f) has a minimum at t and has a value of I there. Use these values to conclude that P"(0) > 0