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# Duration is the Central Measure of IRR to be Introduced Next Week(1)

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Duration is the Central Measure of IRR to be Introduced Next Week(1)

06-4

BOSTON COLLEGE

Edward J. Kane

Financial Institution Management

WEEK 4 EXERCISES

Duration is the Central Measure of IRR to be Introduced in Week 4

The Following Exercises will Be Discussed in Class

1. Please calculate the Macaulay Duration of a 10-year, 6-percent annual-coupon bond

when the market rate is 10 percent. Show all work.

2. The “weighted average life” of a security is found by weighting the “futurity” in years of each promised cash flow by the percentage of the security’s total cash flows that it

represents. Find the weighted average life of the bond described in question #1. Why is the bond’s weighted average life larger than its duration?

3. Using r to represent the relevant market rate of interest, the following formula gives the duration D of a perpetual annuity (or “consol”) that pays C dollars once a year

1;rforever: Dr

a. What is the present discounted value (PDV) of this perpetuity?

b. What is the present discounted value of the part of this perpetuity that accrues

after n years?

c. Why would zero be the present value of a costless forward contract to deliver

1the perpetuity in n years to another party in exchange for receiving the price r

at that time?

d. What is the combined present value to day of the perpetuity and a forward

1contract to sell the bond for in n years? r

e. Using the idea of a “forward sale,” try to explain either formally or intuitively

why the duration of an n-year annuity might turn out to be:

06-4

1;rn.D nn1r(1;r)

4. Why would the duration of a 10-year annuity be less than the duration of a perpetuity?

Why would the duration of a 10-year coupon bond be larger than that of a 10-year

annuity?

Follow-on Question to Be Addressed in Week 6:

5. Extension of Misplaced Trust Company Problem

Remember that, on the day the Misplaced Trust Co. opened as a new bank,

borrowers convinced the bank to make \$100 million in three-year loans, while all

depositors put their money in one-year certificate accounts. The principal and interest

on all loans are compounded at 8 percent and payment is due at maturity. Simple

interest is paid on the deposits once a year. Finally, the bank holds no other assets

and issues \$80 million in deposits at 5 percent per annum, raising the rest of its

loanable funds entirely from stockholders.

Assuming the loans are certain to be repaid on schedule and that all market yields

move up and down with r, please calculate what you regard to be an adequate A

measure of the interest-rate risk exposure the bank’s portfolio passes through to its

stockholders.

The bank’s managers ask you to help them to design a \$10.73 million issue of

subordinated debt that would eliminate interest-rate risk as you chose to measure it in

part d. Suppose that the yield curve facing the bank on such bonds was flat at 5

percent. What choice of bonds would accomplish this goal? Show that the bond

issue has the desired effect.

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