FM for Actuaries

Bond Management 283
8.15 Using Exercise 8.12 or otherwise, show that
(a) the convexity of a n-year coupon bond with annual modified coupon
rate of g is
v 2 { g [
2(Ia)n⌉
(ga n⌉ + v n +2ä
) i
n⌉
− n(n +3)v n] + n(n +1)v n} ,
(b) the convexity of a n-year coupon bond with annual coupons, redeemable
and priced at par, is 2v(Ia) n⌉
.
8.16 Use Exercise 8.15 to calculate the convexity in Example 8.6.
8.17 If the cash-flow stream C t do not have a regular structure, to compute the
present value, the Macaulay duration and the convexity of the cash flows we
may set up a table similar to Table 8.2 using Excel. The following format
may be considered:
Time t C t PV(C t ) t×PV(C t ) (t 2 + t)×PV(C t )
1
2
.
n
Total
After setting up the table, P , D, D ∗ and C can be found using the appropriate
column sums.
(a) Using Excel, calculate the present value, modified duration and convexity
of a decreasing annuity that pays 50 at time 1, 47 at time 2, 44 at
time 3, ···, 2 at time 17 at i =4%.
(b) Using the results in (a), approximate the present value of the annuity if
i increases by 0.5%. Compare the answer with the exact value.
8.18 Harry is the owner of a small grocery store and he has only one employee,
Dick. Dick is going to retire after 4 years and Harry promises to pay Dick
$7,000 once a year for a 5-year period, the first payment starting 5 years from
now. Assume that Dick will be alive for at least 10 more years, that the yield
curve is flat and that the prevailing interest rate is 4% effective.