Krzys’ Ostaszewski, http://www.math.ilstu.edu/krzysio/, Exercise 26, 11/12/5 Author
of the Course FM manual available at:
http://smartURL.it/krzysioFM (paper) or http://smartURL.it/krzysioFMe (electronic)
Instructor for online seminar for exam FM: http://smartURL.it/onlineactuary
Your company has a liability, which calls for a single payment of 1000 exactly due in ten
years, and it funds it with a one-year zero-coupon bond with a price of 200, and a twentyyear zero-coupon bond with a price of 800. Current continuously compounded interest
rate is 4% and the yield curve is flat. Calculate the ratio of the Macaulay duration to the
Macaulay convexity of your company’s surplus.
A. 0.0375
B. 0.1755
C. 0.2985
D. 0.7895
E. 1.0395
Solution.
The present value of the liability is
$1000e!10"0.04 # $670.32.
Therefore, current surplus equals
$1, 000.00 ! $670.32 = $329.68.
Macaulay duration of an individual cash flow is its time to payment. Macaulay duration
of the surplus is the weighted average of the durations of individual cash flows, i.e., here:
200
670.32
800
!1 "
!10 +
! 20 # 28.8061.
329.68
329.68
329.68
Macaulay convexity of an individual cash flow is the square of its time to payment.
Macaulay convexity of the surplus is the weighted average of individual cash flows
convexities, in this case:
200
670.32
800
!12 "
!10 2 +
! 20 2 # 767.9204.
329.68
329.68
329.68
The ratio of the two equals
28.8061
! 0.0375.
767.9204
Answer A.
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