Krzys’ Ostaszewski, http://www.math.ilstu.edu/krzysio/.
Author of a study manual for exam FM available at:
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November 2005 Course FM/2 Examination, Problem No. 2, and Dr. Ostaszewski’s
online exercise No. 257 posted April 17, 2010
Calculate the Macaulay duration of an eight-year 100 par value bond with 10% annual
coupons and an effective rate of interest equal to 8%.
A. 4
B. 5
C. 6
D. 7
E. 8
Solution.
The Macaulay duration of this bond equals, by definition,
1⋅10 ⋅1.08 −1 + 2 ⋅10 ⋅1.08 −2 + … + 8 ⋅10 ⋅1.08 −8 + 8 ⋅100 ⋅1.08 −8
=
10 ⋅1.08 −1 + 10 ⋅1.08 −2 + … + 10 ⋅1.08 −8 + 100 ⋅1.08 −8
a8 8% − 8 ⋅1.08 −8
−8
+ 800 ⋅1.08 −8
10 ⋅
10 ⋅ ( Ia )8 8% + 800 ⋅1.08
0.08
=
=
≈ 5.9891.
10 ⋅ a8 8% + 100 ⋅1.08 −8
10 ⋅ a8 8% + 100 ⋅1.08 −8
You can also calculate for the portion of this bond that pays an 8% coupon, which
represents a bond trading at par, as having the price of 100, and Macaulay duration of
a8 8% ≈ 6.2064, while the additional payment of 2 per year over eight years has the value
of 2a8 8% = 2
a8 8% ⋅1.08 −1 ≈ 11.4933, and the Macaulay duration of (note that we have
m = 1, n = 8, d = 0.08 ⋅1.08 −1 )
n
1.08
8
1
=
−
≈ 4.0985.
n
(m) −
d
(1 + i ) − 1 0.08 1.08 8 − 1
Thus the Macaulay duration of the bond can be calculated as the weighted average of its
two pieces
11.4933
100
⋅ 6.2064 +
⋅ 4.0985 ≈ 5.9891.
111.4933
111.4933
Answer C.
© Copyright 2010 by Krzysztof Ostaszewski.
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Exercises from the past actuarial examinations are copyrighted by the Society of
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