Krzys’ Ostaszewski, http://www.math.ilstu.edu/krzysio/, Exercise 68, 9/2/6
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Casualty Actuarial Society Course 8 Examination, May 2000, Problem No. 14, parts
(b) and (c) (slightly modified)
You are given the following information about a bond:
• The term to maturity is 2 years.
• The bond has a 9% annual coupon rate, paid semiannually.
• The annual bond-equivalent yield-to-maturity is 8%.
• The par value is $100.
Calculate the (regular, not Macaulay) M-squared of the bond.
A. 1.72
B. 6.78
C. 12.75
D. 20.25
E. 101.75
Solution.
The coupon amount is $4.50. The effective interest rate applicable is 4% per half a year,
and if we write i for the effective annual interest rate, we have 1 + i = 1.04 2. There are
four coupon periods until maturity. Given that, the price of this bond is
4.5a4 4% + 100 !1.04 "4 # 101.81.
Macaulay duration of this bond is
4.5
3 4.5
104.5
1 4.5
!
+ 1!
+ !
+ 2!
2
3
1.04
2 1.04
1.04 4 # 1.87574389.
2 1.04
4.5a4 4% + 100 !1.04 "4
Regular duration of the bond is
4.5
3 4.5
104.5
1 4.5
!
+ 1!
+ !
+ 2!
2
3
1
1.04
2 1.04
1.04 4 # 1.73423067.
! 2 1.04
2
"4
1.04
4.5a4 4% + 100 !1.04
Macaulay convexity of this bond is
4.5
4.5
4.5
104.5
0.5 2 !
+ 12 !
+ 1.5 2 !
+ 22 !
2
3
1.04
1.04
1.04
1.04 4 # 3.6492821774.
4.5a4 4% + 100 !1.04 "4
Regular convexity is
1
1
C=
C !
2 DM +
(1 + i )
(1 + i )2 M
1
1
"1.87574389 +
" 3.6492821774 ! 4.722815438.
4
1.04 4
1.04
Therefore, M-squared is
M 2 = C ! D 2 " 4.722815438 ! 1.73423067 2 " 1.715259421.
Answer A.
!
© Copyright 2006 by Krzysztof Ostaszewski.
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