Krzys’ Ostaszewski, http://math.illinoisstate.edu/krzysio/krzys.html
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Spring 2000 SOA Course 6 Examination, Multiple Choice, Problem No. 3, also Dr.
Ostaszewski’s online exercise No. 4 posted June 11, 2005
You are given the following with respect to a bond with semi-annual coupon payments
priced to yield 8% nominal annual compounded semi-annually:
Semi-Annual
Payment
Present Value of
Period (t)
Payment at Yield
1
3.00
2.89
2
3.00
2.78
3
3.00
2.67
4
3.00
2.57
5
3.00
2.47
6
103.00
81.76
(2)
Calculate the convexity of the bond with respect to i , the nominal annual interest rate
compounded semiannually.
A. 1.29
B. 1.58
C. 8.78
D. 9.50
E. 17.56
Solution.
The actual exam question just asked for the convexity. But this author finds the approach
of the problem and its solution problematic, although valuable for learning. Let us study
this exam question a bit. It is unclear from the wording of the problem whether the yield
of 8% is the effective annual interest rate or the nominal annual interest rate compounded
semiannual interest rate (which is suggested by the semi-annual bond coupons). Let us
examine the effective annual interest rate implied by the present values of cash flows
given (note that we count time in years):
0.50
3
⎛ 3.00 ⎞
− 1 ≈ 7.76%. But also note that
• For cash flow at time 0.5, i = ⎜
≈ 2.89,
⎟
⎝ 2.89 ⎠
1.08 0.50
so that the discrepancy between 7.76% and 8.00% can be due to rounding error.
3
3.00
• For cash flow at time 1, i =
− 1 ≈ 7.91%. But also note that
≈ 2.78, so that the
1.08
2.78
discrepancy between 7.91% and 8.00% can be due to rounding error.
2
3
⎛ 3.00 ⎞ 3
− 1 ≈ 8.08%. But also note that
• For cash flow at time 1.5, i = ⎜
≈ 2.67, so
⎟
⎝ 2.67 ⎠
1.081.5
that the discrepancy between 8.08% and 8.00% can be due to rounding error.
1
3
⎛ 3.00 ⎞ 2
− 1 ≈ 8.04%. But also note that
• For cash flow at time 2, i = ⎜
≈ 2.57, so
⎟
⎝ 2.57 ⎠
1.08 2
that the discrepancy between 8.04% and 8.00% can be due to rounding error.
2
3
⎛ 3.00 ⎞ 5
− 1 ≈ 8.09%. But also note that
• For cash flow at time 2.5, i = ⎜
≈ 2.47, so
⎟
⎝ 2.47 ⎠
1.08 2.5
that the discrepancy between 8.09% and 8.00% can be due to rounding error.
1
⎛ 103.00 ⎞ 3
− 1 ≈ 8.00%. This one gives us exactly the yield.
• For cash flow at time 2, i = ⎜
⎝ 81.76 ⎟⎠
Given this, we will proceed under the assumption that 8% is the effective annual interest
rate. Note first that the sum of present values is 95.14. The convexity with respect to the
force of interest is:
2.89
2.78
2.67
2.57
2.47
81.76
0.5 2 ⋅
+ 12 ⋅
+ 1.5 2 ⋅
+ 22 ⋅
+ 2.5 2 ⋅
+ 32 ⋅
≈
95.14
95.14
95.14
95.14
95.14
95.14
≈ 8.10455644.
This is closest to answer C. But the original question asked for convexity. The Macaulay
duration is
2.89
2.78
2.67
2.57
2.47
81.76
0.5 ⋅
+ 1⋅
+ 1.5 ⋅
+ 2⋅
+ 2.5 ⋅
+ 3⋅
≈
95.14
95.14
95.14
95.14
95.14
95.14
≈ 2.78352954.
Convexity with respect to the interest rate i is:
1
1
C=
⋅D ≈
2 ⋅C M +
(1+ i )
(1+ i )2 M
1
1
⋅ 8.10455644 +
⋅ 2.78352954 ≈ 9.33477879.
2
1.08
1.08 2
This is closest to answer D. This is, of course, very confusing. The solution published by
the SOA did the following. It treated fractional times as whole numbers, i.e., counted
time in half years, and used the basic formula for convexity for fixed cash flows
1
C=
∑ w ⋅t (t + 1),
(1+ i )2 t≥0 t
as well as 4% as the effective semi-annual interest rate, so that it calculated convexity in
half-years squared as
1⋅ 2 2.89
2 ⋅ 3 2.78
3⋅ 4 2.67
4 ⋅ 5 2.57
⋅
+
⋅
+
⋅
+
⋅
+
2
2
2
1.04 95.14 1.04 95.14 1.04 95.14 1.04 2 95.14
5 ⋅ 6 2.47
6 ⋅ 7 81.76
+
⋅
+
⋅
≈ 35.1195311.
2
1.04 95.14 1.04 2 95.14
35.1195311
The convexity measure in years squared is therefore
≈ 8.77988278. Division
4
by four is caused by the fact that there are four half-years squared in one year squared.
Or, you can simply observe that the calculation produces convexity with respect to
≈
j=
i(2)
d 2P
1 d 2P
, assuming j = 4%, and that
=
. This gives answer C. There is a
2
2
4 dj 2
d i(2)
( )
problem with this approach. First, the convexity so calculated is with respect to the
nominal annual interest rate i ( 2 ) , and it is not the regular convexity calculated with
respect to the annual effective interest rate i. Moreover, the semiannual effective interest
3.00
rate implied by the cash flow at time 0.5 years is
− 1 ≈ 3.81%, not 4%. Also,
2.89
3
≈ 2.88 ≠ 2.89, so that the discrepancy cannot be explained by rounding error. The
1.04
1
⎛ 103.00 ⎞ 6
− 1 ≈ 3.92%, and
semiannual rate implied by the cash flow at time 3 years is ⎜
⎝ 81.76 ⎟⎠
103.00
≈ 81.40 ≠ 81.76. The values given in the problem are inconsistent with the
1.04 6
interest rate used in the published SOA solution. Let us ask ourselves one more question:
what is the value of convexity with respect to i ( 2 ) ? Convexity with respect to nominal
annual interest rate i ( m ) is
1
1
1
C (m) =
DM .
2 CM +
2
⎛
⎛
i(m) ⎞
i(m) ⎞ m
⎜⎝ 1 + m ⎟⎠
⎜⎝ 1 + m ⎟⎠
Therefore,
C (2) =
1
⎛
i(2) ⎞
1
+
⎜⎝
2 ⎟⎠
2
⋅ CM +
1
1
⋅ ⋅ DM =
⎛
i(2) ⎞ 2
1
+
⎜⎝
2 ⎟⎠
2
1
1 1
⋅ 8.10455644 +
⋅ ⋅ 2.78352954 ≈ 8.79289001.
1.08
1.08 2
This is closest to answer C.
Answer C.
=
© Copyright 2005 by Krzysztof Ostaszewski.
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