CHAPTER 9
5.
Suppose an initially flat term structure with all interest rates equal to 6 percent. Compute
the price change for a one-year par bond if the interest rate increases by 2 percent. Then
compute the price change on a perpetual par bond if the interest rate on this bond
increases by 1 percent. Compute the Durations of each bond. Which bond has a greater
percentage price change and why?
1-Year Par Bond:
P0 
C + Par
1 + R 0,1
100 =
6 + 100
1.06
Spot Rate rises to 8%
P1 =
6 + 100
= 98.15
1.08
DUR = 1
% change = 1.85%
Perpetual Bond:
P0 =
C
R
100 =
6
0.06
Spot Rate rises to 7%
P1 =
6
= 85.71
0.07
DUR =
1 + y 1.06
=
= 17.67
y
0.06
% change = 14.29%
6.
Suppose the one-period spot interest rate is 4 percent and the forward interest rate is 8
percent. Compute the holding period return over the next period for a two-period zerocoupon bond for each of the following values for next period’s spot interest rate: 5
percent, 8 percent, 12 percent.
R0,1 = 4%
P0 =
Par
(1 + R 0,1)(1 + f 0,2)
P1 =
Par
1 + R 0,1
f0,2 = 8%

HPR = P1 P0
P0
P0 =
100
= 89.03
(1.04)(1.0 8)
R1,1 = 5%:
P1 =
100
= 95.24
1.05
HPR =
95.24  89.03
= 6.97%
89.03
R1,1 = 8%:
P1 =
100
= 92.59
1.08
HPR =
92.59 - 89.03
= 4%
89.03
R1,1 = 12%:
P1 =
100
= 89.29
1.12
HPR =
89.29 - 89.03
= .29%
89.03
7.
You observe the following term structure. What term structure theories are consistent
with it?
Maturity (Years)
1
2
3
4
Spot Rate
0.12
0.14
0.05
0.04
f0,2:
(1.14)2 = (1.12)(1 + f0,2)
f0,2 = 16.04%
f0,3:
(1.05)3 = (1.14)2(1 + f0,3)
f0,3 = -10.92%
f0,4:
(1.04)4 = (1.05)3(1 + f0,4)
f0,4 = 1.06%
No term structure theories have negative forward rates.
8.
Assume the combined theory explains the term structure. The one-period spot rate is 5
percent, the expected spot rate next period is 7 percent and the liquidity premium is 2
percent. Compute the forward interest rate.
R0,1 = 5%
E(R1,1) = 7%
L = 2%
f0,2 = 7% + 2% = 9%
9.
Suppose that we observe the following prices on strips with $100 par values: S1 =
$97.09; S2 = $92.46; S3 = $86.38; S4 = $79.21. Which term structure theories are
consistent with these strips prices?
par = 100
S1 = 97.09 S1 =
100
1 + R 0,1
S2 = 92.46 f 0,2 =
R 0,1 = 3%
S1
 1 = 5%
S2
S3 = 86.38 f 0,3 =
S2
 1 = 7.04%
S3
S4 = 79.21 f 0,4 =
S3
 1 = 9.05%
S4
All of the theories are consistent with these strip prices because the forward rates are
increasing.
10.
Suppose that we observe the following prices on strips with $100 par values: S1 =
$97.09; S2 = $92.46; S3 = $86.38; S4 = $85.48. Which term structure theories are
consistent with these strips prices?
S1 = 97.09 S1 =
100
1 + R 0,1
R 0,1 = 3%
S2 = 92.46 f 0,2 =
S1
 1 f 0,2 = 5%
S2
S3 = 86.38 f 0,3 =
S2
 1 f 0,3 = 7.04%
S3
S4 = 85.48 f 0,4 =
S3
 1 f 0,4 = 1.05%
S4
Segmented markets, expectations hypothesis, combined theory.
11.
Suppose that the combined theory explains the term structure of interest rates. The
current one-period spot interest rate is 4.50%. The market expects interest rates next
period to increase by x%. There is a liquidity premium of 1.5%. The two-period spot
interest rate is current 5.74%. What is the expected increase in the spot interest rate over
the next period?
(1  R 0, 2 ) 2  (1  R 0,1 )(1  f 0, 2 )
(1.0574) 2  (1.0450)(1.0450  x  0.015)
(1.0574) 2
 1.06  x
1.0450
(1.0574) 2
x
 1.06
1.0540
x  0.009947  0.99%
12.
Suppose that we observe the following prices on strips with $100 par values. S1 =
$93.46; S2 = $82.64; S3 = $79.38; S4 = $79.21. This is consistent with which term
structure theories?
1 + R0,1 =
100
, R0,1 = 6.9976% ≈ 7.00%
93.46
1 + f0,2 =
93.46
, f0,2 = 13.09%
82.64
1 + f0,3 =
82.64
, f0,3 = 4.1068% ≈ 4.11%
79.38
1 + f0,4 =
79.38
, f0,4 = 0.2146% ≈ 0.21%
79.21
Humped term structure, expectations hypothesis, combined theory, segmented markets.