FM for Actuaries

Bond Yields and the Term Structure 227
Figure 7.2: Estimated spot-rate curve of Example 7.7
5
4.5
4
Spot rate of interest (%)
3.5
3
2.5
2
1.5
1
0.5
0
0 1 2 3 4 5 6 7
Time to maturity (yrs)
Let us assume we have a set of m bonds for which the coupon-payment dates
are synchronized. We denote the prices of these bonds per 100 face value by P j ,
their coupon rate by r j per half-year and their time to maturity by n j half-years, for
j =1, ··· ,m. We denote
1
v h = ⎡
⎣1+
i S h
2
2
⎤h , (7.10)
⎦
which is the discount factor for payments due in h half-years. Thus, the equation
of value for the jth bond is
n j
∑
P j = 100 r j v h + 100v nj .
h=1
If the last payment (redemption plus coupon) of the bonds occur in M periods (i.e.,
M is the maximum of all n j for j =1, ··· ,m), the pricing equations of the bonds
can be written as