FM for Actuaries

Bond Management 247
Using PV(C t ) as the factor of proportion, we define the weighted average of the
time of the cash flows, denoted by D,as
D =
=
n∑
[ ] PV(Ct )
t
P
n∑
tw t , (8.3)
t=1
t=1
where
w t = PV(C t)
. (8.4)
P
Note that w t ≥ 0 for all t and ∑ n
t=1 w t =1, so that w t are properly defined weights
and D is the weighted average of t =1, ··· ,n. We call D the Macaulay duration,
which measures the average period of the investment. The value computed from
(8.3) gives the Macaulay duration in terms of the number of payment periods. If
there are k payments per year and we desire to express the duration in years, we
replace t in (8.3) by t k
. Thus, if the payments are made twice a year, D is the
weighted average of 0.5, 1, ··· , n−1
2 , n 2
years, and the resulting value of D is then
the Macaulay duration in years.
Figure 8.1 illustrates the cash flows of a coupon bond and the calculation of the
Macaulay duration of the bond.
Figure 8.1:
Present values of the cash flows of a coupon bond
Present value
PV(C t )
······
Time (t)
1 2 3 4 ······ n − 1 n