FM for Actuaries

88 CHAPTER 3
We now consider the evaluation of a stream of cash flows at arbitrary time
points, assuming that future payments earn the forward rates of interest. Specifically,
we consider payments of C 1 ,C 2 , ··· ,C n at time (0 ≤) t 1 <t 2 < ··· <t n ,
respectively. Suppose we are interested in evaluating the value of this cash flow at
any time t ∈ (0,t n ) . For the payment C j at time t j ≤ t, its accumulated value at
time t is C j a tj (t − t j ). On the other hand, if t j >t, the discounted value of C j at
C
time t is j
a t(t j −t)
. Thus, the value of the cash flows at time t is (see equation (3.24))
∑
C j a tj (t − t j )+ ∑ [ ]
1
C j = ∑ a
t j ≤t t j >t t (t j − t)
t j ≤t
n∑
=
j=1
= a(t)
[ ] a(t)
C j + ∑ [ ] a(t)
C j
a(t j )
a(t
t j >t j )
[ ] a(t)
C j
a(t j )
n∑
C j v(t j )
j=1
= a(t) × present value of cash flows.
(3.29)
An analogous result can be obtained if we consider a continuous cash flow. Thus,
if C(t) is the instantaneous rate of cash flow at time t for 0 ≤ t ≤ n, so that the
payment in the interval (t, t +∆t) is C(t)∆t, the value of the cash flow at time
τ ∈ (0,n) is
∫ τ
0
∫ n
∫
C(t)
τ
C(t)a t (τ − t) dt +
τ a τ (t − τ) dt = 0
= a(τ)
= a(τ)
C(t)a(τ)
a(t)
∫ n
0
∫ n
0
C(t)
a(t) dt
∫ n
C(t)a(τ)
dt +
dt
τ a(t)
C(t)v(t) dt. (3.30)
Example 3.9: Suppose a(t) =0.02t 2 +0.05t +1. Calculate the value at time
3 of a 1-period deferred annuity-immediate of 4 payments of $2 each. You may
assume that future payments earn the forward rates of interest.
Solution: We first compute the present value of the annuity. The payments of $2
are due at time 2, 3, 4 and 5. Thus, the present value of the cash flows is
[ 1
2 ×
a(2) + 1
a(3) + 1
a(4) + 1 ]
.
a(5)