- Risk-Free Assets 29
2.1.3 Streams of Payments
Anannuity is a sequence of finitely many payments of a fixed amount due
at equal time intervals. Suppose that payments of an amountC are to be
made once a year fornyears, the first one due a year hence. Assuming that
annual compounding applies, we shall find the present value of such a stream
of payments. We compute the present values of all payments and add them up
to get
C
1+r
+
C
(1 +r)^2+
C
(1 +r)^3+···+
C
(1 +r)n.
It is sometimes convenient to introduce the following seemingly cumbersome
piece of notation:
PA(r, n)=^1
1+r+^1
(1 +r)^2+···+^1
(1 +r)n. (2.7)
This number is called thepresent value factor for an annuity. It allows us to
express the present value of an annuity in a concise form:
PA(r, n)×C.The expression for PA(r, n) can be simplified by using the formula
a+qa+q^2 a+···+qn−^1 a=a^1 −qn
1 −q.
In our casea=1+^1 randq=1+^1 r, hence
PA(r, n)=1 −(1 +r)−n
r. (2.8)Remark 2.3
Note that an initial bank deposit of
P=PA(r, n)×C= C
1+r+ C
(1 +r)^2+···+ C
(1 +r)nattracting interest at a ratercompounded annually would produce a stream
ofnannual payments ofCeach. A deposit ofC(1 +r)−^1 would grow toC
after one year, which is just what is needed to cover the first annuity payment.
A deposit ofC(1 +r)−^2 would becomeCafter two years to cover the second
payment, and so on. Finally, a deposit ofC(1 +r)−nwould deliver the last
payment ofCdue afternyears.