- Risk-Free Assets 31
perpetuity can be obtained from (2.7) in the limit asn→∞:
nlim→∞PA(r, n)×C=C
1+r+C
(1 +r)^2 +C
(1 +r)^3 +···=C
r. (2.9)The limit amounts to taking the sum of a geometric series.
Remark 2.4
The present value of a perpetuity is given by the same formula as in Exam-
ple 2.2, even though periodic compounding has been used in place of simple
interest. In both cases the annual paymentCis exactly equal to the interest
earned throughout the year, and the amount remaining to earn interest in the
following year is alwaysCr. Nevertheless, periodic compounding allows us to
view the same sequence of payments in a different way: The present valueCr
of the perpetuity is decomposed into infinitely many parts, as in (2.9), each
responsible for producing one future payment ofC.
Remark 2.5
Formula (2.8) for the annuity factor is easier to memorise in the following way,
using the formula for a perpetuity: The sequence ofnpayments ofC=1can
be represented as the difference between two perpetuities, one starting now
and the other afternyears. (Cutting off the tail of a perpetuity, we obtain
an annuity.) In doing so we need to compute the present value of the latter
perpetuity. This can be achieved by means of the discount factor (1 +r)−n.
Hence,
PA(r, n)=^1
r−^1
r×^1
(1 +r)n=^1 −(1 +r)−n
r.
Exercise 2.17
Find a formula for the present value of an infinite stream of payments
of the formC,C(1 +g),C(1 +g)^2 ,...,growing at a constant rateg.By
the tail-cutting procedure find a formula for the present value ofnsuch
payments.