108156.pdf

38 Mathematics for Finance

admits only time instantstbeing whole multiples of the compounding period
1
m. An argument similar to that in Example 2.5 shows that the appropriate no-
arbitrage value of an initial sumP at any timet≥0 should be

(

1+mr

)tm
P.
A reasonable extension of (2.5) is therefore to use the right-hand side for all
t≥0 rather than just for whole multiples ofm^1. From now on we shall always
use this extension.

In terms of the effective raterethe future value can be written as

V(t)=(1+re)tP.

for allt≥0. This applies both to continuous compounding and to periodic
compounding extended to arbitrary times as in Remark 2.6. Proposition 2.4
implies that, given the same initial principal, equivalent compounding methods
will produce the same future value for all timest≥ 0 .Similarly, a compounding
method preferable to another one will produce a higher future value for allt>0.

Remark 2.7

Simple interest does not fit into the scheme for comparing compounding meth-
ods. In this case the future valueV(t) is a linear function of timet, whereas it is
an exponential function if either continuous or periodic compounding applies.
The graphs of such functions have at most two intersection points, so they can
never be equal to one another for all timest≥0 (except for the trivial case of
zero principal).

Exercise 2.26

What is the present value of an annuity consisting of monthly payments

of an amountCcontinuing fornyears? Express the answer in terms of

the effective ratere.

Exercise 2.27

What is the present value of a perpetuity consisting of bimonthly pay-

ments of an amountC? Express the answer in terms of the effective

ratere.