108156.pdf

32 Mathematics for Finance

2.1.4 Continuous Compounding

Formula (2.5) for the future value at timetof a principalPattracting interest
at a rater>0 compoundedmtimes a year can be written as

V(t)=

[(

1+r

m

)mr]tr

P.

In the limit asm→∞,we obtain

V(t)=etrP, (2.10)

where

e = limx→∞

(

1+

1

x

)x

is the base of natural logarithms. This is known ascontinuous compounding.
The correspondinggrowth factor is etr. A typical graph ofV(t)isshownin
Figure 2.3.

Figure 2.3 Continuous compounding at 10% (r=0.1,P=1)

The derivative ofV(t)=etrPis

V′(t)=retrP=rV(t).

In the case of continuous compounding the rate of the growth is proportional
to the current wealth.
Formula (2.10) is a good approximation of the case of periodic compounding
when the frequencymis large. It is simpler and lends itself more readily to
transformations than the formula for periodic compounding.