28 Mathematics for Finance
Exercise 2.10
Find the present value of $100,000 to be received after 100 years if
the interest rate is assumed to be 5% throughout the whole period and
a) daily or b) annual compounding applies.One often requires the valueV(t) of an investment at an intermediate time
0 <t<T, given the valueV(T) at some fixed future timeT.Thiscanbe
achieved by computing the present value ofV(T), taking it as the principal,
and running the investment forward up to timet. Under periodic compounding
with frequencymand interest rater, this obviously gives
V(t)=(
1+
r
m)−(T−t)m
V(T). (2.6)To find the return on a deposit attracting interest compounded periodically
we use the general formula (2.3) and readily arrive at
K(s, t)=V(t)−V(s)
V(s) =(1+r
m)(t−s)m− 1.In particular,
K(0,^1
m)=r
m,
which provides a simple way of computing the interest rate given the return.
Exercise 2.11
Find the return over one year under monthly compounding withr=
10%.Exercise 2.12
Which is greater, the interest rateror the returnK(0,1) if the com-
pounding frequencymis greater than 1?Remark 2.2
The return on a deposit subject to periodic compounding isnotadditive. Take,
for simplicity,m=1.Then
K(0,1) =K(1,2) =r,
K(0,2) = (1 +r)^2 −1=2r+r^2 ,and clearlyK(0,1) +K(1,2)=K(0,2).