108156.pdf

34 Mathematics for Finance

Exercise 2.21

Find the present value of $1, 000 ,000 to be received after 20 years as-

suming continuous compounding at 6%.

Exercise 2.22

Given that the future value of $950 subject to continuous compounding

will be $1,000 after half a year, find the interest rate.

The returnK(s, t) defined by (2.3) on an investment subject to continuous
compounding fails to be additive, just like in the case of periodic compounding.
It proves convenient to introduce thelogarithmic return

k(s, t)=ln

V(t)

V(s)

. (2.12)

Proposition 2.3

The logarithmic return is additive,

k(s, t)+k(t, u)=k(s, u).

Proof

This is an easy consequence of (2.12):

k(s, t)+k(t, u)=lnV(t)

V(s)

+lnV(u)

V(t)

=ln

V(t)

V(s)

V(u)

V(t)

=ln

V(u)

V(s)

=k(s, u).

IfV(t) is given by (2.10), thenk(s, t)=r(t−s), which enables us to recover
the interest rate

r=

k(s, t)

t−s.

Exercise 2.23

Suppose that the logarithmic return over 2 months on an investment

subject to continuous compounding is 3%. Find the interest rate.