108156.pdf

2. Risk-Free Assets 33

Exercise 2.18

How long will it take to earn $1 in interest if $1, 000 ,000 is deposited at

10% compounded continuously?

Exercise 2.19

In 1626 Peter Minuit, governor of the colony of New Netherland, bought

the island of Manhattan from Indians paying with beads, cloth, and

trinkets worth $24. Find the value of this sum in year 2000 at 5% com-

pounded a) continuously and b) annually.

Proposition 2.2

Continuous compounding produces higher future value than periodic com-
pounding with any frequencym, given the same initial principalPand interest
rater.

Proof

It suffices to verify that

etr>(1 +

r

m)

tm=

[

(1 +

r

m)

mr]rt

.

The inequality holds because the sequence (1+mr)mr is increasing and converges
to e asm↗∞.

Exercise 2.20

What will be the difference between the value after one year of $100

deposited at 10% compounded monthly and compounded continuously?

How frequent should the periodic compounding be for the difference to

be less than $0.01?

The present value under continuous compounding is obviously given by

V(0) =V(t)e−tr.

In this case thediscount factoris e−tr. Given the terminal valueV(T), we
clearly have
V(t)=e−r(T−t)V(T). (2.11)