- Risk-Free Assets 25
just by the original deposit, but also by all the interest earned so far. In these
circumstances we shall talk ofdiscreteorperiodic compounding.
Example 2.3
In the case of monthly compounding the first interest payment of 12 rPwill be
due after one month, increasing the principal to (1 + 12 r)P,all of which will
attract interest in the future. The next interest payment, due after two months,
will thus be 12 r(1 + 12 r)P,and the capital will become (1 + 12 r)^2 P.Afterone
year it will become (1 + 12 r)^12 P,afternmonths it will be (1 + 12 r)nP,and after
tyears (1 + 12 r)^12 tP. The last formula admitstequal to a whole number of
months, that is, a multiple of 121.
In general, ifminterest payments are made per annum, the time between
two consecutive payments measured in years will bem^1 , the first interest pay-
ment being due at timem^1 .Each interest payment will increase the principal
by a factor of 1 +mr. Given that the interest raterremains unchanged, aftert
years thefuture valueof an initial principalPwill become
V(t)=(
1+
r
m)tm
P, (2.5)because there will betminterest payments during this period. In this formula
tmust be a whole multiple of the periodm^1 .Thenumber
(
1+mr)tm
is the
growth factor.
The exact value of the investment may sometimes need to be known at time
instants between interest payments. In particular, this may be so if the account
is closed on a day when no interest payment is due. For example, what is the
value after 10 days of a deposit of $100 subject to monthly compounding at
12%? One possible answer is $100, since the first interest payment would be
due only after one whole month. This suggests that (2.5) should be extended
to arbitrary values oftby means of a step function with steps of durationm^1 ,
as shown in Figure 2.2. Later on, in Remark 2.6 we shall see that the extension
consistent with the No-Arbitrage Principle should use the right-hand side of
(2.5) for allt≥0.
Exercise 2.5
How long will it take to double a capital attracting interest at 6% com-
pounded daily?