The b F (calc-fin-fv)
[fv] command computes the future value of an
investment. It takes three arguments from the stack:
‘fv(rate,
n,
payment)’. If
you give payments of payment every year for
n years, and the money you have paid earns interest at
rate per year, then this function tells you what your
investment would be worth at the end of the period. (The actual
interval doesn't have to be years, as long as n and
rate are expressed in terms of the same intervals.)
This function assumes payments occur at the end of each
interval.
The I b F
[fvb] command does the same computation, but
assuming your payments are at the beginning of each interval.
Suppose you plan to deposit $1000 per year in a savings account
earning 5.4% interest, starting right now. How much will be in
the account after five years? fvb(5.4%, 5, 1000) =
5870.73. Thus you will have earned $870 worth of interest
over the years. Using the stack, this calculation would have been
5.4 M-% 5 <RET> 1000 I b F. Note that the rate
is expressed as a number between 0 and 1, not as a
percentage.
The H b F
[fvl] command computes the future value of an
initial lump sum investment. Suppose you could deposit those five
thousand dollars in the bank right now; how much would they be
worth in five years? fvl(5.4%, 5, 5000) =
6503.89.
The algebraic functions fv and fvb
accept an optional fourth argument, which is used as an initial
lump sum in the sense of fvl. In other words,
fv(rate,
n, payment,
initial) =
fv(rate,
n, payment) +
fvl(rate,
n, initial).
To illustrate the relationships between these functions, we
could do the fvb calculation “by hand”
using fvl. The final balance will be the sum of the
contributions of our five deposits at various times. The first
deposit earns interest for five years: fvl(5.4%, 5, 1000) =
1300.78. The second deposit only earns interest for four
years: fvl(5.4%, 4, 1000) = 1234.13. And so on down
to the last deposit, which earns one year's interest:
fvl(5.4%, 1, 1000) = 1054.00. The sum of these five
values is, sure enough, $5870.73, just as was computed by
fvb directly.
What does fv(5.4%, 5, 1000) = 5569.96 mean? The
payments are now at the ends of the periods. The end of one year
is the same as the beginning of the next, so what this really
means is that we've lost the payment at year zero (which
contributed $1300.78), but we're now counting the payment at year
five (which, since it didn't have a chance to earn interest,
counts as $1000). Indeed, ‘5569.96
= 5870.73 - 1300.78 + 1000’ (give or take a
bit of roundoff error).