### Future Value

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).