Future value, and compound versus simple interest

◈ 8 cards

FV = PV(1 + r)ⁿ by formula and by =FV — grow a reserve for eight years, then measure the gap between compound and simple interest, which is interest on interest.

One deposit, left alone

Future value is what a sum becomes after it earns interest for a number of periods. Under compound interest each period's interest is added to the balance, and the next period's interest is earned on the larger balance — interest on interest. The closed form is

$F V = P V \times (1 + r)^{n}$

and the Excel function is =FV(rate, nper, pmt, pv, type). For a single deposit pmt is 0, and the deposit goes in negative because it leaves your pocket — Excel then returns the future value positive, as money coming back (Lesson 1.2).

Worked example — Halton Dairy's reserve

Halton Dairy Co-op sets aside $5,000 in a reserve account paying 4.5 % a year, compounded annually, and leaves it for 8 years.

$F V = 5, 000 \times 1.04 5^{8} = 5, 000 \times 1.42210 = 7, 110.50$

In Excel: =FV(0.045,8,0,-5000) → 7,110.50; or, by the formula, =5000*(1+0.045)^8. Both give the same number; the paper marks the number.

Simple interest is the road not taken. Interest is paid on the original $5,000 only, every year: 5,000 × 0.045 = 225 a year, so after 8 years

$F V_{simple} = 5, 000 \times (1 + 0.045 \times 8) = 6, 800.00$

=5000*(1+0.045*8). The gap, 7,110.50 − 6,800.00 = 310.50, is interest earned on earlier interest. In year 1 the two balances are identical (5,225.00 each); by year 4 compound is ahead by 62.59; by year 8 by 310.50. Simple interest grows in a straight line; compound interest curves upward, and the curve steepens with time.

Three commandments

FV rises when any of three things rises: the rate $r$ , the number of periods $n$ , and (Lesson 6.3) the frequency of compounding. Re-run the reserve at 6 %: =FV(0.06,8,0,-5000) = 7,969.24, against 7,400.00 simple — the gap widens to 569.24, because a higher rate has more interest to compound. Doubling the rate on a long deposit more than doubles the interest earned; under simple interest it would exactly double.

Predict before you compute

Two deposits: $5,000 at 4.5 % for 8 years, or $5,000 at 4.5 % for 8 years with the interest withdrawn and spent each year. The first is compound (7,110.50); the second is, in effect, simple (6,800.00). Whenever interest is left in the account it compounds — and every TVM function in Excel assumes it is.

The gap is zero at the start and 310.50 at year 8, and it grows faster every year — that is what interest on interest looks like in a table.