Rate per period and periods in total

The one adjustment that is made twice or not at all: divide the rate by p AND multiply the years by p.

Interest is quoted annually, compounded more often

A rate is almost always quoted per year, but the compounding happens $p$ times a year: monthly ( $p = 12$ ), quarterly ( $p = 4$ ), semi-annually ( $p = 2$ ). Excel's functions know nothing about years. They take a rate per period and a number of periods, so before any TVM function is typed two conversions happen:

$rate = \frac{annual rate}{p}, nper = years \times p .$

The letter $p$ appears twice. That is the whole lesson, and it is the error the calculation part of the paper is built to catch.

Worked example — 6 % compounded monthly for four years

What does one dollar grow to at 6 % compounded monthly over four years?

rate per period: $0.06/12 = 0.005$
periods in total: $4 \times 12 = 48$
=FV(0.005,48,0,-1) → 1.2705

Hand check: $1.00 5^{48} = 1.27049$ . In a worksheet you would keep the inputs in cells — annual rate in B2, years in B3, $p$ in B4 — and write =FV(B2/B4,B3*B4,0,-B5), so changing the compounding frequency changes one cell.

The half-adjustment, and why it is obviously wrong

Now the trap. A learner multiplies the years by 12 (48 periods) but forgets to divide the rate, typing =FV(0.06,48,0,-1). Excel happily returns 16.39. Stop and read that number: one dollar cannot become sixteen dollars in four years at 6 %. The half-adjustment applied a 6 % rate forty-eight times, which is 48 years of growth, not four.

The mirror error — dividing the rate but leaving the years alone, =FV(0.005,4,0,-1) — returns 1.0202, four months of growth dressed up as four years. Both wrong answers come from touching one argument and not the other.

A sanity check you can run in your head: at 6 % a year, money roughly doubles in twelve years (the Rule of 72), so over four years the growth factor must sit a little above 1.24. Values of 1.02 or 16.39 fail the check on sight.

The same four years at every frequency

The figure below runs the same 6 %, four-year deposit at four compounding frequencies. Two things to notice: the future value rises with the frequency, because interest starts earning interest sooner; and it rises by very little — from 1.2625 annually to 1.2705 monthly — so an answer far outside that band is a sign that $p$ was applied once, not twice.

The payment must be per period too

When a problem has a payment, the same discipline applies to it: a mortgage with monthly payments needs the monthly rate, the total months, and the monthly payment — all three in the same unit of time, or none of them. Feeding an annual payment into a monthly PMT slot is the third form of the same mistake.

One example, four frequencies. The FV column moves only in the third decimal — a good sanity band for spotting a half-adjusted answer.