Two formulas that look like conventions until they are built. A = P(1 + r/n)^(nt) is one multiplication repeated nt times, and A = P · e^(rt) is the value that expression settles down to as n grows. The second is “The Number e”'s limit with letters in it, and no new fact about money is involved.
Core ideas
One idea builds the first formula, one limit builds the second, and one habit of units keeps both of them right.
One period at rate r/n turns P into P(1 + r/n). A second period multiplies by that same factor again. After nt periods the factor has been applied nt times, which is the entire derivation of A = P(1 + r/n)^(nt).
On $1000 at 5% for 10 years, moving from yearly to monthly gains $18.11; moving from monthly to continuous gains $1.71 more. The sequence increases with n but is bounded, and its limit is the continuous value.
(1 + 1/m)^m → e. Put m = n/r, and (1 + r/n)^(nt) = [(1 + 1/m)^m]^(rt), which tends to e^(rt). e was not introduced for finance; finance is where the limit was first noticed.
The material
One repeated multiplication, one limit, and the units that make both of them wrong when they are wrong.
A principal P is invested at a nominal annual rate r, compounded n times a year, for t years. Every quantity in that sentence has a unit attached, and most errors with these formulas are unit errors rather than algebra errors.
The nominal annual rate r is quoted per year and split evenly among the n periods, so one period carries the rate r/n. A nominal 5% compounded monthly means 5/12 of a percent a month. It is a name for twelve times the monthly rate, not the growth actually achieved over a year.
The first formula is a repeated multiplication and nothing more. Each period multiplies the balance by 1 + r/n, so the balances form a geometric sequence.
The effective annual rate is (1 + r/n)^n − 1: the single yearly rate that would produce the same balance after one year. For a nominal 5% compounded monthly it is (1 + 0.05/12)^12 − 1 ≈ 0.051162, that is 5.1162%. Two accounts quoted at different frequencies can only be compared through this number.
Now increase n and watch. The numbers below are $1000 at a nominal 5% for 10 years, each rounded to the nearest cent.
The column climbs and then stops climbing. Daily and continuous differ by about six cents over ten years on a thousand dollars. "Compounded continuously" is not a claim about speed — it is the ceiling of the process, and the last few frequencies are already sitting against it.
As n → ∞, P(1 + r/n)^(nt) → P · e^(rt).
Substitute m = n/r, so that r/n = 1/m and nt = m · rt. The expression becomes P[(1 + 1/m)^m]^(rt), and m → ∞ as n → ∞. The inner bracket tends to e by “The Number e”, so the whole expression tends to P · e^(rt).
P and A come out by dividing. r and t sit in the exponent, so they need “Solving Exponential Equations”. To double a balance under continuous growth, set 2P = P · e^(rt); the P cancels, 2 = e^(rt), and rt = ln 2, so t = ln 2 / r.
That the principal cancels is the characteristic property of exponential growth from “Growth and Decay”: the time to double does not depend on where you start. At 5% continuous it is ln 2 / 0.05 ≈ 13.86 years, and compounded yearly instead it is ln 2 / ln(1.05) ≈ 14.21 years.
The banker's shortcut follows from ln 2 ≈ 0.6931. Doubling time is roughly 70 divided by the rate as a percent, which at 5% gives 14 years against a true 13.86. It is an approximation of a logarithm, and it drifts at large rates: at 20% the rule says 3.5 years and the continuous answer is 3.47, but at 100% it says 0.7 and the answer is 0.69.
Entering 5 instead of 0.05 for a monthly-compounded account makes the monthly factor 1 + 5/12, a rate of 500% a year. Over ten years $1000 becomes about 1.4 · 10^21 dollars. The formula reports it without complaint, because nothing in the algebra knows what a percent is.
Interactive
Six frequencies against the continuous limit
Method
Four steps, and two of them are about units rather than algebra.
5% is 0.05. Entered as 5 it is a rate of 500%, and every later step will run perfectly and return a number that is wrong by a factor of billions.
n is periods per year and t is years, so the exponent is their product. Quarterly for five years is 20 periods — not 5, and not 4.
Keep 1 + r/n exact if you can: 1 + 0.06/4 = 1.015, while 1 + 0.05/12 = 1 + 1/240 should stay in that exact form rather than be replaced by a truncated decimal. Rounding the base before raising it to the 120th power moves the error up into the dollars.
(1 + r/n)^n − 1 for periodic compounding, e^r − 1 for continuous. Two nominal rates quoted at different frequencies cannot be compared as they stand.
Four numbers go in. The only thinking is which of them belongs where.
Quarterly beats yearly by $8.63 and loses to continuous by $3.00. Both gaps are small next to the $346.86 of interest itself, which is the honest summary of compounding frequency: it matters, and it matters far less than the rate and the term.
The effective rate is what makes the account comparable with anything else. A flat 6.1% paid once a year sounds better than 6%, and it is not: 6% compounded quarterly earns 6.1364% a year.
For a faded model, take $1500 at a nominal 4.8% compounded monthly for two years. The period rate 0.048/12 and period count 24 are supplied. Assemble the periodic formula without truncating the base, compute once at the end, and compare it with the continuous expression 1500·e^(0.048·2). Check that both exceed the simple-interest amount and that the periodic value stays below the continuous ceiling.
Asked for a monthly-compounded balance, you should be able to write 1 + r/12 for one month and 12t for the number of months, and assemble A = P(1 + r/12)^(12t) from those two facts alone. A recalled formula with n and t in the wrong places looks exactly like a correct one until the money is wrong.
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