Simple interest vs compound interest

Both rules start identically. Take a principal, apply a rate, add the interest. They part company at the next period, over a single question: what is the rate applied to now?

Simple interest keeps pointing at the original sum. The principal never moves for the purposes of the calculation, so every period produces the same interest amount and the total is that amount multiplied by the number of periods:

$$A = P(1 + rt)$$

Compound interest points at the balance instead. Last period's interest was added to that balance, so this period charges on a bigger base, and the base keeps rising even though the rate never does:

$$A = P(1 + r)^t$$

Read $r$ there as the rate for one compounding period and $t$ as the number of those periods. Rates are usually quoted by the year and compounded more often than that, and the two do not have to match. Starting from an annual rate $R$ compounded $m$ times a year for $y$ years, the same formula is written:

$$A = P\left(1 + \frac{R}{m}\right)^{my}$$

That is the whole of it. Nobody raises the rate, and the rate is the same letter in both formulas. What changes is the base the rate is applied to, and a rising base is what bends a straight line into a curve.

One consequence is worth naming early, because it explains why the difference is so easy to underrate. Over a single compounding period the two produce exactly the same number. Compounding needs a previous period's interest to work on, and in period one there is none. Everything that separates them happens afterwards.

Compounding pays nothing in the first period and very little in the first several. It is a slow mechanism that turns into a fast one, which is why it looks like a rounding error over any horizon short enough to check casually.

Here is one 5 percent rate under both rules, compounded once a year throughout, written as growth on top of the starting sum:

The last column is the third divided by the second, so it compares interest against interest rather than one ending balance against another. Five years of compounding earns about a tenth more than the straight line. Forty years earns three times as much.

Rate and time work on each other, so a higher rate makes the same horizon far more dramatic. At 8 percent compounded yearly, thirty years of simple interest adds 240 percent while compounding adds 906 percent. Frequency moves the compound column too, which is why every figure above names one: the same 5 percent compounded monthly adds 64.70 percent over ten years rather than 62.89 percent.

The compound interest calculator runs the compounding side at any rate, term and frequency, and the straight line to set beside it is the starting sum times the rate times the years. The point worth watching is not where the two finish but where they visibly separate, which arrives later than most people guess.

The name on the contract settles less than it appears to. Compounding needs unpaid interest to work on, so a debt compounds only when interest is allowed to sit on the balance instead of being cleared as it accrues.

So the honest answer to which rule applies is that it depends, and what it depends on is payment behaviour rather than the product label.

• Pay every period's interest in full and a balance described as compounding behaves exactly like a simple-interest one, because nothing is left over to be charged on next time.
• Miss a payment, pay only a minimum, or defer, and a loan described as simple interest begins compounding the moment that unpaid interest is folded into the balance.

In the United States, that folding-in has a name on student loans: capitalisation. Interest accrues on principal alone, which is a simple-interest rule, and then at certain defined events the accrued interest is added to the principal, after which it accrues interest of its own. The rule did not change. The base did.

Other systems reach the same place under other names. Interest rolled up during a payment holiday, arrears added to a mortgage balance after missed payments, and unpaid trade invoices where the charge is recalculated on the total outstanding all do the identical thing. The wording varies by country and by contract. What decides the arithmetic is whether unpaid interest ends up inside the balance the rate is applied to.

The saving side works the same way in reverse. A bond pays a fixed coupon on face value, which is a simple arrangement, and that coupon compounds only if you reinvest it. Left idle, the return stays linear. Reinvested at a similar rate, the position behaves like a compounding one. The decision is doing the work, not the instrument.

On a debt the same curve runs in the other direction, and it is usually steeper, because revolving credit is priced well above what the same institution pays a depositor and the compounding is monthly or daily rather than yearly.

A card balance is the clearest case. Interest is charged on what you owe, unpaid interest joins what you owe, and the next charge is worked out on the larger figure. Take 20 percent nominal compounded monthly as an illustration rather than as any particular card: a balance carried untouched for a year costs about 21.94 percent of itself rather than 20 percent, and the gap widens for as long as the balance survives.

Which is why the figure to compare between two offers is the effective annual rate, the rate after compounding is accounted for. It is quoted as APY on United States deposits, as AER on British ones, and under other labels elsewhere, but the arithmetic behind all of them is the same. The APR against APY calculator converts a nominal rate into it.

An amortising loan behaves differently again. Its scheduled payment is built to cover each period's interest in full and take a slice off the balance as well, so unpaid interest never accumulates and the debt shrinks on a fixed schedule. That is compound-interest arithmetic used to design a payment, producing a balance that does not compound for as long as the schedule is kept. The loan payment calculator shows the split period by period.

Neither rule is the good one and neither is the bad one. It is a single piece of arithmetic, which is why it describes a growing investment and a growing debt equally well, and the only question is which side of it you are standing on.

Compounding more and more often does not grow without limit. As the compounding period shrinks the growth factor converges on the number $e$, which is where continuous compounding comes from: Euler's number.