Compound interest

With simple interest, the rate is applied to the original amount: $10,000 at 6% earns $600 a year, every year. With compound interest, the interest earned is added to the balance at the end of each period, and the next period's interest is worked out on the new balance. The second year starts from more than $10,000, so it earns more than $600.

How often the interest is added is the compounding frequency. A stated 6% compounded once a year adds $600 after twelve months; compounded monthly, it adds half a percent of the running balance every month, which comes to $616.78 in the first year. More frequent compounding at the same stated rate grows a balance a little faster, and the gap widens with time.

The same arithmetic runs on a debt. A balance that accrues interest which is not paid grows by that interest, and the next month's interest is charged on the larger balance. A payment that covers the month's interest stops the compounding; one that does not leaves the unpaid part to accrue interest of its own.

A $10,000 balance sits at 6% a year for three years, with nothing added and nothing withdrawn. Under simple interest it earns 6% of $10,000, or $600, each year: $1,800 over the three years, and a closing balance of $11,800.

Compounded monthly, the same balance earns half a percent of its running balance each month. The first month adds $50; the last of the 36 months adds $59.54, on a balance that has grown to $11,907. Over the three years the interest comes to $1,966.81, and the closing balance is $11,966.81. Compounding earned $166.81 more than simple interest at the same stated rate.

Balance

$10,000

Simple interest, 6% × 3 years

$1,800.00

Compounded monthly, 0.5% × 36 months

$1,966.81

Difference

$166.81

The difference is small in one year and larger every year after, because each period's interest is worked out on the last period's result. In the example the first year's gap is $16.78; the three-year gap is $166.81. A retirement projection, a mortgage schedule and a debt payoff plan all rest on this arithmetic, so the stated rate and how often it is applied set what they print.

On a debt, interest that goes unpaid becomes part of the balance and is charged interest itself. That is why a payment below the month's interest leaves a balance larger than it started, and why a projection at a stated rate needs to say how the rate is applied.

Taiga Finance's debt, mortgage and retirement planners compound monthly, applying one twelfth of the stated yearly rate to the running balance each month. No screen uses the words “compound” or “simple” for a rate on a debt, a mortgage or a retirement balance; each planner lists what it leaves out under “What this plan doesn't capture”.

The debt payoff planner, “Payoff plan” on the Debt page and “Debt payoff” under Scenarios, works out each month's interest as the balance times the rate in effect, divided by 12. A promo rate applies while its window is open. When a payment does not cover a month's interest, the unpaid part is added to the balance and the planner flags “Balance grows before it falls” and carries the rise forward.

The Mortgage Payoff planner and the public mortgage payoff calculator apply the loan's rate, divided by 12, to the balance each month, and state that the schedule is “Principal and interest only”. Home value, where an appreciation rate is set, grows at that one rate, and the planner's notes say so: “Appreciation at one flat rate”.

The Retirement savings planner's rate is a “Real return” under “Assumptions”, in “%/yr”, captioned “After inflation”: the plan is simulated in real dollars at one expected real return every year, and the “Chance money lasts” figure re-runs it over randomized market paths. The balance grows at one twelfth of that yearly return each month, with the month's contributions and withdrawals applied, in the single projection and in every Monte Carlo path alike.