A nominal rate is not what you actually pay

What this works out

The gap between the rate that is quoted and the rate that is charged. Type a nominal rate and a compounding frequency and it gives you the effective annual rate; type an effective rate and it gives you the nominal one that produces it.

The table underneath applies the same nominal rate at every frequency at once, which is the comparison worth having.

Why there are two numbers at all

A nominal annual rate is a periodic rate multiplied by the number of periods. That is all it is. 1% a month is quoted as 12% a year, and the multiplication throws away the fact that the second month’s interest is charged on a balance that already includes the first month’s.

The effective rate puts it back:

effective = (1 + 0.12/12)^12 - 1
          = 1.01^12 - 1
          = 12.6825%

Two-thirds of a percentage point, on a rate that both parties would describe as 12%. It is not a rounding difference and it is not optional — it is what actually happens to the money.

Almost all of it arrives by monthly

The same 12% nominal, compounded at every frequency this tool offers:

Compounded	Periods	Per period	Effective
Annually	1	12%	12%
Half-yearly	2	6%	12.36%
Quarterly	4	3%	12.5509%
Monthly	12	1%	12.6825%
Weekly	52	0.2308%	12.7341%
Daily	365	0.0329%	12.7475%
Continuously	—	—	12.7497%

Before you read on

A savings account pays 12% nominal. One provider compounds it monthly, another daily. How much more does daily compounding earn you over a year?

• AAbout half a percentage point.

  That is roughly the gap between annual and monthly, not monthly and daily.

• BAbout six hundredths of a percentage point.

  Yes — 12.6825% against 12.7475%.

• CNothing at all — the nominal rate is the same.

  It is a real difference, just a very small one.

0.065 of a percentage point: 12.6825% compounded monthly against 12.7475% compounded daily. Monthly compounding already captures 91% of everything available between annual and the continuous limit, which is why compounding frequency is almost never where a decision should turn. The rate is.

The whole range is 0.7497 percentage points, and monthly compounding captures about 91% of it. Everything from monthly to the continuous limit is six and a half hundredths of a point.

That is the useful conclusion, and it runs against the intuition that daily compounding is a meaningfully better deal. It is not. The rate matters; the frequency almost never does.

Continuous compounding is the limit rather than a product. As the periods grow the expression converges on e^r - 1, and it converges fast. It is here because it is the closed form everything else approaches, and because seeing daily and continuous differ in the third decimal place settles the question.

The gap grows with the rate

At small rates the distinction is academic. At credit card rates it is not:

 2% nominal, monthly  ->   2.0184% effective   a gap of 0.02
12% nominal, monthly  ->  12.6825% effective   a gap of 0.68
24% nominal, monthly  ->  26.8242% effective   a gap of 2.82

The gap grows faster than the rate does, which is exactly why it is safe to ignore on a savings account and expensive to ignore on a debt. A card advertising 22.9% costs an effective 25.4632% on a balance carried through the year.

The method

The two directions are exact inverses, and that is the property the tests lean on: a rate converted one way and back has to arrive where it started. One assertion, across six rates and seven frequencies, catches essentially any algebra slip in either direction — which is worth more than a table of expected answers that has to be maintained alongside the code.

Everything is computed in decimal arithmetic at a single precision, including the continuous branch, which uses decimal exp and ln rather than the floating-point ones so the whole module works to one standard.

Nothing on this page is money. Every field and every output is a percentage, which is why there are no currency symbols anywhere — a rate conversion is identical in every country, and pinning a currency would have made the page wrong for everyone outside it.

A worked example

12% compounded monthly:

Effective annual rate	12.6825%
Nominal annual rate	12%
Rate per period	1%
Periods a year	12

Run the other way — 12.6825% effective, compounded monthly — and the nominal rate comes back as 12%.

The credit card case. 22.9% nominal compounded monthly is an effective 25.4632%.

These are the same figures asserted in this tool’s test file, so the page and the formula cannot drift apart without the build going red.

What it does not do

It does not include fees, and that is the important limitation: a regulatory APR in most jurisdictions folds compulsory charges into the figure as well as the compounding, so a legal APR and the effective rate here are not always the same number. It does not handle a variable rate, which has no single effective figure. It does not model the day-count conventions that decide whether a “daily” rate divides by 360, 365 or 366, which differ by market.

For what a rate does to an actual balance over time, compound interest and the credit card payoff schedule are the tools.

From a nominal rate to the effective one:

  effective = (1 + nominal/n)^n - 1

And back again:

  nominal = n x ((1 + effective)^(1/n) - 1)

Continuously compounded, the limit as n grows:

  effective = e^nominal - 1
  nominal   = ln(1 + effective)

n = periods a year: 1, 2, 4, 12, 52, 365

The two directions are exact inverses, which is
asserted rather than assumed.