Nominal Interest Rate Calculator

One deal, two rates, both correct

Here is a fact that causes more confusion in finance than almost anything else. The same interest arrangement can be described by two different percentages, and neither of them is wrong.

A deposit paying 5.8411 percent compounded monthly and a deposit paying 6 percent a year are the same deposit. Identical money, identical dates, two numbers on the page.

The first is a nominal rate. It is a per-period rate multiplied up to look annual, and it deliberately ignores the fact that the interest it pays will itself start earning interest.

The second is an effective rate. It is what actually happens across a year, compounding included.

This calculator goes from the effective figure back to the nominal one. You give it the annual effect you want and the number of times a year it compounds, and it tells you what per-period rate produces that effect.

Two boxes, and the second one is optional

1. Effective annual interest rate. The rate that describes a full year including compounding. It starts at 6 so there is something to work with.
2. Number of compounding periods per year. Twelve for monthly, four for quarterly, two for half yearly, one for annually, 365 for daily.

Press Calculate and the nominal rate comes back to three decimal places, which is the precision this kind of conversion needs. Round it to two and you can be a few units of currency out over the life of a large loan.

That second box can be left empty on purpose, and doing so gives you a specific and useful answer. There is a section on it below.

Undoing the compounding

Going from nominal to effective is the familiar direction. Take the per-period rate, apply it the appropriate number of times, and see where you end up:

Effective = (1 + nominal ÷ m)^m - 1

This calculator runs that backwards. Which means taking a root instead of a power:

Nominal = m × ((1 + effective)^1/m - 1)

In words, and the words are clearer than the algebra. Ask what a single period has to grow by, so that repeating it m times gets you to your annual target. That is the m-th root. Then multiply that per-period growth back up by m to express it as an annual figure, in the nominal style.

The nominal rate that comes out is always lower than the effective rate you put in, and it has to be. Compounding does part of the work. If your money is going to earn interest on its interest during the year, then the raw rate needed to reach a given annual result is smaller than that result.

The one exception is annual compounding. Put 1 in the second box and the nominal rate equals the effective rate exactly, because with only one period a year there is nothing to compound.

Six percent, six different ways

An effective annual rate of 6 percent, expressed as a nominal rate at each compounding frequency:

Every row describes the same 6 percent a year. The more often it compounds, the less raw rate is needed to get there.

You can check any of these in reverse. Take 5.841 percent, divide by 12 to get the monthly rate, apply it twelve times, and you land back on 6.000 percent.

Two things worth noticing about the shape of that column.

The first step down is the biggest. Going from annual to half yearly costs 0.087 of a percentage point. Going from weekly to daily costs 0.003. The gains from compounding more often shrink rapidly.

It is converging on something. The numbers are heading toward a floor rather than falling forever, and that floor is what the next section is about.

Leave the second box empty and something interesting happens

The number of periods box is not required. Leave it blank, press Calculate, and on a 6 percent effective rate you get 5.827 percent.

That is not an error and it is not a default of daily. It is the answer to a different question: what nominal rate would produce 6 percent a year if it compounded continuously, meaning not twelve times or 365 times but at every instant.

Mathematically it is the limit of the table above as the number of periods runs off to infinity, and it has a clean closed form:

Continuous nominal rate = ln(1 + effective)

The natural logarithm, and nothing else. For 6 percent, ln(1.06) is 0.0582689, which is 5.827 percent.

You can watch it converge. Daily compounding gives 5.8274 percent. Push it to a million periods a year and you get 5.826891 percent, which matches ln(1.06) to six decimal places.

This is worth knowing for two reasons.

It puts a floor under the whole exercise. However often a bank compounds, it can never do better for you than the continuous figure, and daily compounding is already within three ten-thousandths of it. Which is why the difference between daily and monthly compounding on a savings account is real but almost never worth choosing a bank over.

And continuous compounding is not merely a curiosity. It is the standard convention in options pricing and much of quantitative finance, where the mathematics is far tidier with an exponential than with a discrete number of periods. If you have met a rate described as a force of interest, this is that number.

Why loans and savings are quoted on opposite conventions

Here is where this stops being a mathematical exercise and starts costing people money.

In the United States, two different regulations govern how rates are disclosed, and they landed on opposite conventions.

Lending, under Regulation Z, uses the annual percentage rate. The APR is defined as the periodic rate multiplied by the number of periods in a year. That is a nominal construction. It also folds in fees, which the periodic rate does not.

Savings, under Regulation DD, uses the annual percentage yield. The APY is defined as a rate reflecting the total interest paid, based on the interest rate and the frequency of compounding. That is an effective construction.

So the headline figure on your loan is nominal and the headline figure on your deposit account is effective. Two products, two acronyms three letters apart, built on incompatible bases.

The practical consequence: you cannot compare an APR against an APY directly. They are not the same kind of number. A 6 percent APR on a monthly-compounding loan is an effective 6.168 percent, while a 6 percent APY on a savings account is an effective 6 percent flat. The loan is more expensive than the deposit is generous, even though both say six.

This calculator is one way through that. Convert the APY to its nominal equivalent at the same compounding frequency as the loan, and then the two figures sit on a common basis and the comparison means something.

Outside the US the terminology differs and the underlying split does not. The UK quotes AER on savings, which is effective, alongside APR on credit. India quotes effective yields on deposits and reducing balance rates on loans. The names change, the trap does not.

When you actually need this

Feeding a loan calculator. Most payment calculators, including ours, want a nominal rate and apply the compounding themselves. If your lender has given you an effective figure, converting it first is the difference between a right answer and a payment that comes back too high.

Comparing across markets or products. Any time two rates were quoted on different conventions, one of them has to be converted before the comparison is worth anything. This is that conversion.

Setting a rate to hit a target. If you need a facility to return 8 percent a year and it will compound quarterly, you need to quote 7.771 percent, not 8. Working backwards from the effect is exactly what this box does.

Checking a lender's arithmetic. If a document states both a nominal rate and an effective one, run the conversion. They should agree. When they do not, the usual explanation is that fees have been folded into one of them, which is worth knowing about.

Questions people ask

What is the difference between a nominal and an effective rate?

A nominal rate is a per-period rate multiplied up to look annual, and it ignores compounding. An effective rate is what actually happens across a full year, compounding included. The effective figure is always the higher of the two unless compounding is annual, in which case they are equal.

Why is the nominal rate always lower?

Because compounding does part of the work. If interest earns interest during the year, the raw rate needed to reach a given annual result is smaller than that result.

What happens if I leave the periods box empty?

You get the continuously compounded equivalent, which is the natural logarithm of one plus the effective rate. On 6 percent that is 5.827 percent, and it is the floor that no finite compounding frequency can beat.

What number should I put for compounding periods?

Twelve for monthly, four for quarterly, two for half yearly, one for annually, 365 for daily. Match whatever your lender or bank actually uses.

Why does 1 period give the same rate back?

Because with one compounding period a year there is nothing to compound. Nominal and effective are the same number by definition at annual compounding.

Can I compare an APR against an APY?

Not directly. An APR is a nominal figure under the US lending rules, while an APY is an effective figure under the savings rules. Convert one to the other's basis first.

Is daily compounding much better than monthly?

Barely. On a 6 percent effective target, monthly needs 5.841 percent nominal and daily needs 5.827 percent. The gains from compounding more often shrink very quickly.

How do I go the other way, from nominal to effective?

Raise one plus the nominal rate divided by the number of periods to the power of the number of periods, then subtract one. A nominal 5.841 percent compounded monthly gives an effective 6 percent.

References

The relationship between nominal and effective annual rates, and the continuously compounded limit given by the natural logarithm, follow standard financial mathematics as set out in university materials for the actuarial syllabus. The definition of the annual percentage rate as a periodic rate multiplied by the number of periods in a year comes from Regulation Z. The definition of the annual percentage yield as a rate reflecting total interest based on the interest rate and the frequency of compounding, and the formula for calculating it, come from Regulation DD and its Appendix A.

1. Consumer Financial Protection Bureau (CFPB), Regulation DD, Appendix A to Part 1030: Annual Percentage Yield Calculation. https://www.ecfr.gov/current/title-12/chapter-X/part-1030/appendix-Appendix A to Part 1030
2. Consumer Financial Protection Bureau (CFPB), 12 CFR Part 1030, Truth in Savings (Regulation DD). https://www.ecfr.gov/current/title-12/chapter-X/part-1030
3. Consumer Financial Protection Bureau (CFPB), Regulation Z, Appendix J to Part 1026: Annual Percentage Rate Computations for Closed-End Credit Transactions. https://www.consumerfinance.gov/rules-policy/regulations/1026/j/
4. Miguel A. Arcones, Binghamton University, Manual for SOA Exam FM, Chapter 4: Amortization and Sinking Funds, Section 4.1 Amortization Schedules. https://people.math.binghamton.edu/arcones/exam-fm/sect-4-1.pdf