Loan Interest Calculator

Two dropdowns most calculators fold into one

Look at the two selects on this tool. One asks how often the interest compounds. The other asks how often you repay.

Almost every loan calculator on the internet assumes those are the same thing. Monthly payments, monthly compounding, done. And for a great many loans that assumption holds, so nobody notices.

It does not always hold. Interest can accrue annually on a facility you service monthly. It can compound monthly on a loan you settle once a year. Agricultural credit, some business term loans and plenty of arrangements between private parties separate the two, and when they are separate the difference in what you pay is not small.

This tool keeps them apart and does the conversion between them properly. What follows is how that conversion works and what it costs when you get it wrong.

Five fields

1. Loan Amount. What you are borrowing.
2. Loan Term. In whole years.
3. Interest Rate. The annual rate quoted by your lender. There is a section below on exactly which rate this should be.
4. Compound Frequency. How often interest is added to the balance. Monthly, quarterly or annually.
5. Pay Back. How often you make a payment. Every month or every year.

Press Calculate and you get the payment, the total of all payments and the total interest.

The two dropdowns default to annually and yearly, which is the simplest case and a sensible place to start. Change one and watch the results move. Change both to monthly and you get the ordinary loan everybody is familiar with.

Three steps, and why the middle one exists

Getting from a quoted annual rate to a payment takes three steps here, where a simpler calculator uses one.

Step one: turn the quoted rate into an effective annual rate.

EAR = (1 + r ÷ c)^c - 1

where r is the annual rate you typed and c is the number of times a year it compounds. This is the step that captures compounding. A rate of 10 percent compounded monthly is not 10 percent a year, it is 10.4713 percent, because the interest added in January starts earning interest in February.

Step two: turn the effective annual rate into a rate per payment.

periodic rate = (1 + EAR)^1/p - 1

where p is the number of payments a year. This asks what rate, applied p times, compounds up to the same annual effect.

Step three: the ordinary annuity formula, run with that periodic rate and the total number of payments.

The middle step is the one that matters, and it exists because you cannot go straight from a compounding schedule to a payment schedule when the two do not line up. Routing through the effective annual rate gives you a common currency. Compounding goes in, the annual effect comes out, and the payment schedule takes it from there.

Doing it any other way, by dividing the quoted rate by the number of payments and ignoring the compounding, gets the right answer only in the case where the two happen to match.

Six versions of the same ten thousand

Loan of 10,000, quoted at 10 percent, over 5 years. Nothing changes except the two dropdowns.

Same loan, same term, same headline rate. The cheapest arrangement costs 2,621.35 in interest and the dearest costs 3,349.06.

A gap of 727.71, which is more than a quarter of the smaller figure, produced entirely by two dropdowns.

Two patterns in that table are worth naming.

More frequent compounding always costs more. Read the effective annual rate column. Ten percent compounded annually stays 10 percent, quarterly becomes 10.3813, monthly becomes 10.4713. Each time you compound, interest starts earning interest sooner.

More frequent repayment always costs less. Compare the monthly rows against the yearly ones. Paying monthly clears the balance in twelve steps a year instead of one, so there is less debt outstanding for less of the time.

The two dropdowns pull in opposite directions, which is why the best case in the table is annual compounding with monthly repayment and the worst is monthly compounding with annual repayment.

When the clever part disappears

Set both dropdowns to monthly and something satisfying happens.

The effective annual rate comes out at 10.4713 percent, and then the second step converts it back down to a rate per payment of 0.8333 percent. Which is exactly 10 divided by 12.

Not approximately. Exactly, to every decimal place a computer will show you.

The reason is visible in the algebra. Step one raises the monthly rate to the power of twelve, and step two takes the twelfth root of the result. Those two operations undo each other, and you are left with the monthly rate you started from.

So when compounding and repayment match, the three step method collapses into the one step method every ordinary loan calculator uses. The sophistication is still there, it just has nothing to do.

This is worth knowing for two reasons. It means you can use this tool for perfectly ordinary loans without worrying that it is doing something exotic. And it means the simple method everyone uses is not a shortcut or an approximation, it is the correct answer to the particular case where the frequencies agree, which happens to be most of them.

What number the rate box wants

The rate box wants your lender's quoted annual rate, before compounding has been applied to it. In financial mathematics this is called a nominal annual rate, and it is the number a lender means when they say a loan is at 10 percent compounded monthly.

The distinction that matters: the tool takes that figure and applies the compounding itself, in step one. So you should enter the rate as quoted and let the Compound Frequency dropdown do the rest.

What you should not enter is a rate that already has compounding baked into it. If your lender has given you an effective annual rate, an annual equivalent rate, or an annual percentage yield, that figure already reflects a full year of compounding. Feeding it in here and then selecting monthly compounding compounds it a second time, and the payment comes back too high.

If effective is all you have, convert it first. The nominal interest rate calculator does exactly that, taking an effective annual rate and a number of compounding periods and returning the nominal rate to enter here.

Worth adding, since it causes a lot of confusion in both directions: an APR under the US lending rules is not the same thing as an effective annual rate. An APR is a periodic rate multiplied by the number of periods in a year, which makes it a nominal figure, and it additionally folds in fees. An APY on a savings account is an effective figure that reflects compounding. Two similar looking acronyms, built on opposite conventions.

Setting the two dropdowns to match your loan

Both settings should describe your actual loan rather than what feels normal. The trouble is that loan documents are often clearer about repayment than about compounding.

Pay Back is the easy one. It is on your schedule, and you will know it.

Compound Frequency takes asking. The phrase to look for is anything of the form "compounded monthly" or "interest calculated monthly on the reducing balance". If the document is silent, ask directly: how often is interest added to the balance?

Three common patterns worth recognising:

• Monthly and monthly. The ordinary consumer or housing loan across most markets. Both dropdowns to monthly.
• Annual compounding, monthly instalments. Seen on some agricultural and development credit, where interest is reckoned yearly but you pay in instalments. Compounding annually, pay back monthly. This is the cheapest combination in the table.
• Monthly compounding, annual repayment. Facilities settled once a year against a balance that has been compounding throughout. Compounding monthly, pay back yearly. The most expensive combination, and worth checking twice if it is what you have been offered.

If you genuinely cannot find out how often it compounds, set it to match your repayment frequency. That gives you the standard answer, and the sections above tell you which direction the real figure would move if you learn otherwise.

Questions people ask

What is the difference between compounding and repayment frequency?

Compounding is how often interest is added to what you owe. Repayment is how often you pay. They are separate settings on a loan and only the second one is obvious from your schedule.

How much difference does it make?

On 10,000 at 10 percent over five years, the cheapest combination costs 2,621.35 in interest and the dearest costs 3,349.06. That is 727.71 of difference from the dropdowns alone.

Should I enter a nominal or an effective rate?

Nominal, meaning the rate as quoted before compounding is applied. The tool applies the compounding itself. If you only have an effective annual rate, convert it first with the nominal interest rate calculator.

Does more frequent compounding cost me more?

Yes, always. Ten percent compounded annually is 10 percent for the year, quarterly is 10.3813 percent, and monthly is 10.4713 percent, because interest starts earning interest sooner.

Does paying more often save me money?

Yes, always. Each payment reduces the balance, so paying twelve times a year rather than once leaves less debt outstanding for less of the time.

What if both are monthly?

Then the conversion collapses and the rate per payment is exactly the annual rate divided by twelve. The tool gives the same answer any ordinary loan calculator would.

I do not know how often my loan compounds.

Ask the lender directly, since it is not always stated in the paperwork. In the meantime, set it to match your repayment frequency, which gives the standard result.

Is an APR the same as an effective annual rate?

No. An APR under the US lending rules is a periodic rate multiplied by the number of periods in a year, making it nominal, and it also includes fees. An APY on a savings account is an effective figure that reflects compounding.

References

The conversion from a nominal annual rate to an effective annual rate, and from an effective annual rate to an equivalent rate for a different payment frequency, follows standard financial mathematics as set out in university materials for the actuarial syllabus. The treatment of an annual percentage rate as a periodic rate multiplied by the number of periods in a year comes from Regulation Z. The contrasting treatment of annual percentage yield, defined as a rate reflecting the total interest paid based on the interest rate and the frequency of compounding, comes from Regulation DD.

1. 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
2. J. Robert Buchanan, Millersville University, Loan Repayment, MATH 372 Financial Mathematics I. https://sites.millersville.edu/rbuchanan/math372/LoanRepayment-handout.pdf
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. Consumer Financial Protection Bureau (CFPB), Regulation DD, 12 CFR Part 1030, Truth in Savings, and Appendix A to Part 1030: Annual Percentage Yield Calculation. https://www.ecfr.gov/current/title-12/chapter-X/part-1030