Calculating Interest Charged Based On Payments

Solving For APR By Iteration

There's no need to work through this discussion if your interest is limited to calculating basic mortgage payments. However, this discussion of solving for APR by iteration is actually quite handy for determining the interest rate you are really paying on any outstanding loans you may have. Back when the author was in graduate school, he took out a Stafford student loan, and the paperwork sent by the loan servicer turned out to be wrong. The interest rate on the loan was listed as 8%, but the payments requested turned out to be based on an interest rate of 19.85%. It took months of wrangling and complaints to the state banking commission and attorney general's office to get the loan straightened out, so the moral of the story is double check the math on money you're being asked to pay.

The process of iteration is simple to describe, but time consuming to compute. When given an equation with a single unknown, like solving for the interest rate in our compund interest formula, the solution starts with guessing an interest rate and plugging it into the equation. After doing the math, we end up with two different quantities with an equal sign between them. The trick is to increase or decrease our guess until the numbers on each side of the equation really are equal, or close enough that we're ready to stop. As the monthly payment and the original principal won't change for any of these calculations, it makes sense to gather them on the left side of the equal sign, as:

M / P = [ i(1 + i)ⁿ ] / [ (1 + i)ⁿ - 1]

And using the loan amount and final payment from our last example (from the ebook), we can solve for M / P once as:

M / P = $830.33 / $100,000

= 0.0083

And the number of months, n, also remains constant at 180 for fifteen years, giving

0.0083= [ i(1 + i)¹⁸⁰] / [ (1 + i)¹⁸⁰- 1]

So the only work remaining is to guess a value for i and start solving. We know that the increase in the monthly payment by almost $40 a month means that the APR, or effective annual interest rate we are paying has risen from the original 5%, so let's try 6% and see what happens:

i = 6% / # months per year

= 0.06 / 12

= 0.005

And our expression (1 + i)¹⁸⁰

(1 + i)¹⁸⁰ = (1 + 0.006)¹⁸⁰

= (1.006)¹⁸⁰

= 2.4541

And the full equation now reads:

0.0083 = [0.006 (2.451)] / [ (2.451- 1]

= 0.014706 / 1.451

= 0.01014, which we'll round down to 0.0101

Well, 0.0083 clearly doesn't equal 0.0101, so our guess of 6% interest was too high. Let's try another iteration with 5.5% for the APR.

i = 5.5% / # months per year

= 0.055 / 12

= 0.0045833

And our expression (1 + i)¹⁸⁰

(1 + i)¹⁸⁰= (1 + 0.0045833)¹⁸⁰

= (1.0045833)180

= 2.2776

And the full equation now reads:

0.0083 = [0.0055 (2.2776)] / [ (2.776- 1]

= 0.012527 / 1.776

= 0.00706, which we'll round down to 0.0071

Comparing again, 0.0083 doesn't equal 0.0071, so the 5.5% interest guess was too low. So let's try another iteration, splitting the difference by using an APR estimate of 5.75%

i = 5.75% / # months per year

= 0.0575 / 12

= 0.00479167

And our expression (1 + i)180

(1 + i)¹⁸⁰ = (1 + 0.00479167)¹⁸⁰

= (1.00479167)¹⁸⁰

= 2.3642

And the full equation now reads:

0.0083 = [0.00479167 (2.3642)] / [ (2.3642- 1]

= 0.011328 / 1.3642

= 0.008304, which we'll round down to 0.0083

Since 0.0083 = 0.0083, we can stop now, with the calculated APR for this loan being 5.75%. As a check, a using an online calculator for APR yields a rate of 5.7486%, meaning our final guess was accurate within a couple one hundredths of one percent! And remember, for an APR to meet the legal requirement, it can err on the upside or the downside by 0.125% meaning that a bank could legally report the APR on this loan to be anywhere between 5.6236% and 5.8736%.

We had a pretty good idea where to start guessing because we knew that the monthly payment wasn't too far from the original mortgage payment before the fees were added. But if you are simply presented with a monthly payment and a mortgage amount out of the blue, you can waste a lot of time guessing the interest rate to start the calculations going. One trick is to use the table of standard mortgage interest factors we presented back in the section of Tables For Fixed Rate Mortgages. From the example above, our M /P = 0.0083, so the factor we are looking for is:

M / P = [value from table] / 1,000

(0.0083) = value from table / 1,000

8.3 = value from table

Checking the table in the fifteen year column, we find the value 8.30410 for the interest rate 5.75%, so we got lucky and can stop without doing any further iterations. But in general, it won't take many iterations to get within a fraction of a percent of the true interest being charged on your loan, which is all you need if you're worried about out-and-out errors instead of penny skimming frauds. The APR is supposed to express the annual interest rate you end up being charged on the amount of money YOU THOUGHT you would be borrowing. The full text of the regulation from which the following is excerpted can be found on the FDIC website.

The idea behind the APR was to provide an apples-to-apples comparison that consumers could use to shop for loans. The APR interest rate is supposed to express the true cost of the loan to the consumer on an annual basis, but it generally falls short for a variety of reasons that we'll discuss in the next section. The law starts out by giving bankers a quarter point of wiggle room in the accuracy of the APR they report to the borrower. A loan with a stated APR of 4.75 % might actually be identical to a loan with a stated APR of 5% without breaking the law, as long as the true calculated APR for both is 4.875%. It's entirely possible that Congress saw the need to allow an error of 1/8 point in either direction once they saw how complicated the interest calculation was becoming, because they doubled the allowed error for "irregular" loans. On top of that, there are numerous exceptions to the rule that allow for more errors to be introduced.

The APR is supposed to take into account all of the extra fees that add to the cost of a loan, and which may be included into the principal amount. There are at least 25 different legitimate fees that may be heaped onto a mortgage at closing, depending on the bank, including: lender and broker origination fees, loan and broker discount fees, lender and broker underwriting fees, mortgage insurance, credit report and appraisal fees, signing, administration and application fees, attorney fees, charges for couriers, bank transfers, escrow, and the kitchen sink! By including all of the fees into the annualized cost of the mortgage, the APR, you should end up with a simple number that's roughly comparable between banks or other loan originators, allowing you to shop on price.