Loan repayment and the cost of credit

What this works out

Two loans can advertise the same number and cost you hundreds of pounds differently, because “the rate” means three different things. A rate charged on what you still owe is one thing. The same rate charged on the amount you originally borrowed, for the whole term, whether or not you have paid most of it back, is another — that is a flat rate, and it costs about twice as much. And an APR is a rate defined by law to include the fees, which the headline number usually does not.

So this tool asks which of the three you were quoted, builds the schedule from that, and then works the APR back out of the instalments themselves. That last step is the only way to put a flat quote and an APR quote side by side honestly, and it is also how both the American and the European regulators define an APR in the first place.

The figures below carry no currency symbol on purpose: the arithmetic is the same in pounds, euros or dollars, and the tool follows whichever currency your locale uses.

The method

For a rate charged on the falling balance, the instalment comes from the standard annuity formula. The only decision worth explaining is how a yearly rate becomes a monthly one. Divide by twelve and you get the American convention, which is what Regulation Z means by APR and what nearly every loan agreement’s payment table is built on. Take the twelfth root of annual growth and you get the British and European convention, where an APR is an effective yearly rate. On a 7.9% quote the two differ by about half a percent of the monthly instalment. Small, but it is the difference between matching the lender’s paperwork and not, so it is a choice rather than a guess.

For a flat rate there is no falling balance in the arithmetic at all. The whole interest charge is worked out once, on the original amount, for the full term, and then divided across the instalments. That is the entire reason a flat rate is roughly half the APR it corresponds to: you keep paying interest on money you have already handed back.

The APR is then found by bisection rather than by a formula, because there is no closed form for it — both Regulation Z Appendix J and Annex I of the Consumer Credit Directive prescribe iteration. The present value of the instalments is strictly decreasing in the rate, so a sign change brackets exactly one answer and halving the interval forty times cannot land on the wrong root. Notice that the amount being discounted back to is the cash you received, not the balance you owe. A fee added to the loan is repaid with interest but was never lent to you, and that asymmetry is what puts it into the APR.

Every figure is computed in decimal arithmetic and rounded to the cent at each month, the way a ledger does it, rather than in binary floating point. A loan schedule that drifts by a cent a month is wrong by half a pound by the end.

Before you read on

12,000 over 48 months. One lender quotes 7.9% a year on the falling balance; another quotes 3.9% flat. How much cheaper is the flat quote?

• AAbout half — the rate is half.

  A flat rate charges the whole term's interest on the original amount, so half the rate is nowhere near half the cost.

• BBy 162.85 — about 8%.

  Yes. 1,872.00 against 2,034.85, on a quote that looks half the price.

• CIt is not cheaper at all — it costs more.

  It is genuinely cheaper here, just barely. At 7.9% flat against 7.9% on the balance it would cost nearly twice as much.

By 162.85, or about 8%. A flat rate works the whole interest charge out once, on the original amount, for the full term, and then divides it across the instalments — so you keep paying interest on money you have already handed back. That is why 3.9% flat comes out at 7.54% APR: roughly double, which is the rule of thumb for a flat rate, and it is the reason a quote that looks like half the price of 7.9% APR is barely cheaper than it.

• The 12,000 advanced12,00076.0%
• Interest3,79224.0%

A worked example

Borrow 12,000 over 48 months. Here is the same loan under three quotes that all look like small numbers.

Quoted as	Monthly	Total repaid	Cost of credit	APR
7.9% a year, charged monthly	292.39	14,034.85	2,034.85	8.19%
7.9% flat	329.00	15,792.00	3,792.00	15.14%
3.9% flat	289.00	13,872.00	1,872.00	7.54%

The first two rows are the same two digits. They differ by 1,757.15.

The third row is the one worth staring at: 3.9% flat looks like half the price of 7.9% APR, and it is barely cheaper at all.

Figure	7.9% a yearon the balance	3.9% flat	Difference
Monthly instalment	292.39	289.00	−3.39
Total repaid	14,034.85	13,872.00better	−162.85
Cost of credit	2,034.85	1,872.00better	−162.85
APR	8.19%	7.54%better	−0.65%

Half the quoted rate buys 162.85, or 8%. That is the whole point of an APR: it is the figure the two quotes can be compared on, and the two headline rates are not.

Check the first row against the annuity formula:

i = 0.079 / 12 = 0.006583333
n = 48

(1 + i) ^ -48 = 0.7298149
1 - 0.7298149 = 0.2701851

payment = 12,000 x 0.006583333 / 0.2701851
        = 79.00 / 0.2701851
        = 292.3921  ->  292.39

47 instalments of 292.39, then a final 292.52
                                = 14,034.85
less the 12,000 advanced        =  2,034.85 of interest

And the second, where the arithmetic is easier and the answer is worse:

interest = 12,000 x 0.079 x (48 / 12) = 3,792.00
payment  = (12,000 + 3,792) / 48      =   329.00

These are the same numbers asserted in this tool’s test file, so if the formula ever changes without this page changing with it, the build fails.

What it does not do

It models one fixed-rate loan repaid exactly on schedule. It does not handle variable rates, promotional periods that end, payment holidays, overpayments, early settlement rebates, or insurance sold alongside the credit. It also cannot tell you the rate you will actually be offered: an advertised APR is a representative one, which a lender only has to give to 51% of the people who take the loan, so the number on the poster and the number in your agreement are often not the same. Use this to compare offers you have in writing, and to work out what a quote really means before you accept it.

The arithmetic is the one a mortgage uses. Over decades against a property rather than years against a car, mortgage repayment is the same formula with the assumptions that actually fit.

A rate charged on what is still owed:

  i = r / 12                  a yearly rate, interest charged monthly
  i = (1 + r) ^ (1/12) - 1    an effective yearly rate (UK and EU APR)

  payment = P x i / (1 - (1 + i) ^ -n)

A flat rate, charged on the original amount for the whole term:

  interest = P x r x (n / 12)
  payment  = (P + interest) / n

The APR, recovered from the instalments by bisection. Find the monthly
rate j at which:

  advance = sum over k = 1..n of  payment(k) / (1 + j) ^ k

  APR, nominal    = j x 12
  APR, compounded = (1 + j) ^ 12 - 1

where
  r       = the advertised rate, as a decimal
  n       = the term, in months
  P       = the amount borrowed plus any fee added to the loan
  advance = the cash actually handed over, which excludes a financed fee