Why a 5% flat-rate loan costs more than a 7% APR loan over the same four years

1 · Two quotes

A car dealer offers you £10,000 over four years at a 5% flat rate. Your bank offers the same loan at 7% APR.

Same money, same four years. Which loan costs you less?

2 · What the dealer's 5% counts

A flat rate takes three steps. First, one year's interest on the whole loan: £10,000 × 5% = £500 a year.

Then multiply by the years: £500 × 4 = £2,000 of interest.

Add it to the loan and split it into 48 monthly payments: (£10,000 + £2,000) ÷ 48 = £250 a month.

Notice what's missing. None of these steps asks how much you still owe. They treat you as owing the full £10,000 for all four years.

3 · What you actually owe

Here's the dealer's assumption, drawn: £10,000 owed every month for 48 months, with interest charged on all of it.

But that isn't what happens. Every £250 payment chips away at what you owe.

After the first payment you owe £9,827. The flat rate still charges as if you owed £10,000.

By the halfway point you owe roughly half: £5,459.

After 48 payments you owe nothing. You only owed the full £10,000 for the first month.

your turn

4 · Find the real rate

So what is the dealer really charging, measured on what you actually owe? That's the question an APR (annual percentage rate) answers:

Which monthly rate, charged only on what's left, makes 48 payments of £250 bring the balance to exactly £0?

The equation in UK rules can't be rearranged for the rate, so you test rates. The coins stay fixed: 48 × £250.

First try: the dealer's 5% spread over 12 months, 0.42% a month. The loan is paid off at payment 44, with four payments to spare. Too low: the real rate is higher.

Try 1% a month. After all 48 payments you'd still owe £817. Too high.

About 0.77% a month fits: the 48th payment brings the balance to exactly £0. Measured on what you owe, this loan costs more than 5% a year.

At 0.77%, each month's interest is worked out on what's left:

To turn a monthly rate into a yearly one you grow it, not add it: 1.0077 multiplied by itself 12 times is 1.096. That's a 9.6% APR.

It adds nothing to the bill. You still repay £12,000: the 5% flat and the 9.6% APR describe the same deal.

In a spreadsheet: =(1+RATE(48,-250,10000))^12-1

your turn

too low

5 · Why nearly double?

Both ways of counting share out the same £2,000 of interest. The flat rate puts £500 in each year. Charged on what you still owe, more falls in the early years, when you owe more.

The flat rate divides £500 a year by the full £10,000. But across the loan you owe about £5,400 on average. The same interest measured against about half the money needs about twice the rate.

×2

So the APR on a flat-rate loan comes out a little under twice the flat rate: 1.87 to 1.97 times, for flat rates of 3% to 10% over two to five years, with equal monthly payments and no fees.

6 · The fair comparison

Now set 9.6% beside the bank's 7%. The bank's 7% APR is 0.565% a month, so £238.47 a month clears £10,000 in 48 payments.

The bank's loan costs £11,447 in total, the dealer's £12,000. Interest: £1,447 against £2,000. The bank's is £553 cheaper, even though its percentage looked higher.