Credit Card Repayments

Question

I'm trying to calculate how long it would take to pay off a credit card loan. I've used online calculators, most disagree with what I've worked out, but some agree. I have no idea where I'm going wrong.

If I have -£10000 on my card at 18.9% APR and I'm paying £300 a month to pay it off, this is what I get:

 1   £10000.00 + £157.50 - £300.00 = £9857.50
 2   £ 9857.50 + £155.26 - £300.00 = £9712.76
 3   £ 9712.76 + £152.98 - £300.00 = £9565.73
 4   £ 9565.73 + £150.66 - £300.00 = £9416.39
 5   £ 9416.39 + £148.31 - £300.00 = £9264.70
 6   £ 9264.70 + £145.92 - £300.00 = £9110.62
 7   £ 9110.62 + £143.49 - £300.00 = £8954.11
 8   £ 8954.11 + £141.03 - £300.00 = £8795.14
 9   £ 8795.14 + £138.52 - £300.00 = £8633.66
10   £ 8633.66 + £135.98 - £300.00 = £8469.64
11   £ 8469.64 + £133.40 - £300.00 = £8303.04
12   £ 8303.04 + £130.77 - £300.00 = £8133.81
13   £ 8133.81 + £128.11 - £300.00 = £7961.92
14   £ 7961.92 + £125.40 - £300.00 = £7787.32
15   £ 7787.32 + £122.65 - £300.00 = £7609.97
16   £ 7609.97 + £119.86 - £300.00 = £7429.83
17   £ 7429.83 + £117.02 - £300.00 = £7246.85
18   £ 7246.85 + £114.14 - £300.00 = £7060.98
19   £ 7060.98 + £111.21 - £300.00 = £6872.20
20   £ 6872.20 + £108.24 - £300.00 = £6680.43
21   £ 6680.43 + £105.22 - £300.00 = £6485.65
22   £ 6485.65 + £102.15 - £300.00 = £6287.80
23   £ 6287.80 + £ 99.03 - £300.00 = £6086.83
24   £ 6086.83 + £ 95.87 - £300.00 = £5882.70
25   £ 5882.70 + £ 92.65 - £300.00 = £5675.35
26   £ 5675.35 + £ 89.39 - £300.00 = £5464.74
27   £ 5464.74 + £ 86.07 - £300.00 = £5250.81
28   £ 5250.81 + £ 82.70 - £300.00 = £5033.51
29   £ 5033.51 + £ 79.28 - £300.00 = £4812.79
30   £ 4812.79 + £ 75.80 - £300.00 = £4588.59
31   £ 4588.59 + £ 72.27 - £300.00 = £4360.86
32   £ 4360.86 + £ 68.68 - £300.00 = £4129.54
33   £ 4129.54 + £ 65.04 - £300.00 = £3894.58
34   £ 3894.58 + £ 61.34 - £300.00 = £3655.92
35   £ 3655.92 + £ 57.58 - £300.00 = £3413.50
36   £ 3413.50 + £ 53.76 - £300.00 = £3167.26
37   £ 3167.26 + £ 49.88 - £300.00 = £2917.15
38   £ 2917.15 + £ 45.95 - £300.00 = £2663.09
39   £ 2663.09 + £ 41.94 - £300.00 = £2405.04
40   £ 2405.04 + £ 37.88 - £300.00 = £2142.92
41   £ 2142.92 + £ 33.75 - £300.00 = £1876.67
42   £ 1876.67 + £ 29.56 - £300.00 = £1606.22
43   £ 1606.22 + £ 25.30 - £300.00 = £1331.52
44   £ 1331.52 + £ 20.97 - £300.00 = £1052.49
45   £ 1052.49 + £ 16.58 - £300.00 = £ 769.07
46   £  769.07 + £ 12.11 - £300.00 = £ 481.18
47   £  481.18 + £  7.58 - £300.00 = £ 188.76
48   £  188.76 + £  2.97 - £300.00 = £-108.26

Total interest: £4291.74

When I use this calculator, it tells me, it'll cost me £3773.62 in interest and that I'll pay it off in 46 months, can you tell me where I'm going wrong? I've even tried calculating the percentage on a daily basis in case that was the issue, it's not. It affected it marginally, but not by ~£500.

Answer 1

In most countries (see below), including the UK, an APR of 18.9% is equivalent to a monthly rate of 1.45%, not 1.575% that you used.

To calculate monthly from annual, you do:

 M = ((1 + A) ^ 1/12) - 1

Where A and M are in decimal form (ie 0.189 for 18.9%)

Why did I use 1.45% and not 1.575%?

At first glance, you'd think that you'd simply divide the annual interest rate by 12 months to get the monthly rate; 18.9% / 12 months gives you 1.575%.

However, when you apply this monthly rate, once a month for a year to get back to the annual rate, you can see where the problem comes in. Starting with $100 on Jan 1, you'd get $101.575 next month, then $103.175 the next, and so on. At the end of 12 months, you're at $120.63... but this is equivalent to an annual rate of 20.63%, not 18.9%.

The reason is that interest compounds; each month you add interest to the original debt, plus whatever interest you have accrued.

On the other hand, a lower rate of 1.45% build up like

(100 * 1.0145) * 1.0145) * 1.0145...

AKA

$100 * 1.0145^12 = $118.85

Which gives you your 18.9% increase over one year.

What will the bank use?

Things get complicated here. Different countries use different definitions for converting from their posted annual rates to the rate that they use to calculate your interest.

For instance, APR in the US is (generally) just divided by 12 to get the monthly rate. In other countries, the approach above is used.

In some circumstances, interest may be calculated on a daily basis, leading to a different effective monthly rate.

Ultimately, this is laid out in the small print. Here's the take-away:

Read the small print of your credit agreement to know for sure how your interest will be calculated and how frequently.

Answer 2

@Dancrumb is incorrect in presenting their calculation as being the standard meaning of "APR". Assuming no compounding within the month, the standard meaning of APR is that an interest rate of 18.9% is equivalent to 1.575% being charged each month. An interest rate of 18.9% APY is equivalent to 1.45% being charged each month. APY is also sometimes referred to as the "effective APR". The meanings within the UK may be be different however; my googling "UK APR definition" found rather oblique results. So if the numbers match the website's using 1.45%, that means that the website is taking the interest to be in APY rather than APR, or using a non-standard meaning of "APR". You should check your credit card statement to see what terminology is used.