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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.

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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.

  • Ha, that's less than obvious. Thanks though, spot on. Now I know, I've managed to find a few articles explaining why too. Sorry I can't upvote! Thanks for the help. – user82296 Feb 7 at 20:22
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    Note that this is the way the monthly interest rate is calculated in most places in the world except the US, where the APR has a legal definition that is different and results in the monthly rate being 18.9% divided by 12 = 1.575% which is what the OP used. – Dilip Sarwate Feb 7 at 20:53
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    And what's being described here is the difference between Rate and Yield. – quid Feb 7 at 21:35
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    This might be dependent on the exact credit company you have an agreement with, but the agreements that I have read myself state that if the account is not paid in full, then they will be calculating interest at a Daily Rate, for a total APR at the described rate. I suspect that their rate calculation relies on the number of days in the year (rather than over 12 months). Your agreements may vary. The statement should provide the formula for calculating interest. – Shorlan Feb 7 at 23:29
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    Daily compounding also makes repayment time calculations complex since the difference between paying on the day you get the statement vs paying on the last day before a penalty is assessed is several weeks of interest on the principle. Especially at the start of very long period loans you can end up really slowing down your principal reduction with last day payments. – Dan Neely Feb 8 at 5:10
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@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.

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    This is only standard in the US - the UK (where the NatWest bank is based) use the more standard approach that I outlined. – Dancrumb Feb 8 at 0:58
  • @Dancrumb So, it's standard "only" in the largest English-speaking economy in the world? And the usage in the UK is somehow "more standard"? – Acccumulation Feb 8 at 1:08
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    What does English-speaking have to do with it? The OP listed their prices in pounds, not dollars. And in most of the world interest is NOT calculated this way. In the end, going on US rules for an answer for the UK is just wrong. – Sebastiaan van den Broek Feb 8 at 5:05
  • @SebastiaanvandenBroek "What does English-speaking have to do with it?" We're discussing the meaning of "Annual Percentage Rate", an English term. "And in most of the world interest is NOT calculated this way." That statement makes no sense. We're not discussing how interest is calculated, we're discussing what terms are used to refer to different methods of calculating interest. – Acccumulation Feb 8 at 15:55
  • "going on US rules for an answer for the UK is just wrong." Stack Exchange is not a British-only website, and the purpose of the question-answer format is NOT for a single person to get an answer to their specific question. The purpose is to be a general reference where someone who has a question can find a similar question already posted, and find answers that hopefully apply to their own question. Answering a question from a British person with a British-specific answer with no note that it is British-specific is highly inappropriate. – Acccumulation Feb 8 at 15:56

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