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I have really bad credit from when I was younger.

The only credit card I can get is one 49.9% APR.

If I spend £500 in a single transaction and pay back lets say £50 a month, what interest would I be paying monthly?

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    It should be zero. In other words don't do it. – Pete B. Jan 26 '17 at 14:17
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    "Too much" would be the answer here. That you're considering a 49.9% APR for a £500 purchase is a good indication that the bad habits you had "when you were younger" are still around. – ceejayoz Jan 27 '17 at 3:20
  • You can plug in your number in online calculator to see how long it will take and how much you end up paying. myfincal.com/Credit/CreditCalculation – Hakunamatata Jan 27 '17 at 3:40
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Monthly interest wouldn't be consistent as your balance would decrease over time. So, more of your £50 payment would go towards principal and less towards interest with each month that passes. Total interest over the 13 months it will take to pay off would be about £159.

  • tcalc.timevalue.com/all-financial-calculators/… works it out for you. although it requires that you pay more than £50 in the last month. – MD-Tech Jan 26 '17 at 13:11
  • In the UK APR is an effective rate. The calculator takes a nominal rate, standard for the US. 49.9% effective APR = 41.17% APR nominal compounded monthly. – Chris Degnen Jan 26 '17 at 14:46
  • Taking APR as nominal (which is incorrect for the UK) r = 0.499/12, s = 500, d = 50, number of months k = Ceiling[-(Log[1 - (r s)/d]/Log[1 + r])] = 14 so interest = (d + d k r - d (1 + r)^k - r s + r (1 + r)^k s)/r = 159.884 – Chris Degnen Jan 28 '17 at 8:16
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Short answer

total interest = (d+d k r-d (1+r)^k-r s+r (1+r)^k s)/r

where

s = present value of loan
d = periodic payment
r = periodic interest rate
k = number of whole periods (rounded up)
  = Ceiling of -(Log[1-(r s)/d]/Log[1+r])

Detailed answer

APR in Europe and the UK is stated as an effective annual rate rather than a nominal rate compounded monthly, (which is the US standard). For info see EU APR.

To calculate the monthly rate r from an effective annual rate APR:

r = (1 + APR)^(1/12) - 1
  = (1 + 0.499)^(1/12) - 1 = 3.43086 %

The mathematics of a loan can be expressed like so:-

The present value is equal to the sum of the discounted future payments.

s = present value of loan
n = number of periods
d = periodic payment
r = periodic interest rate

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by induction

enter image description here

Rearranging for n

n = -(Log[1-(r s)/d]/Log[1+r])

s = 500 and d = 50

∴ n = -(Log[1-(0.0343086*500)/50]/Log[1+0.0343086])

∴ n = 12.4566

So it will take 13 months to clear the loan.

A quick estimate of the interest would be

n d - s = £122.832

But a full month's interest will be charged on the balance in month 13.

The actual interest paid each month changes as the loan is paid down.

The balance p in month k follows this recurrence equation

p[k + 1] = p[k] (1 + r) - d

So it can be calculated that p and the interest i paid in month k are

p[k] = (d+(1+r)^k (r s-d))/r
i[k] = p[k-1] r

∴ i[k] = d+(1+r)^(k-1) (r s-d)

E.g. in the first month

i[1] = d+(1+r)^(1-1) (r s-d) = 17.1543

which is also equal to 500 r, so that checks out.

In the second month

i[2] = d+(1+r)^(2-1) (r s-d) = 16.0274

Summing over 13 months the total interest is $123.041

An expression for this is

sumi[k] = (d+d k r-d (1+r)^k-r s+r (1+r)^k s)/r

sumi[13] = £123.041

Edit

To use the calculator posted in the comment by MD-Tech the interest rate should be entered as a nominal rate compounded monthly, i.e.

12 r = 41.1703 % nominal APR compounded monthly

enter image description here

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