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?
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.
total interest = (d+d k r-d (1+r)^k-r s+r (1+r)^k s)/r
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])
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
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
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.
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
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 = 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 = 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 = £123.041
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