Interest or APR Query

Question

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?

Answer 1

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.

Answer 2

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

∴ by induction

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