# Interest or APR Query

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?

• It should be zero. In other words don't do it. Jan 26, 2017 at 14:17
• "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. Jan 27, 2017 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 Jan 27, 2017 at 3:40

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. Jan 26, 2017 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. Jan 26, 2017 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` Jan 28, 2017 at 8:16

``````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])
``````

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
``````

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
``````