How has this compound interest been calculated?

Question

I'm trying to recreate a credit account statement from mathematical principles in order to verify its veracity. This is because I suspect I've been charged an additional payment after supposedly settling, but would like to be more confident in this before querying it.

However, no matter what numbers I plug in, I can't reproduce even the first interest payments of the credit agreement:

The rate of interest on your agreement is 12.5% per annum. Interest is calculated on a day to day basis on the balance outstanding under your agreement and is added to your account on the same dates as your repayments are due.

Date       Description           Debits     Credits      Balance
--------------------------------------------------------------------
           Opening Balance                               £0.00
03-Sep-14  Initial Purchase      £2,556.23               £2,556.23
23-Oct-14  Repayment                        £84.70       £2,471.53
23-Oct-14  Interest              £25.66                  £2,497.19
23-Nov-14  Repayment                        £84.70       £2,412.49
23-Nov-14  Interest              £25.07                  £2,437.56
23-Dec-14  Repayment                        £84.70       £2,352.86
23-Dec-14  Interest              £23.68                  £2,376.54
23-Jan-15  Repayment                        £84.70       £2,291.84
23-Jan-15  Interest              £23.85                  £2,315.69
23-Feb-15  Repayment                        £84.70       £2,230.99
23-Feb-15  Interest              £23.24                  £2,254.23

[and so on]

Recreating each line in an Excel spreadsheet and attempting to reproduce the interest figures, I've tried:

• Interest = 0.125/12 * (balance before today's repayment)
• Interest = 0.125/12 * (balance after today's repayment)
• Balance after interest = (balance before today's repayment) * (1 + 0.125/365)^{(days in the month)}
• Balance after interest = (balance after today's repayment) * (1 + 0.125/365)^{(days in the month)}
• Balance after interest = (balance before today's repayment) * (1 + 0.125/365)^{(days since last interest)}
• Balance after interest = (balance after today's repayment) * (1 + 0.125/365)^{(days since last interest)}

Yet none of them give me a result of £25.66 for the first interest payment.

What am I missing? What formula should I be using for each "interest" line? (Once I have that, I can simply "drag it down" in Excel and prove whether the last payments were correct.)

Answer 1

With an extended first period the formula for a loan can be derived like so.

For illustration, only four periods are shown in this diagram. The OP's loan has 36 periods.

pv is the present value of the loan
c is the periodic repayment amount
r is the periodic interest rate
n is the number of periods
x is the fraction of a period by which the first period is extended

All the repayments are discounted to present value by dividing by the interest factor, and summed equalling the present value of the loan. For more detail see Calculating the Present Value of an Ordinary Annuity.

The extension is 20 days, so x = 20/31 since the period the extension falls in would be 23rd August to 23rd September which is 31 days.

pv = 2556.23
n = 36
r = (1 + 0.125)^(1/12) - 1

Using the formula

pv = (c (1 + r)^(-n - x) (-1 + (1 + r)^n))/r

∴ c = (pv r (1 + r)^(n + x))/(-1 + (1 + r)^n)

∴ c = 85.2418

By this calculation the regular repayment should be £85.24.

Answer

In fact the bank's calculation does not count the extension of the first period. It uses the standard loan formula.

pv = 2556.23
n = 36
r = (1 + 0.125)^(1/12) - 1

pv = (c - c (1 + r)^-n) / r

∴ c = pv r (1 + 1 / (-1 + (1 + r)^n))

∴ c = 84.7037

This matches the bank's repayment amount.

Detail

Addressing the OP's question regarding the first interest payment, the periodic (monthly) interest rate is

r = (1 + 12.5/100)^(1/12) - 1 = 0.00986358 = 0.986358 % per month

Checking by compounding: (1 + r)^12 - 1 = 12.5 % per annum

So the first interest payment should be £2,556.23 * r = £25.2136

Oddly this does not match the bank's interest payment, so they have used some other calculation for the interest. Nevertheless, they have charged the correct repayment amount for a 36 month loan and not even charged for the extra 20 days at the start of the loan.

Back-calculating the rate from the balance and interest charges shows some odd figures which presumably even out. Still, it does not appear you have lost out.

Balance      Interest    Rate           Annualised
£2,556.23    £25.66      0.01003822     12.73%
£2,497.19    £25.07      0.010039284    12.74%
£2,437.56    £23.68      0.009714633    12.30%
£2,376.54    £23.85      0.010035598    12.73%
£2,315.69    £23.24      0.010035886    12.73%

Answer 2

If you're carrying a balance interest begins accruing on new charges as they're made, not as of the next statement period.