# help with how a loan repayment is calculated

can anyone shed some light on how my finance company calculated my loan installment to be 4 payments of £163.08 and a final payment of £163.13?

i have used amortization simple interest calculations and cannot get 5 repayments of £163. My understanding is that this only works on the basis that the repayments are for a full month and not part months, like the first month below which is 22 days.

The loan amount is £500 Interest rate is a fixed flat rate of 0.7% per day. Term of loan is 144 days

    Date        Capital Interest    Duration    Total
1   31-07-2015  £86.08  £77.00      22 days     £163.08
2   28-08-2015  £81.96  £81.12      28 days     £163.08
3   30-09-2015  £86.40  £76.68      33 days     £163.08
4   30-10-2015  £111.52 £51.56      30 days     £163.08
5   30-11-2015  £134.04 £29.08      31 days     £163.12


Total: £815.44

I would be grateful if anyone could help as I have spent so much time trying to work it out but with no success.

• So what part is confusing? You've calculated how much of each payment goes to interest and how much to principle. Add up the totals of your principle column and it comes to exactly 500 pounds. I just created a spreadsheet and I get very similar numbers: with no attempt at rounding, I'm off by 3 1/2 pence. Or are you saying that you didn't create that table, that the bank gave it to you, and you don't understand it? If that's the case, simple enough: the interest column is equal to the number of days in the period times 0.7% times the outstanding principle. – Jay Jul 22 '15 at 17:56
• Hi, i'm hoping to find the formula on how the finance company get the monthly repayment of £163. The only information that I have is that the loan amount is for £500, the interest rate (simple interest) is fixed at 0.7% per day and the term of the loan is 144 days. i'm trying to work backwards by finding the monthly repayment and then I can calculate what element is capital and what is interest. – Vicki Jul 22 '15 at 20:54

It appears the interest is not compounded daily. Each period of interest has the loan amount calculated on the "capital" remaining on the start of period, for each day in the period. The Excel finance functions don't handle irregular periods that well, but I can reconstruct the interest calculations:

• 500*.007 * 22 days = 77.00
• (500-86.08)*.007 * 28 days = 81.128
• (500-86.08-81.96)*.007 * 33 days = 76.682
• (500-86.08-81.96-86.40)*.007 * 30 days = 51.567
• (500-86.08-81.96-86.40-111.52)*.007 * 31 days = 29.086

In this case, it looks like the interest is simply the nominal daily interest rate times number of days in the period. From that you can use a spreadsheet to calculate the total payment by trial and error. With the different number of days in each period, any formula would be very complicated.

In the more usual case where the interest charge for each period is the same, the formula is:

m=P*r^n*(r-1)/(r^n-1)

where

* is multiplication

^ is exponentiation

/ is division

(Sorry, don't know if there's a way to show formulas cleanly on here)

P=original principle

r=growth factor per payment period, i.e. interest rate + 100% divided by 100, e.g. 1% -> 1.01

n=number of payments

Note the growth factor above is per period, so if you have monthly payments, it's the rate per month.

The last payment may be different because of rounding errors, unequal number of days per period, or other technicalities.

Using that formula here won't give the right answer because of the unequal periods, but it should be close. Let's see:

r=0.7% times an average of 28.8 days per period gives 20.16% + 1 = 1.2016.

n=5

P=500

m=500*1.2016^5*(1.2016-1)/(1.2016^5-1)

=167.78

Further off than I expected, but ballpark.