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Annual Percentage Rate (APR) & Loans
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I am having trouble understanding the APRs in relation to the total repayable on personal loans as quoted by banks.
For example, RBS is advertising:
Personal Loan of £5,000 over 36 months
36 monthly payments of £177.29
Total amount payable: £6,382.44.
Interest rate: 17.9% p.a. (fixed) [calculated daily]
Representative 17.9% APR.
Because the annual interest rate is the same as the APR we can conclude there are no other fees or charges.
My understanding is that at the end of the first year you will have paid off 12 x £177.29 (ie £2,127.48), so the total annual interest paid in the first year is:
[£2,127.48 – (£5000 / 3)] = £460.81.
But £460.81 equivalent to an annual rate of [£460.81 / (£5000 / 3)] x 100% = 27.6486%
This doesn’t make sense.
Alternative calculation:
17.9% of £5000 = £895 so the total payable would be £5895.00 but RBS quote the payable as £6,382.44. This doesn’t make sense.
Alternative calculation:
The daily interest rate is (17.9% / 365) = 0.049% (rounded) and is calculated daily.
Imagine the loan beginning on 01 Jan Year 1. At the end of the first month (ie after 31 days) the balance including the daily compound interest would be (£5000 x [(1 + 0.00049)^31]) = £5076.58. If payments are made first thing on the first of every month the balance at the close on 01 Jan Year 2 would be £3,668.33 (all months with their correct numbers of days taken into account).
Using the above payment schedule and daily compound interest rate formulae, the ending balance figure after 36 payments would still be £164.52 outstanding (meaning in fact monthly payments of £180.76 would be needed plus £0.22 extra).
Again this doesn’t make sense.
Can anyone please help to explain how, given only a loan amount, annual interest rate and loan period how banks arrive at the :
(1) monthly payment figure and
(2) APR
THANKS!
For example, RBS is advertising:
Personal Loan of £5,000 over 36 months
36 monthly payments of £177.29
Total amount payable: £6,382.44.
Interest rate: 17.9% p.a. (fixed) [calculated daily]
Representative 17.9% APR.
Because the annual interest rate is the same as the APR we can conclude there are no other fees or charges.
My understanding is that at the end of the first year you will have paid off 12 x £177.29 (ie £2,127.48), so the total annual interest paid in the first year is:
[£2,127.48 – (£5000 / 3)] = £460.81.
But £460.81 equivalent to an annual rate of [£460.81 / (£5000 / 3)] x 100% = 27.6486%
This doesn’t make sense.
Alternative calculation:
17.9% of £5000 = £895 so the total payable would be £5895.00 but RBS quote the payable as £6,382.44. This doesn’t make sense.
Alternative calculation:
The daily interest rate is (17.9% / 365) = 0.049% (rounded) and is calculated daily.
Imagine the loan beginning on 01 Jan Year 1. At the end of the first month (ie after 31 days) the balance including the daily compound interest would be (£5000 x [(1 + 0.00049)^31]) = £5076.58. If payments are made first thing on the first of every month the balance at the close on 01 Jan Year 2 would be £3,668.33 (all months with their correct numbers of days taken into account).
Using the above payment schedule and daily compound interest rate formulae, the ending balance figure after 36 payments would still be £164.52 outstanding (meaning in fact monthly payments of £180.76 would be needed plus £0.22 extra).
Again this doesn’t make sense.
Can anyone please help to explain how, given only a loan amount, annual interest rate and loan period how banks arrive at the :
(1) monthly payment figure and
(2) APR
THANKS!
---
100% debt-free!
100% debt-free!
0
Comments
-
Your last estimate was the closest.
The APR is the annual equivalent rate charged on an outstanding balance.
Getting the numbers exact down to the penny is difficult, but I can get within about £3 by doing the following:
APR 1.179
Monthly rate 1.179^(1/12) = 1.013817 (or ~1.38%)
Begin with £5000
1 month later, balance is 5000*1.179^(1/12), then you pay £177.39
So on the first payment, the balance is £4891.79.
1 month later, balance is 4891.79*1.179^(1/12), then you pay £177.39
On the second payment, the balance is £4702.09.
And so on - until the 36th payment, which I calculated as an overpayment of around £3.20. I suspect this can be accounted for by bank holidays, leap years, differences in calendar months etc.
It's just like a savings account in reverse. You have to calculate interest on the current balance.The daily interest rate is (17.9% / 365) = 0.049% (rounded) and is calculated daily
Why?
Because of compounding.
Try doing 1.00049*1.00049 for a year, i.e (1.00049^(365)).
You get ~1.196, which is 19.6% APR.
You need to take the 365th "root" (like a square root, but you want the number that multiplied by itself 365 times gives 1.179) of 1.179. That's the daily rate. Monthly rate is (roughly) the 12th root, etc.Said Aristippus, “If you would learn to be subservient to the king you would not have to live on lentils.”
Said Diogenes, “Learn to live on lentils and you will not have to be subservient to the king.”[FONT=Verdana, Arial, Helvetica][/FONT]0 -
Thanks, EdgEy.
Did you use £177.39 or £177.29 as the monthly payment, as the bank says the payment is £177.29?
Also, can you understand how they arrive £177.29 [ie total payable (£6,382.44)] if you were supplied only with the loan amount (£5000), loan period (36 payments) and fixed annual interest rate of 17.9%?
Thanks!---
100% debt-free!0 -
Thanks, EdgEy.
Did you use £177.39 or £177.29 as the monthly payment, as the bank says the payment is £177.29?
177.29 - my apologies, typo in the first post.Also, can you understand how they arrive £177.29 [ie total payable (£6,382.44)] if you were supplied only with the loan amount (£5000), loan period (36 payments) and fixed annual interest rate of 17.9%?
Thanks!
Analytically I am not sure (something to read up on when I get a bit of free time - thanks), but I can do it via trial and error in Excel very easily.
Set up a spreadsheet, add on monthly interest each month, and vary the monthly payment until Month 36 ends up at a zero balance. You've then got the monthly payment, which you can multiply by 36 to get the total payable.Said Aristippus, “If you would learn to be subservient to the king you would not have to live on lentils.”
Said Diogenes, “Learn to live on lentils and you will not have to be subservient to the king.”[FONT=Verdana, Arial, Helvetica][/FONT]0 -
Yes, there is a well known formula for calculating the monthly payment, which the banks use:
http://en.wikipedia.org/wiki/Amortization_calculator
In your case, P=5000, n=36, i=0.179/12 = 0.0149166667.
Putting this into the formula A = (P*i*(1+i)^n)/((1+i)^n -1) gives A = 5000*1.49166*(2.49166)^36/(((2.46)^36)-1) = 180.51, which is very close to your payment.
Are you sure the APR is 17.9% and not 16.9%? If it's the latter the monthly payment yielded by the formula is almost exactly equal to £177.29 (to within a few pennies)0 -
malcombffc.
If you use i = 1.179 ^(1/12) (12 root) in place of i=.179/12 you will get the ans £177.22.
I have noticed before that Lenders seem to adopt this approach in their calculations , whereas "Excel" with their Finacial functions use Yearly interest rate divided by 12.
This (12th root) method actually appears to benefit the customer.
Regards Mick0 -
This (12th root) method actually appears to benefit the customer.
As far as I am aware, it is legislative - the APR must be equal to the actual rate charged yearly.Said Aristippus, “If you would learn to be subservient to the king you would not have to live on lentils.”
Said Diogenes, “Learn to live on lentils and you will not have to be subservient to the king.”[FONT=Verdana, Arial, Helvetica][/FONT]0 -
I've not really sure what the correct answer is , but if you use the PMT function in Excel to return monthly payments on this loan, then formula looks like this:- =PMT(B1/12,36,-5000) and returns :-£180.5
The Rate over 12 is how you are advised to convert the yearly rate to Monthly.
If you use the same formula , but use the 12 root of 1.179 as:-
=PMT(((B1+1)^(1/12)-1),36,-5000) you get the result £177.22 which equates to the Original Data.
Now 0.179 /12 =0.0149166. If you now add 1 and compound it 12 times the result is 1.1944 = 19.44%
Where as 1.179^ (1/12) =1.013816 ^12 = 1.179 = 17.9%.
Which seems correct
Perhaps what excel returns is not the APR.
Which is the correct way ??
Mick0 -
the excel PMT function needs the simple monthly interest rate
this is NOT the APR/12 as has clearly been shown0
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