AER vs Gross interest

grumbler

isasmurf wrote:

...I've been looking for somewhere to back up my calculation, and struggled for a while, ...

Save this in your favorites:

British Bankers Association Code of Conduct for Advertisers.

Where an interest rate is calculated on a basis other than actual/365, the basis should be stated and the AER should nonetheless be calculated on an actual/365 basis.
Actual/365' means that interest is calculated by taking the balance multiplied by the annual interest rate divided by 100, multiplied by the number of calendar days that the balance is held divided by 365 ...

This is from my old argument with MJSW that I lost . I agree with you regarding ING. This is a rare occurance when the bank clearly states that

Interest on your savings account is accrued daily, compounded monthly and credited to your available balance at the end of each month.

This explains why the above formula works for 12 months, but not for 365 days: (1+0.0465/365)^365-1=4.7595%

tomstickland

Interesting stuff this. So the code of conduct actually sets out a recipe that will slightly benefit the customer, since the daily rate using this method is slightly better than it "should" be.

I remember asking the bank, years ago, why the APR and gross rate were different on a personal loan. They just looked at me and then said "well it's the computer that does it all".

isasmurf

I'm not getting all this business about the 365th power and root. This seems wrong to me. Taking that formula you are assuming interest is being compounded on a daily basis aren't you?

Assume you get 2p interest per day, taking it to the power of 365, means 2*2*2*2*2*2*.... (etc 365 times).

What you actually want is 2+2+2+2+2+2+... (etc 365 times)

To repeat what ING Direct say Interest is calculated daily, paid and compounded monthly. Unless it pays annually in which case it's calculated daily and paid annually. Interest does not get compounded until it is paid, taking to the power of 365 means that interest is compounded daily which is not true (actuallly I'm not even sure it's saying that). I think some of you are trying to make it more complicated then it actually is.

Grumbler's calculation in post 5 is sort of correct. It's fine to use that if you want to work out how much interest you would get over the course of year, or for a rough ball-park figure for a period of months. However, it doesn't work if you want to be accurate in your calculation of how much interest you get over a non-annual period because months unfortunately aren't equal in the number of days. I'm not sure that can be put in a simple formula.

tomstickland

To repeat what ING Direct say Interest is calculated daily, paid and compounded monthly. Unless it pays annually in which case it's calculated daily and paid annually. Interest does not get compounded until it is paid,

OK, but at what rate is is calculated daily <q mark>.

Most of this is new to me, so it's a bit of an exploration.
....this is my understanding so far.

You start off with £x in the account. Every day they calculate the daily interest and keep a reference of the total accumulated. When payment day comes they then move the accumulated amount over into the account and start applying interest to that too.

Hence the monthly interest is compounded.

I assume that the daily rate will be the 365th root of the annual rate.

Limes

tomstickland wrote:

OK, but at what rate is is calculated daily <q mark>.

When payment day comes they then move the accumulated amount over into the account and start applying interest to that too.

That is correct. The interest is calculated daily, credited to the account at the close of business on the last day of the month and then interest is earned on that new balance the following day. Well at least for ING anyway.

Robert_Sterling_3

When I worked for a lender I did actuallly take the three hundred and sixy fifth root of the annual muliplier and use that as the daily multiplier.
E.G. Annual Rate 5%
Annual Multiplier 1.05
Daily Multiplier 1.05^(1/365) = 1.000133681 approx
Daily interest rate 0.0133681%
In leap years I took the 366th root of the annual multiplier.

Quoting monthly rates is to say the least questionable as months can have 28, 29, 30 or 31 days.

I can quite believe that banks use rough and ready approximations which pre-date the introduction of computers when it was not easily possible to extract the 365th root quickly.

tomstickland

I've been doing some first princinple maths with the aid of Excel. It's thrown up some "quite interesting" results....

Taking the 365th root of the annual multiplier gives a result that is very, very close to just dividing the annual interest rate by 365. Even at a 10% annual rate (1.1 mutliplier) then the discrepancy between the two methods is only 0.0013%. ie: true value is 1.00026 and the approximate value is 1.00027. Actually, I think that calculators etc compute the 365th root using a Taylor's expansion and the first term of that is actually the 1/365 multiplier.

All of this works because of the approximation
(1+a)^b = 1 + ab which is true when a is small (and 0.05 is considered quite small).
So in this case:
(1+0.05)^(1/365) = 1 + 0.05/365 approximately

Similar logic applies to daily interest. If you had a daily decimal rate of "a" and the month was "b" days long and you were paid the interest every day then the amount you had at the end would be the starting amount multiplied by (1+a)^b
As a simplification you can just accrue interest at a rate of "a" each day and then pay it at the end. The total amount at the end will then be: (1+ab)

So the question is, what's the difference between (1+a)^b and (1+ab)?
For a 5% annual rate then the daily multiplier is 1.000134, ie: a=0.000134. For an average month with n=30 then those two methods agree within 0.0008% of each other. For a full year then the two methods agree within 0.115% of each other.

So, over a reasonably short time period, like a month, accrueing interest each day and then paying it at the end of the month gives near as dammit the same result as compounding it each day.

I imagine that this method was adpoted as it was very much easier to compute and administer.

lipidicman

Wow, this debate has warmed up! It is interesting to consider how the banks do the calculations. As far a I know some banks do compound daily. Anyhow, this shows the value of the AER!

In response to ISAsmurf the only time you can do x%/365 is if the rate is a gross rate paid daily (just as you can do x%/12 for a gross rate paid monthly. You then raise this result to the power of 12 to get the AER. This explains why the AER and the gross rate are different for monthly interest)

al_yrpal

Nationwide e-savings pays 4.75% AER gross with interest calculated daily, and paid annually. Ing Direct pays 4.75% AER, and pays interest monthly, but when you look closely Ing Direct is actually 4.65% AER gross pa. I thought there must be a catch !("no catches"). Pays to switch to Nationwide then, and, you don't have to wait three days without any interest whilst they transfer your money to a bank account every time. Nationwide transfer is instant. They both do fixed rate bonds at 4.60% - I wonder what the catch is there?

grumbler

al_yrpal wrote:

... They both do fixed rate bonds at 4.60% - I wonder what the catch is there?

See Not allowed to get money out of building society thread ...