AER vs Gross interest

tomstickland

What are the risks with Icici?

isasmurf

tomstickland wrote:

So, my understanding of gross annual rate now is that it's the money that they pay as interest, but not including any gains from compounding.

That is correct sire.

Gross interest is your contractual rate of interest - this is the rate on which your interest is calculated.

Annual Equivalent Rate is a notional rate - this is the rate to demonstrate how much interest you would receive over 12 months if you had the interest paid into your account and made no withdrawals during that period. It takes account of any compounding effect if the interest is paid more frequently then annually.

tomstickland

Good, I'm getting somewhere now.
It all makes good sense now.

Since gross ignores any compounding, then the rate for more frequent payments is indeed the annual rate divided by the frequency of payment. So daily is 1/365th of the annual and monthly is effectively 1/12th of the annual.

Since AER is affected by compounding, then the AER is:
AER% = 100 * ( (1 + (GROSS%/100) / FREQ)^FREQ - 1)

AER = (1 + gross_pa/freq)^freq - 1

Identical, apart from the tedious 100 factors I've used.

where FREQ is frequency of payment per year. ie: 12 for montly and 365 if you were paid daily.

Gross %, AER % for F=12, AER% for F=365
0,0,0
1,1.01,1.01
2,2.02,2.02
3,3.04,3.05
4,4.07,4.08
5,5.12,5.13
6,6.16,6.18
7,7.22,7.25

tomstickland

For my next stage, I will be able to use a common mathematical expansion, the Taylors series, to replace the x^12 or x^365 term in the conversion equation. This will then allow the extra interest rate due to compounding to be understood better. It's interesting to see that daily compounding doesn't actually increase the AER that much over montly compounding. For fun, if you compound twice per day (F=730) then the rate hardly improves any more.

Next question....how is tax calculated? Do they tax the interest payments. So if you were going to be paid £x interest at the end of a month, you just receive 0.8 of that if you are charged 20% tax?

tomstickland

OK, it's 1:50 am and I really should get a life....but via http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node46.html I've remembered enough to tell you that.

(1+x)^n = 1 + nx + n(n-1)/2x^2 + n(n-1)(n-2)/6*x^3 ... and then a load of ever small terms because they contain x^4 and x^5 etc
Express this in a more useful way to the current thoughts:

(1 + a/f)^f = 1 + f*a/f + f(f-1)/2*a^2/f^2 .....
this is
(1 + a/f)^f = 1 + a + a^2/2*(f-1)/f ....

I've not bothered with any more terms on the right hand side because if a is the annual rate (eg:0.05 for 5% ) then this number cubed is very much smaller than this squared, so the terms become ever smaller and smaller.

The right hand side is an approximation to the left hand side. The left hand side is the true annual return. The 1+a on the right hand side is the gross rate. So the extra term to the right of this is the extra you receive because of compounding with frequency f per year.
Take a look at the first extra term:

Extra bit due to compounding more frequently approximately equals:
a^2/2*(f-1)/f

So, it goes up with annual rate squared divided by two and then that's multiplied by a factor of (f-1)/f. Bear in mind that the annual rate number is smaller than 1, so squaring it makes it smaller.
This means that if the gross rate is doubled then the extra due to compounding will increase by a factor of 4.
An interesting thing is that (f-1)/f becomes ever closer to 1 as f gets larger and larger. For monthly rate it's 11/12ths, but for daily payment it becomes 364/365ths. So, monthly compounding actually gives you 11/12ths of what compounding over the smallest time interval possible to use would be. Whereas if you were paid twice per year then you'd get half of that.

Just check this.
I'm saying that the extra bit of interest rate that compounding gives you is:

a^2/2*(f-1)/f

For an annual gross rate of 5% a is 0.05. The a^2/2 term is then 0.00125, so you have the potential for an extra 0.125% per year. Paying monthly gives you 11/12ths of this, so you get 0.001146 extra which is an extra 0.1146%. Applying the original proper equation says that you would get an extra 0.116%, so this approximation (by ignoring higher terms) is wrong by 0.0014 of the % rate.

In summary...
the extra money you receive due to compounding for an x% annual gross rate for daily compounding is
x% of the x% then halved.
For montly compounding you multiply this by 11/12ths (0.92).

tomstickland

I think I'll go to bed now.

lipidicman

tomstickland wrote:

Forgetting about compounding, they pay 5.5% per year, but they pay it daily. So, without compounding, over a year they would pay you 5.5% of the start sum. So, each day they pay 1/365th of this.
Hence, each day they pay 0.055/365 which is 0.000147 of the amount paid in at the start.
If they compound it daily, then the daily multiplier is 1.000147 and they compound this 365 times. Hence the annual multiplier is 1.000147^365 which is 1.054953.
ie: AER is 5.5%.

but 0.055/365=0.000150685, so there is a problem here. Raise 1.000150685 to the power of 365 and you will get a bigger number. The daily rate will be smaller.

My stance is still that you can divide a gross rate by any perod smaller than the payment frequency. ie with a gross rate paid monthly you cannot multiply up to get a 6 month or a yearly rate (here you must raise to a power), but you can divide to get a weekly rate or daily rate (since they appear to compound monthly). This certainly applies to ING but as Isasmurf has highlighted, the actual method a bank uses to work out the interest does vary and does matter. Generally using the AER should put most minds at rest

tomstickland

Yes, what I wrote there is out of date with my thinking, plus contains incorrect maths! (if the maths is corrected then I end up with AER = 5.65%). I also agree tha dividing down from gross is fine, but to work up you need to raise to powers because the payments are compounded.

My understanding has improved during the life of this thread. I'd now say that:

Since gross rate ignores compounding then you can divide down the gross rate to give the daily or monthly rate. Compounding only applies with the payment frequency, in which case you can apply

AER = (1 + Gross/freq)^freq - 1

lipidicman

With the above we may not understand how each bank does its calculations, but we can do basic calculations. Also (in general) the AER allows you to compare rates so everyone should be quite happy. I am inspired to do some research on how each of my accounts is calculated though......

tomstickland

I never thought I'd find interest payments interesting until this last week or so.