compound interest rates

zerocold

Hi, I am new here and not sure if this is in the right section; i hope it is.

I am a bit confused about compound interest rates and apy.
For example:

an account compounds daily and is credited monthly.
I begin with 8,400 $ and the interest rate is 1.35
1.35% of 8400 is about 113 (correct?)

Therefore, if interest is compounded daily, I should be getting about 113 a day? now obviously this is very incorrect. Can someone please explain to me how compound interest is calculated when it is compounded daily?

evenasus

You are actually posting on a UK website, although interest would be calculated in the same way, whether it be in $ or £.
The 1.35% interest rate is per year I take it?
If so, then on $8,400 lump sum the first days interest would be 31 cents.
On day two the interest would be calculated on the $8,400 + the first days interest and so on.
Hope this makes sense.

Reaper

Yes the interest earned over the year will be 113.4, but they won't give you that every day. Instead it will be divided by 365 and further reduced to compensate for compounding.

There is a calculator here which will tell you the daily rate and APY:
http://www.pine-grove.com/online-calculators/interest-calculator.htm

Incidentally this is a UK based site but I believe the APY you refer to is similar to our AER. It is the equivilent of an annual interest rate (regardless of the term or interest payment frequency) which allows you to compare different savings accounts.

zerocold

Oh. I didnt notice that. but yes, i believe apy (annual percent yield) is the same as aer.

Reaper, can you please elaborate on how it is reduced to compensate for compounding?

So basically, the apy or aer is the the amount (based on your investment) that you will earn over a year? So is the frequency of compounding factored into the apy/aer rate or into the actual interest rate? Or something else, maybe?

I sort of understand how this works, but i would prefer to completely understand.

Thank you all for your help!

Reaper

The interest you earn each day is not compouded to start with because it is not added to your account until the interest period is up (* see below). So if the account added interest annually the calculation would be as simple as dividing it by 365 (let's ignore leap years for now!):

$8,400 x 1.35% / 365 = $0.31 per day

However yours adds interest monthly. After the first interest payment has been made it will also start to earn interest. Therefore you will see that if a bank offers you a choice of monthly or yearly interest the monthly rate will be lower to compensate for this.

I don't know of an easy formula to give you to work it out (maybe somebody else will come up with one). At this point I would reach for an online calculator or create a spreadsheet and play with the numbers until I got the correct end figure.

EDIT: * As I recall not every bank works it out the same way. I seem to remember from past threads that one bank in the UK called Egg do actually compound the interest on a daily basis (i.e. do not wait for the interest to be credited to the account) and therefore apply a lower daily rate to compensate.

zerocold

Like that Egg bank you were speaking of, this bank does compound daily (as many banks in the US seem to) Thus, as i understand, you gain interest on the interest of the previous day, the day immediately following. However, if you make a withdrawal before the crediting of said interest (which occurs at the end of the month) then you lose that interest/the interest compounding resets. (I think)

Anyway, thank you again for your help. It is much appreciated.

Reaper

That sounds very odd though I can't comment as I don't know American banks.

Egg was bought by American bank Citibank so perhaps they imported their method of calculating interest.

It leaves the saver fractionally worse off when you make withdrawals as the rate is based on compounding you will not then receive. However we are only talking about a very small amount.

xrjtg

If you earn 5% AER, then after a year your balance has increased by a factor of 1.05, after two years it has increased by a factor of 1.05^2, after three years it has increased by a factor of 1.05^3, ..., and after a day it should have increased by a factor of 1.05^(1/365). This is very close to 1+(0.05 / 365); in fact, (1+(0.05 / 365))^365 = 1.05126... .

I've tried this kind of calculation before though and it didn't seem to agree as well with the actual figures as some simpler methods (like dividing the annual rate by 365). I'd be interested to hear if anybody knows exactly how even one bank handles interest compounded daily.

Reaper

As I say although most banks calculate interest daily they do not compound daily, which will be why you find your formula does not give you the right result. Egg apparently are an exception and you can find a discussion about how they calculate it on this thread, particularly from post 54 to the end.

Milarky

xrjtg wrote: »

If you earn 5% AER, then after a year your balance has increased by a factor of 1.05, after two years it has increased by a factor of 1.05^2, after three years it has increased by a factor of 1.05^3, ..., and after a day it should have increased by a factor of 1.05^(1/365). This is very close to 1+(0.05 / 365); in fact, (1+(0.05 / 365))^365 = 1.05126... .

I've tried this kind of calculation before though and it didn't seem to agree as well with the actual figures as some simpler methods (like dividing the annual rate by 365). I'd be interested to hear if anybody knows exactly how even one bank handles interest compounded daily.

The way I assume Egg calculates it is to apply a 'days-difference' formula to the balance between each transaction affecting the balance

For example, a hypothetical statement lmay look something like...

1 Jan 100 [20 days]
21 Jan 110 [28 d]
18 Feb 140 [13 d]
3 Mar 176
5 May 211
6 Jul 108
11 Aug 138
5 Nov 188
7 Dec 148

We could work out the 'days-difference' for each item and raise our daily rate to this power

5% = (1.05)^(1/365) = 1.00013681 [=0.01336% daily]

1 Jan 100 [20 days] = 100 x [1.05]^(20/265) minus balance [-100]
21 Jan 110 [28 d] = 110 x [1.05]^(28/265) minus balance [-110]
18 Feb 140 [13 d] = 140 x [1.05]^(13/265) minus balance [-140]

Note that if we just work out the exponents as 'decimals' first it makes the calculation more accurate thereafter because excessively small intermediate quantities (e.g '1.00013681^20') are avoided

For instance, this gives a 'direct' exponentiation something like

1 Jan 100 [20 days] = 100 x [1.05]^(0.05479452)
21 Jan 110 [28 d] = 110 x [1.05]^(0.076712328)
18 Feb 140 [13 d] = 140 x [1.05]^(0.035616438)

Egg may even store all 365 (or 366) 'partial year' exponents thoughout the year and use a look up method to apply them directly. [This would also allow for changes in interest rates during the period as a whole]

Eg, suppose the rate changes on 1 Feb from 5% to 4%..

1 Jan 100 [20 days] = 100 x [1.05]^(0.05479452)
21 Jan 110 [10 d] = 110 x [1.05]^(0.076712328)
1 Feb 110 [18 d] = 110 x [1.04]^(0.076712328)
18 Feb 140 [13 d] = 140 x [1.04]^(0.035616438)

Egg could have just stored the equivalent of

5% for each of 1,2,3...29,30,31 days
followed by
4% for each of 1,2,3... [etc] days

The program would then just has to identify the dates [throughout the year] on which a balance [and/or rate] changed, look up the corresponding 'multiplier' for each entry and then do a 'sum product' very similar to a non-compounded interest calculation.

It seems counterintuitive why Egg bothers, but it does save them some money when [eg] mid year deposits earn the 'square-root' of the nominal rate rather than half of it as people might expect to receive.

The limiting case (eg compounding in 'zero time') is actually the base of the natural logarithm, e [2.718...] raised to the time-fraction the balance or rate remain unchanged [see here]

xrjtg

Thanks Milarky. Come the mathematical revolution all interest will be compounded continuously.

Milarky wrote: »

The limiting case (eg compounding in 'zero time') is actually the base of the natural logarithm, e [2.718...] raised to the time-fraction the balance or rate remain unchanged [see here]

Forgive me for adding to the record that you're assuming that the continuous rate is 1.

If anyone's not familiar with this, but does have some latent interest in maths (and the interest in finance implied by reading this forum) you might enjoy looking it up.