Interest calculated, accrued, paid - what they all mean and which is best

TheLittleThings

I've looked at various sources online and still can't get my head around this, so please explain simply if possible! Here are my questions:

1) If interest is 'calculated daily' and paid annually, does this mean that the interest calculated on the previous day is included in the calculation for that day, or is it as if the interest isn't yours at all until it is paid?

2) Is there are difference between 'calculated daily' and 'accrued daily'?

The above answers will hopefully answer this, but:
3) If the interest rate is equal, which is best:
a) accrued daily paid annually
b) calculated daily paid monthly

Thanks!

eskbanker

TheLittleThings said:

1) If interest is 'calculated daily' and paid annually, does this mean that the interest calculated on the previous day is included in the calculation for that day, or is it as if the interest isn't yours at all until it is paid?

The latter.

TheLittleThings said:

2) Is there are difference between 'calculated daily' and 'accrued daily'?

Not really.

TheLittleThings said:

3) If the interest rate is equal, which is best:
a) accrued daily paid annually
b) calculated daily paid monthly

The key measure is the AER, which is designed to allow such comparisons to be made on a like-for-like basis, so two accounts with the same AER will deliver the same amount of interest even if one is paid monthly and the other annually.

AmityNeon

TheLittleThings said:

1) If interest is 'calculated daily' and paid annually, does this mean that the interest calculated on the previous day is included in the calculation for that day, or is it as if the interest isn't yours at all until it is paid?

Interest is calculated on the account balance (closing balance for the day), so until interest is actually paid/received into the same account, compound interest isn't possible.

TheLittleThings said:

2) Is there are difference between 'calculated daily' and 'accrued daily'?

Not in practice. You could say interest must be calculated first before it can accrue.

TheLittleThings said:

The above answers will hopefully answer this, but:
3) If the interest rate is equal, which is best:
a) accrued daily paid annually
b) calculated daily paid monthly

In practice, what's best largely depends on your tax position; monthly interest in the current tax year versus annual interest in the next tax year. If tax is irrelevant, then the differences are minimal and practically insignificant.

Monthly interest only matches annual interest (plus or minus a few pennies due to rounding) when the interest rate remains the same and the account balance is untouched for an entire year. Annual interest yields more if the interest rate changes, or if the account receives additional deposits (no withdrawals) or is closed prematurely. Monthly interest yields more if there are only withdrawals.

But again, the differences (outside of tax) are too minimal to be worth scrutinising, especially for variable rate accounts with multiple deposits and withdrawals throughout the year.

eskbanker

AmityNeon said:
Monthly interest only matches annual interest (plus or minus a few pennies due to rounding) when the interest rate remains the same and the account balance is untouched for an entire year. Annual interest yields more if the interest rate changes, or if the account receives additional deposits (no withdrawals) or is closed prematurely. Monthly interest yields more if there are only withdrawals.

I don't understand the point you're making here - given that interest will be calculated daily, i.e. based on the balance and rate applicable to each day, the frequency with which it's paid shouldn't make any difference to the impact of rate changes, deposits or withdrawals?

Even if it was true that "Annual interest yields more [than monthly interest] if the interest rate changes", I don't see how it could be so universally, as it would surely depend on the direction of the change?

AmityNeon

eskbanker said:

AmityNeon said:

Monthly interest only matches annual interest (plus or minus a few pennies due to rounding) when the interest rate remains the same and the account balance is untouched for an entire year. Annual interest yields more if the interest rate changes, or if the account receives additional deposits (no withdrawals) or is closed prematurely. Monthly interest yields more if there are only withdrawals.

I don't understand the point you're making here - given that interest will be calculated daily, i.e. based on the balance and rate applicable to each day, the frequency with which it's paid shouldn't make any difference to the impact of rate changes, deposits or withdrawals?

The monthly gross rate is lower and therefore interest at a specific rate must compound for 12 months to match the annual gross rate. Unless the interest rate remains the same, and the account balance remains untouched (barring interest credit), there will be differences, albeit minimal.

eskbanker said:

Even if it was true that "Annual interest yields more [than monthly interest] if the interest rate changes", I don't see how it could be so universally, as it would surely depend on the direction of the change?

Whether the interest rate increases or decreases does not affect the outcome; if the interest rate changes, the lack of 12-month compounding makes it mathematically impossible for monthly interest to yield more than annual interest (assuming no withdrawals).

Increase:

Decrease:

The above comparison uses a large balance and change of interest, and does not incorporate any rounding, specifically to highlight the mathematical difference.

In practice, due to rounding to two decimal places (of both the gross monthly rate and interest pennies) as well as the variable number of calendar days, it could be possible for monthly interest to yield insignificantly more.

spider42

eskbanker said:

AmityNeon said:
Monthly interest only matches annual interest (plus or minus a few pennies due to rounding) when the interest rate remains the same and the account balance is untouched for an entire year. Annual interest yields more if the interest rate changes, or if the account receives additional deposits (no withdrawals) or is closed prematurely. Monthly interest yields more if there are only withdrawals.

I don't understand the point you're making here - given that interest will be calculated daily, i.e. based on the balance and rate applicable to each day, the frequency with which it's paid shouldn't make any difference to the impact of rate changes, deposits or withdrawals?

Even if it was true that "Annual interest yields more [than monthly interest] if the interest rate changes", I don't see how it could be so universally, as it would surely depend on the direction of the change?

The direction of change isn't relevant, since both addition (which is the basis of calculation of annual interest) and multiplication (which is the basis of calculation of compound interest) are commutative (in other words the order doesn't matter). So a+b = b+a, and a x b = b x a. Probably best explained with an example (in which I'm assuming a month is exactly a twelfth of a year, but in practice it would be done to the nearest day).

Let's suppose you've deposited £100k at 5% AER. The rate then drops to 4% AER after 6 months.

With monthly interest, at the end of the year you have £100,000 x 1.05^(6/12) x 1.04^(6/12) = £104,498.80.
With annual interest, you have £100,000 x (0.05 x 6/12 + 0.04 x 6/12) = £104,500.00

So annual interest is £1.20 higher than monthly. So there is difference, but it is pretty small.

Now lets suppose this had happened the other way around, i.e. the rate starts at 4% AER, and then rises to 5% AER. You swap 0.05 and 0.04 in the above calculations, but the answer for both the monthly and the annual figure will remain the same as before, i.e. £104,498.80 and £105,000.00, as the order makes no difference.

Or if you want to try with 5% increasing to 6%, then monthly results in a balance of £105,498.82 after a year, whereas annual would be £105,500, so annual is £1.18 higher.

The maths is involved is an application Bernoulli's inequality , which says that (1+x)^r <= 1+rx, if 0<=r<=1 and x>=-1.

x is the interest rate (expressed as a decimal, i.e. 5% = 0.05). r is the fraction of a year for which the interest is being paid (6/12 in my example). The bit on the left of the inequality is essentially how monthly interest is calculated. The balance is multiplied by a root of the interest factor each month. The bit on the right is essentially how annual interest is calculated. The interest is the balance multiplied by the interest rate for the fraction of a year for which that interest rate is paid. They are equal if r=1 (in other words the same interest rate applies for a full year). But if r is less than 1 (which will be the case if the interest rate changes mid-year, and so each rate is paid for less than a whole year), then the bit on the left (monthly) will be smaller than the bit on the right (annual).

AmityNeon is quite correct that any interest rate change, in either direction, will slightly favour annual interest over monthly.

Martico

TheLittleThings said:

I've looked at various sources online and still can't get my head around this, so please explain simply if possible! 

Is Bernoulli's inequality simple enough for you?

spider42

Martico said:

TheLittleThings said:

I've looked at various sources online and still can't get my head around this, so please explain simply if possible! 

Is Bernoulli's inequality simple enough for you?

LOL! Sorry - lost sight of the OP's original request! But AmityNeon's first post in this thread already provides an exceptionally good and concise summary of the main things to consider when choosing between monthly and annual interest (mainly tax), and when differences between the two will arise.

Too many people incorrectly believe that a monthly account with the same AER as an annual account will always pay the same amount of interest as the annual account. In general, that will only be the case if the balance is left untouched for the full year and the interest rate doesn't change. But the differences will generally be fairly
small.

Another opportunity with annual accounts is 'self-compounding'. In other words, if you close the account, all of the accrued interest is added on closure, so you get the interest paid earlier than usual. You can then reopen a similar account, and earn interest on the interest, so you get the higher annual interest rate, plus the benefits of compounding too.

E.g. if you leave £100k earning 5% for a year, you get interest of £5,000, so closing balance is £105,000. If you instead close the account after 6 months, you get £2,500 interest, so have a closing balance of £102,500. Put that in another account paying 5% for the next 6 months, and you've got £102,500 plus 2.5% = £105,062.50 at the end of the year. So the closure has generated an additional £62.50 of interest.

RG2015

spider42 said:

Martico said:

TheLittleThings said:

I've looked at various sources online and still can't get my head around this, so please explain simply if possible! 

Is Bernoulli's inequality simple enough for you?

LOL! Sorry - lost sight of the OP's original request! But AmityNeon's first post in this thread already provides an exceptionally good and concise summary of the main things to consider when choosing between monthly and annual interest (mainly tax), and when differences between the two will arise.

Too many people incorrectly believe that a monthly account with the same AER as an annual account will always pay the same amount of interest as the annual account. In general, that will only be the case if the balance is left untouched for the full year and the interest rate doesn't change. But the differences will generally be fairly
small.

Another opportunity with annual accounts is 'self-compounding'. In other words, if you close the account, all of the accrued interest is added on closure, so you get the interest paid earlier than usual. You can then reopen a similar account, and earn interest on the interest, so you get the higher annual interest rate, plus the benefits of compounding too.

E.g. if you leave £100k earning 5% for a year, you get interest of £5,000, so closing balance is £105,000. If you instead close the account after 6 months, you get £2,500 interest, so have a closing balance of £102,500. Put that in another account paying 5% for the next 6 months, and you've got £102,500 plus 2.5% = £105,062.50 at the end of the year. So the closure has generated an additional £62.50 of interest.

5% monthly equates to 5.12% AER.

£100,000 @ 5.12% = £105,120.

To my knowledge banks don’t pay the same rate for monthly as annual interest. Hence your £62.50 gain is fundamentally flawed.

spider42

RG2015 said:

spider42 said:

Martico said:

TheLittleThings said:

I've looked at various sources online and still can't get my head around this, so please explain simply if possible! 

Is Bernoulli's inequality simple enough for you?

LOL! Sorry - lost sight of the OP's original request! But AmityNeon's first post in this thread already provides an exceptionally good and concise summary of the main things to consider when choosing between monthly and annual interest (mainly tax), and when differences between the two will arise.

Too many people incorrectly believe that a monthly account with the same AER as an annual account will always pay the same amount of interest as the annual account. In general, that will only be the case if the balance is left untouched for the full year and the interest rate doesn't change. But the differences will generally be fairly
small.

Another opportunity with annual accounts is 'self-compounding'. In other words, if you close the account, all of the accrued interest is added on closure, so you get the interest paid earlier than usual. You can then reopen a similar account, and earn interest on the interest, so you get the higher annual interest rate, plus the benefits of compounding too.

E.g. if you leave £100k earning 5% for a year, you get interest of £5,000, so closing balance is £105,000. If you instead close the account after 6 months, you get £2,500 interest, so have a closing balance of £102,500. Put that in another account paying 5% for the next 6 months, and you've got £102,500 plus 2.5% = £105,062.50 at the end of the year. So the closure has generated an additional £62.50 of interest.

5% monthly equates to 5.12% AER.

£100,000 @ 5.12% = £105,120.

To my knowledge banks don’t pay the same rate for monthly as annual interest. Hence your £62.50 gain is fundamentally flawed

That example is comparing an account paying 5% annually and held for a full year to an account paying 5% annually which is closed after 6 months and then put into a new annually paying account. No monthly interest is involved, so I'm not sure how you come to the conclusion it is "fundamentally flawed".

I started it off with "Another opportunity with annual accounts is .... " so I'm not sure why you think I'm talking about monthly paying accounts here?

slinger2

If you're getting 5% gross/year paid monthly, your getting 5%/12=0.417% per month. After one month you'd have £100,417, after two months you'd have 100000*(1.00417)^2=£100,835 and after 12 months you'd have 100000*(1.00417)^12=£105,120. That's how 5% gross becomes 5.12% AER.