Regular invetment future value calculations

There is only one correct way of working it out. Lots of idiots can write "calculators" but the result just depends on what formula they use (it may be the wrong one) and on what assumptions they make (they may be the wrong ones).

Here's where your problem starts. Not enough information given.

You're obviously assuming the investment compounds monthly, and so you need to be clear what the *monthly* interest rate is. Is it

Finally, you need to be clear when your time frame begins and ends. The most natural way for me (and you to, it seems), is that the time frame begins on the day you make your first payment, and that the Future Value you're interested in is to be measured exactly 120 months later, that is to say one month after the last (120th) payment has been made. Some people, though, look at the value immediately after the last payment.

The derivation of this is straightforward. You confuse yourself by referring to the converted value by the letter r again, which suggests it's a rate, whereas in fact it's a factor. To get the rate you need to subtract 1. So what was done is first to convert the [presumed annual equivalent] rate r to an annual equivalent factor f=r+1, then taking the twelfth root to give the monthly factor f'=f^(1/12), and then stopping there and using the factor f' in the same formula as you are using below (formula C).

If in your formula C you replace the rate r with f-1, then whereever you use r+1 you then use f instead, and where you have r on its own, you put f-1, then you get:

Formula C: FV = P(1+r) [(1+r)^n-1] / r becomes: FV = P f [f^n-1] / (f-1)

You can see that the 'f's in this formula correspond exactly to the 'r's in your formula A.

This one gives the value immediately after the 120th payment, i.e.

This one gives the value one month after the 120th payment, i.e.

Thanks for your help with this, ithis is making more sense now with your explanation. Of the Uk calculators I have looked at, 8 out of 12 are using the formula A/C and 4 are using the B version. None of these calculators specify which interest rate they are using, AER or the nominal rate - this is an obvious ommission.

The remaining maths issue, is why in the AER factoring do you take the twelve root and not just divide by 12. Whats the logic behind this? I have found derivations from first priciples of the other maths for methods B and C, but not for A.

Also, for what reas> snipped-for-privacy@bt> > There appears to be three methods of calculating the future value of a

The logic is that it depends on whether the annual interest rate figure you're given is nominal or annual equivalent. To obtain the monthly version, in one case you divide the rate by 12 in the other you take the 12th root of the factor.

If the monthly interest rate is 0.5%, then the nominal interest rate is 6%. But the monthly interest factor is 1.005 and because compounding corresponds to exponentiation, the annual interest factor would be 1.005^12 which is 1.06168, i.e. the AER is 6.168%.

To go backwards you have to reverse the exponentiation, and that is taking the 12th root: Start with AER 6.168%, move to AEF 1.06168, take 12th root 1.06168^(1/12) to get monthly interest factor 1.005 and so that makes the monthly rate 0.5%.

The derivation from first principles is the same for A as for C, once the monthly factor or rate has been determined. The only difference is how that rate is determined. You either assume the annual rate given is nominal or you assume it is AER. I don't know what practice prevails in America, but in the UK it has been traditional always to publish nominal rates, though in recent times AERs have been listed alongside them, and occasionally AERs now appear in the marketing blurb so prominently that the nominal rate is marginalised or hidden. Of course AERs are generally bigger than nominals, so they help to draw in the confused punters (in the case of savings, but of course APRs in the case of loans put them off).

I always use f instead of r (also in loan repayment calcs) because it looks simpler. I find it messy to fiddle around with brackets in the likes of "(1+r)^n" when "f^n" looks more aesthetically pleasing.

And what about the standard loan repayment formula? An often quoted one is: P = A r (1+r)^n / ((1+r)^n - 1) This simplifies to: P = A r / (1 - (1+r)^-n) But I still prefer: P = A (f-1) / (1 - f^-n)

Using factors instead of rates also avoids the sloppy practice of expressing the rate as a "percent number" which means you have to divide by 100 all over the shop.