I'm stuck with a little calculation and hope someone will be able to help me. I need to calculate how much money I will have if I invest a regular amount over a period with a certain rate of return BUT the regular contribution increases over the period.

I know that if I invest £100 a year over 10 years and receive interest of 5% per annum the calculation is

£100.00 * ((1.05^10)-1) * 1/0.05 = £1257.79

But I can't work out what to do if that £100.00 increases each year, say by 2.5% so the first years contribution is £100.00 the next is £102.50 then £105.06 etc.

Any mathemeticians able to help me please?

I'd do it in Excel (by the looks of the formulae, you already are).

First row as year 1 – go Beginning funds | End funds

Beginning funds would be your £100, end funds would be =A1 * 1.05

Second row would begin with =B1 + £100, then second cell (B2) would be =A2*1.05

Then just select the second row, use the little marker in the bottom right of the selection to drag and fill for as many years as you want. You may have to use some ! marks to force some references to be static.

Interest is calculated at set times ie daily, weekly or whatever – it should tell you this. So each day, you get given 0.00324% or whatever of what your balance is.

There's no formula for it – you need to use Excel (or similar) to list the figures for each day/week/month and do the calculation based on the previous row.

My above formula is based on annual interest rates and you adding an extra £100 a year (hence the + £100 for cell A2).

Don't the headline AER rates take into account the way the monthly / daily rates compound these days? I thought there had been a big hoohaa about it…

£1395.24p for your example 😉

formula for a growing annuity is here:
http://en.wikipedia.org/wiki/Time_value_of_money#Future_value_of_a_growing_annuity

I can do it in excel but it would be better to use a formula for what I want to do. The calculation is an estimate for the future so I can ignore interest rate changes as I am assuming they remain constant.

Maybe my explanation wasn't the best….imagine I am saving for a single speed niche wagon and can only afford to save x percent of my salary each month but I am assuming my salary will go up by 2.5 percent each year and my savings will return 5 percent interest each year. I need the formula to calculate how much I will have after a specific period of saving.

I can remember vaguely what sigma notation, differentiation and integration are so I know this can be put into a nice little formula but not how to do it. When I did A levels they were proper exams etc, etc.

Stoner may have it but it's going to take me a few mins to check.

Thanks all, I'll be back when it turns out I'm still lost 🙂

You can trust me. Im clever 😉

The PV of a growing annuity is a nicer forumla IMO.

100*(1.025^(10+1))-1.05^(10+1))/(1.025-1.05) is probably what you're after.

Putting it into sigma notation won't help you work it out. You'll still need to use Excel. I don't think there is a way of doing it with one single calcualtion.

Even loan companies use tables for this.

Bah, I didn't know it was a given problem.
Wouldn't have bothered deriving the formula if I'd known!

molgrips – you dont need iterations to solve. It is a finite sum to term – hence the single forumla Ive linked to above.

and finance companies use tables because they're not very good at the maths! 😉

Stoner – Member
£1395.24p for your example

formula for a growing annuity is here:
http://en.wikipedia.org/wiki/Time_value_of_money#Future_value_of_a_growing_annuity

I agree and that is what I learned when I was doing my actuarial exams.

To be fair, if I was doing it as a one off thing, I'd take the lazy approach and just dump it into a spreadsheet.

Deriving the formula isn't too bad though, if you know how the derivation of a sum of a geometric series works.

Oh yes, I mis-read that link 😳

Thanks all, especially Stoner who is indeed clever. Future value of a growing annuity is exactly what I need. Looks like Aidy came to the same result the hard way so must also be clever.

Stoner is right about finance companies too 🙂

Deriving the formula isn't too bad though, if you know how the derivation of a sum of a geometric series works.

show your workings then 😉

Using induction?

imagine I am saving for a single speed niche wagon and can only afford to save x percent of my salary each month

Or just imagine using a 0% credit card 😉

Yeap, well done stoner. Exactly how this should be calculated.

If you knew how to use those tables though you might have been able to transfer the skills to other areas…. say, ordering windows?

Ahh, but then Id end up with the right number of windows but that they would be getting progressively too big to fit 😉

😆

show your workings then

Using induction?

No, just knowing how to poke summation formulae into line.

Okay, so at the end of the first year, you have:
100 * 1.05 + 100 * 1.025 – your initial investment + interest + your second investment

At the end of the second year:
(100 * 1.05 + 100 * 1.025) * 1.05 + 100 * 1.025^2

Third:
((100 * 1.05 + 100 * 1.025) * 1.05 + 100 * 1.025^2) * 1.05 + 100 * 1.025^3.

Or, to simplify terms:

100 * (1.05^3 + 1.05^2 * 1.025 + 1.05 * 1.025^2 + 1.025^3)

This looks a bit like a geometric progression, but with terms of the type Ar^k s^n-k.

So take S to be the sum:

S = x^3 + x^2 * y + x * y^2 + y^3. (this logically can be expanded to n terms).

Examine, and multiply by x/y.

Sx/y = x^4/y + x^3 + x^2 * y + x * y^2.

Subtract one from the other:

S-Sx/y = -x^4/y + y^3

Multiply by y:

Sy-Sx = y^4 – x^4
S(y-x) = y^4 – x^4.

S = (y^4-x^4)/(y-x).

And for n terms:

S = (y^(n+1) – x^(n+1))/(y-x).

Feed back into original formula to get the answer.

but with terms of the type Ar^k s^n-k.

So take S to be the sum:

this is the step that always used to get me – an inability to recognise what progression form something looks like.

Nice algebra there though.