Compound Interest calculations

hollymcr

My Dad and I were trying to work this one out, but my (long forgotten) maths degree is failing me..

I understand compound interest, at least in theory. What Dad wants to work out is this: if he has sum S1 invested in an account at rate R (AER) for a period of time T1, then part way through that period he adds more money (S2) into the same account (where T2 is the time remaining when S2 is added), then at the end of the period he will have a total sum invested of S (which includes compound interest from both S1 and S2).

Given that he knows the final total S, the initial sums S1 and S2, and the times they were invested for (T1 & T2), can he work out the AER?

The best we've done is work out formulae for the totals then put in estimates for AER and tweak it until it gets the right amount, and that's answered the question he was initially trying to solve, but he's convinced that he should be able to write a general formula for it, and I'm fairly sure he won't be able to.

CLAPTON

I don't have a maths degree but one can't solve the problem out either interatively or by formula because knowing the AER is insufficient


try T1= 2days
and T2 =1 day

you can't even work out the daily interest rate!

MRLX69

I'm guessing you have something like:
S = S1 * (1+R)^T1 + S2 * (1+R)^T2

Then, you must know S1, S2, T1 and T2 to fiddle around with R and see if it matches S? - I'm trying to re-arrange it to work out R but it's proving quite difficult...

sloughflint

I was about to play around but there is no point until we know for definite how the interest is calculated ( daily?) and when the interest is paid within the time frame ( ie has an interest payment been made at any point in the time frames T1 and T2)

MRLX69

It's not possible to find a closed formula even if you have the time.
It has to be solved numerically, since I realised that there are only closed formulas for quadratic, cubic, quadruplet and quintic equations, it's been proven that no formula exists for powers of >5. Also, it works with integers only.

Toastie

S = S1 (1 + r) n

S = The future value of the amount invested
S1 = Original amount invested
r = rate of interest, expressed as a proportion(i.e. 4% = 0.04)
n = the number of year for the investment.(I.e 6months = 0.5)

This is how you work out compound interest, but i was never good at re-arranging algerbra

CLAPTON

sloughflint wrote: »

I was about to play around but there is no point until we know for definite how the interest is calculated ( daily?) and when the interest is paid within the time frame ( ie has an interest payment been made at any point in the time frames T1 and T2)

quite correct

meester

What you are calculating is:
S = s1 * r ^ t1 + s2 * r ^ t2

A simpler version of this is

a = b ^ c + b ^ d, where you want to find b in terms of a, c, and d.

Clearly there is no way to simplify this. An example of this is 10 ^ 2 + 10 ^ 5 = 100,100. What can you do with that?

You end up with results like (10^2)(1+10^3), i.e. b^c(1 + b^(d-c)) (given that d>c), which is no closer to a result.

Another way to prove it is not solvable to plug in a few example numbers. Take t1 = 5, t2 = 1,

E.g.,
1000 = r ^ 5 + r ^1

This is a quintic equation. There is no generalised solution for fifth order equations and higher

http://en.wikipedia.org/wiki/Abel-Ruffini_theorem

meester

sloughflint wrote: »

I was about to play around but there is no point until we know for definite how the interest is calculated ( daily?) and when the interest is paid within the time frame ( ie has an interest payment been made at any point in the time frames T1 and T2)

I think it is reasonable and appropriate to assume that interest is calculated daily (whether it is PAID daily, monthly, or yearly, makes no difference at all), or otherwise that t1 or t2 are integer numbers of years.

MRLX69

meester wrote: »

What you are calculating is:
S = s1 * r ^ t1 + s2 * r ^ t2

A simpler version of this is

a = b ^ c + b ^ d, where you want to find b in terms of a, c, and d.

Clearly there is no way to simplify this. An example of this is 10 ^ 2 + 10 ^ 5 = 100,100. What can you do with that?

You end up with results like (10^2)(1+10^3), i.e. b^c(1 + b^(d-c)) (given that d>c), which is no closer to a result.

Another way to prove it is not solvable to plug in a few example numbers. Take t1 = 5, t2 = 1,

E.g.,
1000 = r ^ 5 + r ^1

This is a quintic equation. There is no generalised solution for fifth order equations and higher

http://en.wikipedia.org/wiki/Abel-Ruffini_theorem

Quite exactly what I was saying. But I mistakenly thought that the quintic is solvable..... Thus you fire up matlab or excel and use numerical methods.

CLAPTON

meester wrote: »

I think it is reasonable and appropriate to assume that interest is calculated daily (whether it is PAID daily, monthly, or yearly, makes no difference at all), or otherwise that t1 or t2 are integer numbers of years.

the OP says he is a maths graduate, so no I don't think it reasonable to make such assumptions