# Math Help - Investment yield

1. ## Investment yield

Say I invested £1,000 and received £100 per month for twelve months (20%APY). If I reinvested those payments at 20%APY (and reinvested the returns on those too, ad infinitum) how much cash would I have after 12 months?

I've spent the past 3 hours trying to figure it out with a variety of home-made spreadsheets and calculations but they've totally lost me. Thanks in advance for any help/tips/pointers , I'm looking forward to seeing the formula behind it!

2. i = 0.20
j = i/12 = 0.0166666...
r = 1+j = 1.0166666...

1) Considering the payments and their reinvestment.

$100*r^{11} + 100*r^{10} + ... + 100*r^{0}$

You should be able to add those up, but you may wish to wait a moment.

2) Considering the base investment.

$(((1000*r - 100)*r - 100)*r - 100)$ ...etc. up to 12 uses of "*r". This simplifies nicely to...

$1000*r^{12} - 100*r^{11} - 100*r^{10} - ... - 100*r^{0}$

What happens if you add the two?

HOWEVER! And this is a VERY IMPORTANT "however", this plan doesn't work!!! Your 12th payment of 100.00 doesn't actually exist. There is only 3.04 in the fund so you have the unexpected 1,222.436, rather than the expected 1,219.391

3. Originally Posted by TKHunny
i = 0.20
j = i/12 = 0.0166666...
r = 1+j = 1.0166666...

1) Considering the payments and their reinvestment.

$100*r^{11} + 100*r^{10} + ... + 100*r^{0}$

You should be able to add those up, but you may wish to wait a moment.

2) Considering the base investment.

$(((1000*r - 100)*r - 100)*r - 100)$ ...etc. up to 12 uses of "*r". This simplifies nicely to...

$1000*r^{12} - 100*r^{11} - 100*r^{10} - ... - 100*r^{0}$

What happens if you add the two?

HOWEVER! And this is a VERY IMPORTANT "however", this plan doesn't work!!! Your 12th payment of 100.00 doesn't actually exist. There is only 3.04 in the fund so you have the unexpected 1,222.436, rather than the expected 1,219.391

Thanks so much for the quick answer.

I tried doing this but I don't know if it is mathematically sound:

m = monthly return
n = number of months
c = initial capital

with

m = 1.015309 (equates to 20% per annum compound)
n = 12
c = 1000

$((((c*0.2)+c)/n)*m)^n$

That gave me 1199993326981518177083440.920005522 which is way off

What did you mean when you said:

HOWEVER! And this is a VERY IMPORTANT "however", this plan doesn't work!!! Your 12th payment of 100.00 doesn't actually exist. There is only 3.04 in the fund so you have the unexpected 1,222.436, rather than the expected 1,219.391
?

Thanks

4. You are correct. I used 20% Nominal. Sorry about that.

1) The same things holds. Please look at my demonstration. Add those two monstrosities together.

2) As this constitutes a slightly lower interest rate, it will have an even bigger problem with the last payment. I didn't do the arithmetic, but I suspect some of hthe second to last payment is missing, too.