# Compounding interest of principal P, where a compounding withdrawal amount W get withdrawn from P before each compounding of  P.

In accordance with the title I need a single formula that expresses the balance of a principal after "t" amount of time (in years), where

• V(t) is the balance after "t" amount of time,
• P is the balance before any withdrawals (subtractions) or compoundings,
• W is the amount to be withdrawn (subtracted) from P at each compounding,
• r is the decimal percentage the principal growths with (each year),
• i is the decimal percentage the withdrawal amount growths with (each year),
• n is the number of times r or i is used to compound W or P (at each period t).
• If compounding takes place each month n would be = 12. Hence the decimal percentage growth would be r/12 and i/12 for principal and withdrawal amount respectively.

Here is what I can make happen in excel, that I want to be able to do in a single formula as well. The following is an example of two following compoundings (from the beginning of a period):
1. W amount is withdrawn from principal P --> P-W,
2. What is left of the principal P is compounded and gives me a new principal value --> (P-W)*(1+r/n),
3. W amount is compounded and thus giving me a new amount to withdraw next month --> (P-W)*(1+r/n), W*(1+i/n)
NEXT MONTH
1. W amount is withdrawn from the latest principal --> (P-W)*(1+r/n)-W*(1+i/n),
2. What is left of latest principal P is compounded and gives me a new principal value --> ((P-W)*(1+r/n)-W*(1+i/n))*(1+r/n),
3. Newest W amount is compounded and thus giving me a new amount to withdraw next month --> ((P-W)*(1+r/n)-W*(1+i/n))*(1+r/n), W*(1+i/n)*(1+i/n)
And so, it can go on expanding until t amount of time has passed.

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• Thanks you for your answer. I tested it up against an online calculator through desmos.com/calculator with success. Any idea how to use the formula in excel? I currently use the following for the W*SIGMA part. =W*SUMPRODUCT((1+r/n)^(m-FACT(-1+ROW(INDIRECT(0+1&":"&m-1+1)))))*SUMPRODUCT((1+i/n)^(FACT(-1+ROW(INDIRECT(0+1&":"&m-1+1))))) But results are slightly off.

• I am confident about the formula we have derived. Make sure the variables in our formula are the same as the online calculator. In particular in our formula W is the amount to be withdrawn after one month. Some calculators may use W_0 (an initial amount set at the initial time). Then W=(1+i/n)W_0.

• For the sigma notation in the formula, you may write all the numbers appearing inside the sigma notation in a column and then add them up. Please watch: https://www.youtube.com/watch?v=eaei7DN6Z9M

• The formula fits with the online calculator where I compared results - Well done NicolasP! I am NOT complaining.

• As for the YouTube video. That is a good starting point which has inspired the current Excel calculation mentioned in my first comment. My results in Excel are slightly off. It needs tweaking. On desmos.com/calculator I have used our formula and was able to match the result on a few compounding interest calculator sites. Once again. Thank you for your help! :-)

• Great! I am glad it worked!

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