Time given future value of reoccurring deposit
I've recently come across the follow formula for calculating future value with reoccurring deposits formula. However, I have been unable to solve for t. I essentially want to find how long it will take to earn a million dollars (the future value). Any ideas on this formula or how to go about finding it?
p = initial value = 2500
n = compounding periods per year = 12
r = nominal interest rate, compounded n times per year = 4% = 0.04
i = periodic interest rate = r/n = 0.04/12 = 0.00333333
y = number of years = 5
t = number of compounding periods = n*y = 12*5 = 60
d = periodic deposit = 100
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Here T is the total ( million in your case). I used natural log but of course you can use log with any base.
The answer to this question on Math SE also provides a derivation, using different variable names and missing the i+1 factor at the end of your formula (basically meaning the first periodic deposit is delayed by one period).
Note that i + 1 = 301/300.
Let X denote (i + 1).
Then you wish to solve :
1,000,000 = 2500X^t + 100X(X^t - 1) / (1/300).
Simplifying and taking logarithms of each side gives :
ln(32.41) = (t + 1)(ln(301/300))
This give a value of about t = 1040.
This agrees with the other answer (posted by BrenBarn).
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