a formula for calculating value of interest bearing account to which I make monthly deposits

Question

I have two sons, ages 6 and 11, for whom I deposit 50 dollars each into separate education accounts. I think it would be more fair to give my older son a larger amount monthly so that by the time they each turn 18, I will have invested the same amount of money into their accounts. I have noodled around with amortization formulas but I'm not a good enough mathematician to produce a convincing answer to my problem. Basically, I want x dollars invested in an interest bearing account monthly for the next 84 months to equal $(100-x) invested in a similar account for the next 144 months. What is a good formula for figuring this out?

Answer 1

Yes, there is a mathematical answer to this. If you invest $x$ dollars per month at the beginning of the month at $i$ per month compound interest for $n$ months, one month after the last deposit the last one will be worth $x(1+i)$ and the first will be worth $x(1+i)^n$. This is a geometric series with sum $x(1+i)\frac {(1+i)^n-1}i$ so your equation is $$x(1+i)\frac {(1+i)^{84}-1}i=(100-x)(1+i)\frac {(1+i)^{144}-1}i$$

Fairness is not a mathematical term. This calculation will result in each son having the same number of then-year dollars one month after your last deposit. Do you want to consider the effect of inflation during the five years? Your younger son will presumably face higher prices in the time he is in college. How much do you think the inflation will be? In the US for the past few decades the cost of college has risen faster than inflation. Maybe this last belongs on personal finance.se