0

I'm figuring out some kind of money saving function where the amount, in the end, should be 100.000 value with a time period of 10 years.

I thought of an exponential growth function since people start with jobs and with work experience, you will earn more with results into a higher amount that you can save.

With some search, I found this formula: $P(t) = P0 e^r*e^t$ (I can't format this propper) Where $P(t) = 100.000$ , $r=0.5298$, $P0=500$ and $t=10$ This will lead to an insane amount of money in year 9 and 10.

I'm looking for a more reasonable function where you start with the least amount in year 0 and the most amount in year 10 so the amount that you saved is $100.000$. But it should be reasonable like you can't save 20k in year 10. Also, $P0$ may differ for each person, but the time and the end goal is fixed. I do not take interest into account.

Helper
  • 27

1 Answers1

0

You can impose the following model:

Save $P(0)$ in year 0, $P(1) = \alpha P(0)$ in year 1, $P(2) = \alpha P(1) = \alpha^2 P(0)$ in year 2, etc. for some $\alpha>1$. So you are geometrically/exponentially increasing the amount you save each year. After 10 years you will have $P(0) + P(1) + \; ... \; + P(10)$ = $P(0)[1+ \alpha + \alpha^2 + \; ... \; + \alpha^{10}] = P(0) \frac{\alpha^{10}-1}{\alpha-1}$.

Say $\alpha=1.5$. At the end you will have $113.33 P(0)$, which means $P(0) = 882.38$ if you want the final amount to be 100,000. With this model, you will have to save $\$882.38$ in year 0, $\$1323.60$ in year 1, ..., and $\$5088.2$ in year 10 (which is still reasonable).

Godfather
  • 2,355
  • care to explain how you go from $113.33P(0)$ to $P(0)=882.38$ as I people start with saving 500 euro in P(0) so I get $113.33 *500 = 56665$. I know you calculate to get 100.000 in the end. But I don't quite get it – Helper Dec 13 '18 at 20:00
  • I simply set $113.33P(0) = 100,000$, which gives $P(0) = 883.28$. I assumed that $\alpha = 1.5$ (just as an example). You can fix $P(0) = 500$, and then find an $\alpha$ such that $P(0)\frac{\alpha^10-1}{\alpha-1} = 100,000$. Is that clearer? – Godfather Dec 14 '18 at 03:33
  • 1
    Yes, now I understand the formula. My thanks to you for the explanation. – Helper Dec 15 '18 at 19:00
  • Although If I calculate year 10 by multiplying all years before with 1.5 I get 50882.48 value. Which I call insane value. – Helper Dec 15 '18 at 19:13