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Things I know:

  • Need to make $\$165,000$ by the end of December to break even.
  • Made $\$48,000$ so far this year (Jan-Apr).
  • Made $\$12,500$ in April.

How would I figure out what my consistent growth would need to be to hit my break even point?

Doing a guess and check method I came up with $11.060176\%$. This means

  • May = $\$13,882.52$
  • June = $\$15,417.95$
  • July = $\$17,123.21$
  • August = $\$19,017.06$
  • September = $\$21,120.38$
  • October = $\$23,456.34$
  • November = $\$26,050.65$
  • December = $\$28,931.89$

I know the answer for this specific scenario is $11.060176\%$ growth month over month, but that was with a lot of guess and check. How would you do this in a formulaic way.

Thanks in advance.

Saket Gurjar
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BulletB
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2 Answers2

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You want an expression like this: $Sr^{k}=E$ where $S$ is the starting amount of money, $r$ is the rate of growth (which is the unknown), $E$ is the ending amount you want to reach and $k$ is the number of months you have to do so.

Then you can find $r$ by

$$Sr^{k}=E \implies r^k =\frac {E}{S}\implies r=\left(\frac{E}{S}\right)^{\frac{1}{k}}$$

In your example:

$$60,500 r^8=165,000\implies r^8=2.727272\implies r=(2.727272)^{1/8}=1.13362$$

which translates to about $13.36 \%$ of monthly growth.

N. Pullbacki
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  • So this gives you the same dollar amount increase month to month. I am looking for the same percentage increase month to month. – BulletB May 20 '20 at 19:25
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It is unclear what problem you are trying to solve: do you want

  1. An interest rate on your existing money (presuming the 48000+12500 is all you have) such that the ending balance becomes 165000? Then see Ramita's answer.
  2. Or do you want a growth rate on your income amount per month such that the remaining income for the 12 months is 165000?

To solve 2:

What you have is the standard geometric series $$br+br^2+br^3+br^{n}=br\frac{r^{n}-1}{r-1}$$ where $b$ is your zeroth value ($12500$ in Apr), $r$ is what you're solving for, and $n$ is the total number of time periods you have (May-Dec, so 8).

You need a total of $165000$ from these 8 months, so $$12500r\frac{r^8-1}{r-1}=165000\quad\Rightarrow\quad r^9-14.2r+13.2=0$$ which unfortunately isn't directly solvable in closed form, but will have a unique solution $r>1$.

You can pull out much faster numerical methods to solve this (Newton's method, etc). WolframAlpha gets $11.60175\%$ as expected.

obscurans
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