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Say you have a cash flow diagram with all the outflow and inflow laid out and need to calculate the break even point/rate.

  1. How would you use the annuity formula to find the number of years?
  2. How would you use the annuity formula to find the interest rate given a certain number of years?

For example:

Break Even is inflow = outflow (Economic Equivalence)

And using this formula.

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What would N be or what would i be?

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Redsam121
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  • thank you for fixing the picture. I don't know why it never works for me it the editing tools. – Redsam121 Sep 17 '19 at 03:27
  • The formula can be used for one (negative) payment (P) and $N$ constant earnings (A) only. $-P+A\cdot \frac{(1+i)^N-1}{(1+i)^N\cdot i}=0$. The equation can be solved for $N$. Is this what you want? Or maybe P means the present value of $N$ constant earnings (A). – callculus42 Sep 17 '19 at 03:49
  • P means present value. but like I said, this is for a cash flow diagram to break even. Something like A⋅(1+i)n−1/(1+i)n⋅i = B⋅(1+i)n−1/(1+i)n⋅i where A is inflow and B is outflow. – Redsam121 Sep 17 '19 at 03:52
  • The equation at last your comment just indicates that $A=B$ since the rest is identical. A more specific, numerical example would give me the chance to help you better. – callculus42 Sep 17 '19 at 03:56
  • I've provided another example, this one is about finding the inflow payment. Please be advised I have very minimal understanding of these principles – Redsam121 Sep 17 '19 at 04:12
  • I don´t think that the equation is right. You need for all summed up payments a common reference date. We can use $t=0$ as reference date. The 2X is OK. But the next 7 $X$-payments has to be dicounted twice ($t=2\to t=0$)-not only once. So the divisor should be $1.12^2$ and not $1.12$. – callculus42 Sep 17 '19 at 04:26
  • Have a look here, here and here There are similar problems with my answers. – callculus42 Sep 17 '19 at 04:58
  • ! before is required for images. but apparently images are discouraged. –  Sep 17 '19 at 11:21

1 Answers1

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$$P=A{(1+i)^N-1\over i(1+i)^N}$$ To solve for $N$, divide by $A$, multiply by $i$: $${Pi\over A}={(1+i)^N-1\over(1+i)^N}=1-(1+i)^{-N}$$ Rewrtie as $$(1+i)^{-N}=1-{Pi\over A}$$ Take logs: $$-N\log(1+i)=\log\left(1-{Pi\over A}\right)$$ Now divide both sides by $-\log(1+i)$.

Gerry Myerson
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  • Will this find the break even point on the cash flow diagram? – Redsam121 Sep 17 '19 at 09:01
  • I don't know what that means. I know you wrote down a formula involving $P,A,i,N$ and asked "What would $N$ be?" and I've shown you how to find $N$ from that formula, in terms of $P,A,i$. If that's not what you wanted, then you'll have to explain all the jargon you use. – Gerry Myerson Sep 17 '19 at 09:08
  • N is the number of years, P is present value, i is interest rate and A is annual revenue or expense. Say you have a cash flow diagram (like the one on top) and had the inflow (bottom) and outflow (top) laid out with an annual interest rate. How many years would it take to break even (meaning both sides are equal). Here's another example https://math.stackexchange.com/questions/1140038/cash-flow-diagram-in-outflow-series. – Redsam121 Sep 17 '19 at 09:14
  • Sorry, I have no idea what's going on in that "cash flow diagram", and I don't know what you mean by "both sides" and I don't know how the formula for $P$ relates to any of it. – Gerry Myerson Sep 17 '19 at 09:47
  • Okay, I've added a written version of what I've done so far. Maybe that'll help. – Redsam121 Sep 17 '19 at 10:11