0
$\begingroup$

Is it possible to calculate the flow of cash in each period given a certain fixed net present value with a numerical algorithm?

I would like to be able to apply a certain weighting that can be applied to the cash flows of each period e.g. 90% will be paid back in the last period, 1/(number of periods-1) in all other preceding periods.

For example, parameters could be: number of periods of the cash flow, discount rate, indexation/inflation, certain weighting that can be applied to the cash flows of each period, etc.

What formula should I start from? Or is there a R/Python/... package that can calculate this for me?

enter image description here

So I want to calculate the uneven cash flows CF 1-n, which has to be done numerically I guess.

$\endgroup$
1
$\begingroup$

If you can express the payment in each period as $CF_i = a_i x$ where $x$ is the sum to be paid. You can easily plug x in easily substitute this in the equation and calculate the value of $x$, once you know $x$ you can calculate $CF_i$

To determine $x$ start with

$$ PV = \sum_{i=0}^n \frac{CF_i}{(1+r)^i} = \sum_{i=0}^n \frac{a_i x}{(1+r)^i}$$

and solve

$$ x = \frac{PV}{\sum_{i=0}^n a_i{(1+r)^{-i}}} $$

For your example you have

$$ a_{i} = \left\{\begin{matrix} 1/(10n)&\textrm{if }i \ne n \\ 9/10 & \textrm{if } i=n \end{matrix}\right.$$

$\endgroup$
19
  • $\begingroup$ Something I forgot to mention, is that I also need to calculate net present value, not just present value. So e.g. in the initial period there is a negative value of e.g. -100, and at the end it is assumed it can be resold for e.g. +95. Or based on some formula that relates the 100 to the 95 which can be complex and dependent on n. But let's assume it's not dependent on n and weights can be give beforehand. All the in between cash flows are dependent on the 100 too, e.g. if they were discounted; they would each equal 1/(number of periods-2) of the initial cashflow.This seems to make it harder? $\endgroup$
    – babipsylon
    Jun 21 at 17:33
  • 1
    $\begingroup$ Ach no you are right, there is no index i for x so you can indeed just get it out of the summation from i to n. $\endgroup$
    – babipsylon
    Jun 21 at 17:49
  • 1
    $\begingroup$ $a_i$ may add to something different than 1, if you have a distribution $a_i$ and another distribution $b a_i$, the second will give $x/b$ instead of $x$, and $b a_i (x / b) = a_i x$. The only restriction is that the discounted values of $a_i$ must add to something different than zero, in the denominator of the formula for x. $\endgroup$
    – Bob
    Jun 21 at 20:27
  • 1
    $\begingroup$ Yes, you got it $\endgroup$
    – Bob
    Jun 22 at 9:24
  • 1
    $\begingroup$ If you get a negative X it means that you are trying to pay a negative amount for a positive PV, or vice-versa, the formula is working as an anti-fraud for you hehe $\endgroup$
    – Bob
    Jun 22 at 10:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.