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Simple Way to Solve an Iterative Equation?

Date: 01/09/2009 at 08:47:39
From: Michael
Subject: Never ending equation

I am a CFO of a Company and we have a series of bonuses paid to sales 
folks and executives that are predicated on profit including a 
deduction for the cost of the bonus.

As an example, a sales person's bonus target is paid based on actual 
profitability as a percentage of a budgeted profitability.  More 
specifically, 100% of their bonus is paid out for reaching 100% of 
budgeted profitability, 90% of their bonus is paid out for reaching 
99% of budgeted profitability, 80% paid for reaching 98% of budgeted 
profitability and 0% paid at 90% of budgeted profitability. 
Importantly, the actual results versus budgeted results includes the 
related bonus expense.  So, obviously, profitability goes down once 
you add the the bonus expense which, in turn, changes the actual vs. 
budget percentage which changes the bonus.  The equation essentially 
goes on forever as the variables change.  However, an intersection 
point is eventually reached where the changes in variables have an 
immaterial effect on the calculation.

Is there an equation that can be applied to easily calculate the 
result of this never ending equation?

I understand the concept and can figure out an answer using a chart 
and manual iterations.  Just looking for an equation to solve the 
riddle more easily.

Here is a real, live example.

2008 Budgeted Profitability:  $45,726

2008 Actual Profitability, before a bonus pool is awarded to certain 
sales people:  $44,201

% of budget attained: 96.7%

At this percentage, 60.7% of a bonus pool of $900 would be recorded 
by the company, which would total $546 and would reduce actual 
profitability from $44,201 to $43,655.  At the new profitability 
level, the budget attained percentage to 95.5%.  In turn, the bonus 
pool percentage would change to 50.5% which would change the bonus 
recorded to $454, which then changes the percentage again and so on 
and so....Again, is there a formula you can use to figure out the 
intersection point between adding bonus pool expense and the % 
attained to an immaterial difference?



Date: 01/09/2009 at 10:36:27
From: Doctor Peterson
Subject: Re: Never ending equation

Hi, Michael.

This sort of infinite process tends to occur when you try to solve 
directly a problem that should instead be solved by algebra.  Let's 
define some variables and write an equation.

Let

  A = Actual profit, before bonus ($)
  B = Budgeted profit ($)
  P = bonus Pool ($)
  x = percent of budget achieved (as a decimal)
  y = percent of bonus given (as a decimal)

The realized profit after the bonus will be A - yP, so the percent 
achieved will be

      A - yP
  x = ------
        B

The bonus is calculated (assuming p is greater than 90%) as

  y = 10(x - 0.90)

since it increases by 10% for every percentage point above 90.  (Check
this: if x=0.99, y=10(0.99-0.90) = 0.90.)  Note that each equation
involves both x and y, which explains the recursion you found:
calculating one changes the other, which changes the one, ... But we
can now combine the equations by plugging the first into the second:

  y = 10x - 9
 
      10(A - bP)
  y = ---------- - 9
          B

Multiplying by B, then collecting terms with y on the left,

  By = 10A - 10Py - 9B

  By + 10Py = 10A - 9B

  (B + 10P)y = 10A - 9B

Dividing by the coefficient of y,

      10A - 9B
  y = --------
      B + 10P

Of course, if this is negative, the bonus will actually be zero.

For your example,

  A = 44201
  B = 45726
  P = 900

so the bonus percentage is

      10A - 9B   10*44201 - 9*45726   30476
  y = -------- = ------------------ = ----- = 0.55688 = 55.69%
      B + 10P      45726 + 10*900     54726

Does that agree with your manual result?

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 01/09/2009 at 11:28:29
From: Michael
Subject: Thank you (Never ending equation)

This is awesome!!!!  Thanks for the help.  This is one of the best 
things I have ever see on the Internet.  Thanks again.
Associated Topics:
High School Basic Algebra

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