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.
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