$Algebra Question$

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$Algebra Question$

A company wants to look at their YoY margin % changes on an absolute basis. (ex. if margin in '22 was 50% and margin in '23 was 80%, break out the drivers of the 30% change). I'm having difficulty getting to an answer because when you isolate revenue Yoy to find COGS driver, you get a different answer than if you isolate cogs driver to find revenue impact first.

I tried to simplify an example below to show the mechanics. In (x-y)/x = z, x=revenue, y=cogs, and z= profit margin %.

In scenario A (of (x-y)/x=z), x=4 and y=2, so z= 2/4 or 50%. In scenario B (of (x-y)/x=z), x=5 and y=1, so z= 4/5 or 80%. Z went from 50% to 80% between scenario A and B. Z increased 30% points. Can you break the 30% change into drivers? How much did the change in x contribute to the 30% change and how much did the change in Y contribute to the 30% change? The answer must add up to 30%. (example x=15% y=15%)

Algebra

Mathiscoolsometimes

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Your question is kind of vague, but I will try to clarify and answer it. We have
\[z=\frac{x-y}{x}=1-\frac{y}{x}\]
Suppose the values of $x$ and $y$ change from $(x_1,y_1)$ to $(x_2,y_2)$. Then the change in value of $z$ is
\[z_2-z_1=(1-\frac{y_2}{x_1})- (1-\frac{y_1}{x_1})=\frac{y_1}{x_1}-\frac{y_2}{x_2}\]
\[=\frac{y_1 x_2-y_2x_1}{x_1 x_2}=\frac{y_1 x_2-y_1x_1+y_1x_1-y_2x_1}{x_1 x_2}\]
\[=\frac{y_1 (x_2-x_1)-x_1(y_2-y_1)}{x_1 x_2}=\frac{y_1}{x_1x_2}(x_2-x_1)-\frac{1}{x_2}(y_2-y_1).\]
So
\[z_2-z_1=\frac{y_1}{x_1x_2}(x_2-x_1)-\frac{1}{x_2}(y_2-y_1).\]
You can measure the change in $z$ due to change in $x$
\[\Delta x=\frac{y_1}{x_1x_2}(x_2-x_1)\]
and the change in $z$ due to change in $y$
\[\Delta y=-\frac{1}{x_2}(y_2-y_1).\]
In your example we have
\[\Delta x=\frac{y_1}{x_1x_2}(x_2-x_1)=\frac{2}{4\cdot 5}(5-4)=\frac{1}{10}=10 \%,\]
and
\[\Delta y=-\frac{1}{x_2}(y_2-y_1)= -\frac{1}{5}(1-2)=\frac{1}{5}=20 \%.\]

This took me about 30 minutes to answer. I would appreciate a tip!

Erdos

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Erdos

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Let me know if you need any clarifications.
- Mathiscoolsometimes
  
  0
  
  This is great, thank you. The only discrepancy I see is that to find the change in z due to the change in y, you multiplied 1/5 by (y1-y2). If you would have multipled by (y2-y1), or 1-2, the answer would have been negative. Which is correct? Multiplying by (y1-y2) or by (y2-y1)?
- Erdos
  
  +1
  
  You are right. I revised my solution to include the minus sign. It should make much more sense now. Thank you for the coffees!
Mathiscoolsometimes

+1

Thank you!

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