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I'm Larry Walther, this is Chapter
18 of principlesofaccounting.com.
In the previous module we considered the definitions of fixed cost and
variable cost.
And some of the implications for
a business in terms of understanding those costs.
In this module, however, we are going to consider.
How we would determine the fixed and variable nature of cost for
a particular business.
In other words, we're going to look at cost behavior.
Now, some costs are not strictly fixed or
strictly variable, they are mixed in nature.
Let me illustrate the concept of a mixed cost with a very simple and clear example.
Butler's Car Wash has a contract for
its water supply that provides a flat monthly fee of $1,000 per month.
Plus $3 for each thousand gallons of water that are used.
And so if we were to look at this in spreadsheet form.
In column A I'm showing the gallons that could be used in a particular month.
Column B is the variable costs that would be incurred, for
example on row 9 we can see $2,100.
That's 700,000 gallons, so 700 times $3 or
$2,100 would be the variable cost at that level of usage.
The fixed cost however is $1000 at each level of usage.
And we have our total cost, the accumulation of columns B and C here.
And I've also plotted that to show the total water cost starting at $1000,
even if we use 0 gallons.
And then it goes up in a linear fashion for
the variable cost as the usage continues to increase along the horizontal axis.
Oftentimes, a cost is not clearly fixed or
variable in nature as it was for Butler's Car Wash.
So we may need to do an analysis to separate the total
cost into presumed fixed and variable components.
One way for doing this is the high-low method.
It involves several steps.
The first of which is to find the highest level of activity and
the lowest level of activity.
And identify the cost for each of those levels of activity.
And see what the difference is between the high cost and low cost.
And see what the difference is in the volume between the high level and
the low level.
And based on that we would calculate a variable cost per unit.
In other words, we have so much change in volume and so
much change in cost that suggests the variability rate.
The remaining amount in each case would be the fixed cost.
Recognize that this method can be very imprecise if we have outliers or
rough data points.
In other words, if our high point is way out of the norm or
our low point is way out of the norm.
Then our calculations could be suspect.
So let's see how this would work.
Here I've got an example.
We've identified our usage, our highest level was 850,000 gallons.
And our lowest level was 340,000 gallons.
A difference of 510,000 gallons.
We compare that to our costs.
The highest cost was $3,550, the lowest cost,
$2,020, a difference of $1,530.
So very simply, we've got a range of 510,000 gallons and
a cost range of $1,530.
Based on that we can calculate a variable cost per unit of $3.
So we're adopting an assumption that our variable cost is $3 per unit.
Notice our total cost at the high level is 3,550, and at the low level, 2,020.
Of that, our variable cost would be 2,550, that is, 850 times 3.
And our low cost would be 1,020, that is, 340 times 3.
The difference of the remaining amount is fixed cost.
And it's shown to be $1,000 in each of the two columns.
So here we would have identified a high and a low point.
To discern that our fixed cost was $1,000 and our variable cost was $3 per unit.
A more statistically valid method for
cost behavior analysis is the method of least squares.
It's based on regression analysis.
Recognize that a straight line can be defined by the formula Y = a + bx.
Where a is the intercept which would infer our fixed cost component.
b is the slope of the line which infers our variable cost component.
And x is the position on the X axis, that is, the volume level.
So the least squares method then
defines a line that fits through a set of points on a graph.
Where the cumulative sum of the squared distances between the points and
the line is minimized.
Spreadsheet programs can be used for this analysis.
This contrasts with a scatter graph method.
An alternative method where points are simply plotted on a graph.
And a line is drawn through the graph to approximate these values.
And so let's look at how the least squares method would work in this particular case.
We have monthly data, we have units, and we have total costs.
So, that's our data set.
We've run a regression analysis using the spreadsheets software.
To determine that intercept is 138,533 and the slope is 10.34.
This suggests that for this cost for this business.
That the fixed cost is $138,000 and the variable cost is $10 per unit.
Now these values were derived from formulas included in the spreadsheet.
For example, in cell B17, I calculated the R Square value.
Which I'll comment more on in just a moment, as 0.798.
The formulation in the spreadsheet, rRSQ(.
For the range of data C2 to C13 regressed on the data B2 to B13.
So that would be the formula for R Square.
We would have a similar formulation within cell B15 and
B16 for our intercept and slope terms.
So let's look closer at the graph here.
If we notice the point that has an arrow drawn.
It appears that that's about 95,000 units on the horizontal axis.
And about $1,500,000 on the vertical axis.
That equates to the data point for December.
And if we had just done a scatter graph technique.
We would have simply drawn the dots on the graph.
And then drawn the line through the graph.
And see where it hits the y-axis and what the slope is as the line moves out.
And we could approximate the formula for the line there.
But what regression analysis does is
it optimizes that line through those data points.
Remember, it's called regression analysis or least squares regression.
And so if we look at this particular point.
And we looked at each of the similar points.
And found the horizontal distance between the point and the line.
The sum of the squared distances would be minimized with regression analysis.
In other words, that's deemed to be the best fit line through those data points.
Let's finally close by thinking about that R squared value.
The R squared value of the least squares line is a statistical calculation.
That characterizes how well the line fits to a particular set of data.
In our example, we saw an R squared value of 0.798.
That simply means that almost 80% of the variation
can be explained by volume fluctuations.
The outliers are fairly small, or
in other words, the points are fairly tightly clustered to the line.
As a generalization, the higher the R squared is,
the closer it is to one, the better fit is the line to the dataset.
If the R squared value was exactly one,
that would mean there is no variation between the dots and the line.
The line would pass exactly through each of the dots precisely.
If the R squared value was very small, 0.2.
That would suggest that the line is not explaining a lot of the variation.
There is a lot of deviation between the points and
the line that R squared finds as the best fit line.
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