slope

In the previous post about mixed cost, we stated that a mixed cost is just the sum of the variable and fixed components. This is fairly easy to deal with when we are dealing with an external cost where we are given the variable rate and the fixed cost. In business, many mixed costs are actually generated internally. Therefore, the variable and fixed components are not clear. We must find a way to calculate the fixed and variable components.

There are a number of ways to calculate the cost formula for a mixed cost. In this post, we will focus on the high-low method. This method is not the most precise method but it is the easiest to calculate. It does not require spreadsheet or graphing software.

Why the high-low method works

Below is some data from an oil change service business.

HL1

The business has fixed and variable costs but wants an easy way to do cost planning for future budgets. The company would like you to write a mixed cost formula for planning purposes. It might seem daunting at first but it’s really a lot easier than you might think.

Step 1 – Find the high point and the low point

Since this is called the high-low method, we first need to determine the highest point and the lowest point in the range. Because the variable rate and fixed costs are not always 100% constant, the cost should not be used. Since the number of oil changes is a consistent, reliable measure, we should use that to determine the high and low points. Looking at the data in the chart above, what would you choose as the high and low points? April is the high point with 2,950 oil changes and January is the low point with 2,200 oil changes.

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Once you have picked the high point and the low point, you  can throw out the rest of the data. You no longer need it.

Step 2 – Find the variable rate

You might be wondering how we are going to jump to solving for the variable rate when it doesn’t seem like we have a whole lot of information. We have more information than you might think.

Let’s look at these two points on a graph.

HL3

If you read the post on variable cost or the post on mixed cost, you might remember that we talked about slope. I know that slope is terribly boring and something that you might be trying to forget from your math classes, but is actually important here and makes this concept much easier to understand.

We said in the earlier posts that variable rate is the slope of the line. That means that for every additional oil change performed, the total cost increases by the variable rate. In January (the low point), the company performed 2,200 oil changes with a total cost of $9,860. In April, the company performed 750 more oil changes. Those additional oil changes cost the company an additional $1,725. Over the course of 750 oil changes, cost increased $1,725. That also means that the variable cost of 750 oil changes is $1,725.

Since we know that the variable cost of 750 oil changes is $1,725, we can divide to calculate the variable rate. The variable rate is $2.30.

Let’s go through the calculation step by step so you can see where I got all the numbers.

First calculate the change in cost and the change in activity.

Change in Cost = $11,585 – $9,860 = $1,725

Change in activity = 2,950 – 2,200 = 750

Next we will divide the change in cost by the change in activity to calculate the variable rate.

 Variable rate = $1,725 / 750 = $2.30

Most textbooks will use the following formula for variable rate using the high-low method:

HL4

If you’ve looked at that formula before and thought “huh?!?”, I agree. Many times in managerial accounting, understanding what is actually happening is much more helpful in solving the problem than trying to memorize the formulas. Just remember that the increase in cost is all variable cost. If you calculate how much the activity changed, you now have the total variable cost for the additional activity. That is enough information to calculate the rate.

Step 3 – Find the fixed cost

The formula for mixed cost is:

Total cost = Rate X Activity + Fixed Cost

We need to fill in all the additional information so that we can solve for the fixed cost. We clearly have the rate. We solved for that above.

Total cost = $2.30 X Activity + Fixed Cost

Where could we get figures for total cost and the activity level for that cost? Wouldn’t it be nice if we had some data for total monthly cost and the activity associated with it?

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Well, it’s a good thing we have the high and low points. The data gives us exactly what we need. We have the total monthly cost for two of the months and the activity associated with those months. Brilliant!

Note: You must use the figures from either the high point or the low point since the variable rate was calculated based on those numbers. The high and low points will give you the same fixed cost (within a few cents if you had to round the variable rate).

Plug either the high point or low point into the cost formula and solve for fixed cost.

$11,585 = $2.30 X 2,950 + Fixed Cost

Fixed Cost = 4,800

OR

$9,860 = $2.30 X 2,200 + Fixed Cost

Fixed Cost = $4,800

Step 4 – Write the cost formula

Now that you have the variable rate and the fixed cost, you can write a cost formula for planning. The monthly cost of oil changes is:

Total Monthly Cost = $2.30 X number of oil changes + $4,800

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Fixed costs can be a tricky business. They might seem simple but if you think too much, you might get tripped up.

Just like with variable cost, fixed costs are named because of how the cost behaves in total. It is fixed. It does not change. Now most students will take that to mean that the cost will never change. If that were the case, there is not a fixed cost on the planet. All costs change over time. Remember, we are taking about how a cost behaves today.

A fixed cost does not have an activity or driver that makes the cost increase as the activity or driver increases. Let’s say you start a business and the rent for 500 square feet is $1000 per month for the first three years. Is there an activity or driver that would increase your rent expense?

Number of hours open? Nope

Number of customers per month? Nope

Amount of sales in units or dollars? Nope

And here is the most important question: if all of your drivers go to zero, does the cost go to zero as well? If you go on vacation for a month and close your business so there are no sales, no customers, nada is your rent expense zero? Nope

With a variable cost, when the driver was zero, the total variable cost was zero. With a fixed cost, that is not the case.

Remember our candy bar example from the post on variable cost? What if, in order to sell the candy bars on campus, you needed to pay a fee of $100 to the college. Is that a fixed cost or a variable cost? It is fixed because it does not change no matter how many (or how few) candy bars you sell.

Here is the graph for fixed cost:

Total Fixed Cost

Notice on this graph, there is no slope. The formula for total fixed cost = fixed cost. If looking at the equation for a line y = 0 + b, where b is equal to fixed cost. As long as you are within the relevant range, the formula is valid.

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When thinking about cost behavior, think about how the cost behaves in total. A variable cost is a cost that varies in total. The cost increases or decreases based on a related activity.

The formula for total variable cost is:

Total Variable Cost = Variable Rate X Activity

Assume a constant rate

For planning and decision making purposes, we assume that the variable rate is constant. This allows for a single variable in the calculations. Only the activity will change. Now, that is not always the case, but as long as we are within the relevant range for our decision, we can assume that the rate will stay the same.

But isn’t it fixed if the rate stays the same?

Remember that a variable cost varies in total. The rate might stay the same but once you multiply the rate by varying levels of activity, the total variable cost will change.

Imagine that you are selling candy bars as a fundraiser for a club to which you belong. Your cost is 50 cents per candy bar and the club sells the candy bars for $1 each. If the club sells 200 candy bars, what is the total variable cost? Is it 50 cents? No, that is the cost of a single candy bar. If you sell 200, you would need to multiply that by 50 cents for each of the candy bars sold.

200 candy bars X 50 cents per candy bar = $100

What if the club sold 500 candy bars? The total variable cost would be $250.

Here is a graph of the total variable cost of candy bars for the fundraiser:

Total Variable Cost

Notice that if no candy bars are sold, there is no cost. The more candy bars that are sold, the higher the cost. The cost line is a straight line. The slope of the line is equal to the variable rate. For each additional unit sold, the line increases at a rate of 50 cents. Think of the formula of a line: y=mx + b, where y is your y coordinate, x is your x coordinate, m is the slope and b is the y-intercept (the point where the line hits the y-axis).

The formula for total variable cost is: y=mx. The y-intercept for a variable cost is always zero because if there is no activity, there is no cost. Therefore, the line will always start at 0,0. The slope of the line, m, is your variable rate. The activity is x. See your math teacher was right when he or she told you you would use this stuff someday!

Frequently, you will see textbooks show the formula for the slope of a line as the formula for cost equations.

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