This is the eigth theory session of the Problem Solving Test Course. I explain a shortcut for problems about covering losses or earning a certain profit given the current revenue and cost structure.

00:20 Example 01

01:00 Usual Approach

02:27 Contribution Margin Approach

03:38 Contribution Margin Theory

05:00 Intuition behind the algorithm

07:37 Example 02

09:55 Summary

This is the eighth theory session of the problem-solving test course.

We will learn how to calculate the share of revenue which goes into incremental profit and apply these techniques to otherwise time-consuming problems.

Let us start with the following example.

It will help us appreciate the usefulness of the contribution margin concept.

You might want to pause the video now and try to solve the question on your own. Assuming the ratio of variable costs to revenues remains the same, what is the minimum revenue in millions of dollars that would allow Instazon to earn 30-million-dollar profit? Notice that right now Instazon is making a loss of 20 million dollars per year on its revenue of a hundred million dollars and 70 percent of its costs are variable and the rest is fixed.

The usual way of taking this question would be to write a generic equation: revenue equals costs plus profit a loss and then solve it.

This is pretty doable but it’s a bit time consuming as it will see in a moment.

So, we start with the generic equation: revenues equal variable costs plus fixed costs plus profit, or loss. If we add the current numbers, this is what will get: a hundred million dollars equals eighty-four million dollars plus thirty-six million dollars minus 20 million dollars.

Obviously, the total costs are revenue plus loss, which is hundred twenty-two million dollars, and then we just take 70 percent of that and get variable costs of 84 million dollars and 30 percent of that and get 36 million dollars in fixed costs.

This is our current state.

The future state will be the following: new revenue equals new costs plus new profit; or X equals zero-point eighty-four X plus 36 million plus 30 million in profit.

Now it can solve this equation. Zero-point sixteen X equals 66 million, 66 million equals zero point sixteen which is roughly 66 million times six or roughly three hundred and ninety-six million.

And then the answer we choose is B. This is a reliable solution, but a bit time consuming.

Let us see if we can find it quicker one. The approach we use is called contribution margin, and the first step we will find the variable cost and their share in revenues.

The second step is finding the contribution margin, which is just one minus the share of variable costs.

And in our case is 16 percent.

Now we can calculate the new revenue which will allow us to make up for the loss and to earn the required profit.

New revenue equals old revenue plus the delta in revenue; or in our case 100 million plus 20 plus 30 over zero point sixteen million, which is about a hundred million plus 50 million times 6 which is about four hundred million.

And so again our answer is B. And we arrived at it much faster this time.

I will explain to you how this algorithm works in a moment. Contribution margin is defined as revenue minus costs over revenue.

In essence it is the part of incremental revenue not consumed by incremental variable costs.

For example, if the of share some variable costs is seventy percent, the contribution margin is 30 percent. If the share of variable costs as in our previous case is 84 percent the contribution margin is 16 percent. In a more complicated example, we take the revenue and variable costs, subtract variable costs from the revenue and divide by revenue. And we end up again with the contribution margin. As we have seen in the previous slide, contribution margin is useful in answering questions like, how much extra revenue do we need to earn a certain extra profit, or how much total revenue do we need to cover fixed costs and earn a certain total profit.

The algorithm for applying it to solving such questions is as follows.

One. Find variable costs and their share of revenue. Two. Find contribution margin as one minus this share.

Three. Divide the extra profit we need to make or total profit plus fixed costs by the contribution margin to find the extra revenue, or total revenue that we need. We will apply this algorithm in a few minutes, but now let us talk about the intuition behind it. As we discussed contribution margin is the share of revenue not consumed by increased variable cost.

If the share of variable costs in a company is 70 percent, then extra hundred dollars in revenue will bring extra 70 dollars in costs and three dollars in profit.

These thirty dollars is how much extra hundred dollars of revenue contributes a profit given the cost structure. The contribution margin is 30 or 100 equals 0.3. another 100 dollars will bring in other seventy dollars in costs and 30 in profit and yet another 100 dollars and another 70 dollars and another 30 in profit and so on. The extra revenue at this point is 300 dollars and extra profit is 300 times point three equals ninety dollars.

Now if you want to know how much extra revenue is need to bring extra 90 dollars in profits we just need to perform the reverse operation and divide 90 by 0.3 and get three hundred dollars.

And this is how the contribution margin algorithm works. In this next example suppose we have a fixed cost of 90 dollars, even when the revenue is 0, so we are facing a loss of 90 dollars.

Just like before extra hundred dollars bring extra seventy dollars in variable costs while thirty dollars translate into reduced loss.

Now the loss is 70 plus 90 minus 100 equals sixty dollars.

Add another hundred dollars and the loss is hundred forty plus ninety minus two hundred equals thirty dollars.

And with another hundred dollars the company breaks even. The extra revenue is 300 dollars and the extra profit is 300 times 0.3 equals ninety dollars. All of which translates into reducing loss.

So, if you want to know how much extra revenue is needed to cover the 90 dollars a fixed cost and breakeven, again we just need to perform the reverse operation and divide 90 by 0.3 and get three hundred dollars.

Here’s another example this time from an official practice test.

I suggest you pause the video now and try to solve the question on your own. Assuming claims costs remain unchanged at 83 percent of premium income and operations costs remain unchanged in million dollars compared to last year in order to make an underwriting profit of zero, InCo would have to increase premium income by approximately what percentage? From the text we know that InCo an insurance company is sustaining an underwriting loss because even though it is earning a premium income of seven hundred million dollars its claims costs represent 83 percent of premium income and operations costs represent 20 percent of premium income, so the loss is 3 percent of the premium income.

And our goal is to find by how much we need to increase revenue to breakeven.

So, I would start with calculating the loss. The losses three percent times seven hundred million equals 21 million. We know that the share of variable costs is 83 percent, then the contribution margin is hundred minus three equals 17 percent. The amount of extra revenue we need is defined as loss divided by the contribution margin. And we don’t really need to calculate this number, because what we’re interested in is the percentage increase in revenue, rather than an absolute number increase.

And this is what we’re going to calculate at the next step: percentage increase equals increase in an absolute value over revenue, which is 21 million 0.17 over 700 million, which equals 3 percent over 0.17, which is roughly 3 percent time 6 which is about 18 percent.

Just because 0.17 is about one sixth.

And so, the answer is C. That it. Let us synthesize what we have learn today. Contribution margin is defined as the revenue minus variable costs over revenue. It is the bridge between incremental revenue and incremental costs.

It is the part of the revenue not consumed by variable costs. To apply in a PST setting, find the share of variable costs and revenue, finds contribution margin as one minus the share and then divide that extra profit needed by contribution margin.

You will end up with the value for the desired extra revenue.

Okay let’s call it a day.

Any questions or comments please leave them below the video.

I’ll be happy to reply.

Thanks for watching the video and until next time.