Consulting Blogs: Master Consulting Math

Andrew Savage

Here are some of the most common topics you’ll come across in consulting math. Below those topics are equations you should familiarize yourself with, along with some practice problems. The answers to the practice problems are at the end.

Topics that will be covered in this blog include:

  • Profitability

  • Indifference Point

  • Break-Even

  • Net Present Value (NPV)

  • Market Sizing

1. Profitability

Revenue = Quantity * Price

Total Costs = Total Variable Costs + Fixed Costs

Profit = Revenue - Costs

Profit = (Price - Variable Costs) * Quantity - Fixed Costs

Contribution Margin = Price - Variable Cost

Profit Margin = Profit / Revenue

 

Example: Sally’s Widgets

Sally wants to calculate the profitability of her widget company. Here’s the information:

Each widget sells for $5

Raw Materials = $1/widget, Labor = $2/widget

Fixed Costs = $1000

Quantity Sold = 1000

What is the profitability? What about the profit margin?

 

2. Indifference Point

 

Indifference Point: Level of volume at which total costs, and hence profits, are the same under both cost structures

 

Example: Paper Towels vs. Hand Dryer

Should our restaurant stock the bathroom with paper towels and pay $0.005 per paper towel or pay $100 a month to maintain a hand dryer?

Find the indifference point. In other words, at what point (in # of paper towels used per month) should we instead invest in a hand dryer?

 

3. Break-Even

Break-Even: the point at which total cost and total revenue are equal (profit = 0)

Break-Even Point (Years) AKA Payback Period = Initial Investment Costs/Annual Profit

Break-Even Point (Units Sold) = Fixed Costs/Contribution Margin

Contribution Margin = Price - Variable Costs

Break-Even Point (Sales Dollars) = Fixed Costs/Contribution Margin Percentage

Contribution Margin Percentage = (Price - Variable Costs) / Price

 

Example: Muhammad’s Surf Shop

Muhammad wants to open up a surf shop in Malibu. His initial costs are $1 million. He predicts his shop will profit $400,000 every year. How long will it take for him to break even? 

 

4. Net Present Value (NPV)

For a growing perpetuity (95% of NPV problems on cases):

NPV = [Annual Profit/(Discount Rate - Growth Rate)] - Initial Investment Costs

If we assume operations go on forever (perpetuity), this is the difference between the present value of cash inflows and the present value of cash outflows over a period of time. It tells us the profitability of a projected investment or project.
 

Example: Sherri’s Gym

Sherri is opening up a gym. It will cost her roughly $2 million, but she expects to make $500,000 in annual profit. What is the NPV of the project?

*Ask for the discount rate and growth rate. 

Discount rate = 10%, Growth Rate = 0%

 

5. Market Sizing

Market sizing problems come up quite often in consulting cases. Market sizing is the process of making a series of reasonable estimates to deduce the market opportunity for a client. Usually, the final answer is given in potential market revenue. Let’s try estimating the market size of the U.S. market for lightbulbs.

Top-Down Market Sizing Example

Let’s start by assuming the U.S. has a population of 300 million. This may not be exactly right, but we want to pick a number that’s reasonably close to the actual number and that makes the rest of the math easy.

Now let’s make another assumption: the average U.S. household has about 3 people. If we break our initial population number into households, we would get 100 million households in the U.S. Let’s assume the average household has about 10 rooms, and each room has 2 light bulbs in it. That would bring us to 2 billion light bulbs. If each light bulb costs $5 on average, we can assume the U.S. market for light bulbs is about $10 billion.

Here’s the math written out:

300 million U.S. population / 3 people per household = 100 million U.S. households

100 million U.S. households * 10 rooms per house * 2 light bulbs per room = 2 billion light bulbs

2 billion light bulbs * $5 average price per light bulb = $10 billion market size for U.S. light bulbs

Although these assumptions may not be completely accurate, this is the process of using reasoning to estimate market size. This is top-down market sizing, where we start with a population and make assumptions about that population to get to a market size. Bottom-up market sizing is a lot less commonly used, but it entails making assumptions about individual customers and smaller towns, and then expanding those assumptions to apply to the entire U.S. market. 

 

Bottom-Up Market Sizing Example

Here’s a more complex example of bottom-up market sizing for the U.S. gas station market. Let’s say I live in a town of about 5,000 people, and we have about 4 gas stations. I get gas once every two weeks and spend about $20 every time I go. If we say there are 50 weeks in a year for simplicity’s sake, we can assume that each person spends about $500 a year on gas. However, let’s also say that only 80% of the town drives. That brings our town population down to 4,000. So in my town, gas stations would make about $2 million per year.

If the total U.S. population is 300 million and my town has 5,000 people, we can assume that the U.S. population is about 60,000 times the size of my town. If we scale this, 60,000 multiplied by the $2 million gas station revenue of my town is $120 billion in estimated total gas station revenue in the U.S.

Here’s the math written out:

Within the town:

$20 average spent per visit * 25 visits a year (once every two weeks) = $500 average consumer spends on gas annually

5,000 town population * 80% of the population the drives = 4,000 town population

4,000 town population * $500 average annual consumer spend = $2 million spent on gas in the town

 

Scaling to the U.S. population:

300 million U.S. population / 5,000 in my town = 60,000 (the U.S. population is 60,000 times the size of my town)

$2 million in town gas station revenue * 60,000 = 120 billion in U.S. gas station revenue

Again, the goal isn’t to get to an accurate estimate; it’s to use logic and make smart assumptions to estimate a relatively reasonable market size.
 

6. Other Important Formulas to Know!

Return on Investment = Profit / Investment Cost

Return on Equity = Net Income / Shareholders’ Equity

Market Share = Company Revenue in the Market / Total Market Revenue

 

Answers
 

1. Profitability: Sally’s Widgets

Profit 

  • (Price - Variable Costs) * Quantity - Fixed Costs
  • ($5-$3) * 1000 - $1000 = Profit
  • =$1000

Profit Margin

  •  $1000/($5*1000) = 20%

2. Indifference Point: Paper Towels vs. Hand Dryer

  • $100 = $.005x
  • Indifference Point (x) = 20,000 paper towels
  • If we predict >20,000 paper towels will be used per month, it would make sense to invest in the hand dryer

3. Break-Even: Muhammad’s Surf Shop

  • $1 million/$400,000 = 2.5-Year Payback Period 

4. Net Present Value: Sherri’s Gym

  • NPV = [$500,000/(.1-0)] - $2 million = $3 million

For a broader overview of the consulting industry, see Consulting Blogs: An Overview of the Consulting Career

For an example consulting case and case interview tips, see Consulting Blogs: How to Ace the Case (with Examples)