Calculating Growth Rates This module teaches how to calculate various measures of growth including simple growth, growth rates based on two changing variables, average annual return, compound average annual return, and converting effective growth rates from one time period to another. Author: Stu James 2011-2017 Stu James and Management by the Numbers, Inc.
Growth Rate Concepts Covered This MBTN Module covers the following concepts: Why are growth rates important? How to calculate growth and growth rates Important business contexts for use of growth rates Combining growth rates Multi-period growth rates Average annual return vs. CAGR Converting growth rates rates from between different time periods GROWTH RATE CONCEPTS COVERED 2
Growth Growth is a measure of change from one time period to another. This is important because growth provides insight into the dynamic nature of what one is measuring, in our context, a business or a market. GROWTH Definition Growth ($ or units) is just the change from time period t -1 to time period t. Growth (or change) = Value t Value t-1 Insight While this is very basic, notice that growth indicates direction (positive or negative i.e. growing or shrinking) which, in itself, has major implications. Consider the difference between a growing business and a shrinking business one might be considering hiring people, the other, laying off people. This is but one example of why growth is so important to us. 3
Calculating Growth Rates (%) While Growth, or change in a value, is helpful, it does leave out the important context of how much that change represents relative to the size of what you are measuring. In our previous example, we might calculate that sales grew by $1 million. If the business is a start-up, that might be a huge change for the organization, but in the case of a Fortune 500 company, it might be considered status quo (or a rounding error!). So, most of the time, we measure growth as percentage change relative to the base time period as shown below: CALCULATING GROWTH RATES (%) Definition Growth Rate (%) is the % increase or decrease in value from the intial time period t -1 to time period t, compared to the initial value. Value t Value t-1 Growth Rate % = or (Value t / Value t-1 ) - 1 Value t-1 4
Growth Rates - Example Question 1: In 2010, Bill s BBQ s sold 25,000 Big Burp sandwiches at 5 retail stores. In 2011, he sold 36,000 of these sandwiches from 6 retail stores. Bill wants to know the growth in sales (units), the growth rate (%) in sales, and the growth rate in number of retail stores (%). Answers: Sales Growth (000s) = 2011 sales 2010 sales = 36 25 = 11 Sales Growth (%) = (36-25) / 25 = 11 / 25 = 44% Store Growth (%) = (6-5) / 5 = 1 / 5 = 20% GROWTH RATES - EXAMPLE Insight Again, just from these simple calculations, Bill learned several things. First that his BBQ business is growing and that he probably needed approximately 44% more ingredients in 2011 than in 2010. He can also say that, on average, sales per retail location grew because the overall sales growth rate (%) is greater than the store growth rate (%). But perhaps there is more to the story as the next example illustrates. 5
Growth Rates - Example Question 2: We know that Bill s BBQ s sold 25,000 Big Burp sandwiches at 5 retail stores. In 2011, he sold 36,000 of these sandwiches from 6 retail stores for 44% overall growth in unit sales. Here are his sales broken down by retail establishment: Based on these values by store location, what is Bill s same store growth rate? Store # 2010 Sales 2011 Sales 1 6,500 5,500 2 3,000 2,500 3 5,000 4,000 4 8,000 7,000 5 2,500 8,000 6 9,000 GROWTH RATES - EXAMPLE Answer: Growth Rate % = 2011 sales from stores open in 2010 / 2010 Sales -1 (Same Stores) = (5.5K+2.5K+4K+7K+8K)/(6.5K+3K+5K+8K+2.5K) = 27,000 / 25,000 1 =.08 or 8% So we could say that 8% of Bill s growth is due to same store sales, and 36% is due to opening the new store. Bill also knew that he expanded store #5 at the end of 2010. With this additional information, the outlook is quite different. 6
Variations on a Theme Given a growth rate and the actual value in one of the time periods, you can also calculate the unknown value in the other time period as shown in the definitions below: Definition VARIATIONS ON A THEME Value t+1 = (1 + Growth Rate %) * Value t Value t = Value t+1 / (1 + Growth Rate %) Question 3: Renee s Tofu Delight sold 10,000 bbq tofu dogs in 2011. This represented a 50% increase over the previous year. How many tofu dogs did she sell in 2010? And, if sales grew by the same % in 2012, how many tofu dogs would she forecast to sell in 2012? Answers: 2010 tofu dogs = 10,000 / (1 +.50) = 6,666 2012 tofu dogs forecast = 10,000 * (1 +.50) = 15,000 Note that the growth % is the same, but the unit growth is not (3,334 vs. 5,000) 7
Context for Growth Rates So, where are growth rates used? The short answer is everywhere! But, in the context of business, growth rates are especially prevalent in finance, marketing and economics. Some typical examples include: Finance/Accounting: Average annual return, revenue, expense or profit growth, earning per share growth, dividend growth rate. Marketing: Market growth rate, same store sales growth, sales growth (overall or by product, segment, salesperson, channel, etc). Economics: GDP, population, per capita growth rates, inflation, unemployment, growth in money supply, etc. CONTEXT FOR GROWTH RATES Insight Often, a calculated growth rate will actually be based on two changing streams of data. Per capita growth rates (growth per person) are an excellent example of such a calculation. Generally, we do this to translate macro trends to a more micro level. Let s try one for fun. 8
Growth Rates - Example Question 4: In 2010, India s GDP was $4.060 trillion. In 2009 it was $3.679 trillion. India s population in 2010 was 1,189,172,906 people which represented a population growth rate of 1.344% over 2009. What was India s growth rate (%) in GDP from 2009 to 2010? What was India s GDP per capita* in 2009 and 2010? What was India s growth (%) in GDP per capita from 2009 to 2010? GROWTH RATES - EXAMPLE Answers: GDP Growth Rate = 4.060 / 3.679 1 = 10.4% GDP Per Capita (2010) = 4,060 / 1.189 = $3,415 (both in billions) GDP Per Capita (2009) First calculate India s population in 2009 = 1,189,172,906 / (1 +.01344) = 1,173.4 mil. people Then divide 2009 GDP by 2009 population = 3,679 / 1.1734 = $3,135 = 2009 GDP per capita GDP Per Capita Growth = (3,415 3,135) / 3,135 = 8.9% * Per capita means per person. So GDP per capita = GDP / population. 9
Multi-Period Growth Rate Calculations There are two primary ways to describe growth that spans several periods, but where it makes sense to normalize the rate to a single period average. The two approaches to consider for a yearly basis are Average Annual Return and Compound Annual Growth Rate (CAGR). While these sound more or less the same, there are important differences between the two. Definition Average Annual Return = Average of a series of yearly annual returns (Y) = (Return Y 1 + Return Y 2 + + Y n ) / n MULTI-PERIOD GROWTH RATE CALCULATIONS Question 5a: Tom s Stock Portfolio had the following yearly returns: 2007 (10%), 2008 (-40%), 2009 (-10%), and 2010 (60%). What is the average annual return of his portfolio? Average Ann. Return = (.10 -.40 -.10 +.60) / 4 =.20 / 4 =.05 or 5% 10
CAGR (Compound Annual Growth Rate) Now let s consider the definition of CAGR. CAGR is used to calculate the annual return (often of an investment) over a given time frame. Definition Compound Annual Growth Rate (CAGR) is the annual return necessary to grow a value from X to Y over a period of N years. Ending Value (Y) ^ (1 / N Years) CAGR = - 1 (don t forget the -1) Beginning Value (X) CAGR (COMPOUND ANNUAL GROWTH RATE) Question 5b: Tom s Stock Portfolio had the following values on Jan 1 st of 2007-2010: 2007 ($1,000), 2008 ($1,100), 2009 ($660), 2010 ($594), and 2011 ($950). What is the CAGR of his portfolio? CAGR = (950 / 1,000) ^ (1 / 4) - 1 =.95 ^.25 1 = -.013 or -1.3% 11
Average Annual Return vs. CAGR What you may not realize is that Tom s portfolio is exactly the same in both questions, the only difference is how we choose to measure the average return. It may help to visualize the value of his portfolio as shown at the right. 1200 1000 800 600 400 200 0 Visualize Actual vs. CAGR AVERAGE ANNUAL RETURN VS. CAGR Actual CAGR Insight The difference between the two rates calculations is one way to get an idea of the variability or volatility in the rates of return, or, in other words, the risk. 12
Converting Growth Rates for Different Periods Often one will have growth rates for two different time periods which makes it difficult to compare apples to apples. For example, one might want to see how a monthly sales increase compares to a target annual sales growth rate. How can we convert the growth rate to a comparable basis? Definitions To convert from an annual rate to a monthly, quarterly, weekly or daily effective rate: Effective rate for period = (1 + annual rate) ^ (1 / # of periods) 1 Example: Monthly rate = (1 + annual rate) ^ (1/12) 1 Quarterly rate = (1 + annual rate ) ^ (1/4) 1 CONVERTING GROWTH RATES FOR DIFFERENT PERIODS Conversely: Annual rate = (1 + monthly rate) ^ (12) 1 Annual rate = (1 + quarterly rate) ^ (4) 1 13
Converting Growth Rates for Different Periods Question 6a: Renee s Tofu Delight has set a target goal to double sales in 2012. If sales increase at a steady rate through the year, what monthly growth rate is required to meet this target? Answer: Need to convert annual rate to monthly rate. So use Monthly rate = (1 + annual rate) ^ (1/12) 1 Monthly increase necessary to double sales equals Monthly growth rate = (1 + 100%) ^ (1/12) 1 = (1 + 1) ^ (1/12) 1 = 2 ^ (1/12) -1 = 5.9% (approx.) Question 6b: At the end of January, Renee calculates her actual growth month for January (1 month) is 10%. If she is able to maintain this rate of growth throughout the year, what will be her annual and 1 st quarter growth rates? CONVERTING GROWTH RATES FOR DIFFERENT PERIODS 14
Converting Growth Rates for Different Periods Answer: Need to convert monthly rate to an annual rate. So use Annual rate = (1 + monthly rate) ^ (12) 1 = (1 + 10%) ^ 12 1 = 1.10 ^ 12 1 = 213.8% Quarterly rate = (1 + annual rate) ^ (1/4) 1 = (1 + 2.138)^.25 1 = 33.1% Or, from the monthly rate directly Quarterly rate = (1 + monthly rate) ^ (3) 1 = (1 + 10%) ^ 3-1 = 1.10 ^ 3-1 = 33.1% Insight Notice how one can convert any rate to a different time period if you know how many of the smaller time periods there are in the longer time period (e.g. months in a year, days in a month, etc.) CONVERTING GROWTH RATES FOR DIFFERENT PERIODS 15
Marketing Metrics by Farris, Bendle, Pfeifer and Reibstein, 2 nd edition, pages 125-129. - And - Further Reference Many MBTN Modules include the use of growth rates as it is a very common business calculation. FURTHER REFERENCE 16