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 Stu James and Management by the Numbers, Inc.

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 G ROWTH RATE CONCEPTS C OVERED 2 Growth Rate Concepts Covered MBTN | Management by the Numbers This MBTN Module covers the following concepts:

G ROWTH 3 Growth MBTN | Management by the Numbers 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 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. 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.

C ALCULATING G ROWTH R ATES (%) 4 Calculating Growth Rates (%) MBTN | Management by the Numbers 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: 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

G ROWTH R ATES - E XAMPLE 5 Growth Rates - Example MBTN | Management by the Numbers Question 1: In 2010, Bill’s BBQ’s sold 25,000 Big Burp sandwiches at 5 retail locations. In 2011, he sold 36,000 of these sandwiches from 6 retail locations. Bill wants to know the growth in sales (units), the growth rate (%) in sales, and the growth rate in retail locations. Answers: Sales Growth (000s) = 2011 sales – 2010 sales = 36 – 25 = 11 Sales Growth (%) = ( ) / 25 = 11 / 25 = 44% Location Growth (%) = (6 - 5) / 5 = 1 / 5 = 20% 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 He can also say that, on average, sales per retail location grew because the overall sales growth rate is greater than the location growth rate. But perhaps there is more to the story as the next example illustrates.

G ROWTH R ATES - E XAMPLE 6 Growth Rates - Example MBTN | Management by the Numbers Question 2: We know that Bill’s BBQ’s sold 25,000 Big Burp sandwiches at 5 retail locations. In 2011, he sold 36,000 of these sandwiches from 6 retail locations for 44% overall growth in unit sales. Here are his sales broken down by retail establishment: 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 With this additional information, the outlook is quite different. Store #2010 Sales2011 Sales 16,5005,500 23,0002,500 35,0004,000 48,0007,000 52,5008,000 69,000 Based on these values by store location, what is Bill’s same store growth rate?

V ARIATIONS ON A T HEME 7 Variations on a Theme MBTN | Management by the Numbers 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 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 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, 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)

C ONTEXT FOR G ROWTH R ATES 8 Context for Growth Rates MBTN | Management by the Numbers 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. 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.

G ROWTH R ATES - E XAMPLE 9 Growth Rates - Example MBTN | Management by the Numbers 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 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? Answers: GDP Growth Rate= / – 1 = 10.4% GDP Per Capita (2010)= 4,060 / = $3,415 (both in billions) GDP Per Capita (2009)First calculate India’s population in 2009 = 1,189,172,906 / ( ) = 1,173.4 mil. people Then divide 2009 GDP by 2009 population = 3,679 / = $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.

M ULTI -P ERIOD G ROWTH R ATE C ALCULATIONS 10 Multi-Period Growth Rate Calculations MBTN | Management by the Numbers 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 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 = ( ) / 4 =.20 / 4 =.05 or 5%

CAGR (C OMPOUND A NNUAL G ROWTH R ATE ) 11 CAGR (Compound Annual Growth Rate) MBTN | Management by the Numbers 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) 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. Question 5b: Tom’s Stock Portfolio had the following values on Jan 1 st of : 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 = or -1.3%

A VERAGE A NNUAL R ETURN VS. CAGR 12 Average Annual Return vs. CAGR MBTN | Management by the Numbers 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. 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.

C ONVERTING G ROWTH R ATES FOR D IFFERENT P ERIODS 13 Converting Growth Rates for Different Periods MBTN | Management by the Numbers 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 Conversely: Annual rate = (1 + monthly rate) ^ (12) – 1 Annual rate = (1 + quarterly rate) ^ (4) – 1

C ONVERTING G ROWTH R ATES FOR D IFFERENT P ERIODS 14 Converting Growth Rates for Different Periods MBTN | Management by the Numbers Question 6a: Renee’s Tofu Delight has set a target goal to double sales in 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 = ( %) ^ (1/12) – 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?

C ONVERTING G ROWTH R ATES FOR D IFFERENT P ERIODS 15 Converting Growth Rates for Different Periods MBTN | Management by the Numbers 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 = ( )^.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.)

Marketing Metrics by Farris, Bendle, Pfeifer and Reibstein, 2 nd edition, pages And - Many MBTN Modules include the use of growth rates as it is a very common business calculation. F URTHER R EFERENCE 16 Further Reference MBTN | Management by the Numbers