Growth Rates - Management By The Numbers

```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.
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
GROWTH RATE CONCEPTS COVERED
Growth Rate Concepts Covered
• Multi-period growth rates
• Average annual return vs. CAGR
• Converting growth rates
between different time periods
rates from
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GROWTH
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.
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.
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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 (%)
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
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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.
GROWTH RATES - EXAMPLE
Growth Rates - Example
Sales Growth (000s) = 2011 sales – 2010 sales = 36 – 25 = 11
Sales Growth (%)
= (36 - 25) / 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 2010. 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.
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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:
Based on these values by store
location, what is Bill’s same store
growth rate?
Store #
1
2
3
4
5
6
2010 Sales
6,500
3,000
5,000
8,000
2,500
2011 Sales
5,500
2,500
4,000
7,000
8,000
9,000
GROWTH RATES - EXAMPLE
Growth Rates - Example
Growth Rate %
(Same Stores)
= 2011 sales from stores open in 2010 / 2010 Sales -1
= (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.
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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
Value t
VARIATIONS ON A THEME
Variations on a Theme
= (1 + Growth Rate %) * 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?
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)
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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
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.
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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
Growth Rates - Example
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.
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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 Y1 + Return Y2 +… + Yn ) / n
MULTI-PERIOD GROWTH RATE CALCULATIONS
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%
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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)
-1
CAGR =
(don’t forget the -1)
Beginning Value (X)
CAGR (COMPOUND ANNUAL GROWTH RATE)
CAGR (Compound Annual Growth Rate)
Question 5b: Tom’s Stock Portfolio had the following values on Jan
1st 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%
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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.
Visualize Actual vs. CAGR
1200
1000
800
600
400
200
0
Actual
AVERAGE ANNUAL RETURN VS. CAGR
Average Annual Return vs. CAGR
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.
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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
Converting Growth Rates for Different Periods
Conversely: Annual rate = (1 + monthly rate) ^ (12) – 1
Annual rate = (1 + quarterly rate) ^ (4) – 1
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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?
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 1st
quarter growth rates?
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CONVERTING GROWTH RATES FOR DIFFERENT PERIODS
Converting Growth Rates for Different Periods
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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.)
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CONVERTING GROWTH RATES FOR DIFFERENT PERIODS
Converting Growth Rates for Different Periods
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Marketing Metrics by Farris, Bendle, Pfeifer
and Reibstein, 2nd edition, pages 125-129.
FURTHER REFERENCE
Further Reference
- And Many MBTN Modules include the use of
growth rates as it is a very common business
calculation.
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