Putting the Mathematics of Percentage Rates, and Percentage Rates of Change, into the Context

of a Marketing Machine

Ted Mitchell

Learning Objectives

• At the end of this lecture students should be able to• 1) Redefine the traditional mathematical context of a

Percent with its base and proportion in terms of the Three Elements in a Two-Factor Marketing Model

• 2) Understand why a Rate and a Percent should not be used as a Whole Number

• 3) Understand that some rates are value-free decimals and need to be stated as a percent to ensure that the rate is not confused with a whole number

• 4) Transform traditional context-free “Math Questions” about percent into problems in a business context

Two-Factor Models

• Are the best context in which to learn that a value-free rate or ratio is reported as a percent, that outputs are considered the final states of a transformation process, and bases are discussed as the inputs and initial states.

Simple Two-Factor Model• Is to visualize basic marketing operations as simple

machines• Marketing Machines have three elements• 1) Output• 2) Input • 3) Conversion Rate• The Marketing Machine is

Output = Conversion Rate x Input

They are called Two-Factor Models• Because the Amount of the Output is determined by

Two Factors• Factor1) The amount of Input• Factor 2) The rate or efficiency of the conversion• Output = Factor 2 x Factor 1• Output = (Conversion Rate, r) x Input• Conversion Rate is defined as the ratio of Output to

Input• Conversion Rate, r = (Output / Input)• and is written as the rate of the Output per Input

Simple Marketing Machine

Input = one of the 4 P’s

Output

Conversion Process Efficiency =

Output/InputCrank Handle

$$$

$$

Common Concrete Rates of Conversion No Need For Percents or Percentage Changes!

• Output = Conversion Rate x Input• Customer Visits =

Customers per Hour x Hours Open• Quantity sold =

Sales per Salesperson x Number of Salespeople• Quantity sold = Sales per Ad x Number of Ads• Sales Revenue =

Price per Unit x Number of Units Sold• Customer Called Upon =

Calls per Day x Number of Days Worked

When Rates of Conversion • Have the same unit measures in their

Input and their Outputs. then the metrics cancel each other out and the conversion rate is a value free rate

• There is room for confusion• $ Sales Revenue =

(conversion from advertising) x $ advertising

Simple Marketing MachineInput is Advertising Dollars

Output

Conversion Process Efficiency = Output/Input = $R/$A =conversion

percentCrank Handle

$$$

$$

$

$

$ $

Output is Dollars of Sales Revenue

There is room for confusion• $ Sales Revenue = (conversion from advertising) x $ advertising• Conversion rate = Dollars of Sales Revenue / Dollars of Advertising

• Observe the advertising machine$500 of Revenue = (conversion rate) x $200 of Advertising

• Conversion rate = $500/$200 = 2.5• $500 of Revenue = 2.5 x $200 of Advertising• To Prevent the decimal from being treated as a Whole Number we convert

it to a percent

• $500 of Revenue = 250% x $200 of Advertising• Some percentage rates of return or efficiency are very common and have

acquired labels to help prevent confusion

• 250% “Sales Revenues Returned on Advertising” • 250% “Return on Advertising” remains ambiguous

Ambiguity Abounds Due to Two Similar Machines

• 1) With an Output measured as Revenue• Sales Revenue, $R = (% conversion from advertising) x $ advertising, $A

• Revenue, R = (R/A) x Advertising, A• Revenue, R = %A x Advertising, A• 2) With an Output measured as Profit from

AdvertisingAdvertising Profit = (Sales Revenue-Advertising) = R-A

• Notation: (R-A) = ∆A means the size of the difference from $A to $R

• Advertising Profit, (R-A) = ((R-A)/A) x Advertising, A • Advertising Profit, ∆A = (∆A/A) x Advertising, A • Advertising Profit = (%∆A) x Advertising, A

Simple Marketing MachineInput is Advertising Dollars, A

Output

Conversion Process Efficiency =

Output/Input = (R–A)/A = size of conversion percent

Crank Handle

$$$

$$

$

$

$ $

Output is Dollars of Profit From AdvertisingProfit = Revenue – Advertising,

Profit = (R-A)

Ambiguity Abounds Due to Two Similar machines

• 1) with a conversion rate of advertising into dollars of sales revenue

• Sales Revenue, $R = (% conversion from advertising) x $ advertising, $A

• Revenue, R = %A x Advertising, A• Vaguely called ‘Return on Advertising’• 2) with a conversion rate of the advertising into dollars of

advertising profitAdvertising Profit = (Sales Revenue-Advertising) = R-A

• Advertising Profit = (% of $ change from Advertising) x Advertising, A

• Advertising Profit, (R–A) = ((R-A)/A) x Advertising, A• Advertising Profit, (R–A) = (%∆A) x Advertising, A• Also Vaguely called ‘Return on Advertising’

Confusion Yes!

• 1) %A is called “Return on Marketing”Should be called

• Sales Revenue being Returned on Advertising• 2) %∆A is called the “Return on Marketing”

Should be called• Marketing Profit being Returned on Marketing

To Reduce Confusion

• 1) Always report and record value-free rates as percents

• 2) be as specific as you can about the context of the conversion process

• Focus on the output! Ensure you have stated the output as either – The size of the output as a proportion of the input– The size of the output as the difference between

the output and the input

Examples of Outputs that are relatively ‘concrete’ amounts of input

• Output, O = (Output, O)/(Input, I) x Input, I• Output, O = %i x Input, I• Output, 3 = a percent of input x Input, 5

Output, 3 = 60% x Input, 5• Output, $3 Cost = 60% x Input, $5 Revenue• Cost is 60% of Revenue• Output, 3 returning customers = 60% x Input, 5 total customers• 60% Retention rate• Output, 3 sales = 60% x Input, 5 total industry sales• 60% Market Share• Output, 3 satisfied customers = 60% x Input, 5 total customers• 60% satisfaction rate• Output, 3 aware customers = 60% x Input, 5 total customers• 60% awareness level

Two Ways to Measure Output

• 1) The amount of output measured in concrete terms, O = dollars, customer, units sold and described as a proportion of the Input, Output, O described as %I

• 2) The measure the output as the size of the difference between output and input, ∆I = (O-I),

• the difference in dollars, customers, units soldand described as a percent difference from the InputOutput, ∆I described %∆I

Examples of abstract outputs that are the size of the difference from input

• Output, (O-I) = ((O–I)/ I) x Input, I• Output, ∆I = (∆I/I) x Input, I• Output, ∆I = (%∆i) x Input, I• Output ∆I, 3–5 = (percent ∆ from input) x Input, 5

Output ∆I, –2 = (3-5)/5 x Input, 5• Output ∆I, –2 = -40% x Input, 5• -2 total customers = -40% x 5 total customers• Customer Loss Rate, (%∆i) = 40%• -$2 from price= -40% x $5 original price• Coupon Discount Rate, (%∆i) = 40%• -$2 from book value = -40% x $5 original book value• Depreciation Rate, (%∆i) = 40%

Examples of abstract outputs that are the size of the difference from input

• Output, (O-I) = (O–I) / (Input, I) x Input, I• Output, ∆I = ∆I / I x Input, I• Output, ∆I = %∆i x Input, I• Output, 23–20 = (percent ∆ from input) x Input, 20

Output ∆I, 3 = (23-20)/20 x Input, 20• Output ∆I, 3= 15% x Input, 20• 3 customer gain = 15% x 20 total customers• Customer gain rate, (%∆i) = 15%• $3 interest= 15% x $20 principal invested• Interest Rate, (%∆i) = 15%• $3 profit= 15% x $20 sales revenue• Return on Sales (Profit Margin), (%∆i) = 15%

The Two Types of Outputs and Rates

• Are confusing because people in the profession know the context of their conversations

• “That is a great return.”• Without context you don’t know

if ‘great return’ means• the size of dollar gain, ∆I = (F-I) or • the percentage rate of the dollar gain,

%∆I = (F-I)/I

Students like

• Concrete rates, customers per hour• There are no percents, %I, and no percent

differences, %∆I• In marketing there are a large number of

performance measure that use concrete rates• However, the strategic performance measures

are invariably, value-free or ‘context -free’ rates using percents to imply a rate, %I, or a rate of difference, %∆I is being used