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 Conversion Process Efficiency = Output/Input Crank Handle Output $ $ $ $ $

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 Machine $ Input is Advertising Dollars $ $ $ Conversion Process Efficiency = Output/Input = $R/$A =conversion percent Crank Handle Output $ $ 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 Advertising Advertising 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 Machine $ Input is Advertising Dollars, A $ $ $ Conversion Process Efficiency = Output/Input = (R–A)/A = size of conversion percent Crank Handle Output Output is Dollars of Profit From Advertising Profit = 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 profit Advertising 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 sold and described as a percent difference from the Input Output, ∆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