First of Two Types of Percent: Relative Percentage Conversion or Efficiency Rates, %i Ted Mitchell

The two types of percents 1) Those dealing with the size of the final state, F, of a variable relative to the original size of the state, I, of the variable. F = F/I x I F= %i x I Your initial weight was I=150 pounds and your current weight is %i=110% of your initial weight. You current weight is F=165 pounds. 2) Those dealing with the size of the change in the original variable relative to the original size of the variable (F-I) = (F-I)/I x I ∆I = %∆i x I Your initial weight was I =150 pounds and you had a change in weight of %∆i =10%. The change from your initial weight is ∆I =15 pounds.

This set of slides deals with the first type 1) Those dealing with the size of the final state, F, of a variable relative to the original size of the state, I, of the variable. F = F/I x I F= %i x I %i is the ratio of the Size of the Output to the Size of the Input and is often called the percent efficiency of the process I have 40% efficiency because 40% of my sales calls result in transactions

Working With Relative Percentage Conversion Rates Relative Percentage Conversion Rates are those dealing with the ratio of the size of the final state, F, of a variable relative to the initial size, I, of the variable. F = F/I x I F= %i x I My firm’s sales are 30% of the total sales in the industry

Working With Relative Percentage Conversion Rates Relative Percentage Conversion Rates can be considered to the percent conversion rate of a Two-Factor Model in which the Input and the Output variable are measured with the same metric F = F/I x I F= %i x I Output: Size of the Final State = (the percent conversion rate or conversion efficiency, %i) x (Input: Size of the Initial State

In Marketing Management There are many popular relative conversion rates that have inputs and output using the same metric: Customer Retention rate: Loyal Customers, L = % T x Total Customers Market Share rate: Firm’s Sales, R = % I x Total Industry Sales, R I Sales Revenue Returned on Advertising rate: $ of Revenue, R = % A x $ of Advertising Expense, A

Marketing Outputs that are Relative amounts of Inputs 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

Relative Percentage Conversion Rate %i = (Output, Final Value, F) / (Input, Initial Value, I) Reflects a reduction or shrinkage process when the relative percentage conversion is between 0% and 100% (0 < %i < 1) Current revenue is 80% of last year’s revenues Reflects a growth or expansion process when the relative percentage conversion is greater than 100% ( %i > 1) Sales Revenue Returned on Promotion ‘Investment’ or expenditure is 500%

Relative Percentage Conversion problems come in Three flavors Percentage Problems are Mathematical Identities You need to know all three components to properly forecast with a Two-Factor Percentage Machine or compare 2 Performances If you know 2 of the 3 components you can calculate the third!

1) Calculate Output or Final State, F Biz-Café has a marketing discount machine. You put the current price tag into the Machine and it creates a new price tag through a reduction process that makes the new tag 90% of the current tag. Your current price tag is $4. What is the value of the Input? Current Price tag = $4 What is the conversion rate (reduction rate)? %i = 90% reduction rate What is the value of the new price tag? New Price tag, F = 0.9 x $4 = $3.60

2) Calculate the Percentage Rate or Ratio of Conversion, %I You have observed the performance of a discount machine. You saw the Output, O, was a new price tag of $4 You saw the Input, I, was a price tag of $5 What is the discount machine’s conversion rate? O/I = $4/$5 = 80 cents per every dollar of price tag Or 80 cents on the dollar Or an 80% relative reduction The conversion rate as a percent is value free and has no context

3) Calculate the Input or Initial state, I Biz-Café reviews last week’s decision to reduce the selling price of its coffee. The newly determined selling price is F=$3.50 a cup. The new price is 70% of the old price. What was the old price, P? Output, F = (conversion rate, %i) x Input, I New Price, F = %p x Old Price, P Old Price, P = (New Price, F) / %p Old Price, P = $3.50/0.70 = $5.00 The Old Price, P, was $5.00 a cup

The Three Types are That can be asked about using simple percentages described as the efficiency or the conversion or the rate factor in a two factor model Output = Conversion Factor x Input Factor Type 1) Given the Rate of the Conversion Factor, %i, and the size of the Input Factor, I, determine the size of the Output, O Type 2) Given the size of the Output, O, and the size of the Input Factor, I, determine the Rate of Conversion as a percent %i Type 3) Given the Rate of Conversion Factor, %i, and the Output, O, determine the size of the Input, I

The Three Types In equation form they are Type 1) find the output, O = %i x I is the classic Slope-Origin form of the percentage equation Type 2) find the rate, percent or ratio of conversion, %i = O / I is the common calculation for the percentage rate from an observation of O and I measured using the same metric ($, customers, transactions, etc.) Type 3) find the input, I = O / %i

We use percentage rates because They are convenient for discussion purposes because they are context free However you can maintain the original context Remember a percent is a ‘per centum’ A 60% customer retention rate can be stated as 60 loyal customers per 100 customers served A 5% interest rate can be stated as a return of $5 for every $100 invested

Questions? Can You 1) Define a traditional Percent in terms of the value of initial base and final output? 2) Describe the Three Elements of a Two-Factor Model of Percentage Conversion Machine? 3) Transform traditional “Math Questions” involving percent into problems in a business context? 4) Can you add context to percentage questions by maintaining the metrics for input and output?