CPAs are rates not costs

My thoughts on Cost per Acquisition.

In marketing analytics there are two numbers that matter:

• ROI (Return on Investment)
• CPA (Cost per Acquisition)

ROI is intuitive to think about: it’s a revenue multiplier on spend (e.g. you’ll spend x you’ll get 1.2x back), CPA is not. The way we talk about CPA in marketing is as a cost: a price-tag on a conversion. If we pay the cost, we’ll get a conversion, just like if we pay $3 we can get an egg at Trader Joe’s.

an egg with a price tag

an acquisition with a price tag

But CPA is really a rate. How many exposures do we need to pay for before we get a conversion? Let’s simplify: an exposure costs $0.10. A CPA of $3 means that on average it took 300 exposures before we got a conversion.

Say we spent $1,000 per day for 30 days and saw 0 conversions. When we think about CPA as a cost we might see the $1,000 in spend per day and 0 conversions and conclude that our CPA is probably higher than $100. It’s like the world’s worst store, we walked up to the till each day, paid $1,000, hoped that was more than the price-tag on a conversion, and left disappointed. But, when we think about CPA as a rate we see $1,000 worth off exposures every day over 30 days with 0 conversions, a much bleaker picture.

Framed as a rate, what we saw was 300,000 exposures ($30,000) with 0 conversions. The cost per conversion is likely more than $30,000 (not $1,000). If the CPA was $1,000 we should see a conversion every 10,000 exposures or so, but across 300,000 exposures we saw 0.

library(ggplot2)
library(patchwork)

plot_binomial <- function(n, p, min, max) {
  k <- 0:n
  prob <- dbinom(k, size = n, prob = p)
  
  df <- data.frame(k = k, prob = prob)
  
  ggplot(df, aes(x = k, y = prob)) +
    geom_histogram(stat = "identity", fill = "#56B4E9") +
    labs(
      title = "Expected Conversions",
      subtitle = sprintf("(total exposures = %.0fK, p(conversion) = %.6f)", n/1000, p),
      x = "number of conversions",
      y = "P(X = k)"
    ) +
    theme_minimal(base_size = 13) +
    theme(
      panel.grid.major = element_blank(),
      panel.grid.minor   = element_blank(),
      plot.title         = element_text(face = "bold"),
      plot.subtitle      = element_text(color = "grey50"),
      axis.text.y = element_blank(),
      axis.ticks.y = element_blank()
    ) + 
    xlim(c(min,max))
}

# example call
x <- plot_binomial(n = 300000, p = 1/1000,240,360)
y <- plot_binomial(n = 300000, p = 1/300000,0,10)
x/y

Figure 1: Expected conversions over 30 days of exposures (10,000 per day: $1,000 worth at $0.10 each)

This framing also explains why we can have a CPA of $100 and still see 3 conversions on $150 of spend (we just got lucky).

library(ggplot2)
library(patchwork)

plot_binomial <- function(n, p, min, max, hl = 3) {
  k <- 0:n
  prob <- dbinom(k, size = n, prob = p)
  
  df <- data.frame(k = k, prob = prob)
  df$highlight <- ifelse(k == hl, "yes", "no")
  
  ggplot(df, aes(x = k, y = prob, fill = highlight)) +
    geom_histogram(stat = "identity") +
      scale_fill_manual(values = c("yes" = "#009E73", "no" = "#56B4E9"), guide = "none") +
    labs(
      title = "Expected Conversions",
      subtitle = sprintf("(total exposures = %.1fK, p(conversion) = %.6f)", n/1000, p),
      x = "number of conversions",
      y = "P(X = k)"
    ) +
    theme_minimal(base_size = 13) +
    theme(
      panel.grid.major = element_blank(),
      panel.grid.minor   = element_blank(),
      plot.title         = element_text(face = "bold"),
      plot.subtitle      = element_text(color = "grey50"),
      axis.text.y = element_blank(),
      axis.ticks.y = element_blank()
    ) + 
    xlim(c(min,max))
}

# example call
x <- plot_binomial(n = 150/0.1, # 0.10 per conversion
                   p = 1/(100/0.1), # expect conversion every 100/0.1 exposures
                  0,10)
x

Figure 2: Expected conversions for $150 of spend with a true CPA of $100

Marketing spend is more complicated than this. We don’t always pay a flat rate for exposures and exposures are not the same quality as we scale spend, but the framing stands. We should encourage people to talk about CPAs as rates, not as costs because it makes interpreting CPAs more intuitive. It also highlights the mechanisms for improving CPA:

• improve the rate of conversions (e.g. better targeting, quality creative)
• reduce the cost of an exposure (idk, bully Meta a little?)

AI disclosure

🤖 Used AI for: AI used for blog template and plotting code.

🧍🏻‍♀️Did not use AI for: ideas and written blog content.