Optimizing a "new customer" bonus is an easy way for a business to increase profits, especially in industries known to exhibit customer loyalty.

At a time in my life when it was worth doing so, I repeatedly took advantage of a bank's $50 bonus offer for new customers. I'd maintain the minimum balance for the required period, collect the reward, close my account, and then wait for the postman to hand me ... the same offer. Same story with a credit card offering airline miles.

If not to sustain my social life one notch above "do not resuscitate" during grad school, then what was this bank trying to achieve? And were they achieving it?

If they were trying to impress their board of directors with an influx of new signups, they almost certainly did. But if they were trying to increase their profits by increasing the number of customers who eventually stay long enough to pay annual fees, I can't tell if they were successful.

## Analyzing "new customer" bonuses

We'll show how a business can increase its profits by adjusting the size of its "new customer" bonus (or discount or offerâ€”the name doesn't matter; each can be translated to dollars).

(True, a business might not be trying to optimize profits at the moment, and might be pursuing market share, or some combination of the two. That's fine, and these situations will be handled in other posts.)

### How the size of a "new customer" bonus affects profit

Intuitively, as the size of the bonus increases, so does the chance that a prospective customer (for example, a visitor to your store) converts and contributes revenue. Great, more revenue! But on the other hand, it would be silly to increase the bonus so high that it completely negated that expected revenue.

So the question is: what size bonus leads to the biggest expected profit, i.e. the largest gap between the expected revenue and the cost of acquiring that revenue?

Let's break this down and state it more concisely.

#### Profit associated to an existing customer

By definition, the profit associated to an existing customer is the difference between their lifetime revenue (CLR) and the cost of acquiring that customer (CAC), part of which is the "new customer" bonus paid upon conversion, and part of which is of marketing costs paid to attract prospective customers prior to conversion:

profit = CLR - bonus - marketing

#### Expected profit associated to a prospective customer

Then, the expected profit for a prospective customer is that profit above, but with the revenue and bonus multiplied by the probability that the prospect converts to an actual customer:

expected profit = P(conversion | bonus) x (CLR - bonus) - marketing

We can see that a prospective customer's expected profit depends on the size of the bonus in two places: the probability of conversion, and as a portion of the acquisition cost. We wish to vary the bonus up or down until the expected profit is maximized.

Technical note: For this round, we'll make these simplifying assumptions:

Customer lifetime revenue is constant. That is, we assume that all customers are essentially cut from the same cloth, no matter the size of the discount that enticed them to convert. This assumption implies that an existing customer would not be moved by the prospect of receiving another "new customer" bonus to quit early. (Unlike with the bank.) It also assumes that the effects of retention bonuses or discounts (aimed to please existing rather than prospective customers) have already reached an equilibrium, and now have only negligible impact on a customer's lifetime revenue.

Conversion probability depends only on the size of the discount offer. This implies that, for example, network effects (social pressures that might also influence a prospective customer to convert) are negligible. It also implies that a churned customer's propensity to convert again later would be no different than the first time around (also unlike with the bank, where I felt increased propensity to sign up again).

Businesses for which customer loyalty is not a phenomenon, such as when the goods being sold are fungible, might not satisfy the first assumption. And businesses that exhibit strong network effects might not satisfy the second assumption. We will account for those phenomena in later posts.

### Finding the bonus size that maximizes expected profit

Here are the steps for increasing your expected profits on prospective customers, just by adjusting the size of the "new customer" bonus.

#### Step 1. Estimate customer lifetime revenue (CLR).

For example, we might estimate that an average customer contributes $1000 in revenue across all time.

Say an average customer is worth $1000 in revenue.

Here's one way to get there.

Since it's impractical to wait for several lifetimes to finish in order to calculate this average, it's common to estimate CLR from a shorter-term proxy, such as average monthly revenue measured after 2-3 months, divided by the monthly churn rate.

For instance, if an active customer averages $100 per month (say, as measured over three months), and an active customer has a 10% chance of being inactive for the next monthâ€”a decent definition of churn for some businesses, then we can apply a formula to calculate the estimated customer lifetime revenue: $100/10% = $1000.

#### Step 2. Measure the conversion rate for three different bonus sizes.

For example, we might measure that prospective customers who were offered

no bonus convert at 5%,

a $50 bonus convert at 7%,

a $100 bonus convert at 8%.

Conversion might start at 5%, and grow to 8% if there's a $100 "new customer" bonus.

Note that we could have started with only two bonus sizes to identify a winner, but the advantage of starting with three is that we will also learn which which direction to explore nextâ€”higher or lower than the winning bonus.

Here are some tips and caveats for collecting this data.

Ideally, we can cleanly divide our pool of prospective customers into groups that look the "same" as one another, and then assign a distinct "new customer" bonus to each group. The groups don't have to be the same size, but each should be large enough that we can confidently measure the conversion rate.

To promote uniformity between the groups, it's best to run the different bonuses concurrently, eliminating the possibility that any group is also affected by seasonality.

Of course, there might be other interference that we're explicitly ignoring for now. For example, there might be network effects, in which a prospective customer's behavior is influenced by external social connections (say, friends who got this or that bonus). Or it might be hard to ensure that a customer is placed only in one group, and shown the same bonus on each successive visit.

#### Step 3. Estimate the expected the profit associated to each bonus.

This step is just math.

Suppose marketing costs, such as ad spend that results in folks coming to your shop, average $15 per prospective customer. Then using the CLR and conversion rates above, the expected profit for a prospective customer who was offered

no bonus is 5% x ($1000 - $0) - $15 = $35,

a $50 bonus is 7% x ($1000 - $50) - $15 = $51.50,

a $100 bonus is 8% x ($1000 - $100) - $15 = $57.

Then expected profit grows from $35 to $57 if there's a $100 bonus.

The $32 increase in expected profit associated to a $100 "new customer" bonus, over no bonus, is as increase of 63%!

#### Step 4. Follow the curve, and optimize.

Why stop there?

The upward trend in our three data points suggests that we can increase expected profit further by increasing the bonus. (Of course, we might have observed that the middle bonus, or the leftmost bonus, was best, and our next choice would be in a different direction.)

Why stop there? Our model suggests expected profit could hit $69.50â€”an increase of 99%â€”if the right bonus is offered!

To perform the optimization, we could explore by stepping incrementally through more bonus size, or we could try to skip directly to an approximately optimal bonus, making use of a rough analytical relationship between bonus and conversion.

For example, if we guessed that beyond $50, each additional $50 added to the bonus increased the conversion rate by one percentage point (until conversion hits 100%), then a bit of math yields

conversion = .0002 x bonus + .06

expected profit = (.0002 x bonus + .06) x ($1000 - bonus) - $15

which is a parabola peaking at bonus = $350 and expected profit = $69.50, accompanied by a conversion of 13%!

Ponder this. Assuming our estimates are sound, we could conceivably expect a 99% increase in expected profit from prospective customers (a jump from $35 to $69.50) and a 160% increase in conversion rate (a jump from 5% to 13%)! We win on profits and conversion.

All that said, given that we're extrapolating this speculative relationship quite far to the right, we might want to get there in stages. Or maybe just go for it. If we're wrong, we'll know as soon as we can confidently estimate conversion in that range. And if we're right, we just increased expected profits on prospective customers by a ton!

## Coda

Oh yeah, about the bank. What should they have done differently? They evidently did not inspire customer loyalty, which was one of the stated assumptions in this article. In fact the way they executed their "new customer" bonus not only incentivized me to churn early thereby contributing less revenue, but also to have an artificially high conversion probability on subsequent visitsâ€”a sort of network effect. Enforcing "one offer per SSN" should've done the trick.

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