Tricky Finance-related Question

Question

Suppose I have acquired 103,304 customers for a transaction-based business at 12 months into the business and their number has increased at a monthly rate of 40% (from month 1, in which 2,551 customers were acquired). Assume 103,304 represents the total number of new customers (active or dead) acquired over the 12 month period.

If there is one fixed-price product/service worth $10 that each customer makes 4 purchases of across a 6 month period (1 every 1.5 months) and then ceases to be a customer, how can I calculate my total revenue by month 12?

I know that I'm something of an interloper in this forum, but I'll absolutely accept an answer and up-vote all good ones.

Answer 1

You don't give enough data to find an answer. Each customer is worth 40 in revenue, so if they all signed up by month 6 (and therefore already spend their 40) you have 4M. If they all signed up in month 11.5 and haven't bought yet you have zero. You need something about the rate of increase of customers and about when in the six months they buy.

It would simplify things to assume they spend evenly over time, so every customer is worth 40/6 per month. Then if n(i) is the number of customers who join in month i, the number of customers active in month i is the sum over the last six months of n(i) and your revenue will be 4M less what you will get from the current customer base in the next months.

Answer 2

Well, the easiest way is to write a program or create a spreadsheet to answer the question. But if you insist on doing it using just math...

If the number of customers you acquire in the zeroth month is $C_0$, and the growth factor is $a$, then in month $n$ you acquire

$$C_n = a^nC_0$$

new customers. Therefore the total number of customers acquired is

$$C_{Tot} = \left( 1+a+a^2+\cdots+a^{11}\right)C_0 = \frac{a^{12}-1}{a-1} C_0$$

You said that $C_{Tot}\approx 100,000$ and $C_0\approx 2,500$, so the numbers work out if you have monthly growth in signups of about 20% (i.e. $a=1.2$).

You're assuming that each customer makes 4 payments of $10, one every 1.5 months, and then leaves. That means that once you've received all the payments, you expect to have received

$$$40 \times C_{Tot} \approx $4,000,000$$

However, the customers that arrive in the final month only make one payment, so you have to subtract

$$$30 \times a^{11}C_0 \approx $550,000$$

The customers arriving in month 10 only get a chance to make two payments (one when they arrive, one after 1.5 months) so you have to subtract

$$$20 \times a^{10}C_0 \approx $300,000$$

The customers in months 8 and 9 only get time to make three payments, so you have to subtract

$$$10 \times (a^8 + a^9) C_0 \approx $230,000$$

Therefore your total expected revenue after 12 months is around $2,900,000.