So this is a bit of a mix of a math question and a valuation one.
I'm trying to place a dollar value on an average customer for business purposes (to determine how much we should pay in ads to acquire them). Simple enough, but the business is a weird one where customers can actually go on to recruit other customers on their own (in theory, infinitely so).
Each customer brings in around $2,000 in purchases (on average), but each customer also has a 1% chance of converting into a recruiter, which on average brings in 4 new clients.
What's a good mathematical approach to doing a valuation here? From a high-level, it's:
- Value of purchases ($2,000) +
- Expected Lifetime Value as a recruiter
I started by simply adding the 2000 to the probability adjusted recruiter value (2,000 x 4 x 1%) but then you run into the fact the 4 potential additional customers generated will generate more customers, and those even more, and so on....
Is there a way to limit this and place a realistic mathematical valuation on a customer in this scenario? Any advice would be greatly appreciated.
The new potential 4 customers have again 1% chance of bringing in 4 new customers each so at this stage you can potentially bring in 16 customers with $(0.01)^2 = 0.0001$ chance. If you keep doing this you expect on average:
$$\begin{eqnarray}1+4\times 0.01 + 4^2\times (0.01)^2 + 4^3\times (0.01)^3 + 4^4\times (0.01)^4 + \ldots \\ = 1+ 0.04 + (0.04)^2 + (0.04)^3 + (0.04)^4 + \ldots \end{eqnarray}$$
which is a geometric series that converges to about 1.04167. This valuates an incoming client at about 2083.33. Note that this is like calculating an annuity.
Presumably, the customer will not bring in those new customers in instantly. So, you should account for the time value of money as well. Suppose the interest rate is now 0.1% monthly. Then the value of the 4 new clients of the next period, provided we assume that the 4 new clients brought in by the preceding client come a month later will be
$$2000 \times 4 \times (0.01)/(1.001) \approx 2000 \times 4 \times 0.00999 \; .$$
Doing the whole computation over from the beginning with the adjusted "recruitement" increase:
$$2000 \times (1+ 0.03996 + (0.03996)^2 + (0.03996)^3 + (0.03996)^4 + \ldots )$$
will give you about 2083.25, which is not so far off from the previous value because I assumed interest rates to be rather low.