Summary Customer Analytics

Contents
Module 1 - CLV in Contractual Settings 1
Geometric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Calculating CLV with the BG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Calculating RLV with the BG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Ruse of heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
CAC and ROI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Shifted Beta Geometric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Calculating CLV with the sBG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Calculating RLV with the sBG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4


Module 1 - CLV in Contractual Settings
Definition: CLV is the PV of future profits of one particular customer
We look at specific cohorts that started at t = 0
Survival function (prob that a customer is still active, thus compared to t = 0)
=
S(t) = P (T > t)

Retention rate (prob that a customer who was active in t − 1 it still active at the end of t, thus compared
to the last period)
=
S(t)
r(t) = P (T > t|T > t − 1) =
S(t − 1)


Geometric Model

Coin-flipping
Probability that a customer leaves (i.e. T ) at period t
P (T = t) = pt−1 (1 − p)
Thus, S(t) = pt
r(t) = p


Calculating CLV with the BG
2 3
E[CLV ] = m + (1+d)
mp
1 + (1+d)2 + (1+d)3 + . . .
mp mp
= m(1+d)
1+d−p
with: retention rate (p) , margin (m) , discount rate (d)




1

,Calculating RLV with the BG

CLV looks at new customers, but RLV looks at customers that did start before t = 0
RLV = already acquired customers = they tend to have lower churn rates than new customers
i.e. for customers with an age > 0 at t = 0
RLV right before renewal:
mp2 mp(1+d)
E[RLV ] = mp + (1+d) + ... = 1+d−p

RLV right after renewal:
You have to discount with one additional /(1 + d) , so it goes away in the upper term
E[RLV ] = mp
1+d−p



Ruse of heterogeneity

In a given cohort, the retention rate increases over time due to heterogeneity = sorting effect


CAC and ROI

Customer Acquisition Cost: to be profitable CAC < CLV =

marketing expenditures
CAC =
customers acquired

Return on Investment = ROI = prof it
cost = CLV −CAC
CAC



Shifted Beta Geometric Model

Model with a distribution of retention rates across customers; we have to write the survival function
in terms of θ
S(t|θ) = pt = (1 − θ)t
With the distribution of θ between 0 and 1 with parameters a and b
Beta Distribution:
θa−1 (1 − θ)b−1
f (θ|a, b) =
B(a, b)

With mean = a
a+b and variance = ab
(a+b)2 (a+b+1)

This gives the new survival function and retention rate:

B(a, b + t) b+t−1
S(t|a, b) = r(t|a, b) =
B(a, b) a+b+t−1

r(t) increases over time, depending on a and b

• small a and b: retention rate rises quickly but levels off quickly
• medium a and b: rate of increase in retention slows down
• large a and b: hardly any increase; almost a constant retention rate
• a and b relatively equal in value: symmetric beta distribution


2

, • b > a: skewed to the right; values are closer to 1
• a > b: skewed to the left; values are closer to 0




In this model, you can estimate using the maximum likelihood function


Calculating CLV with the sBG

Each term contributes less to CLV, because of discounting and the diminishing survival function
You need a suitably large T, to estimate the same E[CLV ] function as before (you can ignore the latest
terms because they are so small)

mS(1) mS(2) mS(3) mT
E[CLV ] = m + + + + ... +
(1 + d)1 (1 + d)2 (1 + d)3 (1 + d)T


3