Customer Lifetime Value Modeling

Customer Life Time Value Modeling

Revision History
Revision Date Status Revised By Revision Details
0.1 12 July 2010 Dr. Asoka Korale Derivation and analysis of existing scheme,
forecasting future revenues / growth rates,
accounting for other relevant revenues,
accommodating social network of subscriber.
0.2 19 July 2010 Dr. Asoka Korale Proposed model also incorporated

1.0 Introduction
At the moment the customer life time value model is based on the discounted cash flows
arising from the average annual revenues contributed by each subscriber (model A).
The second model is also based on the discounted cash flows arising from the average
annual revenues contributed by a subscriber but a constant annual growth rate is also
assumed to govern the rise in the growth in revenues (model B).
The improvements to be considered include giving due consideration to estimating the
future cash flows and growth rates through regression analysis, accounting for the other
revenue streams that the subscriber contributes such as DTV memberships and the value
of the subscriber’s social network.
2.0 Derivation of current models A/B and their
implications
I. Model A is a direct discounting of the cash flows which are represented by the average
revenue per user over the number of years the subscriber is expected to be with the
network. Thus the churn rate applicable to the subscriber’s particular segment is also
taken in to account in determining the expected duration of stay of the subscriber with the
network and thus determines the number of periods over which the cash flows are
discounted.
∑= +
=
N
i
i
i
k
CF
P
1 )1(
where k is the cost of capital and where N is the number periods
typically determined by the churn rate for that package
For equal cash flows at end of each period iCFCF = and setting
)1(
1
k
a
+
=
[ ]N
N
i
i
aaCFaCFP ++== ∑=
....
1
[ ]12
..... +
++= N
aaCFPa simplifying results in
[ ] [ ]
k
k
CF
a
aaCF
a
aaCF
P
N
NN














+
−
=
−
−
=
−
−
=
+ 1
1
1.
1
1.
1
. 1

II. Model B considers the cash flows to grow at a constant rate “g”. Equations (1) and (2)
give the expression used in method B
Terminal Value (TV) = (1+Terminal Growth) x Last Year ARPU (1)
(Cost of Capital – Terminal Growth)
Discounted Terminal Value (DTV) = Terminal Value x Discounted Value x 35% (Profit
Margin) (2)
This expression may be derived as follows using the same principal by summing the
discounted cash flows used earlier with the modification that the cash flows now grow at
a constant rate “g”.
( )
)1(
1
k
g
a
+
+
=
[ ]
gk
k
g
gCF
a
aaCF
P
N
N
−














+
+
−+
=
−
−
=
1
1
1)1.(
1
1.
Under the assumption that k > g and N is sufficiently large we may arrive at equation (1)
above as follows
[ ]
gk
gCF
a
aaCF
P
N
−
+
=
−
−
=
)1.(
1
1.
(3)
• As the terminal value estimated by (3) already includes the discounted cash flows
up to period N, it is unclear why the “discounted terminal value” in (2) is further
discounted.
2.1 Implications of the current methods
• From II above it is unclear why the terminal value as defined in (1) is further
multiplied by the discounted value (which is the sum of the discounting factors
over a certain number of periods). The expression derived (equation 3) shows that
the sum of discounted cash flows is already included.
• The constant growth rate “g” has to be assumed or estimated. If there was
sufficient historical information (several periods over which the cash flows could
be observed) it would be possible to use a least squares regression to estimate this
growth rate for each subscriber. The question then arises as what should be the

order of the relationship, is it a linear, quadratic, exponential ect. One issue is it
will vary from subscriber to subscriber.
• The interest rate “k” is also assumed to be a constant through out the summation,
an unlikely scenario.
3.0 Proposed Model
This model will principally be based on the discounted cash flows that are expected to be
generated. We may also attempt to forecast the expected cash flows occurring up to “N”
periods in to the future either directly or by estimating the growth rate in the cash flows.
3.1 Forecasting cash flows and estimating growth rate
We may attempt to estimate the cash flows up to N periods in advance by a least squares
regression analysis. We can either estimate the growth rate which need not be a constant
or we can directly estimate the cash flows inclusive of the growth rate.
The principal difficulty with this method will be that we will not know before hand if the
regression is linear, quadratic or exponential. In other words we will not know the kind of
relationship that exists between cash flows occurring at end of each period and the period
number. So this relationship may be one of positive growth, decay, constant ect. The
relationship would also change from subscriber to subscriber so we may have a linear
growth in the case of one subscriber where as in others there may be an exponential
growth.
Thus we will have to estimate the cash flow growth rates for each subscriber separately.
Initially, for simplicity and computational efficiency we can assume that the growth is
linear and perform a linear least squares regression to estimate the relationship between
cash flows and the particular period and thereby forecast cash flows in to the future.
The least squares regression analysis representation bellow is flexible in that it can be
used which ever the type of relationship that holds between the independent and
dependent variable – with the only change that the last column of the “A” matrix will
contain the data values from the independent variable (period number in our case).
• This leads to the question whether there are a sufficient number of data points (if
we look at yearly revenues) to estimate the regression equation for all subs.
3.2 Least squares regression for forecasting revenues
This is a method by which the trend values can be estimated by obtaining a line of best fit
in a series. The line of best fit implies it is a line from which the sum of the deviations

from the points on either side of the line is zero. This is why the sum of the squares of
these deviations is least when compared with the sum of the squares of the deviations
obtained by using other lines. Thus
0=−∑ eYY the sum of the deviations of the actual values of Y and estimated
values of Ye is zero
∑ − 2
)( eYY is least
Let a,b (where Ye = a + bX) be the parameters of the line that is to be determined
(estimated, Ye). Thus for points (x1 y1), (x2 y2), ……,(xn yn)
11 bxay e +=
22 bxay e +=
……………
nne bxay +=
Equivalently
Avye = where










=
nx
x
A
1
....
1 1
, 





=
b
a
v and










=
ny
y
y ..
1
Thus minimizing the mean square error between the actual and estimated values
corresponds to
2
min e
v
yy − equivalently
2
min Avy
v
−
( ) ( ) ( ) ( ) ( ) AvyyAvAvAvyyAvyAvyAvy TTTTT
−−+=−−=−
2
AvyAvAvyy TTTT
2−+=
Differentiating w.r.t. v and setting to zero
( ) yAvAAAvy
v
TT
22
2
−=−
∂
∂
( ) 022 =− yAvAA TT
Thus ( ) yAAAv TT 1−
=
• The same approach can be applied when fitting a polynomial to the given data
with v containing the coefficients of the polynomial and A the data points.
Thus the estimated trend line can be used to forecast future revenues or growth rates as
the case may be

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-0.2
0
0.2
0.4
0.6
0.8
1
Least squares forecasting example, linear trend
a = -0.021098
b = 1.3672
Y = a + bX
x
y
3.3 Accounting for other revenues
Just as the cash flows from mobile is considered we may also consider the revenues
earned from other Dialog subsidiaries such as TV and broadband for a particular
subscriber.
The forecasting of these revenues for each subscriber can take the same form as in the
case of revenues from mobile, but the relationship between independent and dependent
variables will be different so a different regression equation will have to be estimated.
• As before, the number of data points available should be sufficient for a fairly
accurate regression and forecasting to be carried out.
4.0 Accounting for the subscribers social network
A subscriber’s social network is a very subjective construction; however we may attempt
to identify those other subscribers over which a particular subscriber has some influence
by considering the call pattern between the two parties. This classification is very
subjective and open to different interpretations.
For this purpose we may use the minutes of use (MOUs) spent by each subscriber to
arrive at a network of subscribers who’s revenue contributions can be considered to be
partly due to the influence of the influencing subscriber.
The principal assumption is that the subscribers (S1-S5) are influenced by A to the degree
of out going MOUs used by A to call S, Figure 1. We may similarly argue that

subscribers U1-U5 are also influenced by A to the degree of degree of MOUs spent by
U1-U5 in calling A, Figure 2.
Figure 1
Figure 2
A
S
1
S
2
S
5
:
:
:
OG MOU rank 1
OG MOU rank 2
OG MOU rank 5
A
U
1
U
2
U
5
:
:
:
IC MOU rank 1
IC MOU rank 2
IC MOU rank 3

The scheme illustrated in Figure1, chooses the five subscribers to whom A has the
highest number of MOUs and considers their revenues as being partly contributed due to
the influence of A.
Similarly the scheme illustrated in Figure 2, chooses the five subscribers with the highest
number of incoming minutes towards A, and considers their revenues as being partly
contributed due to the influence of A.
In both scenarios we may also consider the percentage of out going or incoming MOUs
to each subscriber and scale the respective revenue contribution of that susbscriber by
this factor. Thus a subscriber S to which A has a significant portion of his outgoing
MOUs would have a higher proportion of S’s revenue contribution. Similarly a
subscriber U who has a high proportion incoming minutes when viewed from A, would
have a high proportion of his revenue contribution being considered as resulting from the
influence of A.
If any of the subscribers are off network their contributions wont be counted even though
the rank in terms of MOUs is within the top 5.
This scheme has many limitations imposed on it due to the need for a simple model, one
that can be implemented in software without too much computational complexity and the
inherent inaccuracies of estimating future cash flows / revenues.
The regression method used earlier to forecast future revenues and growth rates can be
used here as well, with future revenues discounted as described above.
5.0 Data Requirement
In order to carry out an initial investigation of the possibility of forecasting revenues and
or growth rates over a number of periods the revenue information and MOUs for each
call indexed by A number and B number needs to be provided in a table.

6.0 The Proposed Model
In current operator scenario
Customer Acquisition: Is not available for pre and post paid separately. Thus an average
value will have to be used by dividing the total acquisition cost by the additions in that
month. The marketing component of acquisition can be estimated separately for both pre
and post paid on a per subscriber average basis.
Monthly margin: ARPU – Direct Cost
Direct Cost: IDD/ Roaming costs calculated for both pre and post paid as international
call charges are heavy. In the case of post paid there is also the billing cost
The ARPUs will have to be forecasted over the expected life time of the subscriber using
one of techniques described in sections 2 and 3.
Lifetime: estimated using the churn rate applicable each subscribers package
Upgrades / Retention: Costs can be estimated for each post paid subscriber
Disconnection: No applicable disconnection fee, however post paid subscribers bad debt
information is available
• As in the previous case the future revenue streams have to be estimated /
predicted from historical data. A regression line may be used for this purpose and
the cash flows discounted using a “suitable” interest rate, to find the present value
of the future cash flows.
=CLV Acquisition–
Upgrades/
Retention–
Dis-
connection–
Disconnection fee
Bad Debt
Number portability
Monthly
Margin
Lifetime
x+
SP support
Equipment
Any channel costs
Marketing (acquisition
only)
Churn:
Predicted
loyalty in months
Discount Rate (sometime
included)
Similar to acquisition but may
occur nought to many
times
+ Average Revenue expected
- Ongoing product costs
All network costs
Interconnection
- Ongoing management costs
Customer Care
Billing

• The revenue future streams may be assumed to be growing at a particular rate and
the estimate of the revenue stream can include this growth or we may estimate the
growth rate separately and apply it to the revenue streams to get a series of cash
flows growing at a particular rate. In both cases regression can be used to in the
estimation.

Customer Lifetime Value Modeling

More Related Content

Similar to Customer Lifetime Value Modeling (20)

More from Asoka Korale (20)

Recently uploaded (20)

Customer Lifetime Value Modeling