Marketing Models
Example 12.12
Marketing Models
Background Information
 DoItQuick is software company that sells
programs to individuals for keeping track of
home finances, home inventory, and other
common tasks.
 The company has done extensive research
into its costs and revenues, and it has
discovered that new customers are much less
profitable on an annual basis than longstanding customers. There are several
reasons for this.
Background Information -continued
 Long-standing customers tend to require less in
overhead costs, they tend to order more merchandise
annually, and they help DoItQuick make money by
referring new customers to the company’s products.
 The company estimates that a customer who has
been loyal for n years – that is has bought from the
company for n consecutive years – contributes a
normally distributed random amount of profit in the
nth year that has mean and standard deviation as
listed on the next slide.
Background Information -continued
Background Information -continued
 DoItQuick is interested in seeing how much
profit a typical customer is worth over his or
her years with the company.
 This depends on the probability of retention.
To model retention, let r(n) be the probability
that a customer who has purchased for n
consecutive years does not purchase the
next year.
 If this occurs, we assume that the customer
switches loyalty and never purchases from
DoItQuick again.
Background Information -continued
 A consultant has suggested to DoItQuick that a
reasonable model of customer retention is to let r(1) =
1-p for some p between 0 and 1, and to use the
equation r (n) = qr(n-1) for n>2, where q is a positive
constant.
 What does this model mean, and how can it and the
data be used to simulate the nest personal value
(NPV) of profit over a 20-year period from a typical
customer who has made his or her first purchase
from DoItQuick this year? Assume an interest rate of
10% for discounting.
Solution
 The solution is broken into several parts.
 First we will explain the consultant’s retention
model.
 Then we will fit curves to the profit data.
 Finally, we will develop the simulation model
and run it with @Risk.
Explaining the Retention Model
 The consultant’s retention model makes sense.
 First, p represents the probability that a customer
who purchases this year for the first time will
purchase again next year.
 Then q is the fraction by which the probability of not
remaining loyal changes year by year.
 The company wants the r(n) values, the probabilities
of losing customers, to be small, so it wants p to be
large and q to be small. We will test several pairs of p
and q when we run the simulation to see how these
parameters affect the NPV of profit.
Finding the Data
 We first use the ideas from Chapter 2 to fit equations
to the means and standard deviations.
 For each, we draw a scatterplot versus year, then
superimpose an appropriate trendline with Excel’s
Chart/Add Trendline menu item.
 As shown in the figure, a logarithmic fit of the means
looks good, whereas a linear fit of the standard
deviation seems appropriate.
 Therefore, in the simulation model we will estimate
the mean and standard deviation of profit from a
customer in her nth year with the company as –
23.285 + 64.941ln(n) and 5.5515 + 1.3505n,
respectively.
LOYALTY.XLS
 The simulation model appears on the next
slide.
 This file contains the model.
The Simulation Model
Developing the Spreadsheet Model
 The model can be developed with the following steps.
 Inputs. Enter the inputs in the shaded cells. These
include the parameter of the fitted equations for mean
and standard deviation, the discount rate, and selected
values of the retention parameters p and q.
 Simulation index. We will use RISKSIMTABLE to run
the simulation 12 times, once for each combination of p
and q. To set up the model to do this, enter the formula
RISKSIMTABLE(SimIndexes) in cell B11. Then obtain
the corresponding values of p and q in cells B13 and
B15 with the formulas
=VLOOKUP(SimIndex,LookupTable,2) and
VLOOKUP(SimIndex,LookupTable,3)
Developing the Spreadsheet Model
-- continued

Profits. We want to simulate profits from a customer for as
long as the customer remains loyal to the company. To do
so, first calculate the appropriate means and standard
deviations in columns B and C of the simulation section with
the formulas =InterceptMean+SlopeMean*LN(A21) and
=InterceptStdev+SlopeStdev*A21 in cells B21 and C21,
and copy them down for all 20 years. Then generate the
actual profits from this customer in column D as long as the
customer remains loyal. Start by generating the first-year
profit in cell D21 with the formula =RISKNORMAL(B21,C21)
Then for succeeding years, enter the formula
=IF(OR(F21=“Yes”,D21=“”),””,RISKNORMAL(B22,C22))
in cell D22 and copy it down. The OR condition checks
whether the customer has discontinued buying from
DoItQuick. If so, a blank is entered. Otherwise, a normally
distributed profit is generated.
Developing the Spreadsheet Model
-- continued


Probabilities of quitting. Calculate the probabilities of
quitting in column E from the retention model. To do so,
enter the formula =1-PrKeepBuying1 in cell E21. Then
for succeeding years, enter the formula
=IF(OR(F21=“Yes”,D21=“”),”” ,RetFactor*E21) in
cell E22 and copy it down.
Quits? We keep track of the customer’s status in
column F. First, enter the formula =IF(RAND( ) <
E21,”Yes”,”No”) in cell F21. Then enter the formula
=IF(OR(F21=“Yes”,D21=“”),””,IF(RAND( )<=E22,
“Yes”, “No”)) in cell F22 and copy it down. This logic
will produce several values of “No”, followed by a
single “Yes” and then blanks.
Developing the Spreadsheet Model
-- continued

Output cells. We will keep track of the NPV of
profit and the number of years remaining loyal
for this customer as @Risk outputs. Calculate
these in cells B43 and B44 with the formulas
=RISKOUTPUT( ) + NPV(DiscRate,Profits)
and =RISKOUTPUT( )+COUNT(Profits). Note
that the COUNT function counts nonblank
cells only.
@Risk Results
 We set the number of iterations to 1000 and
the number of simulations to 12.
 Selected summary results appear on the next
slide.
 For a change, we copied and pasted the
@Risk results to the spreadsheet so that we
could easily see how they vary with p and q.
 The bar charts of the means clearly show
how large values of p and small values of q
are best for the company.
@Risk Results
@Risk Results -- continued
 By increasing the probability of keeping
customers loyal, the company can make a big
improvement in its bottom line.
Example 12.13
Consumer Preference Models with
Correlated Values
Background Information
 There are currently two brands of brownies
on the market.
 The Bisquake Company plans to enter the
brownie market with one of two new brands.
 Each of these existing brands and potential
new brands is characterized by three
attributes: sweetness (measured on a 1 to 10
scale), chewiness (measured on a 1 to 10
scale), and price per box.
 These attributes are shown in this table.
Background Information -continued
Attributes of Brands in Brownie Example
Sweetness
Chewiness
Price
Existing brand 1
8
6
$3.00
Existing brand 2
10
7
$3.80
Potential new
brand 1
8
6
$2.00
Potential new
brand 2
10
9
$4.50
 Each customer is assumed to choose one of these brands over
the other on the basis of a weighted combination of the three
attributes.
 That is, each customer is assumed to calculate a score for each
brand as
Score = ws(Sweetness) + wx(Chewiness) + wp(Price)
where the w’s are weights.
Background Information -continued
 Each customer’s weights are different, depending on
how important sweetness, chewiness, and price are
to the customer. However, we might expect these
weights to be correlated.
 For example, if a customer attaches a lot of
importance to sweetness, she might also attach a
large weight to chewiness; thus they would be
positively correlated.
 We assume that the population of customers assign
normally distributed weights with the means and
standard deviations shown in this table.
Background Information -continued
Means and Standard Deviations of Weights for Brownie
Example
Mean
Standard
Deviation
Sweetness
5.0
1.0
Chewiness
4.0
0.6
Price
-9.0
2.0
 We will also assume that the correlations between
these weights are as given in the table below.
Correlations Between Weights in Brownie Example
Sweetness
Chewiness
Price
Sweetness
1.00
0.80
0.70
Chewiness
0.80
1.00
0.60
Price
0.70
0.65
1.00
Background Information -continued
 Note that all correlations are positive, which
implies that if a customer puts a large weight
on one attribute, he will tend to put a large
weight on the other two attributes.
 Bisquake wants to use simulation to identify
the new brand (from the two possibilities) that
is likely to obtain the larger market share.
Simulation
 A single iteration of this simulation will
simulate the behavior of a single customer.
That is, it will generate this customer’s
weights, find the customer’s scores for each
of these brands, and see whether the
customer prefers new brand 1 or new brand 2
to the existing brands.
BROWNIE.XLS
 This file provides the setup to develop the
model seen on the next slide.
The Simulation Model
Developing the Spreadsheet Model


Inputs. Enter the inputs in the shaded ranges. These
include the given means, standard deviations, and
correlations for customers’ scores on the attributes.
They also include the actual attributes of the two
existing and two new brands.
Simulated weights. @Risk’s method of generating
correlated random numbers is not very intuitive, but it
is quite easy once you see how it works. We want the
weights in the SimWeights range to be normally
distributed with the means and standard deviations in
the range B6:D7, but we also want them to be
correlated. To accomplish this, generate the first weight
(for sweetness) in cell B24 with the formula
=RISKNORMAL(B6,B7,RISKCORRMAT
(CorrMatrix,B23)) Then copy this to the range
C24:D24 to generate the weights for the other two
attributes.
Developing the Spreadsheet Model
-- continued


It simply instructs @Risk to generate a normal random
number but to correlate it with other potential random
numbers, using the correlations in the first column of
the CorrMatrix range. (It uses the first column because
the second argument of RISKCORRMAT is 1.) For
chewiness, this second argument is 2, and for price it
is 3. This second argument essentially designates the
position in the correlation matrix for the particular
random value.
Scores for brands. Calculate this customer’s scores
for the four brands in the range B27:B30 by entering
the formula =SUMPRODUCT(SimWeights,B17:D17)
in cell B27 and copying it to the range B28:B30. This
formula weights the attributes of each brand with the
customer’s weights.
Developing the Spreadsheet Model
-- continued


Is either new brand chosen? One of the new brands will be
chosen if its score is larger than the larger score of the two
existing brands. Therefore, enter the formula
=RISKOUTPUT( )+IF(B29>MAX(ExBrScores),1,0) in cell
B33 to check whether new brand 1 is preferred to the exiting
brands. Then copy it to cell B34 to do the same for new
brand 2.
Summarize output cells. The @Risk output cells, B33 and
B34, contain 1 or 0 depending on whether either new brand
is preferred to existing brands. We want to determine the
fraction of the time these will be 1. To do so, we can run the
simulation for many iterations and calculate the means of the
output cells. This is because the average of a sequence of
0’1 and 1’s is the fraction that are 1’s. We can calculate
these fractions directly in the spreadsheet by entering the
formula =RISKMEAN(B33) in cell B37 and copying it to the
cell B38.
Using @Risk
 We set the number of iterations to 1000 and
the number of simulations to 1.
 After running @Risk, we see from cells B37
and B38 that new brand 1 preferred to
existing brands in 64.7% of the iterations, and
new brand 2 is preferred to existing brands in
76.6% of the iterations.
 Based on this information, new brand 2
appears to be the more promising brand for
BisQuake to market.
@Risk Results
 How do these results depend on the correlation
structure we assumed earlier?
 We note that because price weights are negative, the
positive correlation between the sweetness (or
chewiness) and price is less intuitive. It says that
large weights on sweetness tend to go with “large”
weights on price. But because price weights are
negative, a “larger” weight on price means a less
negative weight for price – for example –5 is “larger”
than –8. So the positive correlation between
sweetness and price really means that if a customer
puts a lot of weight on sweetness, he cares less
about price.
@Risk Results
 For the sake of argument, suppose you think that the
weights a customer assigns to the three attributes are
probabilistically independent.
 Then we should change the correlations in all cells of
the correlations matrix to 0 and rerun the simulation.
 When we did this, the values in cells B37 and B28
changed to 69.8% and 82.2%.
 This is not a dramatic change, but it does show that
correlations can make a difference.
Example 12.14
A Market Share Model
Background Information
 Sweetness and IceT are the two dominant
companies in the bottled iced tea market.
 Each currently possess 49% of the total iced
tea market, with three smaller companies
splitting the remaining 2%.
 At the beginning of any year, a random
number of new small companies enter the
iced tea market.
 The actual number of new entries is assured
to be Poisson distributed with mean 1.
Background Information -continued
 After the new entries enter the market, there is a
random shift in market share among all competitors.
 Essentially, all competitors lose a random percentage
of their market share to other competitors.
 We will assume that each of these percentages is
triangularly distributed with the parameters given in
the table on the next slide.
 Therefore, the more small companies there are in the
market, the more of its market share Sweetness will
tend to lose to them.
Background Information -continued
Parameters of Lost Market Share Percentages
Minimum
Most Likely
Maximum
To IceT
1.0%
5%
10%
To each small company
0.5%
1%
3%
To Sweetness
1.0%
5%
10%
To each small company
0.5%
1%
3%
To Sweetness
5.0%
10%
15%
To IceT
5.0%
10%
15%
From Sweetness
From IceT
From Small Companies
Background Information -continued
 At the end of each year, each of the small companies
has a 50% chance of exiting the ice tea market.
 Each small company that exits will lose its market
share to Sweetness or IceT.
 The percentage of this marketshare that goes to
Sweetness is triangularly distributed with parameters
40%, 50%, and 60%; the rest goes to IceT.
 The dominant companies, Sweetness and IceT, want
to use simulation to see how their market share is
likely to change over the next 10 years.
Solution
 At the beginning of the year we observe the market
shares of Sweetness, IceT, and the small companies
(combined).
 Next, we simulate the number of new entrants. Then
we simulate the shifts in market share during the
year.
 Next, we simulate the number of small companies
that exit at the end of the year, and we simulate the
market shares that go to Sweetness and IceT.
 Finally, we tally the total market share at the end of
the year for all competitors.
ICETEA.XLS
 This file provides the setup to develop the
model seen on the next two slides.
The Simulation Model
The Simulation Model
Developing the Model
 The model can be formed with the following steps:
 Inputs. Enter the inputs shown in shaded ranges.
 Beginning market shares. For year 1 the beginning
market shares are inputs. For example, find the
beginning market share for Sweetness in cell B35 with
the formula =B5. For every other year, the beginning
market shares are the ending market shares from the
previous year. For example, find the beginning market
share for Sweetness in year 2 by entering the formula
=B66 in cell C35. Then copy this to the range C35:K37
for all the competitors over the remaining years.
Developing the Model -- continued

Entries to the market. In year 1 find the number of
small companies before entries, the number of new
entries, and the number of small companies after
entries by entering the formulas =B9,
=RISKPOISSON(MeanEntries), and =SUM(B40:B41)
in cells B40, B41, and B42, respectively. Note that the
RISKPOISSON function, which takes a single
argument, generates the number of new entrants in a
single year. For year 2 the number of small companies
before entries is the remaining number from year 1.
Therefore, enter the formula =B58 in cell C40. Then
copy the formulas in cells C40, B41 and B42 across
the rows.
Developing the Model -- continued

Market shares lost during the year. Generate the
percentage of its market share Sweetness loses to
IceT and to the small companies (combined) in year 1
by entering the formulas
=RISKTRIANG($B$24,$C$24,$D$24)*B35 and
=RISKTRIANG($B$25,$C$25,$D$25)*B35*B42 in
cells B46 and B47 and then copy these across rows 46
and 47. Next, enter similar formulas in rows 49, 50, 52
and 53 for market share lost by IceT and the small
companies. For example, the formula in cell B53 is
=RISKTRIANG($B$31,$C$31,$D$31)*B37.
Developing the Model -- continued

Exiters. Rows 56-59 contain information about small
companies before and after exiting. To calculate this
information, enter the formulas =SUM(B37, B47,B50)SUM(B52:B53),
=IF(B42>0,RISKBINOMIAL(B42,$B$13),0), =B42B57, and =IF(B42>0,(B57/B42)*B56,0) in cells B56,
B57, B58, B59. The copy these across rows 56-59.
The formula in B56 simply tallies the market shares
lost and gained for the small companies before exiting
takes place. The formula in B57 uses the
RISKBINOMIAL function to generate the number of
small companies and the probability that any company
exits. Finally, the formula in B59 finds the amount of
market share possessed by the exiting companies
under assumption that all small companies have an
equal market share.
Developing the Model -- continued


Market share gained by exiters. The assumption of
the model is that the market share of the exiters in row
63 is split randomly between Sweetness and IceT. To
generate the split, enter the formula
=RISKTRIANG($B$20,$C$20,$D$20)*B59 and =B59B62 in cells B62 and B63. Then copy these across
rows 62 and 63.
Year-end market shares. Calculate the year-end
market shares of Sweetness, IceT, and the small
companies (combined) by entering the formulas
=SUM(B35,B49,B52,B62)-SUM(B46:B47),
=SUM(B36,B46,B53,B63)-SUM(B49:B50), and =B56B59 in cells B66, B67, and B68. Then copy these
across rows 66-68. If you like, you can check that the
year-end market shares sum to 100% for each year, as
they should.
Developing the Model -- continued


@Risk outputs. We have not yet designated any cells
as @Risk output cells. There are at least two
possibilities. If we are interested in only the final market
shares after 10 years, we should designate cells K66,
K67, and K68 as output cells.
Alternatively, if we want to see how market shares
move through time, we can specify whole ranges as
output ranges. When you do this, the formulas change
slightly. For example, the formula in cell B66 becomes
=RISKOUTPUT(,”Sweetness”,1)+SUM(B35,B49,B52
,B62)-SUM(B46:B47) to indicate that this is the first
cell in the Sweetness output range.
@Risk Results
 We set the number of iterations to 1000 and the
number of simulations to 1.
 After running @Risk, we obtain histograms of market
share after 10 years. The histograms can be seen on
the next two slides.
 We see that the final IceT market share is essentially
symmetric around its original value of 49%, although
there is considerable variability.
 In contrast, the final market share for the small
companies has a good chance of being 0, although
there is a small probability that it could be
considerably larger – up to 8%, say.
@Risk Results
@Risk Results
@Risk Results -- continued
 Assuming that we designate whole rows as output
ranges, such as row 66 for Sweetness, we can obtain
a summary chart of the company’s market share
though time as shown on the next slide.
 This chart shows that the mean market share for
Sweetness remains approximately constant through
time.
 However, as we stand at the beginning of year 1 and
try to predict the future, there is more uncertainty the
farther out we look.
 This is a general rule. It is almost always harder to
make long-range forecast than short-range forecasts!
@Risk Results