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It is of great benefit for an apartment community to predict the customer lifetime value (CLV) for each tenant. This prediction can be used not only to identify the most profitable residents, but also to make better pricing decisions, especially when optimizing renewal rents for expiring leases. CLV has been studied extensively in many industries such as service and retail. However, to our knowledge, there is no literature specifically addressing the estimation of CLV for apartment tenants. In this study, we propose an approximate approach to predicting the lifetime length and value for apartment tenants as well as their renewal probabilities. The model was estimated and tested based on a real dataset from 68 apartment companies in the US. The resulting prediction accuracy was particularly satisfactory for the tenants who did not renew or only renewed once.
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ORIGINALBEITRAG
DOI 10.1365/s41056-016-0015-0
Z Immobilienökonomie
Predicting and measuring customer
lifetime values for apartment tenants
Jian Wang · Murtaza Das · Amar Duggasani
Received: 3 June 2016 / Accepted: 4 November 2016
© The Author(s) 2016. This article is available at SpringerLink with Open Access.
Abstract It is of great benefit for an apartment community to predict the customer
lifetime value (CLV) for each tenant. This prediction can be used not only to identify
the most profitable residents, but also to make better pricing decisions, especially
when optimizing renewal rents for expiring leases. CLV has been studied extensively
in many industries such as service and retail. However, to our knowledge, there is
no literature specifically addressing the estimation of CLV for apartment tenants. In
this study, we propose an approximate approach to predicting the lifetime length and
value for apartment tenants as well as their renewal probabilities. The model was
estimated and tested based on a real dataset from 68 apartment companies in the
US. The resulting prediction accuracy was particularly satisfactory for the tenants
who did not renew or only renewed once.
Keywords Customer lifetime value · Apartment · Real estate · Prediction · Leases
1 Introduction
Increasingly, companies are viewing customers in terms of their lifetime value. In the
apartment industry, the customer lifetime value (CLV) for a tenant measures the total
value of the tenant for an apartment. CLV calculates the lifetime value that a tenant
contributes during the tenant’s entire stay in the apartment. In addition, the lifetime
length for the tenant’s entire stay is an important complementary measurement to
CLV. It is advantageous for an apartment community to predict the lifetime length
and value for each tenant. This prediction can be used not only to identify the
J. Wang () · M. Das · A. Duggasani
The Rainmaker Group, 4550 North Point Parkway, Suite 400, Alpharetta, GA, USA
E-Mail: jwang@LetItRain.com
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most valuable tenants, but also to make better pricing decisions, especially when
optimizing renewal rents for expiring leases.
Apartments are one of the most important categories of commercial real estate.
In the rental business of real estate, tenancies are created when the landlord and the
tenant enter into a rental or lease agreement that conveys a possessory interest in
the real estate to the tenant (Crane 2016). There are basically two types of tenancy:
commercial tenancy and residential tenancy. Commercial tenants and residential
tenants share some characteristics, e. g., both of them are leasing some property for
a use, but they differ in several aspects. For example, the purposes of commercial
tenants leasing a building are usually associated with a business operation like a store
or an office, while those of residential tenants are tied with a personal residency such
as single- and multi-family housing. The rental square footage for a commercial
tenant tends to be, on average, larger than that for a residential tenant resulting in
different tenant values. Apartments and apartment communities hereafter are limited
to be the residential tenancy for multi-family housing.
Apartment communities belong to the service industry, but they have their own
characteristics (Wang 2008). For example, the duration of a tenant is continuous.
Before a prospective tenant moves in to an apartment, the person signs a new lease
which specifies the unit to be rented, the move-in date, the chosen lease term (i. e.,
the number of months to stay), the monthly rent for the chosen lease term, as well
as other terms. Before the lease is about to expire, the tenant will be offered a set
of renewal options consisting of different lease terms at varying rents. If the tenant
chooses to continue his stay in the apartment, he will renew the lease by signing
another lease from the lease terms and rents of the renewal offer; otherwise, he will
move out of the apartment by the time when the current lease expires. This renewal
process will be repeated until the tenant moves out. Tenants rarely move back to the
same apartment after moving out. This is largely because they may have relocated
to a different city, moved to another apartment community, bought a house, etc.
Because of this unique characteristic of continuous residency, the lifetime of
a tenant for an apartment is defined as the continuous duration between the times
when the tenant first moves in and when he finally moves out. During a lifetime, the
tenant will sign one or more leases, where the first lease is called a new lease, and
any subsequent leases are called renewal leases. Therefore, the lifetime length and
value of the tenant are calculated as the total length of lease terms and the accrued
rents, respectively, from all the leases that the tenant has signed during a lifetime.
Since tenants are not all the same, their leases are not necessarily the same thus
leading to different lifetime lengths and values.
In this study, we propose an approximate approach to predicting and measuring
the lifetime length and value for apartment tenants as well as their renewal prob-
abilities. This paper is organized as follows: Sect. 2 presents a review of related
literature. Sect. 3 describes a dataset of actual leasing transactions. This dataset
is divided into two sample datasets to estimate and test an approximate approach,
which is proposed in Sect. 4. In Sect. 5, we provide an empirical estimation of the
proposed approach, followed by measuring accuracies. Finally, Sect. 6 concludes
this paper with suggestions for further research and improvement.
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2 Literature review
There exist many variants of CLV definitions, but they have similar meanings. For
instance, CLV for a customer is often defined as the present value of all future profits
generated from the customer (Gupta and Lehmann 2003). Research on the prediction
of CLV has been performed mainly for retail and service industries. Because CLVs
are closely related to the consumer behaviors, various methods for estimating CLVs
have been suggested across industries (Fader et al. 2005). In measuring CLV for
a customer, a common approach is to estimate the present value of the net benefit
to the firm from the customer over time (generally measured as the revenues from
the customer minus the cost to the firm for maintaining the relationship with the
customer). Typically, the cost to a firm is controlled by the firm, and therefore is
more predictable than the other drivers of CLV. As a result, researchers generally
focus on a customer’s revenue stream as the benefit from the customer to the firm.
Different models for measuring CLV result in different estimates of the expec-
tations of future customer purchase behavior. For example, some models consider
discrete time intervals and assume that each customer spends a given amount (e. g.,
an average amount of expenditure in the data) during each interval of time. This in-
formation, along with some assumptions about the customer lifetime length, is used
to estimate the lifetime value of each customer by a discounted cash-flow (DCF)
method (Berger and Nasr 1998).
Research on CLV measurement has so far focused on specific contexts. This is
necessary because the data available to a researcher or a firm might be different under
different contexts. Two types of context are generally considered: non-contractual
and contractual (Reinartz and Kumar 2000,2003; Borle et al. 2008). A non-
contractual context is one in which the firm does not observe customer defection
and the relationship between customer purchase behavior and customer lifetime is
not certain (Fader et al. 2005). A contractual context, on the other hand, is one
in which customer defections are observed, and a longer customer lifetime implies
a higher customer lifetime value (Thomas 2001). Our problem tends to fall in the
contractual context. When a prospect moves in or a tenant extends his stay in an
apartment, he will sign or renew a lease with the apartment community. When he
decides to move out, he will inform the apartment community about his decision.
However, the apartment community does not know a tenant’s lifetime length and
value completely before he has physically moved out. This is because at each point
in the tenant’s lifetime, the apartment community is not certain whether the tenant
will default his lease by moving out earlier, how many more times the tenant will
renew, what lease terms thetenant will select, and what rents the tenant is willing to
pay. In this regard, the tenant’s lifetime length and value are uncertain.
There is a wealth of publications related to real estate. However, to our knowl-
edge, none of them specifically addresses the estimation of CLV for apartment
tenants. As defined early, the CLV of a tenant results from the accrued rents from
one or more leases that the tenant signs during his lifetime. How the rents are
set will thus impact the resulting CLV. On the other hand, the aggregate amount
of CLVs from all individual tenants constitutes the gross rental revenue that the
apartment will receive. This amount should reflect some kind of values that the
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apartment property possesses, e. g., the “highest and best value” which, according
to Miller and Geltner (2004), is defined as the reasonably probable and legal use
of vacant land or improved property, that is physically possible, appropriately sup-
ported, financially feasible, and which results in the highest valuekat the date of the
appraisal. As an attempt to obtain some insights about how to approximate CLV, it
is helpful to understand how the rents and values of an apartment are determined.
Rent setting is an important and fundamental decision that apartment owners and
operators have to make frequently. In traditional apartment management, rents are
often set with the objective of maximizing occupancy and return on investment.
Rents are normally determined by such factors as physical characteristics of a prop-
erty, its current vacancy rate and competitive position in the market place, along with
managers’ prior experience (Wang 2008). Sirmans and Benjamin (1991) performed
an extensive literature review about the setting of apartment rents. For example,
Simans et al. (1989) examined the effects of various factors on rent, in which
the factors that they studied include amenities and services like covered parking,
modern kitchen, and maid service, occupancy restrictions such as no pets allowed,
traffic congestion, proximity to work, access to public transportation, and so on.
Pagliari and Webb (1996) built a regression model to set rental rates based on rent
concessions and occupancy rates. In modern apartment management, rents are often
set with the objective of maximizing total revenue. For example, apartment revenue
management (RM) proposes to optimize the rents by optimally balancing demand
and supply in the consideration of competitor rents such that total revenue will be
maximized (Wang 2008).
In apartment valuation, on the other hand, appraisers, investors, tax assessors
and other real estate market participants are mainly the ones who are interested in
estimating the values of properties. For example, when making income or DCF
calculation of an apartment, an appraiser may take into account the lifetime value
of a lease as well as many other factors that influence the values. His objective is
to establish the market value of a property that accounts for the most probable price
that would be paid for the property under competitive condition (Adetiloye and Eke
2014). The valuation of real estate has been studied extensively. There exist various
valuation models such as cost approach, income capitalization approach, hedonic
models, and so on, for estimating different types of values. Adetiloye and Eke (2014)
give a thorough review on the real estate valuation. For instance, an appraiser may
use the cost approach to estimate the market value by systematically estimating
the cost of production (Miller and Geltner 2004). For apartment properties of real
estate, in particular, there are also many researches. Zietz (2003) summarizes the
empirical and theoretical studies for multifamily housing. As an example. Bible
and Grablowsky (1984) observe that multifamily housing complexes located within
restorative zoning neighborhoods increase in value at a higher rate than comparable
complexes in neighborhoods that are not subject to restorative zoning codes.
The valuation of real estate, in general, and apartments, in particular, is sometimes
performed by applying financial theory. Real estate investments comprise the most
significant component of real asset investments. It is argued that real estate and
financial assets share similar characteristics (Damodaran 2012). For instance, the
accrued rents of a lease contract and the return of a bond or a stock represent
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the expected cash flows on a real estate and a financial asset. Their values are
determined by the cash flows they generate, the uncertainty associated with these
cash flows and the expected growth in the cash flows. Since research on financial
instruments is generally far ahead real estate research, many real estate researchers
have therefore applied general financial theory to the valuation of real estate. There
is a stream of literature relating real estate to financial assets. For example, Mc-
Connell and Schallheim (1983) priced leases and lease options using Black Scholes
option pricing techniques. Wendt and Wong (1965) compare the investment perfor-
mance of common stocks and apartment houses. They compare the DCF of FHA-
financed residential projects with 76 randomly selected industrial stocks. They ob-
serve that because of the special real estate tax advantages, after-tax returns on
equity investments in apartment houses are twice that of stock returns, but after-tax
rates of returns vary significantly over different periods of time and across different
properties.
In rent setting and valuation, the variables involved are more “exogenous” in the
sense that they are more generic and related to the characteristics and conditions
of the property and market where tenants reside. To a large extent, these variables
play an important role in estimating the average rent and average value of a tenant,
and thus the corresponding average CLV. Because CLVs are specific to individual
tenants, other variables that are more “endogenous” should be taken into account so
that we can better understand the CLVs on a personal level. Endogenous variables
are those that are more specific and related to the characteristics and behaviors
of individual tenants such as life expectancy, purchasing power, creditworthiness,
household change, product promotion, default risk, lease term, number of renewal
times, and so on. For instance, wealthier tenants may have a higher CLV because
they can better endure rent increases or periods of unemployment. A tenant may
increase his CLV by moving to a larger unit of the same apartment community
when his family is growing. In addition, his CLV could also increase because
larger families move less frequently leading to higher renewal probabilities. Also,
a higher default risk increases the odds of moving out earlier than anticipated thereby
decreasing his expected CLV.
In an attempt to increase the explanatory power of a statistical model, one may
try to include as many variables as desired. In practice, however, there are several
issues by doing so. First, the resulting error of such a fitted model may increase
disproportionately. Second, it is difficult, if not impossible, and expensive to access
all desired data, particularly, the endogenous data about personal demographics.
Third, it is not easy to interpret a model when too many variables are included. As
a result, in this study, we focused our attention on a small subset of variables that
are related to leases. Our intention is to emphasize the underlying methodology of
the proposed approach.
3 The dataset
The dataset consists of 77,536 historical leases from 62,643 tenants with lease dates
ranging from March 4, 1995 to April 22, 2015. It was provided by The Rainmaker
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Group (www.letitrain.com), a revenue management software company providing
pricing solutions for multi-family housing and gaming casino resorts industries. This
anonymized dataset was selected from the historical transactions of 795 communities
belonging to 68 apartment management companies across the United States. For
each tenant in the dataset, we know his complete history of leasing transactions,
i. e., total number of leases that he has signed, and the term, monthly rent, start
date and end date of each signed lease. In this dataset, since there are more leases
than tenants, we can infer that some tenants have signed two or more leases during
their stays. In addition, this dataset also has 77,536 sets of 12 renewal options that
were offered to the tenants when a lease was about to expire. Each renewal option
specifies a rent for every lease term ranging from 1 to 12 months. A tenant has
either chosen one of the 12 renewal options to renew, or he has chosen not to renew
by moving out. From the dataset, we can compute the actual lifetime length and
value for each tenant, which are equal to the total number of months, and the total
amount of rents that he has paid during his stay. Furthermore, we can also compute
the residual lifetime length and value for the remaining lifetime at each point in
a lifetime. This dataset can thus be used to resemble a realistic context under which
at each point in the lifetime of a tenant, we assume that the apartment community
did not know how many more times the tenant would renew, what renewal lease
terms the tenant would choose, and what monthly rents the tenant would pay. By
doing so, we would be able to compare the predicted lifetime lengths and values
with the actual ones.
Two samples are randomly drawn (without replacement) from this dataset. The
first sample, referred to as the estimation sample, contains 62,049 leases from 50,181
tenants representing 80% of all the tenants. This sample will be used to estimate
the parameters of a model to be proposed in Sect. 4. The second sample, referred to
as the validation sample, includes 15,487 leases from 12,462 tenants representing
the other 20% of the tenants. This sample will be used to predict and measure three
dependent variables of primary interest: the residual lifetime length, the residual
lifetime value, and the renewal probability, respectively, for each tenant during
a lifetime.
Table 1summarizes some descriptive statistics for the variables of lifetime length,
lifetime value, number of renewals, and lease terms observed across all of the tenants
in the estimation sample dataset. For example, on average, a tenant stays for about
14.3 months making a revenue contribution of $ 15,352. The average number of
renewals is 0.5 times with an average lease term 9.8 months.
Tab l e 1 Summary Statistics
Lifetime length
(months)
Lifetime value
(dollars)
Number of renewals
(times)
Lease terms
(months)
Mean 14.3 15,352 0.5 9.8
Std. Dev. 7.7 10,576 0.7 3.1
Median 12 12,375 1 12
Min 1 464 0 1
Max 48 138,655 5 12
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Fig. 1 Tenants by Lifetime
Length
Lifeti me length
s
t
n
a
n
etfo
eg
a
t
ne
c
reP
0.0
0.1
0.2
0.3
10 20 30 40 50
Fig. 2 Tenants by Lifetime
Value
Lifetime value (times $1000)
stnanetfoegatnecreP
0.00
0.02
0.04
0.06
0.08
50 100 150
Figs. 1,2and 3display the histogram plots of lifetime length, lifetime value,
and number of renewals, separately. They show significant heterogeneity across the
tenants. Specifically, Fig. 1shows the distribution of lifetime lengths ranging from 1
to 48 months. It can be seen that the peak occurs around 12 months representing 40%
of tenants. This is consistent with the median of 12 months in Table 1, meaning
that the majority of tenants sign a single lease with a lease term of 12 months.
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Fig. 3 Tenants by Number of
Renewals
Number of renewals
stnanetfoegatnecreP
0.0
0.2
0.4
0.6
0.8
123456
Fig. 4 Average Lifetime Length
and Average Lease Term by
Number of Renewals
10
15
20
25
30
35
)shtnom(htgnelemitefilegarevA
Average lease term (months)
4
5
6
7
8
9
10
0123
4
5
Number of renewals
Avg lifeti me length
Avg lease term
Fig. 2shows the distribution of lifetime values. In this Figure, although details are
not shown, the lifetime values range from $ 500 to $138,600, among which more
than 60% of tenants have a lifetime value of $10430 or more. Fig. 3shows that
the number of renewals spans from zero to five times. The details have not been
reported here, but more than 64% of tenants did not renew at all, around 26% of
tenants renewed once, and the remaining 10% of tenants renewed two or more times.
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Fig. 5 Average Lifetime Value
and Average Revenue by Num-
ber of Renewals
15
20
25
30
35
)
00
01
$
sem
i
t
(
eulav
emitef
i
l
e
gare
vA
Average revenue (times $1000)
4
6
8
10
012345
Number of renewals
Avg lifeti me value
Avg revenue
The renewal behavior of tenants determines their lifetime lengths and values as
well as the numbers of renewals. In this paper, two terminologies of number of
renewals and renewal times are used throughout. Number of renewals is defined
as the total number of renewal leases that a tenant has signed in a lifetime, while
renewal times as the number of times that a tenant has been making a renewal
decision at some point time in his lifetime. By this definition, total number of
renewal times for a tenant is equal to the number of renewals plus one, because the
tenant will not renew at the last renewal times.
Fig. 4displays the average lifetime length and average lease term by number of
renewals, where the number of renewals of zero is for the tenants who only signed
a new lease and did not renew at all. It shows that the average lifetime length
increases over number of renewals that is equal to or less than three. This seems
intuitive because we would expect that the larger the number of renewals, the longer
the lifetime length. However, the lifetime length starts to decrease over the number
of renewals that is larger than three. This says that, on average, the lifetime length
of a tenant does not necessarily become larger as the number of renewals increases.
A plausible explanation is that tenants tend to sign shorter lease terms when they
renew more times. The decreasing trend of average lease terms over number of
renewals tends to support this explanation. Fig. 5exhibits the average lifetime value
and average revenue by the number of renewals. It shows that they maintain similar
patterns as the average lifetime length and average lease term.
Fig. 6illustrates the average residual lifetime lengths and values by renewal times.
The residual lifetime length and revenue of a tenant at a renewal times are calculated
as the sums of lease terms and revenues from the current and any future leases. As
expected, both residual lifetime length and revenue decrease over renewal times.
Because the maximum number of renewals is 5 in the estimation sample dataset,
these values become zeros at the renewal times of 6.
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Fig. 6 Average Residual Life-
time Length and Value by Re-
newal Times
0.0
0.5
1.0
1.5
2.0
)
s
h
tnom
( ht
gn
el
emi
t
e
fi
l
l
a
udi
s
er
e
gare
vA
Average residual lifetime value ($)
0
500
1000
1500
2000
123456
Renewal Ti mes
Avg residual lifetime length
Avg residual lifetime value
Fig. 7 Renewal Rates by Re-
newal Times
etarlaweneR
0.00
0.05
0.10
0.15
0.20
12345 6
Renewal ti mes
Exi sting t enants
Original tenants
Finally, Fig. 7displays two renewal rate curves by renewal times. The first
renewal rate curve is defined as the fraction of the tenants who were active (i.e.,
still living in the apartment) at a given renewal times and had decided to renew.
Specifically, at the first renewal times, about 20% of tenants chose to renew; at the
second renewal times, about 18% of the remaining tenants chose to renew; and so
on. By the last renewal times, all of the remaining tenants had moved out. On the
other hand, the second renewal rate curve is defined as the fraction of the original
tenants who chose to renew at a renewal times. It shows that the renewal rates of this
curve decrease monotonically, meaning that the number of remaining active tenants
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becomes smaller as renewal times increases, and that every tenant would move out
by the last renewal times.
4Model
The lifetime length and value, and renewal probability of an active tenant are de-
termined by the renewal decisions of the tenant. When the current lease is about to
expire, the tenant has two choices. He can choose either to renew for a particular
lease term at an offered rent, or to move out. The renewal decision will impact the
tenant’s residual lifetime length and value. Fig. 8illustrates a division of an active
tenant’s lifetime.
In Fig. 8, the lifetime of an active tenant is divided into three periods: past, current
and future. A past period represents the time length of lease terms of all previous
leases that the tenant has ever signed. A current period represents the time length of
the lease term of the current lease. A future period represents the time length of lease
terms of any future leases that the tenant may renew during the remainder of the
tenant’s lifetime. Furthermore, Fig. 8illustrates a partition of the three periods into
a number of consecutive segments, each of which contains a set of leasing options
represented by arrows (directed edges) connected with circles (vertices). A dashed
arrow represents a lease option that was offered or is to be offered, while a solid
arrow represents an actual lease that the tenant has signed. Two circles at the ends of
an arrow denote the start date and end date of a lease. Under this illustration, each
segment in the past and current periods has one and only one solid arrow, while the
segments in the future period have only dashed arrows.
Given any active tenant i, denote Liand Vias the tenant’s lifetime length and
value. At any time point tin the tenant’s current period, Lican be decomposed as
LiDLi;p .t/ CLi;c .t/CLi;f .t/,whereLi;p .t/,Li;c .t/ and Li;f .t / represent the
lifetime lengths for the past, current and future periods, respectively. Accordingly,
Vican also be decomposed as ViDVi;p .t/ CVi;c .t/ CVi;f .t/,whereVi;p .t/,
Vi;c .t/ and Vi;f .t/ represent the lifetime values for the past, current and future
periods at t. For both past and current periods, Li;p .t/,Vi;p .t/,Li;c .t / and Vi;c .t/
are all known. They do not change over t, and can be calculated directly as the sums
Past Current Future Lifeme
Fig. 8 Visual Depiction of the Division of a Lifetime
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of lease terms and revenues (i. e., lease terms times monthly rents) from all the leases
in the previous and current periods, respectively. For the future period, Li;f .t/ and
Vi;f .t/ represent the residual lifetime length and value from any future leases that
might be signed after t. Both of them are unknown, and need to be estimated. For
the sake of brevity, Li;f .t/ and Vi;f .t/ will be called lifetime length and value
at tin the remainder of the paper. We now propose an approach to approximating
Li;f .t/ and Vi;f .t/.
Denote as a discrete random variable corresponding to the renewal times at t,
for which may take values of 1; :::; N ,whereNis the maximum renewal times
to be allowed. Table 1shows that the maximum number of renewals is 5 in our
dataset, implying that ND6 because every tenant had moved out by the sixth
renewal times. Also, denote …./ Dl;j ./1313 as a renewal probability matrix,
or called as transition matrix, at renewal times ,wherel;j ./ is defined as the
renewal probability from the current lease term lto the lease term jof a possible
renewal lease, for l;j D0;1; :::; 12. It is noted that when lease tem jis equal to 0,
it means that the tenant chooses to move out. When lease tem lis equal to 0, there
is no practical meaning. The reason that we introduce the notation of lD0hereis
just a matter of algebraic convenience. Specifically, for <N,wesetl;j ./ 0
for any jwhen lD0. Also, we impose a condition that P12
jD0l;j ./ 1for
any lD1; :::; 12, meaning that the tenant chooses either one of 12 lease terms to
renew or chooses to move out. For DN,wesetl;j ./ 0foranyland j,
i. e., ….N/ 0, to comply with the assumption that Nis the maximum renewal
times. This condition implies that the estimates for Li;f .t/ and Vi;f .t/ at DN
are b
Li;f .N / Db
Vi;f .N / D0.
In addition, denote !
lD.1; :::; 12;0/as a 113 row vector representing all of
the possible lease terms in which the special lease term of zero means a move-out.
Suppose that liis the lease term of the current lease for tenant i.Letli;./ D
li;1. / ; :::; li;12 . / ; li;0./be a 113 row vector of renewal probabilities of
selecting the lease terms !
lat by the tenant. This vector li;. / is the li-th row
from the renewal probability matrix …./.
For any <N, we propose to approximate Li;f .t/ and Vi;f .t / by
b
Li;f ./ Dli;. / 0
@1C
N1
X
sDC1
s
Y
hDC1
….h/
1
A!
lT
And
b
Vi;f ./ Dli;. / 0
@!
vi./TC
N1
X
sDC1
0
@
s
Y
hDC1
….h/
1
A!
vi.s/T1
A
where !
vi.s/ Dvi;1.s / ; :::; vi;12 .s/ ; 0is a 113 row vector representing the rev-
enues to be gained by choosing !
lat renewal time s,forsD; :::; N .That
is, !
vi.s/ D!
lı!
ri.s/, a Hadamard product (or entry-wise product) of the lease
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terms !
land the corresponding renewal rents !
ri.s/ Dri;1.s / ; :::; ri;12 .s/ ; 0.
Specifically, vi;j .s/ Djr
i;j .s/,forjD1; :::; 12.
In the proposed formula, b
Li;f ./ and b
Vi;f ./ are the sums of expected lease
terms and expected revenues, from .N / possible future renewal leases at re-
newal times ;  C1; :::; N 1. Take a simple example to illustrate how b
Li;f ./ is
estimated. Suppose that tenant ihas a lease term of liD12 for the current lease,
It is the first time for the tenant to renew . D1/, and the maximum number of
renewal times allowed is 2 .N D2/. Then, b
Li;f ./ for D1 can be estimated as
b
Li;f .1/D12;.1/!
lT,inwhich12;.1/and 12;.1/!
lTrepresent the renewal
probabilities and the expected lease term from of a possible future renewal lease.
In a similar manner, b
Vi;f .1/D12;.1/!
vi.1/Tare the expected revenue from
a possible future renewal lease.
It is noted that !
vi.s/ is derived from !
ri.s/ Dri;1.s / ; :::; ri;12 .s/ ; 0and !
l.
For sD, the renewal offer rents of !
ri.s/ usually become known within 30 to
60 days before the current lease expires. For s>,!
ri.s/ are not known and need
to be estimated. In the literature review, we mentioned that there exist a number
of ways to estimate !
ri.s/. It is beyond the scope of this paper to discuss how
to estimate !
ri.s/. To simplify our approximation, we will simply replace the val-
ues of !
ri.s/ for s>by the average renewal rents from the historical renewal offers.
Next, we describe the estimation of …./. In practice, the determination of …./
is complicated, and it is influenced by many factors including !
ri.s/. To simplify
our estimation, we assume that …./ satisfied the Markov property. Specifically,
l;j ./ are assumed to depend only the rents and lease terms of the current lease and
a possible lease to be renewed. They are not dependent on the states of any previous
leases that preceded the current lease. In this regard, for any renewal times s>
for which the renewal options !
ri.s/ are not known, we will approximate l;j .s/
by using the empirical estimate of renewal probability; for the renewal times sD
in which the renewal options !
ri./ are available, we will approximate li;j ./
by using Multinomial Logit (MNL) model. MNL is a variant of customer choice
models (Train 2009). The renewal probability li;j . / can be approximated by
bli;j . / DeVi;j .li;ri;ri;j .//
1CP12
j0D1eVi;j0.li;ri;ri;j0. //
where ridenotes the rent of the current lease, and Vi;j li;r
i;r
i;j ./a utility
function (or called representative utility) at obtained by choosing the lease term j.
Vi;j li;r
i;r
i;j ./measures the perceived benefit of renewing a lease term jover
choosing to move out. We assume that Vi;j li;r
i;r
i;j ./is linear-in-parameter.
Namely, Vi;j li;r
i;r
i;j ./Daj. / Cıi;j . / bj./ Chi;j . / cj./,wherethe
coefficients of aj./,bj. / and cj. / are unknown parameters to be estimated,
and ıi;j ./ and hi;j ./ are alternative specific variables derived from li,riand
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Z Immobilienökonomie (2016)
Tab l e 2 Estimates of MNL Parameters for D1 and2
D1D2D1D2
ba1–4.3*(0.17) –4.6*(0.37) b
b7–3.6*(0.48) –5.0*(0.99)
ba2–3.8*(0.11) –3.8*(0.19) b
b8–4.5*(0.59) –4.9*(1.25)
ba3–3.9*(0.10) –3.7*(0.18) b
b9–4.2*(0.60) –5.0*(1.31)
ba4–4.6*(0.14) –4.4*(0.26) b
b10 –2.4*(0.29) –1.6*(0.60)
ba5–5.2*(0.19) –4.8*(0.29) b
b11 –4.2*(0.51) –2.5*(1.23)
ba6–3.2*(0.04) –3.2*(0.09) b
b12 –3.1*(0.20) –2.7*(0.49)
ba7–4.5*(0.07) –4.8*(0.17) bc11.4*(0.42) 2.3*(0.48)
ba8–5.1*(0.09) –5.2*(0.20) bc21.1*(0.16) 1.4*(0.24)
ba9–5.0*(0.09) –5.4*(0.22) bc31.2*(0.14) 1.5*(0.23)
ba10 –3.4*(0.03) –3.3*(0.07) bc41.3*(0.27) 1.2*(0.49)
ba11 –4.9*(0.07) –4.9*(0.15) bc51.3*(0.43) 1.7*(0.74)
ba12 –3.1*(0.04) –3.3*(0.08) bc61.3*(0.07) 1.3*(0.12)
b
b1–4.5*(0.57) –2.9*(1.16) bc71.7*(0.17) 2.5*(0.30)
b
b2–2.7*(0.42) –2.3*(0.74) bc82.1*(0.22) 0.6 (1.02)
b
b3–2.9*(0.41) –2.9*(0.72) bc91.8*(0.23) 2.2*(0.55)
b
b4–3.4*(0.59) –2.8*(1.09) bc10 1.0*(0.07) 0.7*(0.14)
b
b5–3.4*(0.80) –4.6*(1.27) bc11 2.2*(0.10) 2.6*(0.23)
b
b6–3.6*(0.25) –3.7*(0.51) bc12 1.6*(0.04) 1.4*(0.10)
* indicates that the 95% posterior interval for a parameter does not contain 0
ri;j ./. Specifically, ıi;j ./ Dri;j ./
ri1represents the relative rent change of
the renewal rent ri;j ./ over the current rent ri,andhi;j ./ is a habit formation
(or inertia) describing whether a tenant tends to choose the same lease term as the
current one when renewing, that is, if jDli,hi;j ./ D1; otherwise, hi;j ./ D0.
5 Estimation and validation
5.1 Empirical estimations
Renewal probabilities li;j ./ were estimated based on the estimation sample
dataset. As proposed early, when renewal rents !
ri./ are offered for some ,
li;j ./ can be approximated via the estimation of MNL parameters; when !
ri./
are not available, li;j ./ will be approximated empirically. In this estimation sam-
ple, more than 90% of tenants did not renew or just renewed once, which is not
uncommon for apartments. This situation will cause the estimates of li;j . / to be
inaccurate for 3 for which there does not have enough data. To alleviate this
problem, we will approximate li;j . / using MNL model for D1,2, and using
the empirical estimates of probabilities for 3.
Table 2reports the estimates of MNL parameters baj. /,b
bj./ and bcj. / for
jD1; :::; 12 and D1, 2. The numbers in parentheses are the posterior standard
deviations, and the superscript asterisks * indicate that the 95% posterior interval
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Z Immobilienökonomie (2016)
Tab l e 3 Empirical Estimates of …./ for D1
12 3 4 567891011120
1 0.08 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.91
2 0.02 0.06 0.01 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.91
3 0.01 0.01 0.05 0.01 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.92
4 0.02 0.09 0.07 0.02 0.00 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.76
5 0.01 0.05 0.05 0.03 0.01 0.06 0.01 0.00 0.00 0.01 0.00 0.00 0.77
6 0.02 0.04 0.02 0.01 0.00 0.09 0.01 0.00 0.00 0.00 0.00 0.00 0.81
7 0.00 0.07 0.10 0.02 0.01 0.05 0.03 0.01 0.01 0.01 0.00 0.01 0.68
8 0.01 0.03 0.01 0.04 0.03 0.14 0.03 0.01 0.01 0.06 0.01 0.04 0.58
9 0.00 0.00 0.01 0.01 0.00 0.18 0.06 0.01 0.01 0.04 0.01 0.08 0.59
10 0.00 0.00 0.00 0.00 0.00 0.02 0.00 0.02 0.02 0.09 0.00 0.05 0.80
11 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.09 0.05 0.07 0.79
12 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.14 0.86
for a parameter does not contain 0. This is interpreted as an indicator of the estimate
being statistically different from zero. It can be seen that the signs of baj. / are all
negative implying that there was a larger tendency to move out than to renew. This
implication is demonstrated in Fig. 7also, in which, e. g., more than 80% of tenants
chose to move out at the first renewal times. The signs of b
bj./ are negative as
well, meaning that the larger the renewal rent change ıi;j ./, the lesser the utility
Vi;j li;r
i;r
i;j ./. This is consistent with intuitive expectation. Finally, the signs
of bcj. / are positive indicating that there exists a habitual inertia among tenants.
Namely, when renewing, a tenant tends to renew to the same lease term as the
current one.
As an illustration, Table 3shows the empirical estimates of …./Dl;j ./1213
for D1. The 12 rows represent the possible lease terms of a current lease, and
the 13 columns the possible renewal lease terms, in which the renewal lease term
of zero again represents the move-out option. On each row, the summation of the
probability values across the 13 columns is equal to one, meaning that a tenant
would either renew with one of 12 lease terms or choose to move out. It again
shows that, as expected, the probabilities of moving out are larger than those of
renewing. In addition, it can be observed that the diagonal entries of …./ are
not always larger than the off-diagonal entries. This empirical estimation does not
provide strong evidence of the existence of habitual inertia as we saw in coefficient
estimate of bcj./ in MNL model.
5.2 Prediction and validation
We applied the proposed approach to predicting the lifetime length and value as
well as renewal probability for each tenant in the validation sample, which consists
of 12,462 past tenants with a total of 15,487 leases spreading across 4 renewal
times. Because we already know the actual lifetime lengths and values and renewal
outcomes of the tenants, we also tested the prediction performance of our approach.
K
Z Immobilienökonomie (2016)
Tab l e 4 Comparison of actual and predicted lifetime length and value, and renewal probability
1234
Lf./ 12.17 10.69 8.89 6.52
b
Lf./ 11.84 (3%) 10.66 (0%) 9.56 (8%) 7.12 (9%)
Vf./ 13,308.07 11,527.73 9190.05 7313.97
c
Vf./ 13,147.28 (1%) 11,735.46 (2%) 9907.5 (8%) 8001.93 (9%)
f./ 20% 18% 13% 7%
bf./ 20% (0%) 18% (0%) 17% (31%) 17% (143%)
For D1;2;3;4, denote Lf. /,Vf./ and f./ as the averages of the
observed Li;f ./ and Vi;f ./, and renewal probability from the validation sample,
respectively. That is, Lf. / and Vf. / were calculated as the averages of actual
lease terms and revenues from the current and subsequent leases with respect to .
f./ was computed as the percentage of tenants who were active and had renewed
at . Accordingly, denote b
Lf./,c
Vf./ and bf. / as the averages of predicted
b
Li;f ./,b
Vi;f ./ and P12
jD1bli;j . /, respectively. Table 4summarizes the
estimates of Lf./,b
Lf./,Vf. /,c
Vf./,f./ and bf. /. The numbers in
parentheses represent Mean Absolute Percentage Errors (MAPE) between the pairs
of Lf./ and b
Lf./,Vf. / and c
Vf./,andf. / and bf./, separately.
It can be seen that the prediction errors for Lf./,Vf./ and f. / were very
small (i. e., MAPEs 3%) for D1,2. However, for D3,4, the prediction errors
were not that satisfactory, particularly for f. /. The main cause was because, as
described earlier, the data was sparse (there are only less than 10% of tenants who
had renewed two or more times). Therefore, this problem of data sparseness led to
inaccurate estimates for Lf./,Vf. / and f. / for D3,4.
6Conclusion
In this study, we proposed an approximate approach to predicting the lifetime lengths
and values for active tenants. We divided a sample dataset into estimation and vali-
dation samples. Based on the estimation sample dataset, we estimated the renewal
probabilities. We then predicted the lifetime lengths and values as well as renewal
probabilities for the tenants in the validation sample. The resulting prediction accu-
racy seemed to be satisfactory only for the tenants who did not renew or renewed
once. It should be noted that in this article, lifetime lengths and values are forecast
for the active tenants in the apartments of US market. As a consequence of the
specifics of that market, the transferability of the results and applicability of the
proposed model to other jurisdictions and cultures is limited.
To improve the prediction accuracy, the following explorations can be performed:
Inclusion of additional variables: In this approach, we only use four variables:
current lease term, renewal lease term, number of renewal times, and actual (or
estimated) renewal rent offers. If accessible, we can consider more endogenous
K
Z Immobilienökonomie (2016)
and exogenous variables such as demographic information of age, income and
family size, economic condition, market rents, migration tendency between states,
and so on.
Utilization of existing data: The prediction accuracy became unsatisfactory at the
time when data amount was sparse, particularly for higher number of renewal
times. To alleviate this issue, a set of hierarchy rules can be designed to pool
the data of a lower number of renewal time for a higher number of renewal time.
Although there is no rigorous academic proof about how much an improvement
can be gained by doing so, this data pooling technique seems to be prevalent in
practice.
Estimation of future renewal rent offers:!
ri./ are unknown for any future renewal
times , and need to be estimated. We estimated it with the average of renewal
rent offers from historical expiring leases. As an alternative, we can consider to
use other methods such as DCF and RM models.
Alternative customer choice models: The MNL model that is used in estimating
the renewal probabilities is popular in practice. It has many advantages such
as being simple to understand, and easy to use. However, it sometimes suffers
from an inherent assumption of Independence of Irrelevant Alternatives (IIA)
(Meyer and Kahn 1991). This IIA assumption presumes that tenants would
ignore the similarities among alternative lease terms when they make a renewal
decision, which might not be always true. To mitigate this issue, other customer
choice models such as Nested Logit (NL) model (McFadden 1981) can be taken
into account. One of challenges of using NL model, for example, is to cluster
“similar” lease terms into a group. Doing this “right” is not easy in practice.
Some unsupervised learning techniques in data mining field might be needed.
The results may seem rudimentary, but they can still provide apartment commu-
nities with some insightful knowledge about the values of their tenants. A good
estimate of CLV of the current tenants can be an additional key metric in assess-
ing the financial value of the apartment property in comparison to other competing
multifamily assets or other sister properties in an owner’s portfolio. Since the apart-
ment industry has a competitive market environment, tenant behaviors might change
quickly over time. As a consequence, the prediction of lifetime length and value can-
not just be evaluated once and kept unchanged. They need to be updated regularly
to reflect possible changes in tenant behaviors.
Acknowledgments We would also like to thank the three anonymous reviewer for their suggestions and
comments. We would also like to show our gratitude to Ms. Marlene Rinker, our former colleague from
The Rainmaker Group, for her revision on an earlier version of the manuscript, although any errors are our
own and should not tarnish her esteemed reputation.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-
tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
K
Z Immobilienökonomie (2016)
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