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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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Z Immobilienökonomie (2016)

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 Lifeme

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.

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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

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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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