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A generalized model of ground lease pricing
Marcin Anholcer1 and Maria Trojanek2
July 16, 2017
Abstract
In the paper we presented a generalization of Mandell’s Model for the estimation of ground lease
pricing. We adjusted the model so that it fits in particular the Polish legal regulations and situation on
Polish real estate market. Our model involves in particular two issues. The first is the perpetual
usufruct, a form of possessing the ground similar to the long-term lease, but having some specific
features. Another generalization consists in allowing the lease rent adjustments made after some fixed
periods (i.e., we consider the situation where the payments are fixed during some periods defined in
the contract).
Keywords: ground lease, ground rent, perpetual usufruct
1. Intoduction
The public real estate management in Poland is governed by the Act on Real Estate Management
(Journal of Laws of 2014, item 518) and private ones in the Civil Code (Journal of Laws 1964, item 93,
as amended).
Legal regulations provide that real estate may be the subject of sale, lease and perpetual usufruct (this
right applies only to properties owned by the public, i.e. Treasury and local government units).
The right of perpetual usufruct possesses features similar to the perpetual lease rights of other
countries, although these rights vary. In Poland, in addition, there is the right of long-term lease, used
in very limited way. Choice of forms of real estate management (sale, perpetual usufruct or lease)
should be made taking into account the legal provisions and the objectives and criteria for the
1 Poznań University of Economics and Business
Faculty of Informatics and Electronic Economy Al. Niepodległości 10, 61-875 Poznań, Poland Email: [email protected]. Corresponding author.
2 Poznań University of Economics and Business Faculty Management Al. Niepodległości 10, 61-875 Poznań, Poland Email: [email protected].
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management of landowners, as well as the expectations and benefits of potential buyers, perpetual
usufructors and tenants.
The study attempted to identify the parameters determining the choice of management form (sales,
perpetual usufruct or lease) of land, taking into account the interests of each party to the transaction
(owner and tenant / perpetual usufructor). The presented model enables the determination of limit
rents, which equates to updated revenues from perpetual usufruct or sale of real estate.
1.1. Legal basis for perpetual usufruct
Perpetual usufruct is one of the forms of management of public real estate (properties owned by the
State Treasury and local government units or their unions). In the Polish legal system perpetual
usufruct occurred more than half a century ago (1961), when the system of state ownership was the
dominant form of ownership. The legislature intended this right to regulate the ownership relations
between the state and the entities of the private sector. The current regulations of perpetual usufruct
significantly differ from those introduced in 1961 with the Law on the Economy of Land in Cities and
Settlements (Journal of Laws of 1961 no. 32 item 159; see Winiarz 1967, Truszkiewicz 2006 for details).
In addition, despite the decreasing presence of the political reasons for the introduction of this right,
the perpetual usufruct continues to play an important role in the system of rights in rem and affects
various aspects of social and economic life.
Perpetual usufruct is an instrument permitting long-term (40 to 99 years) usage of land without the
need to involve capital in the amount equivalent to the acquisition of the ownership.
For land property put into perpetual usufruct, the first fee and annual fees are collected (Real Estate
Management Act of 21 August 1997, Article 71, item 1). These fees are determined at a percentage of
the land price determined in accordance with the provisions of art. 67 of the cited Act i.e. price
determined on the basis of value. If land property is given for perpetual usufruct by auction, the basis
for the calculation of such fees is the price obtained during the auction. On the other hand, if the right
of perpetual usufruct is established by the open market rules, the price that is the basis for the
calculation of the first and the annual fee is the price of the property determined at a level not lower
than its market value. The amount of first and annual fees varies and depends on the specific purpose
of the contract. The first fee amounts to 15-25% of the land price. On the other hand, the percentage
of annual fees may be e.g. (Art.72, item 3):
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− for land properties committed to the defense and security of the state including fire protection –
0.3% of the price,
− for land properties committed for charity purposes and for non-profit caring, cultural, curative,
educational, scientific or research and development activities – 0.3% of the price,
− for land properties made for agricultural purposes – 1% of the price,
− for land properties completed for residential purposes, for the implementation of technical
infrastructure and other public purposes and sports activities – 1% of the price,
− for land properties for tourism – 2% of the price,
− for other land properties – 2% of the price.
The annual fee may be increased (Article 76, item 1) by the governor (in relation to property owned by
the State Treasury) or by a resolution of the appropriate council or regional government. An increase
in the rate may take place before the property is handed over for perpetual usufruct and is for real
estate for which the annual fee, due to the purpose of putting the property, is 3%. In some cases, the
competent authority may grant a discount (Article 73, Paragraph 2a, item 3).
The first payment is made in the year in which the perpetual usufruct was established, and the annual
fees are paid during the perpetual usufruct, until 31 March of each year, in advance for the given year.
Thus, the amount of proceeds from these charges is dependent on:
− regulations that the municipality does not influence (interest rates),
− the level of prices achieved on auctions or the value of real estate delivered in perpetual usufruct
in the open market mode,
− the adopted resolutions on granting discounts on established fees.
Annual fees for perpetual usufruct vary over time, taking into account changes in the value of land
properties. According to Art. 77 item 1, the annual fee for perpetual usufruct is subject to updating no
more than once every 3 years, provided the value of the property changes. The updated annual fee is
determined using the fixed percentage of the current level from the market value as of the update
date.
A detailed regulation was introduced in two cases. If the value of the property as at the date of
updating the annual fee would be lower than the price of that real estate determined on the date of
its submission for perpetual usufruct, then the update of the fee will not be made. This provision shall
apply to immovable property transferred for perpetual usufruct of property for a period of 5 years
from the date of conclusion of the contract for the transfer of land title to perpetual usufruct (Article
77 item 2). If the updated annual fee is at least twice the current annual fee, the perpetual user pays
an annual fee equal to twice the current annual fee. On the other hand, the remaining amount, more
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than double the current fee, is divided into two equal parts, which increase the annual fee in
subsequent years.
The annual fee is updated ex officio or at the perpetual usufructor application (the reason is usually
the reduction of the value of the property). To reassume the above, the updating of annual fees for
perpetual usufruct is possible:
− no more than once every three years,
− provided that there has been a change in the market value of the land property.
1.2. Legal basis for lease
Unlike perpetual usufruct, the lease institution is a form of ownership known not only in Poland, but
also in other European countries as well as in other continents.
In Poland, legal regulations of lease agreements, as civil law institutions, are contained in the provisions
of the Civil Code Act of 23 April 1964 (Articles 659-709). As stated in Article 693 of the Civil Code by
the lease agreement, the landlord undertakes to give the tenant the property to use and retrieve the
property for a specified or unspecified period of time, and the tenant agrees to pay the rentee the
agreed rent. The lessee has the right to use the property, i.e. the income it earns, but is obliged to take
care of the proper condition of the leased property (e.g. carry out repairs at its own expense) and
return the object at the end of the lease term in a specified contract.
The subject of the lease may be real estate, machinery, equipment, means of transport and other
property. The most often leased property is a public or private real estate. In the case of public
properties, because of the presence of the perpetual usufruct, long-term lease (up to 30 years) is used
in relatively small terms.
Communes usually lease land for a fixed term (up to three years) with the option of renewing the lease.
Leases are usually applied to land whose destination is different (they are reserves of land for future
public purposes) or not specified (no local land use plans). When land is leased, the landlord does not
bear the cost of maintaining property, providing security, and is gaining the rent for the duration of
the lease.
The scope of use of lease or detailed decisions vary from country to country and changes over time.
However, the most common issues that are resolved when entering into a lease agreement include
the following (see e.g. Trojanek 2015):
− property of buildings or objects made by the lessee,
− duration of the contract and the possibility of its renewal,
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− settlements for expenditures incurred on the ground,
− level and updates of lease rental rates,
− basis for determining the rent of the lease,
− possibility to buy land by the lessee,
− possibility of subletting,
− way of land management,
− other arrangements such as the proper maintenance of buildings located on land, real estate
insurance, etc. (Trojanek 2015).
The source of potential problems associated with operating the lease may be (see e.g. Mandell 2001,
Nystrom 2007):
− lack of adequate change in lease rates in relation to changes in land value,
− resistance and opposition of tenants caused by changes in lease rates,
− difficulties and costs involved in valuing and securing the interest of each of the parties to the
lease,
− problems with how to set lease rates (most often are estimates rather than market rates)
− widespread presence of individual negotiations instead of market or auction procedures,
− costs associated with the lease being sometimes higher than the cost of selling the land,
− lack of benefit for the owner in the case of non-economic use of land by the lessee.
1.3. Lease vs. perpetual usufruct
We summarized the basic features of lease and perpetual usufruct in Table 1.
Table 1: The features of perpetual usufruct, perpetual lease (present in chosen countries) and long-term lease of
land properties.
Feature Perpetual usufruct Perpetual lease Long-term lease
Ownership of
buildings
Pepretual usufructor Lessee Lessor
(If the lessee built the
building, they can claim for
reimbursement)
Duration 40-99 years Depending on country, e.g.
20-30 years, 50-99 years,
200 or 999 years
No longer than 30 years
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Table 1: The features of perpetual usufruct, perpetual lease (present in chosen countries) and long-term lease of
land properties (continued).
Feature Perpetual usufruct Perpetual lease Long-term lease
Possibility of
extension
For next 40-99 years Various possibilities,
depending on the contract
Various possibilities,
depending on the contract
A lease concluded for a
period longer than 30 years
is treated as a lease
concluded for an indefinite
period of time (then
termination at any moment
is possible)
Method of
settlement of
expenditures
incurred on the
ground
Remuneration defined as
the market value of
buildings and other
equipment, and if, due to
the type of real estate,
market value cannot be
determined, the
replacement value is
determined
Compensation calculated as
the difference between the
market value of developed
land and the value of land
without buildings
By default, without
reimbursement of
expenditure on the subject
of the lease
Other arrangements must
be included in the contract
The rules of
setting the fee /
rent
First fee
Annual fees as a fixed
percentage of market value
of land
There are contracts in which
the first fee is established
Most usually the rent is a
constant percentage of the
current market value of the
land
Additional fees are often
added to the basic rent
Based on the rental rates of
similar land properties
In a parametric way as the
percentage of market value
through negotiations
(arrangements) between
the owner and the lessee
The amount of
the fee / rent
Differentiation according to
purpose
Differentiation according to
purpose, location, form of
the ownership (public or
private)
Differentiation according to
purpose, location, form of
the ownership (public or
private)
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Table 1: The features of perpetual usufruct, perpetual lease (present in chosen countries) and long-term lease of
land properties (continued).
Feature Perpetual usufruct Perpetual lease Long-term lease
Adjusting the
fee / rent
Not more often than once
every 3 years provided that
the market value of the
property has increased
Periodically update every 3
to 4 years to adjust to
market rents
As claimed in the contract,
e.g. in correspondence with
the CPI
Preemption For perpetual usufructor To the lessee Not present
Determining
the way of using
the land and the
duration of
development
Yes Yes Yes
Law established
on land owned
by
State Treasury or local
government units
Any private or public subject Any private or public subject
As we can see, long-term usage of land being someone other’s property, without the need to purchase
it, as well as the ownership of buildings or facilities by a perpetual usufructor or lessee occurs in many
countries, although the details vary according to social, economic and law determinants (see
Truszkiewicz 2006; Kokot 2009; Dobek, Wajszczuk and Wielicki 2009 for more details).
In the remainder of this paper, considering the legal regulations concerning the lease and perpetual
usufruct of public land in Poland, the circumstances determining the choice of a particular form using
the land, taking into account the interests of each contracting party, have been identified. The
proposed model indicates when (at which rate of lease) it is better to sell or lease perpetual usufruct
or rent (landowner's point of view) or buy, take into perpetual usufruct or lease (the holder of
perpetual usufruct or tenant point of view). The proposed model can be used as a tool for optimizing
decision making in land management, and a fixed stake may be a benchmark in negotiating rates with
a potential lessee. This problem is significant given that real estate generates significant profits to the
owner of the land. On the other hand, the derived lease rates from the point of view of the tenant's
interests indicate the choice of a specific form of land usage (ownership, lease, perpetual usufruct).
The perpetual usufructor or lessee must include the cost of acquiring that right (first and annual fees
or rent) in the profitability account of the business. The negotiated lease rate is part of the cost of
doing business on the leased property. Hence, the negotiated rate of rent should secure the interests
of each party (owner and tenant, owner and perpetual usufructor).
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2. Literature review
Various legal and financial aspects of leasing attracted some attention of the researchers in the past
years. Let us recall some of the works.
Long (1977) studied the relationships between leasing and the cost of capital. Lewis and Schallheim
(1992) studied the question whether leases and debts should be treated as substitutes (which is a
rather common assumption) or rather as complements. The same issue was studied by Yan (2006). Liu,
Liu and Zhang (2016) investigated the relation between the asset liquidation values and real estate
firm’s financing choice. Clapham and Gunnelin (2003) considered the term structure of lease rates,
where both rents and interest rates are were stochastic. In particular, they derived equilibrium
relationships in a general continuous time setting. Lally and Randal (2004) investigated the influence
of a ratchet clause on the rental rate for a lease of periodically re-evaluated ground.
Fallis (1988) discussed the legal issues of lease, in particular he performed the public choice analysis of
the rent control. Mooradian and Yang (2002) studied the influence of the assymetric information on
the choice of lease type. Özdilek (2016) studied the property valuation with separate values of land
and buildings, based on example of Montreal. Pope, Gallimore and Shi (2016) investigated the risk
related to a leasehold property from the investor’s point of view. Similar topic were investigated by
Rabbani et al. (2015). Sevelka (2011) studied the influence of the rent reset clause on the valuation in
the ground lease problem. Yoshida, Seko and Sumita (2016) studied the rent term premium for
cancellable leases, by modelling the lessor’s tradeoff between the costs of leasing and the costs of
cancellation.
Benjamin and Chinloy (2004) developed an option – theoretic model of a retail lease. In particular, they
studied economic tradeoffs between base rent, security deposit and percentage rent provisions.
Capozza and Sick (1991) developed the model for valuing leased properties that involved the option to
upgrade or redevelop. Similar topic was studied by Dale-Johnson (2001). Halim and Jaaman (2013)
developed a mathematical model for a specific type of contract, named AITAB. McCann and Ward
(2004) analyzed the rental payments using a multi-period stock inventory model and show that the
factor that influences most the tenat’s value of the land is the duration of the lease. Stanton and
Wallace (2009) proposed a no-arbitrage based lease pricing model. Steele (1984) considered a lease
valuation model using difference equations. Wheaton (2000) presented a two-period model for retail
rental contracts. Country-specific models and case studies with respect to chosen factors were
presented also e.g. by Walters and Kent (2000), Guntermann and Thomas (2005), Tian (2006),
Kaganova, Akmatov and Undeland (2008), Guarino, Chehrazi and Bohl (2014), Irumba (2015).
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Several papers in this subject were published by Jefferies (1997a, 1997b, 1998, 2005a, 2005b, 2005c).
In particular, he considered Lessor’s Return Model and used it to analyze various aspects of the real
estate market, in particular in New Zealand. The model was further developed by Mandell (2001,
2002). A similar approach, but with simplified formulae, were used by Trybusz (1993). See next section
for further discussion of this topic.
3. The Model of Mandell
As we already mentioned above, the main goal of our paper is presentation of a model for ground
lease pricing that would be applicable in countries having legal situation similar to Poland. Our model
is a generalization of the combined model applied by Mandell (2002). In particular, it involves
additional form of ground rental, called the right of perpetual usufruct (RPU).
The model of Mandell involves two sides of the contract: lessor and lessee. Let us first recall the
Lessor’s Return Model (LRM) presented in this article. It was firstly introduced by Brown. Although not
published, it was referred to by Jeffries (1997) and a similar model was considered by Brown in another
context (Brown 1995).
Let � denote the initial ground rent, that is fixed for � periods and then may be changed according to
a growth rate that lessors expects to be equal to �. Then the present value of the future stream of
perpetual payments equals to (c.f. Mandell 2002, formulae (1-2)):
���� = � 1 − 1 + ���� � �1 + �1 + ������� = � 1 + �� − 11 + �� − 1 + ���.
Here denotes the discount rate assumed by the lessor. In the LRM, the lessor has to choose between
selling and leasing the property. Let us denote the present market value of the property by �. The
ground rent is usually defined as a percentage � of the market value:
� = ��. The lessor prefers leasing the land to selling it if the preset value of all the payments is not less than
the present market price i.e., when
���� ≥ �. This means that the ground rent rate � must satisfy the condition (c.f. Mandell 2002, formula (5)):
� ≥ ���� = 1 + �� − 1 + ���1 + �� − 1 .
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On the other side of the market there is the lessee, who also considers two possible decisions: leasing
or buying the property. In the Lessee’s Affordability Model (LAM) presented by Jefferies (1997), the
present value of leasing the land and the present value of freehold are compared. It is assumed that
the buyer needs to finance the buying by through loans and the interest rate (i.e., the cost of capital)
equals to �. It is also assumed, however, that in the case when there is a choice between buying the
property for � and leasing it for ��, the psychological aspects will make the decision maker to choose
the freehold. For that reason, Jefferies assumes that the discount rate on the leasing side of the
equation should be less than �. To be more specific, it is defined as
� = ��, where � ∈ 0,1" is the reduction ratio (also called �-factor). The present value of leasing for one
period is equal to (c.f. Mandell 2002, formula (8)):
#�$ = %� 1 − 1 + �����
where % is the rent expressed as the percentage of the market value �. The present value of the
freehold financed with loan (for one period) is in turn expressed by the formula (c.f. Mandell 2002,
formulae (9-10)):
#�& = �� 1 − 1 + ����� = �1 − 1 + �����. Obviously, the lessee will prefer the leasing variant if
#�$ ≤ #�&
i.e., when (c.f. Mandell 2002, formula (13)):
% ≤ %� = � 1 − 1 + ����1 − 1 + ����. As indicated by Mandell (2002), this model is not satisfactory. The main disadvantage is that it is not
completely clear how to compute the �-factor. To be more specific, it is necessary to introduce LAM
to derive the correct value of �, but it is necessary to know the value of � in order to introduce LAM
(see Mandell (2002) for details and for the discussion of other disadvantages of Jefferies’ approach).
This is why another model was proposed. We fully agree with Mandell that this model reflects the
reality in a much better way. Let us assume that the lessee expects that the future growth rate (in the
period of length �) of the market value � of the property equals to ℎ. They also expect that the present
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value of the benefits of possessing (not necessarily of being the owner of) the property is equal to ).
Under such assumptions, the present value of the net benefits of the freehold equals to
#�&* = ) + +1 + ℎ��1 + ��� − 1, �, while the present value of the leasing is (c.f. Mandell 2002, formula (14)):
#�$* = ) − %� 1 − 1 + ����� . Since the lessee chooses the lower value, the present value of the benefits from leasing must be not
less than the present value of the benefits from the freehold, which means that the following condition
must hold (c.f. Mandell 2002, formula (16)):
% ≤ %�-� = � 1 + ��� − 1 + ℎ��1 + ��� − 1 . The combined model assumes that the payment rate lies between ���� and %�-� . Let us present its
generalization, involving specific legal conditions present in Poland.
4. The Generalized Model
As mentioned above, third variant of possessing the land occurs in Poland – It is the perpetual usufruct.
In this case not only the regular rent, but also the initial payment . = /� has to be paid (here / is the
initial payment expressed as the ratio of the property’s market price �). Then the yearly rent is
calculated as the ratio of the market price �. The contract ends after 0 years and may be renewed. In
the remainder of this section we are going to present the estimations of the present value of all the
income from the perspective of the property’s owner, then from the perspective of the potential user.
In the last subsection we will compare the results in order to define the intervals to which the payment
rates should belong.
4.1. The Owner’s Perspective
First possible decision of the owner may be to sell the property. In such case, the present value of the
income may be estimated by the market price of the property:
#�1 = �.
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Next possible choice is to lease the property. It may take two forms: the perpetual usufruct and the
regular lease. Let us start with the analysis of the first form of leasing. In the beginning, the initial
payment is made. As mentioned before, it is equal to . = /� i.e.,
. = /#�1. Then, each year, the lessee needs to pay the yearly rent � = �2�, where �2 denotes the rate and �
the market price. The payment is fixed during some period expressed in number of years (denoted by 32) and then is adjusted according to the yearly increment rent 42, often deduced from the increment
of the property’s market price. This means that the yearly payment (in constant prices) increases after
each 32 years by the growth factor
� = �1 + 421 + �56 , where is the expected discount rate. Since the total number of years 0 of the perpetual usufruct does
not need to be divisible by the number of years, where the rent is fixed, we need to distinguish the
number of full periods and the remainder. The number of full periods during which the payment is
fixed, will be denoted by 02� and may be computed from the formula:
02� = 7 0328 = 0 div 32, where ⌊⋅⌋ denotes the floor function i.e., the biggest integer not greater than the argument (e.g., ⌊1.2⌋ = 1, ⌊13.77⌋ = 13), while div is the integer division function. The number of remaining years
(i.e., the number of years in the last period not longer than 32, where the payment remains fixed) is
then equal to
02C = 0 − 3202� = 0 mod 32,
where mod denotes the remainder of the integer division. This means that the value of the yearly
payments during 32 years after 3 changes of the payment rate (3 = 0,1, … , , 02� − 1), at 332 years from
the beginning (i.e., after 3 fixed payment periods) equals to
� �2�1 + �G56
G�C = �2� × 1 − 1 + ��56 . This means that the present value of the stream of all the payments made during all the complete fixed
payment periods is equal to
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�2� × 1 − 1 + ��56 × � �GI6JG�� = �2� × 1 − 1 + ��56 × � �1 + 421 + �56GI6J
G��
= �2� × 1 − 1 + ��56 × 1 − K1 + 421 + L56I6J
1 − K1 + 421 + L56 . The present value of the stream of the payments made during the remaining 02C years equals to
� �2�1 + 42�56I6J1 + �56I6JMGI6NG�C = �2� × 1 − 1 + ��I6N × �1 + 421 + �56I6J .
For simplicity of the further notation, let us define the factor O2 as
O2 = 1 − 1 + ��56 × 1 − K1 + 421 + L56I6J
1 − K1 + 421 + L56 + 1 − 1 + ��I6N × �1 + 421 + �56I6J . The total income of the property’s owner is equal to the sum of the present values of the initial
payment, yearly payments and the discounted future value of the property, including the expected
average increase of its value 4. Taking into account the above considerations, the present value of the
income after choosing the variant of the perpetual usufruct is equal to
#�2 = /� + �2O2� + �1 + 41 + �I �. Note that if the payments may by adjusted each year i.e., when 32 = 1, the formula for O2 simplifies
to the form
O2 = 1 − K1 + 421 + LI − 42 .
Similar considerations allow us to analyze the case of regular lease contract. This time, the number of
years in a fixed payment period will be denoted by 3P. The number of such complete periods equals to
0P� = 7 03P8 = 0 div 3P ,
while the number of years in the remainder is equal to
0PC = 0 − 3P0P� = 0 mod 3P . As before, if we define the factor
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OP = 1 − 1 + ��5Q × 1 − K1 + 4P1 + L5QIQJ
1 − K1 + 4P1 + L5Q + 1 − 1 + ��IQN × �1 + 4P1 + �5QIQJ , then the present value of the total income of the owner is equal to
#�P = �POP� + �1 + 41 + �I �, In particular, if 3P = 1 (i.e., if the yearly payments are adjusted each year), then
OP = 1 − K1 + 4P1 + LI − 4P .
Given the three variants, the owner will choose the lease if #�P ≥ maxT#�1, #�2U. This allows us to
derive the minimum payment rate that is acceptable for the owner. First, let us compare lease with
selling the property:
#�P ≥ #�1, �POP� + �1 + 41 + �I � ≥ �, �P ≥ 1 − K1 + 41 + LI
OP .
Similarly, the comparison with the perpetual usufruct allows us to find the following lower bound:
#�P ≥ #�2, �POP� + �1 + 41 + �I � ≥ /� + �2O2� + �1 + 41 + �I �
�P ≥ / + �2O2OP . Finally, the yearly payment rate must satisfy the condition
�P ≥ max V1 − K1 + 41 + LIOP , / + �2O2OP W.
if the owner is supposed to choose the leasing.
4.2. The User’s Perspective
The calculations similar to the ones performed in the previous subsection allow us to estimate also the
costs of the potential user (buyer or lessee). We assume that the user may cover part of the cost using
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the loan with interest rate 4X, that must be paid back in 0X years. We assume that the loan is paid with
decreasing instalments, 0X fixed principal payments and respective interest payments (in case of
different payment schedule, one may easily transform the formulae given below using the elementary
rules of financial mathematics). We assume that part of the cost, Y1 < 1, must be paid in cash. Then
the total present value of the cost of buying the property is equal to
#�1⋆ = Y1� + 1 − Y1� × �0X × 1 − K1 + 4X1 + LI\1 − 1 + 4X1 + .
We may observe that
#�1⋆ = #�11 + ]1�, where
]1 = Y1 + 1 − Y1�0X × 1 − K1 + 4X1 + LI\1 − 1 + 4X1 + − 1 > 0
(i.e., #�1⋆ > #�1).
Similarly we can estimate the present value of the cost of perpetual usufruct and lease. In both cases
we take into account the fact that when the contract ends, the property goes back to the owner.
Let us consider the perpetual usufruct first. We assume that the potential user may cover part of the
initial payment (Y2) using the bank loan. This means that the present value of the cost of the perpetual
usufruct is equal to
#�2⋆ = /#�11 + ]2� + �2O2�, where
]2 = Y2 + 1 − Y2�0X × 1 − K1 + 4X1 + LI\1 − 1 + 4X1 + − 1 > 0.
Finally, the present value of the total cost corresponding with lease equals to
#�P⋆ = �POP� = #�P − �1 + 41 + �I �.
The reasoning similar as in the previous section leads us to the conclusion, that the user will accept the
form of leasing if #�P⋆ ≤ minT#�1⋆, #�2⋆U. This allows us to estimate the highest payment rate
acceptable by the user. We have
#�P⋆ ≤ #�1⋆, �POP� ≤ �1 + ]1�,
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�P ≤ 1 + ]1OP , and
#�P⋆ ≤ #�2⋆, �POP� ≤ /�1 + ]2� + �2O2�, �P ≤ /1 + ]2� + �2O2OP .
Thus, if the user is supposed to choose the form of leasing, the following condition must be satisfied:
�P ≤ min `1 + ]1OP , /1 + ]2� + �2O2OP a.
4.3. The range of acceptable leasing rates
Taking together the inequalities deduced in the two preceding subsections, we obtain the following
interval for the leasing rate:
�P ∈ bmax V1 − K1 + 41 + LIOP , / + �2O2OP W , min `1 + ]1OP , /1 + ]2� + �2O2OP ac.
Only if the above interval is nonempty, the leasing may occur. This is not an obvious thing. Although it
is straightforward to see that
1 − K1 + 41 + LIOP < 1 + ]1OP
and
/ + �2O2OP < /1 + ]2� + �2O2OP , the relations between all four parts of the formula may be various in the general setting. This means,
that under some specific conditions, the leasing is not acceptable.
4.4. The perpetual usufruct payments
Although the yearly payments in perpetual usufruct are fixed by the law and thus not negotiable, one
could wonder whether they are set at the right level. Of course changing the law is much more difficult
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than signing a contract, but they are still possible. The reasoning similar as above allow us to define
the acceptable range for the yearly payment in the perpetual usufruct. If it is the variant to choose
against the selling/buying the property, then the following must hold:
#�2 ≥ #�1, #�2⋆ ≤ #�1⋆. The first inequality is equivalent to
/� + �2O2� + �1 + 41 + �I � ≥ �, i.e.,
�2 ≥ 1 − / − K1 + 41 + LIO2 .
Similarly, the second one leads us to
/�1 + ]2� + �2O2� ≤ �1 + ]1�, �2 ≤ 1 + ]1 − /1 + ]2�O2 .
It follows that the rate of yearly payment in the perpetual usufruct should belong to the following
interval:
�2 ∈ b1 − / − K1 + 41 + LIO2 , 1 + ]1 − /1 + ]2�O2 c.
5. Conclusion
The generalization of Mandell’s model presented in the article allows to involve various additional
elements that are present e.g. in the Polish real estate market. In particular, we took under
consideration the third possible form of possessing the ground, which is the perpetual usufruct. We
also analyzed the more general setting, where the yearly payments are not adjusted every year, but
nay be fixed for some period, defined in the contract (which is also the case in Polish market).
This theoretical paper was used as a base for computational experiments. Currently we are preparing
a follow-up paper, in which we present the results of simulations performed for various scenarios, with
the data specific for Polish real estate market. The first conclusions from the experiments are proving
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our suppositions that the leasing rates used in Poland in the leasing contracts between local
communities and individuals have been improperly defined.
Another conclusion is that the yearly payment rates in perpetual usufruct are often incorrect. We are
aware of the fact that any changes in the law are a difficult and time consuming process, but maybe
our conclusions could be a starting point to the further considerations in this area.
There are some limitations of the presented model. For instance, we assumed that the durations of
leasing and perpetual usufruct are equal. This could be overcome by introducing two, possibly distinct,
durations. However, since after ending one leasing contract, another may be started immediately, one
could treat such a leasing as a continuous process. Another limitation are the assumptions about the
constant character of the parameters. Of course, the parameters like interest rates change in time, but
it is almost impossible to predict their future values, especially in emerging markets like Poland. For
that reason some simplifications are necessary. On the other hand, we discuss the present situation
and decisions of the owner and possible user, so analyzing the possible scenarios based upon the
current market situation seems to be reasonable, if we assume the symmetry of information. Yet
another problem is how to estimate the values of the parameters, even if we assume that they will be
constant in the future. This is however not a significant problem, since we may assume again that both
sides of the contract have the same knowledge about the market and so if one of the sides
underestimates or overestimates any of the parameters then so will do the other, which finally will
increase or decrease both estimates of present values of income and cost and finally will have a small
influence on the estimation of rates.
As we mentioned in the beginning of this section, we performed some numerical experiments. We
think that also performing simulations for other countries, having similar legal situation, would be
interesting. We also think that the model itself could be further developed by introducing additional
parameters, as expected income of the user resulting from the possession of the property compared
with other possible forms of earning.
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