Valuation of Real Estate Investments through Fuzzy Logic

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Introduction
Non-conventional methods for real estate property valuation have increased in the last several years, albeit most studies looked at neural networks with mixed results [1][2][3][4][5][6]. Others researchers have suggested the use of fuzzy logic-based methods, although there have been only a few works that show the application of Fuzzy Logic to assessing real estate investments.
A Fuzzy Logic framework has been proposed as an alternative to the traditional probability-based methods for property assessment [7][8][9][10][11][12][13][14][15][16][17]. These researchers have demonstrated the application of fuzzy analysis software to real estate appraisal, showing that fuzzy analysis could reduce uncertainty to a limited extent.
Artificial neural networks and Fuzzy Logic are the main two non-conventional approaches that have been often applied or suggested for valuation of real estate investments. Unlike probabilistic methods, Fuzzy Logic allows for representation mathematically, through a calculation system, judgments without exact and univocal definition: the deterministic statement "the value of this input is X" is replaced by the possibilistic assertion "the value of this input is approximately -X". It assumes, therefore, that uncertainty presents possibilistic character rather than probabilistic, and that uncertainty could depend on the perception of eligibility for a certain event, rather than from its degree of statistical confidence.
The fuzzification of a numeric variable (as income or financial costs, but also discount rate, etc.), when its measure is reasonably uncertain, may consist in the assignment of more than one possible value to the variable.
Preliminarily, the measurement (or value) that is believed most probable is assigned to a variable-for example, the best estimate or "most probable" value of the variable.
Other elements of the fuzzy set can be constituted by the highest or lowest possible values; if the same values are not part of the set of values that can be assumed by the variable, it is also possible to choose to give these values a degree of membership to fuzzy set equal to 0.
The "most probable" value, however, could have a degree of membership in the fuzzy set equal to 1, considering that the "best appraisal" identifies a value still belonging to the set of values that the variable in question may take.
In so doing, any other value attributable to the variable between the highest and lowest values will be placed within the fuzzy set with a membership degree between 0 and 1.
The graphical representation of the variability of membership degrees to fuzzy sets, for every value assumable by variable, can satisfy different forms according to knowledge about the possibility of occurence of different values attributable to the variable.
In this way, it is possible to highlight a parallelism with the logical process that leads to the attribution of subjective probabilities to a given variable values.

Literature Review
Hedonic approaches have been utilized extensively in real estate literature to valuate real estate investments or the relationship between selling prices and characteristics owned by properties [18,19].
Advancements in data processing techniques have had a wide impact on the real estate appraisal process. These have led to the use of more complex analytical applications, such as Artificial Neural Networks, Fuzzy Logic and Expert System [10].
Similarly to Multiple Regression Analysis (that show all the limits in the presence of outlier data, or when the analytical function is nonlinear or non-normal) [20,21], Artificial Neural Networks (ANN) models predict the value of a dependent variable (such as house price or rent) through the values of independent variables (real estate features), but they are subjected to some relevant limits as overfitting problems when monotonic is the relationship between input and output data of ANNs [22]. Fuzzy Logic applied to predict real estate prices showed good results when the structure of input and output variables is complex and nonlinear.
According to Zurada et al. [10], the use of Fuzzy Logic is similar to the human brain function in decision making, then considering all quantitative and qualitative input data for the better solution (output). The main aspect and point of strength of Fuzzy Logic in the prediction of property prices or in the valuation of real estate investments is the use of less quantifiable data [10].
Fuzzy Logic was implemented by Zadeh in the 1960s [17] as a technique able to model the improbability between normal spoken and written language. For this reason, Fuzzy Logic included some qualitative linguistic functions.
According to Steele [23], Fuzzy Logic is founded on the "degree of truth" rather than on Boolean logic that uses dichotomic values. For which Fuzzy Logic would be an extended Boolean logic superset introducing the concept of partial truth [24].
As reported, taking into account its original version, Fuzzy Logic may be viewed as a computing technique that uses words rather than numbers. Even if words have more imprecision than numbers, their use is closer to human behaviour exploiting the tolerance for imprecision [25].
In many applications, Fuzzy Logic uses "if/then" rules often applied in the past in Artificial Intelligence models. For approximate reasoning that allows for predicting estimated values with incomplete information, Fuzzy Logic was also noted as useful [23].
With reference to the real estate field, Bagnoli and Smith [33] showed the application of Fuzzy Logic to real estate valuation, providing as a result a fuzzy set output but neglecting some relevant real estate risk factors. Sun et al. [34] have used a fuzzy analytical hierarchy process for the evaluation of risk in residential real estate projects through linguistic variables rather than crisp values. Cui and Hao [35] are interested in the fuzzy cost approach for real estate purposes and for determining the building depreciation over time. Barranco et al. [36] experimented with a web-application based on a fuzzy set and applied to real estate management (with real estate attributes expressed by fuzzy data). In Krol et al. [37], a Mamdani-type model and a Takagi-Sugeno-Kang-type model have been compared (two particular fuzzy models useful for real estate appraisal) in order to determine the fuzzy rule bases for both models, concluding that fuzzy models provide an acceptable solution for real estate appraisals.
Based on a hedonic approach to predict the selling price for a new property, Liu et al. [38] developed with success a fuzzy neural network model, in which database storage for real estate features included numbers, sets and fuzzy operations.
The fuzzy sets are based on the canons of a polyvalent logic that go beyond Aristotle's setting founded on the "non-contradiction" principle and on third exclusion principle [16].
They may to be represented by three modalities: a. Using a set with ordered pairs, where the first term represents the element, while the second term represents its membership degree to subset of a given universe. For example: is the representation of a subset where the three elements 1,2,3 have a different membership degree; b. Defining a membership function. A fuzzy set could be described as follows: = ( , ( )) | ∈ ; where the elements are individuated by a membership function that links each element to a subset of universe.
The membership function of the single element in the subset A could be described, in formal terms, by the following relations: A particular type of fuzzy set is so-called fuzzy numbers, which may be considered as an extension of the concept of probabilistic confidence interval used for inaccurate data.
It is possible, in fact, to define a set of several confidence intervals not only to a level, but with more levels included between 0 and 1.
A fuzzy number is a particular fuzzy subset, belonging to the set of the real numbers, with a membership function continuous ( | ) that can satisfy the following properties: To represent the membership function of a fuzzy number A, the following relation is used: where: The function = ( | ) will be so defined: A trapezoidal fuzzy number is a fuzzy number with linear functions f1 and f2 such that: In this case, the membership function is uniquely identified by means of the quadruple ( , , , ).
If a2 = a3, the fuzzy number is triangular and it could be identified by means of the triple ( , = , ).
As for the fuzzy sets, also for the fuzzy numbers, it is possible to define a normal condition, or if elements present a membership function equal to 1; otherwise, a fuzzy number is abnormal.
Through a defuzzification process, it is possible to transform a fuzzy number into a crisp number [2]. In particular, one of the most common procedures applied to this process consists in calculating the arithmetic mean of the values, which characterize the fuzzy number, or more generally, the possibility distribution. Using the "extension principle" of Zadeh, all possible operations with crisp numbers can be easily extended to the fuzzy numbers [3].
If f is a function defined in R n with values in R, and if A1, A2… An, are fuzzy numbers, then it is possible to define the fuzzy number B = f(A1, A2, …, An) with following membership function: Through this principle, a variety of operations, including reversal, sum, difference, product, etc., have been extended to fuzzy numbers from Dubois and Prade [4].

Difference.
A  B is a fuzzy number such that: 3. Product by a scalar k. k  A is a fuzzy number such that: (x| k  A) = (ka1, kf1(y|A)/ka2,ka3/kf2(y|A),ka4).

Exponentiation scalar k.
A k is a fuzzy number such that:  (x|A k ) = (a k 1, f1(y|A) k /a k 2,a k 3/f2(y|A) k ,a k 4).
The first three operations retain trapezoidal (or triangular) the fuzzy numbers. Product and exponentiation, however, do not have same properties, although for convenience of representation, they are often used as trapezoidal approximations of results.
For the purpose of this work, it was useful to define a further subtraction operation, denoted by symbol ', that would always provide results in a positive fuzzy number (a fuzzy number A is positive if ( | ) = 0, ∀ ≤ 0). The fuzzy number A ' B has, then, a membership function as follows: where ( ) = 0, .

From Fuzzy Numbers to Fuzzy Financial Mathematics
The introduction of fuzzy arithmetic, briefly mentioned in the previous paragraph, has allowed to develop over time a fuzzy financial mathematic [39,40].
In fuzzy financial mathematic, the current value PV(S,n) of an S amount available between n time periods is the amount that, if invested today at a rate r, will become S in n years.
Using crisp mathematic, PV(S,n) is calculated as follows: If S and r are expressed by fuzzy numbers, P = PV(S,n) is calculated with the membership function that follows: if S is positive, it will be: if S is negative, it will be: In both cases: If n is also fuzzy, applying the extension principle of Zadeh [17], it will be: The Net Present Value (NPV) for a cash flow S = S0, S1, …, Sn, with a rate r, is calculated, using crisp mathematic, as follows: If the S amounts and the discount rate r are expressed through fuzzy numbers, the fuzzy Net Present Value can be calculated as follows: where is the fuzzy summation. In any case, the calculation of Net Present Value for projects with uncertain end time can be generalized. If n is a discreet fuzzy set, = ( , ( | ) = ) is also equal to the end of a project that gives a cash flow S = S0, S1, …, Sn, N = NPV(S,n) with a membership function equal to: For cash flows in which all of amounts Si are positive, for i ≥ 1, the calculation of Net Present Value can be simplified using the following membership function: where (x|N,ni) indicates the membership function of NPV(S,ni).
The Internal Rate of Return (IRR) is commonly used for comparing different investment alternatives.
Let S = (S0, S1, S2,…, Sn) be the cash flow for an investment project; then, IRR can be defined as the interest rate such that: If cash flow has only one sign change, the previous equation has only one solution for r > −1; if instead it presents more sign changes, the equation cannot have solutions or have more than one. In last case, the IRR method is considered not applicable for the project evaluation.
Assuming that all projects proposed only have a unique IRR, the best project will be that with the highest IRR.
Assuming that S is a fuzzy cash flow, where S0 is a negative fuzzy number, while all Si for i > 0 are positive fuzzy numbers, the fuzzy IRR can be defined, for extension of relation (1), as the fuzzy interest rate r such that: where  and  are the mathematic operations (sum and product) extended to fuzzy numbers (Zadeh's principle) and 0 is a representation of 0 fuzzy.
As there is no standard definition in the literature for 0 , this approach is difficult to apply; for different definitions of 0 , different interest rates would exist.
The problem, however, can be overcome by using the definition of IRR, considering r as the interest rate (fuzzy) that makes the present value of all future amounts equal to the initial expense.
Consequently, r is a fuzzy number such that: where indicates the fuzzy summation. If the equation terms are positive fuzzy numbers, then f1 (y|r) and f2 (y|r) can be defined with following relations: Therefore, if r is a fuzzy number, it is necessary that f1 (y|r) is growing, f2 (y|r) is decreasing, and f1 (y|r) ≤ f2 (y|r); these conditions not are always verifiable, and, for this reason, generally a valid IRR does not exist for a fuzzy cash flow. However, there are some conditions that guarantee the existence and uniqueness of this rate. In fact, if S is a set of fuzzy cash flow composed by trapezoidal numbers such that: the following properties are verified: and there is a unique number fuzzy r > −1, solution of (2) and obtainable by solving relations (3) and (4). Starting from relations (4) and (5), and placing xk (y) = 1 + fk(y|r), we obtain the following polynomial equations: Given a fixed y, Equations (6) and (7) can be solved by calculating the positive real roots.
If cash flow has only one sign change, there is only one positive real solution xk (y) for each equation, from which fk(y|r) = xk (y) -1.
Varying y between 0 and 1 the curves f1(y|r) and f2(y|r) are obtained, which represent the cutting functions of searched solutions.

Case Study
To highlight the applicability of Possibility Theory, we proceeded to reconsider an example of property investment evaluation through Fuzzy Logic.
The example considered is taken from a publication of Brueggeman and Fisher [41], where an investment analysis was performed with a deterministic approach through crisp numbers. In its original version, the case study of Brueggeman and Fisher had as its main objective to illustrate how to make a projection of income over time, through the possible purchase of an office building by an investor for 8.5 million dollars. Construction of the Monument Office Building was completed in two years (before of the time of purchase). The lead tenant was a bank that signed a five-year lease, which started when the building was completed. A law firm signed a five-year lease and a mortgage broker just signed a five-year lease on the remaining space.
In this paper, the case study has the purpose of comparing results obtained by using, for the same example, Fuzzy Logic and crisp numbers' approaches.
As already mentioned, the case study concerns the purchase of an office building, and it was assumed that the building is leased with rents that increase over time, in nominal terms, of a defined percentage equal to the consumer prices index.
In the investment analysis, the total Gross Income perceivable from an office building was calculated as the sum of annual lease fees paid by tenants.
In a generic year, the net income of the building was derived by subtracting, from the gross income, the amount of operative costs (property taxes, insurance costs, etc.); for the latter, it is assumed that the costs vary over time with a predetermined incremental percentage.
In the definition of operative costs, it was taken into account the amount limit imposed by specific contractual clauses (Operating Expense Stop).
The calculation of net income has been performed considering the variable "rate vacancy"(vacancy), the amount of which was defined in proportion to the gross income of the building, as well as management fees (management), stated as a percentage of gross income purified by vacancy rates (effective gross income).
Other investment variables are represented by funding arrangements, expressed as a combination of equity capital and debt financing, and resale value (scrap value) of the building at the end of investment.
For all variables that cannot be quantified with certainty, the values (or degrees of explicitness) have been defined on the basis of Fuzzy Logic principles.
The range of values, with which the corresponding fuzzy number has been "alibrated", was taken of different sizes in relation to the level of uncertainty associated with the explicitation of the examined variable.
The net present value was calculated for the two situations "before" and "after" tax was levied. In the situation "after", NPV was performed taking into account the effects of taxation on income and capital gains (at the resale time of building).
The variables considered are the following: Of all the investment variables, those affected by uncertainties are expressed by fuzzy numbers.
It is assumed that the building is leased to N renters and ( ) (in square meters) is the area attributed to each of them.
( ) ( ) is also the rent paid by the i-th lessee at the end of the first year and ACPI the percentage of CPI increase.
That being said, the unit rent paid by the i-th lessee at year t is obtained as follows: In turn, the total rent is equal to: The amount of last total rent is a fuzzy number, as calculated using the formula of increase , and the latter also fuzzy.
The unit rental fee for the year t + 1, given that the lease expires in year t, is given by: The Base Rent Income at year t is given by: while the total amount of operative costs related to k-th lessee is given by: It should be noted that some specific contractual clauses may restrict the above amount under a maximum value, which can be indicated with ( ) (Operating Expense Stop). This maximum value is subject to change depending on the type of lease. It is not inconceivable that the lessee is required to repay any difference between total expenditure and maximum contractual value: Total amount of operative costs and amount of any repayments, referring to the entire building, are given by: With regard to "vacancy rate" variable, its function is given by multiplying the percentage fuzzy for total income ( ) : In this case, ( ) (Effective Gross Income) is equal to: Management expenses ( ) can be calculated as fuzzy percentage on ( ) : Consequently, Net Operating Income is equal to: More investment variables are related to the financing arrangements. The latter is a combination of equity capital and debt financing.
If ( ) = ( ) + ( ) is the mortgage payment at year t (with ( ) and ( ) equal, respectively, to interest share and capital share), shortly "Before Tax Cash Flow" is equal to: If property resale value and value of investment period SP (Sales Price) are fuzzy, indicating with MB (T) the mortgage balance at year t, the cash flow becomes: The fuzzy NPV of investment at time t, in the situation "before" of tax being levied, is then given by: where is a fuzzy discount rate. The Before Tax Net Present Value (fuzzy) at year 0 can be obtained with the sum of ( ) amounts, appropriately discounted to time 0.
Similarly, it is possible to calculate the NPV of investment taking into account taxation effects on income and increase in property value at the time of its resale.
Fixed ( ) as the taxable amount given by the difference between net operating income and deductible portion of interest: Tax income is given by following relation: where ( ) is the fuzzy taxation rate.
Then, After Tax Cash Flow ( ) is given by: If ( ) is the increase of property value (capital gain) calculated between its resale price at the year t and the corresponding initial price, eventually depreciated using the coefficient DER [42] =  PP  (1  DER), denoting with the fuzzy tax rate of capital gains for year t, the tax amount on increased property value is equal to: Consequently, the After Tax Present Value is obtained as follows: and After Tax Net Present Value is equal to: = ∑ ( ) .
Considering, however, S as a fuzzy cash flow and So as a fuzzy negative number, each Si for i > 0 being a fuzzy positive number, the IRR can be obtained as a fuzzy interest rate r such that: S0   S1         S2        Sn        0  However, not having a clear and univocal definition of zero fuzzy, it is possible to obtain different measurements of rate r, according to different definitions formulated.
Of considerable interest appears the possibility offered by Fuzzy Logic to contemplate the uncertainty linked to the investment time.
The investment time depends on: characteristics of the individual investor (closed real estate fund, speculation, etc.), object of the investment (real estate commercial, residential, industrial, etc.), macro and micro economic conditions in the real estate market (market liquidity, characteristics of the real estate cycle, possibility to take advantage of financial leverage, etc.) [43,44].
In the current practice, in order to capture the real profile of cash flow, it is considered a long-term scenario (10 years).
NPV can be also calculated with reference to the most likely sales scenarios in the short and medium term (years 4 or 5), reflecting the position of the majority of real estate funds, driven by the logic of profit and high degree of liquidity.
In any case, Fuzzy Logic allows expressing the time variable uncertainly, associating to this variable a distribution of discrete possibilities with a degree of membership equals to 0.6 (year 4), 0.8 (year 5) or 0.7 (year 6).
The result will be a fuzzy NPV composed by the others' three fuzzy NPVs, representative of the horizon time associated to each, and cut in correspondence of possibility degree hypothesized.
In this case study, Fisher and Brueggeman's data were subject to fuzzification through triangular fuzzy numbers.
To simplify the numerical representation, fuzzy numbers that are not triangular have been represented in the case study by a set of three numbers. To facilitate the interpretation of results in the applied model, each fuzzy table is supported by its corresponding deterministic table. Table 1 shows assumptions concerning key parameters involved in the analysis of investment projects (purchase of an office building). To these assumptions are also added the following:

Key parameters Deterministic Fuzzy
Purchase   Table 2 shows the main information relating to each tenant.
The "end of contract" column indicates the expiry of the lease computed from the year in which the valuation is made (present time); the latest column indicates the percentage of rent increase, referring to the index of increase in consumer prices. The percentage for updating the rent year by year depends on market conditions and willingness of tenants to take into account the risk of future inflation, initially unknown.
The updates of the rent consequent to the inflation may eventually be limited by introducing a ceiling to fee increase.
Obviously, if there is an oversupply in the real estate market considered, the bargaining power of property is reduced significantly [45][46][47]. On the other hand, if the percentage increase of rent is not correlated with inflation rate, the owners may indicate a ceiling of expenditure on operative costs, beyond which the same costs are charged to the tenant.
In the proposed fuzzy model, the uncertainty related to the choice of increasing the rate of rent, and to the increased percentage of operative expenses, was considered by triangular fuzzy numbers.
In addition, the market rent was subject to fuzzification, due to the difficulty of finding appropriate information for considering its deterministic value. Table 3 shows the cash flow investment compared to the assumptions on growth rates of each variable.
For each tenant, the rent increase subsequent to renewal of the contract is contemplated. In this case, it was assumed that the future of the market scenario is characterized by uncertainty in the trend. For this reason, it was considered, via fuzzy, a percentage increase in the unit market rent expressed by (3%, 4%, 5%).  Table 3 shows that the central value of fuzzy number is always very close to the corresponding deterministic value. This is due to the values attributed to parameters for fuzzy increase; in fact, these last are centered with respect to the deterministic values in many cases.
It is also clear that the uncertainty of financial amounts grows over time; for this reason, the amplitude of fuzzy numbers is manifested year after year higher and higher.
In turn, the calculation of annual amounts of each operative cost has been executed, considering a prior definition of a percentage of the fuzzy increment for each cost amount, as shown in Table 4. In Table 4, the fuzzy interval is closely related to the uncertainty of increased percentage.  In Table 5, the column of operative costs contains deterministic values as amounts paid at the valuation time.
The percentage of increase relative to the real estate tax load has been assumed constant for the first two years, in accordance with the original example of Brueggeman and Fisher [41], and it was subsequently increased by a fuzzy percentage of about 10% (5% ; 10%; 15%). 498,346 Table 6 shows the costs to be paid to the property when the maximum limit established in the contract is exceeded. This limit was hired fuzzy from the second year of the analysis period. Table 6. Costs to be paid to the property when the maximum limit established in the contract is exceeded.

Tenant
Year   In Table 7 calculates Net Operating Income (NOI). The vacancy rates were calculated assuming a fuzzy rate (4.5%, 5%, 5.5%) for vacancy and collection losses starting from the 3rd year because, in this year, there are the deadlines of some examined leases.
The management costs are calculated by multiplying the Effective Gross Income (EGI) for a fuzzy percentage so defined (4.5%; 5%; 5.5%). It reveals an increase of uncertainty (fuzzyness) connected with the explicitation of NOI, with increasing time.   With regard to the amount of capital financed, for reasons of simplification, it is considered a mortgage with a constant rate, without allowing the possibility of introduction of fuzzy variables or different financing arrangements within the analysis. Before Tax Cash Flow (BTCF) is obtained as shown in Table 8. It is obvious that property sale price at the last year T, or at the end of holding period, will greatly affect the present investment value; however, its appraisal is difficult to assess. For this reason, the property sale price has been expressed in Table 9 through a fuzzy number characterized by an amplitude of ± 15% (−15%Vm, Vm, +15%Vm), with respect to Vm value obtained from an appraisal carried out with conventional methods. This change can easily be expressed by defining a capitalization rate of the fuzzy Net Operating Income.
Usually, as the capitalization rate at the time of property resale, the rate that the real estate market expresses for that particular property at evaluation time is used.
Obviously, the assumption of permanence of economic and financial conditions, as well as the dynamics of the real estate market segment examined, as far as is reasonable in the short term, are considered unlikely in the medium and long term investment scenarios [48]. With these assumptions, the investment cash flow is presented in Table 10, making it possible to calculate Before Tax Net Present Value (BTNPV).
The calculation assumes, of course, the choice of a discount rate for financial amounts. The possibility to express with uncertainty this discount rate is a characteristic of fuzzy approach. In this case, the discount rate was expressed with an asymmetrical triangular number (16.5%; 18%; 18.5%) compared to the deterministic rate of 18%. It can be considered as the sum of a risk-free rate and a risk-premium rate.
The investment analysis allows for verifying the change of cash flow "after" taxation on taxable income, calculated by deducting from NOI the interest share and annual depreciation share, with the latter defined by a fuzzy number (1.5%; 2%; 2.5%).
In particular, assuming a proportional taxation and a fuzzy tax rate (30%; 36%; 40%), the After Tax Cash Flow (ATCF) can be represented as shown in Table 11.  Using the fuzzy discount rate (12%; 13%; 13.5%), it is possible, at this point, to calculate the present investment value "after" taxation on income (see Table 13). In the case considered, it is evident that the discount rate is smaller than the previous rate used for the calculation of before tax present value (16.5%; 18%; 18.5%) because the uncertain variable is outdated and the risk-premium tends to decrease.  Tables 14 and 15 show the fuzzy amounts already calculated and related, respectively, to Before Tax Cash Flow (BTCF) and After Tax Cash Flow (ATCF).  Solving Equations (6) and (7), with reference to the cash flows indicated in Tables 14 and 15, through an iterative process, the fuzzy internal rates of return (IRR) are obtained, respectively, "before" and "after" taxation, assuming, in the example of real estate investment of Brueggeman and Fisher, that property sale takes place in the fifth year: Before Tax IRR: ( This result represents, undoubtedly, a more flexible response to problems of uncertainty as it provides a variation range of the results for changes in input. This means to have a more accurate and flexible description of uncertainty than to crisp results because crisp conditions force having forecasts that are not easier, though accurate but often hardly arguable.
It seems difficult, in fact, be able to justify a taxation growth, or an increase of income, around a precise value that is 5 or 7%: much more flexible is a tool that takes into account a variation range for the phenomena observed giving a range of outputs.

Conclusions
The paper has established a basic framework of Fuzzy Logic for real estate investment, showing a correct correlation between theory and practice. This work used a Fuzzy Logic system applied to the case of purchasing an office building, reasoning, on implications for decision-making processes, that affec the sector of real estate investment when there is imperfect market information. The results showed that, with a correct application of fuzzy logic, operators and investors are able to improve their investment decisions in terms of quality, reducing the risk arising from the uncertainties of inputs.