Annals of Operations Research 138, 79–96, 2005 c 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.
Real Estate Appraisal: A Double Perspective Data Envelopment Analysis Approach MARCOS PEREIRA ESTELLITA LINS ∗
[email protected];
[email protected] Production Engineering Program, Federal University of Rio de Janeiro, Rua Belis´ario T´avora 80 ap 506 Laranjeiras, Rio de Janeiro, RJ, Brazil LUIZ FERNANDO DE LYRA NOVAES
[email protected];
[email protected] CAIXA ECONOMICA FEDERAL, Brazilian Federal Savings Bank LUIZ FERNANDO LOUREIRO LEGEY Energy Planning Program, COPPE, Federal University of Rio de Janeiro
[email protected]
Abstract. This paper proposes a new methodology for the assessment of the value range for real estate units. The theoretical basis of the methodology is built on the Data Envelopment Analysis—DEA approach, which has its original concept adapted to the case where the units under assessment consist of transactions among sellers and buyers. The proposed approach—christened Double Perspective-Data Envelopment Analysis (DP-DEA)—is applied to a database comprising the prices and features of the units under assessment. It is shown that the DP-DEA presents some specific advantages when compared to the usual regression analysis method employed in real estate value assessment. Keywords: data envelopment analysis, double perspective, production frontier, real estate, value assessment
Introduction The Direct Capital Comparison (DCC) is the most conventional method used in real estate value appraisal (Mackmin, 1994). One problem with DCC is that it needs an ex ante selection of residential properties, which should be comparable to the subject property being appraised. Alternatives to DCC include: (i) the income capitalization approach, which estimates the value of the property using the present value of a cash flow based on the income or expenses generated during the expected property’s lifetime (Akerson, 2000); (ii) the cost approach, which values the property as in a budget, taking into account the accumulated labor, materials and equipment, and (iii) the residual or developers approach, which forecasts the difference-ie the residual—between the selling price and the combined costs of buying, holding and selling a property (Mackmin, 1994; Baker, 2001). ∗
Corresponding author.
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DCC requires an “allowance in money terms” (Lewis, 1994) for any differences between the subject property and the comparable properties, for which the appraiser can employ either multiple regression analysis (MRA), expert systems (ES) or artificial neural networks (ANN). MRA statistically models the relationship between a dependent variable Y and many independent variables X 1 . . . X n . In real estate value appraisal, the dependent variable is the property value (sometimes expressed by the sale price) and the independent variables are those attributes that describe a property and its location features. As it is usual in applying regression analysis, when estimating market values of residential properties using this technique, it is advisable to compute several regression equations from different samples of a predetermined population, until the regression coefficients and associated statistics become consistent. Another problem confronting MRA is the need process of model identification. Using linear relationships can have, as Adair et al. (1996) suggest, “wide application in real estate”. On the other hand, Bruce and Sundell (1977) disagree, saying that “the issues involved in property valuation are too complex for the simple additive theory on which it [MRA] is based”, despite the convenience of being available in most statistical analysis packages. The required underlying functional form (e.g. polynomial) renders non-linear multiple regression analysis extremely difficult, especially for multidimensional data (see example in Goulden (1989)). Expert systems are computer-based techniques that emulate the procedures used by an expert in a well-defined domain. ES are able to replicate the actions of an expert, provided the expert can articulate his expertise. This approach can be used to model the comparative adjustment process using a database system, resulting in an ‘expert database’ system (Lewis, 1999). In order to develop an expert valuation system, the knowledge possessed by an expert must be acquired or elicited to construct a set of domain rules. This process can take the form of interviews and questionnaires, from which domain a knowledge engineer formulates rules. Alternatively a team of expert valuers can supply domain rules (Boyle, 1982). Tazelaar (1989) describes ANNs as “humanity’s attempt to mimic the way the brain does things in order to harness its versatility and its ability to infer and intuit from incomplete or confusing information”. ANNs are able to generalize from examples and have the ability to interpolate from previous learning. ANNs often work as pattern classifiers in “areas where problem solutions are complex and difficult to specify, but which have an abundance of data from which a response can be learnt” (DTI Guidelines, 1990). ANNs do not require an array of a priori knowledge, which in many cases is a prerequisite for MRA (Tay and Ho, 1992). ANNs learn by “inducing the latent rules inherent in the training set of input and output patterns” (Tay and Ho, 1992). An artificial neural network, when applied to real estate value assessment, models the relationship between the property value and independent variables describing property’s features. The difference between the two techniques relies on the structure of the model. ANNs are distributed arrays of highly interconnected processing elements.
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Each connection has an associated strength or weight. Processing at each node involves a weighted sum of each connection forming a single input to a non-linear transfer function. The output from each node becomes the input to successive nodes. To their detriment, however, neural networks are akin to a black box, as they have definite inputs and outputs but lack any underlying functional transparency, and reveal little of their processing logic. The ability to furnish users with explanations of the reasoning process or underlying functionality is an important feature of any model (Clancy, 1983). Explanation facilities are required both for user acceptance and the validation of reasoning procedure (Davis et al., 1977). In expert systems, explanations are typically provided by tracing the ‘chain of inference’ during the reasoning process (Southwick, 1991). This is a difficult task when analysing neural networks as they do not have explicit or “declarative knowledge” (Diederich, 1989). Besides; ANN can take a very long time to train and require some expert knowledge to define the network structure and the feature representation (Lu and Lu, 1992). McGreal et al. (1998) conducted a study of ANNs for the purpose of generating residential property appraisal models, concluding that “Whilst some very close predictions are possible, others can deviate appreciably from the sale price. Under such circumstances the use of neural networks for mass appraisal purposes must remain problematic”, although the authors do comment that for homogeneous data there is a “tendency for better results” (McGreal et al., 1998) In this paper, a new methodology is proposed, where the uncertainty in the unit’s value resulting from market transactions was taken into account by explicitly representing the economic agents involved in a transaction, namely the buyer and the seller, whose actions establish a set of accomplished transactions. Because of its non-linear structure, the proposed method performs better than regression analysis, in terms of fitness to data, as we will be shown in Section 4.2. Concerning ES and ANN, the new method surpasses an important weakness, as it offers model parameters, i.e., the weights assigned to each input, easing the cognitive reasoning process, as will be explored in Section 5. However, the major advantage of this method consists in supplying the reference set for comparison with the assessed property, a characteristic directly linked to DCC philosophy and not provided by any of the former methods. The methodology to achieve this was christened Double Perspective-Data Envelopment Analysis (DP-DEA), because it makes use of two encapsulating surfaces that enfold, in an n-dimensional space, all the observed data. Those real estate units that present, from the point of view of either the seller or the buyer, an “efficient” price, define these surfaces. The remaining units can have their value assessed by taking the envelopments as frameworks, under an output-oriented or an input-oriented DEA model. The mathematical formulation of the DP-DEA method makes it possible to obtain an interval for a property’s value as a function of its physical features and location. The DP-DEA method was applied to a database consisting of the market values of apartments established in transactions, which have occurred in several neighborhoods of the Rio de Janeiro’s municipal district. The buy and sell bids for the different units define the
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supply and demand possibilities as a function of the commodities’ attributes (Debreu 1951, 1959). This paper intends to make a contribution to the literature in basically two ways: firstly, by introducing the concept of the DP-DEA; and secondly, with the application of this concept to a real problem situation, namely the real estate value assessment problem, which brings in a meaning to the idea of the two surfaces within the more general context of economic/financial transactions or auctions. The specific application of DP-DEA to real estate value assessment can have wide impact in those institutions, like the Brazilian Savings and Loan Federal Bank CEF (Caixa Econˆomica Federal), which has to deal with many daily decisions concerning property loans. If a more precise tool is available, CEF will be able to produce a much more reliable portfolio of loans and, therefore, decrease its credit risk. 1.
Data envelopment analysis
The DEA method is used to accomplish comparative analyses of a set of observations. For each observed unit, be it a state in a national economy or a simple piece of equipment, it provides a measure of efficiency or productivity. The first classical DEA model was the CCR proposed by Charnes, Cooper, and Rhodes (1978), also known as CRS because it assumes Constant Returns to Scale. The second one was the BCC introduced by Banker, Charnes, and Cooper (1984), or VRS as it postulates Variable Returns to Scale. Shortly, the method works as follows: Starting from the selection of n observed units, with m inputs and s outputs, a DEA model determines a subset composed of k efficient units. These units are considered benchmarks and define the segments of the enveloping surface, thus motivating the envelope form of DEA CCR or BCC models. The contained subset, not belonging to this surface, is formed by n − k inefficient units. The computation of the efficiency of each observed unit requires the solution of a linear programming problem. The formulation of both classical CCR and BCC, in their dual multipliers/envelope output-oriented DEA models is shown in Table 1. In the multipliers model, the optimal values of the decision variables: µ, and u ∗0 are the parameters of the frontier hyperplane defined by the constraints (3) such that the observed DMU “0” appears with the highest possible efficiency in the objective function (1). This efficiency can be defined as the minimum linear combination of its inputs (1) given the normalized linear combination of its outputs (2). Non-Archimedean constrained multipliers (4) can be a substitute for classical positive constrained multipliers (5), where ε is an infinitesimal (non-Archimedean) amount. According to the envelope form, the problem consists of maximizing the objective function (7) on the decision variable h 0 , subject to constraints (8) to (11). These constraints guarantee that the projected efficient unit will be located inside the production possibilities set, which is defined as a linear combination of the outputs (and inputs) vectors, using the coefficient vector λ. In accordance to the BCC assumptions, this linear combination should be subjected to a convex constraint (11), which does not hold in
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Table 1 Output-oriented CCR and BCC models. Multipliers Min
L0 =
Subject to −
s r =1
m i=1 s
i xi0 + u ∗0
(1)
µr yr 0 = 1
(2)
r =1 m
µr yr j +
i xi j + u ∗0 ≥ 0,
j = 1, . . . , n
µr ≥ ε, r = 1, . . . , s i ≥ ε, i = 1, . . . , m i , µr > 0, ∀i, r For CCR: u ∗0 = 0 For BCC: u ∗0 unconstrained Envelopment Max
(3)
i=1
H 0 = h 0 +ε.
Subject to −xi0 +
n
h 0 yr 0 −
s
s y+ +ε.
y=1 n
r
sx
(4) (5) (6) (7)
x=1
λ j yr j +s r+ = 0,
r = 1, . . . , s
(8)
j=1
λ j xi j +s i− = 0,
i = 1, . . . , m
(9)
j=1
λ j, sr+ , si ≥ 0, ∀ k, j, i n λj= 1 For BCC:
(10) (11)
j=1
the CCR model. The inclusion of this latter constraint in BCC corresponds to an unconstrained dual variable (6) in the multipliers model. Analogously, Table 2 shows both input-oriented dual multipliers/envelope CCR and BCC models. The variables and constraints are quite similar; the main difference being that in the multipliers model the objective is to maximize the linear combination of the outputs of the observed unit (12), keeping the inputs normalization constraint (13). As for the envelope form, the variable h is to be reduced, and multiplied by the input of the assessed (observed) unit. 2.
Double perspective DEA model—DP-DEA
Double Perspective DEA (DP-DEA) uses, as an objective measure of the observed units efficiency, two simultaneous perspectives: the maximization of outputs and the minimization of inputs, in such a way that inputs under one perspective are the outputs under the other and vice-versa. The method can employ both classic CRS and VRS DEA models. as figures 1 and 2 illustrate, considering the maximum and minimum value of negotiating apartments whose only attribute consist of their areas. In order to simultaneously use both input and output oriented models, we will transpose the graph of the input-oriented model in figure 2 to obtain in figure 3 the same axes as in the output-oriented model of figure 1. Figure 4 shows the two graphs put together. The space defined by the enveloping surfaces corresponds to a set of accomplished transactions. It results from the intersection
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Figure 1. Output-oriented model.
Figure 2. Input-oriented model.
Figure 3. Input-oriented model (transposed).
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Table 2 Input-oriented CCR and BCC models. Multipliers Max
Z0 =
Subject to −
s r =1
s r =1 m
µr yr 0 + v0∗
(12)
i xi0 = 1
(13)
i=1
µr yr j +
m
i=
i xi j + v0∗ ≥ 0,
j = 1, . . . , n
i ≥ ε, i = 1, . . . , m µr ≥ ε, r = 1, . . . , s µr , i > 0, ∀, i, r For CCR: v0∗ = 0 For BCC: v0∗ unconstrained Envelopment Min
H0 = h 0 − ε.
Subject to yr 0 −
n
(14)
1
h 0 xi0 −
s y=1 n
s y+ −ε.
r
sx
(15) (16)
(17)
x=1 − λ j xi j −s i0 = 0,
i = 1, . . . , m
(18)
j=1
λ j yr j +s r+ = 0,
j=1 λ j , s r+ , si−
For BCC:
r = 1, . . . , s
≥ 0, ∀ j, i, r n λj= 1
(19) (20)
j=1
Figure 4. Double perspective DEA method.
of the set of supply possibilities (F¨are and Grosskopf, 1994; F¨are, Grosskopf, and Lovell, 1996; Shephard, 1953, 1970), and the set of demand possibilities (Novaes, 2002). In other words, the DP-DEA defines supply and demand frontiers. Formally, it is possible to devise the DP-DEA model as a classic DEA output-oriented model together with an input-oriented model with a transposition of axis, as shown in figure 4. Figure 3 shows the input-oriented model transposed.
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Table 3 Input-oriented CCR and BCC models transposed. Multipliers Max
Z0=
Subject to −
s r =1
s r =1 m
vr xr 0 + w0∗
(21)
u i yi0 = 1
(22)
i=1 m
v r xr j +
u i yi j + w0∗ ≥ 0,
j = 1, . . . , n
vi ≥ ε, i = 1, . . . , m u r ≥ ε, r = 1, . . . , s u r ,vi > 0, ∀i, r For CCR: w0∗ = 0 For BCC: w0∗ unconstrained Envelopment Min
H0 = h 0 − ε.
Subject to yr 0 −
(23)
i=1
n
h 0 xi0 −
s y=1 n
s y+ − ε.
r
sx
(24) (25) (26)
(27)
x=1
λ j xi j −s i− = 0,
i = 1, . . . , m
(28)
j=1
λ j yr j +s r+ = 0,
j=1 λ j , s r+ , si−
≥ 0, ∀ j, i, r n λj= 1 For BCC:
r = 1, . . . , s
(29) (30)
j=1
3.
Real estate assessment
This section presents the implementation of the DP-DEA using data from a database (Paiva, 2000) within the Real Estate Information System of the Brazilian Caixa Econˆomica Federal-CEF (Federal Savings Bank). Data corresponding to 1,256 residential units (all inspected), located in the Rio de Janeiro municipal district, were obtained from CEF’s registered transactions along the first quarter of 2000. Sixty-seven variables representing the transactions’ price and several characteristics of each property were considered. From the seller’s perspective, the price is considered as the output and the unit’s features are taken as inputs. The symmetric holds true under the buyer’s perspective, that is, the price is considered as the input and unit’s features as outputs. From the seller’s perspective, an output-oriented VRS model, whose objective is the maximization of the unit’s value, determines the efficiency of a particular unit transaction. An Input-oriented VRS model with the objective of minimizing the unit’s value determines the efficiency under the buyer’s perspective. If the unit is efficient under certain viewpoint, it is on the frontier and has efficiency equal to one. Other inefficient units do not belong to the frontier and therefore present efficiencies less than one. As can be seen in figure 4, the variable 1/h is the largest percentage value that can be applied, from the seller’s perspective, to the output Y10 so as to get the greatest
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selling value of unit (X 10 , Y10 ), corresponding to a projection on a fictitious unit at the efficient frontier. Analogously, from the buyer’s perspective, the variable h is the smallest percentage value that can be applied to the input Y10 , such that it yields the least buying value of unit (X 10 , Y10 ), corresponding to a projection on a fictitious unit at the efficient frontier.
3.1. Distance between frontiers index—IRDF The IRDF—Index of the Relative Distance between Frontiers is adopted here in order to compare the results obtained by DP-DEA with those obtained in a regression analysis. The comparison is made by considering the opinion of real estate market experts. These experts define which range of IRDFs can be considered acceptable. The IRDF is defined through the partition of the set of realized transactions into three subsets: the competitive set; the buyer’s perspective set and the seller’s perspective set, as figure 5 shows. Those units, whose transaction prices favors the seller, belong to the seller’s perspective set and those whose transaction prices favor the buyers, belong to the buyer’s perspective set. Finally, those units where there is a balance between purchase and sale prices belong to the competitive set (Samuelson, 1975). We will focus on the competitive set to assess a units’ value. The seller’s and buyer’s perspectives sets are determined by an iterative process, in which a gradual reduction of the distance between the two enveloping surfaces is achieved. This process—known as “peeling the onion” (Tavares, 1998)—eliminates, at each stage, the efficient units computed by the DP-DEA, and recalculates a new efficient frontier in the next stage. The process stops when an indicator, the IRDF, reaches an acceptable range, thus establishing the competitive as well as the seller’s and the buyer’s perspective sets.
Figure 5. Double perspective DEA frontiers.
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The IRDF is an indicator of the adequacy of the estimated competitive set in terms b of value assessment. It is defined by IRDF = VsV−V , where Vs (Vb ) is the largest selling a b (smallest buying) projected value in the competitive set frontier and Va = Vs +V is the 2 average unit’s price. These projected prices Vs and Vb are calculated through the DP-DEA model. 3.2. Application of the DP-DEA method The DP-DEA efficiency determination process starts with the original set of accomplished transactions. Figure 6 exemplifies this with a graph in which the x and y axes represent, respectively, the units’ total area and location (measured by the reference value used in the computation of the 1998 urban property tax—VR98), both divided by the transaction’s value, in such a way that the upper and lower frontiers that envelop all data represent, respectively, the buyer and seller perspectives. Figure 6 also shows that the efficient frontier from the buyer’s perspective is defined by units 166, 23, 32 and 234; and from the seller’s perspective, by units 90, 55, 56, 112 and 233. The database used in DP-DEA computations consists of 67 variables for each of 1,256 residential units (apartments). These variables include: buy/sell transaction price; total area; unit’s age; number of rooms; parking places; construction standards; neighborhood index and unit’s preservation. In order to select more homogeneous sets of residential units, an integrated multiple criteria method, the capacity model (Cano, 1996), was used. The method made a choice of 242 apartments located in the administrative regions of Botafogo, Copacabana, Leme,
Figure 6. The efficient frontiers.
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Figure 7. Boolean choice procedure.
Flamengo, Catete, Gl´oria, Laranjeiras, Maracan˜a, Santa Teresa and Tijuca. The method is sketched in figure 6, for the variables: average household income; neighborhood; and educational levels. The selection of the variables used in the capacity model was based in the heuristics “IO-Stepwise” (Lins and Mesa, 2000), resulting into the following explanatory features of the units: • Total area; • Location • Number of parking places; • Installed Equipment; • Construction standard; • Preservation; • Floor; • Depreciation; and • Number of units per floor served by the same social access. Specialists in real estate valuation consider it acceptable to have an IRDF range of less than 12.5%. The units that satisfy this constraint were considered to belong to the competitive set.
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Next, a heuristics was implemented aiming at identifying and excluding those units located in extreme regions according to either buyer/seller perspective. This heuristics employs an algorithm—implemented with the help of the software Frontier Analyst— to apply the DP-DEA in successive rounds to exclude those efficient DMUs which are references to a number (defined in each stage) of inefficient units. The algorithm stops when the IRDF is within a pre-specified range. In the case studied in this paper, the number of inefficient units referenced in stages 2, 3 and 4 were respectively 40, 15 and 10. In stage 1 the DP-DEA is applied to the original data set, and the IRDF computed so as to obtain an assessment of its possible range. For those units in the competitive set, the IRDF range in respect to the unit’s price, presented a great reduction. The maximum IRDF (24%) occurred in unit number 16 and the average IRDF was 5%. The same database was used for comparing the frontier’s adjustment process of the DP-DEA and regression analysis. To this purpose, two parallel lines, corresponding to a given confidence interval were added, at each stage, to the multiple linear regression line. The confidence interval was adjusted in such a way as to contain the same number of units as in the DP-DEA model. Results of this comparison are shown in Table 4. Some comments on the results shown in Table 4.1 are due: (1) Through successive rounds and removal of units in the efficient frontiers, which were found with the help of either an input or an output-oriented DEA model, the distance between frontiers was reduced. Consequently, there was a decrease in the maximum IRDFs up to the value of 24%, which happened in the 13th round; (2) This last result defines a competitive set (which is, by definition, a set that has a maximum IRDF lower than 25%) consisting of a total of 119 units; (3) The excluded units amounted to 123 apartments, from which 69 were in the seller’s perspective set and 54 in the buyer’s perspective set. (4) At the initial round, the frontiers of the regression method encapsulated all of the 242 apartments, presenting an average IRDF of 364%; (5) The frontiers of the regression method were adjusted in such a way as to contain, in each stage, the same number of units as the DP-DEA method, therefore decreasing the distance between frontiers and consequently decreasing the maximum IRDF, up to the percentile of 108%, which occurred in the 13th round; (6) Even in the 13th round, the regression method did not reach the competitive set; on the contrary, it was very distant from the established goal of an IRDF less than 25%. As an illustration of the DP-DEA method in real estate value assessment, a simplified example is presented in what follows. Starting from a set of 119 apartments, an interval of value is determined as a function of their neighborhood index and some pre-established
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Table 4 Comparison of DP-DEA and multiple regression results. Real estate evaluation methods compared Stages 1st Stage Original Set Average IRDF Maximum IRDF Frontier Interval 2nd Stage Central Nucleus Set Seller’s Perspective Set Buyer’s Perspective Set Average IRDF Maximum IRDF Frontier Interval 3rd Stage Central Nucleus Set Seller’s Perspective Set Buyer’s Perspective Set Average IRDF Maximum IRDF Frontier Interval 4th Stage Competitive Set Central Nucleus Set Seller’s Perspective Set Buyer’s Perspective Set Average IRDF Maximum IRDF Frontier Interval
DP-DEA 1st Frontier Analyst Round 242 apts 55% 117% (Apt 85) 4th Frontier Analyst Round 215 apts 18 apts 9 apts 25% 67% (Apt 235) 9th Frontier Analyst Round 158 apts 29 apts 28 apts 12.5% 50% (Apt 201) 13th Frontier Analyst Round 119 apts Competitive Set 22 apts 17 apts 5% 24% (Apt 16)
Multiple regression 1st Statgraphics Round 242 apts 364% 10010% (Apt 156) 9.85 × σ = 215.59 (σ = 21.887) 2nd Statgraphics Round 215 apts 15 apts 12 apts 97% 387% (Apt 17) 2.837 × σ = 62 (σ = 21.854) 3rd Statgraphics Round 158 apts 37 apts 20 apts 47% 146% (Apt194) 2.18 × σ = 29.832 (σ = 13.684) 4th Statgraphics Round − 119 apts 20 apts 19 apts 32% 108% (Apt 194) 2.372 × σ = 19.7 (σ = 8.305 )
features, such as: • 1 or 2-bedroom apartments with an area from 60 to 80 m2; • Construction standard index in the interval 260 to 320; • Regular preservation index in the interval 180 to 240; • Apparent age from 15 to 25 years; • One parking place; and • Any floor.
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Table 5 Range of possible units’ values. Unit number 12 28 67 69 78 134 135 148 154 175 201 214 242
Number and street name
City sector
No. of bedrooms
Neighborhood index (R$ 1000)
Value (R$ 1000)
Total area
330/101 Professor Gabizo 30/201 Bambina 51/105 Moraes e Silva 124/202 18 de Outubro 301/401 Muniz Barreto 360/202 Hadock Lobo 79/802 Mariz e Barros 90/506 Minstro Raul Fernandes 764/501 Pereira da Silva 43/301 Benjamim Constant 429/903 18 de Outubro 184/602 General Severiano 862/103/bl.01Conde de Bonfim
Maracan˜a Botafogo Maracan˜a Tijuca Botafogo Tijuca Tijuca Botafogo Laranjeiras Gl´oria Tijuca Botafogo Tijuca
2 2 2 2 2 2 2 1 2 2 2 1 2
63.1 77.5 63.1 29.7 77.5 63.1 63.1 82 34.7 68.4 29.7 77.5 53.3
88 90 95 70 100 70 94 110 75 80 60 105 80
72 69 78 78 72 65 75 65 67 71 69 63 69
When those constraints are applied to the original set, there is a reduction to the list of 13 apartments shown in Table 4.2. This list defines the new units—with the chosen features—range of possible values as a function of location (represented by the proxy neighborhood index). Figure 8 depicts the units shown in Table 4.2, presenting the buyer and seller’s efficient frontier, which define graphically the range of possible new units’ value as a function of neighborhood index. The commodity space between those two frontiers was defined respectively by units 201, 134, 28, 148 and 69, 154, 148.
Figure 8. Buyers and sellers’ efficient frontiers.
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4.
Conclusions
The DP-DEA method introduced in this paper has been developed essentially as a tool for real estate value assessment. As compared to multiple regressions, the DP-DEA presents better results, in terms of narrower intervals for the units’ value. The DP-DEA results were inserted into a Geographic Information System—GIS for the Rio de Janeiro municipal district, as a way to geo-reference data of the assessed units. The assignment of each unit is based on its local address in the map of streets and public areas. The thematic maps generated by the GIS allow for the analyst to identify the participation of each variable in the efficiency of the unit’s transaction under either the buyer or seller’s perspective. An example of this is in figure 9, which shows the location of the competitive set for 119 units in the Rio de Janeiro municipal district. The thematic map identifies the participation of the different features of each unit in the efficiency computation from the buyer’s perspective. It is possible to observe the predominance of the feature “location” in areas close to the seaside, unlikely to what happens in suburban areas (at the left side of the figure). This demonstrates a disposition of the buyer in seaside areas to allow for great variations with respect to other unit’s features, in exchange for small variations when the location is concerned. However, it should be brought up again that the proxy variable for location is the unit’s reference value used in the computation of the urban tax of the year 1998. It can be
Figure 9. Location of the competitive set.
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used to represent the heterogeneity within a location, since it varies according to different aspects of the unit’s site (view, proximity to slums and so on). From the analysis of the frequency of extreme (maximum) participation of the unit’s features (units belonging to the competitive set) in the value assessment obtained by the DP-DEA (Table 5.1) we can draw the following conclusions: a. for the units assessed from the seller’s perspective, the most important variables were firstly, the unit’s age (47 units with maximum weight assigned); secondly, total area; thirdly, number of apartments accessed by a same social entrance; and fourthly, location. b. Considering the units that were projected onto the buyer’s perspective frontier, the most important variables were firstly the number of apartments accessed by a same public entrance (45 units with maximum weight assigned); secondly, the construction standard; thirdly, location; and fourthly, preservation. c. The variables: equipment, number of parking places, and floor level, in that order, presented a higher frequency of zero weights assigned. Therefore, a lower influence in the determination of the maximum or minimum assessed value. The example above reveals the potential of the DP-DEA in the definition of values for new properties, as a function of the similar features encountered in the sample of observed units. It is important to mention that the GIS was used as a tool that complements the analysis making it possible to visualize information, which would otherwise be difficult to grasp. The proximity of a slum, for instance, at a distance of only 139 meters, influences directly on the value of apartments 69 and 201. As figure 10 shows, apartment 69, Table 6 Frequency of occurrences of extreme weights assigned to a feature according to different perspectives. Weights
Features
1. Zero from buyer’s perspective 2. Zero from seller’s perspective 3. Zero from buyer and seller’s perspective 4. Maximum from buyer’s perspective 5. Maximum from seller’s perspective 6. Maximum from buyer and seller’s perspective
Neigh- Total Parking Stand- Preserborhood Area places Equipments ard vation Floor Age Access 28
42
70
98
54
69
64
85
28
17
40
66
59
65
49
42
51
41
1
5
19
44
21
14
9
28
3
17
6
3
0
22
17
2
6
45
15
22
0
0
2
10
1
47
20
0
0
0
0
0
0
0
1
3
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Figure 10. The effect of location.
placed at a greater distance from the slum belongs to the seller’s perspective frontier. On the other hand, apartment 201, which is closer to the slum, belongs to the buyer’s perspective frontier. Therefore, the lowest value assigned to variable neighborhood index seems consistent. Finally, a last word should be said about the DP-DEA method. Despite the fact that its development has been based on the problem of real estate assessment, its potential for other applications seems to be enormous. Actually, it can possibly be applied to any situation—typically those encountered in auctions—in which there is a game situation where one player aims at maximizing an output which is the input that the other player would like to minimize. If this is true, this paper has only “scratched the surface of the iceberg.” Besides the application issues, potential for future theoretical research includes new developments in DEA, like time series indexes (Malmq¨uist, Tornq¨uist), Zero Sum Gains DEA and extension to stochastic frontier analysis (SFA), which could allow for stochastic uncertainties, but require specification of a functional form. References Akerson, C.B. (2000). Capitalization Theory & Techniques Study Guide, 2nd edn. MAI Publisher: Appraisal Institute.
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