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A quick assessment from expert judgements to assist in farmland valuation
Land Use Policy 46 (2015) 324–329
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Land Use Policy journal homepage: www.elsevier.com/locate/landusepol
A quick assessment from expert judgements to assist in farmland valuation M. Franco a,∗ , J.M. Vivo a , R. Herrerías b a b
Department of Statistics and Operations Research, University of Murcia, 30100 Murcia, Spain Department of Quantitative Methods for Economy, University of Granada, 18011 Granada, Spain
a r t i c l e
i n f o
Article history: Received 17 December 2014 Received in revised form 12 February 2015 Accepted 14 March 2015 JEL classification: Q12 Q24
a b s t r a c t Land is the most important agricultural asset, farmland transactions are generally not recurring actions for most buyers and sellers, thus it is often required reliable information from expert judgements. This paper presents a quick asset pricing method under uncertainty, based on the use of two reliability functions, a multidimensional index of characteristics or qualities influencing on the market value and the expert judgments, which overcomes the disadvantages of the valuation method based on two distribution functions for a multidimensional quality index. The usefulness of this method is illustrated through an empirical application on farmland pricing in South-East Spain, using a two-dimensional quality index. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Farmland pricing Expert judgements Bivariate reliability Weighted model
Introduction The asset pricing constitutes an important current issue due to the wide range of economic events that require estimating the value of the assets within the economic development of the countries. At the same time, the increasing complexity of the appraisal problems and the particular characteristics of the markets, it becomes more necessary to make better and more accurate valuations. Specifically, there exist some markets wherein the information is not enough due to not frequent transactions or lack of clarity in the prices. For instance, land is the most important agricultural asset but farmland transactions are generally not a recurring action for most buyers and sellers. In such situations, appraisers have to work in an uncertainty environment where the market value of the asset is obtained from the information provided by an expert, about the asset and its characteristics or qualities. Characteristics affecting farmland prices have been studied by some authors (e.g. Huang et al., 2006; Sills and Caviglia-Harris, 2008; Troncoso et al., 2010; Sklenicka et al., 2013; Awasthi, 2014; and the references therein), as well as the effect of transaction costs and taxes on the farmland prices (e.g. Lence, 2001; De Fontnouvelle and Lence, 2002; Segura
∗ Corresponding author at: Regional Campus of International Excellence “Campus Mare Nostrum”, University of Murcia, 30100 Murcia, Spain. Tel.: +34 868884187. E-mail address: [email protected] (M. Franco). http://dx.doi.org/10.1016/j.landusepol.2015.03.008 0264-8377/© 2015 Elsevier Ltd. All rights reserved.
et al., 2010, 2012). Undoubtedly, economic valuation of these assets is necessary for the development and functioning of these markets. In this context, the asset appraisal, under uncertainty, is frequently analyzed by econometric modelling and hedonic price indexes in both residential properties (e.g. Goodman and Thibodeau, 2003; Sirmans et al., 2005; Páez et al., 2008; Cervelló et al., 2011; Cervelló and Segura, 2011) and land valuations (e.g. Ma and Swinton, 2012; Samarasinghe and Greenhalgh, 2013; Awasthi, 2014; Lavee, 2015), as improvements of the classical synthetic method, but the weakness of these techniques is well-known when the information is not enough. Nevertheless, the valuation method based on the two cumulative distribution functions (VMTDF) has been widely used to approach the market value of an asset from a characteristic or quality index because of their easy implementation and its important advantages with respect to the above comparative techniques, since the VMTDF works well with very little information (Herrerías and Herrerías, 2010; and the references therein). Indeed, the VMTDF allows us valuing an asset under uncertainty, when only the pessimistic, optimistic and most likely values are available to the appraiser, which may be supplied by expert judgments. Ballestero (1971) introduced this technique by using two beta distributions, inspired by PERT methodology. Initially, it was extended to other distribution models by Romero (1977) and Ballestero and Caballer (1982), and the VMTDF has been studied and applied since then (see references in Herrerías and Herrerías,
M. Franco et al. / Land Use Policy 46 (2015) 324–329
2010). For instance, Romero (1977) used the VMTDF from the triangular distribution, which has been considered a standard model in risk analysis and suggested by Williams (1992) and Johnson (1997) as a more intuitive and simpler distribution since it only requires three parameters (pessimistic, optimistic and most likely). Although, the VMTDF was originally proposed for the valuation of an asset from the values of one single index, the appraisal processes under uncertainty generally use the supplied information by two or more quality indexes. Thus, extensions of this method have been analyzed by some authors; see, e.g. García et al. (2002) and Herrerías and Herrerías (2010). Unfortunately, in the multidimensional case, the VMTDF produces a loss with respect to the assessments from each component of a multidimensional quality index, and then weights among its components might be used to approach the asset pricing. In order to overcome these limitations, the reliability or survival function, also called decumulative distribution function by Yaari (1987) in dual theory of choice under risk, provides an alternative measure to the distribution function, and consequently, the valuation method based on two reliability functions (VMTRF) might help to appraise an asset under uncertainty from a quality index, even more when the dimension is reduced by unobserved components of quality. Note that the reliability measures have been widely used in many areas of economics, political science, biology, and engineering. In particular, many interesting results of reliability theory have been applied in risk analysis, and their properties have interesting qualitative implications in these fields (e.g. Bagnoli and Bergstrom, 2005; and the references therein). The main aim of this paper is to display a quick valuation method for land pricing based on expert judgements and illustrate its empirical application through an agricultural land pricing in South-East Spain. In general, the VMTRF allows us to approach the assessment of an asset from a multidimensional quality index by using the reliability functions corresponding to the two probability models. From a practical viewpoint, in this study the focus is placed on a two-dimensional quality index, and its comparison with the assessments given by the VMTDF from the joint twodimensional quality index and from each one of its components, being straightforward its generalization to the multidimensional case. The remainder of the paper is organized as follows. Section “Valuation method based on expert judgements” summarizes the valuation method based on expert judgements and presents the theoretical results to value an asset under uncertainty from a two or multidimensional quality index by using the reliability functions and the information supplied by expert judgements. In Section “An empirical application in South-East Spain”, an empirical application on farmland pricing in South-East Spain is used to illustrate the usefulness of the VMTRF in the valuation based on expert judgements. Section “Discussion” concludes the paper by summarizing key findings.
Valuation method based on expert judgements Valuation method based on two reliability functions In general, it is usual to assume certain logical market rules in economical modelling theory. For instance, the following basic valuation principle may be assumed to estimate the market value of an asset from a quality index: the asset with the highest quality index has the highest market value. It can be expressed as follows: Let j and k be two assets, with values ij and ik of their quality indexes, and vj and vk their market values, respectively. If ij < ik then vj < vk .
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Within this framework, the market value V of the asset by the VMTDF is based on the equality between its distribution function FV and the distribution function FI of its quality index I; and then, for a quality index I = i, it is given by
vD = ˚D (i)
(1) FV−1
where ˚D = ◦ FI . In this respect, it is possible to consider alternative measures to the distribution function where the above basic principle holds as well. In particular, taking into account the reliability function RV (v) = 1 − FV (v) of the market value V of the asset and the reliability function RI (i) = 1 − FI (i) of its quality index, instead of their distribution functions, an alternative valuation method based on the equality of both reliability functions can be considered. Clearly, the VMTRF verifies the basic valuation principle, since both reliability functions are decreasing. Hence, from the VMTRF, the market value of the asset with quality index I = i is given by
vR = ˚R (i)
(2)
where ˚R = RV−1 ◦ RI . Note that, for a single quality index, the VMTRF is only an alternative viewpoint to the VMTDF, since both methods provide the same assessment from (1) and (2):
vR = RV−1 (RI (i)) = (1 − FV )−1 (1 − FI (i)) = FV−1 (FI (i)) = vD
(3)
However, as we will see below, (3) does not hold for a two or multidimensional quality index, i.e. these techniques provide different assessments of the asset when the quality index has two or more components. Two-dimensional extension of the VMTRF In order to approach the market value of the asset when the quality index is two-dimensional, the same basic valuation principle is required. Now, it can be written as follows: Let j and k be two assets, with values (i1j , i2j ) and (i1k , i2k ) of their quality indexes, and vj and vk their market values, respectively. If (i1j , i2j ) < (i1k , i2k ) then vj < vk , where the ordering between two vectors is determined by the orderings between their corresponding components. In this context, the appraisal of the asset with quality index I = (i1 , i2 ) by the VMTDF is
vD = ˚D (i1 , i2 )
(4)
FV−1
where ˚D = ◦ FI , see (2) in Herrerías and Herrerías (2010). Note that (4) provides a market value lower than the valuations obtained with each component of its quality index,
vD ≤ inf {v1 , v2 }
(5)
where v1 and v2 are the assessments of the asset through each one of the components of the quality index, given by (1) from their marginal distribution functions F1 and F2 , respectively. Here, (5) reflects an undervaluation of the asset or loss in its appraisal as long as it is considered further information by more than one component of the quality index. In this setting, the use of reliability functions instead of distribution functions is more relevant if it allows us to avoid the undervaluation by the VMTDF of the asset with a two-dimensional quality index. Therefore, from the equality between the reliability function RV of the market value V of the asset and the joint reliability function RI of its quality index, the assessment of an asset with quality index I = (i1 , i2 ) by the VMTRF is given as
vR = ˚R (i1 , i2 )
(6)
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where ˚R = RV−1 ◦ RI and RI (i1 , i2 ) = P(I > (i1 , i2 )), which is determined by the joint distribution function of the quality index and their marginal distributions RI (i1 , i2 ) = FI (i1 , i2 ) − F1 (i1 ) − F2 (i2 ) + 1
(7)
where 0 < ˛ < 1 represents the weight of the first component of the quality index. This weighted model is equivalent to the weighting of their reversed hazard rate functions. From the weighted model (9), the appraisal vWD of the asset with quality index I = (i1 , i2 ) by the VMTDF holds
Furthermore, taking into account that for any joint reliability function the following inequalities holds:
vD ≤ inf {v1 , v2 } ≤ vWD ≤ sup{v1 , v2 } ≤ vR ,
RI (i1 , i2 ) ≤ R1 (i1 ) and RI (i1 , i2 ) ≤ R2 (i2 ) ⇔ RI (i1 , i2 )
since
≤ inf
inf {F1 (i1 ), F2 (i2 )} ≤ FWD (i1 , i2 ) ≤ sup{F1 (i1 ), F2 (i2 )}.
R1 (i1 ), R2 (i2 )
where Rj (ij ) = 1 − Fj (ij ) for j = 1, 2, are the marginal reliability functions of each component of the quality index, and RV−1 is decreasing, we have
vR ≥ sup{v1 , v2 }
(10)
(8)
Consequently, this alternative VMTRF does not lead to undervaluation of the asset with respect to their marginal assessments. In addition, the VMTRF produces an appraisal of the asset higher than the valuations obtained for each component of its quality index, i.e., an overestimation when more than one component of the quality index is available for asset pricing. Note that, from (5) and (8), it is immediate to obtain that vD ≤ vR , i.e. the appraisals by the VMTRF are higher than ones obtained by the VMTDF when the quality index has more than one component. From a practical viewpoint, the above results have been shown in the two-dimensional case, but they can be easily generalized to an n-dimensional quality index. For instance, by using the same development as (5) and (8), the following comparison is established between the appraisals of the asset with quality index I = (i1 , ..., in ) by both methods, VMTDF and VMTRF,
Analogously, by using the reliability functions R1 and R2 instead of the distribution functions F1 and F2 , the following bivariate reliability function can be considered for modelling the twodimensional quality index RWR (i1 , i2 ) = R1˛ (i1 ) · R21−˛ (i2 )
(11)
where 0 < ˛ < 1 is the weight of the first component of the quality index. This weighted model is equivalent to the weighting of their hazard rate functions, see e.g. García et al. (2002) and Franco and Vivo (2007). Furthermore, it is immediate to check that the bivariate reliability function RWR given by (11) is bounded as follows inf {R1 (i1 ), R2 (i2 )} ≤ RWR (i1 , i2 ) ≤ sup{R1 (i1 ), R2 (i2 )}, and hence, taking into account that RV−1 is decreasing, the assessment vWR of the asset with quality index I = (i1 , i2 ) by the VMTRF holds
vD ≤ inf {v1 , v2 } ≤ vWR ≤ sup{v1 , v2 } ≤ vR
(12)
where vj ’s are the marginal assessments of the asset given by (2) from each marginal reliability function Rj , j = 1, . . ., n. In any case, to approach the market value of an asset from a twodimensional quality index by both methods, VMTDF and VMTRF, a bivariate probability model is required to fit this index. For example, Herrerías and Herrerías (2010) consider the pyramidal probability model to apply the VMTDF. Nevertheless, the above comparison between the assessments obtained by both methods, VMTDF and VMTRF, suggests to consider different procedures to model a twodimensional quality index in order to find more accurate appraisals, as weighted probability models, which are usual in multivariate risk analysis.
In general, the weighted probability models (9) and (11) provide appraisals of the asset from the two-dimensional quality index by the VMTDF and VMTRF, respectively, which are between the valuations obtained from each component of the index. Nevertheless, from a practical viewpoint, both weighted models require to find 0 < ˛ < 1 which represents the importance or weight of the first component of the quality index. This weight ˛ might be also supplied by expert judgments, or it can be determined by using the most likely values of each component of the two-dimensional quality index without the influence of the market value of the asset (see Franco and Vivo, 2007). For instance, by minimizing the distance between the mode of the weighted distribution function (9) and the modal vector (m1 , m2 ) of both quality components provided by the expert,
Weighted models in the VMTDF and VMTRF
FWD (m1 , m2 ) = F1˛ (m1 )F21−˛ (m2 ) =
vD ≤ inf {v1 , . . ., vn } ≤ sup{v1 , . . ., vn } ≤ vR
The weighted models have been a useful procedure to generate multidimensional probability models from distribution functions of each component of a quality index in an uncertainty environment. Specifically, it seems reasonable to deem that each component of the quality index may have a different relevance or weight in the appraisal of the asset, and consequently, the weighted models may be appropriated to reduce the undervaluation in the assessment by the VMTDF and to reduce the overvaluation in the assessment by the VMTRF, as above mentioned. In this setting, from the distribution functions F1 and F2 , by modelling the minimum, most likely and maximum values of each component of the two-dimensional quality index which may be supplied by expert judgments, the following bivariate distribution function has been used by García et al. (2002, 2012) and Franco and Vivo (2007), among others, FWD (i1 , i2 ) = F1˛ (i1 ) · F21−˛ (i2 )
(9)
F1 (m1 ) + F2 (m2 ) . 2
Hence, when F1 (m1 ) = / F2 (m2 ), the weight is determined by ˛=
ln((F1 (m1 ) + F2 (m2 ))/2) − ln F2 (m2 ) ∈ (0, 1). ln F1 (m1 ) − ln F2 (m2 )
(13)
and otherwise, ˛ could take any value in (0,1). Analogously, the mode of the weighted reliability function (11) may be assumed to be equal to the average of the modes of both quality components, RWR (m1 , m2 ) =
R1 (m1 ) + R2 (m2 ) , 2
therefore, when R1 (m1 ) = / R2 (m2 ), ˛=
ln((R1 (m1 ) + R2 (m2 ))/2) − ln R2 (m2 ) ∈ (0, 1), ln R1 (m1 ) − ln R2 (m2 )
and otherwise, ˛ could take any value in (0,1).
(14)
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Table 1 Expert judgments about farmland grapes.
V = Market value (D /ha) I1 = Gross production (kg/ha) I2 = Sand/soil content (%)
Pessimistic
Most likely
Optimistic
8138.70 15,625 15
10,642.92 18,750 25
15,025.30 26,042 50
Source: Own elaboration, from Guadalajara (1996).
An empirical application in South-East Spain In this section, we will carry out an empirical application to evaluate and illustrate the valuation method based on the two reliability functions through a farmland pricing example under uncertainty. The numerical results will show the comparison between the assessments using both valuation methods, VMTRF and VMTDF. Particularly, we consider the second practical case described by Guadalajara (1996), in order to appraise an agricultural plot dedicated to growing grapes, in Vinalopó Medio (Alicante, Spain). The two-dimensional quality index considered to describe the market value (D /ha) is formed by the gross production of grapes (kg/ha) and the percentage of sand in the soil of the plot. The aim is to approach the market value of a plot with an area of 1.2 ha, a gross production of 20, 413 kg/ha and a sand/soil content of 32%; from the minimum (pessimistic), mode (most likely) and maximum (optimistic) provided by the expert judgments for each variable (Table 1). As mentioned above, Herrerías and Herrerías (2010) consider the pyramidal distribution function for modelling the quality index I = (I1 , I2 ), in order to estimate the market value V of the farmland, by the VMTDF, having V a triangular distribution function. They obtain an assessment of 10, 487.86 D /ha, and then vD = 12, 585.44 D for this agricultural plot of 1.2 ha. Note that the main disadvantage of this assessment is that the quality index of the farmland is better than the most likely index, I = (20, 413 ; 32) > (18, 750 ; 25), and however, from the pyramidal model, the VMTDF provides a worse appraisal than the most likely value, 10, 487.86 < 10, 642.92 D /ha, or equivalently, vD = 12, 585.44 < 12, 771.50 D for this vineyard plot of 1.2 ha. Nevertheless, taking into account the same pyramidal model for the quality index at I = (20, 413 ; 32), along with the derived values from its joint and marginal distribution functions given in Herrerías and Herrerías (2010), it is easy to obtain from (7) that RI (20, 413 ; 32) = 0.206822. Now, using the reversed reliability function of the market value having a triangular model, the assessment by the VMTRF is 12, 526.94 D /ha, and consequently, we have vR = 15, 032.32 D for this agricultural plot of 1.2 ha. Thereby, we obtain a more reliable appraisal, since it is better than the most likely value which will be expected as its two-dimensional quality index to be better than the most likely index. Table 2 summarizes these assessments of the farmland for the empirical comparisons in a quick look, displaying the undervaluation by the VMTDF, vD = 12, 585.44 ≤ inf {v1 , v2 } = 13, 536.38, and by contrast, the estimation with the VMTRF is higher than the valuations v1 and v2 derived from each marginal component of the index, having a pyramidal model. Furthermore, to give a more clear illustration of the comparison among these assessments by both methods, VMTRF and VMTDF, from the two-dimensional quality index, Fig. 1 shows some surfaces
Fig. 1. Surfaces of assessments by VMTRF and VMTDF for a pyramidal quality index.
of assessments of agricultural plots corresponding to a triangular model for the market value and a pyramidal model for its quality index. In order to improve the interpretation of these graphs, the surfaces have been trimmed by a transversal region formed by the rectangle of vertexes (i1 , a2 ) = (20, 413 ; 15), (i1 , i2 ) = (20, 413 ; 32), (b1 , a2 ) = (26, 042 ; 15) and (b1 , i2 ) = (26, 042 ; 32). In the cross section, it is easy to see that for any quality index (i1 , i2 ), the white surface vR is always above the grey surface sup{v1 , v2 }, and the light-grey surface vD is always below the dark-grey surface inf {v1 , v2 }. Note that the assessments obtained by Guadalajara (1996) for this agricultural plot in the one-dimensional case, through the VMTDF from each component of the index, are also embraced between the ones provided by both methods, VMTRF and VMTDF, from the pyramidal model. Therefore, it might be considered the average of both assessments in order to appraise the market value with more relative accuracy, it also allows us to reduce the undervaluation by the VMTDF and the highest valuation by the VMTRF. Thus, the average appraisal of both methods is vA = 13, 808.88 D for this agricultural plot of 1.2 ha, which is also higher than the most likely value. Fig. 2 displays the assessments by both methods and their average appraisal from a pyramidal model for the quality index and its marginal appraisals. So, the pairs of both components of the quality index have been simultaneously taken from the bottom (a1 , a2 ) = (15, 625 ; 15) to the top (b1 , b2 ) = (26, 042 ; 50) passing through the most likely point (m1 , m2 ) = (18, 750 ; 25) and the particular value (i1 , i2 ) = (20, 413 ; 32) of the agricultural plot. On the other hand, we recall that the VMTRF and VMTDF require a bivariate probability model for the two-dimensional quality index. In the above case, the pyramidal model has been considered like in Herrerías and Herrerías (2010). However, the comparison between the assessments obtained with both methods, VMTRF and VMTDF, suggests to consider different bivariate probability models
Table 2 Assessments of the farmland grapes of 1.2 ha, for a pyramidal quality index.
Assessments (D )
vD
inf {v1 , v2 }
sup{v1 , v2 }
vR
12,585.44
13,536.46
13,747.25
15,032.32
Source: Author’s own elaboration.
Fig. 2. Assessments for a pyramidal quality index by varying simultaneously from the minimum to the maximum value of its components.
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Table 3 Assessments of the farmland grapes of 1.2 ha, from weighted models.
Assessments (D )
inf {v1 , v2 }
vWD
vWR
sup{v1 , v2 }
13,772.66
13,890.05
13,898.21
14,018.86
Source: Author’s own elaboration.
by weighting the components of the two-dimensional quality index in order to find better benchmarks for the farmland pricing. In this context, we consider again the valuation example of agricultural plot dedicated to growing grapes. A triangular model is assumed for the market value from the expert judgments regarding the minimum, most likely and maximum values given in Table 1. Thereby, triangular models can be also considered for modelling each component of the two-dimensional quality index, since they are more intuitive and simpler model in these scenarios. Within this framework, we apply the VMTDF when the quality index follows the bivariate weighted probability model (9) from triangular components, and analogously, we apply the VMTRF from the bivariate weighted probability model (11) for the quality index formed by triangular components. In order to do this, the weight of the first component of the quality index is required to use the weighted models (9) and (11), respectively, and then to approach the market value of this agricultural plot of 1.2 ha. Thus, as in PERT methodology, the expert might be also asked about the weight parameter ˛, or it may be calculated from (13) or (14) according to the weighting of distribution or reliability functions, respectively, and hence, we obtain ˛ = 0.506094 or ˛ = 0.497476, respectively. Therefore, using the bivariate distribution function given by the weighted model (9) with ˛ = 0.506094, the assessment by the VMTDF is 11, 575.04 D /ha, i.e. vWD = 13, 890.05 D for this agricultural plot of 1.2 ha. Note that this assessment of the farmland with quality index I = (20, 413 ; 32) is higher than the most likely value corresponding to the most likely index (18, 750 ; 25). Furthermore, it is closed to the above average appraisal. Analogously, from the weight model (11) with ˛ = 0.497476, the assessment by the VMTRF is 11, 581.84 D /ha, and then vWR = 13, 898.21 D for this agricultural plot of 1.2 ha. Note that this appraisal is slightly higher than the one obtained by the VMTDF from the previous weighted model, and consequently, it is also higher than the most likely value corresponding to the most likely index and it is closed to the above average appraisal. These assessments for the farmland grapes of 1.2 ha, through each method with its respective weighted model, are shown in Table 3; i.e. VMTDF with weighted distributions and VMTRF with weighted reliabilities. By contrast to the assessments summarized in Table 2, these appraisals obtained by using bivariate weighted models are embraced between the ones provided by both methods in the one-dimensional case from each component of the index. Similarly to Fig. 1, considering bivariate weighted models for the quality index, Fig. 3 displays the surfaces of assessments by the VMTRF and VMTDF, which illustrate their comparison for any quality index (i1 , i2 ). In detail, the white surface vR is always below the grey surface sup{v1 , v2 } and above the light-grey surface vD , which is also above the dark-grey surface inf {v1 , v2 }. Finally, in order to appraise the market value with more relative accuracy for this agricultural plot, it might be considered the average of both assessments obtained from weighted models. Thereby, the average appraisal of both methods from weighted models is vAW = 13, 894.13 D for this agricultural plot of 1.2 ha. Obviously, this average assessment is also higher than the most likely value, which will be expected when its quality index to be better than the most likely index. In this respect, Fig. 4 depicts these appraisals by the VMTRF and VMTDF, and their average value, which are similar
Fig. 3. Surfaces of assessments by VMTRF and VMTDF from weighted models for the index.
and always take between the assessments from each component of the index. As in Fig. 2, the pairs of both components have been simultaneously taken from the bottom to the top of the quality index. Discussion This paper was aimed at enlarging the valuation tools in farmland pricing under uncertainty from expert judgments, as in PERT methodology. In such situations, the VMTDF has been widely applied for estimating the market value from a quality index. However, when the quality index has two or more components, theoretical reasons and empirical evidences were found which suggested alternative or complementary techniques to make more relative accurate valuations. In this setting, the reliability function has worked as an appropriate alternative to the distribution function, arising the VMTRF as a dual approach to the VMTDF, in line with the usual basic valuation principle. The VMTRF presents the same advantages that the VMTDF with regard to other comparative techniques. Although, in general, both methods yield different assessments, it is worth pointing out that the VMTRF is equivalent to the VMTDF when the quality index has only one component. Moreover, another advantage of this technique is its relative simplicity implementation, so it can be easily incorporated into the routine valuation methodology using by appraisers. The main result of the VMTRF is to provide assessments of the asset by avoiding the undervaluation of the asset undergone by the VMTDF with respect to the appraisals produced from each component of the quality index.
Fig. 4. Assessments with weighted models by varying simultaneously from the minimum to the maximum value of the quality components.
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Furthermore, with regard to the use of the weighted models as a smoothed assessment tool, this procedure has provided more refined and better estimates than the ones yielded by the VMTRF and VMTDF, respectively. So, apart from solving situations wherein one may not to know what bivariate probability model fits to the expert judgments of the two-dimensional quality index, these findings are important because of they could minimize the assessment risk from both methods. These advantages of the VMTRF have been also illustrated through an empirical application on farmland pricing. The results about the farmland grapes show the improvements due to the use of the reliability functions in the valuation method for twodimensional quality indexes. In detail, assuming a pyramidal model for the index, the assessment by the VMTRF has been higher than the most likely appraisal, as expected, since the particular agricultural plot has higher quality components than the most likely quality index, and then, it is more reliable than the estimation by the VMTDF, which was lower than the most likely value. However, the VMTRF could provide an overvaluation of the asset with respect to the appraisals obtained from each component of the quality index. In order to solve this situation, the average appraisal of both methods was also more reliable than the assessment by the VMTDF. In this respect, bivariate weighted models were applied in this example of farmland grapes to assess with more relative accuracy by avoiding the possible overestimation of the agricultural plot with respect to the appraisals derived from each quality component. The empirical assessments from bivariate weighted models were similar, higher than the most likely appraisal, as expected, and taken between the appraisals obtained from each quality component. Our conclusion is that reliability function is a feasible and appropriate measure in valuation theory. The results reported here indicate that the VMTRF is applicable and a useful procedure to assess the market value of an asset with more relative accuracy. Although the paper was focused in the two-dimensional case, it is straightforward its extension to the multidimensional case. Acknowledgement This work was partially supported by Fundación Séneca of the Regional Government of Murcia (Spain) under Grant 11886/PHCS/09. References Awasthi, M.K., 2014. Socioeconomic determinants of farmland value in India. Land Use Policy 39, 78–83.
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Land Use Policy journal homepage: www.elsevier.com/locate/landusepol
A quick assessment from expert judgements to assist in farmland valuation M. Franco a,∗ , J.M. Vivo a , R. Herrerías b a b
Department of Statistics and Operations Research, University of Murcia, 30100 Murcia, Spain Department of Quantitative Methods for Economy, University of Granada, 18011 Granada, Spain
a r t i c l e
i n f o
Article history: Received 17 December 2014 Received in revised form 12 February 2015 Accepted 14 March 2015 JEL classification: Q12 Q24
a b s t r a c t Land is the most important agricultural asset, farmland transactions are generally not recurring actions for most buyers and sellers, thus it is often required reliable information from expert judgements. This paper presents a quick asset pricing method under uncertainty, based on the use of two reliability functions, a multidimensional index of characteristics or qualities influencing on the market value and the expert judgments, which overcomes the disadvantages of the valuation method based on two distribution functions for a multidimensional quality index. The usefulness of this method is illustrated through an empirical application on farmland pricing in South-East Spain, using a two-dimensional quality index. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Farmland pricing Expert judgements Bivariate reliability Weighted model
Introduction The asset pricing constitutes an important current issue due to the wide range of economic events that require estimating the value of the assets within the economic development of the countries. At the same time, the increasing complexity of the appraisal problems and the particular characteristics of the markets, it becomes more necessary to make better and more accurate valuations. Specifically, there exist some markets wherein the information is not enough due to not frequent transactions or lack of clarity in the prices. For instance, land is the most important agricultural asset but farmland transactions are generally not a recurring action for most buyers and sellers. In such situations, appraisers have to work in an uncertainty environment where the market value of the asset is obtained from the information provided by an expert, about the asset and its characteristics or qualities. Characteristics affecting farmland prices have been studied by some authors (e.g. Huang et al., 2006; Sills and Caviglia-Harris, 2008; Troncoso et al., 2010; Sklenicka et al., 2013; Awasthi, 2014; and the references therein), as well as the effect of transaction costs and taxes on the farmland prices (e.g. Lence, 2001; De Fontnouvelle and Lence, 2002; Segura
∗ Corresponding author at: Regional Campus of International Excellence “Campus Mare Nostrum”, University of Murcia, 30100 Murcia, Spain. Tel.: +34 868884187. E-mail address: [email protected] (M. Franco). http://dx.doi.org/10.1016/j.landusepol.2015.03.008 0264-8377/© 2015 Elsevier Ltd. All rights reserved.
et al., 2010, 2012). Undoubtedly, economic valuation of these assets is necessary for the development and functioning of these markets. In this context, the asset appraisal, under uncertainty, is frequently analyzed by econometric modelling and hedonic price indexes in both residential properties (e.g. Goodman and Thibodeau, 2003; Sirmans et al., 2005; Páez et al., 2008; Cervelló et al., 2011; Cervelló and Segura, 2011) and land valuations (e.g. Ma and Swinton, 2012; Samarasinghe and Greenhalgh, 2013; Awasthi, 2014; Lavee, 2015), as improvements of the classical synthetic method, but the weakness of these techniques is well-known when the information is not enough. Nevertheless, the valuation method based on the two cumulative distribution functions (VMTDF) has been widely used to approach the market value of an asset from a characteristic or quality index because of their easy implementation and its important advantages with respect to the above comparative techniques, since the VMTDF works well with very little information (Herrerías and Herrerías, 2010; and the references therein). Indeed, the VMTDF allows us valuing an asset under uncertainty, when only the pessimistic, optimistic and most likely values are available to the appraiser, which may be supplied by expert judgments. Ballestero (1971) introduced this technique by using two beta distributions, inspired by PERT methodology. Initially, it was extended to other distribution models by Romero (1977) and Ballestero and Caballer (1982), and the VMTDF has been studied and applied since then (see references in Herrerías and Herrerías,
M. Franco et al. / Land Use Policy 46 (2015) 324–329
2010). For instance, Romero (1977) used the VMTDF from the triangular distribution, which has been considered a standard model in risk analysis and suggested by Williams (1992) and Johnson (1997) as a more intuitive and simpler distribution since it only requires three parameters (pessimistic, optimistic and most likely). Although, the VMTDF was originally proposed for the valuation of an asset from the values of one single index, the appraisal processes under uncertainty generally use the supplied information by two or more quality indexes. Thus, extensions of this method have been analyzed by some authors; see, e.g. García et al. (2002) and Herrerías and Herrerías (2010). Unfortunately, in the multidimensional case, the VMTDF produces a loss with respect to the assessments from each component of a multidimensional quality index, and then weights among its components might be used to approach the asset pricing. In order to overcome these limitations, the reliability or survival function, also called decumulative distribution function by Yaari (1987) in dual theory of choice under risk, provides an alternative measure to the distribution function, and consequently, the valuation method based on two reliability functions (VMTRF) might help to appraise an asset under uncertainty from a quality index, even more when the dimension is reduced by unobserved components of quality. Note that the reliability measures have been widely used in many areas of economics, political science, biology, and engineering. In particular, many interesting results of reliability theory have been applied in risk analysis, and their properties have interesting qualitative implications in these fields (e.g. Bagnoli and Bergstrom, 2005; and the references therein). The main aim of this paper is to display a quick valuation method for land pricing based on expert judgements and illustrate its empirical application through an agricultural land pricing in South-East Spain. In general, the VMTRF allows us to approach the assessment of an asset from a multidimensional quality index by using the reliability functions corresponding to the two probability models. From a practical viewpoint, in this study the focus is placed on a two-dimensional quality index, and its comparison with the assessments given by the VMTDF from the joint twodimensional quality index and from each one of its components, being straightforward its generalization to the multidimensional case. The remainder of the paper is organized as follows. Section “Valuation method based on expert judgements” summarizes the valuation method based on expert judgements and presents the theoretical results to value an asset under uncertainty from a two or multidimensional quality index by using the reliability functions and the information supplied by expert judgements. In Section “An empirical application in South-East Spain”, an empirical application on farmland pricing in South-East Spain is used to illustrate the usefulness of the VMTRF in the valuation based on expert judgements. Section “Discussion” concludes the paper by summarizing key findings.
Valuation method based on expert judgements Valuation method based on two reliability functions In general, it is usual to assume certain logical market rules in economical modelling theory. For instance, the following basic valuation principle may be assumed to estimate the market value of an asset from a quality index: the asset with the highest quality index has the highest market value. It can be expressed as follows: Let j and k be two assets, with values ij and ik of their quality indexes, and vj and vk their market values, respectively. If ij < ik then vj < vk .
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Within this framework, the market value V of the asset by the VMTDF is based on the equality between its distribution function FV and the distribution function FI of its quality index I; and then, for a quality index I = i, it is given by
vD = ˚D (i)
(1) FV−1
where ˚D = ◦ FI . In this respect, it is possible to consider alternative measures to the distribution function where the above basic principle holds as well. In particular, taking into account the reliability function RV (v) = 1 − FV (v) of the market value V of the asset and the reliability function RI (i) = 1 − FI (i) of its quality index, instead of their distribution functions, an alternative valuation method based on the equality of both reliability functions can be considered. Clearly, the VMTRF verifies the basic valuation principle, since both reliability functions are decreasing. Hence, from the VMTRF, the market value of the asset with quality index I = i is given by
vR = ˚R (i)
(2)
where ˚R = RV−1 ◦ RI . Note that, for a single quality index, the VMTRF is only an alternative viewpoint to the VMTDF, since both methods provide the same assessment from (1) and (2):
vR = RV−1 (RI (i)) = (1 − FV )−1 (1 − FI (i)) = FV−1 (FI (i)) = vD
(3)
However, as we will see below, (3) does not hold for a two or multidimensional quality index, i.e. these techniques provide different assessments of the asset when the quality index has two or more components. Two-dimensional extension of the VMTRF In order to approach the market value of the asset when the quality index is two-dimensional, the same basic valuation principle is required. Now, it can be written as follows: Let j and k be two assets, with values (i1j , i2j ) and (i1k , i2k ) of their quality indexes, and vj and vk their market values, respectively. If (i1j , i2j ) < (i1k , i2k ) then vj < vk , where the ordering between two vectors is determined by the orderings between their corresponding components. In this context, the appraisal of the asset with quality index I = (i1 , i2 ) by the VMTDF is
vD = ˚D (i1 , i2 )
(4)
FV−1
where ˚D = ◦ FI , see (2) in Herrerías and Herrerías (2010). Note that (4) provides a market value lower than the valuations obtained with each component of its quality index,
vD ≤ inf {v1 , v2 }
(5)
where v1 and v2 are the assessments of the asset through each one of the components of the quality index, given by (1) from their marginal distribution functions F1 and F2 , respectively. Here, (5) reflects an undervaluation of the asset or loss in its appraisal as long as it is considered further information by more than one component of the quality index. In this setting, the use of reliability functions instead of distribution functions is more relevant if it allows us to avoid the undervaluation by the VMTDF of the asset with a two-dimensional quality index. Therefore, from the equality between the reliability function RV of the market value V of the asset and the joint reliability function RI of its quality index, the assessment of an asset with quality index I = (i1 , i2 ) by the VMTRF is given as
vR = ˚R (i1 , i2 )
(6)
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where ˚R = RV−1 ◦ RI and RI (i1 , i2 ) = P(I > (i1 , i2 )), which is determined by the joint distribution function of the quality index and their marginal distributions RI (i1 , i2 ) = FI (i1 , i2 ) − F1 (i1 ) − F2 (i2 ) + 1
(7)
where 0 < ˛ < 1 represents the weight of the first component of the quality index. This weighted model is equivalent to the weighting of their reversed hazard rate functions. From the weighted model (9), the appraisal vWD of the asset with quality index I = (i1 , i2 ) by the VMTDF holds
Furthermore, taking into account that for any joint reliability function the following inequalities holds:
vD ≤ inf {v1 , v2 } ≤ vWD ≤ sup{v1 , v2 } ≤ vR ,
RI (i1 , i2 ) ≤ R1 (i1 ) and RI (i1 , i2 ) ≤ R2 (i2 ) ⇔ RI (i1 , i2 )
since
≤ inf
inf {F1 (i1 ), F2 (i2 )} ≤ FWD (i1 , i2 ) ≤ sup{F1 (i1 ), F2 (i2 )}.
R1 (i1 ), R2 (i2 )
where Rj (ij ) = 1 − Fj (ij ) for j = 1, 2, are the marginal reliability functions of each component of the quality index, and RV−1 is decreasing, we have
vR ≥ sup{v1 , v2 }
(10)
(8)
Consequently, this alternative VMTRF does not lead to undervaluation of the asset with respect to their marginal assessments. In addition, the VMTRF produces an appraisal of the asset higher than the valuations obtained for each component of its quality index, i.e., an overestimation when more than one component of the quality index is available for asset pricing. Note that, from (5) and (8), it is immediate to obtain that vD ≤ vR , i.e. the appraisals by the VMTRF are higher than ones obtained by the VMTDF when the quality index has more than one component. From a practical viewpoint, the above results have been shown in the two-dimensional case, but they can be easily generalized to an n-dimensional quality index. For instance, by using the same development as (5) and (8), the following comparison is established between the appraisals of the asset with quality index I = (i1 , ..., in ) by both methods, VMTDF and VMTRF,
Analogously, by using the reliability functions R1 and R2 instead of the distribution functions F1 and F2 , the following bivariate reliability function can be considered for modelling the twodimensional quality index RWR (i1 , i2 ) = R1˛ (i1 ) · R21−˛ (i2 )
(11)
where 0 < ˛ < 1 is the weight of the first component of the quality index. This weighted model is equivalent to the weighting of their hazard rate functions, see e.g. García et al. (2002) and Franco and Vivo (2007). Furthermore, it is immediate to check that the bivariate reliability function RWR given by (11) is bounded as follows inf {R1 (i1 ), R2 (i2 )} ≤ RWR (i1 , i2 ) ≤ sup{R1 (i1 ), R2 (i2 )}, and hence, taking into account that RV−1 is decreasing, the assessment vWR of the asset with quality index I = (i1 , i2 ) by the VMTRF holds
vD ≤ inf {v1 , v2 } ≤ vWR ≤ sup{v1 , v2 } ≤ vR
(12)
where vj ’s are the marginal assessments of the asset given by (2) from each marginal reliability function Rj , j = 1, . . ., n. In any case, to approach the market value of an asset from a twodimensional quality index by both methods, VMTDF and VMTRF, a bivariate probability model is required to fit this index. For example, Herrerías and Herrerías (2010) consider the pyramidal probability model to apply the VMTDF. Nevertheless, the above comparison between the assessments obtained by both methods, VMTDF and VMTRF, suggests to consider different procedures to model a twodimensional quality index in order to find more accurate appraisals, as weighted probability models, which are usual in multivariate risk analysis.
In general, the weighted probability models (9) and (11) provide appraisals of the asset from the two-dimensional quality index by the VMTDF and VMTRF, respectively, which are between the valuations obtained from each component of the index. Nevertheless, from a practical viewpoint, both weighted models require to find 0 < ˛ < 1 which represents the importance or weight of the first component of the quality index. This weight ˛ might be also supplied by expert judgments, or it can be determined by using the most likely values of each component of the two-dimensional quality index without the influence of the market value of the asset (see Franco and Vivo, 2007). For instance, by minimizing the distance between the mode of the weighted distribution function (9) and the modal vector (m1 , m2 ) of both quality components provided by the expert,
Weighted models in the VMTDF and VMTRF
FWD (m1 , m2 ) = F1˛ (m1 )F21−˛ (m2 ) =
vD ≤ inf {v1 , . . ., vn } ≤ sup{v1 , . . ., vn } ≤ vR
The weighted models have been a useful procedure to generate multidimensional probability models from distribution functions of each component of a quality index in an uncertainty environment. Specifically, it seems reasonable to deem that each component of the quality index may have a different relevance or weight in the appraisal of the asset, and consequently, the weighted models may be appropriated to reduce the undervaluation in the assessment by the VMTDF and to reduce the overvaluation in the assessment by the VMTRF, as above mentioned. In this setting, from the distribution functions F1 and F2 , by modelling the minimum, most likely and maximum values of each component of the two-dimensional quality index which may be supplied by expert judgments, the following bivariate distribution function has been used by García et al. (2002, 2012) and Franco and Vivo (2007), among others, FWD (i1 , i2 ) = F1˛ (i1 ) · F21−˛ (i2 )
(9)
F1 (m1 ) + F2 (m2 ) . 2
Hence, when F1 (m1 ) = / F2 (m2 ), the weight is determined by ˛=
ln((F1 (m1 ) + F2 (m2 ))/2) − ln F2 (m2 ) ∈ (0, 1). ln F1 (m1 ) − ln F2 (m2 )
(13)
and otherwise, ˛ could take any value in (0,1). Analogously, the mode of the weighted reliability function (11) may be assumed to be equal to the average of the modes of both quality components, RWR (m1 , m2 ) =
R1 (m1 ) + R2 (m2 ) , 2
therefore, when R1 (m1 ) = / R2 (m2 ), ˛=
ln((R1 (m1 ) + R2 (m2 ))/2) − ln R2 (m2 ) ∈ (0, 1), ln R1 (m1 ) − ln R2 (m2 )
and otherwise, ˛ could take any value in (0,1).
(14)
M. Franco et al. / Land Use Policy 46 (2015) 324–329
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Table 1 Expert judgments about farmland grapes.
V = Market value (D /ha) I1 = Gross production (kg/ha) I2 = Sand/soil content (%)
Pessimistic
Most likely
Optimistic
8138.70 15,625 15
10,642.92 18,750 25
15,025.30 26,042 50
Source: Own elaboration, from Guadalajara (1996).
An empirical application in South-East Spain In this section, we will carry out an empirical application to evaluate and illustrate the valuation method based on the two reliability functions through a farmland pricing example under uncertainty. The numerical results will show the comparison between the assessments using both valuation methods, VMTRF and VMTDF. Particularly, we consider the second practical case described by Guadalajara (1996), in order to appraise an agricultural plot dedicated to growing grapes, in Vinalopó Medio (Alicante, Spain). The two-dimensional quality index considered to describe the market value (D /ha) is formed by the gross production of grapes (kg/ha) and the percentage of sand in the soil of the plot. The aim is to approach the market value of a plot with an area of 1.2 ha, a gross production of 20, 413 kg/ha and a sand/soil content of 32%; from the minimum (pessimistic), mode (most likely) and maximum (optimistic) provided by the expert judgments for each variable (Table 1). As mentioned above, Herrerías and Herrerías (2010) consider the pyramidal distribution function for modelling the quality index I = (I1 , I2 ), in order to estimate the market value V of the farmland, by the VMTDF, having V a triangular distribution function. They obtain an assessment of 10, 487.86 D /ha, and then vD = 12, 585.44 D for this agricultural plot of 1.2 ha. Note that the main disadvantage of this assessment is that the quality index of the farmland is better than the most likely index, I = (20, 413 ; 32) > (18, 750 ; 25), and however, from the pyramidal model, the VMTDF provides a worse appraisal than the most likely value, 10, 487.86 < 10, 642.92 D /ha, or equivalently, vD = 12, 585.44 < 12, 771.50 D for this vineyard plot of 1.2 ha. Nevertheless, taking into account the same pyramidal model for the quality index at I = (20, 413 ; 32), along with the derived values from its joint and marginal distribution functions given in Herrerías and Herrerías (2010), it is easy to obtain from (7) that RI (20, 413 ; 32) = 0.206822. Now, using the reversed reliability function of the market value having a triangular model, the assessment by the VMTRF is 12, 526.94 D /ha, and consequently, we have vR = 15, 032.32 D for this agricultural plot of 1.2 ha. Thereby, we obtain a more reliable appraisal, since it is better than the most likely value which will be expected as its two-dimensional quality index to be better than the most likely index. Table 2 summarizes these assessments of the farmland for the empirical comparisons in a quick look, displaying the undervaluation by the VMTDF, vD = 12, 585.44 ≤ inf {v1 , v2 } = 13, 536.38, and by contrast, the estimation with the VMTRF is higher than the valuations v1 and v2 derived from each marginal component of the index, having a pyramidal model. Furthermore, to give a more clear illustration of the comparison among these assessments by both methods, VMTRF and VMTDF, from the two-dimensional quality index, Fig. 1 shows some surfaces
Fig. 1. Surfaces of assessments by VMTRF and VMTDF for a pyramidal quality index.
of assessments of agricultural plots corresponding to a triangular model for the market value and a pyramidal model for its quality index. In order to improve the interpretation of these graphs, the surfaces have been trimmed by a transversal region formed by the rectangle of vertexes (i1 , a2 ) = (20, 413 ; 15), (i1 , i2 ) = (20, 413 ; 32), (b1 , a2 ) = (26, 042 ; 15) and (b1 , i2 ) = (26, 042 ; 32). In the cross section, it is easy to see that for any quality index (i1 , i2 ), the white surface vR is always above the grey surface sup{v1 , v2 }, and the light-grey surface vD is always below the dark-grey surface inf {v1 , v2 }. Note that the assessments obtained by Guadalajara (1996) for this agricultural plot in the one-dimensional case, through the VMTDF from each component of the index, are also embraced between the ones provided by both methods, VMTRF and VMTDF, from the pyramidal model. Therefore, it might be considered the average of both assessments in order to appraise the market value with more relative accuracy, it also allows us to reduce the undervaluation by the VMTDF and the highest valuation by the VMTRF. Thus, the average appraisal of both methods is vA = 13, 808.88 D for this agricultural plot of 1.2 ha, which is also higher than the most likely value. Fig. 2 displays the assessments by both methods and their average appraisal from a pyramidal model for the quality index and its marginal appraisals. So, the pairs of both components of the quality index have been simultaneously taken from the bottom (a1 , a2 ) = (15, 625 ; 15) to the top (b1 , b2 ) = (26, 042 ; 50) passing through the most likely point (m1 , m2 ) = (18, 750 ; 25) and the particular value (i1 , i2 ) = (20, 413 ; 32) of the agricultural plot. On the other hand, we recall that the VMTRF and VMTDF require a bivariate probability model for the two-dimensional quality index. In the above case, the pyramidal model has been considered like in Herrerías and Herrerías (2010). However, the comparison between the assessments obtained with both methods, VMTRF and VMTDF, suggests to consider different bivariate probability models
Table 2 Assessments of the farmland grapes of 1.2 ha, for a pyramidal quality index.
Assessments (D )
vD
inf {v1 , v2 }
sup{v1 , v2 }
vR
12,585.44
13,536.46
13,747.25
15,032.32
Source: Author’s own elaboration.
Fig. 2. Assessments for a pyramidal quality index by varying simultaneously from the minimum to the maximum value of its components.
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M. Franco et al. / Land Use Policy 46 (2015) 324–329
Table 3 Assessments of the farmland grapes of 1.2 ha, from weighted models.
Assessments (D )
inf {v1 , v2 }
vWD
vWR
sup{v1 , v2 }
13,772.66
13,890.05
13,898.21
14,018.86
Source: Author’s own elaboration.
by weighting the components of the two-dimensional quality index in order to find better benchmarks for the farmland pricing. In this context, we consider again the valuation example of agricultural plot dedicated to growing grapes. A triangular model is assumed for the market value from the expert judgments regarding the minimum, most likely and maximum values given in Table 1. Thereby, triangular models can be also considered for modelling each component of the two-dimensional quality index, since they are more intuitive and simpler model in these scenarios. Within this framework, we apply the VMTDF when the quality index follows the bivariate weighted probability model (9) from triangular components, and analogously, we apply the VMTRF from the bivariate weighted probability model (11) for the quality index formed by triangular components. In order to do this, the weight of the first component of the quality index is required to use the weighted models (9) and (11), respectively, and then to approach the market value of this agricultural plot of 1.2 ha. Thus, as in PERT methodology, the expert might be also asked about the weight parameter ˛, or it may be calculated from (13) or (14) according to the weighting of distribution or reliability functions, respectively, and hence, we obtain ˛ = 0.506094 or ˛ = 0.497476, respectively. Therefore, using the bivariate distribution function given by the weighted model (9) with ˛ = 0.506094, the assessment by the VMTDF is 11, 575.04 D /ha, i.e. vWD = 13, 890.05 D for this agricultural plot of 1.2 ha. Note that this assessment of the farmland with quality index I = (20, 413 ; 32) is higher than the most likely value corresponding to the most likely index (18, 750 ; 25). Furthermore, it is closed to the above average appraisal. Analogously, from the weight model (11) with ˛ = 0.497476, the assessment by the VMTRF is 11, 581.84 D /ha, and then vWR = 13, 898.21 D for this agricultural plot of 1.2 ha. Note that this appraisal is slightly higher than the one obtained by the VMTDF from the previous weighted model, and consequently, it is also higher than the most likely value corresponding to the most likely index and it is closed to the above average appraisal. These assessments for the farmland grapes of 1.2 ha, through each method with its respective weighted model, are shown in Table 3; i.e. VMTDF with weighted distributions and VMTRF with weighted reliabilities. By contrast to the assessments summarized in Table 2, these appraisals obtained by using bivariate weighted models are embraced between the ones provided by both methods in the one-dimensional case from each component of the index. Similarly to Fig. 1, considering bivariate weighted models for the quality index, Fig. 3 displays the surfaces of assessments by the VMTRF and VMTDF, which illustrate their comparison for any quality index (i1 , i2 ). In detail, the white surface vR is always below the grey surface sup{v1 , v2 } and above the light-grey surface vD , which is also above the dark-grey surface inf {v1 , v2 }. Finally, in order to appraise the market value with more relative accuracy for this agricultural plot, it might be considered the average of both assessments obtained from weighted models. Thereby, the average appraisal of both methods from weighted models is vAW = 13, 894.13 D for this agricultural plot of 1.2 ha. Obviously, this average assessment is also higher than the most likely value, which will be expected when its quality index to be better than the most likely index. In this respect, Fig. 4 depicts these appraisals by the VMTRF and VMTDF, and their average value, which are similar
Fig. 3. Surfaces of assessments by VMTRF and VMTDF from weighted models for the index.
and always take between the assessments from each component of the index. As in Fig. 2, the pairs of both components have been simultaneously taken from the bottom to the top of the quality index. Discussion This paper was aimed at enlarging the valuation tools in farmland pricing under uncertainty from expert judgments, as in PERT methodology. In such situations, the VMTDF has been widely applied for estimating the market value from a quality index. However, when the quality index has two or more components, theoretical reasons and empirical evidences were found which suggested alternative or complementary techniques to make more relative accurate valuations. In this setting, the reliability function has worked as an appropriate alternative to the distribution function, arising the VMTRF as a dual approach to the VMTDF, in line with the usual basic valuation principle. The VMTRF presents the same advantages that the VMTDF with regard to other comparative techniques. Although, in general, both methods yield different assessments, it is worth pointing out that the VMTRF is equivalent to the VMTDF when the quality index has only one component. Moreover, another advantage of this technique is its relative simplicity implementation, so it can be easily incorporated into the routine valuation methodology using by appraisers. The main result of the VMTRF is to provide assessments of the asset by avoiding the undervaluation of the asset undergone by the VMTDF with respect to the appraisals produced from each component of the quality index.
Fig. 4. Assessments with weighted models by varying simultaneously from the minimum to the maximum value of the quality components.
M. Franco et al. / Land Use Policy 46 (2015) 324–329
Furthermore, with regard to the use of the weighted models as a smoothed assessment tool, this procedure has provided more refined and better estimates than the ones yielded by the VMTRF and VMTDF, respectively. So, apart from solving situations wherein one may not to know what bivariate probability model fits to the expert judgments of the two-dimensional quality index, these findings are important because of they could minimize the assessment risk from both methods. These advantages of the VMTRF have been also illustrated through an empirical application on farmland pricing. The results about the farmland grapes show the improvements due to the use of the reliability functions in the valuation method for twodimensional quality indexes. In detail, assuming a pyramidal model for the index, the assessment by the VMTRF has been higher than the most likely appraisal, as expected, since the particular agricultural plot has higher quality components than the most likely quality index, and then, it is more reliable than the estimation by the VMTDF, which was lower than the most likely value. However, the VMTRF could provide an overvaluation of the asset with respect to the appraisals obtained from each component of the quality index. In order to solve this situation, the average appraisal of both methods was also more reliable than the assessment by the VMTDF. In this respect, bivariate weighted models were applied in this example of farmland grapes to assess with more relative accuracy by avoiding the possible overestimation of the agricultural plot with respect to the appraisals derived from each quality component. The empirical assessments from bivariate weighted models were similar, higher than the most likely appraisal, as expected, and taken between the appraisals obtained from each quality component. Our conclusion is that reliability function is a feasible and appropriate measure in valuation theory. The results reported here indicate that the VMTRF is applicable and a useful procedure to assess the market value of an asset with more relative accuracy. Although the paper was focused in the two-dimensional case, it is straightforward its extension to the multidimensional case. Acknowledgement This work was partially supported by Fundación Séneca of the Regional Government of Murcia (Spain) under Grant 11886/PHCS/09. References Awasthi, M.K., 2014. Socioeconomic determinants of farmland value in India. Land Use Policy 39, 78–83.
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