CC BY-NC-ND
28
DOI: 10.17323/2587-814X.2020.2.48.63
Multidimensional log-normal distribution in real estate appraisals
Michael B. Laskin
E-mail: laskinmb@yahoo.com
St. Petersburg Institute for Informatics and Automation, Russian Academy of Sciences Address: 39, 14 Line, St. Petersburg 199178, Russia
Abstract
The purpose of the research was to develop a market value appraisal methodology based on a set of a joint logarithmically normal distribution of price-forming factors. Joint logarithmically normal distribution means random vector component logarithms are distributed together jointly normally. This article suggests a method for appraising the real estate market value based on the statistical hypothesis of a joint logarithmically normal distribution and conditional distribution of prices with fixed values of pricing factors. The article suggests a method of offer price analysis from the point of view of its relevance to pricing factor values. We consider the features of the coefficient of development depending on the area of the land plot. Additional arguments are given in favor of estimating market value as a mode of conditional laws of price distribution. An example of a multidimensional lognormal distribution of prices and pricing factors such as the area of the improvements (improvements mean buildings and constructions) area and the land area in real data, i.e. for the case of a three-dimensional random vector. We present a formula for determining the absolute maximum density point of a multidimensional logarithmically normal random vector. The proof is given in the Appendix. The results obtained can be used to create information systems to support decision-making in valuation activities for real estate properties.
Key words: market value; logarithmically normal law of price distribution; multidimensional logarithmically normal distribution, valuation of real estate.
Citation: Laskin M.B. (2020) Multidimensional log-normal distribution in real estate appraisals. Business Informatics, vol. 14, no 2, pp. 48-63. DOI: 10.17323/2587-814X.2020.2.48.63
Introduction
One of the most common methods in market value real estate appraisals is the linear regression model of prices with some price-forming factors as regressors. The factors can be qualitative (type of home, encumbrance, floors, window view, the condition of the apartment/room, etc.) and quantitative (area of the object or the land plot, distance to the city center, to the metro, to other infrastructure objects, etc.).
There are various views on division the pricing factors into classes. In the context of this article, we mean splitting into qualitative and quantitative factors in terms of the possibility of representing the values of the factor as a real number (if such a possibility exists, then the factor is quantitative). Combining quantitative and qualitative factors in a single regression model presents a certain difficulty for analysts. However, this problem goes beyond the scope of the present article: here we will limit ourselves only to quantitative (real) factors. Very often such factors, considered as random variables on a set of objects of comparison, can usually be approximated by a logarithmically normal distribution. There are reasons to assume that the prices of properties formed by sequential comparisons follow the logarithmically normal distribution law.
The theoretical reason for the formation of a lognormal general population for prices formed by successive comparisons was given in [1]. The fact of subordination of rental rates in real estate to the logarithmically normal distribution was pointed out by Aitch-inson and Brown in 1963 [2]. More recent researchers have also pointed to the logarithmic distribution of prices in real estate [3]. This approach is not yet traditional from the point of view of the existing practice of real estate valuation, since it requires the use of special applied statistical packages that are not used by practicing appraisers, who use
a small number of objects for comparison. At the same time, the changing information environment encourages researchers to look for new, non-traditional approaches to real estate valuation. As an example, we can cite the works [4—9] devoted to the method of hedonistic pricing, i.e. the identification of a statistical relationship between the average or median cost of housing, internal and external price-forming factors.
Statistical dependence is usually estimated using models of linear, logarithmic, or partially logarithmic dependence. In general, this same ideology is the basis for the report on cadastral value [10] made by the St Petersburg government department "Cadastral assessment" in 2018. A number of works use non-regression models for estimating residential real estate objects: for example, in [11, 12] neural networks are used to predict the value of residential property, in [13, 14] machine learning methods (random forest, support vector method) are used, and in [15] the results of using such methods as decision trees, naive Bayesian classifier, and AdaBoost are compared. These methods require the use of large data samples. Another approach is to use price indices. For example, the Case-Shiller housing price index is considered in [16]. Articles [17—19] study the re-sale index, which predicts changes in the value of a resold property based on the difference in time and changes in its attributes between the initial sale and subsequent resale. The authors of [20—23] consider a hybrid method that combines a hedonistic approach and a method of re-selling.
The main approach to the study of price bubbles is to use variations of auto regression methods applied to average prices, for example, in [24—28]. Thus, the use of multidimensional logarithmically normal distributions is also in line with current trends in the search for non-traditional methods in real estate valuation.
1. Estimation of market value based on conditional distributions with fixed values of price-forming quantitative factors
Let V — be the price of the offer (or transaction), X1, ..., Xn — are the quantitative (real) price — forming factors. Let W = ln(V), Y = ln(X ), i = 1, n (then v = eW, X = eY).
Consider a multidimensional normal random vector (W, Y1, ..., Yn ) with a mean vector (fj.W, /1, ..., nY). Let's write the covariance matrix in block form:
r
CV -
w
cov
(W,f)
cov
(W,f)
COV
\ \ / y
where COV — covariance matrix of a random vector Y = Y, ..., Y,);
cov (w, f) - vector (pwy^ o^,..., p^ a^ayj;
...,ct^ — variances of random variables W, Y., ..., Y ;
* 1* ' n '
Py^,..., — corresponding correlation
coefficients.
Then, the conditional expectation of W, if Y = yp ...,Yn = yn equals to
E{W\Yx=yi,...Jn=yn) = = nw +{C0V1 xcov{w,f)T
where fiY =(/ur1>---»//i;). Conditional variance of W, if Y = y , — ,Y = yn is equal to
D(W\Yl=yl,...,Yn=yn) = = a2w-JCOV-1 xcov,cov{\V,f)J
For fixed values of price-forming factors Xl = x., — ,Xn = xn the most probable value of the offer price (or transaction, depending on what prices were in the source data) V is calculated using the conditional mode formula:
Mode(V\Xl=x1=e*,...,Xn=xn=ey") = exp(Mw+(COV~1xcov(W,Y))T ,(?-&)- (1) -o* + (COV1 x cov{W, f)T, cov(W, f)).
Under the terms of Federal Law No 135 [29], the market value is the most probable price at which the evaluation object can be alienated on the open market in conditions of perfect competition. In practice, appraisers tend to use an average or median estimation. Such estimations can be based on conditional expectation and a conditional median:
W\Xi = *i = exp(^),..., Xn=xn= exp (yn)) = = expi/v +(COV-lxcov(w,Y)T, (2)
+ ^ -^(cOV-'xeovfaff, cov(w,Y) J).
Median (V\Xx = x, = exp(^), ...,Xn = xn = expO>j) = (3)
= expOv + COV1 x cov(W, Yf, (Y - JTy).
Thus, if with respect to some ensemble of quantitative pricing factors and the prices of objects of comparison can be adopted as a working hypothesis on the joint log-normal distribution (joint normal distribution of the logarithms) component of a random vector, then the valuation can be accepted in the evaluation by the formula (1). Estimates according to the formulas (2) and (3) can also be taken; but it should be noted that they do not follow the definition of market value in accordance with Federal Law No 135.
Let's consider an example that uses real data collected by well-known Russian appraisers and was published on the resource [30]. The data set includes 40 real estate objects of industrial and warehouse use with a location in the same region (St Petersburg), on offer for sale in the same time period. Since the authors of the example justified the rejection of a number of adjustments, in our example we will also consider the data compa-
rable and comparable without additional adjustments. Industrial and warehouse purpose real estate is considered as a unit complex consisting of a land plot and improvements (buildings). The data set is presented in Table 1.
The items compared are considered as existing industry and warehouse properties that are offered for sale in the current use. We will build a general method for estimating the market value (without auction discount), if the area
Source
Building area (sq. m) Land area (sq. m) Offer prices (rubles) Price to improvements square ratio (rubles per 1 sq. m of improvements)
400 2 500 20 500 000 51 250
750 5 000 18 000 000 24 000
1 081 3 378 26 000 000 24 052
1 130 6 638 27 500 000 24 336
1 320 4 167 31 500 000 23 864
1 440 10 000 160 000 000 111 111
1 790 3 462 93 000 000 51 955
1 900 13 000 85 000 000 44 737
2 125 5 623 85 000 000 40 000
2 642 5 183 75 000 000 28 388
2 700 6 800 59 000 000 21 852
1 820 2 737 32 000 000 17 582
2 250 9 252 84 000 000 37 333
2 973 5 388 90 000 000 30 272
3 513 10 000 80 000 000 22 773
3 600 5 000 95 000 000 26 389
4 000 13 558 140 000 000 35 000
4 124 12 866 91 000 000 22 066
4 167 5 000 125 000 000 29 998
4 257 6 861 128 500 000 30 186
Table 1.
data
Building area (sq. m) Land area (sq. m) Offer prices (rubles) Price to improvements square ratio (rubles per 1 sq. m of improvements)
5 292 11 143 56 000 000 10 582
5 300 16 000 220 000 000 41 509
6 011 11 319 135 000 000 22 459
6 013 20 781 90 000 000 14 968
6 060 21 790 179 000 000 29 538
6 123 2 390 152 490 000 24 904
6 479 7 337 119 000 000 18 367
6 756 4 220 90 000 000 13 321
10 000 12 000 420 000 000 42 000
10 300 17 000 312 000 000 30 291
10 672 12 194 350 000 000 32 796
10 990 30 000 480 000 000 43 676
12 000 30 000 300 000 000 25 000
13 000 55 000 200 000 000 15 385
14 428 33 000 385 000 000 26 684
15 000 37 000 840 000 000 56 000
18 924 20 600 800 000 000 42 274
22 312 40 162 338 541 000 15 173
34 082 478 000 2 500 000 000 73 353
35 000 160 000 2 400 000 000 68 571
of improvements and land are fixed (of course, at the same time period, same real estate class, and the same region).
In this case, there are random variables V — the offer price per 1 sq. m of improvements, SB — the area of improvements, SP — the area of the land plot. They form a three-dimensional random vector (F, SB, SP ). Let W = = ln( V), Y = ln(SB ), Z = ln(SP ) (then v = e W, SB = eY, SP = eZ). For a three-dimensional normal random vector (ffi, Y, Z) the mean vector is equal to (iW, fip ..., /uZ). The covari-ance matrix looks like:
CV =
or:
f 2 >
&W PWY^W^Y PWZ°W°Z
pmawaY <Ty pyzaYaz
PzwawaZ PzYaYa2 °"2
CV ■
cov
(w.fj
cov
(W,Y)T
COV
where COV = \ ^ pYZ<Jfz
\PZY<JY(JZ ^Z y
; y = {y,z)
cov(w, Y) = (p^^cTy, PwzOwCTz) ;
<tw>°'y><tz — variances of random variables W, Y, Z;
Pwy=Pyw »Pwz=Pzw > Pyz=Pzy — corresponding correlation coefficients.
Conditional expectation of W, if Y, Z = z
E(W\Y = y,Z = z) = = /V + [cOV~1 xcov(x,f)T, (y-^z-to)}.
Conditional variance of W, if Y, Z = z : D{W\Y = y,Z = z) = = a^ -[COV'1 xcov[x,f^, cov [x, f)).
Let's set the values of the area of improvements SB = sb and the area of the land plot
SP = sp. In accordance with the above notation Y = ln(SB), Z = ln(SP), y = ln(sb), z = ln(sp). The most probable value of the offer price V for known values of the area of improvements and land area is calculated using the formula [31]:
Mode(V\SB= sb,SP = sp) = = exp(juw + (COV1 x cov(x, Y^,{y- nY,Z~ fiz)- (4)
-a2w +|COV1 xcov(tr,Y)T, cov(w,Y)J)) .
Conditional expectation:
E(V\SB -sb, SP =E) =
- T
nw + (COV'1 X cov [x, 7) ,(y-MY,z- Mz)) + (5) ~\cOV-1xcov{w,f'), cov{w,f) J).
Conditional median:
Median(V\SB = sb, SP= sp) = exp(nw + + (COV-1 xcov(W,ff, (y-^z-Mz)))- (6)
Before applying formulas (4)—(6) to the data in Table 1, let's check whether there are grounds to assume the lognormality of distributions of components of a random vector (V, SB, SP) (joint normality of logarithms of their components). The Kolmogorov—Smirnov parametric test was used to check marginal distributions. The following p-value figures are received:
V — price of 1 sq. m of improvements: with parameters meanlog = 10.3 and sdlog = 0.43 /7-value is equal 0.7016;
SB — improvements area: with parameters meanlog = 8.45 and sdlog = 1.02 /j-value is equal to 0.9761;
SP— plot of land area: with parameters mean-log = 9.3 and sdlog = 1.01 /j-value is equal to 0.8963.
Let's check the three studied random variables for the joint normality of logarithms. To do this, we use a well-known condition of joint
normality: in order for a multidimensional random vector to have a multidimensional normal distribution, it is necessary and sufficient that any linear combination of its components is distributed normally. The following procedure was implemented in the statistical package R environment:
♦ we take the logarithm from variables
♦ the resulting logarithmic values of the variables are centered and normalized, each with its own standard deviation;
♦ using the standard function R unif(3,0,1), three weight coefficients are generated, and the coefficients are normalized by their sum;
♦ a linear combination of centered and normalized logarithms is formed with random positive coefficients equal to one;
♦ the resulting linear combination is tested using the Kolmogorov-Smirnov normality test, and the test result is written as a /»-value in the array;
♦ the procedure with a random linear combination is repeated a specified number of times, each time the /»-value is written. Then the total /»-value array is compared to the critical level (0.05).
Figure 1 shows a histogram in which /»-values were obtained when the test was repeated 100 000 times.
The minimum of /»-value is equal to 0.2867691; it is more than 0.05. The mentioned procedure of 100 000 time test repeating of random linear combinations of components of random vector (W, Y, Z) seems like a reason for keeping the joint logarithmically component distribution hypothesis as the working one.
For the logarithms of the variables "Ratio of price to area of improvements," "Area of buildings," "Area of land" specified in Table 1, the following values of the mean vector and covari-ance matrix are obtained (Table 2).
The means are the following: nW = 10,2993; HY = 8,4469; nZ = 9,3506, = 0,2381,
Pwy °W °Y= PYW °V °Y = 0,0108;
Pwz °"w az= Pzw aw °"z = 0,1467
a\ = 1,0635, PYZ <rY a= PZY<rY az = 0,8978; o*z = 1,2140.
In the statistical package R, a program code was implemented that allows us to calculate the market value estimation based on the specified values of the parameters "Building (improvements) area" and "Land area" (formula (4)).
Density 3
2
1
0
0.2
0.4
"L KS test result
1ПТПТП11 distribution
0.6
0.4
1.0
Fig. 1. Results of testing random linear combinations of centered and normalized vector components (W, Y, Z) = (ln(^), ln(SS), ln(SP)) on joint normality by the Kolmogorov-Smirnov test
0
Table 2.
Means of the logarithms
Ratio of price to area of improvements (V) Area of buildings (SB) Area of land (SP)
Means of logarithm
10.2993 8.4469 9.3506
Covariance matrix
0.2381 0.0108 0.1467
0.0108 1.0635 0.8978
0.1467 0.8978 1.2140
Similar calculations can be performed for estimates based on median values or on mathematical expectations (formulas (5) and (6)). The results are shown in Table 3.
This table shows the following:
♦ the mode estimate is always lower than the median estimate, and the median estimate is always lower than the mathematical expectation estimate (author's opinion: the market value should be defined as a mode estimate, in accordance with the terms of Federal Law No 135, taking into account the asymmetric distribution of prices, areas and distances observed in the market);
♦ if the area of the land plot is constant, the market value (1 sq. m of improvements) decreases as the area of improvements increases;
♦ if the area of improvements is constant, the market value (1 sq. m of improvements) increases as the land area increases;
♦ the formula (4) can be used to calculate the market value of a property of the same class as the comparison items for any values of improvement areas and land (on the same date, for the same location). Since there is no general consensus in the evaluation community regarding
the numerical characteristics used for estimating market value (mode, median, mathematical expectation), formulas (5) and (6) can be applied, but, strictly speaking, these do not follow the definition of market value in accordance with Federal Law No 135.
2. The ratio of the area of improvements to land square if the price offers are fixed
In [30] the authors considered the question of pricing trends in the property market for industrial and storage purposes, and the dependence of the market value on the "density factor" (development coefficient) of the land, which is defined as the ratio of the area of improvements to the area of land. The model of joint logarithmically normal distribution of components of a random vector (V, SB, SP) considered in this article also allows us to look at the problem of forming price trends. The difference is that all the components of the random vector (V, SB, SP) are distributed on a positive half-axis; for each given value V = v, we can specify the most probable values of the components SB (improvement area) and SP (land area) corresponding to the offer price. In contrast to the previous case (estimation of market value based on the specified values SB and SP), the area of possible deviations from the most probable (median, average) values is not on the numeric axis, but on the plane and consists (as will be shown below) of nested sets obtained from the scattering ellipses of logarithmic values SB and SP in the inverse exponential transformation of the plane.
Let the offer price V = v be known. It is necessary to estimate the ratio of the area of improvements and land for a class of objects with such an initial offer price, i.e. to select objects with lower, middle and upper price trends [30]. Denote the former: V — bid price, SB — area of improvements, SP — area of land, W = ln( V), Y = ln(SB), Z = ln(SP) (then V = eW, SB = eY, SP = eZ).
Table 3.
Estimates of market value per 1 sq. m of improvements for various values of improvement areas, land plots
Moda Plot of land in sq. m.
estimation 2 000 7 000 12 000 17 000 22 000 27 000 32 000 37 000 42 000 47 000
400 26 247 38 298 45 058 50 049 54 096 57 543 60 568 63 279 65 745 68 014
E 2 400 16 938 24 714 29 076 32 297 34 909 37 133 39 085 40 835 42 426 43 890
4 400 14 605 21 310 25 072 27 849 30 101 32 019 33 702 35 211 36 583 37 845
c= 6 400 13 327 19 445 22 877 25 411 27 466 29 216 30 752 32 129 33 381 34 533
CD « 8 400 12 469 18 194 21 406 23 777 25 700 27 337 28 775 30 062 31 234 32 312
"c= o 10 400 11 835 17 269 20 317 22 567 24 392 25 947 27 311 28 533 29 645 30 668
E o > 12 400 11 337 16 542 19 462 21 618 23 366 24 855 26 161 27 332 28 397 29 377
Ci E 14 400 10 930 15 948 18 763 20 842 22 527 23 962 25 222 26 351 27 378 28 323
16 400 10 588 15 449 18 176 20 189 21 822 23 212 24 433 25 527 26 521 27 436
18 400 10 294 15 021 17 672 19 629 21 217 22 569 23 755 24 818 25 786 26 675
M fledian Plot of land in sq. m.
estimation 2 000 7 000 12 000 17 000 22 000 27 000 32 000 37 000 42 000 47 000
400 31 947 46 615 54 843 60 918 65 844 70 039 73 722 77 021 80 023 82 784
2 400 20 616 30 081 35 391 39 311 42 490 45 197 47 573 49 703 51 640 53 421
E CT CO 4 400 17 777 25 938 30 517 33 897 36 638 38 972 41 021 42 858 44 528 46 064
c= ra 6 400 16 221 23 668 27 846 30 930 33 431 35 561 37 431 39 106 40 630 42 032
CD ra 8 400 15 177 22 146 26 055 28 941 31 281 33 274 35 023 36 591 38 017 39 329
1= o E o > CO 10 400 14 405 21 019 24 729 27 468 29 690 31 581 33 242 34 730 36 083 37 328
12 400 13 799 20 134 23 688 26 312 28 440 30 252 31 843 33 268 34 564 35 757
E 14 400 13 304 19 412 22 838 25 368 27 419 29 166 30 700 32 074 33 324 34 473
16 400 12 887 18 804 22 123 24 574 26 561 28 253 29 739 31 070 32 281 33 395
18 400 12 530 18 283 21 510 23 892 25 824 27 470 28 914 30 208 31 385 32 468
Expectation Plot of land in sq. m.
estimation 2 000 7 000 12 000 17 000 22 000 27 000 32 000 37 000 42 000 47 000
400 35 246 51 428 60 506 67 208 72 643 77 271 81 334 84 974 88 285 91 332
E £ 2 400 22 744 33 187 39 045 43 370 46 877 49 864 52 485 54 835 56 971 58 937
4 400 19 612 28 616 33 668 37 397 40 421 42 996 45 257 47 283 49 125 50 820
c= ra 6 400 17 895 26 112 30 721 34 124 36 883 39 233 41 296 43 144 44 825 46 372
CD ra 8 400 16 744 24 432 28 745 31 929 34 511 36 710 38 640 40 369 41 942 43 389
"c= o 10 400 15 893 23 189 27 283 30 305 32 755 34 842 36 674 38 315 39 809 41 182
E o > o 12 400 15 224 22 213 26 134 29 029 31 377 33 376 35 130 36 703 38 133 39 449
ci E 14 400 14 677 21 416 25 196 27 987 30 250 32 178 33 869 35 385 36 764 38 033
16 400 14 218 20 746 24 408 27 111 29 304 31 171 32 810 34 278 35 614 36 843
18 400 13 824 20 170 23 731 26 359 28 491 30 306 31 899 33 327 34 626 35 821
As before, we consider a three-dimensional normal random vector (W, Y, Z) with a mean vector (iW nv nZ) and covariance matrix
CV =
'w
Pyw(tw<Je pzw pzy&y^z
PwYawaY PWZ^W^Z ay pYZaYa2
OÎ
or:
cv=
'w
cov
(■w.f)
COV
(w,ff
where COV =
COV
pYZoYaz
kPzygygz
Y = (Y,Z)
C0V{W, Yj = (p^rCT^Y, p^cr^).
&w,(jy,(j2z — variances of random variables W, Y, Z;
Pwy=Pyw > Pwz=Pzw > Pyz=Pzy — corresponding correlation coefficients.
Conditional expectation of vector Y = (Y,Z) if W = w:
_ , cov(w,f)T
E(Y\W = w) = M +-K-irJ-{w-ME) =
crw
/ \ MY
Pyw—{W~MW)
'w
Pzw—{w~Mw)
'w
(7)
fly+PyW—(w~HW)
w
Mz+Pzw—{w~Vw)
\ "w y
Conditional covariance matrix if W = w:
= COV
COV(Y\W = W) = cov{wf") xcov{w,f)
'w
\
P2Y<Jy<JZ
i 2 2 Pyw0Y
PYWPzw^Y^Z
Pyz<ty(jz
y
Pyw Pzw® y*7z
2 2 Pzwuz ,
(8)
(! - Pyw] °Y°Z{PYZ ~PYWPZW) °Y°Z{PYZ-PYWPZW) ^z^-Pzw) y
Let V = v. In accordance with the notation introduced above W= ln(V), w = ln(v). The most probable combination of SB and SP in the condition if V = v :
Mode(Y \V = v) =
■ exp
cov
M+-
(lW,Y)7
w
cov
{wff xcov(w,Y)
-COV +
Table 4 shows the results: most probable combination of SB and SP in a few cases of bid prices.
It should be noted that for each value of the offer price, the most probable pair of SB, SP values is the only one (the building density coefficient in this case corresponds to the most probable pair SB, SP). Trying to present as the most convenient another pair of components of SB and SP means choosing a point when there are many other equally probable points, with a density less than the maximum, and for which the building density coefficients will obviously be different. Figure 2 shows images of two-dimensional distributions of SB and SP for the offer price of 7.000 rubles, 28.000 rubles (from left to right), 100.000 rubles per 1 sq. m of improvements.
It is possible to see that any other points in plane SP, SB have any set of equal-probability points. The sets of such points shown on Figure 3 (for V = 28 000 rubles per 1 sq. m of improvements).
Table 4.
Most probable combination of SB and SP, density factor in a few cases of bid prices
Bid price on 1 sq.m. of improvements 7 000 12 000 21 000 28 000 40 000 60 000 80 000 100 000
Most probable pair:
Area of improvements 619 634 650 659 669 682 691 698
Plot of land in sq.m. 630 878 1 239 1 479 1 843 2 365 2 824 3 240
Dencity factor 0.98 0.72 0.52 0.45 0.36 0.29 0.24 0.22
Land area
Plot of land
Fig. 3. Equal-probability level lines for SPand SB
3. The density factor (ratio of the improvements area to land area)
Let's assume that the price of the offer (or transaction) is known. Our goal is to estimate what the coefficient of building density should be at a given price and the given area of the land plot. Let's use the formulas (7) and (8). For a fixed offer price ( V = v), formulas (7) and (8) give the calculated values of the conditional mathematical expectations of the improvement area logarithms (SB\V = v), the land area (SB\V = v), and the conditional covari-ance matrix. Additionally let's assume that the area of the land plot is also known. We introduce new notation for conditional logarithms of the improvement area (SB\V = v) and the
land area K = v):
MconäSB =MY+PYW—{W-MW)>
'w
McondSP =Mz+PzJV—{w-Mw)>
'W
°LdSB =<7r(1-Pr>v)> °l>ndSP Pzw\
p = uYaz{pY2 -PywPzw)
("cond" subscripts mean "conditional"). Consider a two-dimensional random vector (SB \ V = v, SP \V= v) with the specified parameters. For a given value of the land plot area and a given value of the price (in the example of the offer price) (SP = sp, V = v), the conditional mode of SB (improvement area) is equal to (by analogy with the proof given in [32]):
Mode{SP\ SP = sp, V = v) = = exV(Pcondsp {sb-pcondSB)) - (9)
alsb
-<7:
condSP
Conditional median of SP is equal to: Median(SP | SP = sp, V = v) =
= expCucondSP+px
condSP
(in {sb-n^ss))-
condSB
Conditional expectation of SP is equal to: E(SP | SP = sp, V = v) = exp(pcondSP +
+p x (In (sb - ncondSB )) + ± *londSP (1 -p1)).
acondSB 1
Let's assume the need to estimate the density factor in a group of items in the lower, middle, or upper price category. Such estimates can be constructed depending on the area of the land plot by modal, median or average values. However, the appearance of the surfaces shown in Figure 2 suggests that the most conservative estimates will be based on modal val-
ues. Estimates for the median or average values seems overestimated (for V = 28.000 rubles/ sq. m approximately 1.4 and 1.7 times, Figure 4). Let's assume that we are interested in the following question: if the offer price is 28.000 rubles per sq. m and if the area of the land plot is equal to 30,000 square meters, then what area of improvements (and, accordingly, what coefficient of building density) should be considered adequate for such a price and land area. Under the development coefficient (density factor), we will understand the ratio of the estimated value of the area of improvements to the area of the land plot, i.e.
Mode(SB\SP = sp) sp
(alternatively, Median (SB\SP = sp)
sp
E(SB\SP = sp)
or
sp
Figure 4 shows that estimates for modal, median, and average values can differ significantly. Applying formula (9) to the results of calculating conditional parameters at the price of 28.000 rubles/sq. m and the value of the land area equal to 30.000 sq. m gives the result of 7.165 sq. m of improvements, and then the building density coefficient (density factor) is 7 165 / 30 000 = 0.24. Thus, based on the example data (table 1), the other coefficient of the building area at the price of 28,000 rubles/sq. m, the land area of 30.000 sq. m may be understood as not appropriate to the price set. The same result could be obtained by applying a formula similar to formula (1). In this section, sequential accounting of conditions (first prices V = v, then land area SP = sp) is used to show that the coefficient of building density is not a constant within one price group or even for one specific price, and has a power-law dependence on the land area. Left part of Figure 4 shows lines of modal, median and average values of the area of improvements, depending on the area of the land plot for the case when
Land area
25000 -
15000 -
5000 _ 0 -
Expectation
Median
Mode
Dencity factor
1,0 "
0,8 -
0,6 -
0,4 -
0,2 -
0 -
— Building coefficient by Expectation
— Building coefficient by Median
— Building coefficient by Mode
2000
4000
6000
8000
2000
4000
6000
8000
Fig. 4. Values of the area of improvements and building coefficients, depending on the area of the land plot
the offer price is equal to 28.000 rubles/sq. m of the area of existing improvements. The right figure shows lines of building coefficients for corresponding estimates of the area of improvements. Figure 4shows that at a given price (price group), the coefficient of development with acceptable accuracy for evaluation purposes can be estimated as a constant only if the land area is large enough. For plots with a small area, the development coefficient cannot be estimated as a constant and must be studied individually taking into account the area of the plot.
4. A note regarding the form of the joint logarithmically
normal distribution of the vector (V, SB, SP)
Multidimensional distribution ofvector components (W, Y, Z) = (ln(V), ln(SB), ln(SP)) it is normal and has symmetry. The scattering clouds of empirical observations will take the form of three-dimensional ellipsoids. The density maximum point has coordinates equal to the mean values of the components W, Y, Z. The distribution of the components of the vector (V, SB, SP) is asymmetric, the density maximum point is not the center of symmetry and can be calculated (see Appendix) using the following formulas:
Vmax = exp(/v -pWYaW°Y ~PwZ°W°z\
SBmax = eXP(>UY ~ PYWUW aY ~ PYZ^ZX -pZYaz<TY-pzw<jw<7z).
Syntax
exP(ßz ~ az
Figure 5 shows the following: the scattering of source data and the scattering of logarithms of source data, the point of maximum density in space ( V, SB, SP) with coordinates Vmax = 20 004 rubles per 1 sq. m, SBmax = 649 rubles per 1 sq. m, SP = 1 202 rubles per 1 sq. m and the point
max r A r
of maximum density in logarithmic space ( W, Y, Z) = (ln(V), ln(SB), ln(SP)) with coordinates fiW = 10.30; fiY = 8.45; //Z = 9.35. Black marks the points of maximum density: on the left — in the space ( V, SB, SP), on the right — in the space ( W, Y, Z) = (ln( V), ln(SB), ln(SP)).
Figure 6 shows the result of 1000 generations three-dimensional random vectors with the same parameters.
It is obvious that (see Appendix) the maximum density point of a multidimensional vector (mode) whose logarithms are normally distributed together is unique. All other density values correspond to the sets described in the logarithmic dimension by hollow three-dimensional ellipsoids, and in the original coordinates, the sets corresponding to a single density value represent the result of distortion (stretch-
0
0
Land area
- - 1 I L - L- L L
i _
1
_
W* J —
a'cf.
Or,
'ce
log(Land area)
- .
U-l--■
■
»»
/
'Or>^ iP*
'ej w
Fig. 5. The scattering of original data (left), the scattering of the logarithms of the original data (right)
Land area
loaiLand area)
t»
c6>
J I
■L L
\<0
-pm ■ '-■ 1 .'.Il
1 I— 1 i _i- i i - o .
4*1 íSi ¡fl» T" i f 1 í !
fi ..i i
o ■> fe '■'■
£ &
-Ç5, %
\09(Wt0VI
emente
atea)
Fig. 6. Result of 1000 generations three-dimensional random vectors
ing) of the hollow ellipsoids during the inverse exponential transformation of space. Thus, it is the modal assessment of the market value that should lead to a correct result that does not create conflict situations. All other (nonmodal) market value estimates are potentially a source of constant disputes about the market value of the object of valuation.
Conclusion
Considering the prices of objects of comparison and the values of price-forming factors as multidimensional random variables opens
up new opportunities in the assessment of real estate. It often turns out that empirical observations of prices and their corresponding values of price-forming factors are well approximated by the logarithmically normal distribution law, including the multidimensional one, which allows us to derive calculation formulas for various estimation problems. The bulkiness of these formulas is compensated by the capabilities of modern applied statistical packages (in particular, R). In addition, the ability to reduce calculations to a well-studied multidimensional normal law by logarithm of components makes this choice of model distribution preferable.
Conditional price distributions with known values of price-forming factors make it possible to estimate the market value in full accordance with its definition fixed in Russian legislation and foreign standards, as the maximum point of the density of the conditional price distribution.
Conditional distributions of price-forming factors at a given offer price allow us to assess the adequacy of the offer price in terms of a set of price-forming factors.
It is hardly to be expected that practicing appraisers are prepared to apply the formulas given in this article in their daily practice of valuation and business analysis. This is not required. Once written and debugged, the script (in the statistical package R or in other specialized packages) will allow is to easily solve such problems practically in real time. It should be recognized that in the period of digital transformation of the economy and business analysis, it is time for the valuation business to move to advanced statistical packages and automatic data processing. ■
Appendix
Statement. The absolute maximum (mode) density of a random logarithmically normal vector x is reached at the point with coordinates exp(¿7 — H x 1), where Ji is the vector of mathematical expectations of the logarithms of the component, H is the covariance matrix of the logarithms of the component, and 1 is a vector consisting of units.
Proof. Consider the density of a multidimensional normal distribution of a centered random vector y:
1
/00=
{ln)i IJdetZ
exp
When replacing variables y = ln(x), the density of the lognormal distribution of the random vector x:
fix)----X---
J \ / n - TT"
(2^)2 -v/detr lL=1xi xexp^-^2'"1ln(jc), ln(jc))
where —*— — coordinate transformation IT i^i Jacobian, E — covariance matrix, ln(jc) — centered random vector. At the point of absolute maximum density of the joint logarithmically normal distribution, the derivative in any direction must be zero, which means that all partial derivatives are equal to zero.
1
1
xexp
#(*)_____
dx¡ {Inf Vdetr nil:*.
x. K i J
exp
2^)2 VdetT
x| -- ^m(x)
X—= 0
After removing the common multipliers from brackets, the condition remains:
-1 + (-2-1 = 0 or (-2T1 ¿(i)) -1,
where 1, —1 — vectors with dimension n, consisting from units/negative units.
Let multiply the last equality on the left by H:
¿^Infx) =-27xl, Ex in(jc) = -Sx 1.
Here E is a unit matrix (on the main diagonal — units, the other elements are zero), 1 — a vector consisting of units. I.e., the values of the vector ln(jc) in which all partial derivatives are zero, are equal to the line-by-line sums of the covariance matrix, taken with the reverse sign.
It remains to remember that y = ln(x) is a centered random vector. If the expectation
vector n contains non-zero values, then the final solution is:
ln(x) = ju-Zxl or x =exp(//-i7xl).
Taking into account negative definiteness of the quadratic form composed of second par-
tial derivatives in point x = exp(^/ — E x 1) (the author omits this bulky record since the result is obvious), the point x = exp(// — I x 1) is a point of maximum density of lognormal random vector x.
The statement is proven.
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About the author
Michael B. Laskin
Cand. Sci. (Phys.-Math.), Associate Professor;
Senior Researcher, St. Petersburg Institute for Informatics and Automation,
Russian Academy of Sciences (SPIIRAS),
39, 14 Line, St. Petersburg 199178, Russia
E-mail: laskinmb@yahoo.com
ORCID: 0000-0002-0143-4164
