OVERVIEW OF APPRAISAL THEORY AND TECHNIQUE

OVERVIEW OF APPRAISAL THEORY AND TECH= NIQUE

Jan Schreiber

President, MicroSolve Corp.

Ass= essing may seem to be a science, appropriate for study at a good university, but in its foundations it is more like a religion, in common with many other human= activities. It rests on three statements, none of which can be conclusively proved. I c= all them articles of faith:

I. &n= bsp; Taxation should be based, at least to some extent, on personal wealth

II. Personal wealth is related to market value of property owned

III Market value of all propertie= s can be determined

The last statement has been elaborated in practice, and = it is now generally accepted that there are three primary methods of determini= ng value:

a. Directly by statistical methods

= i. Clustering algorithms (“comparable sales”)

= ii. Multiple regression analysis

b. Indirectly by assigning component = values to arrive at a replacement cost

= (on the theory that the market trac= ks the replacement cost)

c. Indirectly by inferring market val= ue from property’s ability to produce income

= (applies to commercial properties o= nly)

The= se three options are known respectively as the market approach, the cost approach, a= nd the income approach. I’ll talk a bit about each of them.

The Market Approach

Assumes (1) characteristics of the properties in the community are known, (2) recent sale prices of at least a number of them are known. Two techniques are available: comparable sales and multiple regressi= on.

= 1.&n= bsp; Comparable sales

There must be at least ten sold properties in the databa= se, not counting any property chosen as the “subject.” The assessor decides (based on his knowledge of the market) which factors (property characteristics) are relevant in determining value. He assigns a weight and= an adjustment value to each. The weight indicates how important that factor is= in the selection of a comparable property; the adjustment in the monetary amou= nt that must be added to or subtracted from the sale price of a selected comparable property in order to make it resemble the subject as closely as possible.

As a simple example, consider only two factors, number of rooms and total area, distributed as follows among 11 parcels:

Property	Rooms (wt =3D 100, adj =3D 10000)	Area (wt =3D 1, adj =3D 1000)
S	6	wt X dif²	adj X dif	150	wt X dif²	adj X dif	Q
A	6	0	0	160	100	-10000	100*
B	5	100	10000	160	100	-10000	200*
C	5	100	10000	120	900	30000	1000*
D	4	400	20000	100	2500	50000	2900
E	7	100	-10000	150	0	0	100*
F	7	100	-10000	190	1600	-40000	1700
G	4	400	20000	110	1600	40000	2000
H	9	900	-30000	200	2500	-50000	3400
I	3	900	30000	70	6400	80000	7300
J	5	100	10000	140	100	-10000	200*

The first parcel is considered the subject parcel. The others are to be compared with it. The third column of this table shows the “room” weights multiplied by the squared difference between the number of rooms in a “candidate” parcel and those in the subject parcel (S). The sixth column shows the “area” weights multiplie= d by the squared difference in areas. The final column adds the values in the th= ird and sixth columns to get the total for each parcel, which we call the Q sco= re. Those with the lowest totals are considered most comparable to the subject; they have been marked with an asterisk in this example.

Now let us arrange those properties in order of their totals, starting with the lowest. We have 100, 100, 200, 200, and 1000. Let= us suppose those properties sold respectively for: 200,000, 220,000, 180,000,190,000, and 150,000. If we apply the total adjustments shown in the table, we get the following adjusted sale prices: 190,000, 210,000, 180,000, 190,000, and 190,000. Averaging those values, we get 192,000. It is reasona= ble to think that a number between 190,000 (the adjusted value of the most comp= arable property) and 192,000 (the average) might approximate the market value of t= he subject parcel.

200,000	220,000	180,000	190,000	150,000
-10,000	-10,000	0	0	40,000
190,000	210,000	180,000	190,000	190,000

Of course, in the real world more than two property characteristics would be used to determine comparability, and there would be more than ten sold properties to compare with – at least one would ho= pe.

The comparable sales approach is useful in estimating va= lues for a small number of properties, but it is cumbersome for large numbers (in the thousands), and it forces the assessor to determine weights and adjustm= ents without giving him specific guidance on establishing those numbers. A more sophisticated and efficient approach is multiple regression analysis.<= /o:p>

= 2.&n= bsp; Multiple regression analysis

This is a statistical technique for use in cases where o= ne variable in a population is thought to be influenced by several others. The single, influenced variable is called the dependent variable. In the case of properties, this variable would be selling price. The variables that influe= nce it are called the independent variables. In a property database they would include land size, building size, year built, quality, and a potentially la= rge number of other elements, such as porch area, garage area, air conditioning, and the like.

The variables are assumed to have a linear relationship = to each other. That is, the dependent variable Y can be expressed as a functio= n of several independent variables, each with its own coefficient, plus a consta= nt:

Y =3D c + b₁x₁ + b₂x2 + ... + b_nx_n

If there are enough examples of properties that sold for= a known amount and for which the characteristics are recorded, a good compute= rized regression program can solve the above equation to determine values for the= b and c variables that will give the closest approximation to the actual sale prices of all the properties under consideration. These coefficients can th= en be applied to other similar properties in the same region that did not sell, with some confidence that they will estimate a plausible market value.=

Multiple regression is a powerful tool, even though assessors who use it find it difficult to explain to taxpayers how the resu= lts are derived. But it is only usable when sales information is readily availa= ble to the assessor, and even then it will work best in urban areas where many sales take place in a brief period, such as a year. The reason is that it is fundamentally a sampling procedure, in which information derived from a sma= ll population (sold properties) is generalized to the population as a whole. Statistically, the validity of such a generalization is dubious when the sa= mple population is below approximately 100.

The Cost Approach

Fortunately, there is a widely used technique for derivi= ng property values from property characteristics that does not depend on the existence of sales data. It is called the cost approach, and what it estima= tes is the amount it would cost to replace a building of a certain size and typ= e. If a depreciation is applied to that estimate, to indicate the lessening of value with the passage of time, and if an appropriate value can be found for the land on which the building sits, then the total is taken to approximate= the market value of the property as a whole.

But the devil is in the details. In this case the last t= wo details – depreciation and land value – are full of devils. They are two reasons why we say that assessment is as much an art as a science.<= o:p>

I will talk about those difficult problems in a moment, = but first let me explain how the computer calculates the cost of a building. We must assume there is a database that contains values of many characteristics for many buildings. Here is a simple table illustrating how such a database might look.

Parcel	Living area	Quality	Exterior wall	Porch area	Bathrooms
101-2235	150	3	1	10	1.5
101-2236	190	3	1	12	2
101-2238	110	2.5	1	33	1
101-2245	125	4	2	14	1
101-2249	90	2	1	20	1
101-2344	203	4	3	22	2.5
101-2362	69	3.5	2	0	1
101-2371	142	3	1	18	1.5
101-2375	105	2	1	7	1

Every row is a separate property record. Every column represents a building feature. In a real database there might be over a hun= dred columns and thousands of rows.

Now let us consider a table of values against which the elements in this database might be compared. Here is a table in which the r= ows are living area and the columns are quality:

Quality >	1	2	3	4	5	6
L= iving Area	<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>	<= o:p>
50	485	535	585	665	735	875
60	480	530	580	660	730	870
70	475	525	575	655	725	865
80	470	520	570	650	720	860
90	465	515	565	645	715	855
100	460	510	560	640	710	850
110	455	505	555	635	705	845
120	450	500	550	630	700	840
130	445	495	545	625	695	835
140	440	490	540	620	690	830
150	435	485	535	615	685	825
160	430	480	530	610	680	820
170	425	475	525	605	675	815
180	420	470	520	600	670	810
190	415	465	515	595	665	805
200	410	460	510	590	660	800
210	405	455	505	585	655	795
220	400	450	500	580	650	790

After reading the values of the factors in the database,= the computer locates the row in the lookup table matching the area value, and t= he column matching the quality. If a living area falls between two rows, or a quality rating falls between two columns, the program will interpolate. The value in the cell where the row and column intersect is now multiplied by t= he area to give a base value for the dwelling.

Other property features are valued by means of similar lookups. Often quality is one element of a table. Each value is added to the total, until all property characteristics have been considered. The result = is called the replacement cost new – the amount that would be required to rebuild an identical house.

Now comes one of the devils: the depreciation. Houses depreciate at different rates in different areas, depending on weather conditions, building materials, maintenance, and workmanship. Further, not everyone who examines a house agrees about its condition. So each assessor = is likely to have his own ideas about how much depreciation a twenty-year-old house (for example) has undergone. Often this subjective impression is given the label “effective age.” Each assessor will probably want to = develop his own table of depreciation in relation to quality and effective age. And= no two tables are likely to be identical.

But once a table is developed that the assessor feels is reasonably reliable, it is applied by the computer to every property accord= ing to its effective age and perhaps other parameters such as type of construct= ion. The resulting depreciation is a monetary amount that is subtracted from the replacement cost new. The new total is called replacement cost new less dep= reciation.

We’re getting closer to the market value of a property, but we’re not there yet. We still haven’t taken land = into account. And the value of land is a major contributor to property value overall. In the city where I live, my house could probably be rebuilt for $250,000 to $300,000. And yet my property is valued at nearly three times t= hat amount. Why? Because it’s a neighborhood where lots are not available= for all the people seeking homes, and therefore prices are high. There are almo= st no lots without houses, but even if there were, they would cost almost as m= uch as a lot with a house.

At this point you as an assessor simply have to know something about the market. You have to know how much land is worth, either because vacant land has sold for a known amount or because a property with a building has sold and you know what the replacement cost of the building is= and therefore can infer the land value. You may have to make educated guesses on the basis of aggregate sales information. But somehow you must develop a wa= y of valuing land.

Usually this is done by neighborhood. A typical land tab= le might use a square-meter rate plus a constant for each neighborhood, like t= his.

Land size	Neighborhood 1	Neighborhood 2	Neighborhood 3
Rate	Constant	Rate	Constant	Rate	Constant
1000	9.50	20000	11.50	30000	7.50	35000
1500	9.00	20000	11.30	35000	7.20	35000
2000	8.50	24000	11.00	35000	7.00	40000
2500	9.00	24000	10.80	40000	6.80	40000
3000	7.50	28000	10.50	40000	6.60	45000
3500	7.00	28000	10.00	44000	6.40	45000
4000	6.50	30000	9.80	45000	6.10	52000

Here, if we know the neighborhood and the land size, we = can determine the land value by using a rate per square meter plus a constant. = In some cases the type of land is important as well: Is the land buildable? Is= it shorefront, which is usually more valuable? Is it commercially zoned? But y= ou can see the general idea.

Now we have the components of a complete property cost. = We start with the building replacement cost, then we subtract the depreciation. Finally we add the land value. The number we end up with should approximate= the amount for which the property would sell on the open market.

The Income Approach

One reason the cost approach is popular is that it can be used everywhere, for every type of property. But there is one class of prop= erty for which the cost approach is an inferior means of determining value. That= is the type of commercial property whose value derives mainly from its income-producing potential. A buyer of such a property doesn’t really care what it cost to build the facilities, and he may not plan to sell the property in the foreseeable future. Its value to him depends on how much in= come he can derive from it.

So the assessor takes the same approach. He asks,

= 1.&n= bsp; How much income will this property produce in a y= ear?

= 2.&n= bsp; What expenses much be charged against that income= ?

= 3.&n= bsp; What is the net income after expenses?=

And then he says to himself, “The owner is willing= to recover the cost of this building out of net income over a certain number of years before he sees true profits.” Often that period is about ten ye= ars. That means that in any year, about one-tenth of the building’s total value is paid by income. That fraction is called the “capitalization rate,” or “cap rate” for short. So we can divide the annu= al net income by the cap rate and get an amount that represents the total valu= e of the building to that owner.

Once again, the assessor has some latitude. He may feel = the building actually has a higher value to the investor, so he can reduce the = cap rate somewhat, on the theory that the investor would be willing to recover = his full value over a longer time period. Or he may recognize that the building= is not in good condition, and the owner needs to recover full value in a short= er time. If the income cannot be changed much, that means the total value of t= he building is less as far as the owner is concerned, and probably from the market’s point of view as well.

Summing Up

We’ve looked at the cost, market, and income approaches. Each of these techniques has an environment in which it is espe= cially useful. And all are aimed at the single goal of achieving equity – th= at is, a fair distribution of the tax burden according to property value. So measuring equity amounts to measuring the extent to which the assessment ma= tches the market value of the property. Once again, the standard has to be the ac= tual sale price. The assessor will check the equity of his results by performing= a sales ratio study, in which he isolates the properties that have recently s= old, and divides the assessed value of each property by its sale price.

= &nb= sp; = &nb= sp; Assessed value

&= nbsp; Sales ratio =3D

= &nb= sp; = &nb= sp; Selling price

He wishes to obtain a number as close to 1 as possible. = If he were to plot assessed value against sale price for all the properties in= his database, he would hope to get a scatter graph with dots forming a diagonal path from lower left to upper right. In that case, the value of the assessm= ent on the vertical scale would be very close to the value of the sale on the horizontal scale.

Clearly it is desirable that the points on this graph deviate as little as possible from a straight line, known as the linear reg= ression line, or the line of best fit. This is a&n= bsp; line drawn so as to minimize the average distance from it to each po= int on the graph. Mathematically, the measure of the degree of deviation is the coefficient of dispersion about the median. Here is the formula:=

= &nb= sp; Σ | R – M |

C.O.D. =3D &n= bsp; = X 100

= &nb= sp; N X M

… where R is the ratio being examined, M is the me= dian value of that ratio, and N is the number of cases.

When you have this tool and you have some reliable data = on property sales, you can progressively refine your value estimates. You can search for outliers on your scatter diagram – those properties that m= ay be increasing your C.O.D. by not producing a sales ratio close to the median value. If you focus on them, you may discover the source of the problem. Perhaps it was not a recent sale, or not an arm’s-length sale. (Maybe= it was a property sold by a father to a son for a deliberately low value.) Per= haps an addition was put on the house after it sold. Perhaps it’s in a different, more expensive neighborhood. Perhaps the assessor made a mistake. Eventually, a good assessor should be able to produce a C.O.D. below 15 percent. A very good one may come in under 10 percent.

Not all these approaches will work in every environment.= But they indicate what is possible, and what the really essential advantages of good sales data are. The mass appraisal program from MicroSolve allows you = to do all the things we have been describing, and quite a lot more besides. I = urge you to explore it.