Business, Sauder School of
DSpace
UBCV
Ghert, Bernard Irvin
Master of Science in Business - MScB
University of British Columbia
The tax on real property continues to be a major factor in the fiscal structure of municipal governments. Therefore, justice dictates that the impact of this tax be distributed equitably, however "equitable" may be defined by the legislative authority. This is an administrative problem. In many jurisdictions the administration is directed to distribute the tax impact on an ad valorem basis of current market value. Statistical measures have been devised in order to measure the uniformity of the assessment of current market values. The random variable is defined as the ratio of assessed value to actual sale price, and the sample is the result of property sales which occur under conditions which are circumscribed by the definition of market value. The estimated parameters are then used to make direct statistical inferences regarding the level and uniformity of assessment. These parameters are then compared to pre-selected standard parameters in order to judge the relative uniformity of the assessment roll. First, market generated sales do not produce a random sample of the assessment roll. Therefore, direct statistical inferences from the sample do not apply to the assessment roll. Second, it is demonstrated that the standard parameters are implicitly based on an assumption that the universe of assessment-sale price ratios is normally distributed. The results of empirical investigation upon the Vancouver assessment roll indicate that the normality assumption cannot be justified. Therefore, the pre-selected standards are useless as measures by which to judge the relative uniformity of an assessment roll. Two alternative measures of assessment quality are proposed. First, a binomial model based on correct and incorrect valuation is examined. Though this has some practicable possibilities, it cannot, without some important value judgements, be used to measure degrees of quality or uniformity. A second alternative is proposed in order to overcome the above limitation. The alternative suggests that a good assessment roll have assessment-market value ratios which are normally distributed, and that the actual distribution of assessment-sale price ratios be tested for "goodness-of-fit" to the ideal distribution. The test parameters can be transformed to probability levels which would measure the quality or uniformity of the assessment roll. This measure may indicate that a highly uniform assessment roll on which most of the properties are under-assessed is of lower quality than an assessment roll on which more properties are correctly assessed, but less uniformly assessed. Though it is unlikely that uniformity of assessment can be achieved without correct valuation, it cannot be held that an assessment roll on which more properties are correctly assessed is of higher quality of assessments than a roll on which most properties are under-assessed, if the uniformity of assessment on the former roll is not at least as high as the uniformity of assessment on the latter roll. Since the measures of assessment quality and uniformity examined are not satisfactory, it is concluded that statistical analysis in assessment administration may be more usefully applied to the analysis of the causes of assessment errors than to the measurement of the number and size of the errors.
MEASURES OF THE QUALITY OF REAL PROPERTY ASSESSMENTS: AN EXAMINATION OF THEIR VALIDITY by Bernard I r v i n Ghert B . S c , M c G i l l U n i v e r s i t y , i960 A THESIS IN COMMERCE SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION Me accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1965 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . , I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f C^^^^^^t- \f— (j^^^^a^>< & T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a ABSTRACT The tax on r e a l property continues to be a major f a c t o r i n the f i s c a l s t r u c t u r e of municipal governments. Therefore, j u s t i c e d i c t a t e s that the impact of t h i s tax be d i s t r i b u t e d e q u i t a b l y , however " e q u i t a b l e " may be defined by the l e g i s l a t i v e a u t h o r i t y . This i s an a d m i n i s t r a t i v e problem. In many j u r i s d i c t i o n s the a d m i n i s t r a t i o n i s d i r e c t e d to d i s t r i b u t e the tax impact on an ad valorem b a s i s of current market value. S t a t i s t i c a l measures have been devised i n order to measure the u n i f o r m i t y of the assessment of current market values. The random v a r i a b l e i s defined as the r a t i o of assessed value to a c t u a l sale p r i c e , and the sample i s the r e s u l t of property s a l e s which occur under c o n d i t i o n s which are circumscribed by the d e f i n i t i o n of market value. The estimated parameters are then used to make d i r e c t s t a t i s t i c a l i n f e r e n c e s regarding the l e v e l and u n i f o r m i t y of assessment. These parameters are then compared to p r e - s e l e c t e d standard parameters i n order to judge the r e l a t i v e u n i f o r m i t y of the assessment r o l l . i i i F i r s t , market generated sales do not produce a random sample of the assessment r o l l . Therefore, direct s t a t i s t i c a l inferences from the sample do not apply to the assessment r o l l . Second, i t i s demonstrated that the standard parameters are implicitly based on an assumption that the universe of assessment-sale price ratios i s normally distributed. The results of empirical investi-gation upon the Vancouver assessment r o l l indicate that the normality assumption cannot be justified. Therefore, the pre-selected standards are useless as measures by which to judge the relative uniformity of an assessment r o l l . Two alternative measures of assessment quality are proposed. F i r s t , a binomial model based on correct and incorrect valuation i s examined. Though this has some practicable p o s s i b i l i t i e s , i t cannot, without some important value judgements, be used to measure degrees of quality or uniformity. A second alternative Is proposed i n order to overcome the above limitation. The alternative suggests that a good assessment r o l l have assessment-market value ratios which are normally distributed, and that the actual distribution of assessment-sale price ratios i v be te s t e d f o r "g o o d n e s s - o f - f i t " to the i d e a l d i s t r i b u t i o n . The t e s t parameters can be transformed to p r o b a b i l i t y l e v e l s which would measure the q u a l i t y or u n i f o r m i t y of the assessment r o l l . This measure may i n d i c a t e that a h i g h l y uniform assessment r o l l on which most of the p r o p e r t i e s are under-assessed i s of lower q u a l i t y than an assessment r o l l on which more p r o p e r t i e s are c o r r e c t l y assessed, but l e s s uniformly assessed. Though i t i s u n l i k e l y that u n i f o r m i t y of assessment can be achieved without c o r r e c t v a l u a t i o n , i t cannot be h e l d that an assessment r o l l on which more p r o p e r t i e s are c o r r e c t l y assessed i s of higher q u a l i t y of assessments than a r o l l on which most p r o p e r t i e s are under-assessed, i f the u n i f o r m i t y of assessment on the former r o l l i s not at l e a s t as high as the u n i f o r m i t y of assessment on the l a t t e r r o l l . Since the measures of assessment q u a l i t y and u n i f o r m i t y examined are not s a t i s f a c t o r y , i t i s concluded that s t a t i s t i c a l a n a l y s i s i n assessment a d m i n i s t r a t i o n may be more u s e f u l l y a p p l i e d to the a n a l y s i s of the causes of assessment e r r o r s than to the measurement of the number and s i z e of the e r r o r s . TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION 1 A. Purpose of the Study 1 B. F i s c a l Importance of the Re a l Property Tax 3 C. Method 6 I I . PRELIMINARY CONCEPTS 9 A. B a s i s of Assessment 9 B. Market P r i c e as Evidence of Market Value 15 C. V a l u e r 1 s Margin of E r r o r 21 D. The Tax Base 2k E. Sales P r i c e s and Re a l E s t a t e Finance . 27 F. Sales P r i c e s and Assessment U n i f o r m i t y . 33 G. U n i f o r m i t y of Assessment 39 H. Market A c t i v i t y and Random Sampling . . kk I . U n i f o r m i t y of Assessment and E q u a l i z a -t i o n k7 J . Summary 52 I I I . UNIFORMITY OF ASSESSMENT IN BRITISH COLUMBIA. 55 A. Assessment E q u a l i z a t i o n Act 56 B. C o l l e c t i n g Sales Data f o r E q u a l i z a t i o n . 58 v i CHAPTER PAGE C. Measuring Assessment U n i f o r m i t y 60 D. The Normality Assumption 66 E. Testing the Normality Assumption . . . . 76 F. The Universe of Assessment-Market P r i c e R a t i o s 85 G. S t a t i s t i c a l Tabulations 9k IV. ALTERNATIVE MEASURES OF ASSESSMENT UNIFORMITY AND QUALITY 105 A. An A l t e r n a t i v e Measure 106 B. Standards of Assessment U n i f o r m i t y Using the A l t e r n a t i v e Measure . 107 C. Bi n o m i a l Sampling 11*+ D. Q u a l i t y of the P o p u l a t i o n A c t u a l l y Sampled 119 E. A d d i t i o n a l A l t e r n a t i v e Measures of U n i f o r m i t y and Q u a l i t y 120 V. CONCLUSIONS 129 BIBLIOGRAPHY 135 APPENDIX A 1^ 3 APPENDIX B 166 APPENDIX C I69 LIST OF TABLES TABLE PAGE I. S e l e c t e d F i s c a l S t a t i s t i c s . . . . . . . . . k I I . Assessment Market-Price R a t i o s Frequency-Tabulations and E s t i m a t i o n of Parameters . 95 I I I . H y p o t h e t i c a l Populations of Assessment-Market Value R a t i o s A, B, C, D 108 IV. H y p o t h e t i c a l Populations of Assessment-Market Value R a t i o s E , F , G 110 V. H y p o t h e t i c a l P o p u l a t i o n s of Assessment-Market Value R a t i o s F ^ E 2 , Fg, E^, VI. H y p o t h e t i c a l Populations of Assessment-Market Value R a t i o s G, H 112 V I I . F i t of Observed Frequencies to Standard of Correc t V a l u a t i o n 123 V I I I . F i t of Observed Frequencies to Standard of Uniform Assessment . . . . 126 IX. A c t u a l and Expected Frequencies 131 LIST OF FIGURES FIGURE PAGE 1. Two Skewed D i s t r i b u t i o n s 6k 2. Two Symmetrical D i s t r i b u t i o n s 65 3. "Budget E f f e c t " on D i s t r i b u t i o n of Assess-ment-Sale P r i c e R a t i o s 76 k. F i t of Assessment-Market P r i c e R a t i o s to Assumed Normal D i s t r i b u t i o n 104 5. F i t of Observed Frequencies to Standard of Co r r e c t V a l u a t i o n 124 6. F i t of Observed Frequencies to Standard of Uniform Assessment . . . . . 128 ACKNOWLEDGMENT In the pr e p a r a t i o n of t h i s t h e s i s I have become indebted to many people. I thank them a l l . Very much appreciated i s the access to the M u l t i p l e L i s t i n g records of the Vancouver R e a l E s t a t e Board provided by Mr. A. Creer, and the cooperation of the s t a f f of the U n i v e r s i t y of B r i t i s h Columbia Computing Centre. I am e s p e c i a l l y o b l i g e d to P r o f e s s o r P. H. White without whose advice and encouragement t h i s d i s s e r t a t i o n could not have been s u c c e s s f u l l y completed. The w r i t e r reserves to h i m s e l f the r e s p o n s i b i l i t y f o r a l l the remaining e r r o r s of commission and omission. CHAPTER I INTRODUCTION A. Purpose of the Study The r e a l property tax i s but one tax i n our complex f i s c a l system. In Canada, i t i s one of the taxes on wealth. Therefore, arguments f o r or against the r e a l property tax should not be based on those c r i t e r i a which are used i n the evaluation of the entire tax s t r u c t u r e . 1 2 Many writers have summarized those c r i t e r i a , and there i s l i t t l e point to pursuing them here. On the grounds of j u s t i c e , the rule for a s p e c i f i c tax i s equal treatment of equals, and the d e f i n i t i o n of "equals," or " l i k e circumstances," i s a value judgment to be made by the appropriate l e g i s l a t i v e authority. C r i t i c i s m , favourable or unfavourable, of the r e a l estate tax w i l l f a l l , therefore, on the administration of the tax, and w i l l be actuated by i t s success, or lack of U. K. Hicks, Public Finance (2nd ed. rev.; London: James Nisbet and Company Ltd., 1955)> PP» 115-119; Adam Smith, Wealth of_ Nations. Book V, Chapter I I , Part I I , para. 3. p For example, John F. Due, Government Finance: An Economic Analysis (3rd ed. rev.; Homewood, I l l i n o i s : Richard D. Irwin, Inc., 1963), pp. 102-121. 0 2 success, i n achieving "equal treatment of equals."- 5 A f t e r the r e a l property tax has been adopted as a source of p u b l i c revenue, and given the r e l e v a n t i n t e r p r e t a t i o n of " l i k e circumstances," a d i s c u s s i o n of the tax centres about the s e l e c t i o n of a tax base, the d e f i n i t i o n of the b a s i s of assessment, and the a d m i n i s t r a t i o n of the tax which together w i l l assure "equal treatment of equals." More w i l l be s a i d about the tax base and the basi s of assessment i n Chapter I I . The remarks there w i l l be d i r e c t e d toward the i m p l i c a t i o n of t h e i r s e l e c t i o n and d e f i n i t i o n upon the a d m i n i s t r a t i o n of the tax. I t i s with the a d m i n i s t r a t i o n of the tax that t h i s paper i s concerned. Given a tax base and a b a s i s of assessment which are c o n s i s t e n t with " e q u a l i t y " i n the d i s t r i b u t i o n of the impact of the tax, and which do not block the a d m i n i s t r a t i o n from e f f e c t i n g "equal treatment of equals," then the e q u i t a b l e d i s t r i b u t i o n of the tax impact w i l l r e s t upon the i d e n t i f i c a t i o n of assessable property and the u n i f o r m i t y of i t s assessment. °The a d m i n i s t r a t i o n of a tax i n v o l v e s the i d e n t i f i -c a t i o n of the l i a b i l i t y to tax, measurement of the amount of l i a b i l i t y , and the c o l l e c t i o n of the tax. k H i c k s , op., c i t . . p. 138. H i c k s has used "formal i n c i d e n c e " to d i f f e r e n t i a t e "impact" from " e f f e c t i v e i n c i d e n c e " u s u a l l y c a l l e d " i n c i d e n c e . " , y " U n i f o r m i t y of assessment" r e q u i r e s that every 3 S p e c i f i c a l l y , the purpose here i s to consider absolute u n i f o r m i t y of assessment, a p r a c t i c a b l e standard of assessment u n i f o r m i t y , and to r e l a t e s t a t i s t i c a l l y , or otherwise, a c t u a l u n i f o r m i t y of assessment to the p r a c t i c a b l e standard; to examine the assumptions upon which are based current standards of assessment u n i f o r m i t y ; to suggest an a l t e r n a t i v e s t a t i s t i c a l approach which r e q u i r e s fewer assumptions about the underlying data. B. F j s c a l Importance of the Re a l Property Tax There are at l e a s t two reasons why u n i f o r m i t y of assessment should be of concern. The f i r s t , and i n the view taken here most important, i s j u s t i c e . Even i f the revenue produced and the impact generated by a tax i s almost i n s i g n i f i c a n t the j u s t a d m i n i s t r a t i o n of the tax i s a n e c e s s i t y . I f the y i e l d of the r e a l property tax i s too small to compensate f o r the expenses i n v o l v e d i n the e q u i t a b l e d i s t r i b u t i o n of the impact, then that source of p u b l i c revenue should be replaced by another source.^ assessable property be in c l u d e d on the l i s t at i t s "basis of assessment." Throughout t h i s paper l i a b i l i t y to assessment i s kept d i s t i n c t from l i a b i l i t y to tax. ^Report of a Study Group of the Royal I n s t i t u t e of P u b l i c A d m i n i s t r a t i o n , New Sources of L o c a l Revenue (London: George A l l e n and Unwin L t d . , 1956), pp. 78-9 1*! "Report of the Committee on State E q u a l i z a t i o n of L o c a l Property Tax Assessments," N a t i o n a l Tax A s s o c i a t i o n , Proceedings of the 5 l s t Annual Conference, p. 3M+. k The second reason i s that the revenue produced and the impact generated by the real estate tax i s not insigni-ficant. From the information included in Table I i t i s self-evident that the tax on real property i s , according to several measures, an important part of our f i s c a l system. TABLE I SELECTED FISCAL STATISTICS3 (Millions of Current Dollars) Fi s c a l year ended nearest to December 31* 195*+ 196l Gross National Product 2^ ,0^ 1 36,8M+ A l l tax revenue 5,397 8,777 A l l property tax revenue^ 659 1,4-00 Municipal general revenue 0 885 1,718 Municipal tax revenued 767 1,5^ 0 Municipal property tax revenue** 651 1,391 Gross National Product: National Accounts Income and Expenditure (D.B.S., I95U. and 1961), Table 1; Taxation and Revenue Sta t i s t i c s : A Consolidation of Public Finance Statistics of Municipal. Provincial Governments and the Government p£ Canada (D.B.S., 195*+ and 1961), Table 1. Annual levies except business taxes. Included are taxes on personal property, but i t i s an accepted f i s c a l fact that personal property tax revenue i n Canada i s an in-significant portion of the total figure (J. Stefan Dupre, "The Property Tax in Canada," National Tax Association, Proceedings of the 5lst Annual Conference [1958], p. 77 and Table 3). cDoes not include transfer payments from other governments. ^Includes a l l municipal purpose property tax revenue and a l l school purpose property taxes levied by the municipality or on behalf of the municipality by a school board. 5 Tax revenues alone give the three l e v e l s of govern-ment i n Canada a l a r g e measure of power to i n f l u e n c e the a l l o c a t i o n of productive f a c t o r s . The degree of i n f l u e n c e , as measured by the r a t i o of a l l tax revenue to gross n a t i o n a l product, increased only s l i g h t l y from 195*+ to 1961, w h i l e the share of c o n t r o l a r i s i n g out of property tax revenue increased from 12 per cent i n 195*+ to 16 per cent i n 1961. V i r t u a l l y a l l of the property tax revenue i s c o l l e c t e d at the l o c a l l e v e l of government. At the l o c a l l e v e l the property tax accounts f o r over 80 per cent of muni c i p a l general revenue, and, of course, even a greater percentage of muni c i p a l tax revenue. In 1961 the property tax revenue of the m u n i c i p a l i -t i e s was 3*8 per cent of gross n a t i o n a l product; therefore the c o l l e c t i o n of the property tax and the spending of the revenue produced can have an i n f l u e n c e on the p a t t e r n of expenditures f o r goods and s e r v i c e s . But of more d i r e c t importance i s the f o l l o w i n g f a c t . M u n i c i p a l governments are almost e n t i r e l y dependent upon property tax revenue as the source of t h e i r share of funds f o r p u b l i c expenditure. This f a c t , coupled with the r e a l i z a t i o n that the munic i p a l property tax revenue i s not, over the l a s t few yea r s , a decreasing p o r t i o n of a l l tax revenue, i s the most important f i s c a l aspect of the r e a l property tax. Here \ 6 there i s a pragmatic reason f o r ensuring an equal d i s t r i b u -t i o n of the impact of the r e a l property tax w i t h i n the 7 m u n i c i p a l i t y . ' C. Method This s e c t i o n o u t l i n e s the s t r u c t u r e of the remainder of t h i s paper. Before proceeding with the o b j e c t i v e s of the study some concepts which w i l l provide the foundation f o r the s t a t i s t i c a l d i s c u s s i o n s to f o l l o w w i l l be explained and defined . The s t a t i s t i c a l measure of u n i f o r m i t y c u r r e n t l y used i n B r i t i s h Columbia w i l l be analyzed. The emphasis w i l l be d i r e c t e d towards the b a s i c assumptions and not towards the procedure. The measure of u n i f o r m i t y adopted by the Province of B r i t i s h Columbia i s used because the Assess-o ment E q u a l i z a t i o n Act and the a d m i n i s t r a t i o n of the Assessment Commissioner under t h i s Act are r e p r e s e n t a t i v e of 'C. Ward Macy, "The Property Tax i n the F i s c a l System," N a t i o n a l Tax A s s o c i a t i o n . Proceedings of the 51st Annual Conference. 1958, pp. 57-7°. Macy has concluded that the "property tax as a source of revenue f o r the support of l o c a l government must continue to play a s i g n i f i c a n t p a r t i n the f i s c a l scene" despite the f a c t that " i t has been maligned on both t h e o r e t i c a l and pragmatic grounds." He a l s o s t a t e s that the f u t u r e success of the property tax r e s t s upon the improvement of i t s a d m i n i s t r a t i o n . B r i t i s h Columbia, Revised S t a t u t e s ( i 9 6 0 ) , c. 20. 7 some of the current and past p r o v i n c i a l and st a t e p o l i c i e s 9 of l o c a l assessment s u p e r v i s i o n . For the purpose of an e m p i r i c a l comparison of the c u r r e n t l y used measure of u n i f o r m i t y with measures to be suggested l a t e r i n the paper, a study of the current measure w i l l be conducted using the C i t y of Vancouver assessment r o l l as an example. A l t e r n a t i v e measures of assessment u n i f o r m i t y w i l l be suggested, and from an a b s t r a c t p o i n t of view t h e i r advantages over and disadvantages to the current measure w i l l be examined. The suggested a l t e r n a t i v e measures w i l l be studied e m p i r i c a l l y using the C i t y of Vancouver assessment r o l l as an example. 7Two provinces with s i m i l a r p o l i c i e s are Ontario, The Assessment A c t T and A l b e r t a , A l b e r t a M u n i c i p a l i t i e s Assessment and E q u a l i z a t i o n Act. Ronald B. Welch, "State Tax Department R e s p o n s i b i l i t y f o r L o c a l Assessments: A d m i n i s t r a t i v e Review," N a t i o n a l Tax A s s o c i a t i o n , Pro-ceedings of the ^hth N a t i o n a l Conference« 19*tl. Welch i n d i c a t e s that before 19H1 the f o l l o w i n g s t a t e s attempted e q u a l i z a t i o n among i n d i v i d u a l s ( u n i f o r m i t y of assessment) as w e l l as between d i s t r i c t s . The s t a t e s are: Kansas, A r i z o n a , Kentucky, Montana, South C a r o l i n a , South Dakota, Washington, Alabama, Arkansas, Idaho, Maine, Michigan, Minnesota, Nebraska, New Hampshire, Oregon, Utah, and West V i r g i n i a . See a l s o : N a t i o n a l A s s o c i a t i o n of Tax Administra-t o r s , Guide f o r Assessment Sales R a t i o S t u d i e s , A Report of the Committee on Sales R a t i o Data (Chicago: Fed e r a t i o n of Tax A d m i n i s t r a t o r s , June 195*+). 8 The r e s u l t s of the e m p i r i c a l s t u d i e s w i l l be compared i n l i g h t of whether or not the assumptions underlying the current measure are j u s t i f i e d . The i m p l i c a t i o n s of the r e s u l t s of t h i s study on the development of measures and standards of assessment u n i f o r m i t y w i l l be discussed. CHAPTER I I PRELIMINARY CONCEPTS One cannot over-emphasize the importance of some ba s i c concepts to the problem of assessment u n i f o r m i t y . The explanations and d e f i n i t i o n s which f o l l o w may i n places seem t r i t e , t a u t o l o g i c a l , or t r i v i a l , but a c l e a r understanding of the a p p l i c a t i o n of s t a t i s t i c s to problems of u n i f o r m i t y i n the f i e l d of r e a l property assessment r e q u i r e s more than a mastery of s t a t i s t i c a l technique. The questions that should be kept i n mind are the f o l l o w i n g : what i s being measured; why i s i t being measured; and how i s i t being measured. A. B a s i s of Assessment t The b a s i s of assessment i s the measure used i n determining the l i a b i l i t y to the property t a x . 1 Together wi t h the p h y s i c a l d e s c r i p t i o n of the tax base, the exemp-t i o n s , and the r e l i e f s , the b a s i s of assessment determines the t h e o r e t i c a l d i s t r i b u t i o n of the impact of the tax. At some times and i n some places the b a s i s of assessment has Report of The Royal Commission on Finance and M u n i c i p a l Taxation i n New Brunswick ( F r e d e r i c t o n : 1963), p. 206. 10 p been a p h y s i c a l c h a r a c t e r i s t i c of the property. Cannan r e p o r t s that i n 1250 C.E. each man of Romney Marsh had to maintain that p o r t i o n of the seawalls which equaled the pro p o r t i o n that h i s land area measured of the land protected by the seawalls of Romney Marsh. 3 In 1287 C.E. the p r i n c i p l e was extended to Sussex, but the r e s p o n s i b i l i t y was apportioned by land value r a t h e r than by la n d area. Later i n Romney Marsh apportionment by land value replaced 1+ apportionment by land area. But i t would appear that by 14-30 C.E. the apportionment of the l e v y according to value was g e n e r a l l y accepted. However, as r e c e n t l y as 1964 F. H. F l n n i s has suggested the use of square f o o t area of land and b u i l d i n g s as the ba s i s of assessment. He i s l e d to t h i s unfortunate Edwin Cannan, The H i s t o r y of L o c a l Rates i n England (2nd ed., London: P.A. King and Son, 1927), pp. 10-11. 3 I b i d . , p. 11. ^ I b i d . , pp. 21-22. ^ I b i d . , p. 15. F. H. F i n n i s , "Measuring the L o c a l Tax Base," Canadian Tax J o u r n a l . V o l . X I I , No. 1 (January-February, 1964), p. 48. suggestion because he observes that a tax l i a b i l i t y based on the value of r e a l property does not r e f l e c t a b i l i t y to 7 pay or b e n e f i t s r e c e i v e d . ' Regardless of the accuracy of h i s o b servation, the co n c l u s i o n i s wrong. The c r i t e r i a he has used apply to the e v a l u a t i o n of the e n t i r e tax s t r u c t u r e , and not to one p a r t i c u l a r tax. Other than the above suggestion the author knows of no serious recent suggestions or arguments which support a p h y s i c a l l y o r i e n t e d b a s i s of o assessment. I t i s g e n e r a l l y agreed that value i s the best b a s i s of assessment. "The b a s i s of assessment a f f e c t s the d i s t r i b u t i o n of the burden (impact) of the tax r a t e because there are va r i o u s kinds of value which may be a t t r i b u t e d to r e a l p r o p e r t y . " 7 At t h i s p o i n t the tax experts part company. There are two basic types of v a l u e , and there are many d i f f e r e n t concepts of value. To f u r t h e r complicate the is s u e there i s no general agreement on the d e f i n i t i o n s of J M d . , pp. ^7-^8. F i n n i s t h i n k s p h y s i c a l area i s no worse i n respect of a b i l i t y to pay and b e n e f i t r e c e i v e d , and much e a s i e r to measure than value. g The use of cost manualsby assessors seems to imply a p h y s i c a l l y o r i e n t e d b a s i s of assessment, but the cost manuals are supposedly based on an equation between cost and value. Q 'Report of the Royal Commission on Finance and M u n i c i p a l Taxation i n New Brunswick, OP. c i t . , p. 207. 12 the value c o n c e p t s . 1 ^ The two types of value are annual value and c a p i t a l v a l u e , the c a p i t a l value being a lump sum present e q u i v a l e n t of the f u t u r e annual r e t u r n s . Each of the two types of value can be f u r t h e r subdivided i n t o a value i n c u r r e n t use and a value i n exchange, and these two value concepts can each 11 be subdivided i n t o value to the owner and market value. F u r t h e r , each concept through mutable d e f i n i t i o n s and v a r y i n g terminology has conveyed a d i f f e r e n t meaning i n d i f f e r e n t 12 p l a c e s . I t should be emphasized that value i n the above d i s c u s s i o n i s to be d i s t i n g u i s h e d from c o s t . The l a t t e r term r e f e r s to h i s t o r i c a l c o s t . The only equivalence between cost and value i s under long-run s t a t i o n a r y e q u i l i b r i u m . 1 ^ See, f o r example, James C. Bonbright, The V a l u a t i o n of P r o p e r t y , a T r e a t i s e on the A p p r a i s a l of Property f o r D i f f e r e n t L e gal Purposes (New York: McGraw-Hill Book Company Inc., 1937), V o l . I , P a r t I and V o l . I , P a r t I I I , Chapter X V I I , p. 3 passim. 1*51-510. 1 1 , , V a l u e i n current use" i s not to be confused w i t h the c l a s s i c a l economic concept of "use v a l u e . " This l a t t e r concept i s more c l o s e l y r e l a t e d to "value to the owner." "Value i n exchange" i s not to be confused w i t h the c l a s s i c a l economic concept of "exchange value." This l a t t e r concept i s more c l o s e l y r e l a t e d to "market value." 1 2 B o n b r i g h t , l o c . c i t . : P a u l F. Wendt, Re a l Estate A p p r a i s a l . A C r i t i c a l A n a l y s i s of Theory and Practj.ce (New York: Henry H o l t and Companym 1956), Chapters 1 and 2, p. 1 passim. n -'Alfred M a r s h a l l , P r i n c i p l e s of Economics (8th ed. 5 London: Macmillan and Co. L t d . , 1962), p. 305. 13 A s a t i s f a c t o r y a n a l y s i s and d i s c u s s i o n of the above concepts of value would be a subject f o r both extensive and i n t e n s i v e research beyond the resources a v a i l a b l e f o r one study. With minor exceptions the r e a l property tax i n Ik North America i s a tax on c a p i t a l value. I t i s not the purpose here to debate the r e l a t i v e advantages of c a p i t a l and annual value. The a p p l i c a t i o n of s t a t i s t i c a l method to the measurement of the accuracy with which assessors estimate c a p i t a l value i s the only i n t e r e s t . As noted above there are s e v e r a l d i f f e r e n t concepts of c a p i t a l v alue. In North America assessors are d i r e c t e d 15 to base the assessments on market value; y however t h i s may be d e f i n e d . 1 ^ This i n s t r u c t i o n may be given i n v a r i o u s OP. c i t . New Sources of L o c a l Revenue. p. 68; H a r l e y L e i s t L u t z , P u b l i c Finance ( n t h e d . ; New York: D. Appleton-Century Company Inc . , 19^7)? P« 372; F r e d e r i c H. F i n n i s , R eal Property Assessment i n Canada (Canadian Tax Papers No. 30; Toronto: Canadian Tax Foundation, 1962), p. 9. 15 •'Report of the Committee on State E q u a l i z a t i o n of L o c a l Property Tax Assessments. N a t i o n a l Tax A s s o c i a t i o n , Proceedings of the 51st Annual Conference,1958« p. 318; F i n n i s , p_p_. c i t . T p. 16. ^ T h e r e are numerous d e f i n i t i o n s of market value but there i s an imperfect consensus as to what the d e f i n i t i o n should connote. I t should express the f o l l o w i n g i d e a s : the buyer and s e l l e r are not under duress to buy or s e l l r e s p e c t i v e l y ; the knowledge that the property i s a v a i l a b l e f o r purchase i s to be made a v a i l a b l e to the market; the market p a r t i c i p a n t s — t h o s e who have at l e a s t some l i t t l e i n t e n t i o n of buying or s e l l i n g — h a v e some reasonable p r i c e l it 17 termi n o l o g i e s . ' There are i n s t a n c e s , however, and u n t i l more recent years quite common, where the assessor i s i n s t r u c t e d to base the assessments on market value i n some based p e r i o d . Unless current market value i s s t i p u l a t e d as the b a s i s of assessment the q u a l i t y of assessment cannot be evaluated with the use of observed market t r a n s a c t i o n s . Instead, a s e r i e s of independent v a l u a t i o n s w i l l be re q u i r e d as comparisons to the m u n i c i p a l l y assessed v a l u e s . Since there i s no reason to expect a s i n g l e independent va l u e r to produce more uniform a p p r a i s a l s than the mun i c i p a l assessor each property used i n the evalua-t i o n of the q u a l i t y of the assessment r o l l would have to be valued by at l e a s t two independent a p p r a i s e r s . This could prove to be a c o s t l y procedure. e x p e c t a t i o n s , some knowledge, i n the market p l a c e ; that the exchange value r e a l i z e d w i l l be the r e s u l t of current r e a c t i o n of demand and supply i n the market pl a c e . With the above ideas i n mind, market value could be d e f i n e d as the p r i c e an i n t e r e s t i n land might reasonably be expected to r e a l i z e when s o l d by a w i l l i n g s e l l e r , a f t e r the property has had adequate exposure to the market, to a buyer, both s e l l e r and buyer being subject to the current c o n d i t i o n s of the expected market f o r the property. Refer a l s o to Bonbright, Wendt, l o c . c i t . 17 ' F i n n i s , l o c . c i t . He p o i n t s out that the Supreme Court has i n d i c a t e d that " a c t u a l value," " r e a l value," and " f a i r value" are synonymous w i t h "market value$" Report of the Committee on State E q u a l i z a t i o n of L o c a l Property Tax Assessments, l o c . c i t . : Re Assessment E q u a l i z a t i o n A c t : Rowan's Appeal (1962), kO WWR 627, SCBC. On a case stated by the Assessment Appeal Board, Mootton, J . s a i d that " a c t u a l v a l u e , " " f a i r market value," and " a c t u a l cash v a l u e " are " r e l a t i v e l y the same t h i n g . " 15 Also, i f market value i n a base year i s used, there i s no reason on the basis of equity or convenience to support market value as the basis of assessment over some other c a p i t a l value concept. Most important i s the fac t that current market value as the basis of assessment affords the only oppor-tunity for the taxpayer to know that he i s paying not more 1 8 than h i s share of the property tax burden. Current market value provides the required objective measure of value. B. Market Price as Evidence of Market Value Here, too, i t i s not the purpose to debate the disadvantages and advantages of the use of market value as the measure of c a p i t a l value i n r e a l estate taxation. The endeavour i s to establish s t a t i s t i c a l measures and standards by which to judge the assessor 1s success i n estimating the designated value. Therefore, an objective measure of market value i s needed to evaluate the assessor's performance. The required evidence i s actual sale prices, sale prices established i n transactions which conform to the conditions H a r o l d F. Meyer, "Market Value—Beacon Light of Appraisal for Taxation," Th^ e Appraisal Journal. Vol. XXXIII, No. k (October, 1965), p. 570, 571-572. 16 of the market value d e f i n i t i o n . The immediate problem i s 19 to e s t a b l i s h the r e l a t i o n s h i p between market p r i c e s ' e s t a b l i s h e d i n t r a n s a c t i o n s and, f o r the same p r o p e r t i e s , market values as they should appear on the assessment r o l l . There i s a l e g a l approach and an economic approach to t h i s question. 20 F i r s t consider the economic approach. At t h i s p o i n t another value concept i s introduced. This i s value to the owner. Value to the owner i s the value i n exchange of a p a r t i c u l a r i n t e r e s t i n a p a r t i c u l a r property to a p a r t i c u l a r i n d i v i d u a l at a p a r t i c u l a r time. I t i s an estimate of the present value to him of a l l the f u t u r e u t i l i t y he estimates that he w i l l d e r ive from the c o n t r o l over the i n t e r e s t i n the property. The estimate of the f u t u r e u t i l i t y and i t s present value are cond i t i o n e d by the a l t e r n a t i v e s open to the i n d i v i d u a l . A l l a l t e r n a t i v e s are r e l e v a n t , but the more d i r e c t l y comparable a l t e r n a t i v e s ""Market p r i c e s " w i l l be used to designate "sales p r i c e s e s t a b l i s h e d i n t r a n s a c t i o n s which conform to the c o n d i t i o n s of the market value d e f i n i t i o n . " 20 Ralph Turvey, The Economics of R e a l P r o p e r t y : An A n a l y s i s of Property Values and Patterns of Use (London: George A l l e n & Unwin, L t d . , 1957), PP. 8-21. Turvey has produced a very c l e a r , c o n c i s e , and c o n s i s t e n t a n a l y s i s of the determination of r e a l estate p r i c e s . The d i s c u s s i o n i n the te x t of t h i s study f o l l o w s h i s work. 17 are more r e l e v a n t . For example, to an i n v e s t o r , the f u t u r e income from an apartment b u i l d i n g i s more d i r e c t l y comparable to the f u t u r e income from a second apartment b u i l d i n g than i s the f u t u r e income from a government bond; the p r i c e f o r which he can ob t a i n the second apartment b u i l d i n g i s more re l e v a n t to h i s estimate of the present value of the f u t u r e u t i l i t y to him of the f i r s t apartment b u i l d i n g , than i s the p r i c e f o r which he can obtain the government bond. Turvey has used F l o o r p r i c e to denote the value of an i n t e r e s t to - the present owner and C e i l i n g p r i c e to denote the value of an i n t e r e s t to someone other than the present owner. T h e o r e t i c a l l y every owner has a F l o o r p r i c e and everyone e l s e has a C e i l i n g p r i c e f o r 21 every i n t e r e s t which he does not already own. In p r a c t i c e , of course, only owners with at l e a s t some l i t t l e i n t e n t i o n of s e l l i n g w i l l have a F l o o r p r i c e f o r the i n t e r e s t , and only others with at l e a s t some l i t t l e i n t e n t i o n of buying the i n t e r e s t w i l l have a C e i l i n g p r i c e . I t i s these l a t t e r F l o o r p r i c e s and C e i l i n g p r i c e s which are the determinants of market valu e s . I f , f o r a p a r t i c u l a r i n t e r e s t , there i s at l e a s t one C e i l i n g p r i c e i n excess of the F l o o r p r i c e , a range of Turvey, op. c i t . . p. 19. mutual g a i n , then there w i l l be a t r a n s a c t i o n and a r e s u l t i n g market p r i c e . The a c t u a l p r i c e w i l l be between C e i l i n g p r i c e and F l o o r p r i c e . I f there i s more than one C e i l i n g p r i c e , then the r e s u l t i n g market p r i c e w i l l be "between the two highest C e i l i n g p r i c e s . The r e l a t i v e p o s i t i o n of the market p r i c e i n the range of mutual gain w i l l depend on the bargaining p o s i t i o n and s k i l l of the buyer and s e l l e r . In any case, the r e s u l t i n g market p r i c e w i l l be observed by prospective s e l l e r s and p o t e n t i a l buyers of s i m i l a r i n t e r e s t s . Since the d i f f e r e n t p o t e n t i a l purchasers of a p a r t i c u l a r property w i l l to some extent consider the same a l t e r n a t i v e s , the spread of t h e i r c e i l i n g p r i c e s w i l l not be l a r g e . . . . I t f o l l o w s that i n the case of p r o p e r t i e s f o r which there are s u b s t i t u t e s , the p r i c e which w i l l be r e a l i z e d i s determinate w i t h i n f a i r l y narrow l i m i t s , and can be f o r e c a s t (as market value) f a i r l y w e l l by an expert.2 2 Turvey then concludes that the market p r i c e s are f a i r l y determinate (determinate w i t h i n narrow l i m i t s ) f o r property f o r which there i s a "lar g e market" and there i s adequate market exposure. J Turvey, op., c i t . , p. 19. I b i d . , p. 20. This c o n c l u s i o n means that sale p r i c e s r e s u l t i n g from t r a n s a c t i o n s which occur w i t h i n c o n d i t i o n s s p e c i f i e d by the d e f i n i t i o n of market value are acceptable, w i t h i n narrow l i m i t s , as evidence of market value. The l i m i t s w i l l be more narrow f o r low and medium value range r e s i -d e n t i a l p r o p e r t i e s than f o r commercial and i n d u s t r i a l p r o p e r t i e s , because the former are, as i s common knowledge, more a c t i v e l y traded than the l a t t e r . A b r i e f glance at the j u d i c i a l view of the r e l a t i o n between p r i c e s determined i n the market and market value may be u s e f u l . There are long l i n e s of c o n f l i c t i n g cases, and a d e t a i l e d examination does not re s o l v e the c o n f l i c t , but merely confirms i t s e x i s t e n c e . Wootton, J . i n d i c a t e d that a sale p r i c e i s not n e c e s s a r i l y the best t e s t of market value. He h e l d that 2k the c o n d i t i o n s surrounding the sale must be considered. In other words, f o r a sale p r i c e to be a market p r i c e , the t r a n s a c t i o n must occur w i t h i n the c o n d i t i o n s of market value. The above d e c i s i o n f o l l o w s the statement of A t k i n , 25 L. J . i n Norwich Assessment Committee vs. P o i n t e r . ' He there s a i d that i n a r r i v i n g at market rent ( r e n t a l value) Loc. c i t . : Re Rowan's Appeal. 25 •'Norwich Assessment Committee vs. P o i n t e r , 1922, 2 KB, p. 1+71, Court of Appeal. 20 •'any evidence which i s relevant to that question i s i n law admissible, and i t must depend on the circumstances of the case. . . . " More d i r e c t l y to the point Lord Buck-master^ and Scott, L.J.^' 7 have held that the actual rent being paid by the occupier is' relevant only so f a r as the conditions of the actual tenancy conform to the statutory conditions required of the estimate of gross value. Bonbright quotes from the decisions of two cases which Assessment Committee of the Metropolitan Borough St Poplar vs. Roberts (1922) 38 T.L.R. *+99, House of Lords (see esp. p. 103). 27 'Robinson Brothers (Brewers) Ltd. vs. Houghton and Chester-le-Street Assessment Committee (1937), 2 K.B. k k $ (see esp. p. *+69). "Gross value" f o r the purpose of rating s p e c i f i e s an annual tenancy, the tenant be responsible for tenants* rates and taxes, and the landlord be responsible f o r repairs and maintenance. See the following l e g i s l a t i o n : Great B r i t a i n , Statutes a£ Large 25 & 26 V i c t . (1862), c. 103, "Union Assessment Committee Act," sec. 15(g); Great B r i t a i n , Statutes at Large 32 & 33 V i c t . (1869), c. 67, "The Valuation (Metropolis) Act," sec. H; Great B r i t a i n , Statutes at. Large 15 & 16 Geo. 5,(1925), c. 90, "Rating and Valuation Act," sec. 68. It i s clear that the English d e f i n i t i o n of the basis of assessment r e l a t e s to market rent i n a hypothetical market, i n other words a value i n exchange, but the point to observe i s that the Courts have held that actual rent i s evidence of the hypothetical market rent only so f a r as the conditions surrounding a p a r t i c u l a r tenancy conform to the conditions of the hypothetical market. " b e l i t t l e the d i s t i n c t i o n " between market value and current 29 sale p r i c e . ' The Committee on State E q u a l i z a t i o n of L o c a l Property Tax Assessments i n i t s r e p o r t to The N a t i o n a l Tax A s s o c i a t i o n contends that sale p r i c e at market c o n d i t i o n s i s " h i g h l y persuasive of the market value of the property." Further on they emphasize the p o i n t by saying that the sale produces "conclusive evidence of market v a l u e . " J C. Valuer's Margin of E r r o r With the above weight of evidence, economic and j u d i c i a l , i t can be concluded that an a s s e s s o r ^ estimate of market value should l i e w i t h i n narrow l i m i t s of the observed market p r i c e . The question i s : "How narrow are the l i m i t s ? " The s e t t i n g of the l i m i t s i s an e x e r c i s e i n value judgment. There i s no sharp d e l i m i t a t i o n between a c o r r e c t 'Bonbright, OP. c i t . ? p. k6k. He quotes from Kentucky R i v e r Coal Corp. vs. Knott County. 2$k Ky. 882, 5*+ S.W. (2d) 377 (1932) (Ky. at 828, S.W. at 379); A t l a n t i c S t a t e s Coal Corp. vs. Letcher County. 2ko Ky. 5*+95 55 S.W. (2d) If09 (1932) (Ky. at 551-552, S.W. at ^ - ^ l O ) . ^0 J Report of the Committee on State E q u a l i z a t i o n of L o c a l Property Assessments, N a t i o n a l Tax A s s o c i a t i o n , Proceedings of 5 l s t Annual Conference. 1958, p. 337« I b i d . f p. 3tt2. and i n c o r r e c t v a l u a t i o n . Byrne, J . i n B e l l H o t e l s (19^5) op L t d . vs. Motion and Another- 1 gives some i n d i c a t i o n of where the l i m i t s might l i e . In that case a va l u e r estimated f o r a c l i e n t "that a sale (of h i s p u b l i c house) might be e f f e c t e d at 20,000 pounds, but . . . i f a near o f f e r were forthcoming, i t should be s e r i o u s l y c o n s i d e r e d . T h e owner s o l d the p u b l i c house f o r 17,500 pounds. S h o r t l y a f t e r t h i s s a l e , the purchaser r e s o l d the p u b l i c house f o r 25,000 pounds. Byrne, J . found the v a l u e r n e g l i g e n t and allowed a c l a i m of 5,000 pounds, but the court reasoned that the v a l u e r should be allowed a ten per cent margin on the market p r i c e . L i m i t s of plus and minus ten per cent of market p r i c e should be the most generous l i m i t s f o r the estimate of market value on most types of property, because i t i s recognized that the v a l u a t i o n of l i c e n s e d premises i s more vexatious than the v a l u a t i o n of most other types of J B e l l H o t e l s (1935) L t d . vs. Motion and Another, The E s t a t e s Gazette, May 17, 1952. The d e c i s i o n revolved around whether or not the v a l u e r s were n e g l i g e n t , but the way i n which damages were assessed i s of i n t e r e s t to our problem here. ->->A " p u b l i c house" i s one type of " l i c e n s e d pre-mises." See.D. M. Lawrance, W. H. Rees, and W. B r i t t o n , Modern Methods of V a l u a t i o n of Land, Houses and B u i l d i n g s (5th ed.; London: The E s t a t e s Gazette L i m i t e d , (1962)), P. *+33. ok property.-' Maclean, J . i n Dodds and Dodds ( p l a i n t i f f s ) vs. Millman was not n e a r l y so generous to the defendant i n estimating the amount of the c l a i m . y The defendant, the vendor's agent, produced a v a l u a t i o n of a small apartment b u i l d i n g and caused t h i s v a l u a t i o n to come i n t o the hands of the p l a i n t i f f s , the purchasers. The agent's estimate of market value was $ ^ 2 , 5 0 0 , and i t was accompanied by a p r o j e c t e d "operating statement." L a t e r , the p l a i n t i f f s found the revenue and expense estimates to be g r o s s l y inaccurate and they brought a c t i o n against the vendor and h i s agent f o r f r a u d , or at l e a s t negligence. On the b a s i s of expert witness the Court found negligence against the agent, and awarded the purchasers $ 8 , 5 0 0 i n damages. The expert witness estimated that the market value at the time of purchase would have been $35>000. Maclean, J . allowed the agent no margin of p e r m i s s i b l e discrepancy. This l a s t case i s d i f f i c u l t , i f not impossible to apply. In B e l l vs. M o t i o n s 3 ^ there was a second s a l e , and, there-f o r e , an o b j e c t i v e measure of the e r r o r . In Dodds vs. •V? Millman~" the Court but weighed the testimony of expert w i t n e s s , and accepted the f a c t that the expert v a l u a t i o n J Lawrance, Rees, and B r i t t o n , l o c . p i t . ^Dodds and Dodds vs. Millman 1+5 D.L.R. ( 2 d ) , I 9 6 U , p. h72. fosuora, pp. 2 2 - 2 3 37s upra. p. 2 3 . may have been g r e a t e r , as w e l l as l e s s than, the market p r i c e r e a l i z e d upon sale at the date of the purchase. D. The Tax Base There are s e v e r a l a d d i t i o n a l concepts which r e l a t e to the use of s a l e p r i c e , f i r s t , by the assessor i n producing the v a l u a t i o n l i s t , and second, by the a u d i t o r ^ of the assessment r o l l i n e v a l u a t i n g the performance of the assessor. The tax base i s , i n c o n j u n c t i o n with the b a s i s of assessment, the p r e s c r i p t i o n of the property to be 39 assessed. ' 3 In s e v e r a l places throughout t h i s paper the words " a u d i t " and " a u d i t o r " are used. This r e f l e c t s the a u t h o r T s o p i n i o n that j u s t as the M u n i c i p a l Accounts are each year subject to an e x t e r n a l a u d i t , so should the assessment r o l l be subject to an annual e x t e r n a l a u d i t to examine the u n i f o r m i t y of assessment, the i d e n t i f i c a t i o n of taxable p r o p e r t i e s , and the accuracy of the d e s c r i p t i o n of those p r o p e r t i e s and t h e i r ownership. -^ •Some a u t h o r i t i e s d e f i n e the tax base as "the p r e s c r i p t i o n of the property to be taxed." (Report of The Royal Commission on Finance and M u n i c i p a l Taxation i n New Brunswick, F r e d e r i c t o n , 1963)* The phrase "of the property to be taxed" i n c l u d e s the b a s i s of assessment, but the same phrase does not leave the way open f o r a c l e a r d i s t i n c t i o n between assessed property exempt from t a x a t i o n and s p e c i f i c property which i s a p a r t of the general genus of property to be assessed, but which by s t a t u t e i s exempt from assess-ment . 25 The Vancouver Charter d e f i n e s improvements to in c l u d e " b u i l d i n g s , s t r u c t u r e s , machinery and other things k i so a f f i x e d to land as to make them i n law a p a r t thereof" but f o r the purpose of C i t y t a x a t i o n (as contrasted to school t a x a t i o n ) the p r a c t i c e has been to i n t e r p r e t the d e f i n i t i o n of improvements to have the same meaning as 1+2 t h e i r d e f i n i t i o n i n the M u n i c i p a l Act. Lo The M u n i c i p a l Act J r e q u i r e s that "land and k k improvements s h a l l be assessed, and except f o r school purposes d e f i n e s improvements to exclude " f i x t u r e s . . . i f so erected or a f f i x e d by a tenant would, as between land-l o r d and tenant, be removable by the tenant." J The Vancouver C i t y Charter r e q u i r e s that land values and improvement values be shown separately on the assessment 1+6 r o l l , but each p a r c e l . . . be estimated at i t s a c t u a l B r i t i s h Columbia S t a t u t e s , 1957, c. 1+2. 1+1 x I b i d . . sec. 2. k p Interview with C. Dowling, Deputy Assessment Commissioner, C i t y of Vancouver, J u l y 16, 1965. B r i t i s h Columbia S t a t u t e s , 1957, c. 1+2. ^ I b i d . , sec. 330 ( l ) . ^ I b i d . , sec. 2. 1+6 B r i t i s h Columbia S t a t u t e s . 1953, c. 55 , sec. 3I+I. 26 1+7 v a l u e . " ' The most obvious method to use i n c a r r y i n g out these i n s t r u c t i o n s i s to estimate the market value of the property and then with a b i t of i n t e l l e c t u a l e x e r c i s e separate the "whole" i n t o i t s " p a r t s , " land and improve-ments. F i s h e r has an e x c e l l e n t comment on t h i s s eparation. " F i r s t , the s e r v i c e s rendered are the j o i n t product of land and improvements . . . , the two unite to form a compound, 1+8 not a mixturei' And f o r an i n d i c a t i o n of the d i f f i c u l t y and f u t i l i t y of t h i s separation we can r e f e r to Turvey. He has produced a c l e a r and concise argument proving that the separation of land and b u i l d i n g has no meaning except i n long-run s t a t i c e q u i l i b r i u m , and as such i s useless i n economic a n a l y s i s except when the b u i l d i n g i s new and i n i t s h i g h e s t and best use. ' Where land and improvements are taxed at the same e f f e c t i v e r a t e per cent the separation of the "whole" 50 i n t o the "parts" -' has no e f f e c t on the e q u i t a b l e d i s t r i -b u t i o n of the tax impact. But where the l e g i s l a t u r e i n i t s wisdom has decided to tax land and improvements at d i f f e r e n t e f f e c t i v e r a t e s per cent, then the separation i s of great ^ I b i d . , sec. 3*+2. 1+8 E. M. F i s h e r , Urban R e a l E s t a t e Markets: Charac-t e r i s t i c s and Fina n c i n g (New York: N a t i o n a l Bureau of Economic Research, 1951)5 p. 5» 1+Q 7Turvey, opj. c i t . , pp. 22-2*+. 50 This i s by no means the same process as e s t i m a t i n g the " p a r t s " i n order to construct the "whole." 27 moment. In the l a t t e r s i t u a t i o n , and t h i s i s the s i t u a t i o n i n B r i t i s h Columbia, an a u d i t of the assessment r o l l does not produce a measure of the e q u a l i t y , or i n e q u a l i t y , i n the d i s t r i b u t i o n of tax impact. In order to do that the a u d i t o r would r e q u i r e sales of b u i l d i n g s apart from the l a n d , and s a l e s of land p u l l e d out from under the b u i l d i n g s . Turvey has so a p t l y pointed out that "no o r d i n a r y b u i l d i n g i s ever sold f l o a t i n g i n a i r . " ^ * L Therefore, i n t h i s circumstance an a u d i t of the v a l u a t i o n l i s t using market p r i c e s can but measure the performance of the assessor i n estimating the value of the "whole." E. Sales P r i c e s and Real E s t a t e Finance The phrase "current c o n d i t i o n s of the expected market" used i n the d e f i n i t i o n of market v a l u e ^ 2 a p p l i e s to the f i n a n c i n g of r e a l estate i n t e r e s t s . As the words "expected market" imply, f o r a p a r t i c u l a r property there i s a t y p i c a l expected purchaser. People with high incomes are more l i k e l y to buy high valued housing than are people 51 Turvey, op_. c i t . , p. 2h. Supra, p. 1M-. 28 5^ with low incomes."^ Given a p a r t i c u l a r property and, th e r e f o r e , a general expectation of a t y p i c a l personal covenant, there w i l l be a v a i l a b l e to the p o t e n t i a l purchaser of the property mortgage f i n a n c i n g p i t c h e d upon by the "current c o n d i t i o n s of the expected market." At a given time, i f f o r a p a r t i c u l a r property the f i n a n c i n g used i s d i f f e r e n t from market exp e c t a t i o n s , then the sale p r i c e r e s u l t i n g from a t r a n s a c t i o n w i l l not be a d i r e c t i n d i c a t i o n of market p r i c e . The e f f e c t of the above concept can be i l l u s t r a t e d , w i t h an example. Assume, f o r the purpose of d i s c u s s i o n , that there i s a house with a known market p r i c e of $20,000, that the "expected" mortgage loan w i l l have a loan-value r a t i o of 0 .60, and that the "expected" i n t e r e s t r a t e i s 7 per cent. There are many ways i n which the a c t u a l P. H. White, "Prologue to An A n a l y s i s of the R e s i d e n t i a l Mortgage Market i n Vancouver," Conference of the A s s o c i a t i o n of Canadian Business Schools, Proceedings, 1965 (footnote 2k of White's paper i n d i c a t e s a strong c o r r e l a t i o n between income and value of housings); A l v i n E. Coons and Bert L. Glaze, Housing Market A n a l y s i s and the Growth of Nonfarm Home Ownership (Bureau of Business Research Monograph Number 115; Columbus: Bureau of Business Research, College of Commerce and A d m i n i s t r a t i o n , The Ohio State U n i v e r s i t y , 1963), pp. 29-30 and 132.;. 29 t r a n s a c t i o n may deviate from the expected t r a n s a c t i o n , and we s h a l l explore a few cases and t h e i r e f f e c t s . Case 1. The market p r i c e w i l l be constructed as f o l l o w s : Purchaser's e q u i t y $ k,000 20 year, % — 1st Mortgage Loan 12,000 ( i n s t i t u t i o n a l lender) 10 year, 10$ — 2nd Mortgage Loan (vendor *+,000 Market P r i c e $20,000 But the p r a c t i c e i s to w r i t e the second mortgage a t , or near, 5 per cent and bonus the face value to y i e l d 10 per cent. More oft e n than not a r u l e of thumb i s used to le I 56 c a l c u l a t e the bonus, but a more accurate c a l c u l a t i o n can be provided. 5»f In p r a c t i c e one s t a r t s with an observation of the r e s u l t a n t e f f e c t s and works back to market p r i c e . The i n t e r e s t r a t e s i n the examples are not those of the current market i n a p a r t i c u l a r c i t y . 55 The bonus i s , by the r u l e of thumb, u s u a l l y set between one-quarter and one - t h i r d of the cash amount of the loan. 56 Instead of t r a n s f e r r i n g the t i t l e and the pur-chaser granting a mortgage to the vendor, the vendor could grant a r i g h t to purchase to the buyer. 30 Monthly payment to amortize 1^,000 i n 10 years at 10$ $52.1+2 The present value of $52.^2 per month f o r 10 years discounted at % $ 5 ,000 The sale p r i c e would be constructed as f o l l o w s : Purchaser's e q u i t y $ L , 0 0 0 20 year, % -- 1st Mortgage Loan 12,000 ( i n s t i t u t i o n a l lender) 10 year, % -- 2nd Mortgage Loan 5 5 0 0 0 (vendor) Sale p r i c e $21,000 Case 2. The market p r i c e w i l l be constructed as f o l l o w s : Purchaser's e q u i t y $ k^OOQ 20 year, 6% - - 1st Mortgage Loan?7 16,000 (vendor) Market p r i c e $20,000 A $16,000 loan f o r 20 years at 6% i s equivalent to a $12,000 loan f o r 20 years at % plus a $L , 0 0 0 loan f o r 20 years at 8 i ^ . p o i n t . The f o l l o w i n g c a l c u l a t i o n w i l l i l l u s t r a t e t h i s Monthly payment to amortize: $12,000 i n 20 years at % $ 78.86 *+,000 i n 20 years at 8 ^ 35«10 $16,000 i n 20 years at 6% $113.96 •"Instead of t r a n s f e r r i n g the t i t l e and the pur-chaser granting a mortgage to the vendor, the vendor could grant a r i g h t to purchase to the buyer. In t h i s case the mortgage would be w r i t t e n at 5$> and the face amount bonused to y i e l d 6$. Monthly payment to amortize: $16,000 i n 20 years at 6% $113.96 The present value of $113.96 per month f o r 20 years discounted at % $17,100 The sale p r i c e would be constructed as f o l l o w s . Purchaser's e q u i t y $ *+,000 20 year, 5$ — 1st Mortgage Loan (vendor) $17,000 Sale p r i c e $21,000 Case 3. In some in s t a n c e s there may be against the t i t l e an unfavourable mortgage f o r which the purchaser cannot o b t a i n a r e l e a s e . I n other words, he must "take over" payment on a mortgage the terms of which are not as good as might be expected. To i l l u s t r a t e the e f f e c t of t h i s p o i n t , assume that our example property has a 12-year, 6 1/2$ $10,000 mortgage loan against the t i t l e , and that there are 9 years of the mortgage term remaining. The p r i n c i p a l outstanding at the end of 3 years i s $8,170. Under the assumed present market c o n d i t i o n s the purchaser could expect a $12,000 mortgage at 5$' In order to obtai n the a d d i t i o n a l $3,830 he w i l l have to pay 9 1/2$ on a 58 second mortgage loan. __ ' In two years the purchaser could o b t a i n a r e l e a s e on the f i r s t mortgage and r e f i n a n c e , but he could at that time s t i l l have three years before he could r e l e a s e the second mortgage. Monthly payment to amortize: $12,000 i n 9 years at 6 1/2$ $ 8,170 i n 9 years at 6 1/2$ $ 3,830 i n 9 years at 9 1/2$ $ 9 9 . 7 L 62.62 $163.36 $137.92 The p u r c h a s e r I s foregone opportunity, the excess of expected, a c t u a l payment over the market e x p e c t a t i o n i s $25.M+ per month f o r 9 years. The present value of t h i s o pportunity l o s s at 9 1/2$ i s $1,850. The sale p r i c e would be constructed as f o l l o w s . Case k. I f the vendor has a superior c r e d i t standing to that of the expected purchaser, then the vendor may be able to ob t a i n a more favourable mortgage loan before s e l l i n g the property than could the purchaser a f t e r buying the property. Suppose the vendor obtained a l o a n of $12,000 f o r 20 years at k 1/2$, w h i l e the t y p i c a l purchaser of the property might expect a loan of $12,000 f o r 20 years "Take over" of 1st mortgage Cash to vendor Sale p r i c e $8,170 9,980 $18,150 at 5$. Monthly payment to amortize: $12,000 i n 20 years at 5$ $78.86 $12,000 i n 20 years at k 1/2$ 75-65 Monthly saving $ 3.21 The gain to the purchaser i s $3.21 per month, and the present value of the monthly savings over 20 years discounted at 5$ i s $*+90. The sale p r i c e would be 59 constructed, as f o l l o w s . ' "Take over" of 1st mortgage $12,000 Cash to vendor 8,1+90 Sale p r i c e $20,^90 The above i l l u s t r a t i o n s are not meant to be i n d i c a -t i v e of current market c o n d i t i o n s nor are they a complete co m p i l a t i o n of r e a l estate f i n a n c i n g p r a c t i c e s . They were presented only as some examples of t r a n s a c t i o n s which do not conform to the c o n d i t i o n s as set out i n the d e f i n i t i o n of market va l u e , and, t h e r e f o r e , which might record sale p r i c e s d i f f e r e n t from market p r i c e s . The d i f f e r e n c e may be small or l a r g e depending on the circum-stances surrounding the s p e c i f i c t r a n s a c t i o n . F. Sales P r i c e s and Assessment U n i f o r m i t y That a p e r i o d i c tax on durable goods may be c a p i t a l i z e d , has long been recognized. But almost I n 59 > 7 T h e r e i s of course some s p e c u l a t i o n as to whether or not the a d d i t i o n a l $M-90 i s s u f f i c i e n t to cover the "cost of the contingent l i a b i l i t y of the vendor. 6 0 E d w i n R. A. Seligman, S h i f t i n g and Incidence of Taxation ( 5 t h ed. rev.; New York: Columbia U n i v e r s i t y P r e s s , 1926), pp. I7I+-I83, 328-329. a l l cases the durable good has been considered an investment i n the cash income producing s e n s e . ^ That approach i s reasonable when d i s c u s s i n g commercial and investment r e a l e s t a t e , but not so reasonable when d i s c u s s i n g owner occupied housing. Some economists consider owner-occupied housing 62 a durable consumer good. F i r s t consider the e f f e c t s of over and under assess-ment on the sale p r i c e s of commercial and investment r e a l e s t a t e . Groves s t a t e s that " c a p i t a l i z a t i o n can e x i s t only to the extent that s h i f t i n g does not," and s t i p u l a t e s the c o n d i t i o n s under which c a p i t a l i z a t i o n w i l l occur. J These % i c k s , op_. c i t . , p. 16U; C a r l Shoup, " C a p i t a l i z a -t i o n and S h i f t i n g of the Property Tax," i n H a r o l d M. Groves, ed., Viewpoints on P u b l i c Finance ( [ r e p r i n t e d from Property Taxes, Chap. 13, Tax I n s t i t u t e , New York; New York: Henry H o l t and Company, l ^ l ) , PP. 107-11?; H a r o l d M. Groves, F i n a n c i n g Government (3rd ed.; New York: Henry H o l t and Company, 1950), pp. 126-128; Edwin R. A. Seligman, S h i f t i n g and Incidence of Taxation, l o c . c i t . 6? A l v i n E. Coons and Be r t L. Glaze, Housing Market A n a l y s i s and the Growth of Nonfarm Home Ownership (Bureau of Business Research: Monograph Number 115; Columbus: Bureau of Business Research, College of Commerce and A d m i n i s t r a t i o n , The Ohio State U n i v e r s i t y , 1963). In t h i s monograph the e n t i r e a n a l y s i s i s based on the b e l i e f that housing stock i s a consumer durable. See pp. 16-18; R i c h a r d F. Muth, "The Demand f o r Non-Farm Housing," i n Arnold C. Harberger, ed., The Demand f o r Durable Goods (Chicago: U n i v e r s i t y of Chicago P r e s s , i960). Housing as a consumer durable i s i m p l i c i t i n t h i s study. Groves, pj*. c i t . , pp. 126-127. c o n d i t i o n s apply to income producing goods f o r which the f u t u r e tax p a t t e r n s can be a n t i c i p a t e d , the t o t a l supply i s r e l a t i v e l y f i x e d , and which are taxed unequally i n r e l a t i o n to a l t e r n a t i v e investments. The c o m p l e x i t i e s 6k of t h i s problem are made q u i t e c l e a r by Shoup. He f i r s t c a p i t a l i z e s a change i n tax at current i n t e r e s t r a t e s , but then goes on to recognize that a change i n the tax may cause i n t e r e s t r a t e s to change and hence o f f s e t to some degree the c a p i t a l i z a t i o n . He a l s o argues that i f the tax p a t t e r n i s a n t i c i p a t e d then the c a p i t a l i z a t i o n would be g r a d u a l , the present value of f u t u r e changes i n tax g r a d u a l l y a f f e c t i n g the value of the good as the tax change becomes more imminent. The s i t u a t i o n f o r the present purpose i s not so complex because t h i s study i s not concerned w i t h a change i n tax on one c l a s s of pro-p e r t i e s r e l a t i v e to another, but with the e f f e c t of non-uniform assessment of one property r e l a t i v e to another s i m i l a r property. Therefore, one can s a f e l y assume i n the f o l l o w i n g i l l u s t r a t i o n that the discount r a t e does not change. The maximum e f f e c t w i l l be i n d i c a t e d i f we assume that a l l p r o p e r t i e s except one are uniformly assessed, there are Shoup, p_p_. c i t . , pp. 107-115« 65 s e v e r a l p r o p e r t i e s s i m i l a r i n a l l r e s p e c t s , ' and f u t u r e assessments on the p r o p e r t i e s w i l l bear the same r e l a t i v e r e l a t i o n s h i p s among each other. Suppose that f i v e s i m i l a r p r o p e r t i e s r e c e n t l y s o l d under market c o n d i t i o n s f o r $100,000 each, that each was assessed at $100,000, and that each i s occupied under an annual tenancy at a net annual r e n t of $10,000. The property tax rat e i s 2% of assessed value. A s i x t h s i m i l a r property i s assessed at $130,000. The tax w i l l be $600 more per year on the s i x t h property than on the other f i v e , and the net r e n t a l value w i l l be $600 l e s s . The s i x t h property would s e l l f o r $9^,000. Since the r a t e per cent of tax i s s m a l l , a r e l a t i v e over assessment of $ produces a sa l e p r i c e d i f f e r e n c e which i n a r e a l s i t u a t i o n would be w e l l w i t h i n an allowable margin of 66 e r r o r . The assessment-market p r i c e r a t i o of 1.30 i s but s l i g h t l y smaller than the assessment-sale p r i c e r a t i o of 1.38. I f the assessment on the s i x t h property were 65 ^ " S i m i l a r i n a l l r e s p e c t s " does not mean that a l l are on the same l o t . ^ S u p r a , p. 23. We would expect that where f i v e out of s i x p r o p e r t i e s s i m i l a r i n a l l respects s o l d under market c o n d i t i o n s , that an assessor would do much b e t t e r than assumed i n t h i s example. 37 $110,000 the r a t i o s would be 1.10 and 1.12 r e s p e c t i v e l y . I f on the other hand the assessment on the s i x t h property were $90,000 the r a t i o s would be .90 and .88 r e s p e c t i v e l y . Nowturn to the c o n s i d e r a t i o n of the e f f e c t of over and under assessment on the sale p r i c e s of owner occupied housing. Even i f owner occupied housing i s viewed from an investment p o i n t of view and i f s a l e p r i c e and r e n t a l data were a v a i l a b l e f o r s i n g l e f a m i l y d w e l l i n g s the a n a l y s i s could proceed no f u r t h e r . Coons and Glaze found t h a t , "home ownership and r e n t a l tenure are not p e r f e c t s u b s t i t u t e s , " the l a t t e r being i n f e r i o r i n the consumers' 67 minds to the former. ' Therefore, occupants are w i l l i n g to pay more per month f o r owner-occupied housing than f o r the same q u a n t i t y and q u a l i t y of r e n t a l housing. The same study concludes that housing i s a consumer d u r a b l e , ^ and that housing expenditure i s a budget i t e m . ^ For i l l u s t r a t i v e purpose only, using a "budget approach" the e f f e c t on sale p r i c e of under and over assessment i s demonstrated below. 67 'Coons and Glaze, op. c i t . , p. 130. 6 8 I M d . , pp. 132-136. 6 9 I b i d . . p. 132. 38 Assume that f i v e houses s i m i l a r i n a l l r e s p e c t s r e c e n t l y s o l d f o r $20,000. The sale p r i c e c o n s i s t e d of $8,000 p u r c h a s e r s e q u i t y and a $12,000 i n s t i t u t i o n a l f i r s t mortgage loan f o r 20 years at 5$« The p r o p e r t i e s are assessed at $20,000, and the r a t e of tax i s 2$. The annual cost of occupation to the owners w i l l be as f o l l o w s : Debt s e r v i c e -- 12 months at $78.86 $ 9^6.32 Taxes H-00.00 H e a t i n g , maintenance 600.00 $19^6.32 A s i x t h property s i m i l a r i n a l l r e s p e c t s to the other f i v e i s assessed at $26,000. The taxes w i l l be $520 per year, and i f the purchaser i s to pay no more to occupy t h i s property than the other f i v e he w i l l reduce the debt se r v i c e by $120 per year, or by $10 per month. But payments of $68.86 per month w i l l i n 20 years and at 5$ amortize a loan of only $10,^O. The sa l e s p r i c e w i l l be constructed as f o l l o w s . P u r c h a s e r ^ e q u i t y $ 8,000 20 year, 5$ — F i r s t Mortgage 10,^50 Sale p r i c e $18,1+50 The sale p r i c e i s w i t h i n a 10$ margin of $20,000. The assessment-market p r i c e r a t i o i s 1.30, while the assessment-sale p r i c e r a t i o i s 1.^1. I f the property were assessed at $22,000 the sale p r i c e would be $19,500. 39 The assessment-market value r a t i o and the assessment-sale p r i c e r a t i o would then be r e s p e c t i v e l y 1.10 and 1 .13. The i l l u s t r a t i o n s i n d i c a t e only very g e n e r a l l y what might be the maximum e f f e c t of over and under assessment on s a l e p r i c e s and assessment sale p r i c e r a t i o s . As one would expect, the d i f f e r e n c e between assessment-market value r a t i o and assessment-sale p r i c e r a t i o decreases as the assessed value of the property approaches the assessed value of the s i m i l a r p r o p e r t i e s . The above examples are much too simple f o r p r a c t i c a l a p p l i c a t i o n , but i n r e a l i t y the problem i s too complex f o r an accurate p r a c t i c a b l e s o l u t i o n . The po i n t to remember i s that the d i f f e r e n c e between market value and sa l e p r i c e w i l l i ncrease as the d i f f e r e n c e between the assessed value of the subject property and the assessed value of comparable p r o p e r t i e s i n c r e a s e s ; as the d i s p e r s i o n of the assessment-market p r i c e r a t i o s of the comparable p r o p e r t i e s decreases; as the degree of s i m i l a r i t y among the subject and comparables i n c r e a s e s ; as market knowledge through a c t i v e t r a d i n g i n c r e a s e s ; and as the tax r a t e i n c r e a s e s . G. U n i f o r m i t y of Assessment For absolute u n i f o r m i t y of assessment to p r e v a i l a l l p r o p e r t i e s w i t h i n a f i s c a l j u r i s d i c t i o n must be assessed at the same p r o p o r t i o n of market value. Since market value l i e s somewhere w i t h i n a range of mutual 70 g a i n / reason suggests that a v a l u e r be allowed a range of tolerance i n h i s estimates of market value. With t h i s i n mind one can h a r d l y expect an assessor to achieve absolute u n i f o r m i t y i n h i s v a l u a t i o n l i s t , but f o r the purpose of t h i s s e c t i o n assume that absolute u n i f o r m i t y i s p o s s i b l e . The courts have handed down some i n t e r e s t i n g judgments on the problem of u n i f o r m i t y of assessment, but the c o n f l i c t i n g judgments help l i t t l e i n e s t a b l i s h i n g a p r a c t i c a l s t a t i s t i c a l measure of u n i f o r m i t y or q u a l i t y . One l i n e of B r i t i s h cases agrees w i t h the D i v i s i o n a l Court i n Norwich Assessment Committee vs. P o i n t e r , ^ a l e a d i n g case i n a l i n e of s i m i l a r cases. I t was there h e l d that while evidence of the assessment of s i m i l a r p r o p e r t i e s i s not the best evidence as to the s t a t u t o r y d e f i n i t i o n of v a l u e , i t nevertheless can be considered i n a r r i v i n g at an assessed value. In the case before the court the bench did. accept the assessment of s i m i l a r p r o p e r t i e s as evidence f o r v a l u a t i o n . In other words, 7°Suora. p. 19. 71 ' Norwich Assessment Committee vs. P o i n t e r , 1922, K.B. »+9 ( D i v i s i o n a l C o u r t ) , h71 (Court of Appeal). \ kl u n i f o r m i t y of assessment takes precedence over c o r r e c t v a l u a t i o n . On the other hand, S c r u t t o n , L . J . i n Ladies H o s i e r y and Underwear L t d . vs. West Middlesex Assessment 72 Committee,' was there of the opi n i o n "that the assessing a u t h o r i t y should not s a c r i f i c e c o r rectness i n order to ensure u n i f o r m i t y . " In other words, the duty of the assessor i s to value at the s t a t u t o r y b a s i s of assessment. Bonbright discusses s e v e r a l American cases, but here a l s o there are l i n e s of c o n f l i c t i n g judgements. J 7k H e l l e r s t e i n reaches the same conclus i o n s . Some Canadian courts apply the p r i n c i p l e of u n i -f o r m i t y of assessment, but they hedge t h e i r judgements with statements about c o r r e c t v a l u a t i o n s . Other Canadian courts do the reverse. The Supreme Court of Nova S c o t i a 72 ' Ladies H o s i e r y and Underwear L t d . vs. West Middlesex Assessment Committee, 1932, 2 K.B., 679. 73 '-"Bonbright, op. c i t . , c. X V I I . 7k ' Jerome R. H e l l e r s t e i n , "The Appeal Machinery i n Property Taxation," N a t i o n a l Tax.Association, Proceedings of the 5lst Annual Conference, 1958? PP» *+29-k55» In-cluded i n h i s paper i s an e x c e l l e n t d i s c u s s i o n of st a t u t e and case law i n the United S t a t e s with regard to v a l u a t i o n to the s t a t u t o r y b a s i s of assessment as compared to valua-t i o n to the "tone of the l i s t , " the accepted l e v e l of assessment. .He makes i t q u i t e , c l e a r that the s i t u a t i o n i s a morass of c o n f l i c t i n g p r i n c i p l e s . "Tone of the l i s t " i s a B r i t i s h term synonymous with " l e v e l . o f assessment." 75 i n Re I r v i n g O i l Co. Assessment'^ h e l d that under the Assessment A c t ^ the v a l u a t i o n to be a r r i v e d at i s market value. They then go on to say that the above f i g u r e should be checked by comparing i t with the v a l u a t i o n of other s i m i l a r p r o p e r t i e s . In the A p p e l l a t e D i v i s i o n of the Supreme Court of A l b e r t a , Ewing, J.A. comments on r e a l estate assessments i n a case which i s not d i r e c t l y concerned w i t h municipal 77 assessments. ' A commissioner appointed under the A l b e r t a 78 Succession Duty Act' to a r r i v e at market value f o r the purpose of l e v y i n g succession duty used the mu n i c i p a l assessment. Ewing J.S. s a i d : While the m u n i c i p a l i t y i s bound to assess at the f a i r market value t h i s p r o v i s i o n i s subject to another p r o v i s i o n , v i z . , that the assessment i s uniform w i t h that of s i m i l a r l y s i t u a t e d p r o p e r t i e s . Thus the m u n i c i p a l i t y i s , i n the r e s u l t , compelled only to see that the assessments are uniform . . . . I t i s . . . notorious that the assessment o f t e n bears l i t t l e r e l a t i o n to 'value* of the property.7 9 7^Re I r v i n g O i l Co. Assessment, 19^8, 2 D.L.R., 77^. 7 6Nova S c o t i a , S t a t u t e s , 1938, c. 2. 77 . s i 78 'Re Withvcombe E s t a t e , 19M+, 1 W.W.R, 385 ( A p p e l l a t e D i v i s i o n ) . A l b e r t a , Revised S t a t u t e s . 19^2, c. 57. ^Re Withvcombe E s t a t e , oo. c i t . , p. 397. In order to d e r i v e some standard by which to judge the q u a l i t y of a v a l u a t i o n l i s t i t must be decided whether an assessment r o l l on which a l l p r o p e r t i e s appear at a uniform p r o p o r t i o n of the s t a t u t o r y b a s i s of assessment i s of as good a q u a l i t y as an assessment r o l l on which a l l the p r o p e r t i e s are valued at the s t a t u t o r y b a s i s of assessment. Both r o l l s would produce the e q u i t a b l e d i s t r i b u t i o n of the impact of the tax, but can they, i n theory, be of a d i f f e r e n t q u a l i t y ? The view taken here i s that the second assessment r o l l i s the b e t t e r of the two, and, t h e r e f o r e , the q u a l i t y of the v a l u a t i o n l i s t and a measure of i t s u n i f o r m i t y w i l l be based upon the p r o p o r t i o n of the p r o p e r t i e s on the r o l l which are assessed a t , or w i t h i n a given margin of the s t a t u t o r y b a s i s of assessment. This view i s supported by an i n t e r p r e t a t i o n of S c r u t t o n 1 s 80 8 l remark and by Ewing*s comment. There i s some p o s s i -b i l i t y that S c r u t t o n b e l i e v e d that c o r r e c t assessments would achieve u n i f o r m i t y . Ewing 1s comment can be i n t e r p r e t e d to mean that the court can but give the Supra, p. k2. Supra, p. kk. p r i n c i p l e of u n i f o r m i t y precedence over c o r r e c t v a l u a t i o n when there are so few instances of v a l u a t i o n a t , or near, the s t a t u t o r y b a s i s of assessment. A p e r f e c t l y uniform assessment r o l l w i l l then be a b s o l u t e l y uniform wi t h a l l p r o p e r t i e s on the l i s t valued O p at t h e i r market p r i c e . A p e r f e c t l y uniform assessment r o l l i s an u n r e a l i s t i c o b j e c t i v e . An assessment r o l l , where a l l the p r o p e r t i e s i n the f i s c a l j u r i s d i c t i o n are valued w i t h i n 10 per cent of the b a s i s of assessment, i s a p r a c t i c a b l e g o a l . I f an assessment i s w i t h i n the l i m i t s , then I t i s c o r r e c t . I f i t i s outside the l i m i t s , then i t i s not c o r r e c t . An assessment at 50 per cent above the b a s i s of assessment i s more grievous than an assessment at 15 per cent above the b a s i s of assessment. S t i l l , the f a c t o r c a r r y i n g most weight i n the e v a l u a t i o n of an assessment r o l l should be the p r o p o r t i o n of c o r r e c t assessments. For, except by the most f o r t u i t o u s c o i n c i d e n c e , u n i f o r m i t y i s the r e s u l t of accurate v a l u a t i o n . H» Market A c t i v i t y and Random Sampling Some proponents of assessment-sales r a t i o s t u d i e s assume that i n a given p e r i o d of time market a c t i v i t y Or at the r e q u i r e d s t a t u t o r y p r o p o r t i o n of market value. k5 w i l l generate a random sample of the assessment r o l l . They then make s t a t i s t i c a l i n f e r e n c e s about the r o l l using t h e i r "sample." The Committee on Sales R a t i o Data have made t h i s a s s u m p t i o n . ^ They then suggest that i n order to get a r e l i a b l e measure of the general l e v e l of assessment the sales be s t r a t i f i e d by c l a s s i f i -c a t i o n s of property because the c l a s s i f i c a t i o n s are not Qk p r o p o r t i o n a t e l y represented by s a l e s . The suggestion h a r d l y e x h i b i t s f a i t h i n the assumption. The United S t a t e s Department of Commerce describes a method used to e l i m i n a t e the b i a s , r e l a t i v e to the assessment r o l l , caused because of v a r y i n g turnover r a t e s among d i f f e r e n t 85 c l a s s e s of property. y Because of the heterogeneity of p a r c e l s of r e a l estate no reasonable amount of s t r a t i f i -c a t i o n i s going to remove the e f f e c t of the b i a s . -'Committee on Sales R a t i o Data, Guide f o r Assess-ment-Sales R a t i o S t u d i e s , A Report to the N a t i o n a l A s s o c i a t i o n of Tax A d m i n i s t r a t o r s (Chicago: F e d e r a t i o n of Tax A d m i n i s t r a t o r s , 195*+), pp. 5 and 57-60. 8 1 + I b i d . , p. 2*+. ^United S t a t e s Department of Commerce, Bureau of the Census, Taxable Property Values. Census of Govern-ments, 1962, p. 10. k6 Corbert has constructed an elaborate geometrical model to i l l u s t r a t e that market a c t i v i t y does not generate 86 a random sample. He then adds, "The f i r s t and most obvious s o l u t i o n which might be adopted i n v o l v e s the 87 d e s i g n a t i o n of two separate universes." ' The Committee on State E q u a l i z a t i o n of L o c a l Property Tax Assessments examines the advantages and disadvantages of random sampling of the assessment r o l l as compared to the uses of a market generated sample, and they reach the same 88 c o n c l u s i o n . That the market a c t i v i t y does not generate a random sample of the assessment r o l l cannot be too h e a v i l y emphasized. The d i s t r i b u t i o n of assessment-sales p r i c e r a t i o s so generated i s a separate universe. I t s parameters may be measured d i r e c t l y , or i t s parameters may be i n f e r r e d from a random sample of s a l e s . But from those r e s u l t s alone no i n f e r e n c e , i n the s t a t i s t i c a l sense, can be made about the other universe, the p a r c e l s on the assessment r o l l which d i d not s e l l on the market during the r e l e v a n t p e r i o d . 86 L e s l i e E. Corbert, "An A p p r a i s a l of Sales R a t i o S t u d i e s , " N a t i o n a l Tax A s s o c i a t i o n , Proceedings o£ i h e 5 l s t Annual Conference, 1958, pp. 1 0 7 - l l k . 8 7 I b i d . , p. 116. 88 0p_. c i t . , N a t i o n a l Tax A s s o c i a t i o n , Proceedings of the 51st Annual Conference, 1958, pp. 335-31+1+. **7 I . U n i f o r m i t y of Assessment and E q u a l i z a t i o n Assessment e q u a l i z a t i o n i s the process of assuring c o m p a r a b i l i t y of general l e v e l s of assessment among 89 assessment j u r i s d i c t i o n s . ' An assessment j u r i s d i c t i o n i s the geographical area of the assessment r o l l . Without going i n t o the economic welfare aspects of e q u a l i z a t i o n we can state the two most common purposes of e q u a l i z a t i o n . E q u a l i z a t i o n i s necessary f o r the eq u i t a b l e impact of a property tax where f i s c a l j u r i s d i c t i o n boundaries are not co-terminous with assessment j u r i s d i c t i o n boundaries, or where s e v e r a l assessment j u r i s d i c t i o n s c o n s t i t u t e a l a r g e r f i s c a l j u r i s d i c t i o n . A f i s c a l j u r i s d i c t i o n i s the area over which a p u b l i c body has the power to l e v y r a t e s , or to cause another p u b l i c body to l e v y r a t e s to produce revenue on i t s behalf. M u n i c i p a l i t i e s , c i t i e s , towns, school boards, water boards, c o u n t i e s , townships, s t a t e s , and provinces are f i s c a l j u r i s d i c t i o n s . "The terms " i n t e r n a l e q u a l i z a t i o n , " " v a l u a t i o n e q u a l i z a t i o n , " or " i n t r a - j u r i s d i c t i o n a l e q u a l i z a t i o n " mean " u n i f o r m i t y of assessment" i n some cases and e q u a l i z a t i o n among c l a s s e s of property w i t h i n an assess-ment j u r i s d i c t i o n i n other cases. The terms used f o r " e q u a l i z a t i o n " as i t i s used, i n the tex t are: " e x t e r n a l e q u a l i z a t i o n " ; "apportionment e q u a l i z a t i o n " ; and " i n t e r j u r i s d i c t i o n a l e q u a l i z a t i o n . " 1+8 The second purpose of e q u a l i z a t i o n i s to measure the f i s c a l c a p a c i t y of j u r i s d i c t i o n s where they r e c e i v e g r a n t s , based on t h e i r taxable property value,from superior f i s c a l j u r i s d i c t i o n s , or where t h e i r debt c a p a c i t y i s l i m i t e d by s t a t u t e to some measure of the f i s c a l c a p a c i t y . A more comprehensive l i s t of purposes 90 i s provided by Schwinden. E q u a l i z a t i o n , per se, i s not w i t h i n the subject matter realm of t h i s s t u d y . 9 1 In 1886, Mathew, J . recognized the d i f f e r e n c e between e q u a l i z a t i o n and u n i -f o r m i t y of assessment. 9 2 Commenting on a d j u s t i n g the t o t a l of assessed values on a v a l u a t i o n l i s t without a d j u s t i n g i n d i v i d u a l assessments on the l i s t , so that the t o t a l of i n d i v i d u a l p a r c e l s was not i n agreement w i t h the grand t o t a l he s a i d : " James Schwinden, "Real Property Assessment P o l i c y and P r a c t i c e i n Minnesota",(unpublished Ph.D. d i s s e r t a t i o n , Department of A g r i c u l t u r e , U n i v e r s i t y of Minnesota, 1961), pp. 22-21+. 91 y E q u a l i z a t i o n i s concerned with the l e v e l of assessment, the measure of c e n t r a l tendency, w h i l e u n i f o r m i t y of assessment i s concerned with the measure of d i s p e r s i o n of i n d i v i d u a l assessment-sales r a t i o s about the c e n t r a l tendency. H. vs. The J u s t i c e s of General Assessment Sessions f o r the M e t r o p o l i s , 1886, Q.B. 17, p . 39 r+T" -But the r e s u l t of i n c r e a s i n g t o t a l values l e a v i n g the p a r t i c u l a r assessments untouched would only be to increase the sum to be c o n t r i b u t e d by the p a r i s h (to a M e t r o p o l i t a n government) and to compel the p a r i s h a u t h o r i t i e s to make higher r a t e s . E q u a l i z a t i o n was p r a c t i s e d i n the second quarter of the S i x t e e n t h Century. Under the Poor-Law Rates e s t a b l i s h e d by R i c h a r d I I i n 1388 as amended by Henry V I I i n l>+95 and by Henry V I I I i n 1530-31 and i n 1535-36, the l a t t e r amendment r e q u i r e d t h a t , "the o f f i c e r s of each hundred and corporate town were apparently intended to e x e r c i s e a general s u p e r v i s i o n , and to d i s t r i b u t e the 'overplus 1 of the c o l l e c t i o n s i n wealthy par i s h e s among n93 the poorer parishes."'-' However, since the accuracy of the measure of the l e v e l of assessment on a r o l l depends upon the v a r i a t i o n s among assessment-market value r a t i o of p r o p e r t i e s on the r o l l , the success of an e q u a l i z a t i o n programme i s dependent on the success of achieving u n i f o r m i t y w i t h i n the assess-9^ ment j u r i s d i c t i o n . ^ E d w i n Cannan, The H i s t o r y of L o c a l Rates i n England (2nd ed., London: P. L. King and Son, 1927), pp. 5^-57, esp. p. 57. James Sehwinden, oo. c i t . , pp. 36-l+5« Sehwinden a f t e r an elaborate mathematical deduction concludes that e q u a l i z a t i o n depends on assessment u n i f o r m i t y ; Edwin R. A. Seligman, Essays i n Taxation (6th ed., London: Macmillan and Company L t d . , 1909), p. 26. Seligman says that where Where the e q u a l i z i n g body has, de f a c t o and not j u s t de j u r f i , a u t h o r i t y over the l e v e l of assessment on each v a l u a t i o n l i s t , e q u a l i z a t i o n i s automatic i n the sense that a high measure of u n i f o r m i t y i m p : l i e s that the s t i p u l a t e d l e v e l of assessment i s maintained. Where the e q u a l i z i n g body has no r e s p o n s i b i l i t y 95 f o r the l e v e l of assessment on each v a l u a t i o n l i s t 7 ^ they, i n order to measure and compare f i s c a l c a p a c i t i e s , measure the l e v e l of assessment on each r o l l . Some measure of the assessment-sales r a t i o c e n t r a l tendency i s u s u a l l y used. The Committee on Sales R a t i o Data p r e f e r a assessment r o l l s are in a c c u r a t e "boards of e q u a l i z a t i o n are of no a v a i l " ; N a t i o n a l Tax A s s o c i a t i o n , on. c i t . , Proceedings of the 51st Annual Conference, 1958, pp. 317-321; M. S. Kendrick, P u b l i c Finance, P r i n c i p l e s and Problems (New York: H o u g h t o n - M i f f l i n , 195D, p. 22?. "The p r i n c i p a l need i n the a d m i n i s t r a t i o n of the property tax as an eq u i t a b l e o r i g i n a l assessment of r e a l e s t a t e " ; H. M. Groves, F i n a n c i n g Government (3rd ed.; New York: Henry H o l t and Company, 1950)? P» 59' A f t e r d i s c u s s i n g e q u a l i z a t i o n he s t a t e s , "Much more s e r i o u s , u s u a l l y , are the i n e q u a l i t i e s among taxpayers i n the same d i s t r i c t " ; J . P. Jensen, Government Finance (New York: Lomas, Y. Crow e l l Company, 1937), p. 259. "The only way to eq u a l i z e assessments i s to make them equal i n the f i r s t p l a c e " ; W i l l i a m G. Murray, "Improvement i n Real E s t a t e Taxation through Assessment-Sales S t u d i e s , " N a t i o n a l Tax J o u r n a l . V o l . 5 (1952), p. 86. "Two f a c t s are c l e a r : f i r s t , assessment i n e q u a l i t y i s a major problem; and second, the place to begin i s i n the l o c a l d i s t r i c t — i n t e r n a l e q u a l i z a t i o n should come before e x t e r n a l e q u a l i z a t i o n . •'•'"Valuation l i s t " i s the B r i t i s h term f o r "assess-ment r o l l . " 51 weighted median over the unweighted median, and the 9 6 unweighted median over the a r i t h m e t i c mean. 7 Other 97 bodies p r e f e r the weighted mean.'' The Committee on State E q u a l i z a t i o n of Property Tax Assessments p r e f e r s the weighted mean of a random sample of the assessment 98 r o l l . 7 The accuracy of any measure w i l l depend on the assessment u n i f o r m i t y , because the greater the v a r i a b i l i t y of i n d i v i d u a l members i n a p o p u l a t i o n the l e s s accurate i s the estimate of any measure of c e n t r a l tendency deriv e d from only a p o r t i o n of the universe of assessment-market value r a t i o s . U n i f o r m i t y of assessment i s even more important when the e q u a l i z i n g board uses market generated sales to measure the l e v e l of assessment, because, as pointed out i n the previous s e c t i o n , these s a l e s do not c o n s t i t u t e an unbiased random sample r e q u i r e d f o r s t a t i s t i c a l i n f e r e n c e . I f there i s a high measure of u n i f o r m i t y then the e f f e c t of the sampling b i a s on the estimate w i l l be minimized. 96 7 N a t i o n a l A s s o c i a t i o n of Tax A d m i n i s t r a t o r s , op. c i t . . pp. 2 2 - 2 6 . ^ U n i t e d S t a t e s Bureau of the Census, Taxable Property Values, l o c . c i t . 98 7 N a t i o n a l Tax A s s o c i a t i o n , Proceedings of the 5 l s t Annual Conference, 1958. A measure based on a random sample r e q u i r e s the e q u a l i z i n g body to c a r r y out independent a p p r a i s a l s . 52 J« Summary Even with the narrow approach taken i n the above d i s c u s s i o n , the concepts considered can be brought to bear upon the many problems a s s o c i a t e d with the l e g i s -l a t i o n and the a d m i n i s t r a t i o n of the r e a l property tax. The problem examined i n t h i s paper, measuring q u a l i t y of assessment, i s only one of the a d m i n i s t r a t i v e problems. Therefore i f w i l l be u s e f u l to summarize, from a p o i n t of view to be used i n the f o l l o w i n g chapters, the con-cepts as they bear upon the measurement of assessment u n i f o r m i t y . 1. Assessment o f f i c e r s are i n s t r u c t e d to appraise at market value. 2. Property t r a n s a c t i o n s which occur w i t h i n the condi-t i o n s circumscribed by the d e f i n i t i o n of market value produce market p r i c e s . These market p r i c e s are the u l t i m a t e evidence of market v a l u e , and can be e s t i -mated w i t h i n a reasonable margin of accuracy by a p p r a i s e r s . 3. The opinion here, and t h i s i s a value judgement, i s that a v a l u e r ' s estimate of market value should be w i t h i n ten per cent e i t h e r side of the market p r i c e generated by a sale which occurs under the c o n d i t i o n s circumscribed by the d e f i n i t i o n of market value. Because land and improvements may be assessed and taxed as separate items, the measure of u n i f o r m i t y of assessment i s not a measure of the u n i f o r m i t y of the tax impact d i s t r i b u t i o n . The d e f i n i t i o n of market value i m p l i e s that the f i n a n c i a l arrangements accompanying the t r a n s a c t i o n be those which would not be reasonably expected to d i s t o r t the sale p r i c e from market p r i c e i n the current market. Before they are used as evidence of market va l u e , p r i c e s from sales i n v o l v i n g unusual f i n a n c i n g should be adjusted to remove the e f f e c t s of the unexpected f i n a n c i a l arrangements. An a l t e r n a t i v e would be to r e j e c t e n t i r e l y the sale p r i c e as evidence of market value. This l a t t e r a l t e r n a t i v e i s p r e f e r a b l e when the e f f e c t of the f i n a n c i n g on the sale p r i c e i s not e x a c t l y known. Over and under assessment may a f f e c t the sale p r i c e of a property. Through what mechanism, c a p i t a l i z a -t i o n or "bu d g e t i z i n g , " or to what extent the e f f e c t takes place i s not a l t o g e t h e r evident. But the r e s u l t s of the process at i t s extreme do not a l t e r s i g n i f i c a n t l y the observed assessment-sales p r i c e r a t i o from the assessment-market p r i c e r a t i o . 7. The courts have not made c l e a r whether u n i f o r m i t y of assessment takes precedence over c o r r e c t v a l u a t i o n . The i n t e r p r e t a t i o n here i s that since o v e r a l l c o r r e c t v a l u a t i o n w i l l produce u n i f o r m i t y of assessment, the q u a l i t y of an assessment r o l l should be judged upon the p r o p o r t i o n of assessments w i t h i n ten per cent of market p r i c e . In other words, c o r r e c t v a l u a t i o n r a t h e r than uniform e r r o r i s the c r i t e r i o n by which to evaluate the q u a l i t y of an assessor's v a l u a t i o n l i s t . Yet, some r e c o g n i t i o n must be given to uniform assessment at some other than the s t a t u t o r y l e v e l . 8. I n any given time p e r i o d p r o p e r t i e s traded on the market do not c o n s t i t u t e a random sample of the assessment r o l l . The assessment-market p r i c e r a t i o s so provided, t h e r e f o r e , cannot be used to draw d i r e c t s t a t i s t i c a l i n f e r e n c e s about the assessment r o l l . 9. Correct v a l u a t i o n of a l l p r o p e r t i e s i n a l l assessment j u r i s d i c t i o n s creates e q u a l i z a t i o n . E f f e c t i v e e q u a l i z a t i o n i s s t i l l p o s s i b l e where assessment u n i -f o r m i t y p r e v a i l s w i t h i n each j u r i s d i c t i o n w h i l e the l e v e l of assessment v a r i e s among j u r i s d i c t i o n s . Since u n i f o r m i t y of assessment r e s t s upon c o r r e c t v a l u a t i o n , there can be no s u b s t i t u t e f o r accurate a p p r a i s a l as the go a l of the assessor. CHAPTER I I I UNIFORMITY OF ASSESSMENT IN BRITISH COLUMBIA The stage having been set, an empirical examination of current measures and standards of assessment uniformity can be started. Measures suggested by the Assessment Commissioner of the Province of B r i t i s h Columbia w i l l be used. 1 These measures are similar to those suggested p by the National Association of Tax Administrators, and used elsewhere. 3 But before proceeding to the study of those measures the provisions of the Assessment Province of B r i t i s h Columbia, Appraisal Manual, pp. 308-315. 2 Committee on Sales Ratio Data, Guide f o r Assess-ment-Sales Ratio Studies, Report to National Association of Tax Administrators (Chicago: Federation of Tax Administrators, 195*+), PP« 52-56. ^United States Bureau of the Census, Taxable Property Values, 1962, pp. cjlt. , p. 13; J. Edward Rountrey, "Equalization at Market Value," The Appraisal Journal, Vol. 2k (1956), pp. 219-222; Dudley ¥. Johnson, "A note on Local Administration of the Property Tax: A Case Study of Bethlehem, Pennsylvania," National Tax Journal, Vol. XI (1958), p. 269; Roswell G. Townsend, "Inequalities of Residential Property Taxation i n Metropolitan Boston,'.' National Tax Journal, Vol. IV (195D, P. 36 k . Equalization Act which r e l a t e d i r e c t l y to th i s study w i l l be considered. A. Assessment Equalization Act "There s h a l l be an Assessment Commissioner"^ who i s required, amongst other things, to "give advice and assistance to (Municipal) Assessors f o r the purpose of securing uniformity i n land and improvements within the Province." There can be l i t t l e doubt that the Legis-lature intended to assure uniformity of land assessment and improvement assessment, but as argued i n Chapter II' 7 only the assessed value of the "whole" can be subject of an objective audit. The Gommissioner s h a l l equalize the l e v e l of 8 9 assessment f o r "real-property taxation only" 7 f o r the purpose of school tax, and may d i r e c t an Assessor to equalize assessments among classes of property, or i n d i v i d u a l parcels, within the M u n i c i p a l i t y . 1 0 In other k B r i t i s h Columbia, Revised Statutes, I960, c. 1 8 . I b i d . , sec. 3* 6 I b i d . , sec. 7(b). ?Supra, pp. 27-29. g Assessment,Equalization j££, c i t . , sec. 8 ( 3 ) . 9 I b i d . , sec. 8 ( 5 ) . 1 0 I b i d . , sec. 8(1+). 57 words, e q u a l i z a t i o n i s to be achieved through Province-wide u n i f o r m i t y of assessment, but t h i s u n i f o r m i t y i s , as f a r as the l e g i s l a t i o n under c o n s i d e r a t i o n i s concerned, to apply only f o r r a t e s l e v i e d to r a i s e school revenues. No d i r e c t p r o v i s i o n has been made f o r the P r o v i n c i a l Assessment Commissioner to ensure that m u n i c i p a l general revenue r a t e s w i l l be l e v i e d on uniform assessment, and i t does not appear that the Commissioner can, i n the sense of measuring l e v e l s of assessment, equ a l i z e among c l a s s e s of property, or i n d i v i d u a l p a r c e l s , f o r the general purpose l e v y . But, where a m u n i c i p a l i t y s e l e c t s the same ba s i s of assessment f o r the general purpose l e v y as that used under the school l e v y , any discrepancy between school purpose and general purpose assessments would be 11 untenable. Since i t i s the p r a c t i c e i n B r i t i s h Columbia f o r the m u n i c i p a l i t y to use only the one b a s i s of assessment, the P r o v i n c i a l Assessment Commissioner's de jure a u t h o r i t y over the u n i f o r m i t y of school purpose assessment r o l l becomes de f a c t o a u t h o r i t y over the un i -f o r m i t y of the general revenue assessment r o l l . Some discrepancy would a r i s e because of d i f f e r e n t tax base d e f i n i t i o n s , but these would occur only i n respect of farm p r o p e r t i e s and l a n d l o r d and tenant equipment and f i x t u r e s i n business premises. 58 B. C o l l e c t i n g Sales Data f o r E q u a l i z a t i o n The Assessment Commissioner i s i n t e r e s t e d p r i m a r i l y i n e q u a l i z a t i o n among j u r i s d i c t i o n s , and f o r that purpose he needs assessment-sales r a t i o data to measure l e v e l s of assessment. He and the Assessors can use the same data to measure assessment u n i f o r m i t y . Sales i n f o r m a t i o n i s c o l l e c t e d by the Land R e g i s t r y 12 O f f i c e s . A r e a l Property Transfer Record, Form A.C .2, must be completed with each v o l u n t a r y t r a n s f e r of t i t l e , grant of r i g h t to purchase or assignment of r i g h t to purchase. The questionnaire i s i n c l u d e d as E x h i b i t I i n Appendix A. The A.C. 2 i s then sent to the Assessment Commissioner who examines a l l s a l e s r e j e c t i n g those which 13 he decides w i l l not represent market p r i c e . J The Assessor i s then sent a Request f o r Sales A n a l y s i s Records, Form A.C. 7 ( E x h i b i t I I ) , f o r those s a l e s which are deemed to have taken place w i t h i n the c o n d i t i o n s circumscribed by 1 2 B r i t i s h Columbia, Revised S t a t u t e s , i960, c. 208, "Land R e g i s t r y Act." In B r i t i s h Columbia a modified Torren*s system of land r e g i s t r a t i o n i s used. A l l i n t e r e s t s i n land must be r e g i s t e r e d i n order to p r o t e c t the c l a i m of the owner. 13 -'The Assessor i s a l s o sent a N o t i f i c a t i o n of Forms A.C. 2 Rejected, Form A.C. 19. The s a l e s reported on t h i s form are, at the d i s c r e t i o n of the Assessor, used f o r assessment s a l e s r a t i o s t u d i e s and a p p r a i s a l m a t e r i a l at the l o c a l l e v e l only. The Assessor being c l o s e r to the l o c a l market can more e a s i l y adjust s a l e s p r i c e s to market p r i c e s . 59 iL the d e f i n i t i o n of market value. The Assessor then com-p l e t e s a Sales A n a l y s i s Record, Form A.G.3, sends one copy to the P r o v i n c i a l Assessment Commissioner, and keeps one copy f o r h i s own assessment-sales r a t i o s t u d i e s and a p p r a i s a l records. The Commissioner on the b a s i s of h i s own i n v e s t i g a t i o n , the i n f o r m a t i o n on the A.C .3 Form, or advice from the Assessor may r e j e c t the t r a n s a c t i o n as not being r e p r e s e n t a t i v e of market p r i c e . The Assessor, of course, may make use of the t r a n s a c t i o n f o r h i s own in f o r m a t i o n . The Sales A n a l y s i s Records of t r a n s a c t i o n s not r e j e c t e d are used by the Commissioner f o r the purpose of measuring the l e v e l of assessment f o r the e q u a l i z a t i o n . What measure he uses i s not apparent to us, but i n the A p p r a i s a l Manual he does recommend the median as being I'D the best measure of c e n t r a l tendency. J Measuring the See page l ^ i n Appendix A f o r the d e t a i l s of the e d i t i n g procedure. Appendix A i s an excerpt from the Province of B r i t i s h Columbia, A p p r a i s a l Manual, Assessment E q u a l i z a t i o n A c t, pp. 3 0 1 - 3 2 1 . This p o r t i o n of the Manual i s e n t i t l e d , " S t a t i s t i c a l A n a l y s i s and C o n t r o l . " 15 ' A p p r a i s a l Manual, op. c i t . , p. 3 0 8 . The l e v e l of assessment of a j u r i s d i c t i o n or p a r t i c u l a r c l a s s of p r o p e r t i e s may be measured. In f a c t , the j u r i s d i c t i o n l e v e l may be b u i l t up from the l e v e l s of assessment of any number of c l a s s i f i c a t i o n s of property w i t h i n the j u r i s -d i c t i o n . For an example of t h i s method, see Committee on Sales R a t i o Data, op. c i t . , p. 2k, 60 l e v e l of assessment i s not d i r e c t l y our purpose, but because the accuracy of t h i s measure depends upon 16 u n i f o r m i t y a short comment on t h i s t o p i c w i l l be made. I f the assessments are h i g h l y uniform, and i f the d e v i a t i o n s of assessed value from the median are random, and, t h e r e f o r e , d i s t r i b u t e d evenly on both side s with no b i a s to high or low value p r o p e r t i e s , then the median i s a s a t i s f a c t o r y measure of the l e v e l of assessment. When the above c o n d i t i o n s are not s a t i s f i e d the simple a r i t h -metic mean or the mode are not much more accurate. More accurate measures of the l e v e l of assessment can be made 17 wit h r a t i o estimates, and they do represent the weighted a r i t h m e t i c mean. The weighted mean does account f o r any bi a s between low and h i g h values p r o p e r t i e s . C. Measuring Assessment U n i f o r m i t y While measures of the l e v e l of assessment are based upon c e n t r a l tendency, the measures of u n i f o r m i t y of assessment are based upon d i s p e r s i o n of i n d i v i d u a l r a t i o s about the measure of c e n t r a l tendency. Some measure of the l e v e l of assessment i s of i n d i r e c t i n t e r e s t , because a measure of the u n i f o r m i t y of assessment w i l l depend upon the v a r i a t i o n of the assessment-market p r i c e r a t i o s from t h i s l e v e l . 17 'William G. Cochran, Sampling Techniques (2nd ed.; New York: John Wiley and Sons I n c . , 1963), pp. 15I+-156. 61 The procedures f o l l o w e d by d i f f e r e n t bodies are the same, whi l e the measures of u n i f o r m i t y used vary, but 18 only s l i g h t l y . A f t e r the assessment-sales r a t i o data have been c o l l e c t e d and e d i t e d , the r a t i o s to be used are arranged i n t o a frequency d i s t r i b u t i o n . The c e n t r a l tendency i s then determined, be i t unweighted median, unweighted a r i t h m e t i c mean, or mode. Next the d i s p e r s i o n i s measured. For the purpose of the d i s c u s s i o n , and unless otherwise i n d i c a t e d the designated symbols w i l l be defined as f o l l o w s : M - measure of c e n t r a l tendency s e l e c t e d th f i ~ the frequency of the i i n t e r v a l X j ^ - mid-point of the i t h i n t e r v a l q±- the i t h q u a r t i l e d - the mean d e v i a t i o n s - the standard d e v i a t i o n 1 9 R^- R u s s e l l "Index of Assessment I n e q u a l i t y " 7 q - s e m i - q u a r t i l e d e v i a t i o n i Pi For procedure and methods see A p p r a i s a l Manual, op, c i t . , .pp. 308-321; Committee on Sales R a t i o Data, op. c i t . . pp. 21-29 and 52-56. 19 'Rountrey, op. c i t . , p. 221 has a t t r i b u t e d the development i n 1939 of the "Index" to R u s s e l l ; U n i t e d S t a t e s Bur«eau of the Census, Taxable Property Values, 1962, p. 13. This study uses t h i s measure and has used M as the median. 62 q = (Q3 ~ Qx)/2 R d = 100(d/M) Other measures s i m i l a r to R^ are: R q * 100(q/M) R a = 100(s/M) The f i r s t measure, R d, i s c a l l e d the c o e f f i c i e n t of d i s p e r s i o n and the l a s t measure i s the c o e f f i c i e n t of 20 v a r i a t i o n or the r e l a t i v e standard d e v i a t i o n . The supposed advantage of R d, R^, and R g over d, q, and s r e s p e c t i v e l y as measures of assessment u n i f o r m i t y i s that A p p r a i s a l Manual, op. c i t . , p. 3lh. 63 they a l l o w assessment u n i f o r m i t y comparisons among v a l u a t i o n l i s t s which have d i f f e r e n t average l e v e l s of assessment. The b a s i c p r i n c i p l e i s that an assessment of 150$ market value on a r o l l which has an average assessment l e v e l of 100$ i s r e l a t i v e l y no more grievous than an assessment of 75$ market value on a r o l l where the assessment l e v e l i s 50$ market value. The advantage, unless an assumption about the shape of the frequency d i s t r i b u t i o n s i s made, i s only apparent, and not r e a l . For example, assume M, the simple a r i t h -metic mean, i s the same f o r two d i s t r i b u t i o n s so that the s i z e of i s dependent only upon d. Fi g u r e I shows two d i s t r i b u t i o n s which have the same mean and same mean 21 d e v i a t i o n but are skewed i n opposite d i r e c t i o n s . R^ 22 would be the same f o r both. 21 F r e d e r i c k E. Croxton, Elementary S t a t i s t i c s with A p p l i c a t i o n s i n Medicine and the B i o l o g i c a l Sciences (New York: Dover P u b l i c a t i o n s , I nc., 1959), p. 7%~.The r e l a t i v e p o s i t i o n s of the mean, mode, and median of a skewed frequency d i s t r i b u t i o n are presented. 22 In t h i s example, though the d i s t r i b u t i o n of the tax impacts i s d i f f e r e n t as between A and B, no case can be made f o r a d i f f e r e n c e i n q u a l i t y as between A and B. An examination of Dr. R u s s e l l * s r e s u l t s i n Appendix B w i l l i n d i c a t e the weakness of t h i s type of measure of assessment u n i f o r m i t y . 6k frequency mean FIGURE 1 TWO SKEWED DISTRIBUTIONS The d i s t r i b u t i o n s i l l u s t r a t e d i n F i g u r e 1 w i l l a l s o have equal standard d e v i a t i o n s and equal semi-q u a r t i l e d e v i a t i o n s . F i g u r e 2 presents the nature of two d i s t r i b u t i o n s which are symmetrical and which have the same mode, mean, and median. Only by remote c o i n -cidence w i l l any two of the three measures of d i s p e r s i o n be equal f o r both d i s t r i b u t i o n s . But, i t i s more l i k e l y that any one measure of d i s p e r s i o n w i l l give the same value to the "Index of Assessment I n e q u a l i t y " f o r each d i s t r i b u t i o n . 65 frequency-mean FIGURE 2 TWO SYMMETRICAL DISTRIBUTIONS Without foundation R d, R^, and R g are used as measures of assessment u n i f o r m i t y . In other words, i n order to measure u n i f o r m i t y of assessment and compare the r e s u l t s with a s e l e c t e d minimum standard of u n i f o r m i t y or with the r e s u l t s from previous years of from other j u r i s d i c t i o n s , i t i s necessary to make some assumption about the underlying form of the frequency d i s t r i b u t i o n of assessment-sales p r i c e r a t i o s . F u r t h e r , f o r the a n a l y s i s to be v a l i d the a c t u a l d i s t r i b u t i o n of assessment-s a l e s r a t i o must f i t the assumed d i s t r i b u t i o n on which the measure of u n i f o r m i t y i s based. When n e i t h e r of these c o n d i t i o n s i s s a t i s f i e d R d, R , or R g, or a l t e r n a t i v e l y 66 d, q, or s, may be c a l c u l a t e d but the r e s u l t s have l i t t l e p r a c t i c a l i n t e r p r e t a t i o n . D. The Normality Assumption Though Dr. R u s s e l l d i d not make h i s assumptions e x p l i c i t he d i d suggest i n order to more a c c u r a t e l y judge the u n i f o r m i t y of assessment that the percentage of the sample assessment-sale p r i c e r a t i o s w i t h i n p l u s and minus 20 per cent of the average r a t i o from the average r a t i o be considered along with the "Index of Assessment I n e q u a l i t y . " 2 3 Under the normal d i s t r i b u t i o n the mean and the c o e f f i c i e n t of d i s p e r s i o n f u l l y s p e c i f y the p r o p o r t i o n of the d i s t r i b u t i o n w i t h i n the range of plu s or minus 20 per cent of the mean from the mean. I t i s evident then that R u s s e l l d i d not assume that the d i s t r i b u t i o n of assessment-sale p r i c e r a t i o s would be s t a t i s t i c a l l y normal. I f assessment-market p r i c e r a t i o s were normally d i s t r i b u t e d i t would not be p o s s i b l e f o r an assessment r o l l w i t h a low c o e f f i c i e n t of d i s p e r s i o n to have a smaller p r o p o r t i o n of r a t i o s w i t h i n p l u s and minus 20 per cent of the average r a t i o from the average r a t i o than an assessment r o l l with a higher c o e f f i c i e n t -\John H. R u s s e l l , " I n e q u a l i t y of Real E s t a t e Assessments w i t h i n P o l i t i c a l S u b d i v i s i o n s , " The Common-wealth Magazine ( V i r g i n i a ) , 1939, p. 18. of d i s p e r s i o n . Therefore, under n o r m a l i t y c o n d i t i o n s only the "Index of Assessment I n e q u a l i t y " i s r e q u i r e d to observe the u n i f o r m i t y of assessment. That d i s t r i b u t i o n s of assessment-market p r i c e r a t i o s are not normal i s evidenced by R u s s e l l ' s r e s u l t s 2k i n c l u d e d i n Appendix B. These r e s u l t s are i n c o n s i s t e n t w i t h a normal d i s t r i b u t i o n of assessment-market p r i c e r a t i o s . I t i s eminently c l e a r that B r. R u s s e l l d i d not assume the normality of d i s t r i b u t i o n s of assessment-sale p r i c e r a t i o s , but as w i l l be e x h i b i t e d i n the f o l l o w i n g paragraphs t h i s assumption has been made i n h i s name by other workers i n the f i e l d . The N a t i o n a l A s s o c i a t i o n of Tax A d m i n i s t r a t o r s 2 5 ' have not made e x p l i c i t or i m p l i c i t t h e i r assumptions. The Province of B r i t i s h Columbia Assessment Com-missioner has not made h i s assumptions e x p l i c i t , but there are a few passages i n the A p p r a i s a l Manual which i n d i c a t e h i s assumptions. Whether or not he made the assumptions r e a l i z i n g t h e i r i m p l i c a t i o n s i s not apparent. l i i i d . , p. 18. Committee on Sales R a t i o Data, op. c i t . 68 The Commissioner p r e f e r s the use of the standard ?6 d e v i a t i o n over that of the mean d e v i a t i o n . Since R u s s e l l ' s index, the c o e f f i c i e n t of d i s p e r s i o n , i s expressed i n terms of the mean d e v i a t i o n , the Assessment Commissioner, i n order to r e l a t e the c o e f f i c i e n t of v a r i a t i o n to R u s s e l l ' s suggested maximum value f o r h i s index, expressed the q u a n t i t a t i v e r e l a t i o n s between mean d e v i a t i o n and standard d e v i a t i o n , and between the co-e f f i c i e n t of d i s p e r s i o n and c o e f f i c i e n t of v a r i a t i o n . Comparisons of the r e l a t i o n s h i p s between the va r i o u s absolute measures are as f o l l o w s : — Range: Coarse c a l c u l a t i o n only; no d e f i n i t e r e l a t i o n s h i p to other measures. Q u a r t i l e d e v i a t i o n 0.67k5 standard d e v i a t i o n 0.8k53 average d e v i a t i o n . Average (mean) d e v i a t i o n 0.7979 standard d e v i a t i o n . Standard d e v i a t i o n 1.2533 average d e v i a t i o n 1A826 q u a r t i l e d e v i a t i o n . The c o e f f i c i e n t of d i s p e r s i o n i s the "index of assessment i n e q u a l i t y " r e f e r r e d to by.the l a t e Dr. John H. R u s s e l l , former D i r e c t o r of Research, V i r g i n i a Department of Taxation. H i s recommenda-t i o n was that a c o e f f i c i e n t of d i s p e r s i o n of "20 should be considered a goal d e s i r a b l e of achieve-ment and reasonably a t t a i n a b l e , " and that anything below t h i s i s to be considered as an e x c e l l e n t degree of e q u a l i z a t i o n or u n i f o r m i t y . Conversely, he sta t e d "an index as high as k 5 should be judged cause f o r gravest concern."27 Accepting Dr. R u s s e l l ' s a t t a i n a b l e l i m i t of 20, and knowing that the average d e v i a t i o n on which i t A p p r a i s a l Manual, op. c i t . . p. 312. 27lbid. , p. 31**. i s c a l c u l a t e d i s 0.7979 standard d e v i a t i o n , then the corresponding c o e f f i c i e n t of v a r i a t i o n would be 20 7 0.7979 or 25 per cent. Or, expressed another way, as the standard d e v i a t i o n i s 1.25 average d e v i a t i o n s , then the c o e f f i c i e n t of v a r i a t i o n w i l l be 1.25 times the equivalent c o e f f i c i e n t of d i s p e r s i o n . Thus a c o e f f i c i e n t of v a r i a t i o n of 25 should be " d e s i r a b l e of achieve-ment and reasonably a t t a i n a b l e " and a c o e f f i c i e n t of v a r i a t i o n of 56 should be "cause f o r the gravest concern." With an average assessment l e v e l of 50 per cent a c o e f f i c i e n t of v a r i a t i o n of 25 would mean that about 67 per cent of the items should have r a t i o s between 37»5 and 62.5 per cent.28 F i r s t , i n order to de r i v e the r e l a t i o n between s and d the f u n c t i o n of the frequency d i s t r i b u t i o n must be known. Second, i n the f i r s t quotation above he says that the standard d e v i a t i o n equals 1.2533 times the average d e v i a t i o n . Demonstrated below i s the d e r i v a t i o n of that e q u a l i t y . The c a l c u l a t i o n s are based on an assumption of a normal d i s t r i b u t i o n . For convenience only the normal d i s t r i b u t i o n used i s centred at the o r i g i n . This causes the mean to equal zero. A t r a n s p o s i t i o n of coordinates does not a l t e r the fundamental r e l a t i o n s between the parameters of a f u n c t i o n , but i t o f t e n can he l p emphasize the r e s u l t s of the c a l c u l a t i o n s , r a t h e r than the c a l c u l a t i o n s being emphasized. I b i d . , p. 31*t. 70 For any frequency d i s t r i b u t i o n n d = i = 1 f i ( X i - M) n ~ f, i i =1 where M i s the mean. M i s to equal zero. I f N = t o t a l number of elements i n a l l i n t e r v a l s , then n f i = N i = 1 Now w r i t e f i x i 5^ ( V 1 0 * ! d = i = 1 = i = 1 N n d = <T^ p i x i ' w h e r e i = 1 P ± = f ± / N 71 Now, p.^ ( i = 1, 2, ... n) i s the r e l a t i v e frequency-d i s t r i b u t i o n and d, the mean d e v i a t i o n , i s expressed i n terms of p^ and x^. p^ ( i = 1, 2, ... n) i s a l s o c a l l e d the d e n s i t y f u n c t i o n . The normal d i s t r i b u t i o n i s a d e n s i t y f u n c t i o n of a given form where n tends to i n f i n i t y and the width of the c l a s s i n t e r v a l tends to zero forming a continuous d i s t r i b u t i o n as compared to the d i s c r e t e frequency f u n c t i o n . Therefore, d^, the mean d e v i a t i o n f o r the centred normal d i s t r i b u t i o n i s The normal d e n s i t y f u n c t i o n i s y = where V ' i s the standard d e v i a t i o n . i . e = xydx = j xydx + - oO - o O ^•(x) i s symmetrical about y(0) oo o xydx o since y 72 J~l7= 1.7721*53851 j i r = i . i f i ^ 2 i ^ = i.2533137^9 Therefore, l.2533<% T h i r d , the Assessment Commissioner says, "With an average assessment l e v e l of 50 per cent a c o e f f i c i e n t of v a r i a t i o n of 25 would mean that about 67 per cent of 29 the items should have r a t i o s between 37-5 and 62.5 per cent." In other words R^ =25. Therefore, C'/M = 0.25 M = 0.5 and = 0.125. ^ A p p r a i s a l Manual, l o c . c i t . 73 The Commissioner i n e f f e c t has s a i d that .375 0.125 1 •e -<x - .5 ) 2 /2( . i25) 2 d x i 0 . 6 7 I f the i n t e g r a l of the normal d e n s i t y f u n c t i o n i s evaluated between the i n d i c a t e d l i m i t s the r e s u l t would again demon-s t r a t e that the normal d i s t r i b u t i o n has been assumed. A much more simple approach i s to standardize the normal d i s t r i b u t i o n (M = .5) ^ - .125) to the normal d i s t r i b u t i o n (M = 0.0, 1.0), then determine from already tabulated values of the i n t e g r a l whether or not 67 per cent of the r a t i o s would l i e between 37*5 and 62.5 per cent of market value. Now, f i n d i n t a b l e s of the standardized. Normal D i s t r i b u t i o n the values of z , centred about the mean, and between which 67 per cent of the area l i e s . These values are minus 0.98k to plus .90%. Let z be the standardized normal d e v i a t e . x - M z x - .5 .125 x = .125 + 0.5 7h Therefore the l i m i t s between which 67 per cent of the assessment-sales r a t i o s w i l l l i e are 37«7 per cent and 62.3 per cent. There can be l i t t l e doubt that the assumption underlying the use, i n B r i t i s h Columbia at l e a s t , of the c o e f f i c i e n t of v a r i a t i o n f o r measuring u n i f o r m i t y of assessment i s that the assessment-sales r a t i o s are d i s t r i b u t e d normally. I s t h i s assumption j u s t i f i e d ? Deductive reasoning would not lead to an a f f i r m a t i v e answer. An assessment-s a l e s r a t i o cannot be l e s s than z e r o , but there i s no upper l i m i t . Therefore, t h i s observation would i n d i c a t e a d i s t r i b u t i o n skewed to the r i g h t . A l s o , the " c a p i t a l i -z a t i o n e f f e c t " or "budget e f f e c t " ^ 0 might tend to skew the d i s t r i b u t i o n to the r i g h t . Where p r o p e r t i e s are under-assessed there i s not l i k e l y to be as l a r g e a divergence between market p r i c e with i n c o r r e c t assessment and market p r i c e with c o r r e c t assessment as where p r o p e r t i e s are over-assessed. I t i s reasonable to suppose that purchasers of under-assessed p r o p e r t i e s , i n the expecta-t i o n s of increased f u t u r e assessments, w i l l not "budgetize" the tax saving, while purchasers of over-assessed p r o p e r t i e s Supra, pp. 35-l+l• 75 i n the expectation of not being s u c c e s s f u l i n reducing the f u t u r e assessment or the cost i n v o l v e d i n doing so, w i l l "budgetize" a l a r g e r p o r t i o n of the a d d i t i o n a l taxes. As i l l u s t r a t e d i n Chapter I I 3 1 the " c a p i t a l i z a t i o n e f f e c t " and "budget e f f e c t " tend to accentuate the d i f f e r e n c e , as measured by the r a t i o of assessed value to sa l e p r i c e , between the assessment l e v e l of an i n c o r r e c t l y assessed property and the average l e v e l of assessment. When the property i s over-assessed, the assessment-sale p r i c e r a t i o i s greater than the assessment-market p r i c e r a t i o when the property i s c o r r e c t l y assessed; when the property i s under-assessed, the assessment-market p r i c e r a t i o i s smaller than the assessment-market r a t i o when the property i s c o r r e c t l y assessed. Because i t might be expected that the "budget e f f e c t " occurs more o f t e n and to a greater extent on over-assessed p r o p e r t i e s , the net r e s u l t should be a d i s t r i b u t i o n of assessment-sales r a t i o s skewed to the r i g h t . Assuming f o r the moment that the d i s t r i b u t i o n of assessment-market p r i c e r a t i o s i s normal then the r e s u l t a n t e f f e c t on the d i s t r i b u t i o n of assessment-sales p r i c e r a t i o s i s as i l l u s t r a t e d i n Fi g u r e 3. Supra, pp. 38, ho. 76 assessment -market p r i c e r a t i o s c o r r e c t assessment assessment-sa l e p r i c e r a t i o s i n c o r r e c t assessment FIGURE 3 "BUDGET EFFECT" ON DISTRIBUTION OF ASSESSMENT-SALE PRICE RATIOS E. Testing the Normality Assumption The normality assumption can be tested e m p i r i c a l l y . This w i l l be done below. The method and procedure are o u t l i n e d here. November 1, 196 L to June 30, 1965 s i n g l e and two-family r e s i d e n t i a l property sale p r i c e s were c o l l e c t e d from the Vancouver M u l t i p l e L i s t i n g S e r v i c e r e c o r d s . 3 2 Only C i t y of Vancouver s a l e s were used. The ^ The t r a n s f e r data c o l l e c t e d by the Assessment Commissioner i s c o n f i d e n t i a l and, t h e r e f o r e , not a v a i l a b l e f o r t h i s study. The per i o d November 1, 196^ to June 30, 1965 was se l e c t e d because the assessment r o l l i s "brought i n " e a r l y i n the calendar year. The sales i n the designated p e r i o d should be " t i m e l y " i n respect of the assessed value. The 77 s a l e s were e d i t e d to remove any t r a n s a c t i o n s which d i d not occur under c o n d i t i o n s of market value. Because the p r o p e r t i e s were s o l d by an agent the t r a n s a c t i o n s w i l l a u t o m a t i c a l l y be at arms-length. Therefore, the e d i t i n g process was not r e q u i r e d to r e j e c t the non-b_ona fjLde and oo non-voluntary t r a n s f e r s from the study. -' The assessed values obtained from the C i t y of Vancouver 1965 v a l u a t i o n r a t i o of l i s t i n g p r i c e to s e l l i n g p r i c e may be used as an i n d i c a t o r of market p r i c e trend. For the s a l e s o c c u r r i n g during each month of the study p e r i o d the r a t i o of t o t a l l i s t i n g p r i c e to t o t a l s a l e s p r i c e was c a l c u l a t e d before the data were e d i t e d . The evidence below i s not c o n c l u s i v e , but the l a c k of a sustained change i n the r a t i o does support the contention that market p r i c e s were not changing r a p i d l y over the p e r i o d November 1, 196*+ to June 30, 1965. T o t a l of L i s t i n g P r i c e / Month of Sale T o t a l of Sales November 1.069 December 1.071 January 1.062 February 1.060 March 1.067 A p r i l 1.072 May 1.071 June 1.071 P r i c e -)-)The f o l l o w i n g t r a n s f e r s were not i n c l u d e d i n m u l t i p l e l i s t i n g s a l e s . Changes of executors or t r u s t e e s . Transmissions by death or f o r e c l o s u r e . Q u i t - c l a i m s . Transfers to a b e n e f i c i a r y under a w i l l . T ransfers between r e l a t i v e s or a s s o c i a t e d c o r p o r a t i o n s . Transfers to any church, lodge, school or c h a r i t a b l e , benevolent, f r a t e r n a l , or government o r g a n i z a t i o n . 78 l i s t . The assessment sales r a t i o s were c a l c u l a t e d and arranged i n t o a frequency d i s t r i b u t i o n . J Then the frequency d i s t r i b u t i o n was tested f o r " g o o d n e s s - o f - f i t " to a normal d i s t r i b u t i o n . The Chi Square D i s t r i b u t i o n w i l l be used f o r the t e s t . ' Transfers the circumstances of which are unusual and as a r e s u l t render the t r a n s a c t i o n useless f o r s a l e s a n a l y s i s . Transfers agreed upon some time p r i o r to the year i n which they were recorded. The f o l l o w i n g s a l es were e d i t e d out of the study. Transfers i n v o l v i n g unusual f i n a n c i a l p r o v i s i o n s — c a s h to vendor l e s s 25$ of sa l e p r i c e or take over mortgage l e s s than k0% of sale p r i c e . Transfers i n v o l v i n g trades. ^The c l a s s i n t e r v a l s of the " f i t t e d " frequency d i s t r i b u t i o n are nine i n number and have the f o l l o w i n g l i m i t s and mid-points. C l a s s I n t e r v a l - .13085 0.13085 - .23085 0.23085 - .33085 0.33085 - .$3085 0.^3085 - .53085 0.53085 - .63085 0.63085 - .73085 0.73085 - .83085 0.83085 + Mid-point .0651*25 .18085 .28085 .38085 .58085 .58085 .68085 .78085 .905 K25 Since the assessed values are recorded on the r o l l to the nearest d o l l a r , accuracy i n the c a l c u l a t i o n of the assessment-sale p r i c e r a t i o should be to the nearest ten thousandth. 35 -"^The o b j e c t i v e of t h i s paper i s not to measure the u n i f o r m i t y of a p a r t i c u l a r assessment r o l l . This r e s u l t i s merely a by-product of the a n a l y s i s of a s t a t i s t i c a l measure of u n i f o r m i t y and q u a l i t y of assessment. Therefore, the s e l e c t i o n of the data i s s u b s i d i a r y to the observation of the conclusions about u n i f o r m i t y of assessment which are drawn from the a p p l i c a t i o n of a s t a t i s t i c a l method to the data. 79 Sampling theory and the t h e o r e t i c a l background to the use of X * d e n s i t y f u n c t i o n are not of d i r e c t i n t e r e s t i n the study, but i n order that the r e s u l t s may be p r o p e r l y i n t e r p r e t e d some aspects of these t o p i c s should be con-s i d e r e d here. Included i n the s t a t i s t i c a l a n a l y s i s w i l l be a l l sale t r a n s a c t i o n s which have r e s u l t e d d i r e c t l y i n a market p r i c e during the study p e r i o d . 3 ^ The assessment-s a l e s p r i c e r a t i o s w i l l be treated as a sample from an i n f i n i t e universe that would be generated i f the f o r c e s that gave r i s e to the sample were to operate i n d e f i n i t e l y without change i n cha r a c t e r . There are three a l t e r n a t i v e approaches to t h i s p o i n t . F i r s t , a l l of the assessment-market p r i c e r a t i o s c o uld be treated as a f i n i t e u niverse, and a l l of the a v a i l a b l e r a t i o s could be included i n the a n a l y s i s . No J There are i n e f f e c t three universes of assessed value-market p r i c e r a t i o s . One universe c o n s i s t s of assessed value-market value r a t i o s of p r o p e r t i e s not s o l d on m u l t i p l e l i s t i n g . They may or may not have sold during the study p e r i o d . The second universe c o n s i s t s of assessed value-market p r i c e r a t i o s of p r o p e r t i e s that s o l d on m u l t i p l e l i s t i n g during the study period but which were e d i t e d out of the sample. The t h i r d universe c o n s i s t s of assessed value-market p r i c e r a t i o s which are generated under current market c o n d i t i o n s through m u l t i p l e l i s t i n g s a l e s . A sample from the l a s t universe i s not r e p r e s e n t a t i v e of e i t h e r the f i r s t or second p o p u l a t i o n . Forces such as l o c a t i o n or type of property which generate the t h i r d universe may be d i f f e r e n t from those which generate the f i r s t or second universe. 80 sampling would be i n v o l v e d . Second, using the i n f i n i t e p o p u l a t i o n model, a random sample could be drawn from the a v a i l a b l e assessment-sales p r i c e r a t i o s . T h i r d , the a v a i l a b l e r a t i o s could be viewed as a f i n i t e p o p u l a t i o n from which a random sample i s then drawn. The t h i r d a l t e r n a t i v e i s the sampling model analogous to the f i r s t a l t e r n a t i v e . The second and t h i r d a l t e r n a t i v e s are u s e f u l where the volume of data a v a i l a b l e i s so great that a greater degree of accuracy can be expected from the more d e t a i l e d a t t e n t i o n given to a sample than could be given to a l l the data a v a i l a b l e . f . denotes the observed frequency i n a c l a s s J i n t e r v a l F. denotes the expected or t h e o r e t i c a l frequency i n a c l a s s i n t e r v a l j = 1, 2, ... n-1, n. I f : F. > 5 f o r a l l j n = 1 f . i n the remainder of t h i s paper. then the non-parametric s t a t i s t i c 81 w i l l be d i s t r i b u t e d as ~X 2 with m degrees of f r e e d o m . ^ The number of degrees of freedom i s equal to the number of squared normal v a r i a t e s i n the sum d e r i v i n g the value 2 X l e s s the number k, of l i n e a r c o n s t r a i n t s on that sum. In other words m = n - k. In t h i s study n = 9 and k = 3« There are nine c l a s s i n t e r v a l and three l i n e a r c o n s t r a i n t s . In order to " f i t " •^A random v a r i a b l e has a ' X d i s t r i b u t i o n with m degrees of freedom i f i t has the same d i s t r i b u t i o n as the sum of the squares of m independent standardized normal v a r i a t e s . Where f ^ v a r i e s randomly from F.., as i t should i n an i d e a l l y uniform assessment r o l l : ( f i - F.) i s a normal v a r i a t e ( f i - F.)/F. i s a standardized normal v a r i a t e ( f ^ - F . ) v F . i s a squared standardized normal J J J v a r i a t e . „ Therefore, the p r o b a b i l i t y i n t e g r a l of the "X d i s t r i b u -t i o n can be used to measure the p r o b a b i l i t y l e v e l s of X 2. The d e n s i t y f u n c t i o n of the "X d i s t r i b u t i o n i s given by g ( X , m) = 1 . {y*) 2 € 2 m / 2T(m / 2 ) Values of the p r o b a b i l i t y i n t e g r a l are tabulated. 82 the data to a normal d i s t r i b u t i o n M and. s are c a l c u l a t e d from the data, and . must equal ! ^ F . , therefore J J decreasing the degrees of freedom by three from nine. In order to apply t h i s " g o o d n e s s - o f - f i t " t e s t , F. i s determined from the p r o b a b i l i t y i n t e g r a l of the normal d e n s i t y f u n c t i o n with mean,^/£<_, equal to the estimator M and the standard d e v i a t i o n , *\ , equal to "5* , i t s unbiased estimator. 3" 3 3 P r e v i o u s l y , three a l t e r n a t i v e s to the sampling model used i n t h i s paper were discussed. Since i n the f i r s t a l t e r n a t i v e no sampling i s i n v o l v e d M N = s. Where sampling i s used, such as i n the second and t h i r d a l t e r n a t i v e s , as w e l l as i n the model used i n the t e x t : M i s an e s t i m a t i o n of^CL. ^ ^ s i s used to de r i v e tne es t i m a t o r , 6 , of \ A l s o , i n sampling models where M i s the estimator of _^-cc , the l i m i t s w i t h i n whichyC< w i l l w i t h a given p r o b a b i l i t y be found are of i n t e r e s t . M, the sample mean, can be t r e a t e d as a s i n g l e member of an i n f i n i t e p o p u lation of sample means. The d i s t r i b u t i o n of sample means w i l l have a mean equal toyu. , estimated by M, and a standard d e v i a t i o n "V^j = T^/ JlTfj, estimated by £ m = J2~f ^ i n the case of a l t e r n a t i v e two and i n the sampling model of t h i s paper; and i n the case of a l t e r n a t i v e t h r e e , b y ^ -N i s the number of members i n the f i n i t e p o p u l a t i o n . ^ m and s f f l are often defined r e s p e c t i v e l y as the standard e r r o r of the mean and the estimated standard e r r o r of the mean. Because the d i s t r i b u t i o n of sample means i s often normally d i s t r i b u t e d , even when the p o p u l a t i o n d i s t r i -b u t i o n i s not normal, £ i s used to c a l c u l a t e the confidence of l i m i t s f o r M as an estimator ofyu_ . A S = s - ( x ^ K ) 2 / 2 ^ 1 € 2 .dx where °^ i s the upper l i m i t of c l a s s i n t e r v a l j . In p r a c t i c e the p r o b a b i l i t y i n t e g r a l s of the standardized normal d i s t r i b u t i o n are used. I f z represents the standardized normal v a r i a t e , then = x V and dz Therefore, F^ = 1_ dx I f J" Z 2 /2 2 I— c x 1 . € .d 2 Values of 3-1 z 2 .e /2 .d:z are tabulated and presented i n Appendix C. 8*+ / I f A. -Z2 /2 i . e . d z - O O then where A^ and A^_^ are read from the prepared t a b l e s of the standardized normal d i s t r i b u t i o n . 39 39 Some t a b l e s c o n t a i n e v a l u a t i o n s of Z 2 /2 dz where Since the standardized normal d i s t r i b u t i o n i s symmetrical A j = B j + when ° ^ A. = -B. + 0.5 when . < ^ u -J J J = when • = 85 F. The Universe of Assessment-Market P r i c e R a t i o s The parameters of a p a r t i c u l a r p o p u l a t i o n are to be estimated. They are most oft e n used to measure u n i -f o r m i t y of assessment, but here they are a l s o used to t e s t the "g o o d n e s s - o f - f i t " to a normal d i s t r i b u t i o n . This universe has been described i n general terms as the assessment-market p r i c e r a t i o s which would be generated by m u l t i p l e l i s t i n g s a l e s of s i n g l e and two f a m i l y r e s i d e n t i a l d w e l l i n g s i n the C i t y of Vancouver i f the same f o r c e s which produced the sample i n the s e l e c t e d time p e r i o d were to continue i n d e f i n i t e l y without change i n c haracter. A more d e t a i l e d o p e r a t i o n a l d e s c r i p t i o n i s r e q u i r e d at t h i s p o i n t . F i r s t , the m u l t i p l e l i s t i n g s a l e s are concentrated more i n the lower p r i c e ranges than are a l l of the Vancouver C i t y r e s i d e n t i a l property sales o c c u r r i n g during the same time p e r i o d . Because, as compared to the higher p r i c e ranges, there are more houses i n the lower ranges of value; there i s i n the lower ranges of value l e s s v a r i a t i o n among houses of amenities produced; and there i s a higher turnover r a t e i n the lower ranges of v a l u e , the assessment-sales p r i c e r a t i o s generated by the m u l t i p l e l i s t i n g s a l e s can be expected to be more uniform than the r a t i o s generated by a l l r e s i d e n t i a l s a l e s i n Vancouver C i t y . 86 There are a number of m u l t i p l e l i s t i n g s a l e s which i n v o l v e bonused vendor f i n a n c i n g . There are, a l s o , a number of sales which i n v o l v e b e t t e r than normal f i n a n c i a l arrangements i n that a higher than expected loan to value r a t i o i s accompanied by the i n t e r e s t r a t e expected on a prime or near prime f i r s t mortgage. Because these c h a r a c t e r i s t i c s of a t r a n s a c t i o n may i n f l a t e the sale p r i c e above the market p r i c e , the assessment-sales p r i c e r a t i o s r e s u l t i n g from these t r a n s a c t i o n s w i l l be e l i m i n a t e d from the sample and universe. Only r a r e l y does a purchaser take over from the vendor unfavourable f i n a n c i a l arrange-ments which would tend to e f f e c t a sale p r i c e somewhat l e s s than market va l u e , therefore there w i l l be no f u r t h e r concern w i t h t h i s improbable p o s s i b i l i t y . A two step procedure was used to e l i m i n a t e from the sample assessment-sale p r i c e r a t i o s d e f l a t e d below the assessment-market p r i c e r a t i o s . F i r s t , a l l s a l e s i n which the cash payment to the vendor was l e s s than 25 per Vendor f i n a n c i n g i n v o l v e d a grant of a f i r s t or second mortgage from the purchaser to the vendor or a grant of a r i g h t to purchase or a sub-right to purchase from the vendor to the purchaser. Vendor f i n a n c i n g would o f f e r no d i f f i c u l t y i f the nominal y i e l d on the debt were equal to the e f f e c t i v e y i e l d at which the vendor could s e l l the paper. But almost without exception the p r a c t i c e has been to finance nominally at or near the y i e l d r a t e at which prime f i r s t mortgage loans are a v a i l a b l e . 87 cent of the sale p r i c e were e l i m i n a t e d . Then, f o r a l l the assessed v a l u e - s a l e p r i c e r a t i o s remaining the underlying s ales c o n t r a c t was examined. The f i r s t step i s discussed f i r s t . E l i m i n a t i o n based on the 25 per cent r u l e of thumb appears to be a r b i t r a r y , but the r u l e does have a good foundation. hi U n t i l very r e c e n t l y , and with a few exceptions, i n s t i t u t i o n a l f i r s t mortgage loans were l i m i t e d to 66 2/3 per cent of the lending value of the property. Therefore, a cash payment of l e s s than 33 1/3 per cent of s a l e p r i c e r e f l e c t s vendor f i n a n c i n g i n one of two ways. F i r s t , the s e l l e r may f i n a n c e a l l of the purchase p r i c e save only the downpayment of l e s s than 33 1/3 per cent. This i s e q uivalent to an i n s t i t u t i o n a l f i r s t mortgage loan Loans under the N a t i o n a l Housing Act may be as 95 Per cent of the lending value. "Package" 75 per cent loan-value r a t i o f i r s t mortgage loans have f o r a few years been a v a i l a b l e on superior personal and r e a l covenants. The f i r s t 66 2/3 per cent i s advanced by a mortgage lending i n s t i t u t i o n and the top 18 1/3 per cent i s advanced by a consumer f i n a n c e type of i n s t i t u t i o n . These l a t t e r s m a ll loan companies are not r e s t r i c t e d on mortgage loans i n the same way that are the other i n s t i t u t i o n s . American l i f e insurance companies may, i f t h e i r l e g a l reserve i n Canada exceeds t h e i r Canadian p o l i c y r e s e r v e , I n v e s t part of the excess i n that p o r t i o n of a mortgage loan which exceeds the 66 2/3 per cent l i m i t . The a c t u a l maximum loan to value r a t i o i s l i m i t e d by the s t a t e of i n c o r p o r a t i o n . 88 plus a vendor second mortgage. Second, the purchaser may take over payments on e x i s t i n g debt as part of the purchase p r i c e . Where the t o t a l of the e x i s t i n g debt plus the downpayment i s l e s s than the sale p r i c e there i s vendor f i n a n c i n g . Seldom w i l l the t o t a l of e x i s t i n g debt p l u s vendor f i n a n c i n g be l e s s than the 66 2/3 per cent a v a i l a b l e from a mortgage lending i n s t i t u t i o n . Therefore, the loan from the vendor i s , i n at l e a s t p a r t , e q u i v a l e n t to j u n i o r f i n a n c i n g . The 25 per cent r u l e may r e j e c t from the sample assessment-sale p r i c e r a t i o s which r e s u l t from t r a n s a c t i o n s i n which the vendor f i n a n c i n g i s w r i t t e n at the e f f e c t i v e market y i e l d f o r the j u n i o r s e c u r i t y . As t h i s i s seldom done, few, i f any of these r a t i o s w i l l be e l i m i n a t e d . The purchaser may take over payments on debt which i s i n excess of 66 2/3 per cent of market value. When the i n t e r e s t r a t e i s l e s s than the expected e f f e c t i v e y i e l d , then the sale p r i c e may exceed market p r i c e . Three s i t u a t i o n s i n which t h i s can occur are as f o l l o w s : 1. The value of the property has decreased at a f a s t e r r a t e than the debt outstanding on a conventional The downpayment may be greater than 33 1/3 per cent of sale p r i c e and the vendor may finance the balance, but t h i s i s equivalent to an i n s t i t u t i o n a l f i r s t mortgage loan. 89 f i r s t mortgage loan has been amortized. 2. A house which was b u i l t w i t h i n the l a s t few years may have a N a t i o n a l Housing Act mortgage r e g i s t e r e d against the t i t l e . I f the loan-value r a t i o of the e x i s t i n g mortgage exceeds 66 2/3 per cent then the f i n a n c i n g i s more favourable than might be expected. 3. The house purchaser, who i s vendor f i n a n c e d , may i n a short time a f t e r h i s purchase decide to s e l l h i s i n t e r e s t i n the property. The vendor f i n a n c i n g may become advantageous to the new purchaser. For example, A buys B l a c k Acre valued at $17,000. The sale p r i c e i s constructed as f o l l o w s : Take over f i r s t mortgage {!%) $10,000 7%) 15 year, Bonused vendor second mortgage 8,000 Cash to vendor 2,000 $20,000 Suppose A s e l l s B l a c k Acre to B. B w i l l have the advantage of a high loan-value r a t i o at a lower than expected i n t e r e s t r a t e . The sale p r i c e would be constructed as f o l l o w s : Take over f i r s t mortgage (7$) $10,000 Take over second mortgage (7$) 8,000 Cash to A 2,000 Sale P r i c e $20,000 Yet, B could probably buy back the second mortgage f o r $5,000. Therefore, the market p r i c e would s t i l l be $17,000. The 25 per cent r u l e w i l l i n a quick and convenient f a s h i o n e l i m i n a t e most of assessment-sale p r i c e r a t i o s generated by these t r a n s a c t i o n s . Those that are missed w i l l be e l i m i n a t e d i n the second step, the d e t a i l e d examination of the s a l e s c o n t r a c t s . A second reason f o r a d e t a i l e d examination of the s a l e s c o n t r a c t s i s found p a r t l y i n the e x p l a n a t i o n of why a 25 per cent r u l e was s e l e c t e d r a t h e r than a 33 1/3 per cent r u l e . Why, when the r e g u l a r maximum i n s t i t u t i o n a l c o n v e ntional loan to value r a t i o i s 66 2/3 per cent i s a 25 per cent r u l e used and not a 33 1/3 per cent r u l e ? "Lending value" i s not defined i n the l e g i s l a t i o n , and the pressure of competition may induce lenders to a r r i v e at a lending value i n excess of market p r i c e . Therefore, vendor f i n a n c i n g may not become prevalent u n t i l the property has been financed up to 70 or 75 per cent by a conventional f i r s t mortgage loan. A 33 1/3 per cent r u l e would be too r e s t r i c t i v e , and would not e n t i r e l y e l i m i n a t e the need f o r a d e t a i l e d examination of sale c o n t r a c t s . For example, assume a $10,000 s a l e p r i c e ; $5,000 out-standing on a f i r s t mortgage at 7% and 18 years remaining term; and a vendor second mortgage loan of $1,500 at 7 1/2$ f o r 10 years. Suppose the vendor could s e l l the mortgage paper to y i e l d lk%. 91 The sale p r i c e would be constructed as f o l l o w s : Take over f i r s t mortgage (7$) $ 5,000 7 1/2$, 10 year, Vendor second mortgage 1,500 Cash to Vendor 3,500 110,000 The market p r i c e i s constructed as f o l l o w s : Take over f i r s t mortgage (7$) $ 5,000 1*+$, 10 year, Vendor second mortgage ljl'+O Cash to Vendor 3,500 Market p r i c e $ 9,6kO An assessed value of $ L,500 could r e s u l t i n the f o l l o w i n g r a t i o s . The discrepancy i n t h i s example i s not l a r g e , but the p a r t i c u l a r circumstances w i l l c o n t r o l the s i z e of the d i f f e r e n c e . There i s a t h i r d reason f o r the d e t a i l e d examination of the s a l e s c o n t r a c t s . There are a number of p o s s i b l e cases where the cash payment to the vendor may be as great as 90 per cent or more of the sale p r i c e and s t i l l the t r a n s a c t i o n could i n v o l v e bonused vendor f i n a n c i n g . The purchaser may o b t a i n a s u b s t a n t i a l conventional f i r s t mortgage loan or N a t i o n a l Housing Act f i r s t mortgage loan. The cash payment w i l l equal the amount of the mortgage loan p l u s the purchaser's e q u i t y . Where t h i s t o t a l i s Assessed v a l u e - s a l e p r i c e .1+50 Assessed value-market p r i c e .1+67 92 l e s s than the sa l e p r i c e , vendor j u n i o r f i n a n c i n g i s hi i n v o l v e d . J One f i n a l p o i n t remains to be considered. T h e o r e t i -c a l l y i t i s p o s s i b l e to reduce, as was done i n the example above, every t r a n s a c t i o n p r i c e to i t s equivalent market p r i c e . Therefore, r a t h e r than c l a s s i f y i n g t r a n s a c t i o n s i n v o l v i n g a cash payment of l e s s than 25 per cent of sale p r i c e as being i n e f f i c a c i o u s f o r generating assessment-market p r i c e r a t i o s , should not a l l s a l e p r i c e s be reduced to t h e i r market p r i c e e q u i v a l e n t s ? When the adjustments can be made a c c u r a t e l y t h i s i s a b e t t e r a l t e r n a t i v e to using a d e c i s i o n r u l e . But i n order to a r r i v e at an accurate r e s u l t the property s o l d would have to be inspected and v i s u a l i z e d at i t s date of s a l e . Buyers of j u n i o r paper do not always i n s p e c t the property. They r e l y on the law of averages to p r o t e c t t h e i r c a p i t a l and income. Where d i s p e r s i o n s are being measured the e r r o r s do not "average out." F u r t h e r , the mortgage market, since i t i s more c l o s e l y r e l a t e d to the money and bond markets, i s more v o l a t i l e than the market f o r f e e s . Therefore, i t cannot, without some i n q u i r y , be considered ^This case should not be confused with the case where the purchaser's eq u i t y i s more than 75 per cent of the sale p r i c e and the vendor's f i n a n c i n g l e s s than 25 per cent of the sa l e p r i c e . 93 s t a b l e over the same pe r i o d as that used f o r the study, and the adjustments would have to be based on the s t a t e of the market on the date of the t r a n s a c t i o n . I t i s the o p i n i o n here that the 25 per cent r u l e and the f u r t h e r examination of s a l e s c o n t r a c t s to e l i m i n a t e assessment-sales r a t i o s r e f l e c t i n g bonused vendor f i n a n -c i n g and unusually advantageous f i n a n c i n g to the purchaser w i l l produce r e s u l t s as r e l i a b l e as those a r r i v e d at by approximate adjustments of s a l e s p r i c e s to market p r i c e kk e q u i v a l e n t s . Of the 1,318 property s a l e s s u p p l i e d by the m u l t i p l e l i s t i n g s e r v i c e 139 were r e j e c t e d because the p r o p e r t i e s could not be p o s i t i v e l y i d e n t i f i e d . Of the 1,179 sales remaining i n the sample, 6h7 were r e j e c t e d w i t h the a p p l i c a t i o n of the "25 per cent r u l e . " A sample of 50 was s e l e c t e d randomly from the 6k7 s a l e s . e l i m i n a t e d by the "25 per cent r u l e . " For each one of these 50 t r a n s a c t i o n s the s a l e s c o n t r a c t was examined. Not one of the 50 p r o p e r t i e s traded i n these sales would, on the b a s i s of the s a l e s c o n t r a c t , have been in c l u d e d i n the f i n a l sample. This i s a good i n d i c a t i o n of the e f f e c t i v e -ness of the 25 per cent r u l e of thumb i n e l i m i n a t i n g s a l e s which do not provide a d i r e c t b a s i s f o r the c a l c u l a t i o n of assessment-market p r i c e r a t i o s . A l s o , of the 532 sales remaining a f t e r the a p p l i c a t i o n of the r u l e of thumb, 29 were r e j e c t e d from the sample on the examination of the s a l e s c o n t r a c t . These 29 included s a l e s which i n v o l v e d property t r a d e s , car trades, personal property, i l l e g a l s u i t e s , a d d i t i o n of improvements a f t e r assessment date, a bonused vendor mortgage, and other questionable vendor f i n a n c i n g . Of the 503 remaining three were r e j e c t e d a f t e r an examination of the assessment records i n d i c a t e d improvement a d d i t i o n between November 196h and the date of s a l e . 9h The universe, the parameters of which are being estimated, i s , i n the s t r i c t s t a t i s t i c a l sense, not r e p r e s e n t a t i v e of the assessment r o l l ; a l l p r o p e r t i e s s o l d ; a l l s i n g l e and two f a m i l y r e s i d e n t i a l p r o p e r t i e s s o l d ; and to the extent that bonused vendor f i n a n c i n g i s more prevalent among p r o p e r t i e s of c e r t a i n types or i n given areas i s not r e p r e s e n t a t i v e of a l l s i n g l e and two f a m i l y houses s o l d on m u l t i p l e l i s t i n g . ' G. S t a t i s t i c a l Tabulations The r e s u l t s of the s t a t i s t i c a l t a b u l a t i o n s of the data are e x h i b i t e d i n Table I I . 1. C l a s s i n t e r v a l l i m i t s The l i m i t s of the c l a s s i n t e r v a l s are s p e c i f i e d to midway between the ten thousandth. As the assessment-market p r i c e r a t i o s were to be rounded to the nearest ten thousandth, t h i s s p e c i f i c a t i o n of the c l a s s l i m i t s a i d s i n the i d e n t i f i c a t i o n of the c l a s s i n t o which a p a r t i c u l a r r a t i o w i l l f a l l . 2. A c t u a l frequency The values i n these columns are the number of assessment-market p r i c e r a t i o s f a l l i n g between the s p e c i f i e d l i m i t . ' I t might be p o s s i b l e f o r the Assessment Commissioner to work with samples and a p o p u l a t i o n r e p r e s e n t i n g a l l of the p r o p e r t i e s s o l d i n a C i t y or M u n i c i p a l i t y . TABLE I I ASSESSMENT-MARKET PRICE RATIOS FREQUENCY TABULATIONS AND ESTIMATION OF PARAMETERS S e c t i o n A Se c t i o n B Section C CO > w u co © cd - P i — i a o M CO CO CO O C O -p •H a •H -p i C3 •H O >» o T3 CJ © CD > 3 (H cr CD CD W f n £ f x , O CO > W to u - P CO CD -H co -p a •H q -H O M i-q -p I CJ •H O S O * C J © © f a* ® ©to co > C O C O U - P co © -H co - P B <-\ a -H -p i a •H O ^ © 3 > © C O P>4 o So fl) u X &4 W . l k 9 9 5 -• 2 k 9 9 5 -• 3 k 9 9 5 -. M + 9 9 5 -• 5 L 9 9 5 -. 6 i + 9 9 5 _ 8 . 7 k 9 9 5 -9 . 8 1 + 9 9 5 + . l k 9 9 5 . 2 1 + 9 9 5 • 3 L 9 9 5 ,1+1+995 • 5 L 9 9 5 , 6 1 + 9 9 5 - 7 k 9 9 5 , 8 1 + 9 9 5 . 0 7 k 9 7 5 l . 1 9 9 9 5 0 3 . 2 9 9 9 5 0 1 5 • 3 9 9 9 5 0 l k 8 . 1 + 9 9 9 5 0 2 k 2 . 6 9 9 9 5 0 7 8 . 7 9 9 9 5 0 8 . 8 9 9 9 5 0 3 2 - . 1 3 3 9 5 . 0 6 6 9 7 5 1 • 1 3 3 9 5 - . 2 3 3 9 5 . 1 8 3 9 5 0 2 . 2 3 3 9 5 - . 3 3 3 9 5 . 2 8 3 9 5 0 1 3 . 3 3 3 9 5 - . k 3 3 9 5 . 3 8 3 9 5 0 1 1 6 • L 3 3 9 5 - . 5 3 3 9 5 . k 8 3 9 5 0 2 6 0 . 5 3 3 9 5 - . 6 3 3 9 5 . 5 8 3 9 5 0 8 8 . 6 3 3 9 5 - . 7 3 3 9 5 . 6 8 3 9 5 0 l k . 7 3 3 9 5 - . 8 3 3 9 5 . 7 8 3 9 5 0 h . 8 3 3 9 5 + 2 - . 1 3 0 8 5 . 0 6 5 K 2 5 1 0 .13085-.23085 .18085 2 1 . 2 3 0 8 5 - . 3 3 0 8 5 . 2 8 0 8 5 1 2 2 0 • 3 3 0 8 5 - . 1 + 3 0 8 5 . 3 8 O 8 5 1 1 0 1 2 0 . 1 + 3 0 8 5 - . 5 3 0 8 5 . 1 + 8 0 8 5 265 2 1 8 . 5 3 0 8 5 - . 6 3 0 8 5 . 5 8 0 8 5 8 7 1 2 0 . 6 3 0 8 5 - . 7 3 0 8 5 . 6 8 0 8 5 16 2 0 . 7 3 0 8 5 - . 8 3 0 8 5 . 7 8 0 8 5 5 1 . 8 3 O 8 5 + 2 0 ^(=M) 1+81+00-3 sk _ . k 8 0 8 5 -- 0 8 7 1 2 1 - . 1 0 3 0 2 8 3 . 7 9 5 9 9 6 - . 1 + 7 5 3 0 0 - . 1 + 7 1 8 7 _ . 2 k k 6 0 5 6 . 7 9 5 9 9 6 . 1 + 8 0 5 2 21.95 — .061 vO 96 3 . Expected frequency The values i n t h i s column are the number of assess-ment-market p r i c e r a t i o s which the " f i t t e d normal d i s t r i b u -t i o n " with /^c(=M) and standard d e v i a t i o n ^ ( = s ) would place between the s p e c i f i e d l i m i t s . The numbers are equal to the value of normal p r o b a b i l i t y i n t e g r a l between the l i m i t s m u l t i p l i e d by the t o t a l number of observations. k . Parameters of the frequency d i s t r i b u t i o n s The method of moments has been used i n the c a l c u l a t i o n s of the parameters of the three frequency d i s t r i b u t i o n s . Since the d i s c r e t e frequency d i s t r i b u t i o n i s constructed from an underlying continuous d i s t r i b u t i o n Sheppard*s c o r r e c t i o n has been a p p l i e d to the second and f o u r t h moments. ' The adjusted moments can be denoted as m l ' m 2 ' m 3 ' a n ^ m*+* (a) The mean,^- ? as estimated, by M = m^ has been discussed p r e v i o u s l y . 1+6 The median, M^, was s e l e c t e d as the p o i n t midway between the 250th and 2J?lst l a r g e s t r a t i o s . k 7M. F. Sheppard, "The C a l c u l a t i o n of the Moments of a Frequency D i s t r i b u t i o n , " B i o m e t r i k a , 1907, and "On the C a l c u l a t i o n of the Most Probable Values of Frequency Constants f o r Data Arranged According to E q u i - D i s t a n t D i v i s i o n s of a S c a l e , " London Mathematical S o c i e t y , Pro-ceedings, I898. (b) The standard d e v i a t i o n , \ , as estimated by S has been discussed p r e v i o u s l y . « 5 C f i ( 0 ± ) - i s = m, (c) The skewness i s estimated with the f o l l o w i n g formula. sk = where ft ) f e + 3 2 ( 5 0 1 2 m2/m3 3 2 and (-^ 2 = m^/m2 6 g 9) The s i g n of the skewness i s given by the sig n of the mean minus the median. I f the d i s t r i b u t i o n i s skewed toward the hi g h e r r a t i o s then the median w i l l be l e s s than the mean, and, t h e r e f o r e , the skewness would be p o s i t i v e . (d) The K u r t o s i s measures the excess of the peak of the frequency d i s t r i b u t i o n over that of the normal d i s t r i -b u t i o n w i t h the same mean and standard d e v i a t i o n . The value ^ * a - ft - 3 98 i s used f o r a measure of k u r t o s i s . This measure should be used w i t h care because some c o n f i g u r a t i o n s of data may give a p o s i t i v e value f o r Y 2 ^ e n the k u r t o s i s i s s l i g h t l y negative, (e) The p o s i t i o n of the mode, M', i s i n most in s t a n c e s dependent on the s e l e c t i o n of c l a s s i n t e r v a l l i m i t s . A method of estimating the mode which overcomes t h i s l i m i t a t i o n uses the modal divergence, ^ . ^ = S . sk M o = M -5. Sections A, B, and C The c a l c u l a t i o n of the moments of a frequency d i s t r i b u t i o n i s based on the assumption that the d i s t r i b u -t i o n of the items w i t h i n the c l a s s i n t e r v a l s i s l i n e a r and uniform about the c l a s s i n t e r v a l mid-points. To the extent that t h i s i s not true the s e l e c t i o n of the c l a s s i n t e r v a l l i m i t s a f f e c t s the values of the moments. In Table I I the c l a s s i n t e r v a l l i m i t s of S e c t i o n k were pre-s e l e c t e d . Since the s t a t u t o r y l e v e l of assessment i s . 5 , a frequency d i s t r i b u t i o n which has .5 as the mid-point of the mid-class would be the most obvious choice. Subse-quent c a l c u l a t i o n s produced a mean of O.^1*. The _ The f i r s t moment w i l l not be a f f e c t e d where the n o n - l i n e a r i t y and u n i f o r m i t y are s i m i l a r on both sides of the mid-point of the middle c l a s s i n t e r v a l , and i f the f i r s t moment i s equal to t h i s p r e - s e l e c t e d mid-point. 99 c a l c u l a t i o n s were repeated using OAS 1* as the mid-point of the middle c l a s s i n t e r v a l . This r e s u l t e d i n the t a b u l a t i o n of S e c t i o n B. The mean i n S e c t i o n B was found to be O.lf-8085. Therefore the c a l c u l a t i o n s were repeated using 0.1+8085 as the mid-point of the middle c l a s s i n t e r v a l . This produced S e c t i o n C and a mean of 0.1*8052 which i s i n s i g n i f i c a n t l y d i f f e r e n t from 0. k 8o85, the mean of Se c t i o n B. The process, t h e r e f o r e , was stopped at t h i s p o i n t . In most a p p l i c a t i o n s of "goodness of f i t t e s t s " the process stops at the s e l e c t i o n of c l a s s i n t e r v a l l i m i t s i n S e c t i o n B. These l i m i t s and the parameters, as estimated i n S e c t i o n A, of the d i s t r i b u t i o n are then used 1*9 to d e r i v e the expected f r e q u e n c i e s . 7 The reason f o r extending the process here w i l l be made c l e a r i n Chapter IV. I t r e s t s on the view that u n i f o r m i t y of assessment and c o r r e c t v a l u a t i o n , though interdependent, are not the same t h i n g . I n Chapter I I I the concern i s with u n i f o r m i t y of assessment, t h e r e f o r e , a frequency d i s t r i b u t i o n centred about the mean of the assessment-market p r i c e r a t i o s should be more u s e f u l i n 7 I n some instances the o r i g i n a l c l a s s i n t e r v a l l i m i t s are used i n c a l c u l a t i n g the expected frequ e n c i e s . Where the mean and middle i n t e r v a l mid-point are not equal then the c a l c u l a t i o n i s inconvenient and i t produces an asymmetrical d i s t r i b u t i o n of expected f r e q u e n c i e s . 100 t e s t i n g measures of u n i f o r m i t y than a d i s t r i b u t i o n not centred about the mean assessment-market p r i c e r a t i o . 5 . The Chi Square Test The c l a s s i n t e r v a l l i m i t s as set out i n S e c t i o n C and centred about the mean as c a l c u l a t e d i n S e c t i o n B, and the parameters as estimated i n S e c t i o n B were used to d e r i v e the expected frequencies to be used i n t e s t i n g the a c t u a l frequencies f o r " g o o d n e s s - o f - f i t " to the normal d i s t r i b u t i o n . Because c l a s s i n t e r v a l s one and two had expected frequencies l e s s than f i v e they were combined with c l a s s i n t e r v a l three. S i m i l a r l y i n t e r v a l s e i g h t and nine were combined with i n t e r v a l seven. This reduces the degrees of freedom from s i x to two. A l s o , the power of the t e s t i s reduced. The r e d u c t i o n w i l l not be s i g n i f i c a n t because most of the a c t u a l frequencies i n c l a s s i n t e r v a l s one, two, and nine are sm a l l . The value of c h i square at the 0.001 p r o b a b i l i t y l e v e l i s 13.82. Therefore, there i s l e s s than one chance i n one thousand that random e f f e c t s could have produced the not normally d i s t r i b u t e d sample. The Kolmogorov-Smirnov t e s t may be more powerful than the "X- t e s t i n t h i s case. Where the underlying o t h e o r e t i c a l d i s t r i b u t i o n i s continuous and where the "X-t e s t r e q u i r e s the grouping of c l a s s i n t e r v a l s , the Kolmo-gorov-Smirnov t e s t of "g o o d n e s s - o f - f i t " i s u s u a l l y superior to the 'X? .test. 101 The Kolmogorov-Smirnov t e s t i s based upon the s t a t i s t i c D, where D = max F, •'N " % j F^ i s the cumulative r e l a t i v e frequency at i n t e r v a l N. SJJ i s the cumulative r e l a t i v e frequency at i n t e r v a l N. In S e c t i o n C i t i s found that D = 31/500 at c l a s s i n t e r v a l f i v e . The sampling c h a r a c t e r i s t i c of D i s known where the t h e o r e t i c a l cumulative frequency d i s t r i b u t i o n i s f u l l y s p e c i f i e d . But where the parameters of the t h e o r e t i c a l d i s t r i b u t i o n are estimated from a sample, the sampling d i s t r i b u t i o n of D i s not known. Massey has i n d i c a t e d that where the t e s t i s a p p l i e d i n such cases the c r i t i c a l value f o r D w i l l be l e s s than the c r i t i c a l value f o r the 50 f u l l y s p e c i f i e d case.-' Therefore, when the n u l l -h y p o t h e s i s , that the d i s t r i b u t i o n of assessment-market p r i c e r a t i o s are normally d i s t r i b u t e d , i s r e j e c t e d on the b a s i s of the known sampling d i s t r i b u t i o n of D, the c o n c l u s i o n w i l l be conservative and safe. The c r i t i c a l value f o r D at the .05 p r o b a b i l i t y l e v e l i s 0.0608. 50 y F. J . Massey, "The Kolmogorov-Smirnov t e s t f o r goodness of f i t , " J o u r n a l of the American S t a t i s t i c a l A s s o c i a t i o n . 1951, pp. 68-78. 102 The above observations should be expected because the k u r t o s i s and skewness are h i g h . E. A. Pearson has shown f o r samples of 500 drawn from normal populations that the sampling d i s t r i b u t i o n of has a p r o b a b i l i t y -l e v e l of 0.01 at (2£ = 3.60.51 For a normal po p u l a t i o n | 3 2 » 3.00. The value of ^ 2 f o r the sample of 500 assessment-market p r i c e r a t i o s i s 6.796. He has a l s o shown f o r samples of 500 drawn from a normal po p u l a t i o n that the sampling d i s t r i b u t i o n of (3-^ has a p r o b a b i l i t y l e v e l of 0.02 at = O.O65. The value of the sample of assessment-market p r i c e r a t i o s i s 0.2k5» 7. Conclusion I t i s c l e a r that even f o r p r o p e r t i e s more homo-geneous than the e n t i r e assessment r o l l the universe of assessment-market p r i c e r a t i o s i s not normally d i s t r i b u t e d . Therefore, measures of u n i f o r m i t y of assessment based upon a n o r m a l i t y assumption should be h i g h l y suspect. In c o n s i d e r i n g u n i f o r m i t y of assessment f o r the p a r t i c u l a r universe under c o n s i d e r a t i o n only 265 sample assessment-market p r i c e r a t i o s are w i t h i n plus and minus 0.05 of the mean assessment r a t i o . In other words, 53 per cent of the sample l i e s between 0.k8085 - 0.05 = 0.k3085 and 0.1+8085 + 0.05 - 0.53085. ^ E . A. Pearson, "A Further Development of Tests of Normality," B i o m e t r i k a , 1930, p. 239 f f . Since the mean assessment-market p r i c e r a t i o i s l e s s than 0.50, l e s s than 53 per cent of the sample r a t i o s w i l l l i e between plus and minus 10 per cent of the mean r a t i o from the average assessment-market p r i c e r a t i o . I n other words, l e s s than 53 per cent of the sample w i l l l i e between OA8085 - ( O.l ) C O . k 8085) = .k3276 and 0.k8085 + (0.1)(OA8085) = .5289I*. .13 .73 .<3"3 -V3 5-3 .&3 .73 A/S R a t i o CHAPTER IV ALTERNATIVE MEASURES OF ASSESSMENT UNIFORMITY AND QUALITY A measure of u n i f o r m i t y or q u a l i t y , i f i t i s to be a u s e f u l a d m i n i s t r a t i v e t o o l , should be capable of compari-son to a standard measure. In other words, that the standard d e v i a t i o n of a d i s t r i b u t i o n of assessment-market p r i c e r a t i o s i s .125 means very l i t t l e unless that measure i s r e l a t e d to some preset standard. Where measures of d i s p e r s i o n are to be meaningfully compared some assumption must be made about the underlying d i s t r i b u t i o n . The n o r m a l i t y assumption, as discussed i n Chapter I I I , i s not always j u s t i f i a b l e where d i s t r i b u t i o n s of assessment-market p r i c e r a t i o s are concerned. Sehwinden has used a l i n e a r c o r r e l a t i o n model. 1 He has regressed assessed value on market p r i c e . This does away with a measure of d i s p e r s i o n between the assessment-market p r i c e r a t i o s of d i f f e r e n t valued p r o p e r t i e s , but the u n i f o r m i t y i s a l s o dependent upon the d i s p e r s i o n of the assessed values about the r e g r e s s i o n l i n e at a l l of the values on the a b s c i s s a . Hence, the measure of u n i -f o r m i t y can be based upon the standard e r r o r of estimate "^Sehwinden, op. c i t . , pp. 57-60. 106 or the c o e f f i c i e n t of c o r r e l a t i o n . This use of these measures assumes that the d i s t r i b u t i o n s of both the 2 independent and the dependent v a r i a b l e are normal. There can be some doubt about whether or not market values and assessed values are normally d i s t r i b u t e d . This assumption w i l l not be test e d here. A. An A l t e r n a t i v e Measure An a l t e r n a t i v e measure of assessment q u a l i t y and un i f o r m i t y i s proposed here. This measure r e s t s upon one assumption and one value judgement. The assumption i s that an assessment can be judged c o r r e c t or i n c o r r e c t . The value judgement i s re q u i r e d to determine whether any given assessment i s c o r r e c t or i n c o r r e c t . As argued p r e v i o u s l y , a ten per cent maximum margin of e r r o r should be permitted. I n B r i t i s h Columbia an assessment l y i n g between k 5 and 55 per cent of market p r i c e would be c o r r e c t . In e f f e c t then, i t i s assumed that the assessments form a bin o m i a l d i s t r i b u t i o n of c o r r e c t and i n c o r r e c t v a l u a t i o n s . The higher the p r o p o r t i o n of c o r r e c t Mordecai E z e k i e l and K a r l A. Fox, Methods of C o r r e l a t i o n and Regression A n a l y s i s (3rd ed., New York, John Wiley & Sons Inc., 1959), pp. 279-281. 107 assessments then the greater the u n i f o r m i t y . I f F i s the p r o p o r t i o n of c o r r e c t assessment, Q i s the p r o p o r t i o n of i n c o r r e c t assessments and y " i s the standard d e v i a t i o n of the p o p u l a t i o n , then then as P i n c r e a s e s (Q decreases, and the standard d e v i a t i o n decreases. This i s c o n s i s t e n t with the view that u n i f o r m i t y of assessment i s dependent upon c o r r e c t v a l u a t i o n . B. Standards of Assessment U n i f o r m i t y Usj.ng the A l t e r n a t i v e Measure Assume four assessment r o l l s , A, B, C and D, and that t h e i r assessment-market value r a t i o s are d i s t r i b u t e d as i n Table I I I . I f P .5 Q < .5 108 TABLE I I I HYPOTHETICAL POPULATIONS OF ASSESSMENT-MARKET VALUE RATIOS A,.B, C, D Po p u l a t i o n Assessment-Market Value P r o p 0 r t i o n of P o p u l a t i o n be-L i m l t s tween L i m i t s (a) M - .55 .90 .80 .90 .80 (b) .ko - .60 .95 .95 .93 .95 (c) .35 - .65 .97 .97 .95 .99 Where underlying continuous d i s t r i b u t i o n s are assumed i t i s p o s s i b l e to c a l c u l a t e the standard d e v i a t i o n s from the above i n f o r m a t i o n . The standard d e v i a t i o n s can then be compared. For example, i f the underlying d i s t r i -b u t i o n i s normal, and i f only l i n e (a) above were a v a i l a b l e , then the standard d e v i a t i o n s of A and C would be equal and would be l e s s than the standard d e v i a t i o n s of B and D which are equal. Therefore A and C are more uniform than B and D. I f i n f a c t A were normally d i s t r i b u t e d with mean = 0.5 then = .55 - .5 = .030k , 1.6»+5 and the p r o p o r t i o n of assessment-market value r a t i o s between the l i m i t s of .ko and .60 would be .999 or 99.9 per cent. 109 That type of comparison cannot be made when the underlying d i s t r i b u t i o n i s the proposed b i n o m i a l d i s t r i -b u t i o n . But since the major c r i t e r i o n f o r u n i f o r m i t y of assessment i s c o r r e c t v a l u a t i o n , then with only the in f o r m a t i o n i n l i n e (a) above a v a i l a b l e , A and C are equal and more uniform than are B and D which are e q u a l l y uniform. The secondary c r i t e r i a provided i n l i n e s (b) and (c) above can be used to make a more f i n e measurement. I f the populations are l i s t e d v e r t i c a l l y i n order of decreasing u n i f o r m i t y , then: Using i n f o r m a t i o n i n l i n e (a) only the ranking would be A, C B, D; Using i n f o r m a t i o n i n l i n e s (a) and (b) only the ranking would be A C B, D; Using i n f o r m a t i o n i n l i n e s ( a ) , ( b ) , and (c) the ranking would be A 0 B B. In e f f e c t , l i n e s (b) and (c) are r e p o r t i n g the values of P f o r s u b s i d i a r y b i n o m i a l d i s t r i b u t i o n s of the 110 four assessment r o l l s . The binomial d i s t r i b u t i o n of c o r r e c t and i n c o r r e c t v a l u a t i o n s remains' as the primary c r i t e r i o n by which to evaluate the u n i f o r m i t y of the assessment r o l l . Suppose there are three assessment r o l l s , E, F, and G, and that t h e i r assessment-market value r a t i o s are d i s t r i b u t e d as f o l l o w s . TABLE IV HYPOTHETICAL POPULATIONS OF ASSESSMENT-MARKET VALUE RATIOS E, F, G Population E F G Assessment-Market Value P r o p o r t i o n of Po p u l a t i o n L i m i t s between L i m i t s (e) .U5 - .55 .70 .60 .60 ( f ) .1*0 - .60 .70 .75 .70 (g) -35 - .65 .70 .80 .90 I f the same system of ordering good to bad assess-ment u n i f o r m i t y i s used as was used f o r populations A, B, G and D, then considering l i n e s ( e ) , ( f ) , and (g) the ranking would be as f o l l o w s : . E F G. There can be some doubt as to whether or not the assess-ment q u a l i t y of F i s greater than that of G. A l s o , there I l l i s doubt as to whether or not the assessment q u a l i t y of E i s greater than that of F or G. Except i n extreme cases i t would appear that the s e l e c t i n g of the b e t t e r ordering would be a matter of judgement. Consider two extreme examples shown i n Table V. TABLE V HYPOTHETICAL POPULATIONS OF ASSESSMENT-MARKET VALUE RATIOS E x and F 2 : E 2 and F 2 : E^ and F^ X? TP TP TP TP TP P o p u l a t i o n hi *i fc2 *2 ^3 *3 Assessment-Market Value L i m i t s P r o p o r t i o n of P o p u l a t i o n between L i m i t s Ce) x . k 5 - .55 .20 .05 .20 .05 .20 .05 ( f ) 1 AO - .60 .20 .90 .20 .90 .20 .20 (g) x .35 - .65 .50 .95 .95 .95 .50 .95 I t i s reasonable to argue that the assessment q u a l i t y o f : F^ i s b e t t e r than that of E j j F 2 i s b e t t e r than that of E 2 ; and F^ i s b e t t e r than that of E y E.^ and are equal i n assessment q u a l i t y ; and F 2 are equal i n assessment q u a l i t y . Therefore the ranking of the s i x assessment r o l l s would be as f o l l o w s . 112 F l 0 r F 2 E x or E^. There i s another p o i n t to consider. Suppose Popu l a t i o n s G and H are d i s t r i b u t e d as i n Table VI below. TABLE VI HYPOTHETICAL POPULATIONS OF ASSESSMENT-MARKET VALUE RATIOS G, H Po p u l a t i o n Assessment-Market Value L i m i t s H P r o p o r t i o n of P o p u l a t i o n between L i m i t s (g) (h) ( i ) ( j ) Ck) (1) M - .55 .kO - .60 .35 - .65 .15 - .25 .10 - .30 .05 - .35 .90 .95 .99 .ook5 .0055 .006 .oo»+5 .0055 .006 .90 .95 .99 R o l l H s u r e l y i s not l e s s uniform than r o l l G.J But where the assessor i s d i r e c t e d to assess at a s p e c i f i c p r o p o r t i o n of market value the q u a l i t y of an assessment r o l l has two dimensions. One dimension i s u n i f o r m i t y . -'If the f u l l range of r a t i o s i s considered the d i s t r i b u t i o n of assessment-market value r a t i o s of r o l l H i s more l i k e l y skewed than that of r o l l G. 113 The second dimension i s correctness of valuation. The achievement of the f i r s t dimension rests extensively on the achievement of the second dimension, because, whether or not an assessor who i s directed to assess at 50 per cent of market value, i s attempting to keep an assessment r o l l up to date, only the wildest coincidence would r e s u l t i n a l l properties being assessed at 20 per cent of market value. Therefore uniformity of assessment at any l e v e l rests on correct valuations. Where the assessor i s not directed to assess at any pa r t i c u l a r l e v e l of market value, but the selection of the l e v e l i s l e f t to him, or to chance, uniformity of assessment w i l l s t i l l depend on the achievement of correct valuations. In order to use the binomial model to measure only uniformity i t i s f i r s t necessary to determine the average of the i n d i v i d u a l assessment-market price r a t i o s . In order to measure the quality of an assessment r o l l the average of the i n d i v i d u a l r a t i o s i s not needed. If the assessor arrives at the correct market value and then mul t i p l i e s the assessment by 0.2 then a l l properties can be assessed at 20 per cent of market value. But an assessor who c o r r e c t l y estimates market value, and then when he i s directed to multiply by 0.5? does multiply by some other f a c t o r , i s surely impeachable. Ilk Q u a l i t y of assessment i s dependent upon u n i f o r m i t y of assessment. U n i f o r m i t y of assessment i s to a great extent dependent upon c o r r e c t v a l u a t i o n s . Therefore, from t h i s approach, q u a l i t y of assessment i s i n d i r e c t l y dependent upon c o r r e c t v a l u a t i o n . Then, c o n s i d e r i n g the second dimension, assessment q u a l i t y i s d i r e c t l y dependent upon c o r r e c t v a l u a t i o n . Therefore a bin o m i a l model based on c o r r e c t and i n c o r r e c t v a l u a t i o n can be used to measure d i r e c t l y assessment q u a l i t y and to measure i n d i r e c t l y u n i f o r m i t y of assessment. C. Binom i a l Sampling In the above d i s c u s s i o n the binomial model was discussed i n terms of a p o p u l a t i o n . The p o p u l a t i o n could be a l l the assessment-market value r a t i o s assessment r o l l or i t could be a universe of sales generated assessment-market p r i c e r a t i o s . Remembering that market sales generate a sample of assessment-market p r i c e r a t i o s from a universe which would be generated by the same f o r c e s that generated the sample i f those f o r c e s were to continue i n d e f i n i t e l y without change i n c h a r a c t e r , the universe of sales generated assessment-market p r i c e r a t i o s , unless the context r e q u i r e s otherwise, w i l l be used throughout the r e s t of t h i s chapter. 115 The parameters of the p o p u l a t i o n a re: P - the p r o p o r t i o n of a l l p r o p e r t i e s which would s e l l under market f o r c e s producing d i r e c t evidence of market value and were c o r r e c t l y assessed. Q - the p r o p o r t i o n of a l l p r o p e r t i e s which would s e l l under market f o r c e s producing d i r e c t evidence of market value and were i n c o r r e c t l y ,—. assessed. i/P.Q - the standard d e v i a t i o n of the po p u l a t i o n proportions of c o r r e c t v a l u a t i o n s . The parameters of the sample are: p - the p r o p o r t i o n of a l l p r o p e r t i e s which a c t u a l l y s o l d producing d i r e c t evidence of market value and were c o r r e c t l y assessed, q - the p r o p o r t i o n of a l l p r o p e r t i e s which a c t u a l l y s o l d producing d i r e c t evidence of market value and were i n c o r r e c t l y assessed, s p.q - the standard d e v i a t i o n of the sample proportions of c o r r e c t assessments, n - the number of elements i n the sample. I f n > 50 then the sample p r o p o r t i o n s , p, w i l l be normally d i s t r i b u t e d with mean PP and standard d e v i a t i o n p i s often r e f e r r e d to as the standard e r r o r of a p r o p o r t i o n . Where P i s not known and i s estimated by p then i s the unbiased estimator of p when the universe i s i n -f i n i t e , and 5 P = E l l n-1 i s the unbiased estimator f i n i t e and of s i z e N.^ of \ p p when the universe i s In t h i s paper the universe i s i n f i n i t e . 116 The problem here i s to estimate, given a sample p r o p o r t i o n of c o r r e c t v a l u a t i o n s , the s m a l l e s t reasonably expected p r o p o r t i o n of c o r r e c t assessments i n the popula-t i o n . Remembering that the sample p r o p o r t i o n s , p, are normally d i s t r i b u t e d with mean P m (= P ) , the sampling d i s t r i b u t i o n can be standardized with the f o l l o w i n g formula. 2 = p - P Since P i s not known V Pe~" i s replaced by s p and V i s replaced by p m. P - P m A * = P - P m p.q n-1 A reasonable l e v e l of confidence f o r the estimate of the lower l i m i t of p m i s to be s e l e c t e d . N i n e t y - f i v e per cent would be a reasonable l e v e l of confidence. Therefore, denote the value of pffi at the 95 per cent confidence l e v e l by P ^ (x = .95). z = p - p m .95 m.95 J P.q n-1 117 For example, suppose that out of 901 p r o p e r t i e s that s o l d the assessed values of 720 of them were between k 5 per cent and 55 per cent of market p r i c e . The r e s u l t s are summarized below: p = .8 q = .2 p.q = .16 z 9 5 = 1.6 k5 E L S = .0133 n -1 • 9 5 " ( Z J ( \ P _ L Q ) = 0.022 P F F I = .778 m .95 In e f f e c t the r e s u l t s i n d i c a t e that i f the p r o p o r t i o n of c o r r e c t v a l u a t i o n s i s only 77.8 per cent, then there are only f i v e chances i n one hundred that the sample would have more than 720 c o r r e c t assessments. On the other hand, i t may be judged that the minimum standard of assessment r e q u i r e s 90 per cent of p r o p e r t i e s s o l d to have assessed values w i t h i n *+5 per cent to 55 per cent of market p r i c e . What i s the minimum value of p which would meet t h i s standard at the 95 per cent confidence l e v e l ? -z = p - P • 95 V P ; 118 p - P .95 P.Q n -1.6^5 P = .9 Q = .1 P.Q - .09 p = .010 .95 = -0.016 p = 0 . 8 8 k In other words, i f there were 901 p r o p e r t i e s s o l d , then 796 would be c o r r e c t l y valued i n order to meet the requ i r e d standard. I t might be suggested that the sampling approach i s superfluous, and that a l l the r a t i o s a c t u a l l y generated by sal e s be t r e a t e d as the universe. But the number of r a t i o s , the number of s a l e s , may vary from year to year, and from l o c a l i t y to l o c a l i t y , and the sampling model by c o r r e c t i n g f o r n provides the adjustments f o r r e t a i n i n g c o m p a r a b i l i t y among the years and l o c a l i t i e s . For example, suppose that market a c t i v i t y generated only 1+00 r e l i a b l e assessment-sales p r i c e r a t i o s . Then -2 * 9 5P = 0.025 119 p =0.875 Therefore the standard would i n d i c a t e that 3^9 c o r r e c t v a l u a t i o n s out of kOO assessments i s equivalent to 796 c o r r e c t v a l u a t i o n s out of 900 assessments. Note that 3k9/L00 i s not equal to 796/900. I f only 100 r a t i o s are a v a i l a b l e then p = 0.851, and 85 c o r r e c t v a l u a t i o n s could s a t i s f y the standard of P = 0.90. Note here that (85/100) < (3L9/L00) < (796/900). Q u a l i t y of the P o p u l a t i o n A c t u a l l y Sampled The sample has 2h2 c o r r e c t v a l u a t i o n s . Only the p r o p o r t i o n O.^1* of the assessments are between !+5 and 55 per cent of market value. Therefore the estimated standard e r r o r of the p r o p o r t i o n estimate i s 0.02235, and the 95 per cent confidence l i m i t s f o r p m , the universe p r o p o r t i o n of c o r r e c t v a l u a t i o n s are a minimum of 0. k39 and a maximum of 0.529. This r e s u l t can be s t a t e d d i f f e r e n t l y . I f the p o p u l a t i o n does i n f a c t have 90 per cent of the p r o p e r t i e s c o r r e c t l y valued then there are l e s s than 5 chances i n one m i l l i o n that the sample would have only kB.k per cent c o r r e c t l y assessed p r o p e r t i e s . Only a coincidence has produced p = ,h8k and M « .k8k. *2k2/500 = ,k8k. 120 There are 93»6 per cent of the assessments between 35 and 65 per cent of market value. Only 5l»2 per cent of the r a t i o s are between the l i m i t s of plus and minus 10 per cent of the mean from the mean. This does not i n d i c a t e a high measure of un i f o r m i t y . How high i s the q u a l i t y of the assessment r o l l ? F i r s t , f o r the reasons stated p r e v i o u s l y i t i s not l i k e l y to be as high as the universe being measured. The popu-l a t i o n represented here c o n s i s t s of assessment-market p r i c e r a t i o s of p r o p e r t i e s which form a market more homogeneous than a l l of the p r o p e r t i e s represented on the e n t i r e assessment r o l l . Second, as compared to a standard of 90 per cent c o r r e c t v a l u a t i o n s , the assessment q u a l i t y of the universe measured here i s not h i g h . The degree of the l a c k of q u a l i t y i s not completely measurable against the above standard. For example, would a universe with 65 per cent c o r r e c t v a l u a t i o n s but with only 80 per cent of the assessment-market p r i c e r a t i o s between 35 and. 65 per cent of assessed value be b e t t e r or worse, as compared to the standard, than the universe measured here? E» A d d i t i o n a l A l t e r n a t i v e Measures of U n i f o r m i t y and Q u a l i t y One might suppose that a grading system would be d e s i r a b l e . Such a system i s examined, and i t s l i m i t a t i o n s are noted here. 121 The standard' r e q u i r e s that 90 per cent of the assessment-market p r i c e r a t i o be between 0.^5 and 0.55. I f the d e v i a t i o n s from p e r f e c t assessment, assessment at 50 per cent of market va l u e , are random, then the assessment-market p r i c e r a t i o s w i l l be normally d i s t r i -buted. Therefore, a p o s s i b l e standard by which to evaluate the q u a l i t y of an assessment r o l l would be a normal d i s t r i b u t i o n with /U^ , the mean, equal to 0.5, and the standard d e v i a t i o n , equal to 0.030*+.^ The a c t u a l frequency d i s t r i b u t i o n of r a t i o s could then be te s t e d f o r " g o o d n e s s - o f - f i t " to the expected d i s t r i b u t i o n of r a t i o s under the f u l l y s p e c i f i e d normal d i s t r i b u t i o n . The "X^2 t e s t or the Kolmogorov-Smirnov t e s t can be used. The t e s t s t a t i s t i c can then be t r a n s l a t e d i n t o a p r o b a b i l i t y l e v e l . The lower the p r o b a b i l i t y l e v e l the lower the q u a l i t y of the measured universe. The % t e s t w i l l i n a l l cases have the same number of degrees of freedom, and the t r a n s l a t i o n of the Kolmogorov-Smirnov s t a t i s t i c i n t o a p r o b a b i l i t y l e v e l w i l l adjust f o r va r y i n g sample s i z e s . The use of the p s t a t i s t i c X i n t h i s case reduces the degrees of 2 freedom i n the % d i s t r i b u t i o n by one, i n s t e a d of th r e e , becauseyi*. and *\ are s p e c i f i e d and the only l i n e a r 7 'There i s no suggestion that the a c t u a l d i s t r i b u t i o n of r a t i o s i s normal. 122 c o n s t r a i n t i s a ^ F j / Table V I I below summarizes 2 the c a l c u l a t i o n of X and D. The values i n the column headed "Expected Frequency" are the number of expected observations i n the c l a s s i n t e r v a l under the f u l l y s p e c i f i e d normal d i s t r i b u t i o n . A l l c l a s s i n t e r v a l s except f o u r , f i v e and s i x have 2 zero expected frequency. Therefore, the r e l e v a n t % 2 t e s t has 2 degrees of freedom. X equal to 1053.33 i s w e l l beyond the c r i t i c a l Z>C2 value of 13-82 at the 0.001 p r o b a b i l i t y l e v e l . The D value i s 0 . 2 8 k . This i s w e l l i n excess of 1.63/500, the c r i t i c a l D value at the 0.01 p r o b a b i l i t y l e v e l . The f i r s t l i m i t a t i o n i s that a universe of assess-ment-market p r i c e r a t i o s might have more than 90 per cent c o r r e c t v a l u a t i o n s , and, t h e r e f o r e , a sample from such a universe could r e s u l t i n a low p r o b a b i l i t y l e v e l . This l i m i t a t i o n i s minor, because a v i s u a l examination of the frequency d i s t r i b u t i o n w i l l make evident the high q u a l i t y of t h i s universe. The second l i m i t a t i o n i s more s i g n i f i c a n t . I t e x i s t s because the grading system measures q u a l i t y of assessment p r i m a r i l y . A sample from a universe i n which a l l p r o p e r t i e s are assessed at 25 per cent of market value TABLE VII FIT OF OBSERVED FREQUENCIES TO STANDARD OF CORRECT VALUATION Cl a s s C l a s s i n t e r v a l i n t e r v a l l i m i t s Observed frequency Cumulative r e l a t i v e observed frequency, S^ Expected frequency Cumulative r e l a t i v e expected frequency, F^ 1 FN - % 1 1 0.OOOOO-.1^995 1 1/500 0 0/500 1/500 2 . 1 ^ 9 9 5 - . 2 ^ 9 9 5 3 1+/500 0 0/500 1+/500 3 .21+995-. 3^995 15 19/500 0 0/500 19/500 k • 3 k 9 9 5 - . L k 9 9 5 ll+8 167/500 25 25/500 ik 2/500 5 .M + 9 9 5 - . 5^995 2l+2 1+09/500 1+50 . k 7 5 / 5 0 0 6 6 / 5 0 0 6 . 5 ^ 9 9 5 - . 61+995 78 1+87/500 25 500/500 13/500 7 .61+995-. 75995 8 L 9 5 / 5 0 0 0 500/500 5/500 8 • 7 k 9 9 5 - . 81+995 3 1+98/500 0 500/500 2/500 9 .81+995-1.000 2 500/500 0 500/500 0 / 5 0 0 = . 5 X 2 = 1 0 5 3 . 3 3 < .0301+ D ik2/500 125 A/'S Ratio 125 would, r e s u l t i n a low p r o b a b i l i t y l e v e l . This measure makes no allowance f o r extreme cases such as i n the above example, but again a v i s u a l examination of the data w i l l r e v e a l the cause of the low p r o b a b i l i t y l e v e l . Where the assessment-market p r i c e r a t i o s are t i g h t l y c l u s t e r e d about a mean r a t i o not f a r removed from 0.5 then a v i s u a l examination w i l l not s u f f i c e . The s e l e c t i o n of c l a s s i n t e r v a l s may obscure the nature of the underlying a c t u a l frequency d i s t r i b u t i o n . To some extent the l a t t e r l i m i t a t i o n may be over-come. In a d d i t i o n to the above q u a l i t y grading a u n i f o r m i t y grading can be introduced. I f a universe of assessment-market p r i c e r a t i o s i s h i g h l y uniform then the r a t i o s w i l l be c l u s t e r e d t i g h t l y around and normally d i s t r i b u t e d about the mean r a t i o . In accordance with the measure of q u a l i t y , 90 per cent of the r a t i o s should be between the l i m i t s (M - .10M) and (M + .10M). Table V I I I below summarizes the c a l c u l a t i o n of the " g o o d n e s s - o f - f i t " of the observed frequencies to the expected f r e q u e n c i e s of a normally d i s t r i b u t e d p o p u l a t i o n with mean 0.l+8085 and a standard d e v i a t i o n of 0.02923L. X equals 10k2.L5« The expected frequencies i n a l l c l a s s i n t e r v a l s except f o u r , f i v e , and s i x were z e r o , therefore only the three c l a s s i n t e r v a l s were used f o r the TABLE V I I I FIT OF OBSERVED FREQUENCIES TO STANDARD OF UNIFORM ASSESSMENT Class i n t e r v a l C l a s s Observed i n t e r v a l frequency l i m i t s Cumulative r e l a t i v e observed frequency S^ Expected frequency Cumulative expected r e l a t i v e frequency | FN - h i 1 0.OOOOO-.15522 1 1/500 0 0/500 1/500 2 . 11+1+22-. 25050 3 5/500 0 0/500 5/500 3 . 21+01+0-. 33658 20 2U/500 0 0/500 25/500 1+ .33658-.k3276 106 130/500 25 25/500 105/500 5 .53276-.52895 256 386/500 550 575/500 89/500 6 •5289k-.62512 87 k73/500 25 500/500 27/500 7 .62512-.72130 20 k93/500 0 500/500 7/500 8 . 72130-. 81758 5 *+98/500 0 500/500 2/500 9 .81758-1.0000 2 500/500 0 500/500 0/500 .1+8085 X 2 = 1052.50 <- 0.029235 D = 105/500 +-* ro ON 127 p calculation of X . Because there are two linear con-straints M there is only one degree of freedom. The probability 2 level is below 0.001, as the critical value of 76 at one degree of freedom is 10.83. The value of D is .210, and it is well above the critical value of 1.63/500 at the 0.01 probability level. The D value for the uniformity test is in this case lower than the D value for the "correctness of valuation" test, and, therefore, the use of both tests gives recognizance to both uniformity and quality of assessment. 'IH .3V .5* .7H -H I A/S R a t i o CHAPTER V CONCLUSIONS The f i r s t , and most obvious, c o n c l u s i o n i s that unless an assessment r o l l i s of the highest q u a l i t y i t i s very u n l i k e l y that the assessment-market p r i c e r a t i o s w i l l be normally d i s t r i b u t e d . Therefore, a standard of assessment u n i f o r m i t y or q u a l i t y based upon the assump-t i o n that the d i s t r i b u t i o n of r a t i o s i s normal i s without p r a c t i c a b l e meaning. I f an assessment r o l l has a c o e f f i c i e n t of v a r i a t i o n l e s s than the maximum allowable under the standard, but does not have normally d i s t r i b u t e d assessment-market p r i c e r a t i o s , then the standard cannot be used to i n f e r that the u n i f o r m i t y or q u a l i t y of the r o l l i s acceptable. p Not one of the 76 t e s t , the Kolmogorov-Smirnov t e s t , and the t e s t of the sample moments i n d i c a t e d that the p a r t i c u l a r universe of Vancouver C i t y assessment-market p r i c e r a t i o s i s normally d i s t r i b u t e d . N e i t h e r , t h e r e f o r e , i s i t l i k e l y that the assessment-market p r i c e r a t i o s a v a i l a b l e to the assessor and Assessment Commissioner are normally d i s t r i b u t e d . Regardless of the s i z e of the maximum allowable c o e f f i c i e n t of v a r i a t i o n , any statement that the Vancouver assessment r o l l i s , according to that standard, uniform cannot be accepted. 130 The c o e f f i c i e n t of v a r i a t i o n measured here i s denoted by V. V V V 18.1 per cent. The Assessment Commissioner s t a t e s that a c o e f f i c i e n t of 18.8 should be a t t a i n a b l e and l e s s than 18.8 i n d i c a t e s good r e s u l t s . 1 A c o e f f i c i e n t of 18.8 per cent and a mean of OA8085 imply a standard d e v i a t i o n of O.O90398O. F i t t i n g the observed d i s t r i b u t i o n to a normal d i s t r i b u t i o n iyU= . k 8 0 8 5 , ^ = .0903980) y i e l d s X 2 = 28.37 at 3 degrees of freedom and D = 0.07. The value of ? C 2 (3 degrees of freedom) at the 0.001 p r o b a b i l i t y l e v e l i s 16.27. The value of D, the Kolmogorov-Smirnov s t a t i s t i c , at the .05 p r o b a b i l i t y l e v e l i s .O608. Therefore, the Assessment Commissioner has a paradox. The c o e f f i c i e n t of v a r i a t i o n i n d i c a t e s uniform assessment, but according to the under-l y i n g assumptions of no r m a l i t y the assessments are not uniform. That the a c t u a l d i s t r i b u t i o n does not " f i t " the assumed d i s t r i b u t i o n can be seen i n Table IX. 1 A p p r a i s a l Manual, oju. c J J ^ , p. 315. 131 TABLE IX ACTUAL AND EXPECTED FREQUENCIES A c t u a l Expected frequency frequency 1 -.13085 1 0 2 .13085-.23085 2 1 3 .23085-.33085 12 23 1+ •33085-.k3085 110 121 5 .1+3085-. 53085 265 210 6 .53085-.63085 87 121 7 .63085-.73085 16 23 8 .73085-.83085 5 1 9 .83085 + 2 0 I t would appear from the arguments i n Chapter IV that no s i n g l e measure of q u a l i t y or u n i f o r m i t y of assess-ment can replace j u d i c i o u s use of s e v e r a l a l t e r n a t i v e measures. The measures used must not imply unwarranted assumptions about the d i s t r i b u t i o n of the r a t i o s . The bino m i a l models f o r assessment q u a l i t y and assessment u n i f o r m i t y and the C h i square and Kolmogorov-Smirnov t e s t s f o r assessment q u a l i t y and u n i f o r m i t y used here might not be the only measures a p p l i c a b l e to the problem. They do possess at l e a s t one important q u a l i t y . None of the measures presented i n Chapter IV r e q u i r e s an i n v a l i d assump-t i o n about the a c t u a l d i s t r i b u t i o n of assessment-market p r i c e r a t i o s . 132 According to these measures the q u a l i t y and u n i f o r m i t y of the C i t y of Vancouver Assessment r o l l are very low. The standards used may be h i g h , but as stated p r e v i o u s l y , i f "equal treatment of equals" cannot be achieved then the r e a l property tax as a source of p u b l i c p revenue should be replaced. Further Study The measurement of assessment u n i f o r m i t y and q u a l i t y i s not the only use f o r assessment-market p r i c e r a t i o s . The o v e r a l l o b j e c t i v e of the a n a l y s i s i s to improve assessment q u a l i t y and u n i f o r m i t y . The data used i n measurement of these can a l s o be used to determine the p r o p e r t i e s or c h a r a c t e r i s t i c s of p r o p e r t i e s which are concomitant with over and under assessment. O l i v e r Oldman and Henry Aaron have attempted to analyze the p a t t e r n of •3 assessment d e v i a t i o n from u n i f o r m i t y . They have c l a s s i f i e d p r o p e r t i e s i n t o nine areas, eleven property types, and ten p r i c e ranges f o r the purpose of assessment and measuring u n i f o r m i t y w i t h i n groupings. The average r a t i o was used to measure the assessment l e v e l and the standard d e v i a t i o n was 2 Supra, p. 3. ^ O l i v e r Oldman and Henry Aaron, "Assessment-Sales R a t i o s under the Boston Property Tax," N a t i o n a l Tax J o u r n a l , 1965, PP. 36-49. 133 used to measure assessment u n i f o r m i t y . In order to analyze the i n f l u e n c e of property type and l o c a t i o n on the l e v e l of assessment they c a r r i e d out a two way a n a l y s i s of variance w i t h m u l t i p l e entry i n each c e l l or grouping. The r e s u l t s of t h e i r study are not s i g n i f i c a n t here, because, f i r s t , the use of the standard d e v i a t i o n to compare the u n i f o r m i t y w i t h i n d i f f e r e n t groups assumes s i m i l a r d i s t r i b u t i o n s of assessment-market p r i c e r a t i o s w i t h i n the groups; second, t h e i r variance a n a l y s i s assumes that each group i s an independent random sample and represents a normal popula-t i o n . This c o n d i t i o n does not p r e v a i l i n Vancouver, nor was any i n d i c a t i o n given of the v a l i d i t y of the assumption f o r Boston. A r e l a t e d method can be a p p l i e d without the assump-t i o n of an underlying normal d i s t r i b u t i o n . Contingency t a b u l a t i o n s may be used. P r o p e r t i e s can be c l a s s i f i e d by l o c a t i o n , type, p r i c e range, age, type of c o n s t r u c t i o n , s i z e , use, s t a t e of r e p a i r , and environmental amenities. A system may be used i n d i v i d u a l l y or as p a r t of complex system c o n s i s t i n g of two or more c h a r a c t e r i s t i c s . The p r o p e r t i e s can a l s o be c l a s s i f i e d by type of assessment T F r e d e r i c k C. M i l l s , S t a t i s t i c a l Methods (3rd ed.; New York: Henry H o l t and Company, 1955), p. 571. He a l s o p o i n t s out that when the assumptions are not v a l i d , the a n a l y s i s may be used but that the r e s u l t s are not completely r e l i a b l e . 13 L c o r r e c t , over, and under assessment. Cross t a b u l a t i o n s on the type of assessment against the property c h a r a c t e r i s -t i c s and ~%-2 t e s t s a p p l i e d to these t a b u l a t i o n s w i l l i n d i c a t e whether or not the c h a r a c t e r i s t i c s t a l l i e d are concomitant with over or under assessment. Where there i s a s i g n i f i c a n t r e l a t i o n s h i p between c o r r e c t or i n c o r r e c t assessment the c l a s s i f i c a t i o n a f f e c t e d may be f u r t h e r sub-d i v i d e d by the same or other c h a r a c t e r i s t i c s and the t a b u l a t i o n and t e s t repeated. This w i l l give a more d e t a i l e d i n d i c a t i o n of the incidence of assessment e r r o r s . Where no s i g n i f i c a n t r e l a t i o n s h i p between property c h a r a c t e r i s t i c and c o r r e c t or i n c o r r e c t assessment i s i n d i c a t e d s e v e r a l groups of p r o p e r t i e s may be combined. The o r i g i n a l c l a s s i f i c a t i o n may have been too f i n e to give suggestive r e s u l t s . Since uniform and high q u a l i t y assessment have not been achieved f u r t h e r s t a t i s t i c a l research should be concentrated on the development i n d e t a i l of methods and techniques which can be used to help the assessor i s o l a t e the sources of h i s e r r o r s . The r e s u l t s of Chapter I I I and Chapter IV i n d i c a t e that as i n a l l other problems a s s o c i a t e d with v a l u a t i o n of property the measurement of assessment q u a l i t y and u n i f o r m i t y r e q u i r e s the e x e r c i s e of judgement i n a r e a l world. BIBLIOGRAPHY BIBLIOGRAPHY A. BOOKS Bonbright, James C. The V a l u a t i o n of P r o p e r t y , A T r e a t i s e on the A p p r a i s a l of Property f o r D i f f e r e n t Legal Purposes. New York: McGraw-Hill Book Company, Inc., 1937. Cannan, Edwin. The H i s t o r y of L o c a l Rates jln England. Second e d i t i o n . London: P. A. King and Son, 1927. Cochran, W i l l i a m G. Sampling Techniques. Second e d i t i o n . New York: John Wiley and Sons, Inc., 1963• Croxton, F r e d e r i c k E. Elementary S t a t i s t i c s with A p p l i c a -t i o n s i n Medicine and the B i o l o g i c a l Sciences. New York: Dover P u b l i c a t i o n s , I nc., 1959. Due, John F. Government Finance 1 An Economic A n a l y s i s . T h i r d e d i t i o n , r e v i s e d . Homewood, I l l i n o i s : Richard D. I r w i n , Inc., 1963. E z e k i e l , Mordecai and K a r l A. Fox. Methods of C o r r e l a t i o n and Regression A n a l y s i s . Third e d i t i o n . New York: John Wiley and Sons, 1959. Groves, H a r o l d M. Financing Government. T h i r d e d i t i o n . New York: Henry H o l t and Company, 1950. H i c k s , U. K. P u b l i c Finance. Second e d i t i o n , r e v i s e d . London: James Nisbet and Company L i m i t e d , 1955« Jensen, J . P. Government Finance. New York: Lomas, Y. C r o w e l l Company, 1937. Kendrick, M. S. P u b l i c Finance, P r i n c i p l e s and Problems. New York: H o u g h t o n - M i f f l i n , 1951. Lawrance, D. M., W. H. Rees, and W. B r i t t o n . Modern Methods of V a l u a t i o n of Land, Houses and B u i l d i n g s . F i f t h e d i t i o n . London: The E s t a t e s Gazette L i m i t e d , 1962. L u t z , H a r l e y L e i s t . P u b l i c Finance. Fourth e d i t i o n . New York: D. Appleton-Century Company, In c . , 19 k7. 137 M a r s h a l l , A l f r e d . P r i n c i p l e s of Economics. E i g h t h e d i t i o n . London: Macmillan and Co. L t d . , 1962. M i l l s , F r e d e r i c k C. S t a t i s t i c a l Methods. Third e d i t i o n . New York: Henry H o l t and Company, 1955* Seligman, Edwin R. A. Essays i n Taxation. S i x t h e d i t i o n . London: Macmillan and Company L t d . , 1909. . S h i f t i n g Incidence of Taxation. F i f t h e d i t i o n , r e v i s e d . New York: Columbia U n i v e r s i t y P r e s s , 1926. Smith, Adam. Wealth of Nat i o n s , Book V. Turvey, Ralph. The Economics of Re a l P r o p e r t y : An A n a l y s i s of Property Values and P a t t e r n s of Use. London: George A l l e n and Unwin, L t d . , 1957. Wendt, Pau l F. Real Estate A p p r a i s a l , A C r i t i c a l A n a l y s i s of Theory and P r a c t i c e . New York: Henry H o l t and Company, 1956. B. CASES Assessment Committee of the M e t r o p o l i t a n Borough of Poplar v. Roberts, 38 T.L.R7T-99, House of Lords (1922). B e l l H o t e l s (1935) L t d . v. Motion and Another, The E s t a t e s Gazette, May 17, 1952. Dodds and Dodds v. Millman, k 5 D.L.R. (2d) (I96*f). Ladjes Hpsjery and Underwear L i d . v. West Middlesex Assess-ment Committee, 2 K.B. 679 (1932). Norwich Assessment Committee v. P o i n t e r , 2 K.B. k 9 ( D i v i s i o n a l C o u r t ) , k 71 (Court of Appeal) (1922). R. v. The J u s t i c e s of General Assessment Sessions f o r the M e t r o p o l i s , 2 B.D. 17, 39L (1886). 138 Re Assessment E q u a l i z a t i o n Act: Rowan's Appeal, ho W.W.R 627 (1962). M I r v i n g O i l Cp^. Assessment, 2 D.L.R. 7kk ( 19 L 8) . R o b i n s o n Brothers (Brewers) Lid., v. Houghton and. Chestey-l e - S t r e e t Assessment Committee, 2 K.B. (1937). Re Withycombe E s t a t e , 1 W.W.R. 385 ( A p p e l l a t e D i v i s i o n ) (19M+)7 C. DOCUMENTS Gross N a t i o n a l Product: N a t i o n a l Accounts Income and Expenditure. Dominion Bureau of S t a t i s t i c s , 195? and 1961. Province of B r i t i s h Columbia. A p p r a i s a l Manual. Report of the Royal Commission on Finance and M u n i c i p a l Taxation j n New Brunswick. F r e d e r i c t o n : 1963. Taxation and Revenue S t a t i s t i c s : A C o n s o l i d a t i o n of P u b l i c Finance S t a t i s t i c s of M u n i c i p a l , P r o v i n c i a l Governments and the Government of Canada. Dominion Bureau of S t a t i s t i c s , 195 k and 1961. U. S. Department of Commerce, Bureau of the Census. Taxable Property Values. Census of Governments, 1962. D. MONOGRAPHS Committee on Sales R a t i o Data. Guide f o r Assessment-Sales R a t i o Studjes. A Report to the N a t i o n a l A s s o c i a t i o n of Tax Ad m i n i s t r a t o r s . Chicago: F e d e r a t i o n of Tax Ad m i n i s t r a t o r s , 195? • Coons, A l v i n E. and Bert L. Glaze. Housing Market A n a l y s i s and the Growth of Nonfarm Home Ownership. Bureau of Business Research Monograph #115. Columbus: Bureau of Business Research, Col l e g e of Commerce and Adminis-t r a t i o n , The Ohio State U n i v e r s i t y , 1963. 139 F i n n i s , F r e d e r i c H. Real Property Assessment i n Canada. Canada Tax Papers No. 30. Toronto: Canadian Tax Foundation, 1962. F i s h e r , E. M. Urban Real E s t a t e Markets: C h a r a c t e r i s t i c s and F i n a n c i n g . New York: N a t i o n a l Bureau of Economic Research, 1951. N a t i o n a l A s s o c i a t i o n of Tax A d m i n i s t r a t o r s . Guide f o r Assessment Sales R a t i o S t u d i e s , A Report of the Committee on Sales R a t i o Data. Chicago: F e d e r a t i o n of Tax Ad m i n i s t r a t o r s , June 195*+. Report of a Study Group of the Royal I n s t i t u t e of P u b l i c A d m i n i s t r a t i o n . New Sources of L o c a l Revenue. London: George A l l e n and Unwin L t d . , 1956". E. PERIODICALS Committee on State E q u a l i z a t i o n of Property Tax Assessments, N a t i o n a l Tax A s s o c i a t i o n . Proceedings of the 5 l s t Annual Conference, 1958. Corbert, L e s l i e E. "An A p p r a i s a l of Sales R a t i o S t u d i e s , " N a t i o n a l Tax A s s o c i a t i o n , Proceedings of the 5 l s t Annual Conference, 1958. Dupre, J . Stefan. "The Property Tax i n Canada," N a t i o n a l Tax A s s o c i a t i o n , Proceedings of the 5 l s t Annual Conference, 195°. F i n n i s , F. H. "Measuring the L o c a l Tax Base," Canadian Tax J o u r n a l , V o l . J X I I , N O . I (January-February, 196?). H e l l e r s t e i n , Jerome R. "The Appeal Machinery i n Property Taxation," N a t i o n a l Tax A s s o c i a t i o n , Proceedings of the 51st Annual Conference, 1958. Johnson, Dudley W. "A Note on L o c a l A d m i n i s t r a t i o n of the Property Tax: A,Case Study of Bethlehem, Pennsylvania," N a t i o n a l Tax J o u r n a l , V o l . XI (1958). Macy, C. Ward. "The Property Tax i n the F i s c a l System," N a t i o n a l T a x . A s s o c i a t i o n , Proceedings of the 51st Annual Conference, 1958. iko Massey, F. J . "The Kolmogorov-Smirnov Test f o r Goodness of F i t , " J o u r n a l of the American S t a t i s t i c a l A s s o c i a - t i o n , 1951. Meyer, H a r o l d F. "Market Value—Beacon L i g h t of A p p r a i s a l f o r Taxation,",The A p p r a i s a l J o u r n a l , V o l . X X X I I I , No. h (October, 1965). Murray, W i l l i a m A. "Improvement i n Real E s t a t e Taxation through Assessment-Sales S t u d i e s , " N a t i o n a l Tax J o u r n a l , V o l . 5 (1952). Muth, Richard F. "The Demand f o r Non-Farm Housing," i n Arnold C. Harberger, ed., The Demand f o r Durable Goods. Chicago: U n i v e r s i t y of Chicago P r e s s , i960. Oldman, O l i v e r and Henry Aaron, "Assessment-Sales R a t i o s under the Boston Property Tax," N a t i o n a l Tax J o u r n a l , 1965. Pearson, E. A. "A Further Development of Tests of Normality," Bjometrjka, 1930. Report of the Committee on State E q u a l i z a t i o n of L o c a l Property Tax Assessments. N a t i o n a l Tax A s s o c i a t i o n , Proceedings of th_e_ 5 i?t Annual Conference, 1958. Rountrey, J . Edward. " E q u a l i z a t i o n at Market Value," The A p p r a i s a l J o u r n a l , V o l . 2k (1956). R u s s e l l , John H. " I n e q u a l i t y of Real E s t a t e Assessments w i t h i n P o l i t i c a l S u b d i v i s i o n s , " The Commonwealth Magazine. V i r g i n i a : 1939. Sheppard, W. F. "The C a l c u l a t i o n of the Moments of a Frequency D i s t r i b u t i o n , " B i o m e t r i k a , 1907. . "On the C a l c u l a t i o n of the Most Probable Values of Frequency Constants f o r Data Arranged According to Equ i - D i s t a n t D i v i s i o n s of a S c a l e , " London Mathematical S o c i e t y , Proceeding, 1898. Shoup, C a r l . " C a p i t a l i z a t i o n and S h i f t i n g of the Property Tax," i n H a r o l d M. Groves, ed., Viewpoints on P u b l i c Finance. Reprinted from Property Taxes, Tax I n s t i t u t e , New York. New York: Henry H o l t and Company, 19 k7« Ikl Townsend, Roswell G. " I n e q u a l i t i e s of R e s i d e n t i a l Property-Taxation i n M e t r o p o l i t a n Boston," N a t i o n a l Tax J o u r n a l , V o l . IV (195D. Welch, Ronald B. "State Tax Department R e s p o n s i b i l i t y f o r L o c a l Assessments: A d m i n i s t r a t i v e Review," N a t i o n a l Tax A s s o c i a t i o n , Proceedings of _th_e, V+th N a t i o n a l Conference, 1 9 kl. White, P. H. "Prologue to An A n a l y s i s of the R e s i d e n t i a l Mortgage Market i n Vancouver," Conference of the A s s o c i a t i o n of Canadian Business Schools, Proceedings, 1965. F. STATUTES A l b e r t a , Revised S t a t u t e s , 19 k2. A l b e r t a Succession Duty Act. B r i t i s h Columbia, Revised S t a t u t e s , i960. Assessment E q u a l i z a t i o n Act. B r i t i s h Columbia, Revised S t a t u t e s , i960. Land R e g i s t r y Act. B r i t i s h Columbia, S t a t u t e s , 1 9 5 7 . The Vancouver Charter. B r i t i s h Columbia, S t a t u t e s , 1 9 5 7 . M u n i c i p a l Act. Great B r i t a i n , S t a t u t e s aj. Large, 2 5 and 2 6 V i c t . ( 1 8 6 2 ) . Union Assessment Committee Act. Great B r i t a i n , S t a t u t e s a£ Large, 3 2 and 3 3 V i c t . ( I 8 6 9 ) . S i e V a l u a t i o n ( M e t r o p o l i s ) Ag£. Great B r i t a i n , S t a t u t e s a_£ Large, 1 5 and 1 6 , Geo. 5 ( 1 9 2 5 ) . Rating and V a l u a t i o n Act. Nova S c o t i a , S t a t u t e s . 1 9 3 8 . The Assessment Act. G. UNPUBLISHED MATERIALS Sehwinden, James. "Real P r o p e r t y Assessment P o l i c y and P r a c t i c e i n Minnesota." Unpublished Ph.D. d i s s e r t a -t i o n , Department of A g r i c u l t u r e , U n i v e r s i t y of Minnesota, 196l. APPENDIX A APPENDIX A Included i n Appendix A i s the S t a t i s t i c a l A n a l y s i s and C o n t r o l s e c t i o n of the Province of B r i t i s h Columbia A p p r a i s a l Manual. Except f o r some examples of a p p l i c a -t i o n s the e n t i r e s e c t i o n i s presented below. Also i n c l u d e d are: E x h i b i t I - Form A.C. 2 - Real Property Transfer Record E x h i b i t I I - Form A.C. 7 - Request f o r Sales A n a l y s i s Records E x h i b i t I I I - Form A.C. 3 - Sales A n a l y s i s Record. STATISTICAL ANALYSIS AND CONTROL I n t r o d u c t i o n Assessment of r e a l property creates the base on which i s erected the taxable s t r u c t u r e that produces the greatest p a r t of the funds necessary f o r the operation of the j u r i s d i c t i o n . That s t r u c t u r e i s composed of a multitude of i n d i v i d u a l taxpayers, each of whom forms a part of the whole with an e f f e c t i n p r o p o r t i o n to h i s s i z e . V a l u a t i o n techniques determine the proper s i z e s of the i n d i v i d u a l tax l i a b i l i t i e s , and s t a t i s t i c a l techniques are employed to check the e f f e c t s of the v a l u a t i o n techniques and a s s i s t i n ensuring that the i n d i v i d u a l v a l u a t i o n s and consequent shares of tax l i a b i l i t y are j u s t , uniform, and eq u i t a b l e . " J u s t " as conforming to r e l e v a n t s t a t u t o r y requirements; "uniform" as bearing a f a i r r e l a t i o n s h i p to other comparable p r o p e r t i e s w i t h i n the j u r i s d i c t i o n ; and "e q u i t a b l e " i n being valued s i m i l a r l y to comparable p r o p e r t i e s i n a l l other j u r i s d i c t i o n s i n the Provin c e . Assembly and A n a l y s i s In order to conduct a s t a t i s t i c a l check of v a l u a t i o n r e s u l t s , i t i s e s s e n t i a l to c o l l e c t or assemble the info r m a t i o n which, when analysed, w i l l produce u s e f u l i n f o r m a t i o n . There are three b a s i c methods of checking assessed v a l u e s — v i z . , (1) a p p r a i s a l v e r i f i e d by another s t a f f a p p r a i s e r or by a p r i v a t e a p p r a i s e r , (2) study of the income y i e l d , and (3) assessment-sales a n a l y s i s . Each method has c e r t a i n inherent advantages and disadvantages which must be weighed before d e c i d i n g on the method to be fo l l o w e d i n a p a r t i c u l a r i n s t a n c e . The f i r s t two methods ar e , i n f a c t , v a l u a t i o n s , and the subsequent a n a l y s i s i s a comparison of values determined by d i f f e r e n t i n d i v i d u a l s or by means of d i f f e r e n t methods. Assessment-sales r a t i o s t u d i e s , wherein the assessed values are shown as percentages of the sale p r i c e s , form the most common v e h i c l e f o r determining assessment i n e q u a l i t i e s . E d i t i n g and Screening of Sales The s e l e c t i o n of sales items f o r study purposes r e q u i r e very c a r e f u l e d i t i n g to ensure that only bona f i d e arm's 156 length t r a n s a c t i o n s are considered. The sale v a l u e , or market v a l u e , declared must be the value agreed upon between a true vendor and a true purchaser. The best value obtainable i s that a r r i v e d at by agreement between two p a r t i e s , n e i t h e r of whom i s under pressure to buy or s e l l . What i s r e q u i r e d i s the true market v a l u e , which may be defined as "the p r i c e which the property would b r i n g i n an open market on a f r e e , not f o r c e d , sale between a w i l l i n g buyer and a w i l l i n g and able s e l l e r . " By S tatute " a c t u a l v a lue" i s the b a s i s of r e a l - p r o p e r t y assessment, and i n determining " a c t u a l value" the assessor may, among other t h i n g s , give c o n s i d e r a t i o n to "the p r i c e that such land and improvements might reasonably be expected to b r i n g i f o f f e r e d f o r sale i n the open market by a solvent owner." Every attempt must be made to avoid using s a l e s informa-t i o n based upon unrepresentative s a l e s . Sales may be considered to be unrepresentative f o r many reasons, which may be summarized i n t o the statement that "the p r i c e does not i n d i c a t e the p r i c e at which the property could be s o l d by someone who ,took due time and pains to secure the best market." A sale p r i c e may be unrepresentative a l s o because i t i s abnormally h i g h . Subsequent i n v e s t i g a t i o n may show that the purchaser had a s p e c i a l i n t e r e s t i n the p a r t i c u l a r p r o p e r t y , such as being the owner of an adjacent p a r c e l . No s a l e should be discarded s o l e l y because the p r i c e i s apparently too high or too low. Such a c t i o n should be taken only a f t e r i n v e s t i g a t i o n i n d i c a t e s that the p r i c e i s not r e p r e s e n t a t i v e . A market i n t r a n s i t i o n may produce e i t h e r high or low p r i c e s , which w i l l i n d i c a t e the beginning of a new value phase. Such trends are important, and i r r e s p o n s i b l e e d i t i n g can destroy the evidence of such trends. In B r i t i s h Columbia the o b t a i n i n g of i n f o r m a t i o n about sa l e p r o p e r t i e s i s much e a s i e r than i n many j u r i s d i c t i o n s . S i x Land R e g i s t r y O f f i c e s administered by the Province c o n s o l i d a t e a l l land t i t l e i n f o r m a t i o n i n t h e i r r e s p e c t i v e geographical l o c a t i o n s . T i t l e searches are thus f a c i l i t a t e d . Each assessor i s f u r t h e r a s s i s t e d by being given a copy of every t i t l e change of ownership, complete with the market value of the property as d e c l a r e d to the Land R e g i s t r y O f f i c e . As the fees c o l l e c t e d by the Land R e g i s t r y O f f i c e s f o r r e g i s t r a t i o n are l e v i e d on and ad valorem b a s i s , i t may be accepted that the declared values are reasonably c l o s e to the market value. When p r o p e r l y e d i t e d these "yellowback" copies of t i t l e s may be used i n assessment-s a l e s r a t i o . s t u d i e s , but as s u p p l i e d these "yellowbacks" are unscreened f o r bona f i d e s a l e s . Ik7 Further a i d to the assessors i s provided by the O f f i c e of the Assessment Commissioner, which, i n co-operation with the Department of the Attorney-General and the Land R e g i s t r y O f f i c e s , i n s t i t u t e d a system of s a l e s r e p o r t i n g whereby the Land R e g i s t r y O f f i c e s demand the f i l i n g of a questionnaire t i t l e d "Real Property Transfer Record" and designated as Form A.C. 2. This questionnaire i s r e q u i r e d to be f i l e d f o r each t r a n s a c t i o n where production of a tax c e r t i f i c a t e i s r e q u i r e d to accompany the a p p l i c a t i o n f o r t i t l e change. This requirement a u t o m a t i c a l l y excludes the f o l l o w i n g types of non-bona f i d e t r a n s f e r s : (a) Changes of executors or t r u s t e e s . (b) Transmissions by death or f o r e c l o s u r e . (c) Q u i t - c l a i m s . (d) Transfers to a b e n e f i c i a r y under a w i l l . In order to exclude other non-bona f i d e t r a n s f e r s , s p e c i f i c exemption from f i l i n g the questionnaire i s made f o r a p p l i c a t i o n s to r e g i s t e r mortgages, options to pur-chase, l e a s e s , l i f e e s t a t e s , Crown grants, and tax s a l e s . In t h i s manner, then, questionnaires are received covering only sales which are predominantly bona f i d e t r a n s a c t i o n s . F u r t h e r e d i t i n g i s done by the O f f i c e of the Assessment Commissioner to e l i m i n a t e t r a n s f e r s which, from i n f o r m a t i o n given i n the q u e s t i o n n a i r e , are not bona f i d e s a l e s . These would i n c l u d e f a m i l y or r e l a t e d p a r t i e s t r a n s f e r s , trades, and extremely low down-payment t r a n s a c t i o n s , and others, as shown below: (a) Transfers agreed upon some time p r i o r to the year i n which they were recorded. (b) Transfers i n v o l v i n g unusual f i n a n c i a l p r o v i s i o n s . Cc) Transfers of p a r t l y f i n i s h e d improvements. (d) Transfers i n c l u d i n g personal property, the value of which cannot be separated from the t o t a l c o n s i d e r a t i o n . (e) Transfers i n v o l v i n g trades. ( f ) Transfers between r e l a t i v e s or a s s o c i a t e d corpora-t i o n s . 158 (g) Transfers under f i n a n c i a l duress. (h) Transfers to any church, lodge, s c h o o l , or c h a r i t a b l e , benevolent, f r a t e r n a l , or government o r g a n i z a t i o n . ( i ) Transfers on which the value s t a t e d i s an opi n i o n r a t h e r than an a c t u a l exchange p r i c e . ( j ) Transfers the circumstances of which are unusual and as a r e s u l t render the t r a n s a c t i o n useless f o r s a l e s a n a l y s i s . A f t e r t h i s e d i t i n g process, the sales remaining are reported to the assessor concerned by means of the "Sales A n a l y s i s Record," designated Form A.C. 3 . Because of t h i s p rogressive screening-out of non-bona f i d e t r a n s f e r s , the assessor r e c e i v e s Forms A.C. 3 f o r only a p o r t i o n of the t r a n s f e r s reported to him on "yellowbacks." In a d d i t i o n to the Forms A.C. 3 , the assessor a l s o r e c e i v e s a form l i s t i n g the Forms A.C. 2 r e j e c t e d by the O f f i c e of the Assessment Commissioner. Any t r a n s f e r s reported on "yellowbacks" which are not accounted f o r by a Form A.C. 3 or on the N o t i f i c a t i o n of Forms A.C. 2 Rejected (Form A.C. 19) should be only t r a n s f e r s such as those described above, f o r which no tax c e r t i f i c a t e i s r e q u i r e d to be f i l e d . The assessor may be assured, then, that the sales not reported on Forms A.C. 3 are e i t h e r suspect or d e f i n i t e l y not bona f i d e t r a n s a c t i o n s . As the e d i t i n g by the O f f i c e of the Assessment Commissioner i s p r i m a r i l y to o b t a i n a hard core of bona f i d e sales f o r Province-wide e q u a l i z a t i o n of assessments, i t does r e s u l t i n the a r b i t r a r y r e j e c t i o n of some t r a n s f e r s which the assessor c o u l d , through l o c a l knowledge, salvage and use f o r both v a l u a t i o n and study purposes when the volume obtained by means of Forms A.C. 3 i s too small. Whenever p o s s i b l e the ed i t e d sales i n f o r m a t i o n shown on the Forms A.C. 3 should be confirmed by the assessor through h i s c o n f i d e n t i a l i n q u i r y with the purchaser, vendor, h i s s o l i c i t o r or agent. The assessor must s a t i s f y h i m s e l f that the s a l e s i n f o r m a t i o n i s reasonably accurate before using i t i n the v a l u a t i o n process, or approving i t f o r assessment-sa l e s r a t i o s t u d i e s . The assessor completes the Form A.C. 3 showing the r e l a t i v e assessed v a l u e s , dimensions, type of property, and economic s u b - d i s t r i c t , and a l s o , i f r e q u i r e d , comments as to the s u i t a b i l i t y of the sale f o r a n a l y s i s . 159 From t h i s i n f o r m a t i o n the O f f i c e of the Assessment Commissioner i s able to make a f u r t h e r e d i t , then f o r each of the r e s u l t i n g bona f i d e s a l e s the assessment-sales r a t i o s are c a l c u l a t e d , and by means of mechanical ta b u l a -t i o n f a c i l i t i e s assessment-sales r a t i o s t u d i e s are prepared as r e q u i r e d . P r i n t e d l i s t i n g s of these bona f i d e sales are a v a i l a b l e f o r reference and work-paper use. Assessment-Sales R a t i o A n a l y s i s Having assembled the data r e q u i r e d f o r study, the next step must be to prepare i t i n a manner s u i t a b l e f o r a n a l y s i s . The s t y l e of the Form A.C. 3 i s such that many d i f f e r e n t combinations of i n f o r m a t i o n may be made f o r study purposes. Because every j u r i s d i c t i o n i n c l u d e s areas or zones of d i f f e r i n g a c t i v i t y , the formation or d e s i g n a t i o n of sub-d i s t r i c t s has been f o s t e r e d by the O f f i c e of the Assessment Commissioner. Sales w i t h i n these s u b - d i s t r i c t s may be stud i e d by one s u b - d i s t r i c t alone or by l a r g e r areas formed by combining s a l e s w i t h i n two or more s u b - d i s t r i c t s . Two a p p r a i s e r s may have s l i g h t l y d i f f e r e n t approaches to v a l u a t i o n , and i f they confine t h e i r v a l u a t i o n a c t i v i -t i e s to s p e c i f i c geographic areas, the degree of divergence between t h e i r v a l u a t i o n s may be determined by studying s a l e s of s i m i l a r p r o p e r t i e s o c c u r r i n g i n t h e i r r e s p e c t i v e d i s t r i c t s . This matter of s u b - d i s t r i c t s i s very important, and much thought and study put i n t o the l o c a t i o n of sub-d i s t r i c t boundaries w i l l pay o f f i n more i n f o r m a t i v e evidences of value changes. The boundaries need not be permanent, but may be v a r i e d as the economic need f o r changes becomes obvious. While stud i e s of comparative areas over a long p e r i o d are p r e f e r a b l e to short-term s t u d i e s , i f the b a s i c c h a r a c t e r i s t i c s of the area have changed during the long p e r i o d , then the r e s u l t s of such a study w i l l be misleading and erroneous. Recognition of the need f o r change must be made by a l t e r i n g s u b - d i s t r i c t boundaries, with advice to the O f f i c e of the Assessment Commissioner whenever such changes are made. I t may be that c e r t a i n s u b - d i s t r i c t boundaries w i l l apply f o r land v a l u a t i o n , whereas smaller s u b - d i s t r i c t s would be b e t t e r f o r improvements v a l u a t i o n . There are two methods of handling t h i s s i t u a t i o n — ( l ) small s u b ^ d i s t r i c t boundaries s u i t a b l e f o r improvements study and v a l u a t i o n 150 may be formed, and l a r g e r s u b - d i s t r i c t s comprising the r e q u i r e d number of smaller s u b - d i s t r i c t s may be formed f o r land study and v a l u a t i o n , and (2) c e r t a i n types or c l a s s e s of improvements may be found i n two or more areas i n the j u r i s d i c t i o n , and f o r study and v a l u a t i o n purposes may be considered s i m i l a r . Such s i m i l a r groups may be given the same s u b - d i s t r i c t number wherever they are l o c a t e d i n the j u r i s d i c t i o n . Super-imposed on these may then be l a i d boundaries f o r land study and v a l u a t i o n purposes. As p r o v i s i o n e x i s t s f o r 3 - d i g i t s u b - d i s t r i c t numbers, thus the l a r g e r land s u b - d i s t r i c t s could be designated as 100, 200, and 300, and w i t h i n those could be improvements sub-d i s t r i c t s designated as 1, 2, 3> k> e t c . Thus the complete s u b - d i s t r i c t number could read 102, r e p r e s e n t i n g land sub-d i s t r i c t 100 and improvement s u b - d i s t r i c t 2. In l i k e manner, s u b - d i s t r i c t 302 would represent land s u b - d i s t r i c t 300, but a l s o improvement s u b - d i s t r i c t 2, the improvements t h e r e i n being s i m i l a r to those represented by the 2 i n land s u b - d i s t r i c t 102. A f u r t h e r a n a l y s i s breakdown that should not be overlooked i s the separation of corner l o t s , and waterfront l o t s , from the i n s i d e l o t s . There may be a s i z e a b l e v a r i a t i o n i n p r i c e s p a i d f o r waterfront p r o p e r t i e s as compared with those paid f o r i n s i d e l o t s , and s i m i l a r l y a d i f f e r e n c e may e x i s t f o r corner p r o p e r t i e s . By grouping these v a r i o u s types together i n d i s c r i m i n a t e l y , the i n f l u e n c e s of any s p e c i a l group are l o s t , but by separate a n a l y s i s a proper v a l u a t i o n adjustment may be made where i t i s necessary. Another p r e c a u t i o n to take i n preparing to analyse the assembled i n f o r m a t i o n i s to be sure that the p r o p e r t i e s being s t u d i e d conform i n a c t u a l use to the predominant land use i n the immediate neighbourhood. For example, i f the predominant land use i s r e s i d e n t i a l and the a c t u a l use of a p a r t i c u l a r property i s , say, commercial, then the subject property should not be s t u d i e d w i t h p r o p e r t i e s having a r e s i d e n t i a l a c t u a l use. Study of the sale of a neighbourhood corner s t o r e w i l l not y i e l d much in f o r m a t i o n to help i n v a l u i n g r e s i d e n t i a l p r o p e r t i e s , but i t may be u s e f u l when compared with other sales of s i m i l a r non-conforming p r o p e r t i e s . In so f a r as p o s s i b l e , l i k e must be compared with l i k e . This i s t r u e , too, with the assessed values. Before studying assessment-sales r a t i o s , the assessor must assure h i m s e l f that the assessment r e f l e c t s the value of the 151 property s o l d . I f vacant land was s o l d , an assessment which i n c l u d e s the assessed value of an improvement i s o b v i o u s l y i n c o r r e c t . I t i s not so obvious, however, that l a c k of comparison e x i s t s i n other i n s t a n c e s . I f a water-main, sewer, or road s u r f a c i n g e x i s t s at the time of assessment, but d i d not at the date of s a l e , then the assessment does oot r e f l e c t the property as s o l d . In a s i m i l a r manner, improved p r o p e r t i e s must have assessed values f o r the improvements as they were at the time of sale i n order to be most e f f e c t i v e i n a study. I f a furnace, porch, basement room, or garage has been added and assessed since the date of the s a l e , then the trans-a c t i o n i s not good f o r a n a l y s i s . The O f f i c e of the Assessment Commissioner must r e j e c t such items because of i n s u f f i c i e n t i n f o r m a t i o n , but with care the assessor can salvage such items f o r h i s use. The e f f e c t of the p h y s i c a l change may be e i t h e r very minor or s u b s t a n t i a l , but he must acknowledge i t . He may adjust the assessed value i n many i n s t a n c e s j f o r study purposes only, by deducting the r e l a t i v e assessed value of the furnace, porch, basement room, garage, or whatever e x t r a has been added i n an attempt to o b t a i n an assessment based on the property as i t was at the date of s a l e . This method should be r e s o r t e d to only i f the number of sales i s meagre. S t a t i s t i c a l a n a l y s i s must be approached w i t h great care when the volume of items i s small. When numbers are few and the c o n c e n t r a t i o n of r a t i o s i s widespread, no f a c t u a l s t a t i s t i c a l deductions are v a l i d . In p a r t i c u l a r , no c o n c l u s i o n should be drawn from a s i n g l e item, and no property should be revalued based upon a s i n g l e sale of that property. That p r a c t i c e would destroy r a t h e r than create u n i f o r m i t y because the s a l e p r i c e f o r a given property i s the end r e s u l t of n e g o t i a t i o n between two p a r t i e s and may be high or low, according to the r e l a t i v e persuasive powers of the two p a r t i e s . When the volume of items being analysed i s s m a l l , each item has an appreciable e f f e c t on the r e s u l t , but when the volume i s l a r g e , then the i n d i v i d u a l i n f l u e n c e s are tempered and, as a consequence, extreme items, e i t h e r high or low, are overshadowed by the bulk. The question of v a l u a t i o n based upon the p r i c e p a i d f o r the property being valued was the subject of a Supreme Court d e c i s i o n by the Honourable Mr. J u s t i c e Wootton i n the case of Robert J . Rowan v. C i t y of Vancouver i n 1962, and your a t t e n t i o n i s drawn to that d e c i s i o n . C a l c u l a t i o n of S t a t i s t i c a l Measures 152 1. Measures of Absolute D i s p e r s i o n (a) Mean, Median, Mode, For r e l a t i v e ease of c a l c u l a -t i o n of s t a t i s t i c a l measures, i t i s recommended that the assessment-sales r a t i o s c a l c u l a t e d and assembled as described above be i n the form of an array i n ascending order. I f only a few items are i n v o l v e d they may be set up i n numerical order, such as 20, k 2 , k 2 , kk, M+, k k , k 5 , k 5 , L 6 , k 9 , 50. The a r i t h m e t i c mean of these items i s the sum of each d i v i d e d by the number of items, or k 71 ; 11 = k 2 . 8 per cent. The median i s that value i n the middle, which has the same number of items below i t as above i t ; i n the above inst a n c e i t i s kk per cent. The mode i s that value which i s most numerous. In the above there are three items with kk per cent r a t i o s ; thus kk per cent i s the mode. The use of the median i s recommended as i t i s the measure of the middle item and i s thus t y p i c a l and i s not g r e a t l y a f f e c t e d by extremely high or low r a t i o s . Where a l a r g e number of items are to be s t u d i e d , i t i s e a s i e r to enter the r a t i o s i n the form of a frequency d i s t r i b u t i o n . This c o n s i s t s of the t a l l y i n g of r a t i o s i n predetermined percentage i n t e r v a l s or groups, the r e s u l t a n t number of items f a l l i n g i n a percentage i n t e r v a l being the frequency of occurrence of r a t i o s w i t h i n that i n t e r v a l , and the whole being the d i s t r i b u t i o n of r a t i o s between the v a r i o u s i n t e r -v a l s . A b a s i c assumption i s made that the number of items w i t h i n an i n t e r v a l i s spread evenly from the low to high l i m i t s of that i n t e r v a l . The s i z e of the i n t e r v a l as used by the O f f i c e of the Assessment Commissioner has been 10 percentage p o i n t s , as 30-39 per cent, k0-k9 per cent, etc.* A t y p i c a l frequency d i s t r i b u t i o n on t h i s b a s i s f o l l o w s : 153 EXAMPLE 1.—ASSESSMENT-SALES RATIO ANALYSIS, VACANT LAND SALES Type of Property Median Frequencies of Assessment R a t i o s 0-9 10-19 20-29 30-39 1+0-59 -50-59 60-69 70-79 80-1 89 90-99 100 and oveil (To-t a l R e s i d e n t i a l v a c a n t — Inside l o t s -Corner l o t s -Waterfront l o t s 50 37.5 25 10 1+ 12 3 To c a l c u l a t e the median r a t i o f o r the above i n s i d e l o t s , f o r example, d i v i d e the t o t a l number by 2, thus 50 f 2 - 20, and count from the lowest r a t i o u n t i l that number has been reached; thus i n the i n t e r v a l 20-29 there are 2 items, and 8 i n the i n t e r v a l 30-39? and 10 i n the i n t e r v a l 1+0-59. Adding, 2 + 8 + 10 = 20, i n d i c a t i n g that the median r a t i o i s 50 per cent. This may a l s o be determined by counting down from the h i g h e s t r a t i o s , thus 1 + 1 + 1 + 2 + 3 + 12 = 2 0 , again a r r i v i n g at 50 per cent. Whenever there i s doubt about the accuracy of a c a l c u l a t e d median, i t may be checked i n t h i s manner. To c a l c u l a t e the median r a t i o f o r the c o r n e r - l o t frequency d i s t r i b u t i o n , again d i v i d e the t o t a l by 2 (17 r 2 = 8 . 5 ) , and count from the lower end (10-19) u n t i l 8.5 i s reached. In t h i s case 1 i n the 10-19 i n t e r v a l p l u s 3 i n the 20-29 i n t e r v a l plus 6 i n the 30-39 i n t e r v a l i s too much. Up to the 30-39 i n t e r v a l there are 5 items; another 5.5 are needed. The ba s i c assumption i s that the items i n an i n t e r v a l are evenly spread w i t h i n that i n t e r v a l . Therefore, c a l c u l a t e 5.5 items of the group of 6, m u l t i p l y by the s i z e of the i n t e r v a l (10 i n t h i s case) and add i t to the low l i m i t of the i n t e r v a l (30 per cent h e r e ) . Thus 5 . 5 7 6 i s .75, m u l t i p l i e d by 10 gives 7 .5, which when added to the low l i m i t of 30 gives the c a l c u l a t e d median of 37.5 per cent. This a l s o may be checked from the top end, but i n t h i s example 1.5 4- 6" or 2.5 per cent must be subtracted, from the 50-per-cent l i m i t . The answer i s s t i l l 37.5 per cent. 15? Another measure that may be determined 1 i s the ranee , being the d i f f e r e n c e between the lowest r a t i o and the h i g h e s t . Thus i n the examples above the ranges would be: CD For i n s i d e l o t s , from 20 to over 100, or 80-plus percentage p o i n t s . (2) For corner l o t s , from 10 to 60, or 50 percentage p o i n t s . (3) For waterfront l o t s , from 0 to 60, or 60 percen-tage p o i n t s . I t i s obvious that one estreme item can a f f e c t the range unduly, and f o r that reason the range i s not considered to be a s a t i s f a c t o r y measure. (b) B e c i l e , P e r c e n t i l e , Q u a r t i l e . D e c i l e s , percen-t i l e s , and q u a r t i l e s are other s t a t i s t i c a l measures. D e c i l e s d i v i d e the number of items i n t o 10 equal p a r t s , p e r c e n t i l e s d i v i d e the number i n t o 100 equal p a r t s , and q u a r t i l e s d i v i d e the number of items i n t o k equal p a r t s . These are u s u a l l y used i n such a manner that the number of items r e c e i v i n g i n t e n s i v e study are those remaining a f t e r a deduction of those at the two ends of the array or frequency d i s t r i b u t i o n . Thus the ranges between the 1st and 9th d e c i l e s or the 10th and 90th p e r c e n t i l e s are the same t h i n g , o m i t t i n g the upper and lower 10 per cent. The range between the 1st and 3rd q u a r t i l e s i n c l u d e s only 50 per cent of the t o t a l items. Comparisons of these v a r i o u s measures c a l c u l a t e d f o r one frequency d i s t r i b u t i o n f o l l o w : EXAMPLE 2.—ASSESSMENT-SALES RATIO ANALYSIS 0-9 10-19 20-29 30-39 ?0-^9 50-59 60-69 70-79 80-89 90-99 Over 100 T o t a l — — k 6 15 30 18 15 12 — 100 Per Cent 10th p e r c e n t i l e or 1st d e c i l e ho 1st q u a r t i l e 50 Mode : 55.if Median 58.3 155 Per Cent A r i t h m e t i c mean (common average) 59*5 3rd q u a r t i l e 71.3 90th p e r c e n t i l e or 9th d e c i l e 81.7 Percentage P o i n t s 10 to 90 p e r c e n t i l e range (or 1-9 d e c i l e range) *4l.7 1st to 3rd q u a r t i l e range 21.3 Range (from 20 to 90 per cent) 70 The frequency d i s t r i b u t i o n may als o be p l o t t e d on squared paper to de r i v e a v i s u a l impression of the q u a l i t y of assessment. The above items are shown p l o t t e d on the curve i n the i l l u s t r a t i o n on the next page. (c) Q u a r t i l e D e v i a t i o n , Mean D e v i a t i o n , Standard D e v i a t i o n . Reference to a measure of c e n t r a l tendency, such as the median, as the t y p i c a l or average value i s of l i m i t e d value without f u r t h e r knowledge of the degree of s c a t t e r about that measure. While the range, d e c i l e s , p e r c e n t i l e s , and q u a r t i l e s serve as i n d i c a t o r s of the degree of d i s p e r -s i o n of items, there are other more u s e f u l and d e f i n i t e measures. These are the q u a r t i l e d e v i a t i o n or the semi-i n t e r q u a r t i l e range, the mean d e v i a t i o n , and the standard d e v i a t i o n . As mentioned above, the range i s simply the d i f f e r e n c e between the lowest and h i g h e s t r a t i o s and i s not an i n d i c a t o r of the degree of con c e n t r a t i o n or c e n t r a l tendency of the items. I t s value depends s o l e l y upon the extreme r a t i o s and i s of very l i m i t e d use. ( i ) The q u a r t i l e d e v i a t i o n or the s e m i - i n t e r q u a r t i l e range i s a more i n d i c a t i v e measure. I t i s c a l c u l a t e d by su b t r a c t i n g the 1st from the 3rd q u a r t i l e and d i v i d i n g the r e s u l t by 2. I n the foregoing example the 1st q u a r t i l e was 50 per cent and the 3rd q u a r t i l e was 71.3 per cent, the d i f f e r e n c e between these two being 71.3 - 50 = 21.3. D i v i d i n g t h i s by 2 gives the q u a r t i l e d e v i a t i o n of 21.3 7 2 = 10.65. I f the d i s t r i b u t i o n were p e r f e c t l y symmetrical, the median would l i e half-way between the q u a r t i l e s , and thus one-half of the distance between the q u a r t i l e s may be used, as a measure of the average di s t a n c e of each q u a r t i l e from the FREQUENCY OF ASSESSMENT SALES RATIOS 157 median. As the d i s p e r s i o n of a frequency d i s t r i b u t i o n i s i n c r e a s e d , the distance between the q u a r t i l e s becomes great e r . Hence the smaller the value of the q u a r t i l e d e v i a t i o n , the more t i g h t l y concentrated are the items i n the frequency d i s t r i b u t i o n and the greater the degree of u n i f o r m i t y i n the assessments. A disadvantage of t h i s measure i s that i t i s dependent upon only the c e n t r a l 50 per cent of the cases, while some other measures give c o n s i d e r a t i o n to a l a r g e r percentage of the cases and are thus considered to be more i n f o r m a t i v e . ( i i ) The mean d e v i a t i o n , or average d e v i a t i o n , i s a measure having a value dependent upon the value of every item i n the s e r i e s . This measure c o n s i s t s of the average of the d e v i a t i o n s of each item from t h e i r a r i t h m e t i c mean, or median. Where the number of items i s s m a l l , these d e v i a t i o n s may be c a l c u l a t e d i n d i v i d u a l l y , then averaged, as i s done i n the s o - c a l l e d " R u s s e l l Formula" c a l c u l a t i o n s , but where the volume i s great t h i s becomes a.chore. I f the items are grouped i n the form of a frequency d i s t r i b u t i o n , as i n the e a r l i e r example, the number of c a l c u l a t i o n s may be reduced by means of the f o l l o w i n g procedure: (1) S e l e c t as an a r b i t r a r y o r i g i n the mid-point of the i n t e r v a l c o n t a i n i n g the median. (2) Obtain the d e v i a t i o n s of the mid-points of each i n t e r v a l from the a r b i t r a r y o r i g i n . (3) M u l t i p l y these i n t e r v a l d e v i a t i o n s by the number of items i n the i n t e r v a l . (M) T o t a l a l l the products d e r i v e d i n step (3) and d i v i d e by the t o t a l number of items. The r e s u l t i n g value i s the average d e v i a t i o n i n c l a s s i n t e r v a l s , except f o r an adjustment described below. EXAMPLE 3 I n t e r v a l (Per Cent) Number of items D e v i a t i o n from a r b i -t r a r y o r i g i n i n i n t e r v a l s Frequency X of "deviation 0- 9 10-19 20-29 30-39 ko-k9 50-59 k 6 15 30 3 2 1 0 12 12 15 0 158 EXAMPLE 3 (continued) I n t e r v a l (Per Gent) Number of items D e v i a t i o n from a r b i -t r a r y o r i g i n i n i n t e r v a l s Frequency X of d e v i a t i o n 60-69 70-79 80-89 90-99 100 and over — To t a l s 18 15 12 100 1 2 3 18 36 123 As the median f o r the above, percentage p o i n t s higher than th c e n t ) , or O.33 c l a s s i n t e r v a l s , must be made to compensate. Ass i n t e r v a l are a l l l o c a t e d at the the above case 55 (30 + 15 + 6 + and 1+5 (18 + 15 + 12) items are d e v i a t i o n , or average d e v i a t i o n , i n t h i s manner. 58.3 per cent, i s 3.3 e assumed median (55 per an appropriate adjustment uming the items i n each mid-point, we see that i n 1+) items are understated overstated. The mean then w i l l be c a l c u l a t e d ( T o t a l frequency X d e v i a t i o n from a r b i t r a r y o r i g i n ) d i v i d e d by (the t o t a l number of items) plus an adjustment of (the d i f f e r e n c e i n c l a s s i n t e r v a l s between the assumed and a c t u a l medians) m u l t i p l i e d by (the d i f f e r e n c e between the under- and over-stated i t e m s ) , a l l d i v i d e d by (the t o t a l number of i t e m s ) , t h u s — i f 3 + 0.33 X 55 ~ k ? * 1,263 c l a s s i n t e r v a l s . 100 100 To convert to a c t u a l u n i t s i t i s necessary to m u l t i p l y t h i s f i g u r e by the s i z e of the c l a s s i n t e r v a l , i n t h i s case 10, g i v i n g a mean d e v i a t i o n , or average d e v i a t i o n , of 12.63. For a frequency d i s t r i b u t i o n that i s symmetrical and c h a r a c t e r i z e d by a normal^curve, a measurement of one average d e v i a t i o n on each side of the median w i l l i n c l u d e 57*5 per cent of the items. For moderately skewed d i s t r i -b utions t h i s w i l l be approximately true. 159 ( i i i ) A more e a s i l y c a l c u l a t e d and more u s e f u l measure i s the standard d e v i a t i o n , which i s a s p e c i a l form of average d e v i a t i o n from the mean. The c a l c u l a t i o n of the standard d e v i a t i o n i s as f o l l o w s : (1) S e l e c t as an a r b i t r a r y o r i g i n the mid-point of the i n t e r v a l c o n t a i n i n g the mean. (2) Obtain the d e v i a t i o n s of the mid-points of each i n t e r v a l from the a r b i t r a r y o r i g i n , showing minus signs f o r i n t e r v a l s below and p l u s signs f o r i n t e r v a l s above that o r i g i n . (3) M u l t i p l y these i n t e r v a l d e v i a t i o n s by the number of items i n the i n t e r v a l . (h) M u l t i p l y the products i n step (3) again by the same i n t e r v a l d e v i a t i o n s , thus o b t a i n i n g the frequency (number) times the d e v i a t i o n squared. (5) T o t a l the products i n step ( 3 ) , r e c o g n i z i n g the minus q u a n t i t i e s . (6) T o t a l the products i n step (!+), a l l of which w i l l be p o s i t i v e . (7) D i v i d e the t o t a l i n step (6) by the t o t a l number of items. (8) D i v i d e the t o t a l i n step (5) by the t o t a l number of items, and square the r e s u l t . (9) Take the square r o o t of the d i f f e r e n c e between the r e s u l t s of steps (7) and (8) . (10) M u l t i p l y by the s i z e of the c l a s s i n t e r v a l . C l a s s i n t e r v a l Number of items D e v i a t i o n from a r b i -t r a r y o r i g i n , i n i n t e r v a l s Frequency X d e v i a t i o n ( C o l . 2 X C o l . 3) Frequency X deviatim ( C o l . k X C o l . 3) C o l . 1 f C o l . 2 d* C o l . 3 f d * C o l . k 0-9 — 10-19 20-29 30-39 40 - L 9 50-59 60-69 70-79 80-89 90-99 100 and over Total s -h 6 15 30 18 15 12 100 (3) -12 -12 -15 0 18 30 36 L 5 ( 5 ) 160 The f i g u r e s ( l ) to (6) i n parentheses above represent the steps outlined.above by number. Step (7): 261 7 100 * 2 . 6 l . Step (8): 55 T 100 = .55. .55 X .55 - .2025. Step (9): 2.61 - .2025 - 2.1+075. Square root of 2.5075 s 1.55* Step (10): 1.55 X 10 = 15.5. Thus the standard d e v i a t i o n i s 15»5« The standard d e v i a t i o n measures the d i s p e r s i o n of a s e r i e s : the greater the spread of the s e r i e s , the greater the value of the standard d e v i a t i o n . As a measure of assessment u n i f o r m i t y , the smaller the value of the standard d e v i a t i o n , the greater the degree of u n i f o r m i t y . For a normal d i s t r i b u t i o n , i f a di s t a n c e equal to one standard d e v i a t i o n i s measured o f f on both sides of the a r i t h m e t i c mean, approximately 68 per cent of the values w i l l be i n c l u d e d w i t h i n the l i m i t s thus c a l c u l a t e d . P l u s and minus 2 standard d e v i a t i o n s i n c l u d e s about 95 per cent, and pl u s and minus 3 i n c l u d e s almost a l l of the values (99 Per c e n t ) . As the standard d e v i a t i o n i s measured about the mean, i t would be appropriate to describe how to c a l c u l a t e the mean. From the t a b l e above, the assumed mean i s 55 per cent, but t h i s r e q u i r e s adjustment to d e r i v e the true mean. The adjustment f a c t o r i s simply the i n t e r v a l s i z e m u l t i p l i e d by the t o t a l from step (5) d i v i d e d by the t o t a l number of items. Thus i t i s necessary to add 10 X (*+5 ~ 100), or 5.5, to the assumed mean of 55? g i v i n g a true mean of 59.5 per cent. Both the average d e v i a t i o n and the standard d e v i a t i o n are a f f e c t e d by the value of every item, but greater emphasis i s placed on the extremes i n the standard d e v i a t i o n than i n the mean d e v i a t i o n . Comparisons of the r e l a t i o n s h i p s between the v a r i o u s absolute measures are as f o l l o w s : Range: Coarse c a l c u l a t i o n only; no d e f i n i t e r e l a t i o n -ship to other measures. Q u a r t i l e d e v i a t i o n = 0.6755 standard d e v i a t i o n = 0.8553 average d e v i a t i o n . Average (mean) d e v i a t i o n = 0.7979 standard d e v i a t i o n . 161 Standard d e v i a t i o n = 1.2533 average d e v i a t i o n s l.?826 q u a r t i l e d e v i a t i o n . 2. Measures of R e l a t i v e D i s p e r s i o n To t h i s p o i n t a l l the measures discussed have been measures of absolute d i s p e r s i o n , expressed i n terms of u n i t s of the problem—namely, percentage p o i n t s . I n a d d i t i o n , a measure of d i s p e r s i o n should be compared to the s i z e of the average about which i t i s being measured. This then gives a measure of r e l a t i v e d i s p e r s i o n . (a) C o e f f i c i e n t of v a r i a t i o n i s c a l c u l a t e d by r e l a t i n g the standard d e v i a t i o n to the a r i t h m e t i c mean and expressing i t as a percentage. Thus, f o r the example given above, the mean i s 59*5 per cent and the standard d e v i a t i o n i s 15»55 so 15.5 IC 100 = 26.0 per cent i s the c o e f f i c i e n t of v a r i a t i o n . 59.5 This measure a l s o i s r e f e r r e d to by some a u t h o r i t i e s as the r e l a t i v e standard d e v i a t i o n . (b) C o e f f i c i e n t of dj.spersj.on i s the average d e v i a t i o n of a s e r i e s d i v i d e d by the mean of the s e r i e s . In the example the average d e v i a t i o n i s 12.63, the mean i s 59*5> so the c o e f f i c i e n t of d i s p e r s i o n i s 1 2 , 6 3 X 100 = 21.2 per cent. The smaller the value of 59.5 the c o e f f i c i e n t of d i s p e r s i o n , the greater the degree of un i f o r m i t y . The c o e f f i c i e n t of d i s p e r s i o n i s the "index of assessment i n e q u a l i t y " r e f e r r e d to by the l a t e Dr..John H. R u s s e l l , former D i r e c t o r of Research, V i r g i n i a Department of Taxation. H i s recommendation was that a c o e f f i c i e n t of d i s p e r s i o n of "20 should be considered a goal d e s i r a b l e of achievement and reasonably a t t a i n a b l e , " and that anything below t h i s i s to be considered as an e x c e l l e n t degree of e q u a l i z a t i o n or u n i f o r m i t y . Conversely, he s t a t e d "an index as high as L 5 should be judged cause f o r gravest Concern." Accepting Dr. R u s s e l l ' s a t t a i n a b l e l i m i t of 20, and knowing that the average d e v i a t i o n on which i t i s c a l c u l a t e d i s 0.7979 standard d e v i a t i o n , then the corresponding c o e f f i c i e n t of v a r i a t i o n would be 20 7 0.7979 or 25 per cent. 162 Or, expressed another way, as the standard d e v i a t i o n i s 1.25 average d e v i a t i o n s , then the c o e f f i c i e n t of v a r i a t i o n w i l l be 1.25 times the e q u i v a l e n t c o e f f i c i e n t of d i s p e r s i o n . Thus a c o e f f i c i e n t of v a r i a t i o n of 25 should be " d e s i r a b l e of achievement and reasonably a t t a i n a b l e " and a c o e f f i c i e n t of v a r i a t i o n of 56 should be "cause f o r the gravest concern." With an average assessment l e v e l of 50 per cent a co-e f f i c i e n t of v a r i a t i o n of 25 would mean that about 67 per cent of the items should have r a t i o s between 37*5 a«d 62.5 per cent. With a c o e f f i c i e n t of v a r i a t i o n smaller than 25 the degree of u n i f o r m i t y would be h i g h e r . For example, i f the c o e f f i c i e n t of v a r i a t i o n was 16, then 67 per cent of the items would f a l l between k2 and 58 per cent. The u l t i m a t e , which i s impossible to achieve, i s a c o e f f i c i e n t of zero, i n d i c a t i n g no d e v i a t i o n , and a p e r f e c t assessment i n which every assessed value i s at the same percentage of sales value as every other assessed value. Dr. R u s s e l l * s l i m i t of 20 was determined i n 1939; more recent developments and improved, techniques are b e l i e v e d to have such an e f f e c t that the l i m i t of 20 should now be 15, and that a c o e f f i c i e n t of d i s p e r s i o n of 30 would i n d i c a t e need f o r a general reassessment. These r e v i s e d l i m i t s converted to the c o e f f i c i e n t of v a r i a t i o n would be 18.8 and 37.6 per cent, or 18 and 37. Thus 18 should be a t t a i n a b l e , l e s s than 18 i n d i c a t e s good r e s u l t s , more than 37 i s unacceptable, and c o e f f i c i e n t s of v a r i a t i o n of from 18 to 37 may be acceptable according to e x i s t i n g circum-stances. S p e c i a l Comments 1. Information must be adequately screened or e d i t e d . 2. Sale p r i c e s must be confirmed, i f at a l l p o s s i b l e . 3. Assessed values must be f o r the c o n d i t i o n s and extent of the property as at date of s a l e . h. The degree of u n i f o r m i t y f o r cheap p r o p e r t i e s ( e i t h e r land o n l y , or f o r p o o r - q u a l i t y o l d improvements) w i l l be l e s s than f o r good- and middle-value p r o p e r t i e s . U n i f o r m i t y w i l l s u f f e r more again i n high-value p r o p e r t i e s where whimsey enters i n t o the s a l e p r i c e s . 5. A c t u a l value cannot be determined, s o l e l y on the evidence of a s i n g l e s a l e . EXHIBIT I F o r L a n d R e g i s t r y Office A P P L I C A T I O N N o d ) . U i e O o l y D a i . Received REAL PROPERTY TRANSFER RECORD FOEM A . C . 2 Thli form thill accompany tvtrf application fof (ha rcalttratlon of title tn all CIHI erher* a (ai certificate Ii required under (he proHiiont or lection 130 or Ihe "Land Rcslttr? Act" iictpt applications lo reiltltr mortiates. optioni to purchaxe. kaiei, lilt titatti, Crowo p u u , tai t a i n , tad additional application* r«rtiirad by »lrto» of aacttoa 119 (I) « f tht Act 1. L E O A L D E S C R I P T I O N O F L A N D (Insert l o t , b l o c k , s e c t i o n , d i s t r i c t , p l a n . ) (If iniufTVrltot tpact. UM ret rat tl at of form.) 2 . V e n d o r ' s name „ . _ • Purchaser's n a m e a n d address. . . (If fniuffkient ipacc. v i* tide of fttfm.) 3 , D a t e of t a l c 4 . W h a t was the f u l l sale price agreed u p o n by the V e n d o r and Purchaser? - • $ . _ 3 . W h a t was the a c t u a l cash payment re-ceived by the V e n d o r ? - - - -D i d the P u r c h a s e r assume any e n c u m -brances or other f inancial o b l i g a t i o n s ? Y e s . • N o . • K y e s , state amount o w i n g o n : — la) F i r s t mortgage - -(b) S e c o n d mortgage ( c ) A g r e e m e n t f o r sale • (d) O t h e r ( s p e c i f y ) : ( r ) O t h e r ( s p e c i f y ) : 6 , I i Ibis a t a l c o f — ( a ) V a c a n t l a n d o n l y ? • Or (ft) L a n d a n d improvements ( i . e . , l a n d w i t h b u i l d -inns o r s t r u c t u r e s ) ? * - • - - D 7 . D i d (he f u l l sale price i n c l u d e any p r o p -erty other than l a n d a n d b u i l d i n g s ? Y e s . • N o . Q If yes. itemize the value o f the r e a l a n d personal property. f L a n d . . . . 1 R e a l ] M a i n buildings - - } S -( O t h e r buildings - - J (Uat Itctaa.) To bt computed by Land Rtgistry Offk*. M u n i c i p a l i t y o r P r o v i n c i a l Assessment D i s t r i c t : 8 . D i d the t iamlcT i n c l u d e • trade of any property? Y e s . • N o . • If yes, slate type o f property traded, and esl v a l u e . c a l o r persona) lunate the market 9 . A r e the V e n d o r a n d Purchaser related o r associated c o r p o r a t i o n s ? Y e s . • N o . Q 10. Is this a f o r c e d or distressed sale? Y e s . • N o . O I I . Is this transaction s i m p l y a conveyance issued u p o n c o m p l e -t i o n o f payments under an agreement f o r sale? Y e s . • N o . D 12. D i d the transfer i n c l u d e the assignment of any l a n d h e l d under lease f r o m the C r o w n ? Y e s . • N o . • If yes, state t y p e , acreage, lease n u m b e r . 13. Slate any other details of the sale w h i c h w o u l d Indicate that the tale p r i c e bgreed u p o n is not indicative of a l a i r m a r k e t v a l u e , ' 14. I, the undersigned, d o hereby c e r t i f y that the i n f o n n a t i o a given is complete t a d correct i n a l l respect* . . • Vtnior. • SotkUot tor Vtnior or Fttrrknrr. • Aftnl for Vendor or Farrhttjrr. (* Strtta oot UM words not appucabto.) ZOOM 0 ( I ) All knformalton aobmltttd hanto by l b * applicant *M be (raaiad nrtctl/ cmffdeatUL EXHIBIT I I REQUEST FOR SALES ANALYSIS RECORDS To the Assessor. In accordance * i t h Regulations passed pursuant to the "Assessment Equalization Act," you are requested to complete Sales Analysis Records, Form A. C. 5 for each sale of real property represented in the following numbered copies of Certificates of T i t l e and Agreements-for-sale, etc. Completed Sales Analysis Records, Forms A. C. 3 shall be forwarded within one month of your receipt of this request to the Assessment Commissioner, Parliament Buildings, Victoria, B. C. Certificates of T i t l e Agreementa-For-Sale, etc. FORM A. C. 7 K. E. B. VLLDMAN, ASSESSMENT COMMISSIONER EXHIBIT I I I 165 SALES ANALYSIS RECORD CON1M3L NO. (2) Dola of tola. Municipality or ouotimam dinric' (3) School Oiilrici No_. Mortal value (full .Ola prica|_ (*l Subdinrict N o ^ „ . . . I E G A I DESCRIPTOR lot and kMM>lc ASSESSED VALUES Of PROPERTY TRANSFERRED ASSESSED VAtUES OF Main ftvlW»g If you do not coniider lhi* tola tultabta lor onelyitt. Indicate taoton b<l#Uy.. (S) TYPE OF PROPERTY 14) Mgnnl Clan of Main eutldlna For All Prop*rtl«t Other than "Agricultural" and "Small Holdings" AO* I CULTURAL 4 • S w i l HOLDING j Q OlHll IVECIFYI 6 Q Vocont tond OT • One-lomily dwelling - - •• - • 02 • Two-lomily dwelling 03 Q Row houia 0* Q Conver.ion 05 • Aporimenl M • Shock •• • 07 • Domain'c goroge 08 O Awio t<*xi - - - 09 Q Moial 10 • HOIal 1) Q Holl II • Sror- 13 • 5lOf* ond li»ing.qva/iart or oTI-c«t • Q Oflita building IS Q Bonk building 14 • Th. - U • - 16 • - " • - M a • 31 • re building theolre drive-En Gai and lervice Commercial goro Cold iloroga Storage ond worohouiing-Open . . . . Cloiad 23 • Mofioa locllWe* 2* Q O'her commercial (ipeclfy) - • • 15 Q Water-front 02 • • 2 2 • Sand, grovel, cloy pi', or l Manufacturing (ipecily) • quarry 26 • • • • 27 • • 28 • For All "Agricultural" and "Small Holdings" (Entry raqulrad ii each c< A. B, ond C litlow.I A. E.tenl ol farm building), Voconi, no building! Residence ond outbuilding! • B. Typo of oparaiiofli Groin ond farage -Vegetable ond iruck • >- • • 5 - • Reiidence, no outbuilding! Oulbuildingi, no raildanca. Small Iruin -Colli* • • • C. l» land clouiiied t . . . • • - • -7 a • • • -i a • • -ID i lond'l Ul Q No Q Mi.ad • • <- • • 1- • • a • - 7 D • -»a Other |ip«clfy) • Aga-gra P Safari 1710, Cheap - 1 - • Averoge 2 - • Good -3 - D 1910-1930, Cheap - *- a Averoge s - • Good • 6- Q AHor 1930, Cheap . • Avarage 8 - • Good - 9 - • . Slruelu • Froma - -Brie V-wood - 2 • ftritt.concrete - 3 a Brkt-ileel --* • Siaal • • - 5 • Concrete • -* • log -? a AS5ISSMENT SAIES RATIO CAlCUlATlON (12) Attaiiad Volua (13) Mortal Volua Received ond checked ..•('6l Reiiduol tond Volir mi**M Ooamlitlonar, •atlleiwil ftvlldlnat, Vldada, B.C. Fan* A.C. 3—10 • M.B.F. I O - « APPENDIX B APPENDIX B In 1939, while he was D i r e c t o r of Research with the V i r g i n i a Department of Taxation Dr. John H. R u s s e l l p u b l i s h e d i n "The Commonwealth" an organ of the V i r g i n i a State Chamber of Commerce some r e s u l t s of assessment-sale p r i c e r a t i o s t u d i e s . The a r t i c l e from which the r e s u l t s e x h i b i t e d below were taken was e n t i t l e d " I n e q u a l i t y of Real E s t a t e Assessments w i t h i n P o l i t i c a l S u b d i v i s i o n s . " An examination of the data i n d i c a t e s some r e l a t i o n between the "Index of Assessment I n e q u a l i t y " and the p r o p o r t i o n of assessments w i t h i n a given range of the mean, but i t i s c l e a r that the c o o r e l a t i o n i s not p e r f e c t . Therefore, R u s s e l l could not have assumed that assessment-sales r a t i o s are normally d i s t r i b u t e d . Percentage of County or c i t y Index of Assessment I n e q u a l i t y "Good" assess-ments ( w i t h i n 20$ of average) "Very Bad" assessments (80$ or more away from average) Roanoke C i t y 28 k7.2 Buckingham County 7? 12.6 !+3.0 Amelia County 36 "+0.8 , 9 ' 2 Essex County 75 11.2 kl.l Richmond C i t y 30 1+6.2 7A Hampton C i t y 66 2*+. 2 29.3 Richmond C i t y , 1936 30 1+6.2 7.*+ Richmond C i t y , 1939 25 ^9.1 3.8 E i g h t y per cent i s a r b i t r a r i l y taken as the p o i n t of separation of bad and "very bad" assessments. A c t u a l l y , assessments missing the average by.!+5 per cent w i l l g e n e r a l l y be considered as very bad. PROPORTIONS OF "GOOD" AND "VERY BAD" ASSESSMENTS IN TEN SELECTED VIRGINIA COUNTIES 1936 Percentage of sample w i t h i n -20$ of county*s average I n t e r --mediate assess-ment not h e r e i n considered Percentage of ...,sample 80$ or more away from county* s average Rank of County Scale % county - * 1 • t t t 1 t r t i n I.A.I. 0 10 20 30 *+0 50 60 70 80 90 100 1 Amelia 5 Cumberland 6 C a r o l i n e 8 Lunenburg 10 Rockbridge 93 Mathews 9*+ Alleghany 97 Buchanan 99 Essex 100 Buckingham Imaginary Model County •Index of Assessment I n e q u a l i t y H ON 0 0 APPENDIX C APPENDIX C The l i m i t s of the frequency i n t e r v a l s used i n t h i s study and the values of M and S c a l c u l a t e d from the data produce values of z©<c i, ( j s 1, . . . , 9)> the corresponding frequency i n t e r v a l l i m i t s of the standardized normal d i s t r i b u t i o n , which are not a v a i l a b l e i n the t a b l e s prepared f o r researchers. I t was, t h e r e f o r e , necessary to evaluate the p r o b a b i l i t y i n t e g r a l s . This Appendix b r i e f l y sets out the theory, procedure and r e s u l t s of these c a l c u l a t i o n s . The d e n s i t y f u n c t i o n of the normal d i s t r i b u t i o n i s v'JIrr • 2 v : I f z ss ( x yu)/Sj , then f ( o , l ) = _ - *2 / 2 l / 2 7 T f ( 0 , l ) can be expanded i n a Ma c l a u r i n s e r i e s . 1 f ( o , l ) = 2TT U n where U n = (-1) ( 3 " - D z (2n-2) Since l i m U n+1 n (2n-2)! = 0, U n converges a b s o l u t e l y . The p r o b a b i l i t y i n t e g r a l of the normal d i s t r i b u t i o n i s to be c a l c u l a t e d . 171 F (z) = f ( 0 , l ) . e - X 2 / 2 d z -Z Since which U n I converges there i s some value of n beyond n decreases monotonically* Therefore, U. n n=l can be i n t e g r a t e d term by term to give the f o l l o w i n g r e s u l t . H - - 4 z / n=l / v n n=l where V n •z ( D.dz 2 z n - 2 V - ( - 1) (3 ^ - D . X C2n-2) n Since l i m n 2 C n ' 1 ) . ( 2 n - l ) . ( n - l ) (2n-2) - oo n=l V n converges to F ( z ) f o r |*2 I ^ ° ° Of course the summation to i n f i n i t y i s not performed, but i t was determined f o r the range of values of the argument that F* (^) produced an accurate r e s u l t . 29 F ( Z ) = 2 TT n=l 172 On page 8*+ of the text i t i s noted that A 3 = f (O.l)d-Z whereas • oc / - 1 f ( 0 , l ) d % ) = Therefore i n order to determine A. a f u r t h e r c a l c u l a t i o n •J was performed. A j s F ^ l ^ . I ) + - . 5 , J £ 5 A i 3 - F * ( l z c x . I > + » J ^ 5 F * ( | ^ . | ) - F * ( l ^ o c . I ) , j £ 5 ' * c h o c . | ) + |), j - 5 •1 al 173 The r e s u l t s are tabul a t e d below. M = O.W085 s = 0.087121 2 > 3 = 500 1 2 I 7 8 9 .13085 .23085 .33085 .H3085 .53085 .63085 .73085 .83085 exO -1+.0171* -2.8695 -1.7217 -0.5739 0.5739 1.7217 2.8695 k.017* .9999 .9959 .9150 .91H8 .9958 .9999 1.0000 A 3 0.0000 .0020 .0^25 .2828 .7168 .957*+ .9980 1.0000 1.0000 A -A 0.0000 .0020 .0^05 .2^03 . ^ 0 .2^06 .0H06 .0021 0.0000 0 1 20 120 218 120 20 1 0 Notes, ( i ) °^ i s upper l i m i t of i n t e r v a l j . ( i i ) T h e o r e t i c a l upper l i m i t of i n t e r v a l 9 i s i n f i n i t y . ( i i i ) Since the c a l c u l a t i o n of F * ( ) 29 J i n v o l v e s the summation ^T""' V"n , where n=l v ^ = (_i ) C 3 n-l) # z ( 2 n - 2 ) 2 n * 1 . ( 2 n - l ) . ( n - l ) the computer must handle some very l a r g e and very small numbers. The formation of caused an overflow i n the denominator (and a t r u n c a t i o n to zero. I n the numerator where j = 5 ) . The r e s u l t of t h i s t e c h n i c a l d i f f i c u l t y caused some i n s t a b i l i t y which accounts f o r the i n c o n s i s t e n c y of the f o u r t h decimal place i n the column headed F * ( 2 O v . ) . Because the e r r o r i s so s l i g h t i t had no e f f e c t oh the f i n a l r e s u l t s .
Thesis/Dissertation
eng
Business Administration
Vancouver : University of British Columbia Library
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
Graduate
Measures of the quality of real property assessments : an examination of their validity
Text