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The use of direct conversion ratios and the selection of capitalization rates in residential income property… Farish, William Gordon 1969

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THE USE OF DIRECT CONVERSION RATIOS AND THE SELECTION OF CAPITALIZATION RATES IN RESIDENTIAL INCOME PROPERTY APPRAISAL by William Gordon Farish B.Comm., University of B r i t i s h Columbia, 1968 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION i n the Department of Commerce and Business Administration We accept t h i s thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA May, 1969 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 b e g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y 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 O H K ^ ^ ^ . The 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 , V a n c o u v e r 8, C a n a d a D a t e 311 ABSTRACT Two problems were considered i n this study. The f i r s t was of selecting a c a p i t a l i z a t i o n rate under the t r a d i -t i o n a l approach and the second was the use and accuracy of d i r e c t conversion r a t i o s , p a r t i c u l a r l y gross income multi-p l i e r s . The purpose of the study was to resolve any inconsistencies and i r r a t i o n a l i t i e s which may e x i s t i n appraisal theory and, to the extent that d i r e c t conversion r a t i o s are proven useful, to e s t a b l i s h certain guidelines to f a c i l i t a t e t h e i r use. The various methods of selecting c a p i t a l i z a t i o n rates were discussed and, where j u s t i f i e d , c r i t i c i z e d . Then, t h e o r e t i c a l aspects of gross income m u l t i p l i e r s were con-sidered. After a b r i e f discussion of the data and relevant s t a t i s t i c a l concepts, extensive empirical analysis, using regression and c o r r e l a t i o n models as well as the average m u l t i p l i e r , followed. Assuming that the objective of a c a p i t a l i z a t i o n device i s to predict a market value,, the most probable s e l l i n g p r i c e , i t follows that c a p i t a l i z a t i o n rates should be market deter-mined. I t was found that the t r a d i t i o n a l methods of rate s e l e c t i o n did not meet the c r i t e r i o n of market determination. The i m p o s s i b i l i t i e s of selecting rates from the market were stressed. i i i The o r i g i n a l advantages of gross income multi-p l i e r s were s i m p l i c i t y and data a v a i l a b i l i t y . Despite the t r a d i t i o n a l c r i t i c i s m s , the s t a t i s t i c a l analysis demonstrated another advantage, accuracy. S t r a t i f i c a t i o n by basic struc-t u r a l type, l o c a t i o n , number of suites and income per suite resulted i n average percentage differences between actual and estimated values as low as three per cent. Other results were within acceptable error l i m i t a t i o n s . The findings with regard to c a p i t a l i z a t i o n rates i l l u s t r a t e that inconsistencies and i r r a t i o n a l i t i e s e x i s t i n appraisal theory. The use of the t r a d i t i o n a l methods of selecting c a p i t a l i z a t i o n rates should be terminated as they do not r e s u l t i n market determined rates. The findings with regard to gross income m u l t i p l i e r s i l l u s t r a t e that they are capable of predicting values very accurately i n many cases. Their use i s to be encouraged where i t can be shown that they are accurate. ACKNOWLEDGEMENT The writer wishes to thank Professor Richard U. R a t c l i f f for his very h e l p f u l advice and d i r e c t i o n , Professor Michael A. Goldberg for his s t a t i s t i c a l guidance, the Computing Centre s t a f f and a l l the r e a l t o r s , appraisers and assessors, p a r t i c u l a r l y the o f f i c e r s of the West Vancouver Assessment O f f i c e , who supplied the very valuable sales data. TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION 1 Statement of the Problem . . . . . 1 Purpose of the Study 3 Importance of the Study . 3 Organization of the Study .4 Limitations of the Study 5 D e f i n i t i o n of Terms 7 I I . SELECTION OF CAPITALIZATION RATES 8 C a p i t a l i z a t i o n and Rate Selection 8 Rate Selection by Summation Method 12 Rate Selection by Band of Investment 14 Rate Selection i n the Ellwood Method 22 Rate Selection by Comparison of Quality Attributes 26 Rate Selection by Direct Comparison 29 Some General C r i t i c i s m s 30 Rate Selection from Gross Income M u l t i p l i e r s . 34 I I I . DIRECT CONVERSION RATIOS 36 Market or Income Approach? 37 Single Family or Multiple Unit? 38 Origin and Development 40 Use i n Security Analysis 41 v i CHAPTER PAGE I I I . (Continued)J The Relationship of M u l t i p l i e r s and Ca p i t a l i z a t i o n Rates. 42 Advantages of Use 46 Cri t i c i s m s 48 Selection of Comparables 51 Adjustment of Comparables 57 Use of M u l t i p l i e r s 59 IV. DATA AND METHODOLOGY. 67 Data. 67 Simple Linear Regression and Correlation Model . 69 Average M u l t i p l i e r 81 Multiple Regression and Correlation 82 Application 83 V. RESULTS OF ANALYSIS . 85 Regression Equations 85 Confidence Intervals 85 Predicted Versus Actual Values 110 Simple Regression Versus Average M u l t i p l i e r . . 122 Multiple Regression and Correlation . . . . . . 132 v i i CHAPTER PAGE VI. SUMMARY AND CONCLUSIONS. 136 Restatement 136 Summary of Findings 137 Conclusions and Recommendations. 142 Other Possible Studies 144 BIBLIOGRAPHY 146 APPENDIX . . . . . . . 150 v i i i LIST OF TABLES TABLE PAGE I. Gross Income M u l t i p l i e r s and Gross C a p i t a l i z a t i o n Rates . . . 43 I I . Confidence Intervals - S t r a t i f i c a t i o n by Structural Type 88 I I I . Confidence Intervals - S t r a t i f i c a t i o n by Location . . . . . 92 IV. Confidence Intervals - S t r a t i f i c a t i o n by Size. . 93 V. Confidence Intervals - S t r a t i f i c a t i o n by Number of Suites 95 VI. Confidence Intervals - S t r a t i f i c a t i o n by Number of Suites (Small Samples) 96 VII. Confidence Intervals - S t r a t i f i c a t i o n by Income per Suite 9 8 VIII. Confidence Intervals - S t r a t i f i c a t i o n by Income per Suite (Small Samples) 100 IX. Confidence Intervals - S t r a t i f i c a t i o n by Date of Sale 102 X. Confidence Intervals - Comparison of Large and Small Samples 104 XI. Confidence Intervals - Comparison of Varying Sample Sizes 106 XII. Confidence Intervals - Comparison of Confidence Levels 10 8 XIII. Predicted Versus Actual - S t r a t i f i c a t i o n by Location 112 XIV. Predicted Versus Actual - S t r a t i f i c a t i o n by Size and Type 116 XV. Predicted Versus Actual - S t r a t i f i c a t i o n by Number of Suites and Income Per Suite . . . 117 ix TABLE PAGE XVI. Predicted Versus Actual - Comparison of Varying Sample Sizes 121 XVII. Simple Regression Versus Average M u l t i p l i e r -Basic S t r a t i f i c a t i o n 123 XVIII. Simple Regression Versus Average M u l t i p l i e r -Detailed S t r a t i f i c a t i o n . . . . . . . . . . . . 125 XIX. Best Results-Small Samples . 128 XX. Best Results-Large Samples . 130 XXI. Comparison of Multiple and Simple Regression and Correlation 134 CHAPTER I INTRODUCTION I. STATEMENT OF THE PROBLEM T r a d i t i o n a l l y , appraisal l i t e r a t u r e has discounted d i r e c t conversion r a t i o s , those r a t i o s used i n appraisal to convert income to a c a p i t a l figure, as having only a limited degree of accuracy."'" For instance, although the gross i n -come m u l t i p l i e r , the r a t i o of c a p i t a l value over gross i n -come, i s sometimes suggested as having some limited useful-ness as a rough guide or as a check on the f i n a l c o r r e l a t i o n step, i t i s not considered an accurate enough device to be used singly to calculate the most probable s e l l i n g price of a r e s i d e n t i a l income property. However, even though d i r e c t conversion r a t i o s are said to be unsophisticated and inadequate, there i s some evidence and recognition of them becoming " b u i l t - i n t o " the market for American In s t i t u t e of Real Estate Appraisers (A.I.R. E.A..) , The Appraisal of Real Estate (Chicago: Author, 1967) , pp. 336-337. S.A. Kahn, F.E. Case and A. Schimmel, Real Estate  Appraisal and Investment (New York: Ronald, 1963), pp. 411-13. A.A.Ring, The Valuation of Real Estate (Englewood C l i f f s , New Jersey: Prentice-Hall, 1963), pp. 174-176. P.F.Wendt, Real Estate Appraisal (New York: Holt, 1957), pp. 193-194. 2 2 c e r t a i n classes of property. That i s , as investors and brokers use such r a t i o s with some r e g u l a r i t y , transactions i n the marketplace begin to r e f l e c t these relat i o n s h i p s . To the extent that the foregoing i s true, i t might seem somewhat premature to discount t h e i r use e n t i r e l y . In f a c t , i t would seem more reasonable to attempt to resolve whether d i r e c t conversion r a t i o s can be of any primary value by t h e o r e t i c a l analysis and empirical t e s t i n g . While t r a d i t i o n a l appraisal l i t e r a t u r e has c r i t i c i z e d d i r e c t conversion r a t i o s , i t has continued to promote certa i n somewhat questionable methods of selecting c a p i t a l i z a t i o n rates f o r use i n the conventional income approach to valua-t i o n . Assuming that the objective of a c a p i t a l i z a t i o n device i s to predict a market value, the most probable s e l l i n g p r i c e , i t follows that c a p i t a l i z a t i o n rates should be market deter-mined. However, as w i l l be seen, the conventionally acceptable and well used methods do not pass the t e s t of.market deter-mination. Thus, while c r i t i c i z i n g , perhaps unjustly, d i r e c t conversion r a t i o s as inadequate, t r a d i t i o n a l appraisal l i t e r -ature puts forth certain methods of selecting c a p i t a l i z a t i o n rates which might be even more severely c r i t i c i z e d . R.U. R a t c l i f f , Current Practices i n Income Property A p p r a i s a l — A Cri t i q u e (Berkeley: University of C a l i f o r n i a , 1967) , p. 38. Wendt, op_. c i t . , p. 210. 3 II . PURPOSE OF THE STUDY It i s the intent of t h i s paper to analyze and, where j u s t i f i e d , c r i t i c i z e conventional methods of selecting capi-t a l i z a t i o n rates. Having done so, d i r e c t conversion r a t i o s , p a r t i c u l a r l y - g r o s s income m u l t i p l i e r s , w i l l be studied theore-t i c a l l y and an extensive s t a t i s t i c a l analysis designed to test t h e i r accuracy as appraisal devices w i l l follow. Thus, the purpose i s twofold. F i r s t , any inconsisten-cies and i r r a t i o n a l i t i e s which e x i s t i n appraisal l i t e r a t u r e , p a r t i c u l a r l y i n the theories of selecting c a p i t a l i z a t i o n rates and of using d i r e c t conversion r a t i o s , hopefully w i l l be resolved. Second, to the extent that d i r e c t conversion r a t i o s are proven useful, c e r t a i n guidelines w i l l be established for th e i r use. Hopefully, the appraisal function w i l l be made simpler and more r a t i o n a l while retaining acceptable l i m i t s of accuracy. Thus, th i s paper i s intended to be much more than an academic exercise. Hopefully, i t w i l l r e s u l t i n conclusions which w i l l have p r a c t i c a l value to the appraiser of r e s i d e n t i a l income property. I I I . IMPORTANCE OF THE STUDY To the extent that presently acceptable and employed methods of selecting c a p i t a l i z a t i o n rates r e s u l t i n inconsis-tencies and inaccuracies, r e s u l t i n g appraised values can only be inconsistent and inaccurate. Thus, i t i s important that 4 flaws i n t h i s segment of conventional appraisal l i t e r a t u r e be discovered and resolved. I t i s only by continued questioning and analysis that appraisal methodology w i l l evolve into a thoroughly reasonable basis for estimating the most probable s e l l i n g p r i c e . For the same reasons, any undue c r i t i c i s m s of certain appraisal devices, such as d i r e c t conversion r a t i o s , should be resolved. That i s , i f , say, gross income m u l t i p l i e r s could be demonstrated to have accuracy for use as more than just rough guides and checks, such a demonstration i s important to the improvement of appraisal theory. And i f certain guide-l i n e s can be established for the use of d i r e c t conversion r a t i o s as an accurate, though s t i l l r e l a t i v e l y simple, appraisal device, they should be promoted as being of p r a c t i c a l value to the appraiser. IV. ORGANIZATION OF THE STUDY The study w i l l be organized into chapters as follows. The next two chapters w i l l be largely t h e o r e t i c a l considerations of e x i s t i n g appraisal thought. F i r s t , the various methods of sel e c t i n g c a p i t a l i z a t i o n rates w i l l be discussed and, where j u s t i f i e d , c r i t i c i z e d . Hopefully, the e x i s t i n g inconsistencies already mentioned w i l l be evident, as well as eventually re-solvable. Following that, d i r e c t conversion r a t i o s as mentioned 5 i n t r a d i t i o n a l appraisal l i t e r a t u r e w i l l be considered i n depth. The intent of that chapter w i l l be to provide t h e o r e t i c a l j u s t i f i c a t i o n for pursuing the matter i n the form of s t a t i s t i c a l analysis. Then a chapter on the data and methodology used i n the analysis i s necessary. The nature of the data, p a r t i c u l a r l y i t s extent and i t s l i m i t a t i o n s w i l l be discussed. As well, an introduction to the s t a t i s t i c a l methodology w i l l prove useful. Various s t a t i s t i c a l terms and methods which are relevant to the s t a t i s t i c a l models used must be mentioned. This section w i l l not be p a r t i c u l a r l y extensive as t h i s paper i s considered a consideration of appraisal theory involving s t a t i s t i c a l concepts rather than the reverse. However, t h i s s t a t i s t i c a l section must be s u f f i c i e n t to explain and j u s t i f y the method-ology. A chapter analyzing the results of the s t a t i s t i c a l work w i l l follow. F i n a l l y , a restatement of the study and a summary of the findings and recommendations of the study w i l l be - necessary. The conclusions w i l l be stated and other related problems w i l l be raised. V. LIMITATIONS OF THE STUDY There are several l i m i t a t i o n s which are perhaps obvious, but i n any case should be stated at t h i s point. The th e o r e t i -c a l portion of the study i s l i m i t e d , of course, to only a small 6 segment of appraisal theory. However, the fact that only the methods of c a p i t a l i z a t i o n rate s e l e c t i o n and the use of d i r e c t conversion r a t i o s are being studied does not diminish the signi f i c a n c e of the study. The concern i s with apartment properties, that i s , conventional multiple-unit r e s i d e n t i a l income properties, only. Single-family properties, unconventional revenue proper-t i e s (such as converted houses) and other forms of income prop-erty (such as commercial, i n d u s t r i a l or farm) are not considered. These other forms of income property occur either i n i n s u f f i c i e n t number or with too great a variety to enable generalizations about the application of d i r e c t conversion r a t i o s . As well, the nature of the data suggests some l i m i t a t i o n s . Ideally, a great many more property c h a r a c t e r i s t i c s and fa c t s , p a r t i c u l a r l y net income, would be available for study. However, the data available are very good i n terms of sample size and ce r t a i n l y s u f f i c i e n t i n terms of d e t a i l to produce some i n t e r -esting and useful r e s u l t s . F i n a l l y , i t must be recognized that u n i v e r s a l i t y cannot be claimed from l o c a l data. That i s , although s i g n i f i c a n t r e s u l t s may occur using the Vancouver data, i t cannot be im-mediately assumed that s i m i l a r results w i l l occur i n another urban area. However, s i g n i f i c a n t results should c e r t a i n l y en-courage further investigations elsewhere. 7 VI. DEFINITION OF TERMS Direct conversion r a t i o s are,those r a t i o s used i n appraisal to convert income, usually current, actual or economic ren t a l income, to a c a p i t a l figure representing 3 market value of the property. Gross income m u l t i p l i e r s express the r e l a t i o n s h i p between the annual (or monthly) gross income produced by a property and the c a p i t a l value, be i t actual s e l l i n g price or estimated most probable s e l l i n g p r i c e , of that property. The most probable s e l l i n g p rice i s simply the appraiser's prediction of what would happen i f the subject property were exposed to the market 4 for a reasonable time. C a p i t a l i z a t i o n i s simply the process of converting a series of anticipated future installments into a present value by the application of some rate, the c a p i t a l i z a t i o n rate. Other terms w i l l be defined as the discussion proceeds. R a t c l i f f , op_. c i t . , p. 39. R.U. R a t c l i f f , Modern Real Estate: Valuation (Madison: Democrat Press, 1965), p. 1. CHAPTER II SELECTION OF CAPITALIZATION RATES The t r a d i t i o n a l income approach to value has been c r i t i c i z e d elsewhere as unsuitable for predicting the most probable s e l l i n g price of a property. 1 Although such c r i t i -r cism i s based on several inadequacies, t h i s study s h a l l consider only the question of the selection of an appropriate c a p i t a l i z a t i o n rate i n the t r a d i t i o n a l income approach. I. CAPITALIZATION AND RATE SELECTION The A.I.R.E.A. defines c a p i t a l i z a t i o n as: The process of converting into present value a series of anticipated future annual installments of income by discounting them into a present worth at a rate which i s a t t r a c t i n g purchase c a p i t a l to investments with s i m i l a r c h a r a c t e r i s t i c s , such as r i s k and term. . .2 There may be a number of d i f f i c u l t estimates involved i n t h i s process, including those of gross income, operating expenses and, hence, net income, economic l i f e , value of the R.U. R a t c l i f f , Modern Real Estate Valuation (Madison: Democrat Press, 1965), pp. 3-4. R.U. R a t c l i f f , Current Practices i n Income Property A p p r a i s a l — A Critique (Berkeley: University of C a l i f o r n i a , 1967). R.U. R a t c l i f f , "Capitalized Income i s Not Market Value," Appraisal Journal,(January, 1968), pp. 33-40. 2 . American In s t i t u t e of Real Estate Appraisers, Appraisal Terminology and Handbook (Chicago: Author, 1967) , p. 30. 9 reversion and the c a p i t a l i z a t i o n rate. I t i s with this l a s t estimate that t h i s chapter i s concerned. The A.I.R.E.A. defines rate as: A term expressing a fixed relationship between two magnitudes and used as a means, of measurement. A r a t i o of income to capital.3 The same source defines i n t e r e s t rate as: "The rate 4 of y i e l d earned from an investment" and the c a p i t a l i z a t i o n rate as: The percentage which i s the sum of the i n t e r e s t rate and recapture rate ( i f improved) and which expresses the r e l a t i o n s h i p between the value of the property and i t s share of net income before depreciation.5 Two further d i s t i n c t i o n s are necessary. The composite rate and the o v e r a l l rate are both c a p i t a l i z a t i o n rates. The term "composite rate" simply implies that the rate of c a p i t a l -i z a t i o n i s a composite of the rate of return on the investment ( i . e . p r o f i t ) and the rate of return of investment ( i . e . re-capture of c a p i t a l ) . ^  The o v e r a l l rate of c a p i t a l i z a t i o n i s that rate which i s applied to the income of the entire prop-erty, that i s , both land and improvements. The o v e r a l l rate 3 Ibid., p. 162. 4 I b i d . , p. 109. 5 I b i d . ^A.A. Ring, The Valuation of Real Estate (Englewood C l i f f s , New Jersey: Prentice-Hall, 1963), p. 220. V 10 i s d i s t i n c t from s p l i t rates which are those used i n the land residual and building resid u a l processes; that i s , there i s a separate rate for the portion of income attributable to land and another rate, including a recapture provision, for the 7 portion of income attributable to improvements. According to the t r a d i t i o n a l income approach, part of the periodic net income i s normally allocated to the recapture or amortization of wasting purchase c a p i t a l and the balance to g p r o f i t . Under t h i s approach the projection of s t a b i l i z e d periodic income i s usually divided by a composite rate or m u l t i p l i e d by the r e c i p r o c a l of the composite rate. That i s , Income _ _ . , j- , — = Income x Reciprocal of = Value Composite Rate Composite Rate For example, $16,000 = $16,000 x 12.5 = $200,000. .08 The A.I.R.E.A. text states that the most supportable c a p i t a l i z a t i o n rate i s one abstracted from market data, tested for the probable range of y i e l d s on s i m i l a r investments, and accepted i n the l i g h t of the r e l a t i v e market attractiveness of 7 W.V. Sadesky, "The Relationship of Cost of Borrowed Money to C a p i t a l i z a t i o n Rates," The Appraisal Journal, XXXVI (January, 1968), p. 9. 8 American I n s t i t u t e of Real Estate Appraisers, Appraisal of Real Estate (Chicago: Author, 1967), p. 260. 11 9 the subject property. There are several considerations that are assumed to be included i n the c a p i t a l i z a t i o n rate. The rate should be commensurate with the r i s k . 1 ^ Besides the qu a l i t y of the income, the d u r a b i l i t y of the income must be accounted for as the rate includes both i n t e r e s t on and re-capture of capital."''"'" With these two concepts i n mind, the A.I.R.E.A. text says that the appraiser then selects a capi-t a l i z a t i o n rate on the basis of market experience. Generally, the text suggests, the rate to use i s the rate which inves-tors i n that type of or class of property consider to be 12 required as a condition for purchasing. Under the t r a d i t i o n a l approach there are a variety of methods available to the appraiser for selection of the c a p i t a l i z a t i o n rate. Unfortunately, as Wendt says, . . . the concepts and techniques employed d i f f e r so widely that i t appears that a c a p i t a l i z a t i o n rate must be conjured out of the appraiser 1s vhead.13 For the remainder of t h i s chapter various methods of rate s e l e c t i o n w i l l be considered i n turn. As has been seen, 9 Ibid., p. 263. 1 0 . . , , Ibid. "'""'"Ibid. , p. 265. 1 2 I b i d . , pp. 267-268. 13 P.F. Wendt, Real Estate Appraisal (New York: Holt, 1956), pp. 209-210. 12 the A.I.R.E.A. suggests that, rates be based on market evidence. This i s c e r t a i n l y acceptable; i f i t i s assumed the objective of c a p i t a l i z a t i o n devices i s the prediction of market value, the most probable s e l l i n g p r i c e , i t follows that rates (as well as other factors i n the c a p i t a l i z a t i o n process) should be market determined. What i s of concern i n considering the various methods of rate s e l e c t i o n , i s whether or not i t i s possible to actually determine the rates, i n the manner suggested, from the market. Any method must be possible, l o g i c a l and accurate to be con-sidered acceptable. I I . RATE SELECTION BY SUMMATION METHOD The summation method builds a c a p i t a l i z a t i o n rate by 14 adding several component rates together. The s t a r t i n g point i s the s e l e c t i o n of a base, or "safe," rate of return, some-times c a l l e d a "pure i n t e r e s t " .rate. Usually a long-term government bond or a savings account rate i s used. To t h i s base rate are added components which attempt to evaluate the r i s k s and hazards attendent upon ownership of .a p a r t i c u l a r piece of property, including rates for n o n - l i q u i d i t y , manage-S.A. Kahn, F.E. Case and A. Schimmel, Real Estate  Appraisal and Investment (New York: Ronald, 1963) , p. 132. 13 ment and r i s k . Thus, the c a p i t a l i z a t i o n rate might be "built-up" as follows: Component Rate Base Rate 5.5% Non-liquidity 3.0 Management 1.0 Risk 2.5 Composite 12.0% The summation method f a i l s to meet the c r i t e r i o n of true market determination. There i s no evidence that investors actually use i t i n t h e i r investment decisions; what i s of con-cern i s what they do use, not what they necessarily should use. Even i f i t i s admitted that such components are i n t u i t i v e l y involved i n the c a p i t a l i z a t i o n rate, the market gives no prices for these pure components except for the base rate. These components are not priced i n such s l i c e s since they do not e x i s t i n r e a l i t y i n the market i n any such pure or elemental forms. The degree.of subjective s e l e c t i o n and the ease of manipulation are serious handicaps to the use of th i s method. 15 Many appraisers have condemned i t s use. Thus, whxle the Ibid., p. 131. 14 summation method may be useful as a delineation of the economic functions for which a return must compensate, i t i s t o t a l l y useless as a means of a r r i v i n g at a quantitative measure of the t o t a l per cent return at which a p a r t i c u l a r income flow i s to 16 be c a p i t a l i z e d . I I I . RATE SELECTION BY BAND OF INVESTMENT One reason the summation method i s not considered acceptable by many appraisers i s that i t does not d i r e c t l y consider the borrowed and equity portions of the investment. The composite rate developed by application of the band of investment theory i s a synthesis of mortgage and equity rates which market data supposedly disclose as applicable to com-parable properties. The rate developed i s a weighted average, the weighting being for percentages of the value which would be occupied by the mortgage and equity positions, or band of 4-17 investment. For instance, proponents of the band of investment method would say that in v e s t i g a t i o n of sales of comparable H.B. Dorau, "The C a p i t a l i z a t i o n Rate: Mirage or Will-o'-the-Wisp?" The Appraisal Journal, XXIX (January, 1961), p. 21. 17 A.I.R.E.A., op_. c i t . , p. 269. ( 15 properties may show that financing i s available consisting of a 66-2/3 per cent f i r s t mortgage loan at nine per cent i n t e r e s t and that buyers demand a 12 per cent return to induce them to invest i n 33-1/3 per cent equities. From such data, the i n t e r -est rate i s developed as follows: Component Per Cent of Value Rate Product Mortgage 66-2/3% 9% 6% Equity 33-1/3% 12% 4% Composite 1 Q Q % l f J % Thus, the composite rate used i n the c a p i t a l i z a t i o n process would be 10 per cent. Of course, variations i n finan-cing, such as a second mortgage, could be included e a s i l y . Another application of the band of investment theory i s possible by using composite rates for mortgage debt service and 18 anticipated cash flow to equity. The mortgage requirement rate includes both the i n t e r e s t on and recapture of the mort-gage component of the t o t a l value. The equity rate i s the anticipated cash flow over the equity investment as indicated by comparable sales. The c a p i t a l i z a t i o n rate i s developed under t h i s method as follows: 1 8 I b i d . , p. 270. 16 Component Per Cent of Value Rate Product Mortgage 66-2/3% .1018 .068 Equity 33-1/3% .12 .04 Composite 100% .108 where .1018 i s the annual mortgage component at nine per cent with complete amortization of the mortgage i n 25 years. 66-2/3 per cent of the investment i n 25 years plus a cash flow to equity of 12 per cent per annum on the equity investment, the equity y i e l d w i l l be greater than 12 per cent i f the value of the property does not decline 66-2/3 per cent i n 25 years. If the property does decline more than 66-2/3 per cent i n 25 years, the return on equity w i l l , of course, be less than 12 per cent. Proponents of the band of investment theory say i t r e f l e c t s a rate which accounts for changes i n i n t e r e s t rates, mortgage terms and cash flow requirements for equity and that i t also r e f l e c t s the financing p o t e n t i a l and acknowledges and 19 r e f l e c t s leverage. While the basic concept of the band of investment theory has been described, i t remains to explain just how the Because th i s o v e r a l l rate provides for recapture of R.D. Nelson, "Overall Rate - Band of Investment Style," The Appraisal Journal, XXXVII (January, 1969), p. 25. 17 proportions and i n t e r e s t rates are determined. Some theorists suggest beginning with the mortgage component. I t i s f e l t that the appraiser can determine what mortgage terms are a v a i l -able i n the market. That i s , the appraiser i s said to "know", that, say, 75 per cent of_the value can be borrowed by a t y p i c a l buyer at a given rate of i n t e r e s t with provision for 20 recapture over a given term of years. Thus, i t i s thought that the appraiser has a "fac t u a l " c a p i t a l i z a t i o n rate per-taining to 75 per cent of his appraisal; t h i s i s thought to be not a matter of guesswork, nor even of examining trans-actions i n the marketplace, but of consulting financing 21 sources. In actual cases, however, financing terms such as loan-to-value r a t i o s , the i n t e r e s t rate and the amortization period, a l l of which a f f e c t the c a p i t a l i z a t i o n rate, vary with i n d i v i d u a l investors and lenders. The terms w i l l vary with the investor's preferences and, perhaps to a greater extent, with the lender's evaluation of the r i s k . Thus, the d i f f i -c u l t i e s involved i n generalizing are evident. For instance, there i s the only p a r t i a l question of whether or not secondary 20 L.W. Ellwood, Ellwood Tables (Chicago: A.I.R.E.A., 1967), p. 65. 21 Sadesky, op_. c i t . , p. 11. 18 financing i s involved. One p o t e n t i a l investor might require i t and be able to get i t , another might require i t but be unable to get i t and s t i l l another might have no need for i t . And even i f i t could be generalized as to whether or not secondary financing was involved, how can the appraiser possibly j u s t i f y his judgements as to i t s terms and i t s e f f e c t on the other components? Obviously there are some basic c r i t i c i s m s of the mort-gage component alone. Proceeding to the second component, how i s the appraiser said to determine the equity rate? Searching the l i t e r a t u r e merely results i n finding that the appraiser i s somehow able to judge what prospective equity y i e l d i s necessary to a t t r a c t a buyer to take the equity portion. But just how i s t h i s done? The determination of the y i e l d on the.equity component apparently demanded by purchasers i s said to be accomplished by examining comparable properties and t a l k i n g to t h e i r purchasers, and knowing the quantities of the other ingredients i n the comparable data, that i s , the o v e r a l l rates, mortgage terms, and anticipated property depre-22 c i a t i o n or appreciation. A more detailed suggestion i s that the cash flow rate of return a f t e r payments—the equity yield--can e a s i l y be M.B. Hodges, J r . , "Income C a p i t a l i z a t i o n for Investor C l i e n t s , " The Appraisal Journal, XXXVI ( A p r i l , 1968), p. 182. 19 abstracted from market sales of investment property by deduc-tin g t o t a l financing payments from net income and d i v i d i n g the 23 remainder by the equity or down-payment. However, an immediate problem of data a v a i l a b i l i t y may a r i s e . In the empirical section of t h i s study i t w i l l be seen that d e t a i l s such as r e a l net income and financing were unobtainable. Unless more detailed sources were available, the above method of abstraction of the equity rate would be impossible. While the band of investment method may have the merit of seeking the rate applicable to the debt component of the c a p i t a l i n the money market and s i m i l a r l y seeking an appro-pri a t e rate for the r i s k or equity c a p i t a l component by . reference to market data, i t requires subjective evaluations of s i m i l a r i t y of r i s k and of the optimal proportions i n which 24 debt and equity c a p i t a l are combined. A noticeable inconsistency i s that although the debt financing of comparable properties i s disregarded i n attempting to derive the debt rate, equity rates from these same corn-parables are used. Surely the equity rates so derived w i l l depend on the debt financing terms of the comparables con-sidered. I t would seem to make l i t t l e sense to include some-thing which depends to a considerable extent on something else which has been rejected as unsuitable. 23 Nelson, op_. c i t . , p. 26. 24 Dorau, l o c . c i t . 20 Proponents of the band of investment method would claim that the rates are derived from the market. However, as has been seen, the mortgage component i s derived from f i n a n c i a l sources, not past market evidence, i n an overly generalized way. S i m i l a r l y , the l o g i c behind the derivation of equity rates, which depend to a great extent on debt terms, i s questionable and the necessary data often unavailable. And the combination of the two components, being derived from two d i f f e r e n t sources and r e s u l t i n g . i n a synthetic rate, i s not so l o g i c a l l y j u s t i f i e d as proponents of t h i s method might think. Thus, once again, a t r a d i t i o n a l method f a i l s to meet the t e s t of market determination. Before leaving the band of investment method, an addi-t i o n a l concept should be mentioned. I t has been suggested i n the l i t e r a t u r e that i f , say, a nine per cent debt rate and a 12 per cent equity rate combination could be established by comparable sales as r e f l e c t i n g the market with, say, 75 per cent mortgage and 25 per cent equity r a t i o s , i t could be used as a standard of comparison for c a l c u l a t i n g the r e l a t i v e 25 equity rate with any^other mortgage i n t e r e s t rate. Equity y i e l d s are calculated using the following equation: D + E '• • I Y = E Ellwood, op_. c i t . , p. 66. 21 where Y = comparable equity y i e l d , E = any equity r a t i o , I = any mortgage i n t e r e s t rate, and D = composite rate of standard property - 1 mortgage rate of standard property I t i s claimed that t h i s system would, at least provide a means for explaining the source of an equity y i e l d rate. Despite the fa c t that the actual, ultimate y i e l d i s unknown, the appraiser i s said to need an explicable prospective y i e l d 26 to s t a r t with. I t i s true that t h i s system ensures that the composite rate w i l l always exceed the p r e v a i l i n g mortgage rate and that the equity y i e l d w i l l increase as the r a t i o of equity to value declines because the r i s k of equity increases. These would seem j u s t i f i e d i n t u i t i v e l y . However the system also means that the composite i n t e r e s t rate w i l l always be the same, given the same mortgage rate, regardless of mortgage and 27 equity r a t i o s . This l a s t proposition, which i s claimed as an advantage of the system, cannot be accepted so e a s i l y . I t i s reasonable that the equity rate w i l l vary with the changes i n the mortgage i n t e r e s t (and hence the equity r a t i o ) . But i t i s not so rea-Ibid. •2-7.,. • Ibid. 22 sonable that i t would vary i n the manner i n which i t i s claimed, t i e d to the rel a t i o n s h i p of the mortgage arid com-posite rates of what may be another set of circumstances e n t i r e l y . Such a proposition would have to be empirically proven. IV. RATE SELECTION IN THE ELLWOOD METHOD The determination of the o v e r a l l rate i n the Ellwood, or mortgage-equity, approach to c a p i t a l i z a t i o n might well be considered a v a r i a t i o n of the band of investment theory. This approach to valuation adds several factors to t r a d i t i o n a l c a p i t a l i z a t i o n methods, including consideration of financing, the holding period and the appreciation or depreciation of the property. Under t h i s method recapture i s accomplished through amortization of the mortgage during the investment 2 8 term and the sale of the equity at the end of the period. The mortgage-equity method i s based on the f a m i l i a r formula: Net Income Value = Overall Rate However, the difference between t h i s method and the other c a p i t a l i z a t i o n methods i s that the o v e r a l l rate i s derived d i f f e r e n t l y , supposedly considering a l l the objectives and A.I.R.E.A., op_. c i t . , p. 320. 23 reasoning of the t y p i c a l investor i n income property. The formula for the o v e r a l l rate i s : R = Y - MC - d e P r ' i ^ S n appr. V s n where R = o v e r a l l rate Y = equity y i e l d rate at which the present worth of equity cash flow dividends for the investment period plus the present worth of equity reversion at the end of.the investment period equals o r i g i n a l investment, M = r a t i o of mortgage to t o t a l investment expressed as a percentage, C = mortgage c o e f f i c i e n t (this i s a band of investment type feature to give weight and e f f e c t to mortgage terms, including both i n t e r e s t and amortization), depr = projected depreciation expressed as a percentage of t o t a l property value, appr = projected appreciation expressed as a percentage of t o t a l property value, /^s = sinking fund factor at y i e l d rate (Y) for the i n -come projection period. Thus the o v e r a l l rate, R, i s said to be a composite which includes both equity y i e l d , as influenced by mortgage terms, 29 and the provision for change i n property value. 2 9 I b i d . , pp. 321-322. 24 Amortization recaptures value loss during ownership tenure. When i t exceeds that loss i t contributes to equity y i e l d . The mortgage c o e f f i c i e n t i s the band of investment factor which c a r e f u l l y weighs both i n t e r e s t and amortization of mortgage terms. Hence, Y, equity y i e l d , i s reduced to the extent of MC, or r a t i o of mortgage to t o t a l investment times mortgage c o e f f i c i e n t . The reason depreciation or appreciation i s multi p l i e d by 1/Sn, the sinking fund factor at rate Y for the number of years i n the income projection, i s that mortgage reduction i s expected to recapture any value loss during an investment period, and thi s amount i s a sinking fund operation. As the equity p o s i t i o n i s d i r e c t l y affected by depreciation or appre-c i a t i o n , to make adequate provision for r e a l i z i n g an equity y i e l d , Y, the projected equity reversion must be discounted 30 at rate Y. Of course, depreciation i s added to,.and appre-c i a t i o n subtracted from, the equity y i e l d because to provide for lower value the o v e r a l l rate must be increased; the higher the rate, the lower the value. Thus, Ellwood's o v e r a l l rate i s r e a l l y a v a r i a t i o n of the band of investment rate calculated by using the difference between equity and mortgage rates as a c o e f f i c i e n t of mortgage 31 r a t i o s . J.F. Gibbons, "Mortgage-Equity C a p i t a l i z a t i o n and After-Tax Equity Y i e l d , " The Appraisal Journal, XXXVII (January, 1969), pp. 35-36. 31 Ellwood, op.c i t . , p. 68. 25 That i s , Y I M C r then Y - MC = r . This w i l l produce the same composite i n t e r e s t rate as the band of investment method but offers an advantage i n f a c i l i t a t i n g adjustments for mortgage amortization and conse-quent equity value build-up. Equity y i e l d , Y, can be adjusted to r e f l e c t any equity value build-up by reason of mortgage amortization and the mortgage i n t e r e s t rate, I, can be adjusted to cover the cost of amortization or recapture. When adjusted, I i s subtracted from adjusted Y, the remainder i s a c o e f f i c i e n t , C, which can be used with any mortgage r a t i o , M, to produce a 32 comprehensive composite rate, r. Of course, as has been shown, r becomes R, the o v e r a l l rate, with consideration of depreciation or appreciation. Despite the additional factors, there i s no further explanation of just how the rates and r a t i o s are derived. Thus i f : = Equity y i e l d rate = Mortgage i n t e r e s t rate = Mortgage to value r a t i o = C o e f f i c i e n t ( = Y-I) = Composite i n t e r e s t rate 3 2 I b i d . , p. 69. 26 as i n the band of investment method, i t i s thought that the mortgage terms can be derived from f i n a n c i a l sources and an equity y i e l d selected that w i l l a t t r a c t the necessary equity portion, as judged by the appraiser. The s u b j e c t i v i t i e s and inconsistencies of such an approach are the same as those c r i t i c i z e d i n the other forms of the band of investment method. The Ellwood rate i s a function of assumptions which 33 the appraiser makes; i t i s not a market determined rate. V. RATE SELECTION BY COMPARISON OF QUALITY ATTRIBUTES The selection of an i n t e r e s t rate by the comparison of qu a l i t y attributes method supposedly recognizes the reaction of people i n the marketplace. For instance, i f i t i s found that a number of properties are s e l l i n g for approximately $100,000 and have a net income aft e r depreciation (that i s , af t e r recapture allowance) of $8,000, then i t i s r e a d i l y assumed that the rate of return desired by people who are i n the market for t h i s type of property i s eight per cent. This more or less i s based on the assumption that q u a l i t y i s the 34 same. R a t c l i f f , Current Practices i n Income Property  Appraisal - A C r i t i q u e , op. c i t . , p. 40. A.I.R.E.A., op_. c i t . , p. 270. 27 In comparing properties on a q u a l i t a t i v e basis, i t may be necessary to consider such factors as the r e l i a b i l i t y of the income, r e l i a b i l i t y of the expense prediction, l i k e l i h o o d of competitive construction, s a l a b i l i t y of property, expense-income r a t i o , s t a b i l i t y of value and burden of management. The appraiser may then attempt to assign quality ratings i n percentages of importance to each of these factors. The pur-pose of such a step i s to e s t a b l i s h a chart to rate the net income of properties which have been sold i n an e f f o r t to obtain the probable i n t e r e s t rate of the property which i s the subject of appraisal. That i s , comparable properties are selected and the actual i n t e r e s t rate calculated (for example, the eight per cent above); these rates are m u l t i p l i e d by the o v e r a l l q u a l i t y rating to determine the computed rate for i d e a l income. If the property which i s being appraised were i d e a l i n every respect, t h i s i d e a l rate would be used. If the property were not i d e a l according to the q u a l i t y rating, the rating would be divided into the i d e a l rate to determine the c a p i t a l i -35 zation rate, which would then be higher than the i d e a l rate. The q u a l i t y attributes mentioned overlap somewhat into other steps i n the appraisal process. The r e l i a b i l i t y of 35 Ibid., p. 273. 28 income and expense predictions should have been covered i n the income and expense analysis. Competitive construction and s t a b i l i t y of value tend to overlap into the d u r a b i l i t y of income consideration. However, the A.I.R.E.A. text con-cludes that the q u a l i t y analysis by comparison of attributes 3 6 has the v i r t u e of making the appraiser "think." The band of investment method was c r i t i c i z e d on the basis that i t attempted to generalize from market data cer t a i n components that did not lend themselves to such generalizations. Perhaps the q u a l i t y analysis and comparison technique might be considered an attempt to overcome the band of investment method's f a i l u r e of properly going to the market. However, i t too i s open to serious c r i t i c i s m . F i r s t , as s h a l l be discussed i n d e t a i l l a t e r , recapture of the equity portion, not necessarily the t o t a l value of the property, i s r e a l l y the important con-cern for the investor. Thus, by considering net income after depreciation as a percentage of t o t a l value, the rates being compared are u n r e a l i s t i c measures of actual market behaviour. Second, the s u b j e c t i v i t y involved i n the selection of factors and t h e i r q u a l i t y ratings completely removes any confidence i n the f i n a l c a p i t a l i z a t i o n rate being a market determined f a c t , rather than a subjective preconception. Thus, l i k e the other t r a d i t i o n a l methods, the comparison of q u a l i t y attributes method f a i l s to pass the t e s t of market determination. 3 6 I b i d . , p. 274. 29 VI. RATE SELECTION BY DIRECT COMPARISON With s u f f i c i e n t data, c l o s e l y comparable with respect to type and q u a l i t y , the appraiser, i t i s said, w i l l develop his i n t e r e s t rates by comparison and i n connection with an assumed recapture period. For instance, given that the o v e r a l l rate (net income before recapture divided by s e l l i n g price) and the recapture rate (100 per cent divided by remaining l i f e i n years) are known, the i n t e r e s t rate may be calculated by d i v i d i n g the net i n t e r e s t on land and buildings (net before 37 recapture less recapture installment) by the sale p r i c e . Thus, by investigating market evidence o v e r a l l rates are obtained f i r s t and then allocated, on the basis of judge-ment, into i n t e r e s t rates and recapture rates. The i n t e r e s t rates so demonstrated by these sale properties may be analyzed for a l l categories of r i s k ; and from such analysis a rate may be established for the subject property, r e f l e c t i n g i t s r e l a t i v e attractiveness as an investment, according to the A.I.R.E.A. 4- 4- 38 text. While t h i s method begins with market fa c t s , i t soon degenerates as assumptions about depreciation are introduced. S t r a i g h t - l i n e depreciation i s usually u n r e a l i s t i c . Even i f depreciation could be predicted accurately, as has been mentioned Ibid., pp. 274-275. 'ibid. , p. 277 . 30 the investor i s interested i n recapture of his equity and may never expect to recapture the portion covered by debt. Thus, the eventual application, although not necessarily the o r i -g i n a l base, f a i l s to meet the test of market determination. VII. SOME GENERAL CRITICISMS B a s i c a l l y there are two ways of determining c a p i t a l i -zation rates. One i s to simply ask participants i n p a r t i c u l a r transactions how they arrived at t h e i r investment decisions generally and, hopefully, how they determined t h e i r c a p i t a l i -zation rate s p e c i f i c a l l y . Appraisers do not do t h i s , and even i f they could and did, they might get each individual's sub-j e c t i v e value, or value to the investor, rather than the most probable s e l l i n g p r i c e . The other basic way i s to determine the c a p i t a l i z a t i o n rate from available facts about transactions of comparables. Supposedly, the t r a d i t i o n a l methods just c r i t i c i z e d allow the appraiser to derive the rate i n t h i s way. But as has been seen t h i s i s generally impossible. The methods have f a i l e d to meet the t e s t of market determination. Unfortunately they are used to estimate c a p i t a l i z a t i o n rates which can only be considered guesses based on assumptions rather than market fa c t s . Besides the c r i t i c i s m s which have been made of each of the t r a d i t i o n a l methods, i t i s necessary to make some general 31 c r i t i c i s m s and comments. I t was mentioned e a r l i e r that some of the methods discussed involve an u n r e a l i s t i c assumption about the recapture of the entire depreciable portion of the i n i t i a l c a p i t a l investment over the economic l i f e of the property. In f a c t , investors are generally those who hold properties for periods less than the economic l i f e and are 39 more concerned with the recapture of t h e i r equity investment. Another f a i l u r e of the t r a d i t i o n a l methods of s e l e c t i n g c a p i t a l i z a t i o n rates i s that they e n t i r e l y disregard income taxes, an important consideration i n r e a l estate investment. Investors have d i f f e r e n t tax rates and some invest i n r e a l estate s o l e l y for tax shelter. The question of income taxes i s an example of the t r a d i t i o n a l methods*! f a i l u r e to properly consider investor objectives. As w e l l , there i s no e f f o r t to analyze r i s k i n p r o b a b i l i s t i c terms. Thus, many of the general c r i t i c i s m s about the methods are also c r i t i c i s m s of the t r a d i -t i o n a l c a p i t a l i z a t i o n formulae, which f a i l to consider a l l of the relevant market variables. In summary, i t would appear that i t i s impossible for an appraiser to extract from an actual transaction the i n t e r e s t or discount rate of which the s e l l i n g price was a function or which the buyer or the s e l l e r employed i n his private c a l c u l a -t i o n s . Not only i s the transaction price a function of a R a t c l i f f , op. c i t . , p. 35. 32 market s i t u a t i o n i n which many buyers and s e l l e r s w i l l i n -fluence the s e l l i n g price through competitive forces, but also several other variables, including the l e v e l and pattern of income, economic l i f e , and residual value, which are used i n the c a p i t a l i z a t i o n formula along with the i n t e r e s t rate 40 and which cannot be determined from the s e l l i n g p r i c e . In a recent study of appraisal reports, an e f f o r t was made to discover how appraisers actually arrived at t h e i r i n t e r e s t rate assumptions on the basis of the "market," as claimed. I t was concluded that i n no case was i t l o g i c a l l y demonstrated that the i n t e r e s t or discount rate was objectively 40i extracted from comparable sales. Although the A.I.R.E.A. text presents the t r a d i t i o n a l methods of selecting c a p i t a l i z a t i o n rates which have been d i s -cussed, i t also suggests that, when the appraiser can accu-mulate a s u f f i c i e n t number of comparables to e s t a b l i s h the contemporary o v e r a l l rates acceptable to t y p i c a l investors for various types of property, factual support i s given to the high degree of judgement involved i n the less d i r e c t methods of developing rates, Thus, the appraisal body concludes, the 42 correct rates may be found i n the market. ^ I b i d . , p. 39. 4 L , , , Ibid, 42 A.I.R.E.A., op. Icjtv:, pi J 2 75 33 Two things are worth noting. The appraisal body-admits that there i s a high degree of judgement involved i n the less d i r e c t methods. I t would seem that there i s l i t t l e need, and less value, i n using these methods as long as d i r e c t data i s a v a i l a b l e . And i f data i s not available, i t would seem even more d i f f i c u l t to j u s t i f y rate selection based on premises that are not market determined facts but merely unsupported judgements and preconceptions. The second thing worth noting i s that the o v e r a l l rates the text suggests can be derived are not s t r i c t l y c a p i t a l i z a t i o n rates but simply earnings-price r a t i o s . The r e l a t i o n s h i p between the two w i l l be discussed i n d e t a i l i n the next chapter. I t may be concluded that there e x i s t serious weaknesses and inconsistencies i n the t r a d i t i o n a l methods of selecting c a p i t a l i z a t i o n rates. As c a p i t a l i z a t i o n rates are often based on d i f f e r e n t methods of s e l e c t i o n and f i n a l s e l e c t i o n within these methods can only r e s u l t from unsupported judgements and preconceptions, inaccuracies and inconsistencies can only r e s u l t i n the f i n a l value estimates. The i m p o s s i b i l i t i e s of deriving c a p i t a l i z a t i o n rates from market facts must be stressed. Something which i s possible to extract i s an earnings-price r a t i o , of which the gross income m u l t i p l i e r i s a type, arid which s h a l l now be considered. 34 VIII. RATE SELECTION FROM GROSS INCOME MULTIPLIERS One of the d i f f i c u l t i e s of appraisal which has already-been alluded to i s the determination of expense figures. Another A.I.R.E.A. suggestion of obtaining the o v e r a l l rate, i n the absence of expense figures, i s to use a gross c a p i t a l i -43 zation rate, or i t s r e c i p r o c a l , the gross income m u l t i p l i e r . Beginning with either figure, the net i n t e r e s t rate may be obtained by abstraction, provided reasonable estimates of long-term operating r a t i o s , land-to-building r a t i o s , and c a p i t a l recapture periods for the comparable sales can be made. Thus using t h i s method, i f the gross income m u l t i p l i e r for a par-t i c u l a r class of property were determined to be 6.66, the gross c a p i t a l i z a t i o n rate would then be 15 per cent (100/6.66), from which would be subtracted an estimated gross expense r a t i o , say 44 40 per cent, to determine a net o v e r a l l rate, nine per cent. Thus, i f a standardized expense r a t i o i s used, the same subject property value could be obtained by simply applying the gross income m u l t i p l i e r to the income. For example, using the figures above, a property with a gross income of $100,000 would be valued at $666,666 by dividing i t s net income of $60,000 ($100,000 less 40 per cent) by the o v e r a l l rate of nine per cent or by simply multiplying the Ibid., p. 275. Ibid., p. 276 . 35 gross income by the gross income m u l t i p l i e r of 6.66. The above would seem to indicate that the gross income m u l t i p l i e r might well be an accurate means of ca l c u l a t i n g value. The p o s s i b i l i t y that t h i s i s true s h a l l be f i r s t con-sidered t h e o r e t i c a l l y and then by extensive s t a t i s t i c a l analysis. CHAPTER III DIRECT CONVERSION RATIOS Direct conversion ra t i o s have already been defined as those r a t i o s used i n appraisal to convert income, usually current actual or economic ren t a l income, to a c a p i t a l figure representing market value of the property."'" Although such r a t i o s include both gross and net income m u l t i p l i e r s , the former type i s probably the most common and the one which s h a l l be considered here i n depth. I t has already been stated that the gross income multi-p l i e r expresses the re l a t i o n s h i p between the annual (or monthly) gross income produced by a property and the c a p i t a l value, be i t actual s e l l i n g price or estimated most probable s e l l i n g p r i c e , of that property. For example, a property s e l l i n g for $67,000 and producing $10,000 i n gross income per annum would have a gross income m u l t i p l i e r of 6.7 ( i . e . $67,000/10,000). R.U. R a t c l i f f , Current Practices i n Income Property  Appraisal - A Critique (Berkeley: University of C a l i f o r n i a , 1967), p. 39. 37 I. MARKET OR INCOME APPROACH? Are gross income m u l t i p l i e r s a device used i n the market or the income approach to valuation? Wendt states that they are, i n the l a s t analysis, a market phenomenon, and t h e i r use i s as much an adaptation of the market comparison as of the c a p i t a l i z a t i o n of income method. Market sales prices are, of course, estimated from gross rents by reference to relationships between sales prices and rents of comparable p r o p e r t i e s . 2 The A.I.R.E.A. text book c l a s s i f i e s the gross income m u l t i p l i e r under the market data approach, while net income m u l t i p l i e r s are said to be more properly considered as part of 3 the income approach. In a study of actual appraisal reports, many had the application of m u l t i p l i e r s as part of the market approach; 4 otherwise they appeared under the income approach. Although there are c o n f l i c t i n g opinions on t h i s question, i t i s not of v i t a l importance. Gross income m u l t i p l i e r s are P.F. Wendt, Real Estate Appraisal (New York: Holt, 1956) , pp. 209-210. 3 American I n s t i t u t e of Real Estate Appraisers, Appraisal  of Real Estate (Chicago: Author, 1967), p. 336. 4 R a t c l i f f , op_. c i t . , p. 40. 38 market determined, and, therefore, might be considered as part of the market approach, but convert income productivity to the most probable s e l l i n g price just as the conventional income approach i s supposed to do, and, therefore, might be considered as part of the l a t t e r approach. What i s important i s that the device's use and accuracy be understood. I I . SINGLE FAMILY OR MULTIPLE UNIT? Although th i s study i s concerned with the application of the d i r e c t conversion r a t i o s to r e s i d e n t i a l income property, and s p e c i f i c a l l y to multiple unit r e s i d e n t i a l income property, such r a t i o s have also been applied to single family dwellings. Shattuck claimed i n 1938 that the gross income multi-p l i e r had i t s most general use i n the appraisal of the single-family r e s i d e n t i a l property, although he stated i t was also commonly used i n the appraisal of multi-family proper-5 t i e s . As w e l l , the Federal Housing Administration i n the United States has long employed gross income m u l t i p l i e r s i n g the valuation of single-family r e s i d e n t i a l property. By custom, the m u l t i p l i e r for s i n g l e - and two-family 7 homes i s applied to the monthly rent. For instance, a dwelling that rents for $150 per month and which i s sold for 5 C.B. Shattuck, "Income Approach-Capitalization Processes," Selected Readings i n Real Estate Appraisal (Chicago: American Ins t i t u t e of Real Estate Appraisers, 1953), pp. 1060-1061. ' g Wendt, op_. c i t . , p. 20 4. 7 A.I.R.E.A., op. c i t . , p. 337. 39 $15,000, shows a monthly gross income m u l t i p l i e r of 100. For apartments of more than eight units, the m u l t i p l i e r i s applied to the annual income. Thus, the m u l t i p l i e r i n the above case would be 8.33 ($15,000/1,800). The custom varies for smaller apartments; either the monthly or the annual m u l t i p l i e r i s used. The application of gross income m u l t i p l i e r s to single-family, owner-occupied housing may be questioned. Single-family dwellings are not usually developed as income-producing property, but rather are b u i l t for owner-occupancy. Thus, owner-occupants may be concerned with such amenities, or intangible benefits of home ownership, as l i v a b i l i t y of the dwelling, pride of ownership, functional u t i l i t y , convenience of location and prestige of ownership. Although some of these factors might r e f l e c t i n r e n t a l income i f the property were rented, others are more d i r e c t l y related to home-ownership. Generally speaking, amenities increase with the market value of the house. However, monthly rents do not increase at the same rate as r e n t a l occupants may be unwilling to pay for the amenities associated with owner-occupancy. They price services on grounds of functional u t i l i t y almost e n t i r e l y while, i t i s claimed, prospective owner-occupiers weigh more heavily ce r t a i n amenities. Thus, the gross income m u l t i p l i e r increases as amenities increase; or, put another way, as the 40 the s e l l i n g price increases, the monthly gross income multi-p l i e r tends to increase.^ In many areas the re n t a l market for single-family housing i s not large. The r e s u l t i s a lack of adequate com-parable re n t a l information. Gross income m u l t i p l i e r s are a market phenomenon involving income, as well as sales, data. Therefore, classes of property lacking a s u f f i c i e n t market measurable i n terms of income data, such as the market for single-family homes, cannot be valued by using gross income m u l t i p l i e r s . The remainder of the paper w i l l be concerned with multiple unit r e s i d e n t i a l income property. I I I . ORIGIN AND DEVELOPMENT Wendt presents an excellent summary of the development 9 of the use of income m u l t i p l i e r s i n Real Estate Appraisal. He traces t h e i r use to as early as 1924 when Benson and North wrote: I t i s generally recognized that i f property i s suitably improved i t s r e n t a l i s a safe guide to value. . . . The net re n t a l i s , of course, the amount of net income received by the owner and th i s sum c a p i t a l i z e d at an appropriate percentage gives value. . . . A shorter method than the foregoing i s that of estimating the value from the amount of gross rents... W.M. Shenkel, "Characteristics of Gross Income M u l t i -p l i e r s , " The Real Estate Appraiser, XXXIV (January-February, 1968), p. 25. 9Wendt, op. c i t . , pp. 189-194. 41 Sometimes a rough estimate i s made by traders by multiplying the gross rent by a certain number. Thus the value of an ordinary apartment i s approximated by f i v e , f i v e and one-half or six times the (annual) rent.10 As s h a l l be seen the use of m u l t i p l i e r s p e r s i s t s . I t i s well known that they are widely employed i n the market by investors, brokers, lenders and other r e a l estate market par-ticipants."'""'" IV. USE IN SECURITY ANALYSIS M u l t i p l i e r s are widely used i n the evaluation of common stocks. The m u l t i p l i e r that i s commonly used i s c a l l e d the price-earnings r a t i o and i s simply the price of a p a r t i c u l a r share divided by the earnings-per-share. The s e c u r i t i e s analyst may devise various rules of thumb for s e l e c t i n g an approriate price-earnings r a t i o which can be applied to a company's e x i s t i n g l e v e l of earnings per share to derive an estimated value. The basis for these rules of thumb may range from the purely i n t u i t i v e to elaborate s t a t i s -t i c a l analysis."*" 2 1 0P.A. Benson and N.L. North, Real Estate P r i n c i p l e s and  Practices (New York: 1924), pp. 170-171. "'""''Wendt, op_. c i t . , p. 192. 12 J.B. Cohen and E.D. Zinburg, Investment Analysis and  P o r t f o l i o Management (Homewood, I l l i n o i s : Dow Jones-Irwin, 1967), p. 239. 42 While i t i s true that such r a t i o s for i n d i v i d u a l stocks are market determined f a c t s , the rules of thumb m u l t i p l i e r s may be highly a r b i t r a r y according to Graham. Supposedly they are based on the "quality" of each issue, largely as i t appears i n the eyes of investors and has been r e f l e c t e d i n the 13 market's own valuations over some years past. V. THE RELATIONSHIP OF MULTIPLIERS AND CAPITALIZATION RATES I t has often been said that the use of the gross income m u l t i p l i e r i s a short cut to the c a p i t a l i z a t i o n method of e s t i -mating value and that the m u l t i p l i e r i s simply the r e c i p r o c a l of the o v e r a l l c a p i t a l i z a t i o n rate applied to gross income i n 14 perpetuity. This r e l a t i o n s h i p , which was b r i e f l y mentioned i n the previous chapter, i s i l l u s t r a t e d by the following table;: (Table I) . Most of the l i t e r a t u r e claims that gross income multi-p l i e r s are merely reciprocals of gross c a p i t a l i z a t i o n rates. . The consideration of gross income m u l t i p l i e r s as a c a p i t a l i -zation device i s sometimes implied. 13 B.J. Graham, The I n t e l l i g e n t Investor (New York; Harper, 1949), pp. 159-160. 14 Wendt, op_. c i t . , pp. 189-191. TABLE I GROSS INCOME MULTIPLIERS AND GROSS CAPITALIZATION 15 RATES Gross Y S e l l i n g Price Annual Gross Income M u l t i p l i e r Gross Capitali-zation Rates $10,000 $ 50,000 5 20.00 10,000 60,000 6 16.67 10,000 70,000 7 14.29 10,000 80,000 8 12.50 10,000 90,000 9 11.11 10,000 100,000 10 10.00 10,000 110,000 11 9.09 10,000 120 ,000 12 8.33 In 1938 Shattuck held that the "gross income multiple or conversion factor" was one of four basic methods i n the c a p i t a l i z a t i o n of income process: The gross income multiple, which i s sometimes c a l l e d a conversion factor, i s merely a r a t i o of rent to sales p r i c e . . . This method of converting income into a c a p i t a l value estimate i s the simplest of many c a p i t a l i z a t i o n processes. . . x^ 15 Shenkel, Op. c i t . , p. 23. "^Shattuck, Toe. c i t . 44 The Federal Housing Administration, having long employed gross income m u l t i p l i e r s i n the valuation of r e s i -d e n t i a l property, shares the view that the use of the gross income m u l t i p l i e r i s a c a p i t a l i z a t i o n process: Monthly ren t a l value translated into an estimate of c a p i t a l i z e d amenity income by the use of rent m u l t i p l i e r s , which r e f l e c t the probable q u a l i t y and duration of the amenity returns i n future years, i s i n accord with the general c a p i t a l i z a t i o n theory.1? In f a c t the use of d i r e c t conversion r a t i o s cannot be properly considered c a p i t a l i z a t i o n . In the previous chapter c a p i t a l i z a t i o n was found to involve the future income stream, usually net, rather than the current income to which multi-p l i e r s are normally applied. Thus, while a c a p i t a l figure i s being derived, discounting of future returns i s not considered under the application of d i r e c t conversion ra t i o s .and therefore c a p i t a l i z a t i o n i n i t s defined sense cannot be properly implied. While th i s d i s t i n c t i o n i s f a i r l y obvious and acceptable, i t might s t i l l be thought that the gross income m u l t i p l i e r i s simply the r e c i p r o c a l of the gross c a p i t a l i z a t i o n rate. C l e a r l y , i f the gross c a p i t a l i z a t i o n rate i s 12.50 and the annual gross income i s 8, then they are each other's r e c i p r o c a l . Federal Housing Administration, Underwriting Manual (rev. ed.; Washington: Author, 1952), Sec. 1202(4). 45 At least one author f a i l s to agree that conceptually the gross income m u l t i p l i e r i s necessarily the r e c i p r o c a l of 18 the gross c a p i t a l i z a t i o n rate. Rather, gross income multi-p l i e r s are said to be the reciprocals of return, or income over p r i c e , r a t i o s . Dorau claims that while return r a t i o s may look l i k e and may contain c a p i t a l i z a t i o n rates, they are but crude mixtures of other elements. Thus, while return r a t i o s may look l i k e c a p i t a l i z a t i o n rates and are found i n the normal mathematical rela t i o n s h i p of a c a p i t a l i z a t i o n rate, i t i s a rare coincidence i f , In f a c t , the return r a t i o i s the c a p i t a l i z a t i o n rate. This concept i s based on the idea that to assume that an earnings or income .to price r a t i o i s the i m p l i c i t c a p i t a l i -zation rate by which the price i s determined i s only j u s t i f i e d when the indicated present income i s the only economic advan-tage r e f l e c t e d i n the p r i c e ; that i s , when the price obtained and the indicated income are exclusively related factors. I t i s claimed that such i s seldom the case, because of the numer-ous other factors r e f l e c t e d i n p r i c e ; and even when i t may be, how could i t be proven? v H,B. Dorau, "The C a p i t a l i z a t i o n Rate: Mirage of Will-o'-the Wisp? The Appraisal Journal, XXIX (January, 1961), pp. 19-29. 46 The d i s t i n c t i o n between present and future income pointed out e a r l i e r would seem to mean that m u l t i p l i e r s cannot properly be considered c a p i t a l i z a t i o n devices and, thus, i s reason alone for the m u l t i p l i e r not to be the re c i p r o c a l of the c a p i t a l i z a t i o n rate. They could only be equal when present income i s assumed to be constant into the future. Thus, so long and so often as the price i s the value of any anticipated economic advantage other than the i n d i v i -dual present income, the nearnings or income to price r a t i o 19 xs not the i m p l i c i t capxtalxzatxon rate. Although Dorau distinguishes between c a p i t a l i z a t i o n rates and m u l t i p l i e r s , he concludes: So now i t seems that income or future benefit valu-ations may be made without knowing or finding the c a p i t a l i z a t i o n r a t e - i n - f a c t by the use of income m u l t i p l i e r s or t h e i r reciprocals the earnings or income to price ratios.. Such earnings or income to price r a t i o s are s i g n i f i c a n t relationship which may be derived from the market and transposed for use i n a valuation which the market has not made. 2© VI. ADVANTAGES OF USE The two p r i n c i p a l advantages of the gross income multi-p l i e r , which no doubt o r i g i n a l l y encouraged i t s use, are the a v a i l a b i l i t y of information and s i m p l i c i t y . There i s notusually a data a v a i l a b i l i t y problem as the value data required, namely Ibid., p. 22. Ibid., p. 29. 47 present gross income and s e l l i n g p r i c e , are usually r e a d i l y a v a i l a b l e . The application of the device i s simpler and more e a s i l y comprehended than are many variations of the c a p i t a l i -zation of income method.. As well, assuming comparability of properties, the m u l t i p l i e r method eliminates some of the more subjective processes of estimation i m p l i c i t i n the use of the t r a d i t i o n a l income method. Of course, the advantage of s i m p l i c i t y has also been used as a c r i t i c i s m of the m u l t i p l i e r approach. In any case the use of gross income m u l t i p l i e r s omits thenneed to estimate depreciation as i t i s measured by the prices at which buyers and s e l l e r s are w i l l i n g to exchange income property. As well, forecasts of future income , streams and expense estimates, both 21 of which may be highly subjective opinions, are unnecessary. Of course, there can be no advantage to such an appraisal device unless i t i s , at l e a s t , reasonably accurate. The very fa c t that gross income m u l t i p l i e r s are widely used i n p r a c t i c a l valuation may mean that there i s a tendency for actual sales prices to adapt themselves to the relationships presumed by these m u l t i p l i e r s . To the extent that the foregoing i s true, gross income m u l t i p l i e r s should be an accurate measure. Follow-ing the discussion of the l i t e r a t u r e , t h i s hypothesis s h a l l be tested. 21 Shenkel, op. c i t . , p. 30. 48 VII. CRITICISMS The gross income m u l t i p l i e r i s not always well received i n t r a d i t i o n a l appraisal l i t e r a t u r e . This section discusses some of the basic c r i t i c i s m s which have been levied against the device. One of the basic c r i t i c i s m s of the gross income multi-p l i e r i s simply that i t i s based on gross, rather than net, income. I t i s said to be derived from the p r i n c i p l e that an average percentage rel a t i o n s h i p exists between the sum of d o l l a r allowances for i n t e r e s t on investment, property taxes, management expense, depreciation and maintenance costs, and 22 the c a p i t a l value for p a r t i c u l a r classes of property. That i s , the appraiser using the gross income m u l t i p l i e r i s said to assume standard expense r a t i o s for the comparables used. Or, at l e a s t , i t i s implied that the r a t i o of operating expenses to gross income i s s u f f i c i e n t l y uniform to r e s u l t i n f a i r l y standard relationships between gross income and c a p i t a l value. As early as 1932, Babcock c r i t i c i z e d the gross income m u l t i p l i e r for the above reason: Another form of short-cut sometimes used i n p r a c t i c a l valuation i s the multiplying of estimated gross revenues by selected factors. . . . The method should not be used even for quick estimating for very t y p i c a l proper-Wendt, op_. c i t . , p. 189. Shenkel, op_. c i t . , p. 23. 49 t i e s . . . . The p r i n c i p a l objection to the method i s , then, that i t treats gross revenues rather than net incomes.24 Those who c r i t i c i z e the use of gross income m u l t i p l i e r s on these grounds do so as they f e e l that the variable q u a l i t y of management and the operating expenses w i l l have a s i g n i f i -cant e f f e c t on the annual expense allowance and resultant net income. For instance, they f e e l ithat i t i s e n t i r e l y possible that a property which produces a comparable gross income may y i e l d inadequate or even no net income because of excessive operating or maintenance cost due to faulty construction or inequitable contractual commitments written into long-term lease agreements. In either case, the existence of gross i n -come i s said to give an i l l u s i o n of value that could not be 25 j u s t i f i e d by an "expert" appraiser. Another c r i t i c i s m that i s levied i s that various con-cepts of gross income are employed i n the development of gross income m u l t i p l i e r s . In some cases actual gross rentals for the current year are employed, while i n others, perhaps less frequently, a forecast of so-called " s t a b i l i z e d gross rents" 2 6 i s used. Obviously, these variations i n the precise d e f i -n i t i o n of gross rentals make i t necessary to exercise caution 24 F.M. Babcock, The Valuation of Real Estate (New York: McGraw-Hill, 1932), p. 180.. 25 A.A. Ring, The Valuation of Real Estate (Englewood C l i f f s , N.J.:. Prentice-Hall, 1963), pp. 175-176. 2 6 Wendt, op_. c i t . , p. 201. i n the comparison of gross income m u l t i p l i e r s ; to some, these variations are further reason to advocate complete discourage-ment of use. A related c r i t i c i s m i s that since the gross income multi-p l i e r s are taken from actual incomes, there i s the p o s s i b i l i t y that current s e l l i n g prices do not account for variations i n vacancy rates. Thus, to apply indiscriminately the annual gross income m u l t i p l i e r against properties showing a v a r i a t i o n i n 27 vacancy rates would d i s t o r t the indicated market value. Some c r i t i c s f e e l that gross income m u l t i p l i e r s are often used i n f l e x i b l y . For instance, the number seven has long been held to be the proper m u l t i p l i e r for many classes of property i n 2 8 many areas. Should an appraiser merely learn a common multi-p l i e r by rote, or to the extent i t i s based on past, rather than current, experience, poor valuations w i l l r e s u l t . I t i s the essence of m u l t i p l i e r s that they are subject to change and, thus, 29 must be used with f l e x i b i l i t y . A further c r i t i c i s m i s that error frequently occurs from the application of m u l t i p l i e r s to properties with varying l i f e 30 expectancies. Indiscriminate use of the gross income 27 Shenkel, op_. c i t . , p. 27. 2 8 L.H. Scane, "The Mystic Seven," The Appraisal Journal, XXVI (July, 1958), pp. 390-392. 29 L. Winnick, "Long-Run Changes i n the Valuation of Real Estate by Gross Rents," The Appraisal Journal, XX (October, 1952), p. 485. "^F.F.Sonnenschien, "The E f f e c t of L i f e Estimates on Capi t a l Values," The Appraisal Journal, XIV (January, 1964). 51 m u l t i p l i e r would ascribe equal value to properties of equal income even though one may be i n the l a s t stages of i t s 31 economic l i f e and the other i n new condition. This i s , of course, the same argument that i s used against simple c a p i t a l i -zation i n perpetuity. There are numerous c r i t i c i s m s that have been levied against gross income m u l t i p l i e r s . Most of them involve the fact that the gross income m u l t i p l i e r , by i t s simple nature and past use, does not consider a l l those factors the theorists consider relevant i n the sel e c t i o n of comparable income proper-t i e s . I t might be suggested that there are certain ways of circumventing these c r i t i c i s m s by s t r a t i f y i n g comparables by relevant property c h a r a c t e r i s t i c s . Such s t r a t i f i c a t i o n i s discussed i n the following section. VIII. SELECTION OF COMPARABLES In order to determine a gross income m u l t i p l i e r to be applied to a p a r t i c u l a r property, or class of property, the appraiser must, of course, consider the gross incomes and s e l l i n g prices.of comparable properties. Obviously, the multi-p l i e r w i l l vary widely for d i f f e r e n t types, areas, classes or grades of property, as well as with the type of neighbourhood, Ring, op_. c i t . , p. 176. 32 depending on the amount of r i s k involved i n the investment. However, i t has also been claimed that data often show that the m u l t i p l i e r s vary sub s t a n t i a l l y at any p a r t i c u l a r time for 33 s i m i l a r types of property i n the same l o c a l i t y . If t h i s i s the case, just what i s a comparable property for purposes of s e l e c t i o n and application of gross income mu l t i p l i e r s ? The f i r s t step i n answering t h i s important ques-tio n i s to consider those factors with which gross income m u l t i p l i e r s may vary between properties. The purpose of such considerations i s to i d e n t i f y factors that might be used as guidelines for se l e c t i n g and applying gross income m u l t i p l i e r s One of these factors, i n answer to the c r i t i c i s m men-tioned e a r l i e r that commonly used m u l t i p l i e r s are often based on gross, rather than net terms, i s simply the operating expense r a t i o . Thus, although at f i r s t glance i t w i l l be noted that the m u l t i p l i e r varies according to the gross annual income and the sales p r i c e , i t i s claimed that the operating expense r a t i o cannot be t o t a l l y ignored, assuming buyers and 34 s e l l e r s act r a t i o n a l l y and are concerned with net income. By t h i s reasoning, an apartment building with an expense r a t i o 32 J.J . Carney, "The Development and Use of Gross Income M u l t i p l i e r s , " The Appraisal Journal, XXXI ( A p r i l , 1963), p.225 33 Wendt, op_. c i t . , p. 202. 34 Shenkel, op_. c i t . , p. 24. 53 of 50 per cent may be expected to show a lower gross income m u l t i p l i e r than a s i m i l a r property with an expense r a t i o of 30 per cent. Thus,.it i s often urged that only those proper-t i e s with s i m i l a r operating expense r a t i o s should be used as comparables when se l e c t i n g and applying gross income m u l t i p l i e r s . I t has been suggested i n the l i t e r a t u r e that for accur-ate value estimates the appraiser must use highly comparable conditions for properties within each class of property from 35 which he w i l l develop a gross income m u l t i p l i e r . One of the very obvious c l a s s i f i c a t i o n s that should be made at the outset i s between general classes of property. Of course, r e t a i l store properties cannot be compared with o f f i c e buildings or r e s i d e n t i a l properties. S i m i l a r l y , single-family dwellings cannot be compared with multiple-unit r e s i d e n t i a l properties. And, within the multiple-unit r e s i d e n t i a l c l a s s i f i c a t i o n with which we are concerned, low-rise, frame buildings should pro-bably be distinguished from high-rise concrete buildings. Too many variables are associated with income and i t s quality drawn from these d i f f e r e n t classes of property. The rule i s to apply 3 6 gross income m u l t i p l i e r s to comparable properties. There are many ways broad classes of properties can be broken down. For instance, the qu a l i t y of construction and Carney, op_. cit.-, p. 223. Shenkel, op_. c i t . , p. 28. 54 workmanship, as well as the character of construction (frame, concrete, brick, stone), are more detailed factors with which 37 the appraiser might be concerned. The age of the property has been distinguished as a 3 8 s i g n i f i c a n t f actor. Besides changing construction methods and materials, d i f f e r i n g levels of depreciation may a f f e c t value. Property nearing the end of i t s economic l i f e may be expected to show greater expenses for repairs and maintenance and lower r e l a t i v e gross incomes. Newer properties are more l i k e l y to command higher rent than older properties, subject to higher levels of depreciation. Thus, m u l t i p l i e r s may vary with age of property and, i t has been said, should be compared only to property showing s i m i l a r levels of deprecia-. . 3 9 t i o n . Size i s a further factor sometimes deemed worthy of consideration. That i s , apartment properties may be grouped i n terms of number of suites for purposes of selection and a p p l i -cation of gross income m u l t i p l i e r s . Another important consideration i s lo c a t i o n . This factor 40 has both physical and economic implications. Within a given 37 Carney, l o c . c i t . 3 8 I b i d . 39 Shenkel, op. c i t . , p. 29. 40 . Ibid., p. 28 . 55 metropolitan area there may e x i s t a number of apartment building d i s t r i c t s which may be characterized, at least to some degree, by t h e i r l l o c a t i o n s . This factor may overlap some of the others just discussed to the extent that i n d i v i -dual areas are made up largely of one clas s , s t y l e , size and age of property. Further considerations concern the dependability and qua l i t y of the income stream. I t has been suggested that i t should be determined whether the rents are going to be main-tained or i f they have been increased or decreased as the 41 r e s u l t of some temporary condition. A related factor i s the qu a l i t y of the income stream. I t has been suggested that gross income m u l t i p l i e r s w i l l vary, given the same l e v e l of income, by the r e l a t i v e q u a l i t y or r i s k s associated with the receipt of that income. Thus, t h i s factor recommends the refinement of gross income data to cover properties comparable with respect to income character-. .. 42 i s t i c s . I t has already been mentioned that when income data i s actual, variations i n vacancy rates may d i s t o r t the indicated market value when using gross income m u l t i p l i e r s . As well, actual income data should be distinguished from any sort of Carney, op_. c i t . , p. 225. Shenkel, op_. c i t . , p. 29. 56 standardized "normal." These are s t i l l more reasons for using properly comparable data. The l e v e l of tenant services has also been considered a relevant factor by some appraisers. As a simple comparison of gross incomes and s e l l i n g prices w i l l not show differences i n the l e v e l of services, high comparability i s said to require 43 the same l e v e l of tenant services. Obviously, properties selected should be of a recent and comparable time of sale. Since the m u l t i p l i e r i s derived from at least several observations, and since p a r t i c u l a r care must be taken to confine the m u l t i p l i e r to si m i l a r property types, i t i s not uncommon to r e l y on sales completed over a 44 span of several years. To the extent that a trend, or any fl u c t u a t i o n , occurred over the years considered, the r e l i a b i l i t y of the gross income m u l t i p l i e r might be reduced. I t has also been suggested that conditions of sale are a relevant factor. Comparables should represent "arm's length" transactions. As we l l , any unusual terms of sale, such as those caused by unusual financing terms, should be investigated as they may have caused price d i s t o r t i o n and, consequently, 45 misleading m u l t i p l i e r s . 43T, . , Ibid. 4 4 I b i d . , p. 29. 45 Ibid., p. 28. 57 Thus, t r a d i t i o n a l l y the l i t e r a t u r e has suggested that there are an almost unlimited number of factors which must be considered, either o b j e c t i v e l y or subjectively, i n the selec-t i o n and application of gross income m u l t i p l i e r s . I t should be noted that for valuation purposes, the gross income m u l t i p l i e r i n i t i a l l y suggests i t s e l f for use on the basis of i t s s i m p l i c i t y . Detailed consideration of numer-ous factors i s seemingly at odds with t h i s i n i t i a l j u s t i f i c a -t i o n . Thus i t i s possible to c r i t i c i z e attempts to add sophis-t i c a t i o n to something that i s considered to be a simple valu-ation device, or, by some, a crude rule of thumb. IX. ADJUSTMENT OF COMPARABLES Some theorists d i s t i n g u i s h between adjusted and unadjusted gross income m u l t i p l i e r s , as they f e e l that the uniformity of unadjusted gross income m u l t i p l i e r s i s not to be expected since 46 'there are too many market imperfections to cause v a r i a t i o n s . Thus, besides detailed consideration of the relevant factors just discussed, i t i s sometimes suggested that multi-p l i e r s be "adjusted" when v a r i a t i o n occurs i n certain factors so that they might be used rather than rejected as uncomparable. For instance, i t has been suggested that i f the appraiser i s ^Ibid., p. 26. 58 i n doubt about whether the rents are going to be maintained or i f they have been increased or decreased as the r e s u l t of some temporary condition, i t may be better for him to use the esta-blished f a i r rents per square foot or per room or per apartment from a s i m i l a r b u i l d i n g , rather than erroneous i n f l a t e d or deflated rents from the property as a r e s u l t of a temporary s c a r c i t y or oversupply of rent a l units i n thi s p a r t i c u l a r area. Another adjustment, supposedly necessitated by the fact that gross income figures may not r e f l e c t vacancy rates, involves standardizing income figures used at f u l l economic rent or at a given vacancy rate. As well, time of sale d i f f e r -ences could supposedly be adjusted by correcting for changes i n c a p i t a l i z a t i o n r a t e s . 4 8 A further suggestion for adjustment concerns the fac t that gross income m u l t i p l i e r s do not d i r e c t l y consider varying l i f e expectancies. Under th i s suggestion adjustment factors would be applied to m u l t i p l i e r s based upon the percentage of value calculated as remaining for various periods i n a proper-49 ty's l i f e . The c r i t i c i s m that can be levied against a l l of these "adjustments" i s the same as that levied against detailed Carney, op_. ext., p. 225. Shenkel, op_. c i t . , p. 28. Sonnenschien, l o c . c i t . 59 consideration of too many factors; that i s , oversophistication i s being introduced to what i s meant to be a simple, unsophis-t i c a t e d device. The very nature of the device i s being changed. As w e l l , i f the m u l t i p l i e r s and t h e i r supporting data are adjusted to produce a m u l t i p l i e r that i s believed reasonably t y p i c a l , the objective of the analysis i s defeated. 50 Obviously the r e s u l t can only be based on some preconception. X. USE OF MULTIPLIERS Although the use of gross income m u l t i p l i e r s has re-ceived much c r i t i c i s m as an unsophisticated appraisal device, t h e i r use i s not always e n t i r e l y discouraged. Having considered the various factors that have been deemed necessary of consider-ation i n the l i t e r a t u r e , i t i s also necessary to d i s t i n g u i s h between the uses to which the m u l t i p l i e r can be said to be put. There are those who claim that i t may only be used as a rule of thumb or check or analysis device of some sort. Others t e n t a t i v e l y suggest that given consideration of the factors mentioned and great care, the m u l t i p l i e r may actually be used for valuation, with the further q u a l i f i c a t i o n that i t i s only one of several approaches. T r a d i t i o n a l l y , the gross income m u l t i p l i e r and other d i r e c t conversion r a t i o s have been thought of as rules of thumb. Shenkel, op_. c i t . , p. 26. 60 I t has been stated that there may be many occasions when an appraiser or investor w i l l want a r e l a t i v e l y simple, easy-to-use method of establishing a rough value range for a property Rule of thumb guides may be used to s e l e c t from a number of properties the p a r t i c u l a r ones about which more detailed i n -formation and analysis may be desired, to e s t a b l i s h a range within which the f i n a l estimate of value w i l l be found, to conduct mass appraisals or short, quick appraisals or to t e s t for changes i n the market value of properties. The A.I.R.E.A. textbook suggests that gross income m u l t i p l i e r s are useful as general guides i n testing compara-52 b i l i t y i n the market data approach. For example, suppose that s i m i l a r apartment buildings generally sold for a multi-p l i e r of seven i n a given area. If a recent sale revealed a m u l t i p l i e r of 10, one of several facts might be indicated: the e x i s t i n g gross income allowance was below economic rent, the property was r e l a t i v e l y overpriced or the operating expen 53 were uncommonly high. In any case, the appraiser selecting comparables for use i n the t r a d i t i o n a l methods would probably 51 S.A. Kahn, F.E. Case and A, Schimmel, Real Estate  Appraisal and Investment (New York: Ronald, 1963), p. 41. 52 A.I.R.E.A., op_. c i t . , p. 337. 53 Shenkel, op_. c i t . , p. 24. 61 be prompted to r e j e c t the property with a m u l t i p l i e r of 10 as uncomparable. I t i s well known among r e a l estate brokers, mortgage lenders and appraisers, that the d i r e c t conversion r a t i o s are the basis for at least f i r s t approximations of value for cer-54 t a i n classes of property i n many markets. However, supposedly more sophisticated means are t r a d i t i o n a l l y thought necessary for accurate f i n a l value estimates. Another suggested secondary use i s to provide a check upon the r e s u l t obtained by "other and professionally recom-mended" methods under the three t r a d i t i o n a l approaches to 55 . . . . value. Should the m u l t i p l i e r indicate results inconsistent with those obtained under supposedly more refined and a n a l y t i c a l procedures, i t i s considered the appraiser's duty to explain the v a r i a t i o n and give reasons for the cause of the exception to the rule of thumb findings. Gross income m u l t i p l i e r s also have secondary uses i n the analysis of investment properties. They avoid the more compli-cated analysis of net income statements and serve as useful guides to judge gross incomes and s e l l i n g prices where operating expense r a t i o s are thought to be f a i r l y standardized for the R.U. R a t c l i f f , "Capitalized Income i s Not Market Value," The Appraisal Journal, XXXVI (January, 1968), p. 38. Ring, op_. cit.-, pp.. 174-175. 62 56 same property types. A c r x t i c a l revxew of unadjusted gross income m u l t i p l i e r s r e s u l t i n g i n an explanation of why multi-p l i e r s depart from the p r e v a i l i n g or t y p i c a l m u l t i p l i e r may be useful i n the valuation process. Certainly the r e l i a b i l i t y of the gross income m u l t i p l i e r i s explained and a better under-standing of those factors which a f f e c t value may be achieved. Others have ventured further i n suggesting that gross-income m u l t i p l i e r s may be actually used to predict value accurately. For instance, Wendt concludes that when taken together, the advantages of gross-income m u l t i p l i e r s furnish a strong basis for advocating t h e i r use for many classes of 57 valuation. But he q u a l i f i e s t h i s by stating that they can only be used when income to value relationships are known to be si m i l a r for major classes of property and the subject property i s s i m i l a r i n a l l e s s e n t i a l respects to the properties used i n developing m u l t i p l i e r s . Even those who point out the t r a d i t i o n a l c r i t i c i s m s and dangers i n the use of gross income m u l t i p l i e r s admit that t h e i r use pe r s i s t s as a widely employed method i n the market 5 8 to estimate c a p i t a l value from gross income. I t has been ^Shenkel, op. c i t . , p. 24. 57 Wendt, op_. c i t . , pp. 210-211. 5 8 I b i d . , p. 192. 63 seen that t h e i r o r i g i n a l use was prompted by t h e i r inherent s i m p l i c i t y . Gross income i s much easier to ascertain than net income. And the use of such m u l t i p l i e r s involves no predic-tions or estimates by the appraiser as i n the case of the use of conventional c a p i t a l i z a t i o n formula. Besides t h e i r inherent s i m p l i c i t y there may be another very important reason for t h e i r continued use f o r cert a i n segments of the market; that i s , t h e i r accuracy. Any accuracy that they possess i s largely due to the fact that, because they are widely employed, they may become " b u i l t - i n t o " the market. To the extent that r e a l estate market participants use m u l t i p l i e r s to evaluate investment opportunities and, indeed, make decisions, they become r e f l e c t e d i n the market. I t has been suggested that gross income m u l t i p l i e r s are so generally r e l i e d upon for cert a i n classes of property that market prices tend to c l u s t e r around a r e l a t i v e l y narrow range 59 i n the gross'income m u l t i p l i e r . Thus, i n many cases there i s a tendency for actual sales prices to adapt themselves to the rel a t i o n s h i p presumed by gross income m u l t i p l i e r s . ^ 59 Winnick, op. c i t . , pp. 484-485. 6 0 Wendt, op_. c i t . , p. 210. 64 I t may be true that appraisers use these devices more than i s actually r e a l i z e d or admitted. A recent study of actual appraisal reports showed that i n about o n e - f i f t h of the reports, the only " c a p i t a l i z a t i o n device" employed was the d i r e c t conversion r a t i o , either the gross or net m u l t i p l i e r , and that the appraiser used some form of d i r e c t conversion 61 r a t i o i n 77 per cent of the reports. The gross and net i n -come m u l t i p l i e r s were about equal i n incidence with a few cases of d i r e c t conversion r a t i o s based on net income afte r depreciation and net income afte r debt service. The high incidence of the use of d i r e c t conversion r a t i o s i n income property appraisal creates the presumption that these r a t i o s play an important role i n the appraiser's ? value estimates In addition to, or instead of, the c l a s s i c c a p i t a l i z a t i o n formula. This presumption i s supported by additional facts that resulted from the above mentioned study of appraisal reports. Where values produced by the use of d i r e c t conversion r a t i o s and the f i n a l appraised values of the properties could be compared, they were i d e n t i c a l i n nearly 40 per cent of the cases; i n three-quarters of the cases the difference was less than three per cent with an 6 2 average difference of 0.1 per cent. 61 R a t c l i f f , Current Practices i n Income Property  Appraisal - A C r i t i q u e , op. c i t . , pp. 40-41. 6 2 I b i d . , p. 42. 65 I t might be inf e r r e d that many appraisers place primary reliance on the d i r e c t conversion r a t i o s and that, i n the c o r r e l a t i o n step, they adjust the other approaches to produce value figures of respectable consistency. In any case, i t can c e r t a i n l y be i n f e r r e d that most appraisers use d i r e c t conver-sion r a t i o s and that i f they used nothing else as the basis f o r t h e i r f i n a l value conclusions the f i n a l appraisal value would 6 3 be v i r t u a l l y the same figure as actually reported. From the foregoing survey of the appraisal l i t e r a t u r e on gross income m u l t i p l i e r s i t can be seen that while they have been widely c r i t i c i z e d as unsophisticated and suitable only f o r rough rules of thumb, they continue to be used i n the marketplace. As they become " b u i l t - i n " market guides, they become powerful factors i n the establishment of market pr i c e s , 64 a f a c t which imparts great p r e d i c t i v e value. The appraiser can make use of t h i s phenomenon. That i s , to the extent that these r a t i o s influence investment decisions, they have useful-ness i n predicting market behaviour and market p r i c e . Having discussed much of the theory and suggesting that the r e l a t i o n s h i p between gross incomes and s e l l i n g prices may Ibid., Ibid. pp. 42-43. 66 be useful i n the valuation of certa i n classes of property, the emphasis must now s h i f t to some empirical t e s t i n g . The goal, of course, i s to measure the accuracy and r e l i a b i l i t y of predicting the most probable s e l l i n g price of an income property by using a gross income m u l t i p l i e r and to esta b l i s h some guidelines for the use of such a device. CHAPTER IV DATA AND METHODOLOGY I. DATA Data was col l e c t e d on 385 recent apartment property sales i n the Greater Vancouver Area from l o c a l r e a l t o r s , appraisers and assessors. In each i n d i v i d u a l case the follow-ing information was obtained: 1. Gross income 2. S e l l i n g price 3. Gross income m u l t i p l i e r s (2/1) 4. Number of suites 5. Year of sale 6. Basic s t r u c t u r a l type, i . e . - frame, low r i s e or concrete high r i s e 7. Basic location by area, e.g. Marpole, West End, Burnaby, etc. 8. Basic age, e.g. b u i l t 1940-59, 1958-67, etc. 9. Other, including: elevator or waTkup; i n 36 cases a qu a l i t y indicator of A-plus to C; i n 7 cases, more detailed information, including net income; as well, gross income per suite was calculated. The size of the sample i s r e l a t i v e l y large. In fac t , i t probably represents a large proportion of the apartment property sales i n the area considered over the l a s t few years. 68 Although the d e t a i l of the data i s not quite as favour-able, i t i s s u f f i c i e n t to produce some int e r e s t i n g and worth-while r e s u l t s . Certainly, more d e t a i l s might have been useful. For instance, considering l o c a t i o n , sub-areas within some of the larger areas might have been useful. Or land value i n f o r -mation could have been used. S i m i l a r l y , although the number of suites was available i n each case, number of rooms might have permitted better s i z e s t r a t i f i c a t i o n s . Of course, actual net income would probably have been the most i n t e r e s t i n g of the unavailable d e t a i l s . I t would have enabled a comparison with gross income information as to the accuracy of the value estimate. The general lack of information i s not untypical of a market as imperfect as that of r e a l estate. Depending on the state and si z e of his f i l e s , the i n d i v i d u a l appraiser might have more detailed information i n s u f f i c i e n t numbers than was immediately available for the following analysis. The often c a l l e d for data bank would presumably have more complete data for extensive analysis. The data could be summarized i n 23 groups as o r i g i n a l l y obtained: 1. High-rise, West End, A+, b u i l t since 1960, 5. 2. High-rise, West End, A, b u i l t since 1960, 6. 3. High-rise, West End, B, b u i l t since 1960, 22. 4. High-rise, West End, C, b u i l t since 1960, 3. 5. High-rise, Suburban, 9. 69 6. Frame-Walkup, West End, b u i l t 1954-60, 18. 7. Frame-Elevator, South G r a n v i l l e , Fairview, K i t s i l a n o , 26 8. Frame-Elevator, Marpole, 8. 9. Frame-Elevator, Kerrisdale, 8. 10. Frame-Walkup, West End, 20+ suites, b u i l t 1958-67, 9. 11. Frame-Walkup, South G r a n v i l l e , Fairview, K i t s i l a n o , 20+ suites, b u i l t 1958-67, 18. 12. Frame-Walkup, East End, 20+ suites, b u i l t 1958-67, 8. 13. Frame-Walkup, Marpole, 20+ suites, b u i l t 1958-67, 7. 14. Frame-Walkup, South G r a n v i l l e , Fairview, K i t s i l a n o , -20 sui t e s , b u i l t 1958-67, 22. 15. Frame-Walkup, East End, -20 suites, b u i l t 1958-67, 23. 16. Frame-Walkup, Marpole, -20 suites, b u i l t 1958-67, 19. 17. Frame-Walkup, West End, a l l s i z e s , b u i l t 1940-59, 20. 18. Frame-Walkup, South G r a n v i l l e , Fairview, K i t s i l a n o , a l l s i z e s , b u i l t 1940-59, 52. 19. Frame-Walkup, East End, a l l s i z e s , b u i l t 19 40-59, 9. 20. Frame-Walkup, Kerrisdale, a l l s i z e s , b u i l t 1940-59, 9. 21. Frame-Walkup, Marpole, a l l s i z e s , b u i l t 1940-59, 18. 22. Frame-Walkup, Burnaby, 59. 23. High-rise, West Vancouver, 7. Of course, as analysis proceeded various other groupings were experimented with. I I . SIMPLE LINEAR REGRESSION AND CORRELATION MODEL The gross income m u l t i p l i e r i s a device used to predict one var i a b l e , c a p i t a l value, from knowledge of another, gross income. Regression and c o r r e l a t i o n analysis i s a tool for studying the s t a t i s t i c a l r e l a t i o n s h i p between two or more v a r i -ables , so that the value of one variable can be predicted on the basis of the other, or others. 1 The variable that i s to be J . Neter and W. Wasserman, Fundamental S t a t i s t i c s for  Business and Economics (Boston: A l l y n and Bacon, 1961), pp. 549-550. 70 predicted ( i . e . c a p i t a l value) i s said to be the dependent variable (Y),. while the variable (or variables) used to predict 2 ( i . e . gross income) i s said to be the independent variable (X). If only two variables, one dependent and one independent, are involved, the r e l a t i o n s h i p i s termed simple; i f more than one independent variable i s used to predict the single dependent variable, the r e l a t i o n s h i p i s termed multiple. And i f there i s a tendency for the dependent variable to change by a constant amount when the independent variable changes by a given absolute 3 amount, the rela t i o n s h i p i s termed l i n e a r . I t i s customary to mathematically f i t a regression, or estimating, l i n e to the data (e.g. sales and income data for various comparables) by a method known as the least squares method which minimizes the sum of the squared differences of the observed Y values and the corresponding calculated Y value 4 on the l i n e . That i s , for each X, there w i l l probably be a difference between the observed value of Y and the corresponding value, denoted f v , which f a l l s on the estimating l i n e . The A l e a s t squares method minimizes the sum of the squares of these differences. Thus, the l i n e so f i t t e d i s said to y i e l d the 2 B. P a r i , Basic S t a t i s t i c s (Garden C i t y , New York: Doubleday, 1967), p. 224. 3 I b i d . 4 I b i d . , pp.226-227. 71 closest f i t of any l i n e to the observations under th i s accepted 5 method. A s t r a i g h t - l i n e r e l a t i o n s h i p can be described mathe-g matically by the equation: Y v = a + bX where Y = the estimated or computed l i n e value of Y, given X, x a = estimate of the constant which represents the • value of Y when X = 0, b = estimate of the c o e f f i c i e n t of regression which shows the amount of change i n Y which i s associated with a one-unit change i n X, X = independent v a r i a b l e . I t should be noted that the values of a and b are e s t i -mates based on the sample data of the two corresponding para-7 meters which are assumed to e x i s t i n the universe. Hence, the value of Y v i s also ansestimate of the corresponding universe x value. Thus, i t i s possible to calculate a regression or e s t i -mating equation for sets of data with two variables, such as ^Neter and Wasserman, op_. c i t . , p. 563. .Pa r i , op_. c i t . , p. 227.. 7 Neter and Wasserman, op_. c i t . , p. 559. 72 gross income and c a p i t a l value, and to use the equation to estimate the dependent va r i a b l e , c a p i t a l value, from the independent variable, gross income. However, as Y for any X i s only an estimate, something must be said about i t s r e l i -a b i l i t y . The regression l i n e represents only a measure of the average re l a t i o n s h i p between the dependent and independent g varia b l e s . This rel a t i o n s h i p may not provide completely accurate estimates for i n d i v i d u a l observations. Thus, some estimating errors w i l l be involved unless a l l the observations l i e exactly on the estimating l i n e . The greater the dispersion or scatter of the observations around the regression l i n e , the greater the estimating error. The magnitude of t h i s estimating error i s measured by the standard deviation about the estimating l i n e termed the standard error of estimate. I t i s calculated by 9 the following formula: S y x where S y x = estimate of standard error of estimate Y = observed value of Y Y x = estimated l i n e value of Y n = sample s i z e . g P a r i , o p . c i t . , p. 228. 9 Neter and Wasserman, op_. c i t . , p. 565. 73 Before continuing i t i s necessary to introduce a basic s t a t i s t i c a l concept, that of the normal curve. The normal curve i s a s p e c i f i c bell-shaped and symmetrical frequency curve, defined by a mathematical formula."'"0 The t o t a l area under the normal curve, or the t o t a l frequency, i s considered as 100 per cent. I t can be ascertained from s p e c i a l l y prepared normal curve tables that i n the case of a normal d i s t r i b u t i o n 68.27 per cent of the items f a l l within one standard deviation or standard error above or below themmean, 95.45 per cent within the two standard deviation or standard error range and 99.73 per cent within the three standard deviation or standard error range. Such ranges are c a l l e d confidence intervals."''"'' One of the most frequently used ranges i s the 95 per cent confidence i n t e r v a l , which corresponds to a range of 1.96 standard errors. At such a confidence l e v e l there i s only a f i v e per cent, or one i n 20 chance, of an estimate f a l l i n g 12 outside the s p e c i f i e d confidence i n t e r v a l . Although very few d i s t r i b u t i o n s i n any f i e l d are ex-a c t l y normally d i s t r i b u t e d , the normal curve serves as a s a t i s f a c t o r y approximation of the actual d i s t r i b u t i o n i n the "*"°Parl, op. c i t . , pp. 73-74, ^ I b i d . , p. 112. 1 2 I b i d . , pp. 142-143. 74 case of large samples. Thus, i t i s possible to state a formula fo r the approximate confidence, or prediction, i n t e r v a l for an i n d i v i d u a l value at, say, the 95 per cent l e v e l of confidence as f o l l o w s : 1 3 Y v - 1.96 S y x . The value 1.96 i s termed the standard normal deviate 1.4 corresponding to the 95 per cent l e v e l of confidence. This value, which varies for d i f f e r e n t levels of confidence, i s ob-tained from normal curve tables. In the case of small samples the normal d i s t r i b u t i o n approximation i s not adequate. Another d i s t r i b u t i o n , the t-d i s t r i b u t i o n , i s used instead to derive the i n t e r v a l estimate. Values of t, which vary according to the number of degrees of freedom (which i n the regression c a l c u l a t i o n i s n-2) and the l e v e l of confidence, may be obtained from s p e c i a l l y prepared t a b l e s . 1 ^ When the sample size i s small the methods of regression 16 analysis must be modified. The concept of the standard error of estimate has already been introduced. In addition, because the values of a and b are estimates, they too have standard 13 Neter and Wasserman, op. c i t . , pp. 570. 1 4 I b i d . , p. 569. 1 5 I b i d . , p. 596. 1 6 I b i d . , p. 594. 75 error s . I f a l l three sources of estimation errors are con-sidered together the combined error i n estimating an i n d i v i -17 dual item i s obtained. The formula for the confidence 18 i n t e r v a l i s as follows: Actually this formula y i e l d s the appropriate i n t e r v a l s 19 for any size sample, not just those for small samples. If the sample size should exceed about 32, the t value i n the formula simply can be replaced by the appropriate value for the normal d i s t r i b u t i o n (e.g. 1.96). This i s because the t - d i s t r i -bution i s clos e l y approximated by a normal d i s t r i b u t i o n when the 20 degrees of freedom are 30 or more. If the sample size i s much larger, about 100, or more, then the approximate in t e r v a l s given e a r l i e r (Y"x ± 1.96 S y x at the 95 per cent l e v e l of confidence) w i l l give almost the same 17 P a r i , op, ext., p. 240. 18 Neter and Wasserman, op_. c i t . , p. 597, 1 9 I b i d . , pp. 598-599. 2 0 I b i d . , p. 599. 76 re s u l t s as the exact i n t e r v a l as calculated i n the above formula. 2 1 Certain factors a f f e c t the width of the confidence i n t e r v a l and are, therefore, worth noting. The larger the sample size n, the smaller w i l l be the value of the square root i n the formula and, consequently, the smaller w i l l be the 22 width of the i n t e r v a l . Obviously, as n becomes large, the value of the second term ( jjj- ) under the square root sign be-comes very small. The value of +_ i n the i n t e r v a l i s also affected by sample s i z e . The larger the sample, the smaller 23 the value of t and, therefore, the narrower the i n t e r v a l . The smaller the standard error of estimate, the narrower the i n t e r v a l generally w i l l be. The magnitude of the standard error of estimate i s inherent i n the p a r t i c u l a r 24 problem and does not depend upon the sample s i z e . Of course, should there be one p a r t i c u l a r variable that results i n a p a r t i c u l a r l y great v a r i a t i o n from the l i n e , the e f f e c t on the standard error of estimate w i l l be lessened as the sample size increases since the square root of the average v a r i a t i o n i s taken. Ibid. 2 2 I b i d . , p. 600. 23 Ibid. 24 ^ I b i d . 77 The farther X i s from the mean of X (X), the larger o w i l l be the term (X-X)^, the larger the square root and, therefore, the wider the i n t e r v a l at that X. 25 As the standard error of the regression c o e f f i c i e n t , b, i s expressed as a per one unit change i n X, the error due to an estimating error i n the b value would increase as the estimated value moves away 26 from the mean value of X. Thus, the confidence i n t e r v a l i s not a constant band p a r a l l e l to the regression l i n e . I t i s at i t s minimum where X equals the mean of X (as the t h i r d term under the square root sign equals zero) and curves outwards as X 27 moves away from the mean. The greater the v a r i a b i l i t y of the observed X's, the 2 ( $ X) 2 greater w i l l be IX — — — - • , the smaller the square root and consequently, the narrower the width of the confidence interval.28 dard errors of a and b, i n the construction of the confidence i n t e r v a l becomes increasingly less important as the sample size increases, since the sampling errors decline i n proportion to Y n , while the standard error of estimate i s expected to remain unchanged, when r e f e r r i n g to a relationship within a given The omission of the sampling errors, that i s , the stan-25 Ibid. 26 P a r i , op_. c i t . , p. 238. 27 Neter and Wasserman, lo c . c i t . 28 Ibid. 78 universe. Consequently, i n p r a c t i c a l problems, except when the sample i s very small, researchers usually disregard the samp-l i n g errors of the regression constants as being of r e l a t i v e l y minor importance, while t h e i r i n c l u s i o n may increase the compu-estimate w i l l be considered to be the predominant source of the estimating error. There are several additional concepts which must be introduced as they are both useful and necessary i n the analysis. One i s an abstract measure c a l l e d the c o e f f i c i e n t of determination (R^) which varies from zero to one and represents the proportion of the v a r i a t i o n i n the dependent variable that has been explained by the employment of the relationship between 30 the dependent and independent variables. The c o e f f i c i e n t of determination, i f i t i s not close to zero, indicates that X may be h e l p f u l i n making a useful prediction of Y. However, the usefulness of a prediction of Y can only be evaluated when the appropriate prediction i n t e r v a l has been computed so that i t can be determined i f the prediction meets the requirements for . . 31 p r e c i s i o n . t a t i o n a l burden noticeably. 29 Then only the standard error of 29 P a r i , op_. c i t p. 242. 30 Ibid pp. 232-233. 31. Neter and Wasserman, op. c i t p. 585. 79 Another relevant concept i s the significance test of the regression c o e f f i c i e n t (b). An F-value i s calculated by the following formula: — — 2 F ( l , n-2) = 2 ( YX Y ) ( S y x ) 2 A p r o b a b i l i t y i s associated with t h i s value of F. I t i s the p r o b a b i l i t y of obtaining an F-value greater than or equal to the one calculated, given that the actual regression c o e f f i c i e n t equals zero. I f t h i s p r o b a b i l i t y i s less than .05, i t i s usually concluded that b i s s i g n i f i c a n t l y d i f f e r e n t from 32 zero. I t should be pointed out that using a regression model, or even the simple gross income m u l t i p l i e r , implies using past experience, i n the form of the data used to calculate the regression l i n e , to predict events i n the future, most probable s e l l i n g p r i c e s . Therefore, the regression analysis w i l l be relevant to the future only i f the same fundamental conditions existent i n the past s t i l l p r e v a i l at the time for which pre-33 dictions are being made. To the extent that these basic conditions change, misleading results w i l l occur. Thus, i n 32 J.H. Bjerring and R.H. H a l l , Triangular Regression  Package (Vancouver: University of B.C., Computing Centre, 1968), pp. 32-33. 33 Neter and Wasserman, op_. c i t . , p. 576. 80 using such models for appraisal purposes, data must be kept as up-to-date as possible. The computed regression l i n e i s based on observations where X values are within a given range. Interpolation assumes that the basic conditions which determine the ex i s t i n g r e l a -tionships between variables remain unchanged. Making predictions by extrapolation, that i s , by extending the regression l i n e beyond the actual data range, and using the extended l i n e values 34 as estimates of Y, i s going beyond s t a t i s t i c a l evidence. No rel a t i o n s h i p may e x i s t beyond the observed data range, or the relati o n s h i p may e x i s t i n some form of a curve, rather than a str a i g h t l i n e . Thus, i t i s important that the gross income of the subject property f a l l within the range used to calculate the equation. The above has implications for the value of the constant term (a). While i n t u i t i v e l y i t may seem reasonable that an apartment property would have some residual value when gross income equals zero, the value of a cannot be considered as an accurate estimate of that value. The a value i s simply an ex-tension of the li n e a r r e l a t i o n s h i p outwards from the data range to the Y axis. There i s no proof that such l i n e a r i t y i s j u s t i -f i e d outside that range. Thus, conclusions as to the value of P a r i , op. c i t . , p. 231. 81 a, other than as a mathematical necessity i n the equation, can only be lim i t e d . I I I . AVERAGE MULTIPLIER Generally, when the gross income m u l t i p l i e r i s to be applied to a p a r t i c u l a r property i t i s derived by determining m u l t i p l i e r s for comparable properties and averaging them. Despite the s i m p l i c i t y of the re l a t i o n s h i p , i t i s useful to express i t i n s i m i l a r terms as the simple l i n e a r regression model just discussed. That i s , a s t r a i g h t - l i n e r e l a t i o n s h i p may be expressed mathematically by the equation: Y x = bX where Y v = the estimated or computed l i n e value of Y, given X b = the average m u l t i p l i e r expressed as a c o e f f i c i e n t which shows the amount of change i n Y which i s associated with a one-unit change i n X X = the independent v a r i a b l e . Although the basic forms of the simple l i n e a r regression and c o r r e l a t i o n model and the average m u l t i p l i e r are s i m i l a r , i t i s necessary to distinguish two differences between them. One difference i s that the b values i n each case are calculated d i f f e r e n t l y . The b value i n the average m u l t i p l i e r i s simply the arithmetic mean of the m u l t i p l i e r s of comparable properties. 82 The regression c o e f f i c i e n t (b) i s calculated i n such a way that the sum of the squared differences of actual and calcu-lated Y's i s minimized. Thus, i t should not be assumed that the average m u l t i p l i e r i s the same as the regression model (where a equals zero). The second difference, of course, i s the very f a c t that the constant term (a) i s not present i n the average m u l t i p l i e r . IV. MULTIPLE REGRESSION AND CORRELATION Although the lack of detailed data precludes the exten-sive use of multiple variable regression and c o r r e l a t i o n , a b r i e f discussion of the concept i s relevant to the present discussion. A regression l i n e may be computed by the method of l e a s t squares i n such a way that i t simultaneously f i t s a l l 35 of the variables considered. When each of the b values for the various independent variables i s computed, a l l the other independent variables are held s t a t i s t i c a l l y constant. That i s , the r e l a t i o n s h i p of each one of the independent variables to the dependent variable i s measured, as f a r as i s s t a t i s t i -c a l l y f e a s i b l e , without interference from the other independent • 36 variables. Of course, the p r i n c i p a l reason for using multiple re-gression i s to attempt to improve estimates by adding additional Ibid., p. 253. 3 6 I b i d . , pp. 253-254. 83 variables. This can be checked by plugging figures into both equations and seeing i f they vary s u b s t a n t i a l l y . Another method i s to calculate the standard error of estimate for the multiple regression equation and compare i t with the same s t a t i s t i c i n the simple (one-variable) case. Hopefully, the standard error of estimate w i l l be reduced. If not, nothing has been added by i n c l u s i o n of the additional variables; that 37 xs, relxabxlxty of the estxmates i s not improved. One of the reasons that substantial improvement may not occur i s that there i s a close r e l a t i o n s h i p between the added variables and the o r i g i n a l independent variable. Because of the close r e l a t i o n s h i p , the explanation offered i n terms of one variable i s merely duplicated by the other. There i s no additional explanation introduced by the use of the addi-t i o n a l v a r i a b l e . Additional explanation can only occur i f a variable i s introduced which i s not c l o s e l y correlated to the o r i g i n a l independent variable but i s to the dependent vari a b l e . V. APPLICATION Now that s u f f i c i e n t s t a t i s t i c a l background has been introduced, the models can be applied to the available data. The r e s u l t s of such applications w i l l be presented i n the f o l -lowing chapter. Ibid., p. 257. Ibid., p. 260. 84 I t should be remembered that the goal i s twofold. F i r s t , the accuracy of gross income m u l t i p l i e r s must be determined; second, guidelines for the use of such m u l t i p l i e r s must be established. Such goals necessitate preliminary experimentation. Data w i l l be grouped i n various ways through s t r a t i f i c a t i o n by the d i f f e r e n t parameters. Application of the models and an analysis of the s t a t i s t i c a l r esults then follows. The f i r s t phase of the analysis w i l l involve the calcu-l a t i o n of confidence l i m i t s . Ideally, the confidence band should be narrow as too wide a band would be of l i t t l e use to an appraiser. Secondly, estimates can be made and compared to actual s e l l i n g p r i c e s . Thus, i t w i l l be possible to say that had the appraiser used a c e r t a i n equation he would have been able to predict s e l l i n g price within so many d o l l a r s , as well as so many per cent, of the actual p r i c e . In addition, of course, i t i s possible to analyze the various equations on the basis of some of the regression and c o r r e l a t i o n s t a t i s t i c s . For instance, the c o e f f i c i e n t of determination which i s a measure of the percentage of the t o t a l v a r i a t i o n which i s explained, for a simple equation, could be compared with that of a multiple equation calculated on the same data to see whether there i s something to be gained by the use of more variables i n the equation. CHAPTER V RESULTS OF ANALYSIS I. REGRESSION EQUATIONS The f i r s t step i n the s t a t i s t i c a l analysis was to c a l -culate a number of regression equations. The r e s u l t s , along with the appropriate regression and co r r e l a t i o n s t a t i s t i c s , are presented i n the Appendix. :iSpeaking generally, two things are immediately appar-ent. In almost a l l of the equations the regression c o e f f i c i e n t b i s s i g n i f i c a n t l y d i f f e r e n t from zero, i n d i c a t i n g a d e f i n i t e r e l a t i o n s h i p between c a p i t a l values and gross incomes. The second thing i s that the c o e f f i c i e n t s of determination are, i n most cases, very high, i n d i c a t i n g that a large portion of the va r i a t i o n i n c a p i t a l values i s explained by the v a r i a t i o n i n gross incomes. Aside from these general observations, i t i s necessary to go into more d e t a i l as to the r e l i a b i l i t y and accuracy of the equations. I I . CONFIDENCE INTERVALS As seen i n the previous chapter a regression equation can do more than just make a single estimate of c a p i t a l value. I t expresses a prediction i n t e r v a l into which predicted values may be said to f a l l at a given l e v e l of confidence. Thus, at the 95 per cent l e v e l of confidence, the confidence, 86 or pr e d i c t i o n , i n t e r v a l may be calculated with the assurance that predicted values w i l l f a l l within this i n t e r v a l 95 times out of one hundred. Cl e a r l y , any estimator such as an appraiser, would desire as narrow a confidence i n t e r v a l as possible at a given l e v e l of confidence. If the i n t e r v a l i s too wide i t would be of l i t t l e use to him. For instance, i f a confidence i n t e r v a l for a p a r t i c u l a r class of property was calculated to be $100,000 to $200,000, the prediction would be of l i t t l e use as more pr e c i s i o n i s required i n estimating the most probable s e l l i n g price of income property. As was seen e a r l i e r , the confidence i n t e r v a l i s not s t r i c t l y p a r a l l e l to the regression l i n e , except for large samples. From a minimum width at the mean value of X i t widens towards the outer ranges of the data from which the l i n e was calculated. Thus, to s t r i c t l y define the confidence i n t e r v a l i t would be necessary to make many laborious calcu-lations (most accurately, one for each X). For the immediate purpose at hand,-to gain an i n d i c a t i o n of the r e l a t i v e widths of the confidence i n t e r v a l s for various equations and how, by s t r a t i f i c a t i o n , they may be narrowed, i t i s s u f f i c i e n t to calculate the width of the i n t e r v a l at the mean of X. Because the d o l l a r amount of the mean of X w i l l vary i t i s useful to express the amount that i s added or substracted from the Y y at the mean of X as a percentage of 87 that mean. Clearly an amount of, say, $20,000 at a mean of $200,000 i s d i f f i c u l t to compare with the same amount at a mean of $400,000 unless percentages are also used. Of course, i t would be f o o l i s h to ignore the d o l l a r amounts completely as acceptable i n t e r v a l widths may vary as the size of the means of the gross incomes of groups of property increase or decrease. One d i f f i c u l t y that arises i n comparison i s that sample sizes d i f f e r for many groupings. For samples over 32 the v a r i a t i o n i n sample size i s not p a r t i c u l a r l y important as the standard errors of estimate are a l l mul t i p l i e d by the same standard normal deviate (1.96 i n the case of the 95 per cent l e v e l of confidence used). However, for smaller samples the t-value, which varies with the number of degrees of freedom, must be used. In any case i t proves useful to proceed des-p i t e t h i s d i f f i c u l t y . As w e l l , equations with the same sample size w i l l be compared l a t e r . The f i r s t basic s t r a t i f i c a t i o n of the data i s a basic one, d i f f e r e n t i a t i n g between low-rise frame properties and high-rise concrete properties. I t i s well accepted that these basic s t r u c t u r a l types constitute submarkets of the o v e r a l l apartment property market. To the extent t h i s i s true, i t would seem reasonable to group such properties separately. The results of such a grouping i s presented i n Table I I . Because this i s the f i r s t of several s i m i l a r tables, i t i s worthwhile to f u l l y explain i t at t h i s point. The 88 TABLE II CONFIDENCE INTERVALS - STRATIFICATION BY STRUCTURAL TYPE (1) (2) _ (3) (4) (5) Description n Y (where •+ t _ Sy'x (4) % (3) X X=X) " n " 2 ($) ($) A l l Properties 385 288 ,369 (1 .96)(42,990) 29. 22 = 84,260 Frame Low R i s e - A l l 333 196 ,134 (1 .96) (18,970) 18. 96 = 37,181 Concrete High 52 879 ,099 (1 .96) (88,420) 19. 71 R i s e - A l l =173,303 f i r s t column, "Description," merely describes the grouping of the properties used to compute a p a r t i c u l a r regression equation. The equation i t s e l f i s not shown i n the table but i s available, along with the appropriate regression and c o r r e l a t i o n s t a t i s -t i c s , i n the Appendix. The second column, "n," i s simply the sample s i z e . I t i s useful to show t h i s figure when comparing confidence i n t e r v a l s , p a r t i c u l a r l y when the sample sizes be-come small. The next column, " Y X ' " i s simply the estimated l i n e value of Y at the mean of X; that i s , i t i s calculated by substituting the mean of the X values used to calculate the 89 equation into that equation and deriving the corresponding Y x value. This figure is.an estimate about which the c o n f i -dence i n t e r v a l i s calculated. The next column, " - t .Syx," ' n-2 •* contains the figure which i s added to or substracted from the estimate to compute the confidence i n t e r v a l . I t i s simply the product of the t-value at n-2 degrees of freedom and the stan-dard error of estimate f o r the equation. Of course, as n reaches 32, the value of ^-n_2 1 S approximated by the standard normal deviate. F i n a l l y , the l a s t column expresses the amount i n column four, the amount added or subtracted, as a percentage of the amount i n column three, the single estimate. This i s done for purposes of comparison on a percentile basis as the absolute estimates and standard errors of estimate may vary considerably and, therefore, be d i f f i c u l t to compare. Thus, the confidence i n t e r v a l could be expressed as so many per cent of the estimate above or below that estimate. Having interpreted the table headings, what does Table II demonstrate? The equation for a l l of the properties y i e l d s a very wide confidence i n t e r v a l . This i s understandable as so many d i f f e r e n t types, sizes and locations are involved. I n t u i t i v e l y , i t would seem unreasonable that an equation based on diverse properties could be used to predict precisely the most probable s e l l i n g price of an i n d i v i d u a l property. This statement i s borne out by Table I I . A confidence i n t e r v a l as wide as the one for the f i r s t equation i s obviously so lacking 90 i n p r e c i s i o n as to be almost useless. I t can be c l e a r l y seen that s t r a t i f y i n g by general s t r u c t u r a l type, that is,.frame low-rise versus concrete high-rise, reduces the width of the confidence i n t e r v a l con-siderably. However, the confidence i n t e r v a l s are s t i l l r e l a t i v e l y wide. Although the p r e c i s i o n has been improved, the p r a c t i c a l usefulness i s s t i l l l i m i t e d . This r e s u l t i s not e n t i r e l y unexpected. Even within these two general struc-t u r a l types, there may be considerable variety i n the nature of the properties and hence, the nature of the r e l a t i o n s h i p between gross income and c a p i t a l value. Thus, i t i s necessary to proceed further with s t r a t i f i c a t i o n i n order to discover guidelines for groupings of p r a c t i c a l use i n appraisal. The next basic c h a r a c t e r i s t i c used to s t r a t i f y i s locat i o n . To the extent that cer t a i n areas i n a c i t y c o n s t i -tute submarkets d i s t i n c t from one another on the basis of lo c a t i o n , useful r e s u l t s should be gained by l o c a t i o n a l s t r a t i f i c a t i o n . Differences i n location may r e f l e c t a number of other factors that d i f f e r between areas. For instance, c e r t a i n areas may simply have been b u i l t up at d i f f e r e n t times. Or, cer t a i n areas may have better qu a l i t y apartment buildings generally. Or, cer t a i n areas may have bigger apartment buildings generally. Any number of differences may be r e f l e c t e d i n l o c a t i o n a l differences. Although the data are not d e t a i l e d enough to investigate a l l of these i n d i v i d u a l 91 differences, location i t s e l f can be considered. Table III shows a breakdown of the " A l l Frame Low-Rise" equation into general l o c a t i o n a l area equations. Although one of the groupings results i n a small sample necessitating the use of the t - d i s t r i b u t i o n , an improvement i s s t i l l achieved. The o v e r a l l improvement i n the width of the confidence i n t e r v a l i s obvious i n each of the l o c a t i o n a l subgroupings. However, despite the improvement, the i n t e r v a l s are s t i l l quite wide. Hopefully, they can be improved somewhat more. Thus, some further s t r a t i f i c a t i o n s must be considered. The size of the apartment property i s perhaps thennext l o g i c a l s t r a t i f i c a t i o n . This i s best measured by the number of s uites. Like basic s t r u c t u r a l type and location, the number of suites further s t r a t i f i e s apartment properties into groupings which can be considered more and more comparable. Table IV shows a rough s t r a t i f i c a t i o n i n terms of s i z e . When frame walkup apartments b u i l t between 1958 and 1967 are separated into two groups, those above and those below 20 suites, an improvement i s achieved. But the equation for frame walkups of a l l sizes b u i l t between 1940-59 actually y i e l d s a r e l a t i v e l y wider confidence i n t e r v a l . Although the e f f e c t of the date of construction i s involved i n the comparison (and cannot be removed because of the nature of the data), t h i s equation would seem to confirm that s t r a t i f i c a t i o n by number of suites i s a worthwhile exercise. 92 TABLE III CONFIDENCE INTERVALS - STRATIFICATION BY LOCATION CD-Description- (2) n (3) Y (where A X=X) ($) (4) t t 0 S y x n-2 2 ($) (5) (4) % (3) Frame Low-Rise - A l l 333 196,134 (1.96)(18,970) = 37,181 18. 96 Frame-West End 47 292,881 (1.96)(25,510) = 50,000 17. 07 Frame-South G r a n v i l l e , Fairview, K i t s i l a n o 118 179,791 (1.96) (15,840) = 31,046 17. 27 Frame-Marpole 52 152,014 (1.96) (10,230) = 20,051 13. 19 Frame-Kerrisdale 17 244,525 (2.13) (19,490) = 41,514 16. 98 Frame-East End 40 129,285 (1.96) (23,226) = 23,226 17. 96 Frame-Burnaby 59 221,871 (1.96) (17,750) 15. 68 = 34,790 93 TABLE IV CONFIDENCE INTERVALS - STRATIFICATION BY SIZE (1) (2) _ (3) (4) (5) Description n Y v (where - t ,Syx (4) % (3) X=X) n—z ($) ($) (%) Frame Low-Rise 333 196,134 (1.96) (18,970) 18.96 - A l l = 37,181 Frame Walkups (20+) 42 202,360 (1.96) (16,590) 16.07 (1958-67) = 32,516 Frame Walkups (20~) 64 112,938 (1.96)(8,831) 15.33 (1958-67) = 17,309 Frame Walkups ( a l l sizes) 108 119,316 (1.96)(12,240) 20.11 (1940-59) = 23,990 The res u l t s i n Table IV do not include any e f f e c t of l o c a t i o n a l s t r a t i f i c a t i o n . Thus, to be progressive i n the analysis i t i s necessary to further analyze the e f f e c t of s t r a t i f i c a t i o n by number of suites within the more general s t r a t i f i c a t i o n s which have already been discussed, those by s t r u c t u r a l type and locat i o n . 94 Table V shows the res u l t s of such an analysis. Problems i n terms of sample size begin to arise as the data i s pro-gressively broken down. As can be seen i n the table, a l l of the groupings by number of suites r e s u l t i n sample sizes which necessitate the use of the t - d i s t r i b u t i o n . Thus, the problem of comparability of small samples that was mentioned e a r l i e r a r i s e s . I t i s encouraging to note that, i n most cases, improvements are achieved. I t i s probably true that these improvements would have been greater had large sample sizes been possible. That i s , the improvements which did occur are limi t e d somewhat by the decrease i n sample s i z e . As well, i t could be suggested that those few which increased i n r e l a t i v e width might have improved or, at lea s t , stayed the same had the sample size been larger. For purposes of more accurate comparability, several equations for d i f f e r e n t groupings were calculated on the basis of 15 randomly selected properties i n each case. Thus, com-parisons are enabled without having to attempt to account for the e f f e c t of d i f f e r e n t sample s i z e s . Table VI shows just such a comparison. Because a l l the samples are of the same s i z e , the width of the confidence i n t e r v a l s can be compared very e a s i l y . I t should be noted that; there i s an improvement i n each of the s t r a t i f i c a t i o n s . The results of these tables seem to indicate c l e a r l y that the number of suites i s a relevant variable for purposes 95 TABLE V CONFIDENCE INTERVALS - STRATIFICATION BY NUMBER OF SUITES (1) (2) + (4) (5) Description n Y (where W 3yx (4) Q. *o (3) A X=X) ll £-.($) ($.) (%) -High-Rise-All 52 879,099 (1 .96) (88,420) 19. 71 - 173 ,303 No.of Suites 24 1,380,833 (2 .07) (129,400) 19. 40 ? 45 = 267 ,858 No. of Suites 28 449,147 (2 .06) (21,890) 10. 04 <45 = 45 ,093 Frame-West End 47 292,881 (1 .96) (25,510) 17. 07 - 50 ,000 No. of Suites 24 413,713 (2 .07) (30,980) 15. 50 » 25 = 64 ,129 No.of Suites 23 166,745 (2 .08) (16,060) 20. 03 <25 = 33 ,405 Frame-Burnaby 59 221,871 (1 .96) (17,750) 15. 68 = 34 ,790 No.of Suites 25 390,150 (2 .07) (25,830) 13. 70 »20 = 53 ,468 No.of Suites 34 98,115 (1 .96) ( 6,602) 13. 19 <20 = 12 ,940 Frame-East End 40 129,285 (1 .96) (11,850) 17. 96 = 23 ,226 No. of Suites 23 164,511 (2 .08) (14,700) 18. 59 = 30 ,576 No. of Suites 17 81,613 (2 .13) ( 4,827) 12. 60 <15 = 10 ,282) 96 TABLE VI CONFIDENCE INTERVALS - STRATIFICATION BY NUMBER OF SUITES (SMALL SAMPLES) (1) Description (2) n (3) Y (where A X=X) ($) ±t ] (4) n_ 2Sy.x (4) ($) (5) "O (%) (3) High Rise-West End 15 1,076,061 (2 .16) (74,350) 160,596 14 .92 No.of Suites <45 15 429,730 (2 .16) (20,080) 43,373 10 .09 No.of Suites . >, 45 15 1,490,622 (2 .16)(80,580) 174,053 11 .68 Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o 15 146,438 (2 .16)(17,840) 38,534 26 .31 No.of Suites = 8-11 15 96,128 (2 .16)(11,460) 24,754 25 .73 No.of Suites = 18-24 15 195,725 (2 .16)(16,090) 34,754 17 .76 No.of Suites = 25-35 15 341,852 (2 .16) (21,080) 45,533 13 .32 97 of s t r a t i f i c a t i o n . I t was suggested previously that size c l a s s i f i c a t i o n s would i n t u i t i v e l y seem a reasonable method of s t r a t i f y i n g data af t e r i t was c l a s s i f i e d by basic struc-t u r a l type and l o c a t i o n . This suggestion i s supported by the results i n Tables IV to VI. Thus, i t would be worthwhile to give some consideration to the number of suites i n selec-t i n g comparables. Another variable which i s considered i n the analysis i s gross income per s u i t e . Obviously, t h i s variable can be r e a d i l y calculated from the available data. A measure of apartment q u a l i t y was desired i n order to attempt to s t r a t i f y the data into even more comparable groups. Admittedly, income per suite i s somewhat of a make-shift qu a l i t y variable. However, i t was a l l that was available and thus was used i n several experiments with the data. Table VII shows the same f i r s t equations as Table V, except the breakdown i s by income per suite rather than number of su i t e s . Once again achievements are made i n a l l but two cases despite the fact that considerably smaller sample sizes r e s u l t from the s t r a t i f i c a t i o n , necessitating the use of the higher t-values. Thus, although the r e l a t i v e improvements are diminished somewhat by the smaller sample s i z e , they are s t i l l improvements. A s i m i l a r check to the one done on the number of suites analysis which involved comparisons of samples of the same 98 TABLE VII CONFIDENCE INTERVALS - STRATIFICATION BY INCOME PER SUITE (1) Description (2) n '_ (3) Y„(where A X=X ) ($) +t (4) n = 2 S y x (4) ($) (5) o. (%) (3) Concrete High-Rise - A l l 52 879,099 (1. 96) ( 173, 88,420) 303 19 .71 Income per Suite> $1600 23 1,207,187 (2. 08) ( 176, 84,920) 634 14 .73 Income per Suite <$1600 29 619,060 (2. 05) ( 77, 37,910) 716 12 .55 Frame-West End 47 292,881 (1. 96) ( 50, 25,510) 000 17 .07 Income per Suite > $1250 24 349,337 (2. 07) ( 47, 22,850) 300 13 .54 Income per Suite < $1250 23 233,973 (2. 08) ( 59, 28,680) 654 25 .50 Frame-Burnaby 59 221,871 (1. 96) ( 34, 17,750) 790 15 .68 Income per Suite > $1400 32 319,226 (2. 04) ( 44, 21,570) 003 14 .10 Income per Suite <$1400 27 106,438 (2. 06) ( 16, 7,804) 076 15 .10 Frame-East End 40 129,285 (1. 96) 1 23, 1,850) 226 17 .96 Income per Suite \ $1300 19 132,953 (2. 11) 1 21, 0,190) 501 16 .17 Income per Suite < $1300 21 126,001 (2. 09) 23, [11,340) 701 18 .81 99 size was done for the income per suite variable and appears i n Table VIII. Once again, there was an improvement i n each of the cases when the sample size variable was eliminated. That i s , by s t r a t i f y i n g by income per suite, which i s con-sidered a q u a l i t y i n d i c a t o r , the confidence i n t e r v a l s were narrowed r e l a t i v e l y . Thus, i t would seem that there would be some value i n considering t h i s variable as well i n the s e l e c t i o n of comparables. I t was mentioned e a r l i e r that comparables of s i m i l a r dates of sale should be used. This i s c e r t a i n l y reasonable as i t i s well known that trends e x i s t i n the r e a l estate market. As well, predictions based on regression equations must assume, by the very nature of the t o o l , that conditions are the same today as for the data used to calculate the equation. Thus, i t would be f o o l i s h indeed to attempt use properties with diverse dates of sale as comparables. None of the data dates back l a t e r than 1964 and, i n f a c t , the majority of the transactions occurred i n the past two years. Thus, i t would seem r e l a t i v e l y safe to select comparables from the available data without anymore consider-ation of the date of sale. Had more h i s t o r i c data been available perhaps some in t e r e s t i n g analysis of t h i s variable could have been made as an aside. 100 TABLE VIII CONFIDENCE INTERVALS - STRATIFICATION BY INCOME PER SUITE (SMALL SAMPLES) (1) (2) _ (3) (4) (5) Description n Y v (where t t .Syx (4) % (3) X X=X••') n " 2 ($) ($) (%) High-Rise-West 15 1,076,061 (2.16) (74,350) 14.92 End = 160,596 Income per 15 467,678 (2.16)(20,070) 9.27 Suite < $1400 . = 43,351 Income per 15 1,440,191 (2.16) (96,950) 14.54 Suite >/ $1400 = 209,142 Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o 15 146,438 (2.16) (17,840) 26.31 = 38,534 Income per Suite = 15 $1100-1299 Income per Suite = 15 $1300-1449 96,218 (2.16)(11,460) = 24,754 25.73 195,725 (2.16)(16,090) 17.76 = 34,754 Income per Suite -$1450-1600 15 341,852 (2.16)(21,080) 13.32 = 45,533 101 As i t was, one limited analysis was made of the e f f e c t of the date of sale variable on confidence i n t e r v a l s . Within two groupings i t was possible to further s t r a t i f y the data into sub-groups by the date of sale v a r i a b l e . Once again the e f f e c t of sample size obscures the results s l i g h t l y . However, as can be seen from Table IX no substantial improve-ments occurred. The results probably support the e a r l i e r suggestion that, for the range of data available, the date of sale variable i s not so diverse as to cause any undue concern. Recapitulating somewhat, i t seems that several variables are relevant for purposes of selecting comparables which w i l l y i e l d regression equations with more precise confidence i n t e r -v a l s . S t r a t i f i c a t i o n by basic s t r u c t u r a l type, location, number of suites and income per suite a l l proved worthwhile. A question that remains to be s a t i s f i e d i s that of sample s i z e . I t i s evident that for small samples, the larger the sample the smaller the value of t and, therefore, the narrower the i n t e r v a l s . As the value of t approaches a l i m i t around a sample size of 32, the e f f e c t of sample size w i l l not be important for large samples. I t has already been stated that the magnitude of the standard error of estimate i s inherent i n the p a r t i c u l a r problem and does not depend d i r e c t l y upon the sample s i z e . 102 TABLE IX CONFIDENCE INTERVALS - STRATIFICATION BY DATE OF SALE (1) Description (2) ... n (3) Y (where A X=X) (?) ± t n - 2 S y - X ( 4 ) ($) (5) Q. (3) Concrete High Rise - A l l 52 879,099 (1.96)(88,420) = 173,303 19 .71 Date of Sale >, 67 30 774,808 (2.05) (61,130) = 125,316 16 • 17 Date of Sale <67 22 1,021,018 (2.09) (118,200) = 247,038 24 .20 Frame-West End Date of Sale >/67 47 32 292,881 214,987 (1.96) (25,510) = 50,000 (2J34) (17,560) = 35,822 17 16 .07 .66 Date of Sale < 67 15 458,954 (2.16) (38,250) = 82,620 18 .00 103 Table X presents comparison of large and small samples within the same groupings. Equations and confidence i n t e r v a l s were calculated for a l l of the properties i n each grouping and then for 15 randomly selected properties i n each case. In a l l but one comparison the width of the confidence i n t e r v a l i s r e l a t i v e l y narrower, largely due to the f a c t that the standard normal deviate i s smaller than the t-value. In the one excep-t i o n , the random sel e c t i o n must have resulted by chance i n more close l y comparable properties than i n the whole. In Table XI a series of confidence i n t e r v a l s based on d i f f e r e n t sample sizes within the same grouping i s shown. From sample sizes 5 to 25 the percentages decline, largely as the r e s u l t of the t-values also declining with increasing sample s i z e . The increase i n the percentage at sample size 30 i s due to the s e l e c t i o n , though random, of several properties which are somewhat diverse from the norm, thus causing a larger standard error of estimate. The percentages of the samples above 30 are quite constant as the standard normal deviate i s constant. The s l i g h t differences are due to the small v a r i a -t i o n i n the standard error of estimate from sample to sample. By studying confidence i n t e r v a l s , then, i t i s possible to see how improvements can be made; that i s , how the inte r v a l s can be narrowed, by consideration of basic s t r u c t u r a l type, location, number of suites and income per s u i t e . Despite this TABLE X CONFIDENCE INTERVALS - COMPARISON OF LARGE AND SMALL SAMPLES (1) (2) Description n '_ (3) Y (where X x = x ) (?) - t n _ 2 s y . V x (4) ($) (5) % (3) (%) Frame-With Elevator 42 353,195 (1.96) (19,880) = 38,965 11.03 Frame-With Elevator 15 346,576 (2.16)(18,910) = 40,846 11.78 Frame-Walkups(20+) (1958-67) 42 202,360 (1.96) (16,590) = 32,516 16.07 Frame-Walkups (20+) (1958-67) 15 213,802 (2.16) (20,810) = 44,950 21.02 Frame-Walkups(20~ (1958-67) ") 64 112,938 (1. 96) ( 8,831) 17,309 15 .33 Frame-Walkups (20") (1958-67) 15 123,331 (2. 16) (11,250) 24,300 19 .70 Frame-Walkups ( a l l sizes) (1940-59) 108 119,316 (1. 96)(12,240) 23,990 20 .11 Frame-Walkups ( a l l sizes) (1940-59) 15 110,886 (2. 16) ( 8,606) 18,589 16 .76 Frame-West End 47 292,881 (1. 96) (25,510) 50,000 17 .07 Frame-West End 15 261,560 (2. 16) (33,630) 72,641 27 .77 105 TABLE X (Continued) (1) Description.. (2) n : (3) Y (where x X=X) ($) + (4) t n _ 2 S y x (4) ($) (5) "O (%) (3) Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o 118 179,791 (1 .96)(15,840) 31,046 17 .27 Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o 15 146,438 (2 .16)(17,840) 38,534 26 .31 Frame-Marpole 52 152,014 (1 .96)(10,230) 20,051 13 .19 Frame-Marpole 15 180,059 (2 .16) (14,730) 31,817 17 .67 Frame-East End 40 129,285 (1 .96)(11,850) 23,226 17 .96 Frame-East End 15 150,206 (2 .16)(15,240) 32,918 21 .92 Frame-Burnaby 59 221,871 (1 .96)(17,750) 34,790 15 .68 Frame-Burnaby 15 244,994 (2 .16)(26,270) 23 .16 = 56,743 106 TABLE XI CONFIDENCE INTERVALS - COMPARISON OF VARYING SAMPLE SIZES ( 1 ) Description.. (2) n (3) " Y (where A X=X) ($) (4) =t -Syx n-2 2 ($) (5) (4) % (%) (3) Frame-South G r a n v i l l e , Fairview K i t s i l a n o 5 217,099 (3.18) (20,030) = 63,695 29. 34 10 231,649 (2.31) (14,610) = 33,749 14. 57 15 200,883 (2.16)(12,950) = 27,972 13. 92 20 211,163 (2.10) (11,830) = 24,843 11. 76 25 192,770 (2.07)(11,190) = 23,163 12. 02 30 211,765 (2.05) (19,640) = 40,262 19, 01 35 205,869 (1.96) (15,980) = 31,320 15. 21 40 215,635 (1.96)(15,280) = 29,948 13. 89 45 207,543 (1.96)(16,310) = 31,967 1.5. 40 50 205,778 (1.96)(15,270) = 29,929 14. 54 55 204,535 (1.96) (15,580) = 30,536 14. 93 60 203,506 (1.96)(15,850) = 31,066 15. 26 107 conclusion i t i s s t i l l somewhat disturbing that some of the i n t e r v a l s remain r e l a t i v e l y wide. The ultimate width rests on the inherent v a r i a t i o n i n the grouping as measured by the standard error of estimate, the sample size when i t i s small, and the l e v e l of confidence. As regards the former, no more can be done than try to s t r a t i f y i n such a way that cl o s e l y comparable properties are chosen and> hopefully, the v a r i a t i o n w i l l be minimized. Sample sizes over 32 would be encouraged as there would then be no need for the higher t-values, but t h i s immediately poses a problem when the data are li m i t e d i n number. When this i s the case, detailed s t r a t i f i c a t i o n often results i n small sample s i z e s . A dilemma i s evident. The t h i r d factor, the l e v e l of confidence used, can of course be varied. Because the standard normal deviate, and the t-values, are lower for lower levels of confidence, the confidence i n t e r v a l s can, of course, be narrowed by using a l e v e l of confidence less than 95 per cent. Table XII i l l u s t r a t e s the difference between the 95 per-cent and the 6 8.26 per cent levels of confidence. At the former l e v e l the standard normal deviate i s 1.96, the value which has been used, while at the 68.26 per cent l e v e l i t i s 1.00; that i s , for large samples, the confidence i n t e r v a l extends above and below Y x by the value of the standard devi-ation at t h i s lower l e v e l of confidence. From Table XII i t TABLE XII CONFIDENCE INTERVALS - COMPARISON OF CONFIDENCE LEVELS (1) Description (2) n (3) (X ($) Y x =X) 95% Level . 66.26 Level (4) (5) (6) (7) =1.96 S y x (4) % (3) =1.00 S y x (6) % (3) ($) (%) ($) (%) High R i s e - A l l 52 879,099 (1.96)(88,420) 19.71 (1.00)(88,420) 10.05 = 173,303 = 88,420 Frame-West End 47 292,881 (1.96) (25,510) 17.07 (1.00) (25 ,510) 8.79 = 50,000 = 25,510 Frame-Burnaby 59 221,871 (1.96)(17,750) 15.68 (1.00)(17,750) 7.99 = 34,790 = 17,750 Frame-East End 40 129,285 (1.96) (11,850) 17.96 (1.00) (11,850) 9.16 = 23,326 = 11,850 o 00 109 can be seen that the width of the i n t e r v a l i s almost halved by using the 66.26 per cent instead of the 95 per cent l e v e l of confidence. While the confidence i n t e r v a l i s narrowed considerably, the p r o b a b i l i t y q u a l i f i c a t i o n i s reduced. That i s , the appraiser can express that i n only.two out of three times w i l l an estimate f a l l within the i n t e r v a l , while previously 19 out of 20 estimates f e l l within the larger i n t e r v a l . A l e v e l of confidence must be decided upon by the i n d i v i d u a l appraiser for his p r a c t i c a l use. Certainly, the 66.26 per cent l e v e l would not be p a r t i c u l a r l y accurate. While the appraiser might f e e l that 95 per cent i s unnecessarily high, i t would seem that the confidence l e v e l could not be too f a r below before the p r o b a b i l i t y q u a l i f i c a t i o n becomes of l i t t l e p r a c t i c a l ; v a l u e . Thus, changes i n confidence levels to narrow confidence i n t e r v a l s may be of only limited use. Having, thoroughly analyzed the confidence i n t e r v a l question, i t i s useful to proceed to another step i n analysis, that of comparing predicted with actual values. Hopefully, the relevant s t r a t i f i c a t i o n s that were determined i n this sec-t i o n w i l l be confirmed and further indications of accuracy derived. 110 I I I . PREDICTED VERSUS ACTUAL VALUES Having calculated various confidence i n t e r v a l s has pro-vided some in t e r e s t i n g r e s u l t s . Another useful step i n the s t a t i s t i c a l analysis i s to compare predicted values with actual values. That i s , having calculated regression equations i t i s useful to "test" them by substituting the gross incomes of other properties which are considered comparable into an equa-t i o n and c a l c u l a t i n g the predicted c a p i t a l value figure. The predi c t i o n may then be compared to the actual values and the deviations measured. This i s a very p r a c t i c a l t e s t as i t i s analogous to an actual appraisal s i t u a t i o n were an appraiser to use regression equations. That i s , the question i s being asked: had the appraiser used a c e r t a i n regression equation, how close would he have come i n his prediction to what the property actually sold for? This step can be taken as an additional one to the c a l -c u lation of confidence i n t e r v a l s . Once again the same goals are being sought: how accurate are the estimates for cer t a i n groupings and how can they be improved? Of course, i t i s r e a l i z e d that i n the case of t h i s step, a single estimate of value i s being used to compare to the actual s e l l i n g price which, of course, i s a single value. This may be somewhat repugnant to modern appraisal theory which stresses that properties do not have single inherent values and I l l appraised values are best expressed as a range with a pr o b a b i l i t y q u a l i f i c a t i o n . Of course, many appraisers continue to express t h e i r valuations i n terms' of a single value. In any case, the test exercise i s worthwhile as another step of s t a t i s t i c a l analysis. I t was mentioned b r i e f l y that some of the regression equ-ations were tested by substituting gross incomes of properties into them, c a l c u l a t i n g the estimated c a p i t a l value and comparing i t with the actual s e l l i n g p r i c e . In each case ten properties were used to tes t the equation. The difference between the e s t i -mated and actual values was then expressed as a percentage of the actual value. Although the differences were both p o s i t i v e and negative, signs were ignored when the ten percentage d i f f e r -ences were averaged. What was of main i n t e r e s t was how far the estimate was from the actual value, not necessarily whether i t was high or low. Thus, the average percentage difference i s an average of percentages, not the average absolute difference expressed as a percentage of the mean value. Had the l a t t e r method been used, larger properties would have had a weighted e f f e c t . Averaging the percentages avoids such an e f f e c t . Table XIII i s the f i r s t of several i n thi s section and time should be taken to explain i t thoroughly. The average percentage difference, the c a l c u l a t i o n of which was just ex-plained, appears i n column three. Because i t i s useful to have some idea as to the d o l l a r figures i n each case, two 112 TABLE XIII PREDICTED VERSUS ACTUAL - STRATIFICATION BY LOCATION (1) (2) (3) (4) " (5) Description n Ave. % Actual Mean (3)x(4) Difference of Y Average % (%) ($) Difference i n Dollars ($) Frame-West End 47 3.88 264,798 10,274 Frame-West End 15 5.16 264,798 13,663 Frame-South G r a n v i l l e , Fairview, 118 6.87 136,550 9,381 K i t s i l a n o Frame-South G r a n v i l l e , Fairview, 15 7.56 136 ,550 1(3 y 32-31 K i t s i l a n o Frame-Marpole 52 6.26 143,050 8,955 Frame-Marpole 15 6.39 143,050 9,141 Frame-East End 40 3.08 145,900 4,494 Frame-East End 15 3.64 145,900 5,311 Frame-Burnaby 59 4.88 191,050 9,323 Frame-Burnaby 15 3.68 191,050 7,031 High-Rise-West End 36 6.48 640,319 41,493 High Rise-West End 15 7.23 640,319 46,295 113 addition columns appear. Column four shows the mean of the actual s e l l i n g prices of the properties used to t e s t the equation. And column f i v e i s simply the average percentage difference expressed i n terms of d o l l a r s , which i s the mean of the actual s e l l i n g prices m u l t i p l i e d by the average per-centage difference. Column f i v e , i t should be noted, i s not the actual average d o l l a r difference between the estimated and actual c a p i t a l values. I t i s included only to give some absolute i n d i c a t i o n of the percentages i n each case. Table XIII shows the results of tests on several equa-tions s t r a t i f i e d by l o c a t i o n . For instance, a regression equation for "15 randomly selected frame properties i n the West End was calculated. Then ten other properties were ran-domly selected to t e s t the equation. Thus, none of the test properties were used i n the determination of the equation. The average percentage difference was calculated as above. An additional step was taken for each grouping; that i s , the equation for a l l of the properties i n each case was also tested with the same test properties. For instance, a l l of the 47 frame properties i n the West End were used to c a l -culate a regression equation which then was tested with the same properties used to t e s t the corresponding equation based on 15 properties from the same group. In this case, of course, the test properties are included i n the group used to calculate the regression equation. However, i t could probably 114 be assumed not at a l l u n l i k e l y that the ten test properties could just as well have been ten completely new properties. Such an assumption i s probably more supportable as the sample size becomes quite large. In any case the comparison of the two tests i s i n t e r e s t i n g . Several things are to be noted from Table XIII. The f i r s t relevant thing i s the size of the average percentage differences. Considering that the equations are s t r a t i f i e d only by basic s t r u c t u r a l type and location and that the equations are based s o l e l y on the one independent variable, gross income, the average percentage deviations are remarkably close. The second thing to notice i s that i n a l l but one case the equation for a l l of the properties i n a given class y i e l d s a lower average percentage difference than the equation for the 15 properties. That t h i s i s true may simply be due to the f a c t mentioned e a r l i e r , that the t e s t properties are included i n the regression equation c a l c u l a t i o n . However, i t might also be suggested that the equations for larger sample sizes are more representative, and hence, more accurate. This could be true simply because the e f f e c t of any extraordinary transactions w i l l be lessened as the sample size increases. The one excep-t i o n where the smaller sample si z e equation y i e l d s a better r e s u l t may, perhaps, be explained by suggesting that the 15 115 randomly selected properties may be, by chance, more comparable to the test properties than are a l l of the properties. Given that these results are remarkably good, consider-ing the basic s t r a t i f i c a t i o n , could they possibly be improved by more detailed s t r a t i f i c a t i o n ? Table XIV includes the re-sults of three groupings that are s t r a t i f i e d roughly by number of suites but not by locat i o n . I t i s noted that the equations based on groupings with number of suites less than or greater than 20 y i e l d better results than does the equation based on properties of a l l s i z e s . Although the equations are not s t r i c t l y comparable as they are based on properties b u i l t at d i f f e r e n t times, the suggestion that number of suites i s a s i g n i f i c a n t variable should c e r t a i n l y be investigated further. Before doing so, however, the remaining two equations i n Table XIV should be mentioned. They represent some very accurate r e s u l t s , based on only rough s t r a t i f i c a t i o n . This lends further encouragement to the idea that gross income can be used to accurately predict c a p i t a l values. By taking the grouping which had the highest average percentage difference and analyzing i t i n more d e t a i l , the e f f e c t of more detailed s t r a t i f i c a t i o n can be i l l u s t r a t e d . Table XV presents the results of consideration of two variables suggested as relevant i n the previous section, number of suites and income per s u i t e . F i r s t , the properties were s p l i t into 116 TABLE XIV PREDICTED VERSUS ACTUAL - STRATIFICATION BY SIZE AND TYPE (1) (2) (3) (4) Description n Average % Actual Mean (5) = (3)x(4) Difference of Y Average % Difference i n (%) ($) Dollars Frame Walkups'-' (20+) 42 7 .52 170,019 12,785 (1958-67) Frame Walkups (1958-67) 15 8 .69 170,019 14,775 Frame Walkups (20") 64 6 .92 114,500 7,923 (1958-67) Frame Walkups (20") 15 7 .33 114,500 8,393 (1958-67) Frame Walkups ( a l l sizes) 108 9 .49 99,950 9,485 (1940-59) Frame Walkups ( a l l sizes) 15 9 .63 99,950 9,625 (1940-59) Frame with 42 3 .12 341,435 10,653 Elevator Frame with Elevator 15 3 .16 341,435 10,789 High-Rise-All 52 3 .48 579,569 20,170 High-Rise-All 15 3 .50 593,796 20,783 TABLE XV PREDICTED VERSUS ACTUAL - STRATIFICATION BY NUMBER OF SUITES AND INCOME PER SUITE (1) (2) (3) (4) (5) = (3)x(4) Description • n Average % Actual Mean Average % Difference of Y Difference (%) - ($) i n Dollars Frame-South > G r a n v i l l e , Fairview, 15 8.02 88,640 7,109 K i t s i l a n o Frame-South G r a n v i l l e , Fairview, K i t s i l a n o 15 6.45 88,640 5,716 No.of Suites = 8-11 Frame-South G r a n v i l l e , Fairview, 15 7.88 192,285 15,152 K i t s i l a n o Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o 15 6.49 192,285 12,479 No.of Suites = 18-24 Frame-South Gr a n v i l l e , .Fairview, 15 5.63 94,080 5,297 K i t s i l a n o Frame-South G r a n v i l l e , Fairview, K i t s i l a n o , Income per 15 10.13 94,080 9,530 Suite = $1100-1299 118 TABLE XV (Continued) (1) (2) (3) (4) (5) = (3)x(4) Description n Average % Actual Mean Average % Difference of Y Difference (%) ($) i n Dollars Frame-South Gr a n v i l l e , Fairview, 15 5.55 180,765 10,033 K i t s i l a n o Frame-South G r a n v i l l e , Fairview K i t s i l a n o Income, per 15 5.18 180,765 9,364 Suite = $1300-1449 Frame-South G r a n v i l l e , Fairview, 15 5.06 267,475 13,534 K i t s i l a n o Frame-South G r a n v i l l e , Fairview, K i t s i l a n o , Income per 15 4.17 267,475 11,154 Suite = $1450-1600 119 two large groups by number of suites. F i f t e e n properties were selected from each group and two regression equations calculated. Then, from the remainder i n each group, ten properties were selected and tested. The average percentage differences were below those for the previous test of the equation without number of suites s t r a t i f i c a t i o n . As well, the equation based on 15 randomly selected properties from the group as a whole, that i s , without con-sideration of number of suites, was tested with the test properties as selected above. The r e s u l t s , as shown i n Table XV, were that the o v e r a l l equation did not provide as accurate estimates as the equation based on properties that were s t r a t i -f i e d i n the same manner as the te s t properties. Thus, i t could be concluded that consideration of the number of suites would improve the accuracy of the estimates. Income perssuite, the variable used as a qu a l i t y i n d i -cator, was analyzed i n the same manneraas the number of suites va r i a b l e . The results are presented i n Table XV as we l l . In two out of three cases the equation based on properties that were s t r a t i f i e d i n the same manner as the test properties yielded better results than an o v e r a l l equation tested with the same properties, as well as better results than the o v e r a l l equation tested with properties not s t r a t i f i e d by income per sui t e . I t would seem, then, that consideration of income per suite also results i n more accurate estimates. 120 The l a s t question i n th i s section i s that of sample s i z e . The same series of equations with sample sizes f i v e to s i x t y which was introduced i n the previous section was used to tes t differences i n accuracy of estimates r e s u l t i n g from differences i n sample s i z e s . Once again ten properties that were not used to calculate the regression equations were randomly selected for testing purposes. The results are shown i n Table XVI. As can be seen i t i s d i f f i c u l t to i d e n t i f y even the s l i g h t e s t of trends. Thus, l i t t l e can be said about the s i g -nificance of increasing sample siz e under the test conditions. For the estimates made i t would not seem to matter a great deal about how large a sample was used to calculate the regres-sion equation as long as the r e s u l t i n g equation i s s i g n i f i c a n t . The results of th i s section are g r a t i f y i n g . They sup-port the conclusions reached i n the previous section. Just as s t r a t i f i c a t i o n by basic s t r u c t u r a l type, location, number of suites and income per suite was shown to narrow confidence i n t e r v a l s , the same s t r a t i f i c a t i o n was shown to improve the accuracy of single estimates. At the end to the l a s t section i t was mentioned that, i n many cases, confidence i n t e r v a l s were s t i l l quite wide. I t i s g r a t i f y i n g , therefore, to note the remarkable accuracy of the estimates i n th i s section. 121 TABLE XVI PREDICTED VERSUS ACTUAL - COMPARISON OF VARYING SAMPLE SIZES (1) Description (2) (3) n Average % Difference (%) (4) Actual Mean of Y ($) (5) = (3)x(4) Average % Difference i n Dollars Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o 5 6.62 178,625 11,825 10 6.50 11 11,6.11 15 6.90 I I 12,325 20 6.58 I I 11,753 25 6.50 I I 11,611 30 6.44 I I 11,506 35 6.44 I I 11,506 40 6.46 II 11,539 45 6.49 I I 11,593 50 6.51 I I 11,628 55 6.58 I I 11,753 60 6.51 I I 11,628 118 6.49 II 11,593 122 IV. SIMPLE REGRESSION VERSUS AVERAGE MULTIPLIER An experiment which seemed worthwhile was to calculate estimated values by using the average m u l t i p l i e r and comparing them to the actual values as was done i n the previous section using the regression model. The results could then be com-pared. The somewhat surprising r e s u l t was that the average m u l t i p l i e r often yielded closer predictions than did the more sophisticated regression model. In the previous chapter some differences between the c o e f f i c i e n t s of the regression and average m u l t i p l i e r equations were pointed out. Obviously, the constant term (a) i s not present i n the average m u l t i p l i e r equation. However, the average m u l t i p l i e r equation cannot be considered the same as the least-squares regression l i n e through the o r i g i n ( i . e . the constant term (a) equals zero) because the values of b are calculated d i f f e r e n t l y . In the previous chapter the least squares l i n e was said to give the l i n e of best f i t under the assumptions of that method: a normal d i s t r i b u t i o n of the observations around the l i n e and the reduction of the squared differences. Yet, as can be seen i n Tables XVII and XVIII, the average m u l t i p l i e r (actually the average i s based on n-1 m u l t i p l i e r s , one being randomly dropped, so that the m u l t i p l i e r and regression equa-tions both have n-2 degrees of freedom) results i n a lower 123 TABLE XVII SIMPLE REGRESSION VERSUS AVERAGE MULTIPLIER -BASIC STRATIFICATION Description n Average Per Cent Difference Simple Regression Average M u l t i p l i e r Frame with Elevator 42 3.12 4.77 Frame with Elevator 15 3.16 5.02 Frame Walkups(20 +) (1958-67) 42 7.52 6.20 Frame Walkups(20+) (1958-67) 15 8.69 6.60 Frame Walkups (20-) (1958-67) 64 6.92 7.09 Frame Walkups (20•") (1958-67) 15 7.33 7.05 Frame Walkups ( a l l sizes) (1940-59) 108 9.49 9.00 Frame Walkups ( a l l sizes) (1940-59) 15 9.63 9.42 High-Rise - A l l 52 3.48 4.63 High-Rise - A l l 15 3.50 3.99 Frame-West End 47 Frame-West End 15 3.88 5.16 4.27 4.88 124 TABLE XVII (Continued) , ; Average Per Cent Difference Description r. n Simple Regression Average M u l t i p l i e r Frame-South Granville 2.I8 6 87 6 79 Fairview, K i t s i l a n o Frame-South G r a n v i l l e , Fairview, K i t s i l a n o . 15 7.56 7.56 Frame-Marpole 52 6.26: 6.36 Frame-Marpole 15 6.39 6.02 Frame-East End 40 3.08 3.22 Frame-East End 15 3.6 4 3.22 Frame-Burnaby 59 4.88 3:68 Frame-Burnaby 15 3.68 3.63 High Rise-West End 36 6.48 6.80 High Rise-West End 15 7.23 6.16 125 TABLE XVIII SIMPLE REGRESSION VERSUS AVERAGE MULTIPLIER-DETAILED STRATIFICATION- . Description Average Per Cent Deviation n Simple Regression Average M u l t i p l i e r Frame-South G r a n v i l l e , Fairview, K i t s i l a n o 15 8.02 7.90 Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o , No.of Suites=8-ll 15 6.45 7.04 Frame-South. Gr a n v i l l e , Fairview, K i t s i l a n o 15 7.88 8.00 Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o , No.of Suites=18-24 15 6.49 6.39 Frame-South G r a n v i l l e , Fairview, K i t s i l a n o 15 5.63 5.48 Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o , Income per Suite = $1100-1299 15 10.13 9.21 Frame-South G r a n v i l l e , Fairview, K i t s i l a n o 15 5.55 5.71 Frame,South Gr a n v i l l e , Fairview, K i t s i l a n o Income per Suite = $1300-1449 15 5.18 5.67 126 TABLE XVIII (Continued) Average Per Cent Deviation Description n Simple Regression Average M u l t i p l i e r Frame-South G r a n v i l l e , Fairview, K i t s i l a n o 15 5.06 5.26 Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o Income per Suite = $1450-1600 15 4.17 4.13 average percentage difference i n more cases than does the a p p l i -cation of the regression model. Why i s t h i s so? I t i s possible, of course, that just by chance the test data are closer to the average m u l t i p l i e r l i n e than to the regression model l i n e . However, th i s seems an inadequate explanation considering the frequency of the super-i o r i t y of the average m u l t i p l i e r . As w e l l , i t i s true that the least-squares l i n e picks up other v a r i a t i o n , besides that ex-plained by gross income, i n c a p i t a l value, while the average multiplier, considers only the r a t i o of c a p i t a l value over gross income. I t i s true that the c o e f f i c i e n t s of determination are very high, i n d i c a t i n g a large percentage of v a r i a t i o n i s ex-plained, and therefore, i t might be thought that the l a t t e r 127 explanation i s also inadequate. I t should be remembered, however, that the results of the two methods are, i n most cases, only s l i g h t l y d i f f e r e n t . I t may be too that by only considering r a t i o s , and not the absolute d o l l a r v a r i a t i o n s , the e f f e c t of the extreme var-ia t i o n s i s not the same i n the average m u l t i p l i e r method as i n the regression model. In any case, the results would seem to support the theory that because market participants ac-t u a l l y use m u l t i p l i e r s they become b u i l t into the market. What i s r e a l l y the concern i s the best prediction, not the method used. To the extent that the average m u l t i p l i e r r e s u l t s i n the best prediction only confirms that t h i s i s the way market participants actually predict values. Table XIX presents the lowest average percentage d i f -ference for each grouping of 15 randomly selected properties. Where the regression model resulted i n the lowest difference, the figures i n Table "XIX'are the same as i n Tables XIII, XIV and XV. Where the average m u l t i p l i e r resulted i n the lowest difference the appropriate changes have been made. It can be seen that i n several cases, s t r a t i f i c a t i o n by basic s t r u c t u r a l type and location r e s u l t i n very close estimates. And for those that were not quite so close, the additional experimentation would seem to indicate that further s t r a t i f i c a t i o n by number of suites and income per suite im-proved the estimates. TABLE XIX 128 BEST RESULTS-SMALL SAMPLES (1) (2) (3) (4) (5) = (3)x(4) Description Best Aver. % Actual Mean Average % Model Differences of Y Difference (%) ($') i n Dollars Frame Walkups(20+) (1958-67) Mult. 6 .60 170 ,019 11 ,221 Frame Walkups(20~) (1958-67) Frame Walkups(all sizes) (1940-59) Mult. Mult. 7 9 .05 .42 114 99 ,500 ,950 8 9 ,072 ,415 Frame with Elevator Reg. 3 .16 341 ,435 10 ,789 High-Rise - A l l Reg. 3 .50 579 ,569 20 ,783 Frame-West End Mult. 4 .88 264 ,798 Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o Reg.or Mult. 7 .56 136 ,550 10 ,062 Frame-Marpole Mult. 6 .02 143 ,050 8 ,612 Frame-East End Mult. 3 .22 145 ,900 4 ,698 Frame-Burnaby Mult. 3 .63 191 ,050 6 ,935 High-Rise-West End Mult. 6 .16 640 ,319 39 ,444 Frame-South G r a n v i l l e , Fairview, K i t s i l a n o No.of Suites = 8-11 Reg. 6 .45 88 ,640 5 ,716 No.of Suites = 18-24 Mult. 6 .39 192 ,923 12 ,328 129 TABLE XIX (Continued) (1) - (2) (3) (4) (5) = (3)x(4) Description Best Average % Actual Mean Average % Model Difference of Y Difference (%) ($) i n Dollars. Income per Suite = ; $1100-1299 Mult. 9 .21 94,080 8,665 Income per Suite = $1300-1449 Reg. 5.18 180,765 9,364 Income per Suite = $1450-1600 Mult.. 4.13 267,475 11,047 Table XX i s a s i m i l a r table which shows the test re-sults of the equations based on a l l of the properties, not just 15, i n each grouping. I f the assumption that the ten properties which were included i n the equation c a l c u l a t i o n were also used to test the equation can be ignored on the basis that these ten random properties could be assumed the same as ten other properties outside the equation c a l c u l a t i o n , then i t can be seen that estimates can be improved further by larger sample s i z e s . Just how good are these estimates? Are the estimates close enough to actual values to be acceptable? To answer these questions some sort of acceptable error factor must be 130 TABLE XX BEST RESULTS - LARGE SAMPLES (1) Description (2) (3) (4)' (5) = (3)x(4) Best Average % Actual Mean Average % Model Difference of Y Difference (%) ($) i n Dollars Frame Walkups(20+) (1958-67) Mult. Frame Walkups(20 -) (1958-67) Reg. Frame Walkups(all sizes) (1940-59) Mult. Frame with Elevator Reg. High Rise - A l l Reg. Frame-West End Reg. Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o : Mult. Frame-Marpole Mult. Frame-East End Reg. Frame-Burnaby Mult. High Rise-West End Reg. 6.20 6.92 9.00 3.12 3.48 3.88 6.79 6.26 3.08 3,6 8 6.48 170,019 114,500 99,950 341,435 579,569 264,798 136,550 143,050 145,900 191,050 640,319 10,541 7,923 8,996 10,653 20,170 10,274 9,272 8,955 4,494 7,031 41,493 131 determined. Unfortunately, no empirical work comparing approaches and actual s e l l i n g prices i s available. As well, there i s very l i t t l e i n appraisal l i t e r a t u r e on the subject of error, possibly because t r a d i t i o n a l appraisal theory does not wish to recognize the p o s s i b i l i t y of error. One b r i e f mention was discovered i n the survey of the l i t e r a t u r e done for the e a r l i e r t h e o r e t i c a l sections. Carney states that the range of error i n the appraisal process nor-mally w i l l validate a l a t i t u d e of 10 per cent i n the f i n a l conclusion of value."'' The results shown are c e r t a i n l y well within such l i m i t s . I t would seem then that accurate predictions of value are possible although only gross income, with some s t r a t i f i -cation by property c h a r a c t e r i s t i c s , i s considered. The re s u l t s have a great deal of s i g n i f i c a n c e . They support the preliminary conclusion reached i n the t h e o r e t i c a l section of t h i s paper that gross income m u l t i p l i e r s should not be so quickly dismissed as has been the case i n t r a d i t i o n a l appraisal l i t e r a t u r e . Indeed, they r e s u l t i n accurate estimates, given adequate data, which would normally take the appraiser many more hours i f he used the t r a d i t i o n a l approaches. Thus, i t J . J . Carney, "The Development and Use of Gross Income M u l t i p l i e r s , " The Appraisal J o u r n a l , ( A p r i l , 1963), pp. 221-227. 132 can be concluded that the gross income m u l t i p l i e r can be a very useful t o o l . V. MULTIPLE REGRESSION AND CORRELATION I t was mentioned i n the previous chapter that multiple regression and c o r r e l a t i o n analysis may be used to attempt to improve estimates by adding additional variables. Thus, rather than attempt to s t r a t i f y i n a very detailed way,: cer-t a i n property c h a r a c t e r i s t i c s can be included i n the actual regression equation. It was pointed out that substantial improvement may not occur i f there i s a close rel a t i o n s h i p between the a d d i t i o n a l variables and the o r i g i n a l variable because the explanation offered i n terms of one variable may be merely duplicated by the other. Additional explanation can only occur i f a variable i s introduced which i s not cl o s e l y correlated'to the o r i g i n a l independent variable but i s to the dependent variable. Considering the above, i n t u i t i v e l y i t did not seem that multiple regression and c o r r e l a t i o n would add much more pre-c i s i o n to the estimates. This opinion was based on two fact s . One was that a very high percentage of the v a r i a t i o n i n c a p i t a l value was being explained by the v a r i a t i o n i n the single independent variable, gross income. This i s e a s i l y seen i n the very high c o e f f i c i e n t s of determination of most of the equations. Thus, there did not seem to be a great deal 133 of room for improvement. The other reason that i t was i n i t i a l l y doubted whether multiple variable analysis would add much explanation was that the number of additional variables which could be used was l i m i t e d i n number as well as related c l o s e l y to gross income. The two variables used previously i n the more de-t a i l e d s t r a t i f i c a t i o n were number of rooms, as a measure of s i z e , and gross income per s u i t e , as a make-shift measure of q u a l i t y . I t i s quite obvious that these two variables are cl o s e l y related to gross income. To check the e f f e c t of multiple regression and corre-l a t i o n two equations, one simple and one multiple, were calculated on the same randomly selected 15 properties. This was done for several groupings and the relevant results are shown i n Table XXI. Two measures of improvement, the d i f f e r -ences i n the standard errors of estimate and the c o e f f i c i e n t s of determination, are used. It can be seen that while some improvement was made, the differences are not great i n most of the cases. This observation i s supported by examining the actual c o e f f i c i e n t s associated with the two additional variables i n the multiple regression equation. In a l l but two cases the c o e f f i c i e n t s are found to be not s i g n i f i c a n t l y d i f f e r e n t from zero at the 95 per cent l e v e l of s i g n i f i c a n c e . This would indicate that the relationships of these additional variables alone and the TABLE XXI COMPARISON OF MULTIPLE AND SIMPLE REGRESSION AND CORRELATION Description Sy Multiple •x ($) Simple Difference Multiple R 2 (%) Simple Difference Frame-With Elevator 19,840 24,160 4,320 .9574 .9349 .0225 Frame-Walkups (20+) (1958-67) 13,970 15,750 1,780 . 8782 .8114 .0668 Frame-Walkups (20 -) (1958-67) 8,803 9,490 687 .9554 .9386 .0168 Frame-Walkups(all sizes) (1940-59) 9,289 9,510 221 .9576 .9567 .0009 Frame-Marpole 9,466 9,700 234 .9948 .9944 .0004 Frame-East End 16,900 17,270 370 .9493 .9373 .0120 Frame-Burnaby 8,470 10,510 2,040 .9975 .9929 .0046 High-Rise - A l l 67,920 68,440 520 .9963 .9954 .0009 High-Rise-West End 93,870 94,930 1,060 .9964 .9956 .0008 Average Improvement 1,248 .0099 135 dependent variable i s not r e l i a b l e . Thus, i t would seem that the i n t u i t i v e opinions that l i t t l e would be added by the addition of number of rooms and income per suite were largely borne out with empirical t e s t -ing. However, as long as l i t t l e extra cost were involved i n gathering the data and making the multiple c a l c u l a t i o n these variables could be included as they do tend to increase the percentage of v a r i a t i o n that i s explained. In the case of an appraiser without the benefit of computer f a c i l i t i e s the more extensive computations involved may not with worth the s l i g h t increase i n explanation. The foregoing i s not to deny that precision might be increased with the consideration of more detailed data than was available for th i s study. This statement i s q u a l i f i e d , however, by the fact that l i t t l e could be added to the equa-tions which explain percentages i n the high nineties, as some of the simple l i n e a r equations do. CHAPTER VI SUMMARY AND CONCLUSIONS This f i n a l chapter includes a restatement of the study, a summary of the findings, conclusions of the study and sug-gestions of related topics that might be pursued i n the interests of further understanding. I. RESTATEMENT The problem which has been considered i n t h i s study was r e a l l y twofold. Both aspects involved a certain amount of questioning of the t r a d i t i o n a l thought i n appraisal l i t e r a t u r e . F i r s t , c e r t a i n t r a d i t i o n a l l y acceptable methods of selecting c a p i t a l i z a t i o n rates for use i n the conventional income approach were questioned. The intent was to determine whether V such methods were both possible and reasonable. That i s , t h e o r e t i c a l consideration was given to whether they were based on consistent and acceptable reasoning and whether i t was possible to implement them i n the manner suggested. Throughout the discussion i t was always necessary to keep i n mind the concept of market determination and whether the methods could determine c a p i t a l i z a t i o n rates from the marketplace i n a r a t i o n a l fashion, without inconsistencies and inaccuracies. Second, d i r e c t conversion, r a t i o s , p a r t i c u l a r l y gross income m u l t i p l i e r s , were considered as appraisal devices. 137 T r a d i t i o n a l l y , such r a t i o s have been said to have only a limited degree of accuracy. Although i t has sometimes been suggested that they have limited usefulness as rough guides or checks, they are almost always thought inadequate as accurate appraisal devices." Despite such t r a d i t i o n a l thought, i t i s well recognized that these r a t i o s become " b u i l t - i n t o " the market for c e r t a i n classes of property as they are widely used by market pa r t i c i p a n t s . The c r i t i c i s m s that have been t r a d i t i o n a l l y levied against d i r e c t conversion r a t i o s were considered t h e o r e t i c a l l y . Despite t h e i r s i m p l i c i t y , i t was recognized that because they are widely used, they might be useful i n predicting values. This proposition was then empirically tested. Thus, the purpose of t h i s study was to attempt to re-solve any inconsistencies and inaccuracies i n appraisal theory, by c r i t i c a l l y considering the t r a d i t i o n a l thought on the s e l e c t i o n of c a p i t a l i z a t i o n rates and the use and accur-acy of d i r e c t conversion r a t i o s , and to establish guidelines for the use of such r a t i o s as simple and accurate appraisal devices. I I . SUMMARY OF FINDINGS I t was found that cer t a i n t r a d i t i o n a l methods for selecting, c a p i t a l i z a t i o n rates are questionable. That i s , by considering themmethods i n turn, certain i r r a t i o n a l i t i e s , which could only r e s u l t i n inaccuracies and inconsistencies, 138 were revealed. Assuming, that the objective of a c a p i t a l i z a -t i o n device i s to predict a market value, the most probable s e l l i n g p r i c e , i t follows that capitalization, rates should be market determined. Thus, i n order for a method to be acceptable, i t must pass the t e s t of market determination. The components used to b u i l d up a rate under the sum-mation method simply do not e x i s t i n any such pure or elemental forms. Thus, a c a p i t a l i z a t i o n rate cannot be derived d i r e c t l y from the market under t h i s method; the high degree of sub-j e c t i v i t y and the ease of manipulation can be severely c r i t i c i z e d . While the band of investment method i s said to have merit i n that i t recognizes the mortgage and equity portions of the market, the manner of derivation of the rates for the two components and t h e i r combination into a synthetic r a t i o i s questionable. The mortgage rate i s said to be determined from financing sources and the equity rate i s derived somehow from comparable sales or simply from the appraiser's head. Once again, a t r a d i t i o n a l method f a i l s to meet the test of market determination. The rate selection i n the Ellwood method i s b a s i c a l l y a band of investment method and, therefore, can be c r i t i c i z e d on the same basis. The comparison of q u a l i t y attributes method may be c r i t i c i z e d for the s u b j e c t i v i t y involved i n assigning q u a l i t y ratings to c e r t a i n factors. Once again the c a p i t a l i z a t i o n 139 rate cannot be considered to be derived d i r e c t l y from the market. These t r a d i t i o n a l methods do not r e s u l t i n market determined rates; to so determine c a p i t a l i z a t i o n rates from available data i s an impossible task. More d i r e c t methods of going to the market were found to be reasonable and less subject to the c r i t i c i s m s of i r r a t i o n a l i t y and s u b j e c t i v i t y levied against the less d i r e c t methods. Because one of these more d i r e c t methods involved the use of a market determined r e l a t i o n s h i p , the gross income m u l t i p l i e r , which i s often r e a d i l y available and possibly useful i n the appraisal process, the discussion turned to more detailed consideration of such m u l t i p l i e r s . I t was seen that, t r a d i t i o n a l l y , gross income multi-p l i e r s , and other d i r e c t conversion r a t i o s , have been c r i t i -cized and dismissed as inadequate for accurate appraisal purposes. They are rejected because they are based on gross, rather than net, income, they involve various concepts of income, they are often used i n f l e x i b l y and they do not consider a l l those factors thought relevant i n the selection of com-parable properties. Their advantages are s i m p l i c i t y and data a v a i l a b i l i t y . Some of the factors which are judged necessary of consideration are the operating r a t i o , general property c l a s s , basic s t r u c t u r a l type, age, s i z e , location, dependa-b i l i t y and q u a l i t y of the income stream, vacancy, rate, l e v e l of tenant services, and time and conditions of, sale; I t was 140 seen that some theorists even suggest adjustments i n the multi-p l i e r s to the extent properties are not s t r i c t l y comparable i n a l l of the above factors. As has been mentioned, i t was pointed out that despite the c r i t i c i s m s , gross income m u l t i p l i e r s are widely used and to the extent they begin to be r e f l e c t e d i n market transactions, i t would seem they might be r e l i a b l e and accurate indicators of value. Extensive s t a t i s t i c a l analysis was necessary to t e s t t h i s proposition. Three methods of analysis were considered, the simple and multiple l i n e a r regression and c o r r e l a t i o n models and the average m u l t i p l i e r . I t was seen that such models could re-s u l t i n single estimates or i n confidence i n t e r v a l s of value. Two basic goals were involved i n the empirical te s t i n g , to determine how accurate the models were and to determine how, by consideration of property c h a r a c t e r i s t i c s , the estimates could be improved. Two separate tacks were used. F i r s t , confidence i n t e r -vals were derived for various equations and t h e i r width noted. Then, by s t r a t i f i c a t i o n , i t was possible to determine whether these i n t e r v a l s could be narrowed and, hence, made more precise. Second, the results of the f i r s t tack could be checked by deriving estimates and comparing them to actual values. Once again, i t was possible to determine whether the 141 estimates moved closer to the actual values when more detailed property c h a r a c t e r i s t i c s were considered. The findings were that confidence i n t e r v a l s could be improved by s t r a t i f i c a t i o n by basic s t r u c t u r a l type, location, number of suites and income per su i t e . The variable date of sale was also investigated. Because the data was limited i n the range of date of sale to only several years, i t was i n -t u i t i v e l y thought that s t r a t i f i c a t i o n by date of sale would not r e s u l t i n any appreciable improvements. This was empiri-c a l l y confirmed.. Of course, had there been a wider range i n the data, date of sale s t r a t i f i c a t i o n probably would have been more important. The question of sample size was also considered. Of course, for small samples the confidence i n t e r v a l w i l l be wider simply because r e l a t i v e l y high values of t must be used. The standard error of estimate does not depend d i r e c t l y on the sample siz e but, rather, on the inherent- v a r i a b i l i t y of the sample. Of course, to the extent extreme v a r i a t i o n i n any one transaction occurs, i t s e f f e c t w i l l be averaged out more i n large samples. Narrower confidence inte r v a l s can be achieved by lowering the l e v e l of confidence, but only at the expense of a poorer p r o b a b i l i t y q u a l i f i c a t i o n . Empirical testing demonstrated that except for very small sample sizes and aside from the use of r e l a t i v e l y large t values for small samples, no trend of precision was evident as sample sizes varied. 142 These findings were confirmed by the second phase of the analysis, the comparison of actual and estimated values. S t r a t i f i c a t i o n by basic structural"type, location, number of suites and income per suite resulted i n improvements i n estimates. I t was demonstrated that by using simple l i n e a r regression and c o r r e l a t i o n , or, i n the cases where better estimates were achieved, the average m u l t i p l i e r , average percentage differences between actual and estimated values could be as low as three per cent. In those cases i n which the percentages were somewhat higher, the results were s t i l l within acceptable error l i m i t a t i o n s . Once again i t was d i f f i -c u l t to perceive any trends i n accuracy of varying sample s i z e s . Multiple regression and c o r r e l a t i o n did not add a great deal to the analysis, although i t possibly could i f more variables were ava i l a b l e . I I I . CONCLUSIONS AND RECOMMENDATIONS The implications of the findings of t h i s study are very important. F i r s t , the findings with regard to the t r a d i t i o n a l methods of se l e c t i n g c a p i t a l i z a t i o n rates i l l u s t r a t e that . i r r a t i o n a l i t i e s e x i s t i n modern appraisal theory. Second, the findings with regard to the use and accuracy of gross income m u l t i p l i e r s i l l u s t r a t e that although such m u l t i p l i e r s have been discounted as inadequate, they are capable of predicting values, very accurately i n many cases. 143 Since the goal i n the s e l e c t i o n of c a p i t a l i z a t i o n rates i s to derive them from market experience, the t r a d i t i o n a l i n d i r e c t methods are largely a waste of time. They are ri d d l e d with i r r a t i o n a l i t y and s u b j e c t i v i t y and can only r e s u l t i n inaccuracies and inconsistencies i n the rates selected and, hence, i n the c a p i t a l values derived. Their use should not be encouraged; indeed, i t should be terminated. To base estimates on market evidence means that data must be available from market transactions. One of the most rea d i l y available pieces of information i s the gross income m u l t i p l i e r . This device was shown to be an extremely useful t o o l . By simply considering gross income and some rather basic s t r a t i f i c a t i o n i t i s possible to make accurate predic-tions of value for r e s i d e n t i a l income properties. The obvious advantages are the s i m p l i s t i c nature of the device, the application of which requires f a r less time than the t r a d i -t i o n a l methods, and the f a c t that acceptable degrees of accuracy are achieved by a rea d i l y available market determined r e l a -tionship, rather than by the somewhat questionable methods of the t r a d i t i o n a l approach. The use of gross income m u l t i p l i e r s on a f l e x i b l e and always current basis i s to be encouraged. Further ramifications of t h i s recommendation are that encour-agement should be given to data c o l l e c t i o n and experimentation with c e r t a i n econometric models which are r e a l l y only exten-144 sions of the simple, but often very accurate, m u l t i p l i e r model. IV. OTHER POSSIBLE STUDIES Although there are numerous aspects of appraisal theory which require study, the questions considered here might well be studied i n even greater depth as more detailed data i s avai l a b l e . B I B L I O G R A P H Y 146 A. Books and Monographs American I n s t i t u t e of Real Estate Appraisers. Appraisal Terminology and Handbook. Chicago: Author, 1967. The Appraisal of Real Estate. Chicago: Author, 1967. ' " " Babcock, F.M. The Valuation of Real Estate. New York: McGraw-Hill, 1932. Benson, P.A. and N.L. North. Real Es tate P r i n c i p l e s and  Practices. New York: 19231' Bjerring, J.H. and R.H. H a l l . Triangular Regression Package. Vancouver: U.B.C. Computing Centre, 1968. Cohen, J.B. and E.D. Zinberg. Investment Analysis and P o r t f o l i o Management. Homewood, I l l i n o i s : Dow-Jones, 1967. Ellwood,' L.W. Ellwood Tables. Chicago: A.I.R.E.A., 1967. E z e k i e l , M. and K.A. Fox. Methods of Correlation and  Regression Analysis. New York: Wiley, 1959. Federal Housing Administration. Underwri11rig Manual. Washington: Author, 1952. Freund, J.E. and F.J. Williams. Modern Business S t a t i s t i c s . Englewood C l i f f s , New Jersey: Prentice-Hall, 1958. Graham, B.J. The I n t e l l i g e n t Investor. New York: Harper, 19 Grebler, L., D.M. Blank, and L. Winnick. Capital Formation i n Residential Real Es tate. Princeton: Princeton University Press, 1956. Kahn., S.A., F.E. Case and A. Schimmel. Real Estate Appraisal  and Investment. New York: Ronald, 1963. Lewis, F.E. Methods of S t a t i s t i c a l Analysis i n Economics; and Business. Boston: Houghton M i f f l i n , 1953. Nemmers, E.E. and J.H. Myers. Bus!riess Research. New York: McGraw-Hill, 1966. 147 Neter, J . and W. Wasserman. Fundamental S t a t i s t i c s for Business and Economics. Boston: "Allyn and Bacon, 1961. P a r i , B. Basic S t a t i s t i c s . Garden C i t y , New York: Doubleday, 1967. -R a t c l i f f , R.U. Current Practices In Income Property Appraisal: A C r i t i q u e . Berkeley: University of C a l i f o r n i a , 1967. .. Modern Real Estate Valuation. Madison: . Democrat Press, 1965." A Restatement of Appraisal Theory. Madison: University of Wisconsin, 1963. Ring, A.A. The Valuation of Real Estate. Englewood C l i f f s , New Jersey: Prentice-Hall, 1963. Shattuck, C.B. "Income Approach-Capitalization Processes," Selected Readings i n Real Estate Appraisal. Chicago: A.I.R.E.A., 1953. Wendt, P.E. Real: Estate Appraisal. New York: Holt, 1957. B. Periodicals • Brigham, E.F. and D.M. M c A l l i s t e r . "Applying Econometric Models," The Appraisal Journal, XXXVI (October, 1968), pp. 541-548. Burkey, M.G. "Gross M u l t i p l i e r s , " The Review, XXI (June, 1955), pp. 9-10. Carney, J . J . "The Development and Use of Gross Income M u l t i p l i e r s , " The Appraisal Journal, XXXI (January, 1963), pp. 221-227. Derbes, M.J., J r . "Gross Income Multiple of the Apartment," The: Real Estate Appraiser, XXIX (October, 1963) , pp. 29-30. Dorau, H.B. "The C a p i t a l i z a t i o n Rate: Mirage or Will-o'-the-Wisp?" The Appraisal Journal, XXIX (January, 1961), pp. 19-29. 148 Gibbons, J.F. "Mortgage-Equity C a p i t a l i z a t i o n and After-Tax Equity Y i e l d , " The Appraisal Journal, XXXVII (January, 1969), pp. 196-202. Glaze, B.T. "Relationship of Market Value and Rent: a Market Sample of Single-Family Houses," The. Appraisal. Journal, XXXIV (October, 1966), pp. 574-580. Healy, F.J. "Musing on M u l t i p l i e r s , " The Real Estate Appraiser, XXIX (January, 1963), pp. 21-22. Hodges, M.B. "Income C a p i t a l i z a t i o n for Investor C l i e n t s , " Tho: Appraisal Journal, XXXVI ( A p r i l , 1968), pp. 175-200. Lostetter, E.K. "Income Residential Properties," The Residential  Appraiser, XXVII (March, 1961), pp. 2-6. Nelson, R.D. "Overall Rate - Band of Investment Style," The Appraisal Journal, XXXVII (January, 1969)pp.25-30. Pendleton, W.C. " S t a t i s t i c a l Inference i n Appraisal and Assessment Procedures," The Appraisal Journal, XXXIII (January, 1965), pp. 73-82. R a t c l i f f , R.U. "Capitalized Income i s Not Market Value," The Appraisal Journal, XXXVI (January, 1968), pp. 33-40. Sadesky, W.V. "The Relationship of Cost of Borrowed Money to C a p i t a l i z a t i o n Rates," The Appraisal Journal, XXXVI (January, 1968), pp. 9-14. Scane, L.H. "The Mystic Seven," The Appraisal Journal, XXVI (July, 1958), pp. 390-392. Shenkel, W.M. "Characteristics of Gross Income M u l t i p l i e r s , " The Real Estate Appraiser, XXXIV (January-February, 1968) , pp. 23-30. Sonnenschien, F.F. "The E f f e c t of L i f e Estimates on Capital Values," The Appraisal Journal, XIV (January, 1946), pp. 62-68. Winnick, L. "Long-Run Changes i n the Valuation of Real Estate by Gross Rents," The Appraisal Journal, XX (October, 1952), pp. 484-495, A P P E N D I X 150 The following table contains those regression and co r r e l a t i o n s t a t i s t i c s which are referred to i n the text of the study. The f i r s t column i s simply the description of the grouping of the properties used to calculate the regression equation. The equation, i n the form Y = a + bX, follows i n the second column. The figure i n brackets under the b value i s the standard error of b. The c o e f f i c i e n t of determination i s i n the t h i r d column. And the fourth column i s the F-Ratio, with the F-probability below i t i n brackets. The standard error of estimate and the sample size follow i n the f i f t h and six t h columns. Description Equation R 2 F-Ratio S y x n A l l Properties V -24,880+7.521X (.0376) .9905 39,950 (0.0) 42,990 385 Frame Low Rise - A l l 5,573+6.4883X (.0484) .9819 17,970 (0.0) 18,970 333 Concrete High Rise - A l l V -50,760+7.7295X (.1002) .9917 5,955 (0.0) 88,420 52 Frame-West End -8,242+6.9065X (.1364) .9828 2,665 (0.0) 25,510 47 Frame-South Granville, Fairview, K i t s i l a n o V 2,208+6.7754X (.0932) .9785 5,287 (0.0) 15,840 118 Frame-Marpole V 5,717+6.4848X (.0955) .9893 4,611 (0.0) 10,230 52 Frame-Kerrisdale 19,460+6.1426X (.1688) .9888 1,324 (0.0) 19,490 17 Frame-East End V 9,284+6.076X (.2288) .9489 705 (0.0) 11,850 40 Frame-Burnaby V 10,520+6.0803X (.0858) .9888 5,023 (0.0) 17,750 59 High Rise-West End V -52,650+7.7394X (.1225) .9915 3,989 (0.0) 105,200 36 U l H Description Equation R 2 F-Ratio Sy «x n Frame-Walkups (20+) (1958-67) 38,250+5.3736X (.3865) .8286 193 (0.0) 16,590 42 Frame-Walkups (20-) (1958-67) V 7,194+6.3093X (.2091) .9362 910 (0.0) 8,831 64 Frame-Walkups ( a l l sizes) (1940-59) \= 9,078+6.214lX (.1522) .9402 1,668 (0.0) 12,240 108 Frame-With Elevator 53,900+5.7701X (.2083) .9505 768 (0.0) 19,880 42 High-Rise - A l l No.of Suites t 45 V -76,520+7.7975X (.1707) .9896 2,087 (0.0) 129,400 24 High-Rise - A l l No.of Suites <45 -36,150+7.6751X (.2637) .9702 847 (0.0) 21,890 28 High-Rise - A l l Income/Suite >/$1600 T39,100+7.8531X (.1084) .9960 5,251 (0.0) 84,920 23 High-Rise - A l l Incone/Suite < $1600 32,470+6.5271X (.1174) .9913 3,094 (0.0) 37,910 29 High-Rise - A l l Date of Sale >, 67 -55,480+7.8403X (.1274) .9927 3,787 (0.0) 61,130 30 High Rise - A l l Date of Sale <67 V -55,540+7.6952X (.1617) .9912 2,266 (0.0) 118,200 22 Frame-West End No.of Suites >, 25 V -387,+6.8277X (.2175) .9782 985 (0.0) 30,980 24 Ul - - . • , - - . _ - _ . Description Equation R 2 F-Ratio s y x n Frame-West End No.of Suites < 25 45,880+4.6847X (.7681) .6392 37 (0.0) 16,060 23 Frame-West End Income/Suite >$1250 -13,470+6.9905X (.1496) .9900 2,183 (0.0) 22,850 24 Frame-West End Income/Suite <$1250 -562+6.7125X (.3151) .9558 454 (0.0) 28,680 23 Frame-West End Date of Sale »67 -5,494+6.7446X (.1997) .9744 1,141 (0.0) 17,560 32 Frame-West End Date of Sale <67 V 3,858+6.8067X (.3090) .9739 485 (0.0) 38,250 15 Frame-Burnaby No.of Suites >/ 20 30,070+5.7928X (.2707) .9522 458 (0.0) 25,830 25 Frame-Burnaby No.of Suites <20 V 1,306+6.6262X (.1954) .9729 1,150 (0.0) 6,602 34 Frame-Burnaby Income/Suite >, 1400 V 20,720+5.8935X (.1518) .9805 1,507 (0.0) 21,570 32 Frame-Burnaby Income/Suite <1400 V 210+6.6726X (.1107) .9932 3,634 (0.0) 7,804 27 Frame-East End No.of Suites >15 26,300+5.4629X (.5115) ,8445 114 (0.0) 14,700 23 Frame-East End No.of Suites <15 3,790+6.358lX (.3513) .9562 328 (0.0) 4,827 17 CO Description Equation R 2 F-Ratio Sy-x n Frame-East End Income/Suite >$1300 Y = 17,660+5.5859X X (.2564) .9654 474 (0.0) 10,190 19 Frame-East End Income/Suite <£$1300 Y =-3,772+6.8482X X (.3459) .9538 392 (0.0) 11,390 21 High-Rise-West End No.of Suites < 45 Y = 15,930+6.8544X X (.9240) .8089 55 (0.0) 20,080 15 High-Rise-West End No.of Suites >45 Y = -70,820+7.8425X A (.1133) .9973 4,793 (0.0) 80,580 15 High-Rise-West End Income/Suite <1400 Y — -9,209+7.1983X X (.2536) .9841 806 (0.0) 20,070 15 High-Rise-West End Income/Suite 5-1400 Y = -54,170+7.8403X (.1307) .9964 3,596 (0.0) 96,950 15 Frame-South Gr a n v i l l e , Fairview, K i t s i l a n o No.of Suites=8-ll Y = 9,856+6.3408X (1.194) .6845 28 (0.0002) 11,460 15 Frame-South Granville, Fairview, K i t s i l a n o No.of Suites=18-24 Y = 35,270+5.5272X X (1.2278) .6092 20 (0.0006) 16,090 15 Frame-South Granville, Fairview, K i t s i l a n o Y =137,800+4.2090X X (1.0045) .5746 17 (0.0011) 21,080 15 No.of Suites=25-35 U l Description Equation R 2 F-Ratio Sy •X n Frame-South Granville, Fairview, K i t s i l a n o Income/Suite =$1100-1299 Yx= 9,406+6.5794X (.4282) .9478 263 (0.0) 11 ,260 15 Frame-South G r a n v i l l e , Fairview, K i t s i l a n o Income/Suite=$1300-1449 v - •9,778+7.266X (.3171) .9758 525 (0.0) 16 ,940 15 Fairview-South Granville, Fairview, K i t s i l a n o Income/Suite=$1450-1600 V -9,753+7.1918X (.4166) .9582 298 (0.0) 24 ,200 15 Frame-Walkups(20+) (1958-67) V s 44,570+5.2885X (.7240) .8041 53 (0.0) 20 ,810 15 Frame-Walkups(20-) (1958-67) V 7,529+6.4085X (.4911) .9291 170 (0.0) 11 ,250 15 Frame-Walkups ( a l l sizes) (1940-59) V -1,404+6.6287X (.3656) .9620 329 (0.0) 8 ,606 15 Frame-With Elevator 52,390+5.8324X (.3867) .9459 227 (0.0) 18 ,910 15 High-Rise - A l l V -51,370+7.7604X (.0484) .9995 25,720 (0.0) 32 ,260 15 Frame-West End. V -2,119+6.8577X (.3685) .9638 346 (0.0) 33 ,630 15 Un U l Description Equation R 2 F-Ratio S y x n Frame-South Gra n v i l l e , Fairview, K i t s i l a n o V -671+6.5586X (.6146) .8975 114 (0.0) 17,840 15 Frame-Marpole 8,694+6.3443X (.2319) .9829 749 (0.0) 14,730 15 Frame-East End 20 ,350+5.6410X (.4424) .9260 163 (0.0) 15,240 15 Frame-Burnaby 7,481+6.0359X (.2584) .9767 546 (0.0) 26,270 15 High Rise-West End -40,150+7.7839X (.1100) .9974 5,009 (0.0) 74,350 15 Frame-South Gra n v i l l e , Fairview, K i t s i l a n o V 13,744+6.3710X (.5900) .9749 116 (0.0) 20,030 5 II V 17,164+6.3285X (.3120) .9808 410 (0.0) 14,610 10 V 15,542+6.3460X (.2330) .9827 743 (0.0) 12,950 15 I I 12,530+6.4185X (.1570) .9893 1,671 (0.0) 11,830 20 I I 6,472+6.5354X (.1400) .9895 2,188 (0.0) 11,190 25 II V s 2,277+6.7340X (.2030) .9748 1,087 (0.0) 19,640 30 c n Description Equation R 2 F-Ratio Sy *x n Frame-South Granville Fairview> K i t s i l a n o Frame-With Elevator Frame-Walkups (20+) (1958-67) V YX= Y X = 3,337+6.7004X (.1540) 3,022+6.7096X (.1300) 2,851+6.7582X (.1300) 2,625+6.7754X (.1150) 2,686+6.7869X (.1150) 3,871+6.7197X (.1100) Y =74,470+2531.15X± (1258.93) -18.85X2 (21.75) +4.4042X3 (.8331) Y = 894,600-38,000X-, A (21,950) -680.24 X 2 (358.93) +35.71 X 3 (16.56) .9829 .9859 .9843 .9863 .9849 .9845 .9574 8782 1,899 (0.0) 2,662 (0.0) 2,705 (0.0) 3,045 (0.0) 3,480 (0.0) 3,686 (0.0) 4.04 (.0672) 0.75 (.408) 27.95 (.003) 2.99 (.1084) 3.59 (.0820) 4.65 (.0521) 15,980 15,280 16,310 15,270 15,580 15,850 19,840 13,970 35 40 45 50 55 60 15 15 Description Equation R 2 F-Ratio S y x n Frame-Walkups(20-) (1958-67) YX= -71,680+4732.15X1 (6364.78) +60.74 X 2 (47.88) +2.9179 X 3 (4.8966) ,9554 0.55 (.4784) 1.61 (.2294) 0.39 (.5531) 8,803 15 Frame-Walkups(all sizes) (1940-59) Y x= 12,650-290.04 X, (3,029.81) -3.91 X„ (26.63) +6.4516 X 3 (2.4053) .9576 91.64 (.8874) 0.02 (.8571) 7.19 (.0206) 9,289 15 Frame-Marpole YX= " 95,870+3,503.76 X, .9948 (2.556.55) +74.64 X-(60.83) +3.9822 X 3 (1.9343) 1.88 (.1957) 1.51 (.2445) 4.24 (.0618) 9,466 15 Frame-East End Y x = -126,700+9,509.23 X,.9493 2.49 (6,021.72) 1 (.1397) +104.91 X 2 2.15 (71.63) (.1684) -1.2594 X 3 0.08 (4.5127) (.7754) 16,900 15 U l CO Description Equation Frame-Burnaby Yx= -41,840+5.708.19 X, .9975 (2,510.13) +31.03 X 0 (27.18) +2.5595 X 3 (1.7013) High Rise - A l l YX= -368,300+3,916.94 X,.9963 (4,573.60) +182.08 X 2 (199.19) +5.8219 X 3 (2.4520) High Rise - West End YX= 487,700-8,627.12 X-, .9964 (5,909.83) -322.89 X2 (284.07) +12.8958 X 3 (3.5290) F-Ratio Sy «x n 5.17 8,470 15 (.0423) 1.30 (.2779) 2.26 (.1579) 0.73 67,920 15 (.4142) 0.84 (.3836) 5.64 (.0355) 2.13 93,870 15 (.1697) 1.29 (.2799) 13.35 (.0038) i—1 VO 

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