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UBC Theses and Dissertations

Real estate portfolio diversification 1984

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REAL ESTATE PORTFOLIO DIVERSIFICATION BY TODD H. KURTIN B.S., The U n i v e r s i t y of Wisconsin, Madison 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS OF SCIENCE in Business A d m i n i s t r a t i o n in THE FACULTY OF GRADUATE STUDIES Commerce and Business A d m i n i s t r a t i o n We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA May 1984 (Cc)TODD H. KURTIN, 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 II ABSTRACT Th i s t h e s i s examines the p o t e n t i a l b e n e f i t s of d i v e r s i f i c a t i o n in r e a l e s t a t e . By c a l c u l a t i n g a set of re t u r n s f o r apartment blocks in Vancouver, B r i t i s h Columbia, two issu e s of d i v e r s i f i c a t i o n are d e a l t with: the p o t e n t i a l of d i v e r s i f y i n g w i t h i n r e a l e s t a t e , and the b e n e f i t s of i n c l u d i n g r e a l e s t a t e i n mixed-asset p o r t f o l i o s . To examine the p o t e n t i a l of d i v e r s i f y i n g w i t h i n r e a l e s t a t e , the study looks at the r e l a t i v e p r o p o r t i o n s of systematic and unsystematic r i s k of r e a l e s t a t e . A l s o , the paper i n v e s t i g a t e s the r a t e at which v a r i a t i o n s of re t u r n s f o r randomly s e l e c t e d p o r t f o l i o s are reduced as a f u n c t i o n of the number of p r o p e r t i e s in a p o r t f o l i o . To i n v e s t i g a t e the b e n e f i t s of i n c l u d i n g r e a l e s t a t e i n mixed- asset p o r t f o l i o s , two types of e f f i c i e n t p o r t f o l i o s are c o n s t r u c t e d : one that hedges a g a i n s t i n f l a t i o n , and the other that i s mean-variance e f f i c i e n t . By s e l e c t i n g these two types of e f f i c i e n t p o r t f o l i o s , the paper c o n s i d e r s two major investment o b j e c t i v e s of i n v e s t o r s : (1) that t h e i r p o r t f o l i o p r o v i d e s a r e t u r n to combat i n f l a t i o n ; (2) that t h e i r p o r t f o l i o have minimum r i s k f o r a given expected r a t e of r e t u r n . The f i n d i n g s of the study show that p o r t f o l i o s c o n s i s t i n g s o l e l y of r e a l e s t a t e ( o f one property type in one l o c a l market) are not w e l l d i v e r s i f i e d . The i n v e s t i g a t i o n found that only 29 percent of t o t a l r i s k i s u n s y s t e m a t i c ( d i v e r s i f i a b l e ) . However, a l a r g e p o r t i o n of the unsystematic r i s k can be d i v e r s i f i e d away I l l by h o l d i n g a p o r t f o l i o which c o n t a i n s only a few p r o p e r t i e s . The f i n d i n g s a l s o i l l u s t r a t e that r e a l e s t a t e i s a u s e f u l a d d i t i o n i n mixed-asset p o r t f o l i o s . Real e s t a t e c o n t r i b u t e s to the e f f e c t i v e n e s s of both the i n f l a t i o n - h e d g e d p o r t f o l i o and the mean-variance e f f i c i e n t p o r t f o l i o . In the i n f l a t i o n - h e d g e d p o r t f o l i o , r e a l e s t a t e does not c o n t r i b u t e as s t r o n g l y as expected, but the r e s u l t s s t i l l demonstrate that r e a l e s t a t e should be i n c l u d e d i n p o r t f o l i o s that are designed to hedge i n f l a t i o n . In the mean-variance e f f i c i e n t p o r t f o l i o , r e a l e s t a t e i s found to have a low or negative c o r r e l a t i o n with other a s s e t s , making the p o t e n t i a l to d i v e r s i f y very h i g h . IV TABLE OF CONTENTS A b s t r a c t II Table of Contents IV L i s t of Tables VI L i s t of F i g u r e s VII Acknowledgement VIII Chapter 1 I n t r o d u c t ion 1.1 C o n s t r a i n t s of the Study 4 1.2 Importance of Study 6 Chapter 2 L i t e r a t u r e Review 2.1 The E f f i c i e n t P r i n c i p l e s and the Basic Models of P o r t f o l i o Theory 9 2.2 E m p i r i c a l Research in the Stock Market 13 2.3 P o r t f o l i o A n a l y s i s with Real E s t a t e 19 Chapter 3 Data Base 3.1 The Apartment Market i n Vancouver 35 3.2 Apartment Block.Sample 39 3.3 Other Investment Assets and t h e i r Rate of Returns 46 V Chapter 4 Procedures 4.1 Return and Risk 49 4.2 Procedures to Test D i v e r s i f i c a t i o n w i t h i n Real E s t a t e .55 4.3 Procedures to C a l c u l a t e E f f i c i e n t P o r t f o l i o s 57 Chapter 5 V a l u a t i o n Model 5.1 T h e o r e t i c a l S p e c i f i c a t i o n 61 5.2 Development and A n a l y s i s of the Regression Model 67 Chapter 6 R e s u l t s 6.1 Return and Risk Measures of Apartment Blpcks 79 6.2 Answer and A n a l y s i s of Question One 80 6.3 Answer and A n a l y s i s of Question Two 88 Chapter 7 P i s c u s s i o n 7.1 I m p l i c a t i o n of F i n d i n g s to I n v e s t o r s 101 B i b l i o g r a p h y 108 Appendix A 113 Appendix B 115 Appendix C 183 VI LIST OF TABLES Table Page 2.1 Summary S t a t i s t i c s f o r P r o p e r t i e s with 20 Quarters of Data 31 2.2 D e s c r i p t i o n of P o r t f o l i o S i z e and Reduction i n Return V a r i a t i o n by Property Type 32 2.3 N o n - D i v e r s i f i a b l e Return as a P r o p o r t i o n of T o t a l Risk...33 3.1 Vancouver Apartment Data 37 3.2 Summary S t a t i s t i c s f o r the Apartment Block Sample 41 3.3 An Average Rent Index f o r Vancouver by Area 43 3.4 Apartment Operating Expense R a t i o Equation 45 5.1 Number of Sa l e s T r a n s a c t i o n s / Y e a r 69 5.2 The Average P r e d i c t i v e E r r o r f o r the GIM and the Market Value Model 73 5.3 The Annual V a l u a t i o n Equations 74 6.1 The Return and Risk Measures f o r the Set of Apartment Blocks 81 6.2 The Return and Risk Measures f o r the Sub-Sample of Apartment Blocks 83 6.3 D e s c r i p i o n of P o r t f o l i o S i z e and Reduction in Return V a r i a t i o n 85 6.4 The I n f l a t i o n Rate, the Mean Returns and Standard D e v i a t i o n s f o r the Investment A s s e t s 90 6.5 C o r r e l a t i o n Matrix of I n f l a t i o n and the Investment Assets 91 6.6 The Weighted P r o p o r t i o n s f o r Each Asset in an I n f l a t i o n - H e d g e d P o r t f o l i o 93 6.7 A Set of P o r t f o l i o s along the E f f i c i e n t F r o n t i e r (by d e c r e a s i n g r a t e of ret u r n ) 97 6.8 Comparisons of the Risk (Variance) of the I n d i v i d u a l Assets to E f f i c i e n t P o r t f o l i o s with the Same Mean Return.98 VII LIST OF FIGURES F i g u r e Page 5.1 The Long Run Supply and Demand Schedule of Apartment Blocks i n Vancouver 62 5.2 The Short Run Supply and Demand Schedule of Apartment Blocks i n Vancouver 62 6.1 Graph of the E f f i c i e n t F r o n t i e r 96 VIII ACKNOWLEDGEMENTS I would l i k e to thank the committee members, Dr. Dominique Anchor, Dr. Lawrence Jones, and Dr. Michael Tretheway, f o r t h e i r c o o p e r a t i o n i n h e l p i n g me to complete t h i s t h e s i s p r o j e c t . I am p a r t i c u l a r l y g r a t e f u l to M o l l i e Creery, Dennis Jackson, and Esther Lee f o r t h e i r a s s i s t a n c e r , without which, t h i s study would never have been completed. 1 INTRODUCTION 1 .0 The d i s c u s s i o n of p o r t f o l i o s e l e c t i o n has occupied the pages of f i n a n c i a l j o u r n a l s f o r over t h i r t y y e ars. The breadth of the d i s c u s s s i o n has extended from the e f f i c i e n c y p r i n c i p l e s of Markowitz[40] and the s i m p l i f y i n g model of Sharpe[55] to is s u e s of p o r t f o l i o s i z e , s t r a t e g y , and to the degree of d i v e r s i f i c a t i o n both i n domestic and i n t e r n a t i o n a l markets. Nev e r t h e l e s s , r e s e a r c h on p o r t f o l i o s e l e c t i o n has been l i m i t e d to the i n v e s t i g a t i o n of only a few investment instruments, with the major emphasis'on s e c u r i t i e s . Real e s t a t e as an investment has been ignored. Why? Researchers have been r e l u c t a n t to assess r e a l e s t a t e because i n f o r m a t i o n on r e t u r n s i s not r e a d i l y ava i l a b l e . T h i s paper extends the re s e a r c h of p o r t f o l i o s e l e c t i o n by c o n s i d e r i n g r e a l e s t a t e i n investment p o r t f o l i o s . The paper examines r e a l e s t a t e as an investment through a sample of apartment p r o p e r t i e s i n Vancouver, B r i t i s h Columbia. With these p r o p e r t i e s , q u a r t e r l y r e a l i z e d r e t u r n s are generated over a 10- year p e r i o d ending i n 1979 f o r the purpose of conducting an e m p i r i c a l a n a l y s i s . The a n a l y s i s c o n c e n t r a t e s on two qu e s t i o n s of relevance to i n v e s t o r s : (1) Can i n v e s t o r s d i v e r s i f y t h e i r p o r t f o l i o s s o l e l y w i t h i n a r e a l e s t a t e market? (2) Can the i n c l u s i o n of r e a l e s t a t e r e s u l t i n more e f f i c i e n t mixed-asset p o r t f o l i o s ? To answer the f i r s t q u e s t i o n , the paper i n v e s t i g a t e s the 2 r a t e at which v a r i a t i o n s of r e t u r n s , f o r randomly s e l e c t e d p o r t f o l i o s , are reduced as a f u n c t i o n of the number of p r o p e r t i e s i n a p o r t f o l i o . We w i l l t e s t the hypothesis that i n v e s t o r s can d i v e r s i f y t h e i r p o r t f o l i o s w i t h i n a r e a l e s t a t e m a r k e t - - i . e. that the r i s k of r e a l e s t a t e f o r the most pa r t i s d i v e r s i f i a b l e r i s k and that i n v e s t o r s can b e n e f i t from d i v e r s i f i c a t i o n . P r e v i o u s r e s e a r c h has shown that i n v e s t o r s can d i v e r s i f y w i t h i n investment markets. Using the stock market to t e s t d i v e r s i f i c a t i o n , Evans and A r c h e r [ l 7 ] , Latane and Young[36], E l t o n and G r u b e r [ l 6 ] , and B r e a l y [ 7 ] have found that the r e l a t i o n s h i p between the number of s e c u r i t i e s i n c l u d e d i n a p o r t f o l i o and the l e v e l of v a r i a t i o n takes the form of a d e c r e a s i n g asymtotic f u n c t i o n . In the only study using the r e a l e s t a t e market, M i l e s and McCue[44] have come to s i m i l a r c o n c l u s i o n s . They c o n s i d e r e d the r e l a t i o n s h i p between p o r t f o l i o s i z e and the r e t u r n v a r i a t i o n to f o l l o w the 1/n r u l e of McEnally and Boardman [43]. 1 An h y p o t h e s i s d i r e c t e d to q u e s t i o n 2 p o s t u l a t e s that the i n c l u s i o n of r e a l e s t a t e can improve the e f f i c i e n c y of i n v e s t o r s ' p o r t f o l i o s . E f f i c i e n c y i n t h i s context i n c l u d e s p o r t f o l i o s that hedge a g a i n s t i n f l a t i o n as w e l l as mean-variance e f f i c i e n t p o r t f o l i o s - - i . e. those p o r t f o l i o s which o f f e r s the highest expected r e t u r n f o r a given nominal v a r i a n c e , or which minimizes nominal v a r i a n c e f o r a given expected r e t u r n . Support f o r t h i s h y p o t h e s i s stems from p r i o r academic works. R e s u l t s from Friedman[22] and Hoag[28] have i l l u s t r a t e d that because a 3 low or negative c o r r e l a t i o n e x i s t s between r e a l e s t a t e and other investment a s s e t s , the i n c l u s i o n of r e a l e s t a t e improves the performance of mean-variance e f f i c i e n t p o r t f o l i o s . Fama and Schwert[l9] and Ha l l e g r e n [ 2 7 ] have concluded that r e a l e s t a t e i s a good hedge a g a i n s t i n f l a t i o n . In approaching the i n v e s t i g a t i o n of the paper's two q u e s t i o n s , we f i r s t review the l i t e r a t u r e a s s o c i a t e d with p o r t f o l i o s e l e c t i o n . In Chapter 2, works which r e i n f o r c e the hypotheses or are r e l e v a n t to the i s s u e s r a i s e d i n t h i s paper are d i s c u s s e d . Next, i n Chapter 3 the data used i n the study i s d e s c r i b e d . The process of s e l e c t i n g the f i n a l p r o p e r t y sample and the assumptions necessary to complete the i n f o r m a t i o n f o r the sample are d e a l t with here i n some d e t a i l . A f t e r the d e s c r i p t i o n of the data, the methodology used to answer the two qu e s t i o n s i s presented i n Chapter 4. Included i n t h i s s e c t i o n are the procedures used to measure the d i v e r s i f i c a t i o n c a p a b i l i t i e s w i t h i n r e a l e s t a t e and the techniques used to compute e f f i c i e n t p o r t f o l i o s . Chapter 5 i n t r o d u c e s the v a l u a t i o n f u n c t i o n needed to determine q u a r t e r l y market values f o r the apartment p r o p e r t i e s . Since r e a l e s t a t e does not have the continuous market t r a n s a c t i o n s of most e q u i t i e s , a v a l u a t i o n model must be developed to estimate q u a r t e r l y v a l u e s f o r the p r o p e r t i e s . E m p i r i c a l t e s t i n g and a n a l y s i s i s d i s c u s s e d i n Chapter 6; here a l s o the v a l i d i t y of the hypotheses i s assessed. L a s t l y , i n Chapter 7 the paper reviews the i m p l i c a t i o n s of the f i n d i n g s with respect to r e a l e s t a t e i n v e s t o r s . 4 1.1 CONSTRAINTS OF THE STUDY T h i s paper l i k e most s t u d i e s examining r e a l e s t a t e s u f f e r s from l e s s than adequate i n f o r m a t i o n . I d e a l l y , the data base should c o n s i s t of a t i m e - s e r i e s of re t u r n s f o r the n a t i o n a l r e a l e s t a t e market; a market that i n c l u d e s p r o p e r t i e s of a l l types from ac r o s s Canada. With such a data base, the p o t e n t i a l f o r d i v e r s i f i c a t i o n w i t h i n r e a l e s t a t e c o u l d be f u l l y t e s t e d , and an a p p r o p r i a t e r e a l e s t a t e r e t u r n index c o n s t r u c t e d to compute e f f i c i e n t mixed-asset p o r t f o l i o s . However, s i n c e r e a l e s t a t e l a c k s observable market t r a n s a c t i o n s and r e l a t e d investment r e t u r n i n f o r m a t i o n , i . e. cash flows from p r o p e r t i e s , reseachers i n v e s t i g a t i n g r e a l e s t a t e have o f t e n e i t h e r to narrow the scope of t h e i r a n a l y s i s , or to pl a c e l e s s weight on t h e i r f i n d i n g s . T h i s paper chooses to narrow the scope of the a n a l y s i s . Although the sample used i s as complete as - p o s s i b l e , c e r t a i n aspects of the data base impede a f u l l y adequate a n a l y s i s of the two q u e s t i o n s . F i r s t , the sample i s c o n f i n e d to one l o c a l r e a l e s t a t e market. Obviously, having data f o r only one l o c a l market i n h i b i t s the i n v e s t i g a t i o n of g e o g r a p h i c a l d i v e r s i f i c a t i o n , thus reducing the p o s s i b l i t y of c r e a t i n g an e f f i c i e n t p o r t f o l i o of r e a l e s t a t e p r o p e r t i e s . I f , however, we f i n d that d i v e r s i f i c a t i o n i s o b t a i n a b l e even w i t h i n a l o c a l market, we would have strong evidence i n support of our f i r s t h y p o t h e s i s . Secondly, the sample was intended to c o n t a i n commercial as w e l l as apartment p r o p e r t i e s , thereby i n c r e a s i n g the p o s s i b i l i t y 5 f o r d i v e r s i f i c a t i o n . However, s u f f i c i e n t i n f o r m a t i o n to provide r e l i a b l e r a t e s of r e t u r n f o r the commercial p r o p e r t i e s was not a v a i l a b l e , and so these p r o p e r t i e s were dropped from the study. T h i r d l y , the data base i s a l s o c o n s t r a i n e d by the number of assumptions and e s t i m a t i n g procedures needed to complete i t . Because r e a l e s t a t e p r o p e r t i e s are traded i n f r e q u e n t l y , q u a r t e r l y p r i c e s must be estimated by means of a fundamental v a l u a t i o n f u n c t i o n . T h i s f u n c t i o n i s c r i t i c a l to the r e s u l t s of the paper s i n c e the c a p i t a l g a i n ( l o s s ) on the p r o p e r t i e s i s the major f a c t o r determining the ra t e of r e t u r n . In a d d i t i o n , e s t i m a t i o n s are necessary fo r cash flows and debt. Thus a l l f a c t o r s c o n t r i b u t i n g to the r e t u r n of the p r o p e r t i e s are i n some way estimated. As a r e s u l t of these c o n s t r a i n t s , the paper w i l l t e s t a l i m i t e d v e r s i o n of i t s hypothesis that i n v e s t o r s can d i v e r s i f y s o l e l y w i t h i n a r e a l e s t a t e p o r t f o l i o . The problems s t a t e d above have f o r e s t a l l e d an i n v e s t i g a t i o n of geographic and p r o p e r t y - t y p e d i v e r s i f i c a t i o n , two of the ways in which i n v e s t o r s spread t h e i r r i s k w i t h i n r e a l e s t a t e . As a r e s u l t , the hypothesis should be r e s t a t e d that i n v e s t o r s can d i v e r s i f y t h e i r p o r t f o l i o s w i t h i n a l o c a l r e a l e s t a t e market. If there i s e i t h e r geographic d i v e r s i f i c a t i o n w i t h i n the c i t y or the e x i s t e n c e of high property s p e c i f i c r i s k , then t h i s h y p o thesis w i l l be accepted. The problems mentioned above do not a f f e c t to any s e r i o u s degree the a n a l y s i s of q u e s t i o n two. The i s s u e as to whether 6 r e a l e s t a t e can improve the e f f i c i e n c y of , i n v e s t o r s ' p o r t f o l i o s d e a l s with the co v a r i a n c e of r e a l e s t a t e r e t u r n s to the re t u r n s of other investment a s s e t s . The r e t u r n s generated from the study of apartment blocks in Vancouver should r e f l e c t the movement of the n a t i o n a l r e a l e s t a t e market, i f we agree with Sharpe's argument that s t o c k s , or i n t h i s case p r o p e r t i e s , move together because of macroeconomic events. 2 The v a r i a n c e of these apartment r e t u r n s may be gr e a t e r than they would be elsewhere s i n c e the Vancouver r e a l e s t a t e market i s c o n s i d e r e d q u i t e v o l a t i l e , but the p a t t e r n s t i l l r e f l e c t s the a c t i o n of the n a t i o n a l r e a l e s t a t e market. As a r e s u l t , the measure of cov a r i a n c e between r e a l e s t a t e and the the other investment a s s e t s should be reasonable. 1.2 IMPORTANCE OF STUDY Even though data problems a f f e c t the a n a l y s i s , the paper p r o v i d e s v a l u a b l e i n f o r m a t i o n to r e s e a r c h e r s and to i n v e s t o r s . F i r s t , i n view of the dea r t h of e m p i r i c a l s t u d i e s i n v o l v i n g r e a l e s t a t e r e t u r n s , t h i s paper p r o v i d e s much-needed evidence on the performance of r e a l e s t a t e . Second, the study can be of value to i n d i v i d u a l or small c o r p o r a t e i n v e s t o r s on the s u b j e c t of how to s t r u c t u r e t h e i r p o r t f o l i o s more e f f e c t i v e l y . In r e a l e s t a t e , i t i s not uncommon to f i n d l i m i t e d c a p i t a l i n v e s t o r s r e s t r i c t i n g t h e i r p o r t f o l i o s to one l o c a l market or even to one property type. These i n v e s t o r s c o n f i n e t h e i r p o r t f o l i o s to a l i m i t e d 7 number of h o l d i n g s due to high t r a n s a c t i o n c o s t s , and because r e a l e s t a t e i s a lumpy and i n d i v i s i b l e a s s e t , making i t d i f f i c u l t f o r them to own j u s t a small percentage of the a s s e t . The r e s u l t s of the study w i l l i n d i c a t e to these small c a p i t a l i n v e s t o r s whether they can d i v e r s i f y while h o l d i n g a narrow p o r t f o l i o based on a l o c a l market, or whether they should c o n s i d e r the c o s t / b e n e f i t s of f u r t h e r d i v e r s i f y i n g t h e i r p o r t f o l i o s i n t o other r e a l e s t a t e markets or i n t o a mixed asset p o r t f o l i o . T h i r d , the study g i v e s a l l i n v e s t o r s , l a r g e or sm a l l , i n f o r m a t i o n on how r e a l e s t a t e c o v a r i e s with other investment a s s e t s . Because the study i s intended f o r use by members of the l a y p u b l i c , we w i l l f r e q u e n t l y e x p l a i n some terms at l e n g t h and r e i t e r a t a spects of investment procedures f o r the purposes of a d d i t i o n a l c l a r i t y and understanding. 8 FOOTNOTES The 1/n r u l e of McEnally and Boardman i s expressed in equation form as: V p = V s + !/n(V u) where Vp i s the expected average v a r i a n c e of a p o r t f o l i o , Vs i s systematic r i s k and Vu i s unsystematic r i s k . Sharpe, W i l l i a m F., "A S i m p l i f i e d Model for P o r t f o l i o A n a l y s i s " , Management Science, Vol.9, January 1963, pp. 277-293 9 2.0 LITERATURE REVIEW Th i s chapter reviews a s e l e c t i o n of academic s t u d i e s concerning p o r t f o l i o theory and d i v e r s i f i c a t i o n , beginning with a d i s c u s s i o n of the e f f i c i e n c y p r i n c i p l e s of Markowitz and the basic models of p o r t f o l i o theory as developed by Markowitz and Sharpe. The chapter then presents a number of e m p i r i c a l s t u d i e s which use s e c u r i t i e s to examine the q u e s t i o n of d i v e r s i f i c a t i o n . Although the a r t i c l e s that are examined c o n s i d e r the q u e s t i o n of d i v e r s i f i c a t i o n i n the context of the stock market, they have been i n c l u d e d because of t h e i r relevance to the a n a l y s i s of the paper. The f i n a l s e c t i o n d e a l s with work that has been done on p o r t f o l i o theory and d i v e r s i f i c a t i o n w i t h i n the context of r e a l e s t a t e . In a d d i t i o n , the theory u n d e r l y i n g the v a l u a t i o n models used in the paper i s reviewed. 2.1 THE EFFICIENT PRINCIPLES AND THE BASIC MODELS OF PORTFOLIO THEORY Harry Markowitz proposed the e f f i c i e n c y p r i n c i p l e s of p o r t f o l i o theory over t h i r t y years ago. 1 In 1952, he introduced the e f f i c i e n c y p r i n c i p l e s as p a r t of a new h y p o t h e s i s on investment behavior. The new hypothesis s t a t e d that "the i n v e s t o r does(or should) c o n s i d e r expected r e t u r n a d e s i r a b l e t h i n g and v a r i a n c e of r e t u r n an u n d e s i r a b l e t h i n g . " 2 Before Markowitz proposed t h i s h y p o t h e s i s , t h e o r i e s and models 10 i n t e r p r e t e d investment behavior as that behavior of an i n v e s t o r to maximize the di s c o u n t e d value of f u t u r e r e t u r n s . In h i s i n i t i a l a r t i c l e on p o r t f o l i o s e l e c t i o n Markowitz c o n s i d e r e d the hypothesis of maximizing the dis c o u n t e d value of fu t u r e r e t u r n s , but r e j e c t e d i t : If we ignore market i m p e r f e c t i o n s the for e g o i n g r u l e never i m p l i e s that there i s a d i v e r s i f i e d p o r t f o l i o which i s p r e f e r r a b l e to a l l n o n - d i v e r s i f i e d p o r t f o l i o s . D i v e r s i f i c a t i o n i s both observed and s e n s i b l e ; a r u l e of behavior which does not imply the s u p e r i o r i t y of d i v e r s i f i c a t i o n must be r e j e c t e d as both a hypothesis and as a maxim. 3 In p l a c e of the maximum r e t u r n h y p o t h e s i s , Markowitz presented what he termed as h i s mean-variance r u l e . The hypothesis s t a t e d that an i n v e s t o r does(or should) s e l e c t p o r t f o l i o s that have a maximum expected r e t u r n f o r a given v a r i a n c e of r e t u r n or that an i n v e s t o r does(or should) s e l e c t p o r t f o l i o s that have a minimum va r i a n c e f o r a given expected r e t u r n . 4 P o r t f o l i o s that f i t the d e s c r i p t i o n e l a b o r a t e d by h i s hypo t h e s i s , Markowitz c a l l e d e f f i c i e n t ; such p o r t f o l i o s make up the e f f i c i e n t f r o n t i e r . I n v e s t o r s would choose from among t h i s set of e f f i c i e n t p o r t f o l i o s a c c o r d i n g to t h e i r u t i l i t y p r e f e r e n c e . In h i s a r t i c l e , Markowitz d i d not i l l u s t r a t e the techniques necessary to c a l c u l a t e the set of e f f i c i e n t p o r t f o l i o s , but he d e s c r i b e d the components which made up the model: the measurements of expected r e t u r n and v a r i a n c e ( r i s k ) of a p o r t f o l i o . The measurement of the expected r e t u r n of a 11 p o r t f o l i o i s f a i r l y s t r a i g h t f o r w a r d and i s c a l c u l a t e d as f o l l o w s : N E = Z X.M- 1 1 i = i where X^ i s the percentage of the i n v e s t o r ' s a s s e t s a l l o c a t e d to the i t h s e c u r i t y , and i s the expected value of the i t h s e c u r i t y . The measurement of the v a r i a n c e of a p o r t f o l i o i s more complex as i t i n c l u d e s the v a r i a n c e of the i n d i v i d u a l a s s e t s as w e l l as the c o v a r i a n c e between the a s s e t s . The c a l c u l a t i o n of the v a r i a n c e of a p o r t f o l i o i s as f o l l o w s : N N V = Z Z X.X.o- • i=1 j=1 where X^, Xj are the percentage of the i n v e s t o r ' s a s s e t s a l l o c a t e d to the i t h and j t h s e c u r i t y and a. • i s the c o v a r i a n c e between ass e t i and j . I t i s the c o v a r i a n c e between the a s s e t s which enables i n v e s t o r s to d i v e r s i f y t h e i r p o r t f o l i o s . I f i n v e s t o r s s e l e c t a s s e t s i n t h e i r p o r t f o l i o which have a low or negative c o v a r i a n c e , then the o v e r a l l v a r i a n c e of the p o r t f o l i o i s reduced. Markowitz's mean-variance r u l e d i f f e r e d from p r e v i o u s hypotheses in that i t c o n s i d e r e d the i n t e r r e l a t i o n s h i p of r e t u r n s . Markowitz's f i n d i n g s and investment model a l t e r the concept of p o r t f o l i o theory. However i n v e s t o r s and r e s e a r c h e r s soon 1 2 r e a l i z e d that the model was not p r a c t i c a l ; i t needed too much info r m a t i o n to be u s e f u l . The next step i n p o r t f o l i o a n a l y s i s had to be the development of a more s i m p l i f i e d model. W i l l i a m Sharpe provided t h i s s i m p l i f i e d model i n 1963. 5 In c o n s i d e r i n g h i s model, r e f e r r e d to as the Diagonal Model, Sharpe had two o b j e c t i v e s : to make i t p r a c t i c a l so that i n v e s t o r s c o u l d perform p o r t f o l i o a n a l y s i s at a very small c o s t , and to c o n s t r u c t a model that would not assume away the e x i s t e n c e of the i n t e r r e l a t i o n s h i p among s e c u r i t i e s . Sharpe achieved h i s o b j e c t i v e s by proposing a model that allowed f o r two important assumptions. The f i r s t of these c o n s i d e r e d the r e t u r n s of v a r i o u s s e c u r i t i e s to be r e l a t e d only through a common r e l a t i o n s h i p with some ba s i c u n d e r l y i n g f a c t o r . Sharpe i n c o r p o r a t e d t h i s assumption d i r e c t l y i n h i s model: R. = A . + j3 • I + C. l l F i l where: R^ i s the r e t u r n on the i t h s e c u r i t y ; A^ and are parameters; i s a random v a r i a b l e with an expected value of zero; and I i s the l e v e l of some index f o r the u n d e r l y i n g f a c t o r . The second assumption that Sharpe made, which must h o l d true f o r the f i r s t assumption to be t r u e , i s that the c o v a r i a n c e between the random v a r i a b l e s of any two s e c u r i t i e s i s zero. With these two assumptions, the r e t u r n f o r any s e c u r i t y i s determined by the r e l a t i o n s h i p of the s e c u r i t y to the u n d e r l y i n g f a c t o r and by 1 3 random f a c t o r s . Through h i s model, Sharpe decomposed the r i s k of a p o r t f o l i o i n t o s y s t e m a t i c ( n o n - d i v e r s i f i a b l e ) and u n s y s t e m a t i c ( d i v e r s i f i a b l e ) r i s k . Systematic r i s k i s a s s o c i a t e d with the u n d e r l y i n g f a c t o r and a f f e c t s a l l s e c u r i t i e s ; unsystematic r i s k r e l a t e s to the i n d i v i d u a l s e c u r i t i e s , and i s represented by the random f a c t o r s i n the model. The unsystematic r i s k can and should be d i v e r s i f i e d away. 2.2 EMPIRICAL RESEARCH IN THE STOCK MARKET The i n i t i a l q u e s t i o n of t h i s paper asks whether i n v e s t o r s can d i v e r s i f y s o l e l y w i t h i n a r e a l e s t a t e market. To answer t h i s q u e s t i o n , we examine the r e l a t i o n s h i p between the r i s k of a p o r t f o l i o and the number of p r o p e r t i e s i t c o n t a i n s . Most of the l i t e r a t u r e i n v e s t i g a t i n g the e f f e c t of p o r t f o l i o s i z e and the r e d u c t i o n of r e t u r n v a r i a t i o n has focused on s e c u r i t i e s . E m p i r i c a l s t u d i e s by Evans and A r c h e r [ l 7 ] , Latane and Young[36], E l t o n and G r u b e r [ l 6 ] , and B r e a l y [7] have examined e x h a u s t i v e l y the e f f e c t of p o r t f o l i o s i z e on the r e d u c t i o n of r e t u r n v a r i a t i o n with respect to s e c u r i t i e s . Since a l l of these s t u d i e s have reached s i m i l a r c o n c l u s i o n s , we w i l l only d i s c u s s one of the a r t i c l e s here: the a r t i c l e examined i s the Evans and Archer paper. Evans and Archer argue that i f p o r t f o l i o s i z e has an e f f e c t on the r e d u c t i o n of r e t u r n v a r i a t i o n , the r e s u l t must be a f u n c t i o n of the r e d u c t i o n of the unsystematic p o r t i o n of the 1 4 t o t a l • v a r i a n c e . They a l s o argue that as the number of s e c u r i t i e s i n a p o r t f o l i o approaches the number of s e c u r i t i e s i n the market, the v a r i a t i o n of the p o r t f o l i o r e t u r n w i l l approach the l e v e l of systematic v a r i a t i o n , suggesting a r e l a t i o n s h i p t h a t behaves as a d e c r e a s i n g asymptotic f u n c t i o n . 6 To prove t h e i r p o i n t , they c o n s t r u c t e d randomly s e l e c t e d p o r t f o l i o s of s i z e s 2 to 40. The p o r t f o l i o s were then regressed by the equation: Y = A + B(1/X) where Y i s the computed mean p o r t f o l i o standard d e v i a t i o n ( t h e measure of r i s k ) , and X i s the p o r t f o l i o s i z e . The r e s u l t s of the r e g r e s s i o n a n a l y s i s were q u i t e p o s i t i v e : the c o e f f i c i e n t of d e t e r m i n a t i o n ( a n o t h e r term f o r R 2) f o r the equation was .9863. When the average standard d e v i a t i o n of r e t u r n was p l o t t e d a g a i n s t the number of s e c u r i t i e s i n the p o r t f o l i o , the p l o t t e d graph formed a d e c r e a s i n g asymptotic f u n c t i o n . Evans and Archer then conducted one more experiment. T h i s one i n v o l v e d t - t e s t s on s u c c e s s i v e mean p o r t f o l i o standard d e v i a t i o n s to determine at what p o i n t s i g n i f i c a n t r e d u c t i o n of v a r i a t i o n ( a t the .05 l e v e l ) took p l a c e . The r e s u l t s of the t e s t i n d i c a t e d that the a d d i t i o n of one s e c u r i t y to a p o r t f o l i o of s i z e 2 caused s i g n i f i c a n t r e d u c t i o n in the mean p o r t f o l i o standard d e v i a t i o n . For p o r t f o l i o of s i z e 8, the necessary i n c r e a s e was 5 s e c u r i t i e s ; f o r a p o r t f o l i o of s i z e 16, the necessary i n c r e a s e was 19 s e c u r i t i e s . Evans and Archer concluded that there was probably l i t t l e economic j u s t i f i c a t i o n f o r i n c r e a s i n g p o r t f o l i o s i z e beyond 10 or so s e c u r i t i e s , and 1 5 suggested that i n v e s t o r s i n c l u d e some form of marginal a n a l y s i s in t h e i r p o r t f o l i o s e l e c t i o n models. Moving from the i n v e s t i g a t i o n of d i v e r s i f i c a t i o n , we now review s t u d i e s that examine e f f i c i e n t p o r t f o l i o s . The paper uses two d e f i n i t i o n s f o r e f f i c i e n c y : Markowitz's mean-variance d e f i n i t i o n and i n f l a t i o n - h e d g e d p o r t f o l i o s . To i l l u s t r a t e the p o t e n t i a l advantages of d i v e r s i f i c a t i o n under the d e f i n i t i o n of mean-variance e f f i c i e n c y , Robichek, Cohn and P r i n g l e r [ 5 0 ] presented a study on r e t u r n s of a l t e r n a t i v e investment instruments. The paper computed ex post r a t e s of re t u r n and c o r r e l a t i o n c o e f f i c i e n t s f o r twelve a l t e r n a t i v e investment media f o r the p e r i o d 1949-1969. The authors' aim was to i d e n t i f y the degree to which investment a l t e r n a t i v e s , other than common stock and r i s k l e s s one-period bonds, i n f l u e n c e d the c o n s t r u c t i o n of e f f i c i e n t p o r t f o l i o s . The investment media i n c l u d e d common stocks from the United S t a t e s , Canada and Japan; U.S. government and cor p o r a t e bonds; r e a l e s t a t e ; and commodity f u t u r e s . The data used to compute ret u r n s on r e a l e s t a t e was the U.S. Department of A g r i c u l t u r e index of value per acre of farm r e a l e s t a t e . Though farm land r e t u r n s are a dubious i n d i c a t o r of the re t u r n s on r e a l e s t a t e , the authors were not able to d i s c o v e r a b e t t e r one. The paper found that the c o r r e l a t i o n c o e f f i c i e n t s among the va r i o u s a s s e t s were g e n e r a l l y low, and that the sign s of the c o e f f i c i e n t s were almost e q u a l l y d i v i d e d between p o s i t i v e and neg a t i v e . Of the 66 c o r r e l a t i o n c o e f f i c i e n t s between a l l p a i r s 1 6 of a s s e t s , only 4 i n d i c a t e d p o s i t i v e c o r r e l a t i o n s i g n i f i c a n t at the .05 l e v e l . For r e a l e s t a t e , a l l the c o r r e l a t i o n s with the other a s s e t s were negative except fo r the p o s i t i v e c o r r e l a t i o n with U.S. Treasury B i l l s and with Japanese s t o c k s , which was s i g n i f i c a n t . The i m p l i c a t i o n of the f i n d i n g s i s that d i v e r s i f i c a t i o n among the twelve investment media leads to improved p o r t f o l i o e f f i c i e n c y in the mean-variance co n t e x t . To demonstrate which a s s e t s are e f f e c t i v e hedges a g a i n s t i n f l a t i o n and t h e r e f o r e u s e f u l i n a i n f l a t i o n - h e d g e d p o r t f o l i o , we review a paper w r i t t e n by Fama and S c h w e r t [ l 9 ] , "Asset Returns and I n f l a t i o n " . Fama and Schwert developed a model to t e s t the e f f e c t i v e n e s s of such a s s e t s based on the work of I r v i n g F i s h e r , 7 who had hypothesized that the nominal i n t e r e s t rate can be expressed as the sum of an expected r e a l r e t u r n and an expected i n f l a t i o n r a t e . From F i s h e r ' s p r o p o s i t i o n , Fama and Schwert designed t h e i r model so that the expected nominal r e t u r n on an a s s e t from t-1 to t i s the sum of the expected r e a l r e t u r n and the best p o s s i b l e assessment of the expected and unexpected i n f l a t i o n r a t e from t-1 to t . Fama and Schwert's model, which they t e s t e d by r e g r e s s i o n a n a l y s i s , appeared as: R j t = a + fljEU^W + V V E ( V * t - 1 ) ] + N j t where: R.. i s the nominal r e t u r n on a s s e t j from t-1 to t ; E ( A t / $ t _ 1 ) i s the best p o s s i b l e assessment of the expected value of the i n f l a t i o n r a t e A., that can be made on the 17 b a s i s of the set of in f o r m a t i o n * t _ 1 a v a i l a b l e at t-1; [A f c - E ( A t / $ t _ 1 ) ] i s u n a n t i c i p a t e d i n f l a t i o n ( i n f l a t i o n at time t minus expected i n f l a t i o n made on the base of $ t_ 1; N j t i s the random term f o r as s e t j at time t ; 0_j and 5j are the l i n e a r c o e f f i c i e n t s to be estimated; and the t i l d e s denote random v a r i a b l e s . If /3j = 1.0, in the model, the asse t i s a complete hedge a g a i n s t expected i n f l a t i o n , and the expected r e a l r e t u r n on the asse t i s u n c o r r e l a t e d with expected i n f l a t i o n . If 5^=1.0, the asset i s a complete hedge a g a i n s t unexpected i n f l a t i o n and when /3 j = 6 j= 1 .0 then the asse t i s a complete hedge ag a i n s t both aspects of i n f l a t i o n . The r e g r e s s i o n model was t e s t e d using a number of a s s e t s : (1) T - B i l l s with one-to six-month m a t u r i t y (2) common stocks from the New York Stock Exchange (3) U.S. government bonds (4) human c a p i t a l (the r a t e of change of labor income per c a p i t a ) (5) p r i v a t e l y h e l d r e s i d e n t i a l r e a l e s t a t e ( t h e r a t e of i n f l a t i o n of the Home Purchase P r i c e component of the CPI). 18 Fama and Schwert f i r s t a n a lyzed how we l l the s e l e c t i o n of a s s e t s hedged a g a i n s t expected i n f l a t i o n d u r i n g three time h o r i z o n s : monthly, q u a r t e r l y , and semi-annually. The estimates of B j ( t h e c o e f f i c i e n t f o r the expected i n f l a t i o n ) , were c l o s e to one f o r t r e a s u r y b i l l s , government bonds and r e a l e s t a t e f o r a l l three p e r i o d s . Human c a p i t a l was p o s i t i v e l y r e l a t e d to the monthly and q u a r t e r l y expected i n f l a t i o n r a t e but was n e g a t i v e l y r e l a t e d to the semiannual expected i n f l a t i o n r a t e . Common stock r e t u r n s showed a negative r e l a t i o n s h i p f o r a l l time h o r i z o n s , with the c o e f f i c i e n t i n c r e a s i n g i n magnitude with time. The r e s u l t s from the t e s t of unexpected i n f l a t i o n showed only r e a l e s t a t e to be a complete hedge a g a i n s t unexpected i n f l a t i o n f o r a l l time h o r i z o n s . The c o e f f i c i e n t f o r human c a p i t a l was moderately p o s i t i v e with monthly unexpected i n f l a t i o n , but turned negative f o r q u a r t e r l y and semiannual unexpected i n f l a t i o n . Government bonds and common stock had i n c r e a s i n g l y negative c o e f f i c i e n t s as the time h o r i z o n i n c r e a s e d . The study of Fama and Schwert i m p l i e s that r e a l e s t a t e i s the only a s s e t which a c t s as a complete hedge a g a i n s t both expected and unexpected i n f l a t i o n . Real e s t a t e r e t u r n s move i n high correspondence with both components of the i n f l a t i o n r a t e . If t h e i r f i n d i n g s h o l d up, then r e a l e s t a t e should prove a hedge ag a i n s t i n f l a t i o n and c o n t r i b u t e to the i n f l a t i o n - h e d g e d p o r t f o l i o developed i n t h i s study. In Fama and Schwert's study, the r e g r e s s i o n equation that 19 in c l u d e d r e a l e s t a t e had an R 2 of roughly 60 percent, implying that the i n f l a t i o n a d j u s t e d r e t u r n of r e a l e s t a t e i s not c e r t a i n and that r e a l e s t a t e has a c o n s i d e r a b l e amount of r e a l r e t u r n v a r i a t i o n . 8 2.3 PORTFOLIO ANALYSIS WITH REAL ESTATE The examination of p o r t f o l i o s e l e c t i o n d i d not extend to r e a l e s t a t e u n t i l 1970, when H a r r i s Friedman a p p l i e d p o r t f o l i o theory to e q u i t y investment i n r e a l e s t a t e . 9 Friedman's i n i t i a l work i n v e s t i g a t e d the concept of s e l e c t i n g r e a l e s t a t e p o r t f o l i o s , through the a p p l i c a t i o n of mathematical models used to s e l e c t and eval u a t e common stock p o r t f o l i o s . In a d d i t i o n , he evaluated the r e l a t i o n s h i p of r e a l e s t a t e to common stock by comparing r e a l e s t a t e p o r t f o l i o s to common stock p o r t f o l i o s and by c o n s t r u c t i n g a p o r t f o l i o c o n t a i n i n g both r e a l e s t a t e and common stock. To b u i l d the i n d i v i d u a l r e a l e s t a t e and common stock p o r t f o l i o s i n the study, Friedman employed Sharpe's d i a g o n a l model; 1 0 when he combined the two investment a s s e t s i n t o one p o r t f o l i o , he used the Cohen-Pogue m u l t i - i n d e x model. 1 1 In w r i t i n g t h i s f i r s t paper, Friedman i n i t i a t e d the procedures to r e s o l v e data problems a s s o c i a t e d with r e a l e s t a t e r e t u r n s . To c o n s t r u c t h i s r e a l e s t a t e p o r t f o l i o s , he needed the ho l d i n g p e r i o d r e t u r n s f o r each p r o p e r t y i n the p o r t f o l i o . Using f i v e one-year h o l d i n g p e r i o d s f o r the study, he had in f o r m a t i o n on the y e a r l y cash flow from the p r o p e r t i e s , but knew the market values f o r only the beginning and ending years 20 of the study. Friedman estimated the intermediate values by assuming that the p r o p e r t i e s a p p r e c i a t e d at the compound growth r a t e . Thus, Friedman understated the r i s k i n e s s of r e a l e s t a t e , and provided r e a l e s t a t e with an added advantage i n i t s comparison to common stock. Friedman encountered another d i f f i c u l t y when he t r i e d to s e l e c t an a p p r o p r i a t e index to use i n Sharpe's d i a g o n a l model for h i s r e a l e s t a t e p o r t f o l i o s . He employed an average of the Boeckh c o n s t r u c t i o n c o s t indexes f o r h o t e l s , r e s i d e n c e s , apartments, commercial p r o p e r t i e s , and f a c t o r i e s with the American A p p r a i s a l I n d e x . 1 2 T h i s "hodgepodge" of an index would be expected to have a low a s s o c i a t i o n with the r e t u r n s of the p r o p e r t i e s in the sample; hence most r e a l e s t a t e r i s k would appear to be d i v e r s i f i a b l e . 1 3 Again the r i s k a s s o c i a t e d with r e a l e s t a t e was understated, making r e a l e s t a t e appear undeservedly a t t r a c t i v e . The l a s t major d i f f i c u l t y Friedman faced was the choice of a super-index to use i n the Cohen-Pogue model when combining r e a l e s t a t e and common stock i n t o one p o r t f o l i o . Friedman used the GNP index - an index r e a l l y not adequate to e x p l a i n the v a r i a t i o n of r e t u r n s f o r r e a l e s t a t e and common s t o c k . 1 4 D e s p i t e i t s problems, Friedman's paper presented some notable f i n d i n g s . F i r s t , both on a before-and a f t e r - t a x b a s i s , e f f i c i e n t r e a l e s t a t e p o r t f o l i o s dominated common stock p o r t f o l i o s except i n the range of unusually 4 high r e t u r n s . Friedman q u a l i f i e d t h i s f i n d i n g by p o i n t i n g out that the sample 21 used i n the study was not r e p r e s e n t a t i v e of the un i v e r s e of r e a l e s t a t e a s s e t s . Second, taxes had more impact on common stock r e t u r n s than they d i d on r e a l e s t a t e . The reason f o r t h i s was that tax s h e l t e r b e n e f i t s of r e a l e s t a t e help l e s s e n the e f f e c t of taxes on the r e t u r n s of r e a l e s t a t e as compared to common stock. T h i r d , r e a l e s t a t e appeared as the dominant ass e t i n the mixed asset p o r t f o l i o , e s p e c i a l l y on an a f t e r - t a x b a s i s . L a s t l y , the co v a r i a n c e between r e a l e s t a t e and common stock was negati v e , which g r e a t l y reduced the t o t a l mixed ass e t p o r t f o l i o r i s k . In c o n c l u s i o n , Friedman s t a t e d that models developed to s e l e c t common stock p o r t f o l i o s can be adapted to the s e l e c t i o n of r e a l e s t a t e p o r t f o l i o s , and that r e a l e s t a t e dominates common stock as an investment a s s e t . A more recent paper by Hoag[28] attempted to c o r r e c t some of the problems that Friedman encountered. Hoag's o b j e c t i v e was not to improve on Friedman's work, but to provide i n f o r m a t i o n on r i s k and r e t u r n of r e a l e s t a t e investments in order that c u r r e n t investment management technology c o u l d be a p p l i e d to r e a l e s t a t e . Hoag t r i e d to accomplish t h i s o b j e c t i v e by c o n s t r u c t i n g an index of r e a l e s t a t e value and r e t u r n f o r non owner- occupied i n d u s t r i a l p r o p e r t y . The importance of the Hoag paper to t h i s study i s i n the method he uses to determine p r o p e r t y v a l u e . Because c a p i t a l g a i n ( l o s s ) i s the major f a c t o r f o r the re t u r n on r e a l e s t a t e , the v a l u a t i o n model p l a y s a c r i t i c a l r o l e on the estimate of the 22 r e t u r n on r e a l e s t a t e . Hoag employed a p r o p e r t y v a l u a t i o n f u n c t i o n based on fundamental c h a r a c t e r i s t i c s of the p r o p e r t i e s : p r o p e r t y type, s i z e , age, economic and demographic f a c t o r s , cash flows and t r a n s a c t i o n p r i c e s . Hoag argued that t h i s v a l u a t i o n model was e q u i v a l e n t to income c a p i t a l i z a t i o n a p p r a i s a l , except that as a p p r a i s a l i s s u b j e c t i v e the v a l u a t i o n model makes an o b j e c t i v e judgement. Hoag f u r t h e r argued that t h i s type of fundamental a n a l y s i s i s accomplished by s e c u r i t y a n a l y s t s i n the stock market where macroeconomic v a r i a b l e s and f i r m - s p e c i f i c data are used to assess a f i r m ' s v a l u e . 1 5 Hoag estimated h i s v a l u a t i o n f u n c t i o n by using a c t u a l t r a n s a c t i o n p r i c e s from the sample of i n d u s t r i a l p r o p e r t i e s . In h i s model, a value fo r each n o n t r a n s a c t i n g p r o p e r t y , at any given time, i s estimated from the v a l u a t i o n f u n c t i o n a p p l i e d to the fundamental c h a r a c t e r i s t i c s at that time. The macroeconomic c h a r a c t e r i s t i c s of the model t r y to capture the supply and demand f u n c t i o n s of the i n d u s t r i a l p r o p e r t y market through time, while the microeconomic and p h y s i c a l c h a r a c t e r i s t i c s of the p r o p e r t i e s d e s c r i b e the b u i l d i n g , surroundings and l o c a t i o n . Hoag c o n s i d e r e d the r e s u l t s of the model to be q u i t e reasonable, with an a d j u s t e d R 2=.89. However, the standard e r r o r was unacceptably high, being $352, 000 or 30 percent of the mean s a l e s p r i c e . From the v a l u a t i o n f u n c t i o n , Hoag c a l c u l a t e d the i n d i v i d u a l p r o p e r t i e s r a t e of r e t u r n s and the o v e r a l l market r a t e of r e t u r n . T h i s o v e r a l l market rate of r e t u r n represented h i s r e t u r n 23 index fo r r e a l e s t a t e . The r e t u r n on the index was high(.0338/quarter) as was the the r i s k ( a standard d e v i a t i o n of .0861/quarter). Hoag concluded that the two measures were comparable to those o b t a i n a b l e on stocks and bonds. When Hoag c a l c u l a t e d the c r o s s c o r r e l a t i o n of r e a l e s t a t e to other a s s e t s and i n f l a t i o n , the r e s u l t s i l l u s t r a t e d that r e a l e s t a t e c o u l d h e l p i n v e s t o r s d i v e r s i f y t h e i r p o r t f o l i o s and i n a d d i t i o n allow them to use r e a l e s t a t e as a hedge ag a i n s t i n f l a t i o n . These r e s u l t s support the h y p othesis of q u e s t i o n two i n t h i s paper that r e a l e s t a t e can improve the e f f i c i e n c y of i n v e s t o r s ' p o r t f o l i o s . In h i s implementation of a fundamental v a l u a t i o n f u n c t i o n Hoag d i d not f u l l y d e t a i l the theory u n d e r l y i n g h i s model. Hoag argued that s i n c e stock a n a l y s t s use fundamental techniques to value s t o c k s , i t would be reasonable to develop a fundamental v a l u a t i o n f u n c t i o n f o r r e a l e s t a t e . Since a v a l u a t i o n f u n c t i o n p l a y s a major r o l e i n t h i s paper i n determining the ra t e of r e t u r n on the sample of p r o p e r t i e s , r e f e r e n c e to two papers which d i s c u s s the theory behind fundamental v a l u a t i o n f u n c t i o n s i s i n o r d e r . Both papers c o n s i d e r the v a l u a t i o n of p r o p e r t i e s from the p o i n t of view an a p p r a i s e r . "The V a l u a t i o n of M u l t i p l e Family Dwellings by S t a t i s t i c a l I n f e r e n c e , " by W i l l i a m Shenkel[59] i s the foundation f o r the v a l u a t i o n model developed f o r t h i s paper. Shenkel i n i t i a t e d h i s paper with the p r o p o s i t i o n that income p r o p e r t i e s are bought and s o l d on the b a s i s of a n t i c p a t e d net income. 24 However, he argued t h a t , i n p r a c t i c e , a p p r a i s e r s d e v i a t e from the p r o p o s i t i o n that value i s determined by net income s i n c e i t i s d i f f i c u l t to estimate net income. Instead they o f t e n use gross income as a proxy for net income, and thus assume a r e l a t i o n s h i p between gross income and v a l u e . T h i s r e l a t i o n s h i p i s i l l u s t r a t e d by the gross income m u l t i p l i e r : V = f(GIM). To f i n d the gross income for a p r o p e r t y , a p p r a i s e r s o f t e n c a l c u l a t e the average or median GIM from a sample of r e c e n t l y s o l d p r o p e r t i e s . Shenkel contended that the s t a t i s t i c a l technique of simple r e g r e s s i o n can serve as a s u b s t i t u t e f o r the standard GIM and that r e g r e s s i o n can be a more p r e c i s e t o o l i n e s t i m a t i n g v a l u e : " The r e g r e s s i o n d e r i v e d m u l t i p l i e r i s produced with s t a t i s t i c a l measures of r e l i a b i l i t y and an estimate of the expected e r r o r . " 1 6 Shenkel admitted that the e r r o r from simple r e g r e s s i o n i s o f t e n too great to determine value; he argued r a t h e r that to value p r o p e r t y a c c u r a t e l y , r e l i a n c e must be p l a c e d on m u l t i p l e r e g r e s s i o n . In advocating m u l t i p l e r e g r e s s i o n , he presented a second p r o p o s i t i o n which s t a t e d that i f i t c o u l d be shown that net income and, t h e r e f o r e , value were r e l a t e d to a set of common property c h a r a c t e r i s t i c s , then p r o p e r t y c h a r a c t e r i s t i c s c o u l d p r e d i c t v a l u e . Shenkel wanted to demonstrate that market value c o u l d be estimated d i r e c t l y from v a l u e - s i g n i f i c a n t p r o p e r t y c h a r a c t e r i s t i c s , and that a p p r a i s e r s c o u l d dispense with the c a p i t a l i z a t i o n p r o c e s s . 1 7 25 Shenkel, to c o n f i r m h i s p r o p o s i t i o n , ran a stepwise m u l t i p l e r e g r e s s i o n a n a l y s i s on a sample of 47 apartment houses over a f i v e - y e a r p e r i o d . The sample of apartment houses were l o c a t e d i n a s i n g l e m e t r o p o l i t a n a r e a . He s e l e c t e d 69 p r o p e r t y c h a r a c t e r i s t i c s through which to e x p l a i n v a l u e . These c h a r a c t e r i s t i c s c o u l d be a s s o c i a t e d with three groups: those a s s o c i a t e d with area or s i z e ; those a s s o c i a t e d with l o c a t i o n a l a t t r i b u t e s ; and those c o v e r i n g a m enities and s e r v i c e s of a given apartment house. In Shenkel's i n i t i a l run, the c o e f f i c i e n t of d e t e r m i n a t i o n was .9719 with 20 s i g n i f i c a n t v a r i a b l e s . The average p r e d i c t i v e e r r o r was 6.85 percent. Shenkel reran the r e g r e s s i o n a n a l y s i s e l i m i n a t i n g gross income as a v a r i a b l e . The r e s u l t s from t h i s run were very s i m i l a r (a c o e f f i c i e n t of d e t e r m i n a t i o n of .9776 and a p r e d i c t i v e e r r o r of 7.20 p e r c e n t ) . Shenkel suggested from t h i s second model, that reasonable accuracy might be obtained without r e f e r e n c e to gross income, net income, c a p i t a l i z a t i o n r a t e s or the usual c a p i t a l i z a t i o n procedures. He f u r t h e r p o i n t e d out that the model c o u l d have been even more accurate i f the time p e r i o d had been s h o r t e r : " I d e a l l y , s a l e s should be c o n f i n e d to the s h o r t e s t p o s s i b l e time p e r i o d . . . t h e s h o r t e r the time i n t e r v a l the l e s s the i n f l u e n c e of time on the s a l e s p r i c e . " He suggested a one year time frame. From the r e s u l t s of the t e s t , Shenkel confirmed that market value c o u l d be determined by a set of property c h a r a c t e r i s t i c s . He a l s o d e c l a r e d that m u l t i p l e r e g r e s s i o n a n a l y s i s i s more o b j e c t i v e than c o n v e n t i o n a l c a p i t a l i z a t i o n , t h at m u l t i p l e 26 re g e s s i o n d e a l s d i r e c t l y with those f a c t o r s important to net income and to v a l u e . A second a r t i c l e that p r o v i d e s a t h e o r e t i c a l argument f o r using s t a t i s t i c a l r e g r e s s i o n models i s A l b e r t Church's, " An Econometric Model f o r A p p r a i s e r s " . 1 8 Church opened the d i s c u s s i o n of h i s model by d e r i v i n g the s t r u c t u r a l supply and demand f u n c t i o n f o r i n d i v i d u a l p r o p e r t i e s . The q u a n t i t y demanded i s a f u n c t i o n of p r i c e , P, and a set of c h a r a c t e r i s t i c s , X, that possess value to the buyer: Qd^ = f (P^, X^) i=1...n the number of p r o p e r t i e s The q u a n t i t y s u p p l i e d i s a f u n c t i o n of p r i c e , P, and a set of c h a r a c t e r i s t i c s , Y, that are valued by the s e l l e r : Q S i = g (p., y.) A f t e r having d e r i v e d the supply and demand f u n c t i o n , Church presented the methodology f o r determining market value. He c o n s i d e r e d the supply and demand f u n c t i o n to be d i s c o n t i n u o u s , s i n c e a pro p e r t y i s e i t h e r s o l d or not s o l d and s i n c e the p r i c e may not be uni q u e l y determined by the supply and demand f u n c t i o n . He says there i s a range of c o i n c i d e n c e between the supply and demand f u n c t i o n s where the buyer and s e l l e r bargain on p r i c e . Because of the c o i n c i d e n c e of the supply and demand f u n c t i o n s when a property i s s o l d , the model can only determine the expected value of the s e l l i n g "price [E(Pj, ,X^ , Y^ ) ] ,given a set of c h a r a c t e r i s t i c s f o r the buyer and s e l l e r . The a c t u a l p r i c e f o r the property i s a f u n c t i o n of the 27 expected s e l l i n g p r i c e and a random v a r i a b l e , N ^ . The random v a r i a b l e denotes the b a r g a i n i n g range of the buyer and s e l l e r . The "most probable s e l l i n g p r i c e " f o r a p r o p e r t y not s o l d can be i n f e r r e d from a property which i s s o l d d u r i n g the time i n t e r v a l and which possesses i d e n t i c a l c h a r a c t e r i s t i c s and i d e n t i c a l supply and demand f u n c t i o n s . T h e r e f o r e Church assumed that s a l e s data c o u l d be used to determine the expected or probable s a l e s p r i c e f o r a l l p r o p e r t i e s c l a s s i f i e d by type of c h a r a c t e r i s t i c . From t h i s assumption that the supply and demand f u n c t i o n holds f o r a l l p r o p e r t i e s , Church s i m p l i f i e d the model. The new equation reduced to i t s simplest form i s : P. = e(X.,Y.,N.) 1 1 ' 1 ' 1 where p r i c e equals the f u n c t i o n , e ,which c o n t a i n s the c h a r a c t e r i s t i c s important to the buyer and s e l l e r and the random v a r i a b l e . I t i s t h i s f u n c t i o n , e ,which should be employed i n r e g r e s s i o n a n a l y s i s . In the r e g r e s s i o n a n a l y s i s the value of Ni i s assumed to be equal to z e r o . Church concluded h i s a r t i c l e by p o i n t i n g out a number of problems that a r i s e when a p p l y i n g the model i n r e g r e s s i o n a n a l y s i s . The f i r s t problem i s that l i n e a r l e a s t - s q u a r e s r e g r e s s i o n r e q u i r e s the s p e c i f i e d equation to be l i n e a r i n c o e f f i c i e n t . To accomplish t h i s the f u n c t i o n , e , i s l i n e a r i z e d for m observable c h a r a c t e r i s t i c s : 28 P. a + + m im + A m+1 + N. 1 o 1 i = l i •'• n f o r p r o p e r t i e s m f o r the c h a r a c t e r i s t i c s i s a constant . • • • where: Aj i s the l i n e a r c o e f f i c i e n t to be estimated from data on p r operty s a l e s ; Z^j i s the s p e c i f i c c h a r a c t e r i s t i c s or combination of c h a r a c t e r i s t i c s f o r p r o p e r t i e s ( d e r i v e d from the X^,Y^); and i s the random term. The second problem i s the s e l e c t i o n of c h a r a c t e r i s t i c s d e r i v e d from X^, to be i n c l u d e d in the equation. Church reasoned that a t t r i b u t e s which v a r i e d from p r o p e r t y to p r o p e r t y and which e x p l a i n e d s a l e s p r i c e d i f f e r e n c e s should be i n c l u d e d . C h a r a c t e r i s t i c s which were s i m i l a r between p r o p e r t i e s need not be i n c l u d e d . He c a t e g o r i z e d the v a r i a b l e s that should be i n the equation: p h y s i c a l , l o c a t i o n a l , market, and p r i o r knowledge. The l a s t problem Church mentioned i s the i n t e r a c t i o n e f f e c t of the c h a r a c t e r i s t i c s . I n t e r a c t i o n occurs when a j o i n t occurrence of two or more v a r i a b l e s ( c h a r a c t e r i s t i c s ) produces an e f f e c t which i s d i f f e r e n t from the i n d i v i d u a l occurrences of two separate events. For example a den adds X d o l l a r s to a house and a f i r e p l a c e adds Y d o l l a r s ; together t h e i r worth i s g r e a t e r than or l e s s than X and Y. 29 A f i n a l a r t i c l e , which has been of great b e n e f i t to t h i s work, i s an e m p i r i c a l study of q u e s t i o n one: Can an i n v e s t o r d i v e r s i f y w i t h i n a r e a l e s t a t e market? Only one study has examined d i v e r s i f i c a t i o n with regard to r e a l e s t a t e p o r t f o l i o s ; i t was performed i n 1980 by M i l e s and McCue[44]. M i l e s and McCue conducted t h e i r study on a l a r g e commingled r e a l e s t a t e fund with over 300 p r o p e r t i e s , d i s p e r s e d across the United S t a t e s and c o n t a i n i n g f i v e d i f f e r e n t p r o p e r t y types. The m a j o r i t y of p r o p e r t i e s were o f f i c e b u i l d i n g s , and i n d u s t r i a l p r o p e r t i e s . The o b j e c t i v e of the study was to t e s t r e a l e s t a t e p o r t f o l i o s a g a i n s t the 1/n r u l e of McEnally and Boardman, where the expected average v a r i a n c e of the p o r t f o l i o equals the systematic r i s k p l u s 1/n unsystematic r i s k : V = V + l/n(V ) p s ' u M i l e s and McCue began t h e i r study by c a l c u l a t i n g q u a r t e r l y r e t u r n s on the sample of p r o p e r t i e s over a f i v e - y e a r p e r i o d . Just as we have done i n the present study, the authors had to estimate v a l u e . To do t h i s , M i l e s and McCue accepted the annual a p p r a i s e d value of the p r o p e r t i e s as market v a l u e . 1 9 To determine the q u a r t e r l y value of the p r o p e r t i e s , they s e l e c t e d two methods: the f i r s t g e o m e t r i c a l l y smoothed the changes i n value over the intermediate q u a r t e r s ; the second assumed that p r i c e d i d not change from q u a r t e r to q u a r t e r , but only on an annual b a s i s . Since the authors u t i l i z e d two methods to estimate v a l u e , they needed two r e t u r n measures(both were on a 30 before tax b a s i s ) . Summaries of the re t u r n s and v a r i a n c e s f o r the sample are shown i n Tables 2.1 and 2.2. 2 0 The r e s u l t s from Table 2.2 show that p o r t f o l i o s i z e does have an e f f e c t on the r e d u c t i o n of r e t u r n v a r i a t i o n . These r e s u l t s are c o n s i s t e n t f o r each pr o p e r t y type. When M i l e s and McCue d i v i d e d the sample i n t o four geographic regions, the r e s u l t s were s t i l l the same. Return v a r i a t i o n decreased s u b s t a n t i a l l y with p o r t f o l i o s i z e . M i l e s and McCue conducted one more experiment. They compared the average t o t a l v a r i a n c e to the market r e l a t e d v a r i a n c e . Table 2.3 prese n t s the r e s u l t s . Except i n one case(unsmoothed r e t u r n s i n the West),the r a t i o of market r e l a t e d v a r i a n c e to average t o t a l v a r i a n c e i s below 15 percent. Thus the non-market r i s k of r e a l e s t a t e i s q u i t e high, demonstrating that p o t e n t i a l gains from d i v e r s i f i c a t i o n i n r e a l e s t a t e are q u i t e l a r g e . I t i s of p a r t i c u l a r i n t e r e s t to t h i s study that M i l e s and McCue repeated t h i s experiment f o r one pro p e r t y type, over each of the r e g i o n s . The hi g h e s t r a t i o of market v a r i a n c e to average t o t a l v a r i a n c e i n any region was 16 percent. T h i s r e s u l t suggests that the present study,though r e s t r i c t e d i n i t s f i n a l a n a l y s i s to one pro p e r t y type i n one l o c a l market, can s t i l l show the p o s s i b i l i t y of d i v e r s i f i c a t i o n . 31 TABLE 2.1 SUMMARY STATISTICS FOR PROPERTIES WITH 20 QUARTERS OF DATA BREAKDOWN BY TYPE - T o t a l Sample Industr i a l O f f i c e Other N 166 1 18 29 19 Unsmoothed Returns .0386 .0393 .0402 .0319 Smoothed Returns .0364 .0370 .0382 .0303 Variance Unsmoothed Returns .0048 .0048 .0067 .0021 Varaince Smoothed Returns .0013 .0012 .0023 .001 1 Mean Beta 1 .0 .973 1 . 1 38 .938 - BREAKDOWN BY Region - T o t a l Sample East Midwest South West 166 1 3 78 42 33 Unsmoothed Returns .0386 .0449 0340 .0335 .0535 Smoothed Returns .0364 .0422 0326 .0321 .0488 Variance Unsmoother Returns .0048 .0063 0034 .0034 .0092 Variance Smoothed Returns .001 3 .0032 0010 .001 3 .001 6 Mean Beta 1 .0 1.713 91 83 .61 76 1 .399 Source: M i l e s and McCue[44] 32 TABLE 2.2 DESCRIPTION OF PORTFOLIO SIZE AND REDUCTION IN RETURN VARIANCE BY PROPERTY TYPE (MEAN OF VARIANCE x 10 ) - Smoothed Returns - T o t a l Sample - Unsmoothed Returns T o t a l Sample A l l P r o p e r t i e s I n d i v i d u a l l y 12.739 Random P o r t f o l i o s of P r o p e r t i e s : 2 P r o p e r t i e s 8.647 4 P r o p e r t i e s 3.942 6 P r o p e r t i e s 2.713 8 P r o p e r t i e s 1.900 10 P r o p e r t i e s 1.999 12 P r o p e r t i e s 1.659 14 P r o p e r t i e s 1.432 16 P r o p e r t i e s 1.332 18 P r o p e r t i e s 1.297 20 P r o p e r t i e s 1.182 30 P r o p e r t i e s 1.042 A l l P r o p e r t i e s .627 48.670 23.359 15.433 12.084 .10.529 7.985 7.690 7.051 6.400 6.398 6.815 5.771 4. 1 77 Source: M i l e s and McCue[44] 33 TABLE 2.3 Vpl V p a l l R a t i o NON-DIVERSIFIABLE RETURN AS A PROPORTION OF TOTAL RISK By Type Smoothed Returns T o t a l I n d u s t r i a l O f f i c e 12.739 11.098 21.596 .627 .674 2.571 .049 .061 .119 Other 10.953 1 . 1 94 .109 Vpl V p a l l Rat i o Unsmoothed Returns 48.760 49.211 63.594 4.166 5.945 6.708 .086 .121 .105 22.532 2.171 .096 Vp1 V p a l l Rat i o T o t a l Sample 12.739 .627 .049 By Region Smoothed Returns East Midwest South West 31.849 9.096 12.193 14.516 3.494 .764 .677 -1.815 .110 .084 .056 .125 Vp1 V p a l l Rat i o Unsmoothed Returns 48.670 64.209 35.015 4.177 9.223 8.057 .086 .144 .230 33.921 2.312 .068 93.593 25.419 .271 Source: M i l e s and McCue[44] 34 ENDNOTES 1. Markowitz, Harry M., " P o r t f o l i o S e l e c t i o n " , J o u r n a l of Finance, Vol.12, March 1952, pp.77-91 2. i b i d 3. i b i d 4. i b i d 5. Sharpe, W i l l i a m F., "A S i m p l i f i e d Model f o r P o r t f o l i o A n a l y s i s " , Management Science, Vol.9, January 1963, pp.277- 293 6. Evans, John L. and Archer, Stephen N., " D i v e r s i f i c a t i o n and the Reduction of Dispersion:An E m p i r i c a l A n a l y s i s " , J o u r n a l of Finance, December 1968, pp.761-767 7. F i s h e r , I r v i n g , The Theory of I n t e r e s t , M a c M i l l i a n P u b l i s h e r s , New York, 1930 8. Fama, Eugene F., and Schwert, W i l l i a m G., "Asset Returns and I n f l a t i o n " , J o u r n a l of F i n a n c i a l Economics, June 1977, pp.115-146 9. Friedman, H a r r i s C , "Real E s t a t e Investment and P o r t f o l i o Theory", J o u r n a l of F i n a n c i a l and Q u a n t i t a t i v e A n a l y s i s , A p r i l 1970 10. Sharpe, W i l l i a m F., "A S i m p l i f i e d Model f o r P o r t f o l i o A n a l y s i s " , Management Science, Vol.9, January 1963,pp.277- 293 11. Cohen,K.J., and Pogue,J.A., "An E m p i r i c a l E v a l u a t i o n of A l t e r n a t i v e P o r t f o l i o S e l e c t i o n Models", J o u r n a l of Business, Vol.40, A p r i l 1967, pp.166-196 12. Friedman, H a r r i s C., "Real E s t a t e Investment and P o r t f o l i o Theory", J o u r n a l of F i n a n c i a l and Q u a n t i t a t i v e A n a l y s i s , A p r i l 1970 13. F i n d l a y III , Chapman M., Hamilton, C a r l W., Messner Stephen, D., and Yormark, Jonathan S., "Optimal Real E s t a t e P o r t f o l i o s " , J o u r n a l of American Real E s t a t e and Urban Economics A s s o c i a t i o n , Vol.7, No.3, F a l l 1979, pp.298-317 14. i b i d 15. Hoag ,James W., "Towards I n d i c e s of Real E s t a t e Value and Return", J o u r n a l of Finance, May 1980 16. Shenkel, W i l l i a m M., "The V a l u a t i o n of M u l t i p l e Family Dwellings by S t a t i s t i c a l I n f e r ence", The Real E s t a t e A p p r a i s e r , January-Febuary 1975, pp.25-36 17. i b i d 18. Church, A l b e r t M., "An Econometric Model f o r A p p r a i s i n g " , American Real E s t a t e and Urban Economics A s s o c i a t i o n J o u r n a l , Vol.3, No.1, Spring 1975, pp.17-29 19. M i l e s , Mike and McCue, Tom, " C o n s i d e r a t i o n s i n Real E s t a t e P o r t f o l i o D i v e r s i f i c a t i o n " , Working Paper, U n i v e r s i t y of North C a r o l i n a , 1980 20. The unsmooth r e t u r n s and v a r i a t i o n s assume no p r i c e change durin g a year. The smooth r e t u r n s are g e o m e t r i c a l l y compounded on a q u a r t e r l y basis.The t o t a l sample i s 166 p r o p e r t i e s , s i n c e the authors only had complete data on these p r o p e r t i e s f o r the f i v e year study. 35 3.0 DATA BASE The p r e v i o u s chapter e x p l a i n e d , i n some d e t a i l , the l i t e r a t u r e which has provided a p l a t f o r m f o r t h i s work. T h i s chapter e x p l a i n s the data base u t i l i z e d i n the paper. The data c o n s i s t of a set of apartment p r o p e r t i e s l o c a t e d i n Vancouver, B r i t i s h Columbia, and a set of r e t u r n s from a number of other investment instruments which are r e q u i r e d to answer q u e s t i o n two of the paper. In order to d i s c u s s the data base, the chapter d i v i d e s i n t o three s e c t i o n s . The f i r s t p resents an overview of the apartment market in Vancouver, so as to f a m i l i a r i z e the reader with t h i s market and to h e l p him b e t t e r understand the r e s u l t s of the paper. S e c t i o n 3.2 d e s c r i b e s the sample of apartment p r o p e r t i e s and t h e i r c h a r a c t e r i s t i c s along with the assumptions and e s t i m a t i n g procedures necessary to complete the i n f o r m a t i o n on the p r o p e r t i e s . The chapter concludes with a p r e s e n t a t i o n of the other investment instruments, and e x p l a i n s the methods used f o r c a l c u l a t i n g the r a t e s of r e t u r n s f o r t h i s group of a s s e t s . 3.1 ,THE APARTMENT MARKET IN VANCOUVER The apartment market in Vancouver p r i m a r i l y developed over a ten-year span from 1961-1971. During t h i s p e r i o d , the t o t a l number of apartment u n i t s i n Vancouver almost t r i p l e d , the major c o n c e n t r a t i o n of growth o c c u r r i n g i n the high d e n s i t y zoned area 36 of the West End. 1 The i n c r e a s e was s t i m u l a t e d by a str o n g demand f o r r e n t a l u n i t s , the r e s u l t of the coming of age of the post - war baby boom g e n e r a t i o n . The members of t h i s g e n e r a t i o n were young, and with a good economic c l i m a t e were able to form new households. The types of housing they sought were r e n t a l apartments. A f t e r t h i s ten-year p e r i o d of expansion, c o n s t r u c t i o n of new apartment u n i t s slowed c o n s i d e r a b l y . The supply of r e n t a l u n i t s even s l i p p e d s l i g h t l y over the next nine years, 1971- I979(see Table 3.1). The f a c t o r s f o r t h i s turn-around can be a s s o c i a t e d with the c o n s i d e r a b l e change i n c o n d i t i o n s on the supply s i d e of the market. The supply s i d e had s t a r t e d to encounter c o n s t r a i n t s unfamilar to the i n d u s t r y , c o n s t r a i n t s that began to appear i n 1970 when mortgage r a t e s reached double d i g i t s . Developers, b e l i e v i n g that the high c o s t of c a p i t a l was short term, decided to wait on the s i d e l i n e s u n t i l r a t e s decreased. However, when mortgage r a t e s d i d not recede, these developers looked f o r other investment o p p o r t u n i t i e s i n r e a l e s t a t e . They switched to the condominium market, an a t t r a c t i v e investment s i n c e the pay-back p e r i o d f o r condominiums was s h o r t ( u n t i l the condominuium u n i t s were s o l d o f f ) , while the pay-back p e r i o d f o r r e n t a l apartments extended over a much longer p e r i o d of time. In Vancouver, condominiums s t a r t s represented 90 percent of a l l m u l t i - u n i t s t a r t s i n the sevent i e s . 2 Another c o n s t r a i n t that a f f e c t e d supply was the change i n 37 TABLE 3.1 VANCOUVER APARTMENT DATA CITY OF VANCOUVER YEAR BUILDINGS SUITES VACANCY RATES 1971 2,135 51 ,128 2. 1 1972 - - 1973 1,983 49,930 0.2 1974 - 1975 1,969 48,899 0.1 1976 - 1977 1,973 49,077 1.0 1978 - 1979 1,955 50,982 0.2 Note: Data l i m i t e d to privately-owned r e n t a l u n i t s i n apartment b u i l d i n g s c o n t a i n i n g s i x or more u n i t s Source: Canada Mortgage and Housing C o r p o r a t i o n 38 the f e d e r a l tax laws. E f f e c t i v e January 1, 1972 a l o s s c r e a t e d by c a p i t a l c o s t allowance on the r e n t a l of r e a l p r o p e r t y c o u l d no longer be a p p l i e d to n o n - r e n t a l income. In a d d i t i o n , the r e v i s e d law d i s c o n t i n u e d the p o o l i n g of r e a l e s t a t e a s s e t s , so that a d i f f e r e n t pool had to be c r e a t e d f o r each b u i l d i n g over $50,000. The e f f e c t of these tax law changes to i n v e s t o r s who were l o o k i n g f o r tax s h e l t e r b e n e f i t s was to discourage them from i n v e s t i n g i n the apartment market. The f i n a l c o n s t r a i n t impeding new c o n s t r u c t i o n was the combined e f f e c t of high i n f l a t i o n and rent c o n t r o l s imposed by the p r o v i n c i a l government. High i n f l a t i o n was a new phenomenon in the s e v e n t i e s , and f o r c e d the c o s t of c o n s t r u c t i o n to soar as land, labor and m a t e r i a l c o s t s a l l rose. To recover the higher c o s t s , developers began to charge higher r e n t s . But as the rents began to r i s e , r e n t e r s c r i e d out to the government to stop the higher c o s t of l i v i n g . So, i n 1974, the Province of B r i t i s h Columbia e s t a b l i s h e d c o n t r o l s over rent i n c r e a s e s f o r e x i s t i n g apartment b u i l d i n g s . The f o l l o w i n g l i m i t s were i n e f f e c t d u r i n g the time p e r i o d of the study: January 1, 1974 - December 31, 1974 8.0 percent/year January 1, 1975 - A p r i l 30, 1977 10.6 percent/year May 1, 1977 - June 30, 1980 7.0 percent/year The rent c o n t r o l s imposed on e x i s t i n g b u i l d i n g s kept p r e v a i l i n g market r e n t s low, making i t d i f f i c u l t f o r new apartment b u i l d i n g s to compete. The r e n t s that developers c o u l d r e c e i v e on new apartment b u i l d i n g s were too low for them to recover 39 t h e i r c o s t s , with the r e s u l t that developers r e f r a i n e d from p a r t i c i p a t i n g i n the market. The Canadian Government t r i e d to step i n to s t i m u l a t e c o n s t r u c t i o n a c t i v i t y i n the m u l t i - f a m i l y housing market. The same supply c o n s t r a i n t s that were a f f e c t i n g the c o n s t r u c t i o n a c t i v i t y i n Vancouver were a f f e c t i n g c i t i e s throughout Canada. The F e d e r a l Government decided to i n i t i a t e two s u p p l y - s i d e programs: one was s t a r t e d i n 1974 and the other i n 1976. In 1974, the government developed the MURB Program, a program that t r i e d to r e t u r n p r i v a t e c a p i t a l to the apartment market by once again p e r m i t t i n g c a p i t a l c ost allowance to be a p p l i e d to non- r e n t a l income f o r a l l new c o n s t r u c t i o n a f t e r January 1, 1974. The program the government i n i t i a t e d i n 1976 was the ARP(Assisted Rental Program). T h i s program encouraged developers to c o n s t r u c t moderately p r i c e d r e n t a l housing by g i v i n g them i n t e r e s t - f r e e loans f o r 10-15 years with a maximum l i m i t on the loa n s . U n f o r t u n a t e l y , as seen i n Table 3.1, Vancouver d i d not have an i n c r e a s e i n r e n t a l u n i t s , suggesting that n e i t h e r of the two programs f u l l y a chieved the e x p e c t a t i o n s of the government. 3.2 APARTMENT BLOCK SAMPLE I'n •'• s t a t i s t i c a l terms, the un i v e r s e which t h i s sample i s drawn from i s a l l the apartment blocks l o c a t e d i n the c i t y of Vancouver and b u i l t before I970(the s t a r t i n g time of the s t u d y ) . From t h i s u n i v e r s e , those apartment b l o c k s s o l d d u r i n g 1979 and 40 1980 were s e l e c t e d as the sampling base. The d e c i s i o n to sample apartment b l o c k s that were s o l d d u r i n g 1979 and 1980 was due to the a v a i l a b i l i t y of i n f o r m a t i o n ( p r o v i d e d by the B r i t i s h Columbia Assessment A u t h o r i t y ) . A l s o , the two years of s a l e s , 1979 and 1980, c o i n c i d e d with the time p e r i o d which the data were c o l l e c t e d . The number of apartment blocks s o l d d u r i n g the two year p e r i o d t o t a l e d 347. The sample base of 347 p r o p e r t i e s was reduced i n s i z e by e l i m i n a t i n g apartment blocks which l a c k e d s u f f i c i e n t i n f o r m a t i o n for the study. The study r e q u i r e d s a l e s t r a n s a c t i o n s and income, debt and p h y s i c a l c h a r a c t e r i s t c s of the p r o p e r t i e s . In the e l i m i n a t i o n process 87 p r o p e r t i e s were dropped from the sample base, to produce a f i n a l sample c o n s i s t i n g of 260 apartment b l o c k s . Of the 87 p r o p e r t i e s e l i m i n a t e d , 58 were thrown out f o r lack of income information,17 were dropped due to the u n a v a i l a b i l i t y of e i t h e r debt or t r a n s a c t i o n information,and 12 were d i s c a r d e d because of m i s s i n g p h y s i c a l c h a r a c t e r i s t i c s . General s t a t i s t i c s f o r the f i n a l sample appear i n Table 3.2. Even though the f i n a l sample contained apartment blocks with a l l the r e q u i r e d i n f o r m a t i o n , c e r t a i n assumptions and e s t i m a t i n g procedures s t i l l had to be c a r r i e d out to c a l c u l a t e q u a r t e r l y r e t u r n s . Assumptions and estimates were r e q u i r e d on the income, o p e r a t i n g expenses, debt,and q u a r t e r l y market values for the p r o p e r t i e s . A l l of the 260 apartment b l o c k s i n the f i n a l sample had some income i n f o r m a t i o n , but very few had income f i g u r e s f o r a l l 41 TABLE 3.2 SUMMARY STATISTICSS FOR THE APARTMENT BLOCK SAMPLE CHARACTERISTICS MEAN STANDARD DEVIATION Number of S u i t e s 19.71 1 5.08 Gross F l o o r Area(square f e e t ) 13,542.78 9978.77 Average S u i t e S i z e ( s q u a r e f e e t ) 715.72 219.58 Age(as of 1983) 37. 19 21 .02 Number of S t o r i e s 3.10 1 .89 Lot S i z e ( s q u a r e f e e t ) 8,547.08 4061.50 Number of P r o p e r t i e s / A r e a West End 73 Ki t s i l a n o 29 K e r r i s d a l e 4 Marpole 23 South G r a n v i l l e 43 East Hastings 75 Rest of the C i t y 1 3 42 ten years of the s t u d y . 3 E s t i m a t i n g procedures were t h e r e f o r e necessary to f i l l i n the years when in f o r m a t i o n on income was m i s s i n g on the p r o p e r t i e s . The primary method f o r e s t i m a t i o n was i n t e r p o l a t i o n . If a p r o p e r t y had no more than three c o n s e c u t i v e years of m i s s i n g income, then the compound growth ra t e was a p p l i e d over the intermediate p e r i o d . For almost a l l the p r o p e r t i e s , t h i s procedure was employed over some p o r t i o n of the ten-year p e r i o d . In cases where the spread between income years was g r e a t e r than three years, e x t r a p o l a t i o n was u t i l i z e d . Two methods were used in e x t r a p o l a t i n g income, depending on the time p e r i o d . The f i r s t method, a p p l i e d to the time p e r i o d 197 0- 1973, e x t r a p o l a t e d by means of a y e a r l y growth r a t e model, based on the average rent f o r a given area of the c i t y . The c i t y was d i v i d e d i n t o seven areas; from each area the average rents f o r s t u d i o , one bedroom and two bedrooms s u i t e s was found." 5 The average rent f o r the three d i f f e r e n t types of s u i t e s was then weighted by the p r o p o r t i o n of that s u i t e type to the t o t a l number of s u i t e s i n the area to d e r i v e an o v e r a l l average rent for each area. Table 3.3 presents the growth r a t e s f o r the v a r i o u s a r e a s . The second method of e x t r a p o l a t i o n , a p p l i e d d u r i n g the time p e r i o d 1974-1979, used the maximum a l l o w a b l e rent i n c r e a s e s p e r m i t t e d under the rent c o n t r o l s of B r i t i s h Columbia(see S e c t i o n 3.1). Since the r e n t a l market was extremely t i g h t at the time, we assume that l a n d l o r d s would have i n c r e a s e d r e n t s by the maximum amount granted by law. Our assumption seemed 43 TABLE 3.3 AN AVERAGE RENT INDEX FOR VANCOUVER BY AREA YEAR WEST END KITSILANO KERRISDALE 1 970 100.00 100.00 100.00 1971 109.74 100.00 1 03.89 1 972 112.16 101.50 111.12 1 973 119.35 113.04 1 20.65 1 974 125.65 122.56 127.99 YEAR SOUTH GRANVILLE EAST HASTINGS MARPOLE REST OF THE CITY 1 970 100.00 100.00 100.00 100.00 1 97 1 105.09 106.59 1 10.39 106.96 1 972 110.77 110.17 116.51 109.91 1 973 117.93 117.07 123.75 118.68 1974 126.21 129.05 135.91 130.59 44 j u s t i f i e d by the r e s u l t s of the i n t e r p o l a t i o n computations made fo r t h i s same time p e r i o d , which showed that the compound growth r a t e s i n rent s were very s i m i l a r to the maximum allowed rent i n c r e a s e s . To determine the o p e r a t i n g expenses f o r the p r o p e r t i e s , the s t a t i s t i c a l technique of m u l t i p l e r e g r e s s i o n was used.- Since the p r o p e r t i e s themselves d i d not have s u f f i c i e n t o p e r a t i n g expense i n f o r m a t i o n to run the r e g r e s s i o n a n a l y s i s , the paper made use of the a n a l y s i s performed by Gau. 6 Table 3.4 provi d e s a complete d e s c r i p t i o n of the r e s u l t s . The reader should note that the e s t i m a t i o n i s an expense r a t i o ( o p e r a t i n g expenses to gross income) and not an a c t u a l estimate of o p e r a t i n g expenses. By l o o k i n g at the t a b l e , the reader can see that the only p h y s i c a l c h a r a c t e r i s t i c that has a p o s i t i v e s i g n i s age. T h i s i m p l i e s that o l d e r b u i l d i n g s r e s u l t i n higher o p e r a t i n g expenses. The other two p h y s i c a l c h a r a c t e r i s t i c s , number of s t o r i e s and gross f l o o r area, have negative s i g n s i n d i c a t i n g economies of s c a l e . The complete debt background on the p r o p e r t i e s was gathered from the B r i t i s h Columbia Land T i t l e O f f i c e . Assumptions were r e q u i r e d to determine what debt on the p r o p e r t i e s was property s p e c i f i c . Since r e a l e s t a t e i s an asset which i s o f t e n used as c o l l a t e r a l , the p r o p e r t i e s contained many debt o b l i g a t i o n s which were not prop e r t y s p e c i f i c . The a d d i t i o n a l debt on the p r o p e r t i e s c o u l d have been f o r the purpose of f i n a n c i n g other investments or f o r pe r s o n a l needs, and as such the leverage on 45 TABLE 3.4 APARTMENT OER EQUATION AOER = 47.992 (16.058)* .297 AGE (6.768)* 511 LOC1 - 2.282 LOC2 - (.295) - .582 D68 (.226) -2.317 D72 (.949) -1.883 D76 ( .832) (1.299) + .857 D69 (.414) - 3.500 D73 (1.689) + .086 D77 (.029) .194 STOR (1 .894)* 1 .666 LOC3 ( .988) 1.740 D70 (.869) 2.768 D74 (1.296) 3.884 D78 (1.599) .008 GFA (2.616)* 4.605 LOC4 (2.018)* .291 D71 (2.018) 1.759 D75 (.771 ) 1.177 D79 (.500) R .302 SE = 6.694 n = 263 t - s t a t i s t i c in parentheses * = c o e f f i c i e n t s i g n i f i c a n t at .05 l e v e l AOER = o p e r a t i n g expense r a t i o of apartment p r o p e r t i e s (X100) AGE = age i n years of apartment b u i l d i n g STOR = number of s t o r i e s of b u i l d i n g GFA = average gross f l o o r area per s u i t e i n square feet LOC1...LOC4 = dummy, 0-1 v a r i a b l e f o r s p e c i f i c g e o g r a p h i c a l l o c a t i o n s D68...D78 = dummy, 0-1 v a r i a b l e f o r year of r a t i o from 1968 to 1979, Source: Gau[23] 46 the p r o p e r t i e s was o f t e n o v e r s t a t e d . Two assumptions were employed to l i m i t the debt s o l e l y to property s p e c i f i c debt: (1) Debt c o u l d not be g r e a t e r than the value of the property at the time of purchase. (2) Debt o b l i g a t i o n s r e l e a s e d and not r e f i n a n c e d were not c o n s i d e r e d p r o p e r t y s p e c i f i c u n l e s s the r e l e a s e d o b l i g a t i o n o c c u r r e d at the time of a s a l e s t r a n s a c t i o n . Under the f i r s t assumption, we b e l i e v e that l e n d e r s would have been u n w i l l i n g to lend funds g r e a t e r than the worth of the p r o p e r t y ; t h e r e f o r e the l o a n - t o - v a l u e r a t i o had to be l e s s than one at time of purchase. Under the second assumption, we reason that funds from other investments must have r e t i r e d the debt o b l i g a t i o n , suggesting that the f i n a n c i n g must have i n i t i a l l y been used f o r these other investments too. The e s t i m a t i n g procedure to determine the q u a r t e r l y market valu e s of the p r o p e r t i e s i s d i s c u s s e d i n Chapter 5. 3.3 OTHER INVESTMENT ASSETS AND THEIR RATE OF RETURNS The s e l e c t i o n of investment instruments chosen f o r the study i n c l u d e s a s s e t s of the kind most l i k e l y to be i n c o r p o r a t e d i n t o an i n v e s t o r ' s p o r t f o l i o . Each of these a s s e t s , which are l i s t e d below, can be c o n s i d e r e d to have a d i f f e r e n t investment o b j e c t i v e f o r the i n v e s t o r , i . e . f i x e d income, growth p o t e n t i a l , hedge a g a i n s t i n f l a t i o n : (1) COMMON STOCK - The t o t a l r e t u r n index of the Toronto Stock Exchange 300 r e p r e s e n t s t h i s a s s e t . 47 (2) GOVERNMENT OF CANADA TREASURY BILLS - The 91-day t r e a s u r y b i l l s s o l d by the government represent t h i s a s s e t . The y i e l d on the T - B i l l was used as the rate of r e t u r n . (3) LONG-TERM GOVERNMENT BONDS - The t o t a l r a t e of r e t u r n on long-term government bonds was c a l c u l a t e d f o r the paper. 7 (4) Gold - The r e t u r n on go l d i s measured by the q u a r t e r l y p r i c e change. The source of in f o r m a t i o n was the I n t e r n a t i o n a l Monetary Fund. The consumer p r i c e index f o r Canada,as s u p p l i e d by the Bank of Canada Review, i s the measure used f o r i n f l a t i o n . 48 ENDNOTES 1. M i t c h e l l , E . C . , " T h e Apartment Rental Market i n M e t r o p o l i t a n Vancouver", Real E s t a t e Trends i n M e t r o p o l i t a n Vancouver,197 6,pp.B-1 2. M i t c h e l l , E . C . , " M u l t i p l e Housing A c t i v i t y i n Metropoltan Vancouver:Quo V a d i s ? " , Real E s t a t e Trends i n M e t r o p o l i t a n Vancouver,1977,pp.B-1 3. There were two sources from which income was c o l l e c t e d : T h e B.C. Assessment A u t h o r i t y , and The Greater Vancouver Real E s t a t e Board M u l t i p l e L i s t i n g S e r v i c e . 4. The seven areas are: 1.West End 2 . K i t s i l a n o 3 . K e r r i s d a l e 4.Marpole 5.South G r a n v i l l e 6.East Hastings 7.Remaining areas of c i t y 5. The source f o r the average rent was Real E s t a t e Trends i n M e t r o p o l i t a n Vancouver,1970- 1 979 6. Gau,George W.,"Determinants of Return i n Real E s t a t e Investment and the Role of Real E s t a t e Management", I n s t i t u t e of Real E s t a t e Management Foundation,July 1981,pp.1-46 7. The t o t a l r a t e of r e t u r n was c a l c u l a t e d as f o l l o w s : Pt+1 + J b + Z t - P t P t where: P&t+1 i s the bond p r i c e at the end of the q u a r t e r ; I k i s the i n t e r e s t p a i d on the bond f o r the p e r i o d ; I f c i s the i n t e r e s t c o l l e c t e d from r e i n v e s t i n g the bond coupons at the T - B i l l r a t e ; and P. i s the bond p r i c e at the beginning of the q u a r t e r . 49 4 . 0 PROCEDURES Chapter 3 d i s c u s s e d the data chosen to t e s t d i v e r s i f i c a t i o n ; t h i s chapter presents the methodology r e q u i r e d to answer the two qu e s t i o n s posed at the outset of t h i s paper. It begins by d e l i n e a t i n g r e t u r n and r i s k : the two parameters used to measure the performance of the apartment p r o p e r t i e s as w e l l as of the randomly s e l e c t e d p o r t f o l i o s . Next, the chapter d e s c r i b e s the procedures used to examine d i v e r s i f i c a t i o n w i t h i n r e a l e s t a t e . L a s t l y , the chapter presents the methods used to c a l c u l a t e e f f i c i e n t p o r t f o l i o s , mean-variance and i n f l a t i o n hedged p o r t f o l i o s . 4. 1 RETURN AND RISK In v e s t o r s i n s e l e c t i n g or ranking a l t e r n a t i v e investment c h o i c e s e v a l u a t e these investment c h o i c e s by t h e i r expected r e t u r n and v a r i a n c e of r e t u r n ( r i s k ) . 1 The most a p p r o p r i a t e way to c h a r a c t e r i z e t h i s expected r e t u r n i s i n terms of a p r o b a b i l i t y d i s t r i b u t i o n . T e s t s have shown that the p r o b a b i l i t y d i s t r i b u t i o n s of r e t u r n s on investments(common stock) are normally or logn o r m a l l y d i s t r i b u t e d . 2 Since they are d i s t r i b u t e d in t h i s manner, i n v e s t o r s can d i s t i n g u i s h them from one another by two parameters: mean or expected r e t u r n , and the standard d e v i a t i o n ( t h e squared d e v i a t i o n i s the v a r i a n c e ) . The standard d e v i a t i o n or v a r i a n c e measures the d i s p e r s i o n of the p r o b a b i l i t y 50 d i s t r i b u t i o n around the mean(expected r e t u r n ) . These measures of d i s p e r s i o n d i s c l o s e the r i s k i n e s s of an investment. For the purpose of the study, r e t u r n s on the apartment p r o p e r t i e s are not expected r e t u r n s but r e a l i z e d ( ex post ) r e t u r n s . The study looks h i s t o r i c a l l y at these p r o p e r t i e s to examine the d i v e r s i f i c a t i o n p o t e n t i a l of r e a l e s t a t e . Two measures of r e t u r n are c a l c u l a t e d . The f i r s t , o f t e n r e f e r r e d to as r e t u r n on c a p i t a l , i s c a l c u l a t e d as f o l l o w s : Equation 1. R i f c = f ( M V i t + 1 + C i f c) - MV i f c] MV. . i t where: R^t i s the q u a r t e r l y h o l d i n g p e r i o d r e t u r n of the i prope r t y i n p e r i o d t ; MV\ t + 1 i s the ending market value estimate; i s the net cash flow d u r i n g the p e r i o d t ; and MV^t i s the beginning market value e s t i m a t e . T h i s r e t u r n measure i s c a l c u l a t e d f o r each of the 260 apartment p r o p e r t i e s i n the study. The other r e t u r n measure computed for each pr o p e r t y i s the re t u r n on.equity, which takes i n t o .consideration any f i n a n c i n g a p p l i e d to the pr o p e r t y . The re t u r n on eq u i t y i s determined .as f o l l o w s : 51 Equation 2. R.t = HMV i t + 1 - D i t + , ) + C ^ ) - (MV.t - D . t ) ] ( M V i t - ° i t ) where R-,, MV. . , C , , MV.. are the same as i n Equation 1, i t I t+1 I t i t M ^it+1 * S t * i e ^ e D t o u t s t a n d i n g at the end of the p e r i o d and i s the debt outstanding at the beginning of the p e r i o d . Equation 2. can be f u r t h e r s i m p l i f i e d : Equation 3. Rj = [ ( B T E R i t + C i f c) - Eo] Eo where BTER^ and Eo are the before-tax e q u i t y r e v e r s i o n of pro p e r t y i at the end of p e r i o d t and the i n i t i a l e q u i t y r e s p e c t i v e l y . In both cases, the r e t u r n measures are before tax. Using a before - t a x r a t e of r e t u r n r a i s e s the q u e s t i o n of whether these r e t u r n measures have any relevance for i n v e s t o r s , who are u s u a l l y more concerned with an a f t e r - t a x r a t e of r e t u r n . A befor e - t a x r e t u r n f a c i l i t a t e s the comparison of the r e a l e s t a t e r e t u r n s with the r e t u r n s of the other investment instruments(which are c a l c u l a t e d on a be f o r e - t a x b a s i s ) . However, the reader can argue that the r e l a t i o n s h i p between r e a l e s t a t e r e t u r n s and those of other a s s e t s might be one t h i n g on a bef o r e - t a x b a s i s and q u i t e another on an a f t e r - t a x b a s i s , because of the tax s h e l t e r b e n e f i t s a s s o c i a t e d with r e a l e s t a t e , i . e . the b e n e f i t s from c a p i t a l cost allowances. Gau[23] using 52 almost the same data base as t h i s paper found that the tax s h e l t e r b e n e f i t s were not a major determinant of the r e t u r n . 3 He noted that the lack of r e l a t i v e importance of the tax s h e l t e r was due to the high l a n d - t o - t o t a l - v a l u e r a t i o of the p r o p e r t i e s . T h e r e f o r e using b e f o r e - t a x r a t e of r e t u r n measures should not p r e j u d i c e the r e s u l t s of the a n a l y s i s . Another i s s u e should be c l a r i f i e d . Often two r e t u r n measures e x i s t f o r r e a l e s t a t e , r e t u r n on c a p i t a l and r e t u r n on e q u i t y , while only one i s used f o r the other a s s e t s , r e t u r n on c a p i t a l . The r e t u r n on e q u i t y measure i s i n c l u d e d f o r r e a l e s t a t e , because of the importance of leverage to a r e a l e s t a t e i n v e s t o r . Since r e a l e s t a t e i s a lumpy and an i n d i v i s i b l e a s s e t , small c a p i t a l i n v e s t o r s o f t e n must o b t a i n f i n a n c i n g i n order to purchase r e a l e s t a t e . The r e a l e s t a t e i n v e s t o r i s concerned not only with the r e t u r n on the p r o p e r t y , but a l s o with how h i s e q u i t y r e t u r n i s a f f e c t e d by l e v e r a g e . With other investments, f i n a n c i n g i s not as c r i t i c a l ; i n v e s t o r s can u s u a l l y a c q u i r e e q u i t i e s without the need of l e v e r a g e . In t h i s study, the r e t u r n on e q u i t y w i l l not be compared to the r e t u r n on c a p i t a l of the other investments, but i s i n c l u d e d i n order to provide r e a l e s t a t e i n v e s t o r s and r e s e a r c h e r s with i n f o r m a t i o n on how f i n a n c i n g a f f e c t s the r e t u r n on c a p i t a l . Given these c o n d i t i o n s , the b e f o r e - t a x r e t u r n on the market and on randomly s e l e c t e d p o r t f o l i o s can be c a l c u l a t e d . T h e r e t u r n on the market i n c l u d e s a l l p r o p e r t i e s i n the sample, and i s c a l c u l a t e d as f o l l o w s : 53 K Equation 4. R = I R-^/K M m i t ' i= 1 where: R i s the re t u r n on the market at time t ; m R^t i s the r e t u r n of the i t h pr o p e r t y at time t ; and K i s the number of p r o p e r t i e s in the market. For the r e t u r n on the market, each property i s e q u a l l y weighted. The r e t u r n on a randomly s e l e c t e d p o r t f o l i o i s : M Equation 5. R = I R i t / M i = 1 where R t i s the r e t u r n on the p o r t f o l i o at time t , and M i s the number of p r o p e r t i e s i n the p o r t f o l i o . A f t e r c a l c u l a t i n g the d i f f e r e n t r e t u r n measures, the average q u a r t e r l y v a r i a n c e ( r i s k ) f o r each property i s determined. The v a r i a n c e f o r each pr o p e r t y can be computed as f o l l o w s : Equation 6. V i = (R i - R ^ ) 2 — - j where: V\ i s the v a r i a n c e f o r property i ; R. i s the mean q u a r t e r l y r e t u r n f o r the p r o p e r t y ; and 54 n i s the number of q u a r t e r s i n the study. With the v a r i a n c e f o r each property known, the average t o t a l v a r i a n c e f o r the r e a l e s t a t e market can be c a l c u l a t e d . " The average t o t a l v a r i a n c e r e p r e s e n t s the upper boundary for r i s k , systematic r i s k as w e l l as the unsystematic r i s k of r e a l e s t a t e . The average t o t a l v a r i a n c e i s computed as: K Equation 7. V = L V^/K i= 1 where V"t i s the average t o t a l v a r i a n c e and K i s the number of p r o p e r t i e s i n the market. Next the market v a r i a n c e i s c a l c u l a t e d f o r the t o t a l sample: Equation 8. V m = (R - R )2 ^ m mt m n- 1 where: V i s the v a r i a n c e of the market; m R . i s the r e t u r n on the market in p e r i o d t ; and mt R i s the mean r e t u r n of a l l p r o p e r t i e s in the market over m c c the p e r i o d of the study, n. V r e p r e s e n t s a completely d i v e r s i f i e d p o r t f o l i o and serves as a proxy fo r systematic r i s k . The d i f f e r e n c e between Vfc and V m 55 r e f l e c t s the unsystematic or d i v e r s i f i a b l e r i s k w i t h i n the market. The measure of v a r i a n c e i s a l s o r e q u i r e d f o r the randomly s e l e c t e d p o r t f o l i o s . The average q u a r t e r l y v a r i a n c e f o r these p o r t f o l i o s i s computed as f o l l o w s : Equation 9. V = (R - R ) 2 n-1 where V p i s the average q u a r t e r l y v a r i a n c e , Rp̂ _ and R p are the re t u r n of the p o r t f o l i o in p e r i o d t and the mean r e t u r n of the p o r t f o l i o r e s p e c t i v e l y . The v a r i a n c e f o r a p o r t f o l i o , V , l i k e P the average t o t a l v a r i a n c e of the market, can be decomposed i n t o s y s t e m a t i c ( n o n - d i v e r s i f i a b l e ) and unsystematic ( d i v e r s i f i a b l e ) r i s k : Equation 10. V = V + V M p s us where V s i s the systematic and V u g i s the unsystematic r i s k . 4.2 PROCEDURES TO TEST DIVERSIFICATION WITHIN REAL ESTATE Is there s u f f i c i e n t unsystematic r i s k w i t h i n r e a l e s t a t e to allow i n v e s t o r s to reduce r i s k by purchasing a c r o s s - s e c t i o n of p r o p e r t i e s ? We have approached t h i s q u e s t i o n by measuring the e f f e c t of p o r t f o l i o s i z e on r e t u r n v a r i a t i o n . I f r e t u r n 56 v a r i a t i o n i s reduced as a d d i t i o n a l p r o p e r t i e s are added, then the p o t e n t i a l to d i v e r s i f y w i t h i n r e a l e s t a t e e x i s t s . The exact method used to answer t h i s q u e s t i o n f o l l o w s a number of st e p s . F i r s t the r e t u r n and v a r i a n c e s for a l l the p r o p e r t i e s w i l l be c a l c u l a t e d . From t h i s set of p r o p e r t i e s ( w h i c h w i l l be termed the market), the re t u r n of the market and the r i s k of the m a r k e t ( t o t a l ( V . ) and market(V )) w i l l t m be computed. Next, on a p r e l i m i n a r y b a s i s , a comparison of market r i s k to t o t a l r i s k i s made, (V"m/V^) . T h i s comparison w i l l i n d i c a t e the extent to which r i s k can be d i v e r s i f i e d away. The lower the r a t i o of ^ m ^ t ^ ' fc^e 9 r e a t e r the p o s s i b i l i t y of d i v e r s i f i c a t i o n w i t h i n r e a l e s t a t e . We repeat the comparison by d i v i d i n g the sample i n t o two sub-samples by l o c a t i o n : one f o r p r o p e r t i e s l o c a t e d i n the West End, the urban s e c t i o n of the c i t y , and the other c o v e r i n g the o u t l y i n g p a r t s of the c i t y . T h i s t e s t w i l l check f o r geographic d i v e r s i f i c a t i o n w i t h i n the c i t y . The next step i n measuring d i v e r s i f i c a t i o n w i t h i n r e a l e s t a t e i s to generate random samples of p o r t f o l i o s from s i z e 2 to 30 p r o p e r t i e s . For each property size,30 random p o r t f o l i o s are c r e a t e d , so that a t o t a l of 870 p o r t f o l i o s are formed. By having 30 random p o r t f o l i o s f o r each p o r t f o l i o s i z e , the d i s t r i b u t i o n of re t u r n s and v a r i a n c e of r e t u r n s f o r each p o r t f o l i o s i z e should be normal. Therefore the mean r e t u r n and va r i a n c e f o r the d i f f e r e n t p o r t f o l i o s i z e s can be used in the 57 a n a l y s i s without great concern f o r o u t l i e r s or abnormal r e s u l t s . The set of mean r e t u r n v a r i a n c e s f o r the d i f f e r e n t p o r t f o l i o s i z e s w i l l f i r s t be perused to see i f the v a r i a n c e s are reduced as p o r t f o l i o s i z e i n c r e a s e s . If r e t u r n v a r i a n c e i s reduced, then t - t e s t s w i l l be employed to f i n d out at what p o r t f o l i o s i z e s i g n i f i c a n t r e d u c t i o n in v a r i a t i o n take p l a c e . F i n a l l y , a simple r e g r e s s i o n a n a l y s i s i s run to determine how much r e d u c t i o n i n v a r i a t i o n can be e x p l a i n e d by p o r t f o l i o s i z e . The r e g r e s s i o n equation i s : where Y equals the r e t u r n v a r i a n c e of the p o r t f o l i o and X i s the p o r t f o l i o s i z e . 5 The R 2 w i l l p r o v i d e the answer f o r how much re d u c t i o n in v a r i a t i o n i s e x p l a i n e d by p o r t f o l i o s i z e . To f i n d the e f f i c i e n t p o r t f o l i o s under c o n d i t i o n s of mean- v a r i a n c e , r e c a l l that under Markowitz's d e f i n i t i o n of mean- v a r i a n c e , e f f i c i e n t p o r t f o l i o s are the set of p o r t f o l i o s which o f f e r s the hig h e s t expected r e t u r n f o r a given v a r i a n c e . M a t h e m a t i c a l l y t h i s o b j e c t i v e f u n c t i o n i s w r i t t e n as: Y= a + b(l//x) 4.3 PROCEDURES TO CALCULATE EFFICIENT PORTFOLIOS Equat ion 11. maximize N I X. R. l i X N N I Z i=1 i=1j=1 where: 58 X^,X_j are the p r o p o r t i o n a l weights of the a s s e t s i n the p o r t f o l i o ; P w i s the r e t u r n on ass e t i ; o^j i s the c o v a r i a n c e between asse t i and j;and X i s a Lagrangian m u l t i p l i e r . N The f i r s t s e c t i o n ( ^L^ x i R i ^ o f t h e equation c a l c u l a t e s the N N hig h e s t p o s s i b l e r e t u r n ; t h e second s e c t i o n ( X Z Z i=1j=1 X^Xj ) c o n s t r a i n s the hi g h e s t r e t u r n by minimizing the v a r i a n c e of the p o r t f o l i o . Added to t h i s o b j e c t i v e f u n c t i o n i s the c o n s t r a i n t that the sum of the weights of the a s s e t s in the p o r t f o l i o equals one: Equation 12. N N N N maximize I X . R . - X Z Z X . X . a . . - M ( Z X.-1) 1 1 I J I J l i=1 i=1j=1 i=1 N where M i s another Lagragian m u l t i p l i e r and ( ,Z X^-1) i = 1 c o n s t r a i n s the p o r t f o l i o weights to one.To d e r i v e t h i s o b j e c t i v e f u n c t i o n , a computer program has been w r i t t e n ( s e e Appendix A). The design of the computer program permits the weights of the 59 a s s e t s to be n e g a t i v e , implying that the a s s e t s can be s o l d s h o r t . I f r e a l e s t a t e i s found to have a negative weight i n the p o r t f o l i o s , a c o n c l u s i o n can be drawn that r e a l e s t a t e does not c o n t r i b u t e to the e f f i c i e n c y of the p o r t f o l i o , s i n c e r e a l e s t a t e cannot be s o l d s h o r t . The procedure used to compute an i n f l a t i o n - h e d g e d p o r t f o l i o i s o r d i n a r y l e a s t squares r e g r e s s i o n a n a l y s i s . By r e g r e s s i n g i n f l a t i o n (the dependent v a r i a b l e ) a g a i n s t the v a r i o u s investment returns(independent v a r i a b l e s ) , a l i n e a r equation i s d e r i v e d which r e p l i c a t e s i n f l a t i o n . To c o n s t r a i n the p o r t f o l i o so that the sum of the weights of the a s s e t s equals one, the r e g r e s s i o n c o e f f i c i e n t s are added and each c o e f f i c i e n t i s then d i v i d e d by the sum of those c o e f f i c i e n t s . L i k e the mean- va r i a n c e p o r t f o l i o s , the i n f l a t i o n - h e d g e d p o r t f o l i o can have a s s e t s with negative weights. A l l a s s e t s that have a p o s i t i v e weight c o n t r i b u t e as a hedge a g a i n s t i n f l a t i o n . Those a s s e t s that have a negative weight should be s o l d short s i n c e they are not e f f e c t i v e hedges a g a i n s t i n f l a t i o n . 6 60 ENDNOTES 1. Markowitz, Harry M., " P o r t f o l i o S e l e c t i o n " , J o u r n a l of Finance, Vol.12, March 1952, pp.77-91 2. Fama, Eugene F., Foundations Of Finance, Basic Books Inc., 1 976 3. Gau, George W., "Determinants of Return i n Real E s t a t e Investment and the Role of Real E s t a t e Management", I n s t i t u t e of Real E s t a t e Management Foundation, 1981, pp.1- 46 4. M i l e s , Mike and McCue, Tom, " C o n s i d e r a t i o n s i n Real E s t a t e P o r t f o l i o D i v e r s i f i c a t i o n " ,Working Paper, U n i v e r s i t y of North C a r o l i n a , 1980 5. Latane, H. and Young, W., "Test of P o r t f o l i o B u i l d i n g Rules", J o u r n a l of Finance, Vol.24, September 1969, pp.595- 612 6. A c o r r e l a t i o n matrix of the a s s e t s and i n f l a t i o n can a l s o c o n f i r m which a s s e t s are hedges a g a i n s t i n f l a t i o n . 61 5.0 V a l u a t i o n Model In t h i s chapter a v a l u a t i o n model i s developed to estimate q u a r t e r l y market val u e s f o r the apartment p r o p e r t i e s . The f i r s t s e c t i o n of the chapter d e s c r i b e s the t h e o r e t i c a l s p e c i f i c a t i o n s of the model. Then S e c t i o n 5.2 presents the estimated r e g r e s s i o n equation f o r the apartment p r o p e r t i e s and c o n s i d e r s the e f f e c t i v e n e s s of the model. 5.1 THEORETICAL SPECIFICATION In the marketplace, the value of apartment blocks in Vancouver i s determined by the i n t e r a c t i o n of the supply and demand schedules. Since we need to estimate the value f o r these p r o p e r t i e s , we must d e r i v e t h e i r supply and demand schedules. To do t h i s , two assumptions are made: that a l l apartment blocks have the same supply and demand curves, 1 and that the market i s in e q u i l i b r i u m so that p r i c e i s determined where the q u a n t i t y demanded equals the q u a n t i t y s u p p l i e d . To examine the supply and demand curves, we f i r s t c o n s i d e r the apartment block market i n the long run. The supply and demand curves are n e i t h e r p e r f e c t l y e l a s t i c nor i n e l a s t i c ( s e e F i g u r e 5.1). The supply, the stock of apartment b l o c k s , can a d j u s t i n response to the demand. 62 \ / Supply \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ Demand / \ \ \ Supply \ \ \ \ \ \ \ \ \ \ Demand \ Q Figu r e 5.1 F i g u r e 5.2 The v a r i a b l e s that are important to the developers who provide the supply and- to the i n v e s t o r s who are the demand are c h a r a c t e r i z e d as f o l l o w s : Supply = f ( p r i c e of apartment b l o c k s , c o n s t r u c t i o n c o s t s , i n t e r e s t r a t e s , r e n t a l income, land p r i c e s , i n f l a t i o n , taxes, a v a i l a b l i t y of zoned s i t e s , i n c r e a s e i n non-family households, vacancy r a t e ) Demand = f ( p r i c e of apartment b l o c k s , f u t u r e r e n t s , r i s k premium of apartment investments, i n f l a t i o n , i n t e r e s t r a t e s , p o t e n t i a l of new supply, taxes, expected r e t u r n on a l t e r n a t i v e investment o p p o r t u n i t i e s ) If we reduce the time span to examine the apartment block market in the short run, the supply curve becomes i n e l a s t i c ( s e e F i g u r e 63 5.2). The time p e r i o d i s too short f o r any new stock to be added to the market; hence, market value i s p r i m a r i l y determined by the demand v a r i a b l e s . 2 I n v e s t o r s i n c o r p o r a t e the i n f o r m a t i o n from the supply and demand schedules i n t o mathematical models which analyze the investment. The models which i n v e s t o r s o f t e n use f o r a n a l y s i s are d i s c o u n t e d cash flow models. The most popular of these i s the net present value model, NPV, a model which e v a l u a t e s an investment through a comparison of the e q u i t y i n v e s t e d i n a property at the time of purchase(Eo) and the present value of the a f t e r - t a x e q u i t y cash f l o w s ( C t ) a c c r u i n g to the r e a l e s t a t e i n v e s t o r s d u r i n g the h o l d i n g p e r i o d ( t = 1 , ...n) di s c o u n t e d at the r e q u i r e d r a t e of r e t u r n ( r ) . 3 N Equation 1. NPV = Z Cfc -Eo i = 1 (1+r) f c The d e c i s i o n c r i t e r i o n i s to accept the r e a l e s t a t e investment i f NPV > 0 and r e j e c t i t i f NPV<0. From t h i s equation, we f i n d the value of a property by s e t t i n g the equation equal to i t s present v a l u e , PV, by adding Eo to each s i d e of the equ a t i o n . In t h i s new equation, the dis c o u n t e d f u t u r e b e n e f i t s equal the present value of the pr o p e r t y . N Equation 2. PV. = Z Cfc i = 1 . ( l + r ) f c 64 Shenkel has shown by use of m u l t i p l e r e g r e s s i o n a n a l y s i s that the f u t u r e b e n e f i t s of a pro p e r t y are r e l a t e d to a set of common prop e r t y c h a r a c t e r i s t i c s , and that these pr o p e r t y c h a r a c t e r i s t i c s can be used to p r e d i c t market v a l u e . " The set of pro p e r t y c h a r a c t e r i s t i c s that Shenkel used are grouped i n t o three c a t e g o r i e s : a r e a ( o r s i z e ) , l o c a t i o n , and s e r v i c e s and am e n i t i e s . Church, in h i s use of m u l t i p l e r e g r e s s i o n a n a l y s i s , argued that p r o p e r t y c h a r a c t e r i s t i c s are r e l a t e d to the supply and demand schedules. He c o n s i d e r e d as important those pr o p e r t y c h a r a c t e r i s t i c s that " e x p l a i n e d " s a l e s p r i c e d i f f e r e n c e s from pro p e r t y to p r o p e r t y , and c a t e g o r i z e d these c h a r a c t e r i s t i c s under p h y s i c a l , l o c a t i o n a l , market( economic and f i n a n c i a l ) and p r i o r knowledge c l a s s i f i c a t i o n s . T h i s paper f o l l o w s the work of Shenkel and Church by using l e a s t - s q u a r e s r e g r e s s i o n a n a l y s i s to e x p l a i n the value f o r the apartment p r o p e r t i e s . The v a r i a b l e s we judged to be p e r t i n e n t for t h i s study are l i s t e d below, together with reasons f o r s e l e c t i o n and some d e s c r i p t i v e d e t a i l : Market Value The market v a l u e ( s a l e s p r i c e ) of the apartment b l o c k s i s the dependent v a r i a b l e f o r the i n i t i a l runs i n the r e g r e s s i o n model. The market value i s taken as the a c t u a l s a l e s p r i c e of the p r o p e r t y as f i l e d with the B r i t i s h Columbia Land T i t l e O f f i c e . 65 Gross Income M u l t i p l i e r The gross income m u l t i p l i e r ( G I M ) i s the dependent v a r i a b l e for the f i n a l r e g r e s s i o n model. I t re p r e s e n t s the r e l a t i o n s h i p between the purchase p r i c e of the pr o p e r t y and i t s gross income. The GIM was used as a proxy f o r s a l e s p r i c e because the GIM model i n c r e a s e s the s i g n i f i c a n c e of many of the independent var i a b l e s . Gross Income The gross income f o r each property i s the f i r s t independent v a r i a b l e , and i s estimated from the procedure d e s c r i b e d i n Chapter 3, S e c t i o n 2. The gross income r e f l e c t s the present b e n e f i t s of the property and i n d i c a t e s the p o t e n t i a l f o r f u t u r e b e n e f i t s . The expected sig n of the v a r i a b l e i s neg a t i v e , because of the i n v e r s e r e l a t i o n s h i p between income and the GIM; as income i n c r e a s e s , the GIM decreases. A£e The age of the b u i l d i n g i s taken as the number of years from the year of c o n s t r u c t i o n to the year of v a l u a t i o n . The age v a r i a b l e r e l a t e s to the net o p e r a t i n g income as w e l l as to the r e v e r s i o n v a l u e . The expected sign i s n e g a t i v e . Gross F l o o r Area The gross f l o o r area(measured i n square f e e t ) i s a s i z e f a c t o r , and r e l a t e s to the p r e s e n t ( f u t u r e ) gross income and the 66 o p e r a t i n g expenses of a pr o p e r t y . The expected s i g n i s p o s i t i v e . F l o o r Area Per S u i t e The f l o o r area per s u i t e ( a l s o measured in square f e e t ) r e f l e c t s on average the type of s u i t e s i n the b u i l d i n g s . The f l o o r area per s u i t e r e l a t e s to income and o p e r a t i n g expenses. The expected s i g n f o r the f l o o r area per s u i t e i s p o s i t i v e . Number of S t o r i e s The number of s t o r i e s of the apartment block i s assumed to have an e f f e c t on gross income and o p e r a t i n g expenses. The expected s i g n i s p o s i t i v e . Lot S i z e The s i z e of the l o t i s c a l c u l a t e d i n square f e e t and i s assumed to have an e f f e c t on the r e v e r s i o n v a l u e . The expected sign i s p o s i t i v e . Locat ion The l o c a t i o n a l v a r i a b l e s are dummy v a r i a b l e s and four are inc l u d e d i n the model. The areas f o r the dummy v a r i a b l e s a r e : (l)West End, ( 2 ) K i t s i l a n o , (3)South G r a n v i l l e , (4)East S i d e . 5 Dummy v a r i a b l e s were used f o r l o c a t i o n , to attempt to p i c k up the d i f f e r e n t f a c t o r s r e l a t i n g to l o c a t i o n , i . e . p r o x i m i t y to downtown, vacancy r a t e s , d e s i r a b i l i t y of area, e t c . The 67 expected s i g n s f o r the West End, K i t s l a n o , and South G r a n v i l l e are p o s i t i v e . The expected s i g n f o r the East Side i s e i t h e r p o s i t i v e or n e g a t i v e . Q u a r t e r l y Dummy V a r i a b l e s The q u a r t e r l y dummy v a r i a b l e s are used to capture the change i n economic c o n d i t i o n s as w e l l as s h i f t s i n the supply and demand cur v e s . The q u a r t e r l y dummy v a r i a b l e s are a l s o i n c l u d e d i n the r e g r e s s i o n equation to determine the q u a r t e r l y p r i c e changes of the p r o p e r t i e s . The expected s i g n w i l l vary over the time p e r i o d . There are no f i n a n c i a l v a r i a b l e s i n c l u d e d i n the r e g r e s s i o n model. Church suggested that a v a r i a b l e that r e f l e c t s the du r a t i o n of the debt on the prop e r t y or the i n t e r e s t r a t e weighted by the s i z e of the remaining p r i n c i p a l of each mortgage should be i n c l u d e d . 6 These v a r i a b l e s were excluded because of data which was unable to be processed. 5.2 DEVELOPMENT AND ANALYSIS OF THE REGRESSION MODEL The f i r s t step i n the development of the r e g r e s s i o n model was to see i f the model c o u l d be separated i n t o annual equations. Table 5.1 presents the t o t a l number of s a l e s t r a n s a c t i o n s by year. The r u l e of thumb f o r e s t i m a t i n g our model on an annual b a s i s i s that the number of 68 o b s e r v a t i o n s / e q u a t i o n be g r e a t e r than the degrees of freedom. The t a b l e shows that there were enough o b s e r v a t i o n s ( s a l e s t r a n s a c t i o n s ) f o r each year to permit annual estimated r e g r e s s i o n e q u a t i o n s . These annual equations were u s e f u l , because, as Shenkel p o i n t e d out by l i m i t i n g the estimated equation to a short p e r i o d of time, the i n f l u e n c e of time on the equation i s reduced and the accuracy of the model i s i n c r e a s e d . As a next step, a summary of s t a t i s t i c s was generated on the v a r i a b l e s . The s t a t i s t i c s were u s e f u l in a n a l y z i n g the model, and i n i n s u r i n g that the model conformed to the assumptions of r e g r e s s i o n a n a l y s i s . One assumption the model needed to conform to was that there be l i n e a r i t y i n the c o e f f i c i e n t s of the independent v a r i a b l e s . 7 8 To i n c r e a s e the l i k e l i h o o d t h a t the model s a t i s f i e d t h i s assumption, the d i s t r i b u t i o n s of the v a r i a b l e s were examined f o r n o r m a l i t y , by a s s e s s i n g the skewness of the d i s t r i b u t i o n s on an annual b a s i s . The d i s t r i b u t i o n s f o r the p r i c e , GIM, number of s t o r i e s , f l o o r area per s u i t e and l o t s i z e were a l l p o s i t i v e l y skewed. To normalize these d i s t r i b u t i o n s , l o g r a r i t h m i c t r a n s f o r m a t i o n s were a p p l i e d . The l o g t r a n s f o r m a t i o n s c o n s t r i c t e d the i n t e r v a l s of the data as the values i n c r e a s e d i n s i z e . The consequences to the d i s t r i b u t i o n s were that the r i g h t t a i l was drawn i n while the values of the l e f t t a i l of the d i s t r i b u t i o n were moved away from the mean, thus tending to normalize the d i s t r i b u t i o n s . 9 A f t e r t r a n s f o r m i n g the v a r i a b l e s , a s e r i e s of r e g r e s s i o n equations were run to i d e n t i f y the best p o s s i b l e model: we TABLE 5.1 NUMBER OF SALES TRANSACTIONS PER YEAR YEAR NUMBER OF TRANSACTIONS 1 970 28 1971 35 1 972 39 1 973 56 1974 31 1 975 34 1 976 37 1 977 44 1 978 1 02 1 979 1 32 70 needed a model that minimized p r e d i c t i v e e r r o r and a model that i n c l u d e d enough v a r i a b l e s so as to d i s t i n g u i s h p r i c e d i f f e r e n c e s from property to p r o p e r t y . The f i r s t run of the model had the l o g of the p r i c e as the dependent v a r i a b l e , with the v a r i a b l e s d e c r i b e d i n S e c t i o n 5.1 as the independent v a r i a b l e s . The f i r s t run produced q u i t e s u r p r i s i n g l y good r e s u l t s with the c o e f f i c e n t of d e t e r m i n a t i o n fo r each of the ten annual r e g r e s s i o n equations above .90. The standard e r r o r of the estimate ranged from .10 to .28. These r e s u l t s were s u p e r i o r to those obtained by Hoag[28] but not as strong as Shenkel's r e s u l t s . The problem with t h i s i n i t a l r e g r e s s i o n model was that only one e xplanatory v a r i a b l e , the log of income, was s i g n i f i c a n t f o r a l l ten e quations. Even though the estimated equations achieved the o b j e c t i v e of a strong p r e d i c t i v e model with minimal e r r o r , the model d i d not c o n t a i n enough s i g n i f i c a n t v a r i a b l e s to e x p l a i n p r i c e d i f f e r e n c e s between the p r o p e r t i e s . By having only income as a s i g n i f i c a n t v a r i a b l e , p r i c e would be e s s e n t i a l l y estimated by simple r e g r e s s i o n , a method r i g h t f u l l y c r i t i z e d f o r i t s l a c k of a c c u r a c y . 1 0 Moreover, by only having a s i n g l e v a r i a b l e to p r e d i c t p r i c e , the c o r r e l a t i o n s of the p r o p e r t i e s would be so s t r o n g l y p o s i t i v e that there would be l i t t l e p o . s s i b l i t y of f i n d i n g p o t e n t i a l d i v e r s i f i c a t i o n w i t h i n r e a l e s t a t e . So s i n c e t h i s model f a i l e d to achieve the o b j e c t i v e of i n c l u d i n g p r o p e r t y c h a r a c t e r i s t i c s that vary from p r o p e r t y to p r o p e r t y and that e x p l a i n s a l e s p r i c e d i f f e r e n c e s , 71 the model was d i s c a r d e d . The model was then a l t e r e d by dropping income as an explanatory v a r i a b l e . T h i s procedure had been t r i e d by Shenkel with great s u c c e s s . 1 1 In the c u r r e n t study, the r e s u l t s from the r e g r e s s i o n run were a l s o q u i t e reasonable. The c o e f f i c i e n t of d e t e r m i n a t i o n f o r the ten estimated equations ranged from .70 to .96. The standard e r r o r s of the estimate were higher than in the f i r s t run, ranging from .11 to .34. t h i s second run a l s o brought out the s i g n i f i c a n c e of many of the independent v a r i a b l e s . As a r e s u l t , t h i s model was adequate f o r use in the study; i t had p r e d i c t i v e power and c o u l d e x p l a i n s a l e s p r i c e d i f f e r e n c e s of the study. Even though t h i s model was s a t i s f a c t o r y , another approach was taken to assure that i t was the a p p r o p r i a t e model. The new approach s u b s t i t u t e d GIM f o r market value as the dependent v a r i a b l e . With GIM as the dependent v a r i a b l e , a r e g r e s s i o n was run l e a v i n g the log of income out as an explanatory v a r i a b l e . The r e s u l t s from t h i s run were poor, with the c o e f f i c i e n t of de t e r m i n a t i o n f o r the ten equations ranging from .17 to .68. Most of the independent v a r i a b l e s were not s i g n i f i c a n t . The only good s t a t i s t i c was that the standard e r r o r of the estimate was low, from .11 to .22. Another run was attempted, keeping the GIM as the dependent v a r i a b l e , but i n t h i s equation the l o g of income was i n c l u d e d as an independent v a r i a b l e . By i n c l u d i n g income, the s i g n i f i c a n c e of the other independent v a r i a b l e s i n c r e a s e d . The t - v a l u e s f o r 72 these v a r i a b l e s were l a r g e r in t h i s equation than i n the three p r e v i o u s equations. The c o e f f i c e n t s of det e r m i n a t i o n were mixed, v a r y i n g from .404 to .761, but the standard e r r o r s of the estimates were q u i t e good, ranging from .10 to .21. To compare the p r e d i c t i v e accuracy of t h i s model to the model with market value as the dependent v a r i a b l e , the average r e s i d u a l e r r o r , i n ab s o l u t e terms, was c a l c u l a t e d . Table 5.2 r e v e a l s that the average p r e d i c t i v e e r r o r was lower f o r nine of the ten annual equations with the GIM model. The average r e s i d u a l e r r o r was 11.3 percent in the GIM model as compared to 15.8 percent i n the market value model. As a r e s u l t , s i n c e the GIM model appeared the s t r o n g e s t p r e d i c t i v e model with minimal e r r o r and i t had more s i g n i f i c a n t p roperty c h a r a c t e r i s t i c s to e x p l a i n s a l e s p r i c e d i f f e r e n c e s , t h i s model was used to p r e d i c t market v a l u e . The ten annual equations of the model appear i n Table 5.3. An attempt was made to keep only those v a r i a b l e s which had a t - value g r e a t e r than 1.0, so as to minimize the standard e r r o r . The q u a r t e r l y dummy v a r i a b l e s were an exc e p t i o n ; these v a r i a b l e s were always kept i n the equation even i f the t - v a l u e s were below 1.0. Since a l l the other independent v a r i a b l e s were constant throughout the year, the q u a r t e r l y dummy v a r i a b l e s were needed to c a l c u l a t e the change i n value on a q u a r t e r l y b a s i s . The t - values of many of these variables(DM2, DM3, DM4) were low, implying that f o r many p e r i o d s of time the change i n value was not s i g n i f i c a n t . As a r e s u l t of keeping i n a l l the dummy 73 TABLE 5.2 THE AVERAGE PREDICTIVE ERROR FOR THE GIM AND THE MARKET VALUE MODEL (BY PERCENTAGE) YEAR GIM MODEL MARKET VALUE MODEL 1970 1 1 .25 17.10 1 971 7.54 9.67 1 972 11.41 17.72 1973 12.16 1 9.68 1 974 23.73 1 0.93 1975 7.34 12.10 1 976 13.58 1 3.94 1977 8.87 1 3.92 1 978 7 .44 23.76 1 979 10.04 19.39 AVERAGE 11.33 15.79 TABLE 5 3 1970 THE ANNUAL VALUATION EQUATIONS GIM = ( - - . 6 4 4 + . 0 3 7 L I N C - . 0 0 5 A G E + . 1 5 4 L 0 C 1 . 5 9 1 ) ( . 4 9 3 ) ( - 2 . 3 ) ( 1 . 6 ) + .184L0C5+.120LOCG ( 1 .S) ( 1 . 2 1 + . 3 4 5 L F A S T ( 3 . 0 ) R 2 = . 6 0 3 S . E . = . 1 8 4 0 4 F=3 .042 0BS=28 - . 1 4 1 D M 2 - . 1 5 7 D M 3 - ( - 1 . 1 ) ( - 1 . 2 ) (• 048DM4 . 35 1 ) 1971 GIM = 5 . 6 4 5 - . 4 7 6 L I N C - . O O 7 A G E + . 6 3 3 L 0 C 1 + . 6 17L0C2+.585L0C5+. '58OLOC6+.405E -O4FLAR ( 6 . 0 ) ( - 4 . 2 ) ( - 4 . 4 ) ( 3 . 1 ) ( 3 . 3 ) ( 3 . 0 ) ( 3 . 2 ) ( 3 . 6 ) ^ .0780M2+.155DM3+.062DM4 ( 1 . 1 ) ( 2 . 0 ! ( . 7 2 5 1 R2=.634 S . E . 12099 F = 4 . 1 5 9 0BS=35 1972 GIM =1.4 1 5 + . 0 0 7 L I N C - . 0 0 4 A G E + . 1 3 G L 0 C 1 + . 1 9 2 L 0 C 2 ( 1 . 3 ) ( . 0 7 1 ) ( - 2 . 3 ) ( 1 . 0 ) ( 1 . 3 ) •.204L0C6 ( 1 .9 ) + . 2 9 4 L F A S T ( 1 . 9 ) - . 1 6 5 L L 0 T - . 0 6 1 D M 2 - 022DM3- .111DM4 ( - 1 . 2 ) ( - . 5 9 4 ) ( - . 2 7 0 ) ( - 1 . 3 ) R 2 = . 3 8 2 S . E . = . 1 8 1 6 8 F = 1 . 5 2 0 0BS=39 1973 GIM = 4 . 185- - . 2 5 2 L I N C - . 007AGE+. 097L0C 1 + . 223L0C2+. 1 19L0C5 ( 4 . 6 ) ( - 2 . 7 ) ( - 4 . 4 ) ( 1 . 1 ) ( 2 . 2 ) ( 1 . 0 ) + . 1 1 4 E - 0 4 F L A R ( 2 . 3 ) >• . 0 3 0 L L 0 T - . 0 0 2 D M 2 - .097DM3+ .0620M4 ( 1 . 1 ) ( - . 0 2 1 ) ( - 1 . 2 ) ( . 7 2 6 ) R2=.444 S . E . = . 2 0 8 6 7 F = 3 . 5 8 9 0BS=56 1974 G I M = 2 . 4 3 9 - . 1 3 7 L I N C - ( 2 . 5 ) ( - 2 . 3 ) .006AGE ( - 4 . 0 ) • . 1 O 5 L 0 C 5 - . 15OL0C6 ( - 1 . 1 ) ( - 1 . 8 ) .19GLFAST ( 2 . 3 ) - . 0 4 2 D M 2 - . 0 8 5 D M 3 - . 1 B 0 D M 4 ( - . 6 7 4 ) ( - . 8 3 4 ) ( - 2 . 2 ) R 2 = . 7 1 0 S . E . = . 1 4 0 5 8 F = 5 . 7 0 6 0BS=31 1975 G I M = 4 . 1 9 4 - . 3 7 1 L I N C - . 0 0 4 A G E + . 1 3 O L 0 C 1 + 258L0C2+ 0 6 7 L 0 C 5 ( 6 . 6 ) ( - 5 . 7 ) ( - 3 . 8 ) ( 2 . 4 ) ( 3 . 2 ) ( 1 . 1 ) + . 8 8 6 E - 0 5 F L A R - ( 2 . 4 ) 185LL0T- ( 2 . 7 ) 020DM2< ( - 295) 007DM3+.056DM4 ( . 1 1 2 ) ( . 8 4 2 ) R 2 = 7 6 1 S . E . = . 1 0 3 0 2 F = 7 . 3 2 6 0BS=34 1976 GIM = 4 . 4 7 7 - . 2 2 8 L I N C - . 0 0 4 A G E ( 6 . 3 ) ( - 3 . 1 ) ( - 3 . 1 ) 1 5 9 L O C 6 * . 8 1 3 E - 0 5 F L A R ( - 2 . 8 ) ( 1 . 7 ) - 026DM2* 0 0 3 0 M 3 - . 1 3 2 D M 4 ( - . 4 0 0 ) ( . 0 4 0 ) ( - . 1 . 9 ) R 2 = . 5 6 5 S . E . = . 1 4 5 7 4 F ' 5 . 3 8 8 0BS'37 1977 G I M = 5 . 0 6 6 - . 2 8 1 L I N C - . 0 0 5 A G E + . 0 5 9 L O C 1 + . 0 8 1 L 0 C 2 + . 0 8 4 L 0 C 5 - . 112L0C6+. 1 1 8 E - 0 4 F L A R - . 1 2 9 D M 2 - . 0 8 5 0 M 3 - . 2 1 1 D M 4 ( 6 . 4 ) ( - 3 . 5 ) ( - 4 . 0 ) ( 1 . 0 ) ( 1 . 1 ) ( 1 . 4 ) ( - 1 . 9 ) . ( 2 . 0 1 ( - 1 . 9 6 ) ( - 1 . 3 ) ( - 3 . 2 ) R 2 = . 5 9 2 S . E . = . 1 2 7 2 0 F=4 .784 0BS=44 1978 G I M = 3 . 3 1 8 - . 1 4 6 L I N C - . 0 0 4 A G E + . 0 3 1 L 0 C 1 + . 1 2 9 L 0 C 2 + . 0 9 6 L 0 C 5 ( 1 3 . 2 ) ( - 6 . 0 ) ( - 7 . 2 ) ( 1 . 0 ) ( 3 . 7 ) ( 3 . 3 ) R2 = .543 S . E . 10056 F = 1 0 . 8 4 0 0BS=102 • 0 3 1 L F A S T + . 0 9 8 L N O S T ( 2 . 1 ) ( 3 . 0 ) - .06 1DM2- .015DM3+.049DM4 ( - 2 . 0 ) ( - . 4 8 0 ( 1 . 8 ) 1979 G I M = 2 . 8 4 8 - . 1 9 7 L I N C - . 0 0 2 A G E + . 1 0 7 L 0 C 1 + . 0 7 4 L O C 2 + . O 8 3 L 0 C 5 - . O 4 4 L 0 C 6 • . 130LN05T+. 1 3 4 L L 0 T - . 0 0 3 D M 2 + . 0 0 7 D M 3 + . 0 5 3 D M 4 ( 1 0 . 0 ) ( - 4 . 6 ) ( - 3 . 0 ) ( 2 . 6 ) ( 1 . 6 ) ( 2 . 0 ) ( - 1 . 3 ) ( 3 . 0 ) ( 2 . 4 ) ( - . 0 7 7 ) ( . 2 0 2 ) ( 1 . 4 ) R 2 = 3 0 4 S . E . = . 1 2 8 3 7 F = 4 . 7 8 3 0BS=132 D e f i n i t i o n s o f V a r i a b l e s T - S t a t i s t i c i n P a r e n t h e s e s GIM - G r o s s Income M u l t i p l i e r LINC - Log o f g r o s s income AGE - Age o f A p a r t m e n t B l o c k LOCI - West End L0C2 - K i t s i l a n o L0C5 - S o u t h G r a n v i l l e L0C6 - E a s t S i d e o f V a n c o u v e r F l a r - G r o s s F l o o r A r e a LFAST - L o g o f F l o o r A r e a / S u i t e LNOST - L o g o f t h e Number of S t o r i e s LLOT - Log of t h e L o t S i z e DM2 - E c o n o m i c V a r i a b l e f o r 2nd O u a r t o r DM3 - E c o n o m i c V a r i a b l e f o r 3 r d Q u a r t e r DM4 - E c o n o m i c V a r i a b l e f o r 4 t h Q u a r t e r 76 v a r i a b l e s , the v a r i a b l i t y of value may be o v e r s t a t e d , making the v a r i a n c e of r e t u r n of the p r o p e r t i e s o v e r s t a t e d . Looking again at Table 5.3, we see that most of the s i g n s f o r the v a r i a b l e s were c o n s i s t e n t with the expected s i g n s . Income and age had negative s i g n s and gross f l o o r area, f l o o r area per s u i t e , number of s t o r i e s , l o t s i z e and the l o c a t i o n a l v a r i a b l e s were p o s i t i v e . There were two equations, 1970 and 1972, where the sig n s f o r income, l o t s i z e , and the dummy v a r i a b l e f o r the East Side were the reverse of t h e i r s i g n s i n other equations. These reverse s i g n s along with the high standard e r r o r s of the estimate i n the equations suggest that these equations maybe the weakest of the ten. In terms of problems that are a s s o c i a t e d with r e g r e s s i o n a n a l y s i s : m u l t i c o l i n e a r i t y , h e t e r o s c e d a s t i c i t y , and o u t l i e r s , the equations showed l i t t l e evidence of t h e i r e f f e c t s . With respect to m u l t i c o l i n e a r i t y the c o r r e l a t i o n m a t r i c e s ( s e e Appendix B) i l l u s t r a t e that the v a r i a b l e s a s s o c i a t e d with s i z e ( l o g of l o t s i z e , gross f l o o r area, and l o g of the number of s t o r i e s ) had a high c o r r e l a t i o n with income. The hi g h c o r r e l a t i o n s , though, d i d not a l t e r any of the expected s i g n s . A l s o , the standard e r r o r s of the c o e f f i c i e n t s f o r these v a r i a b l e s were not s i g n i f i c a n t l y g r e a t e r than the standard e r r o r s of the other v a r i a b l e s . A p o s s i b l e reason that m u l t i c o l i n e a r i t y d i d not have an impact i s that o f t e n only one of the v a r i a b l e s r e f l e c t i n g s i z e appeared i n an equation with income at a time. 77 In checking f o r h e t e r o s c e d a s c t i c i t y , the r e s i d u a l e r r o r s were p l o t t e d versus the p r e d i c t e d values f o r GIM(see Appendix B). The r e s u l t s show there to be some h e t e r o s c e d a s t i c i t y . However, the standard e r r o r s f o r the equations are low enough that the equations can t o l e r a t e some overstatement of the r e l i a b i l i t y because of h e t e r o s c e d a s t i c i t y . The l a s t problem to check f o r i s o u t l i e r s . O u t l i e r s e x i s t when a r e s i d u a l i s extremely l a r g e ( p o s i t i v e or negative) compared with other r e s i d u a l s . There were some o u t l i e r s i n the equations. T r i a l runs were made throwing out these o b s e r v a t i o n s , but there were no d i f f e r e n c e s i n the r e s u l t s . Hence a l l o b s e r v a t i o n s were kept i n the study. In c o n c l u s i o n the weakness of using t h i s model i s that i t employs q u a r t e r l y dummy v a r i a b l e s to determine the q u a r t e r l y p r i c e changes. As a r e s u l t , a l l p r o p e r t i e s i n c r e a s e in value by the same percentage, making the c o r r e l a t i o n s between the p r o p e r t i e s 100 percent, from q u a r t e r to qu a r t e r and h i n d e r i n g the t e s t to f i n d d i v e r s i f i c a t i o n . On the whole, the model to p r e d i c t market value i s reasonable. On average, the p r e d i c t i v e e r r o r i s 11 per c e n t . A l s o , the model a l s o c o n t a i n s enough v a r i a b l e s to e x p l a i n s a l e s p r i c e d i f f e r e n c e s . 78 ENDNOTES 1. Church, A l b e r t M., "An Econometric Model f o r A p p r a i s i n g " , American Real E s t a t e and Urban Economics A s s o c i a t i o n J o u r n a l , Vol.3, No.1, Spring 1975, pp.17-29 2. Grether, D. and Mieskowski, P., "Determinants of Real E s t a t e Values", J o u r n a l of Urban Economics, A p r i l 1974, pp.47-52 3. Gau, George W., "Risk A n a l y s i s and Real E s t a t e Investment: T h e o r e t i c a l and Me t h o d o l o g i c a l Issues", Working Paper, U n i v e r s i t y of B r i t i s h Columbia, 1982 4. Shenkel, W i l l i a m M., "The V a l u a t i o n of M u l t i p l e Family Dwellings by S t a t i s t i c a l I n f e r e n c e " , The Real E s t a t e A p p r a i s e r , January-Febuary 1975, pp.25-36 5. For an area to be i n c l u d e d as a dummy v a r i a b l e , at l e a s t 10 percent of the sample to be l o c a t e d i n that a r e a . 6. Church, A l b e r t M., "An Econometric Model f o r A p p r a i s i n g " , American Real E s t a t e and Urban Economics A s s o c i a t i o n J o u r n a l , Vol.3, No.1, Spr i n g 1975, pp.25-36 7. In l e a s t - s q u a r e s r e g r e s s i o n , the most e f f i c i e n t estimator of a c o e f f i c i e n t i s a l i n e a r l e a s t squares e s t i m a t o r . 8. Rummel, R.J., A p p l i e d F a c t o r A n a l y s i s , Evanston: Northwestern Press, 1970 9. i b i d 10. Shenkel, W i l l i a m M., "The V a l u a t i o n of M u l t i p l e Family Dwellings by S t a t i s t i c a l I n f e r ence", The Real E s t a t e A p p r a i s e r , January-Febuary 1975, pp.25-36 11. i b i d 79 6.0 RESULTS Th i s chapter presents the e m p i r i c a l r e s u l t s of the study and analyzes the two q u e s t i o n s proposed in the i n t r o d u c t i o n of the paper. The chapter begins with a d e s c r i p t i o n of the r a t e s of r e t u r n s , the standard d e v i a t i o n s , and the v a r i a n c e s f o r the set of apartment p r o p e r t i e s . In S e c t i o n 6.2, we present the a n a l y s i s of the answer to q u e s t i o n one: can i n v e s t o r s d i v e r s i f y t h e i r p o r t f o l i o s s o l e l y w i t h i n r e a l e s t a t e market? L a s t l y i n S e c t i o n 6.3, we frame our response to q u e s t i o n two: can r e a l e s t a t e improve the e f f i c i e n c y of i n v e s t o r ' s p o r t f o l i o s ? 6.1 RETURN AND RISK MEASURES OF APARTMENT BLOCKS In the l a s t chapter, a v a l u a t i o n model was developed to estimate market v a l u e . Using the p r e d i c t e d s a l e s p r i c e s from the model and the cash flow i n f o r m a t i o n d e s c r i b e d i n Chapter 3, r a t e s of r e t u r n s were c a l c u l a t e d on the apartment p r o p e r t i e s . These r a t e s of r e t u r n s are set out i n Appendix C. Most of the p r o p e r t i e s e x h i b i t a mean r e t u r n on c a p i t a l of between 4 and 6 p e r c e n t / q u a r t e r . The r e t u r n s on e q u i t y are more d i s p e r s e d , with a number of p r o p e r t i e s having a negative mean r e t u r n . G e n e r a l l y , though, most p r o p e r t i e s have a p o s i t i v e r e t u r n on e q u i t y which i s g r e a t e r than the r e t u r n on c a p i t a l . These higher r e t u r n s on e q u i t y i l l u s t r a t e the b e n e f i t s of leverage to an i n v e s t o r . The standard d e v i a t i o n s and v a r i a n c e s ( t h e measures of r i s k ) , are much more d i s p e r s e d f o r the r e t u r n s on e q u i t y , as 80 compared to the r e t u r n s on c a p i t a l . . The m a j o r i t y of p r o p e r t i e s have a standard d e v i a t i o n f o r the r e t u r n on c a p i t a l that f a l l w i t h i n a range of 11.00 percent to 15.00 p e r c e n t / q u a r t e r , and v a r i a n c e of 1.50 percent to 3.25 p e r c e n t / q u a r t e r . In respect to the standard d e v i a t i o n and v a r i a n c e f o r the r e t u r n on e q u i t y , no such d e f i n e d range e x i s t s . The vast d i s p e r s i o n of the standard d e v i a t i o n and v a r i a n c e f o r the r e t u r n s on e q u i t y demonstrates the high r i s k f a c t o r of le v e r a g e . Table 6.1 d i s p l a y s the mean r e t u r n of the market(R m), the average t o t a l r i s k ( V . ) , and the market r i s k ( V ). The mean t m r e t u r n on c a p i t a l i s 5.00 p e r c e n t / q u a r t e r and the r e t u r n on e q u i t y i s 15.81 p e r c e n t / q u a r t e r . In terms of r i s k , the market r i s k ( V m ) and the average t o t a l r i s k ( V ) a s s o c i a t e d with the re t u r n on c a p i t a l i s 1.50 p e r c e n t / q u a r t e r and 2.10 pe r c e n t / q u a r t e r r e p e c t i v e l y . The market and average t o t a l r i s k s a s s o c i a t e d with the r e t u r n on e q u i t y are f a r g r e a t e r at 28.21 percent and 169.27 p e r c e n t / q u a r t e r r e s p e c t i v e l y . 1 The a d d i t i o n a l r i s k caused by leverage seems to outweigh the b e n e f i t of a higher r e t u r n . 6.2 ANSWER AND ANALYSIS OF QUESTION ONE In the i n t r o d u c t o r y chapter of the paper, the f o l l o w i n g q u e s t i o n was proposed: can i n v e s t o r s d i v e r s i f y t h e i r p o r t f o l i o s s o l e l y w i t h i n r e a l e s t a t e ? The only other study to i n v e s t i g a t e 81 TABLE 6.1 THE RETURN AND RISK MEASURES FOR THE SET OF APARTMENT BLOCKS(PERCENTAGE/QUARTER) Return on C a p i t a l Return on E q u i t y Mean Return on Market(Rm) 5.01 15.81 Variance of Market(Vm) 1.50 28.21 Average T o t a l V a r i a n c e ( V t ) 2.10 169.28 Ratio(Vt/Vm) .71 82 t h i s q u e s t i o n so f a r was conducted by M i l e s and McCue[44]; they found that d i v e r s i f i c a t i o n was p o s s i b l e w i t h i n r e a l e s t a t e . 2 Given the r e s u l t s of M i l e s and McCue, t h i s paper has t e s t e d the hyp o t h e s i s that r e a l e s t a t e i n v e s t o r s can d i v e r s i f y t h e i r p o r t f o l i o s w i t h i n a l o c a l r e a l e s t a t e market. Beginning the a n a l y s i s of q u e s t i o n one, we c a l c u l a t e d the r a t i o of ^ v m / v t ^ ^ o r t * i e r e t u r n on c a p i t a l . 3 T h i s r a t i o i n d i c a t e s the p r o p o r t i o n of t o t a l r i s k accounted f o r by the market , i . e . n o n - d i v e r s i f i a b l e r i s k . The more important systematic or market i n f l u e n c e s are, the c l o s e r t h i s p a r t i c u l a r r a t i o w i l l be to 1.0." The r a t i o appearing i n Table 6.1 i l l u s t r a t e s that market r i s k i s 71.43 percent of average t o t a l r i s k . In comparison to other e q u i t i e s , market r i s k was 54.40 percent of average t o t a l r i s k f o r bonds and 37.80 percent f o r s t o c k s . 5 Thus i t appears that the p o t e n t i a l to d i v e r s i f y w i t h i n r e a l e s t a t e i s q u i t e s m a l l ; only 28.57 percent of the t o t a l r i s k i s d i v e r s i f i a b l e . To determine whether g e o g r a p h i c a l d i v e r s i f i c a t i o n w i t h i n the c i t y i s p o s s i b l e , the sample was d i v i d e d i n t o two subsamples by l o c a t i o n ( s e e Chapter 4). The r e s u l t s of t h i s t e s t appears i n Table 6.2. The r a t i o of (V /V.) f o r the West End was 79.80 m t percent, and f o r the o u t l y i n g areas the r a t i o was 71.59 per c e n t . These r a t i o s show l i t t l e p o t e n t i a l to d i v e r s i f y w i t h i n the c i t y , not a s u r p r i s i n g f i n d i n g given the r e s u l t s above. When comparing the two r a t i o s , the o u t l y i n g areas c o n t r i b u t e more to 83 TABLE 6.2 THE RETURN AND RISK MEASURES FOR THE SUB-SAMPLE OF APARTMENT BLOCKS(PERCENTAGE/QUARTER) WEST END REST OF THE CITY Mean Return on Market(Rm) 4.48 5.09 Variance of Market(Vm) 1.71 1.49 Average T o t a l V a r i a n c e ( V t ) 2.15 2.08 Ratio(Vt/Vm) 79.80 71.59 84 d i v e r s i f i c a t i o n than does the West End. The l a t t e r , an area with more v a r i e d types of apartment b l o c k s , from garden apartments to h i g h - r i s e apartments, does not c o n t r i b u t e s t r o n g l y to p o r t f o l i o d i v e r s i f i c a t i o n . The next step of the a n a l y s i s was to examine the r a t e at which v a r i a t i o n of r e t u r n f o r randomly s e l e c t e d p o r t f o l i o s was reduced as a f u n c t i o n of the number of p r o p e r t i e s i n c l u d e d i n the p o r t f o l i o . T h i s examination looked at p o r t f o l i o s from s i z e 2 to 30 p r o p e r t i e s . The r e s u l t s of the t e s t appear in Table 6.3. The v a r i a t i o n s of r e t u r n show a downward but i n c o n s i s t e n t t r e n d . From p o r t f o l i o s of s i z e 2 to 13, a l l but four p o r t f o l i o s had a v a r i a n c e of r e t u r n g r e a t e r than 1.60 percent, but from p o r t f o l i o s i z e 14 to 30 a l l v a r i a n c e s of r e t u r n were below 1.60 percent, i n d i c a t i n g that some r e d u c t i o n i n v a r i a t i o n of r e t u r n was o c c u r r i n g with d i v e r s i f i c a t i o n . The t a b l e a l s o i l l u s t r a t e s that most of the unsystematic r i s k was d i v e r s i f i e d away through the h o l d i n g of only a few p r o p e r t i e s : at two p r o p e r t i e s , approximately 50 percent of the t o t a l unsystematic r i s k had been d i v e r s i f i e d away, while at 29 p r o p e r t i e s ( t h e lowest v a r i a n c e of return) only 75 percent of the unsystematic r i s k was d i v e r s i f i e d away, an improvement of a mere 25 percent f o r a p o r t f o l i o of f o u r t e e n times the s i z e . To analyze the r e s u l t s i n more d e t a i l , we ran t - t e s t s on s u c c e s s i v e p o r t f o l i o s to i n d i c a t e which p o r t f o l i o s i z e s cause s i g n i f i c a n t r e d u c t i o n i n r e t u r n v a r i a t i o n . The r e s u l t s of these t e s t s showed that the a d d i t i o n of one property to a p o r t f o l i o of 85 TABLE 6.3 DESCRIPTION OF PORTFOLIO SIZE AND REDUCTION IN RETURN VARIATION PORTFOLIO SI ZE VARIANCE OF RETURN (percentage/quarter) 2 P r o p e r t i e s 1 .79 3 P r o p e r t i e s 1 .73 4 P r o p e r t i e s 1 .61 5 P r o p e r t i e s 1 .56 6 P r o p e r t i e s 1 .78 7 P r o p e r t i e s 1 .56 8 P r o p e r t i e s 1 .66 9 P r o p e r t i e s 1 .70 10 P r o p e r t i e s 1 .56 11 P r o p e r t i e s 1 .65 12 P r o p e r t i e s 1 .59 13 P r o p e r t i e s 1 .62 14 P r o p e r t i e s 1 .58 15 P r o p e r t i e s 1 .53 16 P r o p e r t i e s 1 .56 17 P r o p e r t i e s 1.57 18 P r o p e r t i e s 1 .56 19 P r o p e r t i e s 1 .57 20 P r o p e r t i e s 1 .57 21 P r o p e r t i e s 1 .54 22 P r o p e r t i e s 1 .53 23 P r o p e r t i e s 1 .58 24 P r o p e r t i e s 1 .57 25 P r o p e r t i e s 1.57 26 P r o p e r t i e s 1.57 27 P r o p e r t i e s 1.54 28 P r o p e r t i e s 1.52 29 Propert i e s 1.51 30 P r o p e r t i e s 1.55 87 s i z e s 3, 6, and 9 cause s i g n i f i c a n t r e d u c t i o n at the .05 l e v e l . However the r e s u l t s should be q u a l i f i e d . Since the v a r i a t i o n s of r e t u r n show an i n c o n s i s t e n t downward tr e n d , i t seems unreasonable to conclude that c e r t a i n p o r t f o l i o s i z e s do s i g n i f i c a n t l y reduce v a r i a n c e of r e t u r n . With regard to p o r t f o l i o s i z e s 6 and 9, the v a r i a n c e of r e t u r n i n c r e a s e d , thus r a i s i n g the p o s s i b i l i t y that a s i g n i f i c a n t r e d u c t i o n i n va r i a n c e would occur with the a d d i t i o n of another p r o p e r t y to the p o r t f o l i o . As a f i n a l t e s t on the set of random p o r t f o l i o s , a simple r e g r e s s i o n a n a l y s i s was run on the v a r i a t i o n s of r e t u r n to analyze the r e l a t i o n s h i p of d e c r e a s i n g p o r t f o l i o v a r i a t i o n as d i v e r s i f i c a t i o n i n c r e a s e s . Regression a n a l y s i s was performed f i t t i n g by l e a s t squares the r e g r e s s i o n f u n c t i o n : Y = a + b(1/fx) where Y equals the r e t u r n v a r i a n c e of the p o r t f o l i o s and x i s the p o r t f o l i o s i z e . The f u n c t i o n d i d not produce an extremely good f i t , as i n d i c a t e d by the low c o e f f i c i e n t of d e t e r m i n a t i o n , .36310. Only 36 percent of the v a r i a n c e of r e t u r n can be e x p l a i n e d by d i v e r s i f i c a t i o n . T h i s r e s u l t diff.er.s from the c o n c l u s i o n s reached by Evans and A r c h e r [ l 7 ] whose r e g r e s s i o n equation had a f i t of .986,3. Our study's comparatively poor r e s u l t i s due to the i n c o n s i s t e n t t r e n d seen i n the v a r i a t i o n s of r e t u r n , and the f a c t that much of the r e d u c t i o n of va r i a n c e o c c u r r e d w i t h i n a very few p r o p e r t i e s . T h e r e f o r e , the r e s u l t s from the t e s t s on the random p o r t f o l i o s are s i m i l a r to the 88 r e s u l t s comparing market r i s k to average t o t a l r i s k ; the p o t e n t i a l to d i v e r s i f y i s m arginal. With the a n a l y s i s concluded, we can now answer the f i r s t q u e s t i o n proposed i n the paper: can i n v e s t o r s d i v e r s i f y t h e i r p o r t f o l i o s s o l e l y w i t h i n a r e a l e s t a t e market? The answer to the q u e s t i o n i s no, i n v e s t o r s cannot d i v e r s i f y s o l e l y w i t h i n a r e a l e s t a t e market i f that market i s c o n f i n e d to one l o c a l e and one p r o p e r t y type. The r e s u l t s demonstrate that l e s s than 30 percent of the r i s k i s d i v e r s i f i a b l e . Since the answer to the q u e s t i o n i s no, the hypothesis that i n v e s t o r s can d i v e r s i f y t h e i r p o r t f o l i o s w i t h i n a l o c a l r e a l e s t a t e market must a l s o be r e j e c t e d . The r e j e c t i o n of the h y p o t h e s i s might be reversed i f d i f f e r e n t p roperty types were i n c l u d e d i n the p o r t f o l i o . A d i s c u s s i o n on the e f f e c t s these c o n c l u s i o n s have on r e a l e s t a t e i n v e s t o r s i s presented i n the next chapter. 6.3 ANSWER AND ANALYSIS OF QUESTION TWO The second q u e s t i o n of the study asks i f r e a l e s t a t e can improve the e f f i c i e n c y of i n v e s t o r s ' p o r t f o l i o s . To d e a l with t h i s q u e s t i o n , two d i f f e r e n t types of e f f i c i e n t p o r t f o l i o s are c o n s i d e r e d . The f i r s t type of e f f i c i e n t p o r t f o l i o r e f e r s to an i n f l a t i o n - h e d g e d p o r t f o l i o : a p o r t f o l i o that has a r e t u r n which keeps pace with i n f l a t i o n and has a high c o r r e l a t i o n with the rate of i n f l a t i o n . The second type of e f f i c i e n t p o r t f o l i o f o l l o w s Markowitz's d e s c r i p t i o n of e f f i c i e n t p o r t f o l i o s , that set of p o r t f o l i o s which o f f e r the h i g h e s t expected r e t u r n f o r a given v a r i a n c e of r e t u r n . Past r e s e a r c h has shown that r e a l e s t a t e does improve the 89 e f f i c i e n c y of i n v e s t o r s ' p o r t f o l i o s i n r e s p e c t to both d e f i n i t i o n s . Since past l i t e r a t u r e demonstrated the u s e f u l n e s s of r e a l e s t a t e in mixed a s s e t p o r t f o l i o s , the paper proposed the h y p o t h e s i s that r e a l e s t a t e w i l l improve the e f f i c i e n c y of i n v e s t o r s ' p o r t f o l i o s under both d e f i n i t i o n s of e f f i c i e n t . Given t h i s h y p o t h e s i s , we s t a r t the a n a l y s i s by examining the e f f e c t s of r e a l e s t a t e on an i n f l a t i o n - h e d g e d p o r t f o l i o . Table 6.4 p r e s e n t s the mean rate of r e t u r n s and standard d e v i a t i o n s f o r the v a r i o u s investment a s s e t s to be i n c l u d e d i n the p o r t f o l i o , and the i n f l a t i o n r a t e . 6 A l l the a s s e t s except t r e a s u r y b i l l s , as the t a b l e i n d i c a t e s , have a r a t e of r e t u r n that surpasses i n f l a t i o n . Treasury b i l l s have a s l i g h t l y lower r a t e of r e t u r n but a l s o a lower standard d e v i a t i o n . In the case of r e a l e s t a t e , the r e t u r n i s h i g h , 5.00 p e r c e n t / q u a r t e r , with a q u a r t e r l y standard d e v i a t i o n of 8.61 percent. The r e t u r n and r i s k are comparable to those o b t a i n a b l e on the other investment a s s e t s . Turning to Table 6.5, we can view the c r o s s c o r r e l a t i o n s of the a s s e t s to i n f l a t i o n . The c r o s s c o r r e l a t i o n s i n d i c a t e which a s s e t s might be u s e f u l i n an i n f l a t i o n - h e d g e d p o r t f o l i o . The t a b l e shows that t r e a s u r y b i l l s have the s t r o n g e s t c o r r e l a t i o n with i n f l a t i o n , .50, and that r e a l e s t a t e and g o l d are s l i g h t l y p o s i t i v e l y c o r r e l a t e d , .24 and .10 r e s p e c t i v e l y . Bonds and stocks have a negative c o r r e l a t i o n of -.28 and -.13 r e s p e c t i v e l y . So t r e a s u r y b i l l s , r e a l e s t a t e and gold, being p o s i t i v e l y c o r r e l a t e d , appear u s e f u l in an i n f l a t i o n - h e d g e d 90 TABLE 6.4 THE INFLATION RATE, THE MEAN RETURNS AND STANDARD DEVIATIONS FOR THE INVESTMENT ASSETS(PERCENTAGE/QUARTER) MEAN RETURN STANDARD DEVIATION INFLATION 1.85 0.86 TREASURY BILLS 1.76 0.66 BONDS 2.22 4.53 GOLD 7.77 15.40 COMMON STOCK 2.93 8.28 REAL ESTATE 5.00 12.25 91 TABLE 6.5 CORRELATION MATRIX OF INFLATION AND THE INVESTMENT ASSETS CPI TBILLS BONDS GOLD TSE RE CPI 1 .000 0.500 -0.283 0. 100 -0.129 0.241 TBILLS 0.500 1 .000 -0.116 -0.006 0.001 0.049 BONDS -0.283 -0.116 1 .000 -0.121 0.332 -0.343 GOLD 0. 1 00 -0.006 -0.121 1 . 000 0.078 0.223 TSE -0.129 0.001 0.332 0.078 1 .000 -0.243 RE 0.241 0.049 -0.343 0.223 -0.243 1 .000 92 p o r t f o l i o . To determine the mixture of the a s s e t s i n an i n f l a t i o n - hedged p o r t f o l i o , the r e t u r n s of the a s s e t s were regressed a g a i n s t i n f l a t i o n . From the equation, we determined which a s s e t s would be i n c l u d e d and which a s s e t s would be s o l d s h o r t . A l s o from the equation, we c a l c u l a t e d the weights of the a s s e t i n the p o r t f o l i o . Table 6.6 presents the weights of the a s s e t s in the i n f l a t i o n - h e d g e d p o r t f o l i o . As seen from the t a b l e , t r e a s u r y b i l l s dominate the p o r t f o l i o and appear to be the only v a l u a b l e a s s e t i n i t . Real e s t a t e and gold are i n c l u d e d in the p o r t f o l i o , but only a small percentage i s a l l o c a t e d to these a s s e t s . Bonds and stocks would be s o l d s h o r t . At the bottom of the t a b l e i s the r a t e of r e t u r n and standard d e v i a t i o n that c o u l d have been obtained from t h i s p o r t f o l i o over the p e r i o d of the study. The r a t e of r e t u r n i s s l i g h t l y l e s s than the r a t e of i n f l a t i o n , 1.82 p e r c e n t / q u a r t e r compared to 1.85 p e r c e n t / q u a r t e r f o r i n f l a t i o n . However, the v a r i a b i l i t y of the p o r t f o l i o i s a l s o lower than that of i n f l a t i o n . The low r e t u r n and v a r i a b i l i t y i s a r e f l e c t i o n of the dominance of t r e a s u r y b i l l s i n the p o r t f o l i o . When the c o r r e l a t i o n between i n f l a t i o n and the p o r t f o l i o was c a l c u l a t e d , the c o r r e l a t i o n was .55, not much l a r g e r than the c o r r e l a t i o n of i n f l a t i o n to t r e a s u r y b i l l s . T h e r e f o r e t h i s i n f l a t i o n - h e d g e d p o r t f o l i o i s not a p e r f e c t hedge. The r e s u l t s i l l u s t r a t e d that r e a l e s t a t e does c o n t r i b u t e to an i n f l a t i o n - h e d g e d p o r t f o l i o . However, t r e a s u r y b i l l s are the 93 TABLE 6.6 THE WEIGHTED PROPORTIONS FOR EACH ASSET IN AN INFLATION-HEDGED PORTFOLIO Treasury B i l l s 1.0368 Bonds - .0503 Gold .0053 Common Stock - .0083 Real E s t a t e .0165 94 dominant asset i n the p o r t f o l i o , and even by i n c l u d i n g r e a l e s t a t e and gold, the hedge a g a i n s t i n f l a t i o n does not improve g r e a t l y over a p o r t f o l i o c o n s i s t i n g s o l e l y of t r e a s u r y b i l l s . T urning to the second d e f i n i t i o n of e f f i c i e n t , we begin by r e c a l l i n g that Markowitz demonstrated that through d i v e r s i f i c a t i o n the o v e r a l l v a r i a b l i t y of the p o r t f o l i o can be reduced, thereby making i t more e f f i c i e n t . The r e d u c t i o n of r i s k occurs when a s s e t s are combined that have a n e g a t i v e ( o r low p o s i t i v e ) c o r r e l a t i o n with other a s s e t s in the p o r t f o l i o . The r e s u l t of combining such a s s e t s i s that the i n d i v i d u a l r i s k of the a s s e t s i s d i v e r s i f i e d away, while only the i n t e r r e l a t i o n s h i p of the a s s e t s c o n t r i b u t e s to the p o r t f o l i o r i s k . To see i f r e a l e s t a t e improves the e f f i c i e n c y of an i n v e s t o r ' s p o r t f o l i o , we should then i n s p e c t the c o r r e l a t i o n s of r e a l e s t a t e to the other investment a s s e t s . The c o r r e l a t i o n matrix in Table 6.5 r e v e a l s that the c o r r e l a t i o n of r e a l e s t a t e to the other a s s e t s i s low p o s i t i v e f o r gold and t r e a s u r y b i l l s and s l i g h t l y negative with common stock and bonds. I t appears that r e a l e s t a t e can improve the e f f i c i e n c y of i n v e s t o r s ' p o r t f o l i o s . The low c o r r e l a t i o n with the other a s s e t s should help d i v e r s i f y away i n d i v i d u a l r i s k of the a s s e t s . To a c t u a l l y a s c e r t a i n i f r e a l e s t a t e improves the e f f i c i e n c y of an i n v e s t o r s ' p o r t f o l i o , we employed the o b j e c t i v e f u n c t i o n d e s c r i b e d i n Chapter 4: 95 maxlmize N Z X • R. N N - X Z Z - M ( 2 ) N l l i = l 1=1j=i i = 1 To d e r i v e t h i s o b j e c t i v e f u n c t i o n a computer program was w r i t t e n which computes the e f f i c i e n t f r o n t i e r ( s e e Appendix A). F i g u r e 6.1 presents a graph of the e f f i c i e n t f r o n t i e r . The s c a t t e r e d l i n e r e p r e s e n t s the e f f i c i e n t f r o n t i e r with r e a l e s t a t e i n c l u d e d i n the p o r t f o l i o s , while the s o l i d l i n e denotes p o r t f o l i o s that c o n t a i n s a l l investment a s s e t s except r e a l e s t a t e . The graph i l l u s t r a t e s that the p o r t f o l i o s which i n c l u d e r e a l e s t a t e s t r o n g l y dominate the p o r t f o l i o s without r e a l e s t a t e . The dominant p o s i t i o n of the r e a l estate-augmented p o r t f o l i o s decrease as the r e t u r n s of the p o r t f o l i o decrease. T h i s i s because at the lower r a t e s of r e t u r n s r e a l e s t a t e becomes a d e c r e a s i n g percentage of the p o r t f o l i o s . Table 6.7 shows the a s s e t mixture fo r p o r t f o l i o s (that i n c l u d e r e a l e s t a t e ) along the e f f i c i e n t f r o n t i e r . If we d i v i d e the t a b l e in two, we see that the p o r t f o l i o s with high r e t u r n s (a r e t u r n above 3.94 p e r c e n t / q u a r t e r ) s e l l t r e a s u r y b i l l s s h o r t . T h i s r e f l e c t s the need of leverage to o b t a i n these h i g h r a t e s of r e t u r n s . Of the other a s s e t s , bonds are dominant; r e a l e s t a t e and gold approximately have the same weight in the p o r t f o l i o s ; and common stock c o n t r i b u t e s s l i g h t l y l e s s than that of r e a l e s t a t e and g o l d . Bonds are a major f a c t o r because of t h e i r low r i s k r e l a t i v e to the other p o s i t i v e weighted a s s e t s . Looking 96 F i g u r e 6.1 THE EFFICIENT FRONTIER O ( p e r c e n t / q u a r t e r ) 97 TABLE 6.7 A SET OF PORTFOLIOS ALONG THE EFFICIENT FRONTIER (BY DECREASING RATE OF RETURN) RETURN VARIANCE OPTIMAL PROPORTIONS FOR EACH ASSET ARE: p e r c e n t / p e r c e n t / q u a r t e r q u a r t e r T-BILLS. BONDS GOLD TSE R.E. 13.73 5.99 - 3.97 2.03 1.10 0.70 1.14 11.16 3.66 - 2.90 1 .60 0.86 0.55 0.89 9.12 2.23 - 2.05 1 .26 0.67 0.43 0.70 7.52 1 .37 - 1 .40 0.99 0.52 0.33 0.55 6.27 0.84 - 0.88 0.78 0.41 0.26 0.43 5.30 0.51 - 0.48 0.62 0. 3,2 0.20 0.34 4.53 0.31 - 0.17 0.49 0.25 0.16 0.26 3.94 0.19 - 0.08 0.40 0.20 0.12 0.21 3.47 0.12 0.27 0. 32 0.15 0.10 0.16 3.11 0.07 0.42 0.26 0.12 0.08 0.13 2.83 0.05 0.53 0.21 0.10 0.06 0.10 2.61 0.03 0.63 0.17 0.07 0.03 0.08 2.43 0.02 0.70 0.14 0.06 0.04 0.06 2.30 0.01 0.75 0.12 0.05 0.03 0.05 2.19 0.01 0.80 0.10 0.04 0.02 0.04 98 TABLE 6.8 COMPARISONS OF THE RISK (VARIANCE) OF THE INDIVIDUAL ASSETS TO EFFICIENT PORTFOLIOS WITH THE SAME MEAN RETURN VARIANCE (p e r c e n t / q u a r t e r ) ASSET ASSET EFFICIENT PORTFOLIO PERCENTAGE DIFFERENCE REAL ESTATE 1 .50 0.43 - 71 .33 BONDS 0.20 0.01 - 95.00 GOLD 2.37 1 .49 - 37. 13 COMMON STOCK 0.69 0.06 - 91.30 Note: Treasury B i l l s s i n c e they have are not i n c l u d e d i n the lowest p o s s i b l e the comparisons, r i s k o b t a i n a b l e . 99 a t the low r e t u r n p o r t f o l i o s , the t a b l e i l l u s t r a t e s t h a t t r e a s u r y b i l l s a r e the most s i g n i f i c a n t a s s e t ; bonds a r e a minor p o r t i o n of the p o r t f o l i o s ; w h i l e r e a l e s t a t e , g o l d and common s t o c k c o n t r i b u t e o n l y m a r g i n a l l y t o the p o r t f o l i o s . To f u r t h e r examine the b e n e f i t s of d i v e r s i f y i n g i n mixed- a s s e t p o r t f o l i o s , T a ble 6.8 compares the r i s k of the i n d i v i d u a l a s s e t s t o the r i s k of e f f i c i e n t p o r t f o l i o s w i t h the same mean r e t u r n . The comparisons c l e a r l y i n d i c a t e the b e n e f i t s of d i v e r s i f y i n g i n mi x e d - a s s e t p o r t f o l i o s . The r i s k of the mixed- a s s e t p o r t f o l i o s i s s i g n i f i c a n t l y l e s s than the r i s k of the i n d i v i d u a l a s s e t s . For example, an e f f i c i e n t p o r t f o l i o w i t h the same mean r e t u r n as r e a l e s t a t e has a p p r o x i m a t e l y 70 p e r c e n t l e s s v a r i a b i l i t y than r e a l e s t a t e (.43 p e r c e n t / q u a r t e r v e r s u s 1.50 p e r c e n t / q u a r t e r ) . In c o n c l u s i o n , r e a l e s t a t e does improve the e f f i c i e n c y of i n v e s t o r s ' p o r t f o l i o s . Our second h y p o t h e s i s can be a c c e p t e d under both d e f i n i t i o n s of e f f i c i e n c y . R e a l e s t a t e does h e l p an i n v e s t o r hedge a g a i n s t i n f l a t i o n ; r e a l e s t a t e has a n e g a t i v e ( o r low p o s i t i v e ) c o r r e l a t i o n w i t h o t h e r i nvestment a s s e t s , which e n a b l e s i n v e s t o r s t o f u r t h e r reduce t h e i r d i v e r s i f i a b l e r i s k . 1 00 ENDNOTES 1. The d i s t r i b u t i o n s f o r the market and average t o t a l v a r i a n c e are s t r o n g l y skewed p o s i t i v e . A few p r o p e r t i e s t h a t have v e r y l a r g e v a r i a n c e s g r e a t l y i n f l u e n c e the market and average t o t a l r i s k . 2. M i l e s , Mike and McCue, Tom, " C o n s i d e r a t i o n s i n R e a l E s t a t e P o r t f o l i o D i v e r s i f i c a t i o n " , Working Paper, U n i v e r s i t y of N o r t h C a r o l i n a , 1980 3. S i n c e the r e t u r n on e q u i t y i s i n f l u e n c e d by i n v e s t o r ' s l e v e r a g e , i t can not i n d i c a t e the r i s k t h a t i s s t r i c t l y a s s o c i a t e d t o r e a l e s t a t e . T h e r e f o r e i t i s unnecessary t o c a l c u l a t e (Vm/Vt) f o r the r e t u r n on e q u i t y . 4. Evans, John L. and A r c h e r , Stephen N., " D i v e r s i f i c a t i o n and the R e d u c t i o n of D i s p e r s i o n : An E m p i r i c a l A n a l y s i s " , J o u r n a l of F i n a n c e , December 1968, pp.761-767 5. M i l e s , Mike and McCue, Tom, " C o n s i d e r a t i o n s i n R e a l E s t a t e P o r t f o l i o D i v e r s i f i c a t i o n " , Working Paper, U n i v e r s i t y of N o r t h C a r o l i n a , 1980 6. Only the r e t u r n on c a p i t a l f o r r e a l e s t a t e i s i n c l u d e d i n t h e e f f i c i e n t p o r t f o l i o s . I t i s o n l y r e a s o n a b l e t o use s i m i l a r r a t e s of r e t u r n measures f o r b oth r e a l e s t a t e and the o t h e r investment a s s e t s . A l s o the c o r r e l a t i o n s f o r b o th measures f o r r e a l e s t a t e , the r e t u r n on c a p i t a l and the r e t u r n on e q u i t y , t o the o t h e r i nvestment a s s e t s are so s i m i l a r t h a t t h e i r e f f e c t on a m i x e d - a s s e t p o r t f o l i o i s about the same. 101 7.0 DISCUSSION Chapter 6 p r e s e n t e d the e m p i r i c a l r e s u l t s and answered the two q u e s t i o n s proposed i n t h i s paper. T h i s c h a p t e r b r i e f l y r e v i e w s the r e s u l t s , i n the c o n t e x t of e x p l a i n i n g t h e i r i m p l i c a t i o n s t o i n v e s t o r s . 7.1 I m p l i c a t i o n s of F i n d i n g s t o I n v e s t o r s The p o p u l a r i t y of r e a l e s t a t e , as an i n v e s t m e n t , i n c r e a s e d s u b s t a n t i a l l y t h r o u g h the s e v e n t i e s . The demand f o r r e a l e s t a t e s o a r e d , as i n v e s t o r s p e r c e i v e d r e a l e s t a t e t o be the investment t o combat i n f l a t i o n . 1 But what was the r e t u r n on r e a l e s t a t e d u r i n g t h i s decade? No one r e a l l y knows. S i n c e r e a l e s t a t e l a c k s a " c e n t r a l i z e d " exchange, 2 i t i s d i f f i c u l t t o c o m p i l e i n f o r m a t i o n on r e t u r n s . As a r e s u l t , l i t t l e r e s e a r c h has been conducted on the b e h a v i o r of r e a l e s t a t e r e t u r n s , a l t h o u g h s t u d i e s i n v e s t i g a t i n g the b e h a v i o r of o t h e r a s s e t s are q u i t e e x t e n s i v e . T h i s study c a l c u l a t e d a s e t of r e a l e s t a t e r e t u r n s f o r apartment b l o c k s l o c a t e d i n Vancouver, B r i t i s h Columbia, from 1970-1979. The paper used t h e s e r e t u r n s t o f o c u s on the p o t e n t i a l b e n e f i t s of d i v e r s i f i c a t i o n i n r e a l e s t a t e . Two i s s u e s of d i v e r s i f i c a t i o n were d e a l t w i t h : the p o t e n t i a l of d i v e r s i f y i n g w i t h i n r e a l e s t a t e , and the b e n e f i t s of i n c l u d i n g r e a l e s t a t e i n m i x e d - a s s e t p o r t f o l i o s . 1 02 The mean r e t u r n c a l c u l a t e d on the apartment b l o c k s was 5.00 p e r c e n t / q u a r t e r and the s t a n d a r d d e v i a t i o n was 12.25 p e r c e n t / q u a r t e r . The r e t u r n and r i s k of r e a l e s t a t e were second h i g h e s t t o g o l d , w i t h r e a l e s t a t e r e t u r n s o u t p a c i n g those of t r e a s u r y b i l l s , bonds, and common s t o c k . I n v e s t o r s r e c e i v e d a r e t u r n from r e a l e s t a t e t h a t not o n l y matched i n f l a t i o n , but a l s o p r o v i d e d a r e a l r e t u r n of a p p r o x i m a t e l y 3.20 p e r c e n t / q u a r t e r . I n v e s t o r s who a p p l i e d l e v e r a g e on t h e i r p r o p e r t i e s , on average , t r i p l e d t h e i r r e t u r n ; however the r i s k c o n t r i b u t e d by l e v e r a g e might outweigh the b e n e f i t s of the h i g h e r r e t u r n . In comparing the average t o t a l v a r i a n c e ( V t ) f o r the r e t u r n on c a p i t a l t o the r e t u r n on e q u i t y , the average t o t a l v a r i a n c e f o r the r e t u r n on e q u i t y was overwhemingly g r e a t e r . T h i s i n f o r m a t i o n i l l u s t r a t e s t o i n v e s t o r s the importance of c o n d u c t i n g some form of a n a l y s i s ; s u c h an a n a l y s i s w i l l make them aware of cash f l o w d i f f i c u l t i e s t h a t might r e s u l t from the added f i x e d c o s t s of l e v e r a g e . A f t e r c a l c u l a t i n g the r e t u r n s on the p r o p e r t i e s , the paper examined the p o t e n t i a l of d i v e r s i f i c a t i o n w i t h i n r e a l e s t a t e . The f i r s t p a r t of the e x a m i n a t i o n l o o k e d a t the r e l a t i v e p r o p o r t i o n s of s y s t e m a t i c and u n s y s t e m a t i c r i s k . The i n v e s t i g a t i o n found t h a t o n l y 29 p e r c e n t of t o t a l r i s k i s u n s y s t e m a t i c ( d i v e r s i f i a b l e ) . In c o n t r a s t , M i l e s and McCue[44] found t h a t between 87 and 95 p e r c e n t of t o t a l r i s k i s u n s y s t e m a t i c ( t h e y used a sample c o n t a i n i n g d i f f e r e n t p r o p e r t y t y p e s throughout the U n i t e d S t a t e s ) . M i l e s and McCue c o n s i d e r e d 1 03 the i m p o r t a n t f a c t o r s f o r the h i g h u n s y s t e m a t i c r i s k t o be the r e s u l t of a p r o p e r t y ' s unique c h a r a c t e r , i . e . l o c a t i o n , cash f l o w , and l e a s e on p r o p e r t y . There were f o u r reasons why our study had such d i f f e r e n t r e s u l t s from t h a t of M i l e s and McCue: (1) the da t a were c o n f i n e d t o a s i n g l e p r o p e r t y t y p e , (2) the p r o p e r t y type was l i m i t e d t o one l o c a l e , (3) the v a l u a t i o n model was not a b l e t o i n c o r p o r a t e enough of the c h a r a c t e r i s t i c s M i l e s and McCue c o n s i d e r e d t o be i m p o r t a n t , (4) the method t o e s t i m a t e v a l u e o v e r s t a t e d the c o r r e l a t i o n of the p r o p e r t i e s ( s e e Chapter 5) . In e v a l u a t i n g the r e s u l t s of t h i s paper, r e a l e s t a t e i n v e s t o r s s h o u l d d i s c o u n t the problems of the v a l u a t i o n model, and r e c o g n i z e the f a c t t h a t p o r t f o l i o s c o n f i n e d t o one p r o p e r t y type i n one l o c a l market a r e not w e l l d i v e r s i f i e d . I f i n v e s t o r s want a d i v e r s i f i e d p o r t f o l i o h o l d i n g o n l y r e a l e s t a t e , then they need t o i n c l u d e a range of p r o p e r t y t y p e s throughout v a r i o u s m a rkets. A f a c t o r t h a t i n v e s t o r s s h o u l d c o n s i d e r i f they t r y t o f u l l y d i v e r s i f y w i t h i n r e a l e s t a t e , i s the c o s t of d i v e r s i f i c a t i o n . By h a v i n g t o d i v e r s i f y a c r o s s p r o p e r t y t y p e s and g e o g r a p h i c a l r e g i o n s , they may f i n d the c o s t s of o b t a i n i n g i n f o r m a t i o n too h i g h and the q u a l i t y of t h a t i n f o r m a t i o n of u n c e r t a i n v a l u e . For the next p a r t of the e x a m i n a t i o n , the paper i n v e s t i g a t e d the e f f e c t of p o r t f o l i o s i z e on the r e d u c t i o n of r e t u r n v a r i a t i o n . The r e s u l t s of t h i s i n v e s t i g a t i o n were weak w i t h o n l y 36 p e r c e n t of the v a r i a t i o n of r e t u r n b e i n g e x p l a i n e d 1 04 by d i v e r s i f i c a t i o n . The r e t u r n v a r i a t i o n of p o r t f o l i o s of i n c r e a s i n g s i z e showed a downward but i n c o n s i s t e n t p a t t e r n . When t - t e s t s were run t o see i f any of the p o r t f o l i o s caused s i g n i f i c a n t r e d u c t i o n i n v a r i a t i o n , t h r e e p o r t f o l i o s were found to have caused s i g n i f i c a n t r e d u c t i o n , p o r t f o l i o s i z e s 4, 7, and 10. But, because of the i n c o n s i s t e n t p a t t e r n i n r e t u r n v a r i a t i o n , t h e s e r e s u l t s s h o u l d not be c o n s i d e r e d f u l l y r e l i a b l e . However, i n v e s t o r s s h o u l d note t h a t i t i s p o s s i b l e t o d i v e r s i f y away a l a r g e p o r t i o n of the t o t a l u n s y s t e m a t i c r i s k by h o l d i n g p o r t f o l i o s which c o n t a i n o n l y a few p r o p e r t i e s . I n v e s t o r s do not have t o i n c u r l a r g e t r a n s a c t i o n c o s t s t o e l i m i n a t e d i v e r s i f i a b l e r i s k i n a l o c a l market; t h r o u g h two or t h r e e p r o p e r t i e s , i n v e s t o r s can t a k e advantage of most of the d i v e r s i f i c a t i o n p o t e n t i a l . Even though the paper d i d not f i n d the p o t e n t i a l t o d i v e r s i f y e f f i c i e n t l y w i t h i n a p o r t f o l i o c o n s i s t i n g s o l e l y of r e a l e s t a t e , i t d i d d i s c o v e r t h a t i n v e s t o r s can b e n e f i t by i n c l u d i n g r e a l e s t a t e i n m i x e d - a s s e t p o r t f o l i o s . The study found t h a t the i n c l u s i o n of r e a l e s t a t e i n an i n f l a t i o n - h e d g e d p o r t f o l i o was b e n e f i c i a l . In t h i s p o r t f o l i o , r e a l e s t a t e , t r e a s u r y b i l l s , and g o l d a l l c o n t r i b u t e d t o i t s e f f i c i e n c y . The most v a l u a b l e a s s e t i n the p o r t f o l i o was t r e a s u r y b i l l s . T r e a s u r y b i l l s had a c o r r e l a t i o n of .50 w i t h i n f l a t i o n , w h i l e the i n f l a t i o n - h e d g e d p o r t f o l i o o n l y had a c o r r e l a t i o n of .55. In o t h e r s t u d i e s , Fama and S c h w e r t [ l 9 ] and H a l l e n g r e n [ 2 7 ] obser v e d t h a t r e a l e s t a t e was the most e f f e c t i v e hedge a g a i n s t 105 i n f l a t i o n . Even though r e a l e s t a t e d i d not c o n t r i b u t e as s t r o n g l y i n t h i s s t u d y ' s i n f l a t i o n - h e d g e d p o r t f o l i o , the r e s u l t s s t i l l demonstrate t o i n v e s t o r s t h a t they s h o u l d i n c l u d e r e a l e s t a t e i n p o r t f o l i o s t h a t a r e d e s i g n e d t o hedge i n f l a t i o n . The study a l s o found r e a l e s t a t e t o have a low or n e g a t i v e c o r r e l a t i o n w i t h o t h e r a s s e t s , making the p o t e n t i a l t o d i v e r s i f y v e r y h i g h i n a mean-variance e f f i c i e n t p o r t f o l i o . For example, an e f f i c i e n t m i x e d - a s s e t p o r t f o l i o w i t h the same r e t u r n as one c o n s i s t i n g s o l e l y of r e a l e s t a t e ( 5 . 0 0 p e r c e n t / q u a r t e r ) had over 70 p e r c e n t l e s s r i s k . So i n v e s t o r s can enjoy the h i g h r e t u r n a s s o c i a t e d w i t h r e a l e s t a t e w i t h o u t t a k i n g on a g r e a t d e a l of r i s k . A l s o , they can d i v e r s i f y i n a m ixed-asset p o r t f o l i o w i t h o u t i n c u r r i n g g r e a t c o s t s . I f i n v e s t o r s s e l e c t mutual funds which r e f l e c t the r e t u r n b e h a v i o r of o t h e r e q u i t y m arkets, then t r a n s a c t i o n c o s t s ( i n c l u d i n g i n f o r m a t i o n c o s t s ) s h o u l d be low, and the i n v e s t o r ' s p o r t f o l i o w i l l be w e l l d i v e r s i f i e d . In a d d i t i o n , the s t u d y found t h a t the e f f i c i e n t p o r t f o l i o s which had h i g h r a t e s of r e t u r n s o l d t r e a s u r y b i l l s s h o r t , i l l u s t r a t i n g the need of l e v e r a g e i n o b t a i n i n g h i g h r a t e s of r e t u r n . The i m p l i c a t i o n of t h e s e f i n d i n g s a r e t h a t : (1) s m a l l i n d i v i d u a l i n v e s t o r s who own t h e i r home s h o u l d c o n c e n t r a t e t h e i r r e m a i n i n g funds i n o t h e r investment a s s e t s , i n o r d e r t o t a k e advantage of d i v e r s i f i c a t i o n ; (2) i n v e s t o r s who i n v e s t s t r i c t l y i n r e a l e s t a t e s h o u l d c o n s i d e r the b e n e f i t s of i n c l u d i n g o t h e r a s s e t s i n t h e i r p o r t f o l i o . The c o s t t o d i v e r s i f y i n a m i x e d - a s s e t 1 06 p o r t f o l i o may be l e s s than the c o s t s of d i v e r s i f y i n g w i t h i n r e a l e s t a t e . (3) i n v e s t o r s concerned w i t h the i l l i q u i d i t y of r e a l e s t a t e can e n j o y the b e n e f i t s of d i v e r s i f i c a t i o n w i t h o u t h a v i n g t o f e e l t h a t a l a r g e p o r t i o n of t h e i r p o r t f o l i o i s t i e d u p ( i l l i q u i d ) ; In c o n c l u s i o n , the paper d i s c o v e r e d t h a t r e a l e s t a t e was b e n e f i c i a l i n mixed-asset p o r t f o l i o s . R e a l e s t a t e i s a u s e f u l a d d i t i o n t o almost any p o r t f o l i o no matter what the investment o b j e c t i v e s a r e . The amount of r e a l e s t a t e t o be i n c l u d e d i n a p o r t f o l i o depends on the i n v e s t o r , h i s investment o b j e c t i v e s , and h i s b e l i e f s on the r e t u r n of r e a l e s t a t e and how i t c o v a r i e s w i t h o t h e r a s s e t s . THAT'S ALL FOLKS 1 07 ENDNOTES 1. I n v e s t o r s b e l i e v e d t h a t the a f t e r - t a x r a t e of r e t u r n on r e a l e s t a t e would be g r e a t e r than o t h e r i n v e s t m e n t s i n an i n f l a t i o n a r y environment, because of l e v e r a g e and the t a x advantages of r e a l e s t a t e . 2. M i l e s , Mike and McCue, Tom, " C o n s i d e r a t i o n s i n R e a l E s t a t e P o r t f o l i o D i v e r s i f i c a t i o n " , Working Paper, U n i v e r s i t y of N o r t h C a r o l i n a , 1980 108 BIBLIOGRAPHY 1. Anderson, L l o y d A., Energy Economics i n O f f i c e B u i l d i n g s , Master of S c i e n c e T h e s i s , U n i v e r s i t y of B r i t i s h C olumbia, 1 982 2. B a l l a r d , CM., " P e n s i o n Funds i n R e a l E s t a t e : New C h a l l e n g e s / O p p o r t u n i t i e s f o r P r o f e s s i o n a l s " , The A p p r a i s a l J o u r n a l , October 1978 3. 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Wonnacott, Thomas H. and Wonnacott, Ronald J . , I n t r o d u c t o r y S t a t i s t i c s f o r B u s i n e s s and Economics John W i l e y and Sons, 1977 1 1 3 APPENDIX A REAL SIGNVR,SIGNVJ,JSGNVJ,JSGNVR REAL ER,VLAM,VAR,SUM C PROG TO FORM PORTS CROM COUNTRY DATA DIMENSION VMN(5),VARC(5,5), 1CORR(2 0,20),SIG(20),RP(20) DIMENSION B(5,5),SIGNVR(5),SIGNVJ(5),IPERM(15),X(100) C READ VMEAN DO 10 1=1,5 READ(1,7)(VMN(I)) 7 FORMAT(5X,F7.3) 10 CONTINUE C READ VARC DO 9 KK=1 , 5 READ(2,12)(VARC(KK,J),J=1 , 5 ) 12 FORMAT(5X,5F9.3) 9 CONTINUE CALL FINV(5,5,VARC,I PERM,5,B,DET,JEXP,COND) DO 14 K=1,5 SIGNVR(K)=0.0 SIGNVJ(K)=0.0 DO 15 J=1 , 5 SIGNVR(K)= SIGNVR(K)+B(K,J)*VMN(J) SIGNVJ(K)=SIGNVJ(K)+B(K,J)*1 15 CONTINUE 14 CONTINUE JSGNVR=0 JSGNVJ=0 DO 16 K=1,5 JSGNVR=JSGNVR+SIGNVR(K) JSGNVJ=JSGNVJ+SIGNVJ(K) 16 CONTINUE VLAM=0.010 DO 18 KL=1,15 VMV=(JSGNVR-2*VLAM)/JSGNVJ W1=1.0/(2.0*VLAM) W2 =VMV/(2.0 *VLAM) DO 19 K=1,5 X(K)=W1*SIGNVR(K)-W2*SIGNVJ(K) 19 CONTINUE SUM=0 DO 731 1=1,5 SUM=SUM+X(I) 731 CONTINUE ER=0.0 VAR=0.0 DO 20 K=1,5 ER=ER+X(K)*VMN(K) DO 21 J=1 , 5 VAR=VAR+X(K)*X(J)*VARC(K,J) 1 1 4 21 CONTINUE 2 0 CONTINUE WRITE(6,282) 282 FORMAT(///,5X,17X,'EXPECTED RETURN',13X,'VARIANCE'///) WRITE(6,283)ER,VAR WRITE(6,997) 997 FORMAT(/'THE OPTIMAL PROPORTIONS FOR EACH ASSET ARE:'//) WRITE(6,998) 998 FORMAT(3X,'T~BILLS',3X,'BONDS',5X,'GOLD',5X,'TSE',7X,'R.E.' WRITE(6,284)(X(l),I=1,5) 284 FORMAT(//,10F9.2,/10F9.2) 283 FORMAT(5X,16X,F13.9,15X, F 1 3 . 9) VLAM=VLAM*1.28 18 CONTINUE STOP END 1970 CORRELATION LINC9 AGE L0C1 L0C2 L0C5 LOC LINC9 1 . . o o o -0. ,666 0, . 186 0, ,044 -0, . 100 -0, , 37 AGE -0. 66G 1 . .000 -0 . 264 -0. .118 0, .039 0, , 39 L0C1 0. . 186 -0. ,264 1 , .000 -0. , 269 -0, , 236 -0. 33 L0C2 0. .044 - o . 1 18 -0, , 269 1 . 000 -0. , 190 -0, 26 L0C5 -0. . 100 0. ,039 -0 . 236 -0, , 190 1 , ,000 -0, 23 L0C6 -0. 379 0. 390 - o . , 333 -0. 269 -0. 236 1 . 00 FLAR 0. .884 -0. 467 0, , 140 0, 024 -0. 176 -0, 18 LF AST 0. 042 -0. 015 -0, 119 -0. 013 0. 050 -0. 14 LN05T 0. 581 -0. 422 -0. 105 0. 051 -0, 155 0. 12 LLOT 0. 868 -0. 6 12 - o . 095 0. 169 -0. 223 -0. 21 DM2 -0. 134 0. 190 0. 132 -0. 321 0. 156 0. 13 DM3 0. 328 -0. 184 0. 048 0. 162 -0. 000 -0. 14 DM4 -0. 330 0. 088 -0. 221 0. 278 -0. 062 0. 13 LGIM 0. 377 -0. 563 0. 211 -0. 129 0. 1 1 1 -0. 24 LGIM LINC9 AGE LOC 1 L0C2 L0C5 L0C6 FLAR LF AST LNOST LLOT DM2 DM3 DM4 LGIM 0. 377 -O.563 21 1 129 1 1 1 -0.249 0. 290 0. 0. 0. 436 297 428 0. 224 0.046 0.086 1 .000 MULTIPLE R 0.77675 R SQUARE 0.60333 ADJUSTED R SQUARE 0.40500 STANDARD ERROR 0.18404 ANALYSIS OF VARIANCE REGRESSION RESIDUAL DF 9 18 F = 3.04202 VARIABLES IN THE EQUATION VARIABLE B SE B BETA T SIG LINC9 0 03702 0 07504 0 1 1 183 0 493 0 62 LF AST 0 34540 0 11518 0 47333 2 999 0 00 LOC5 0 1841 1 0 1 1725 0 27498 1 570 0 13 DM2 -0 14072 0 13406 -0 28051 - 1 050 0 30 LOC1 0 15377 0 09778 0 28420 1 573 0 13 DM3 -0 15679 0 13126 -0 28978 - 1 194 0 24 L0C6 0 12010 0 10438 0 22196 1 151 0 26 AGE -0 00514 0 00220 -0 49508 -2 338 0 03 DM4 -0 0475 1 0 13518 -0 0947 1 -0 351 0 72 (CONSTANT) -0 64434 1 08959 -0 591 0 56 CASEWISE PLOT OF STANDARDIZED RESIDUAL -3.0 0.0 3.0 SEONUM 0 • . . . : 0 LGIM 1 2 0953 2 * 1 2790 3 * 1 9354 4 1 8076 5 1 7489 6 * 1 78 17 7 1 9439 8 * 1 8050 9 1 6726 10 2 0093 1 1 * 1 8399 12 * 1 9917 13 * . 1 8659 14 * 2 0527 15 * 1 7577 16 1 9 124 17 1 8759 18 * 1 9872 19 2 0130 20 0 9282 2 1 1 6156 22 * 1 7546 23 * 1 9703 24 1 9833 25 * 1 795 1 26 1 8999 27 * 1 926 1 28 1 829 1 SEONUM o • . . . : 0 LGIM -3.0 0.0 3.0 FILE NONAME (CREATION DATE = 02/06/84) * * * * M U L T I P L E DEPENDENT VARIABLE.. LGIM RESIDUALS STATISTICS: MI N MAX MEAN STD DEV N *PRED 1 . , 2686 2 . 1225 1 .8242 0. 1686 28 *ZPRED -3 . 2959 1 . 7699 -0. .0000 1.0000 28 *SEPRED 0. .0367 0. 1379 0 .0543 0.0190 28 *ADJPRED 1 . . 5601 2 . 1357 1 . 8434 0.1365 28 *MAHAL 0 .2182 15 . 7172 1 . 9286 2.8479 28 *COOK D O .OOOO 5 . 3060 0 . 2247 0.9992 28 TOTAL CASES = 28 DURBIN-WATSON TEST = 2.38276 OUTLIERS - STANDARDIZED RESIDUAL SEONUM SUBFILE *ZRESID 2 NONAME -2 .57561 20 NONAME -1 . 93982 3 NONAME 1 . .75529 23 NONAME 1 . .71903 1 NONAME 1 . 33676 22 NONAME 0. .97819 8 NONAME -0. .96630 26 NONAME 0. .94388 25 NONAME -0. .71622 1 1 NONAME -0. .69269 FILE NONAME (CREATION DATE = 02/06/84) HISTOGRAM - STANDARDIZED RESIDUAL = 1 CASES, N EXP N ( * 0 0 03 OUT 0 0 02 3 00 0 0 02 2 87 0 0 03 2 75 0 0 04 2 62 0 0 06 2 50 0 0 08 2 37 0 0 1 1 2 25 0 0 15 2 12 0 0 19 2 00 0 0 24 1 87 2 0 30 1 75 0 0 37 1 62 0 0 45 1 50 1 0 54 1 37 0 0 64 1 25 0 0 74 1 12 2 0 85 1 00 : * 0 0 95 0 87 0 1 05 0 75 3 1 15 0 62 • * 0 1 23 0 50 2 1 30 0 37 : * 1 1 35 0 25 1 1 38 0 12 3 1 40 O 00 : * 1 1 38 -0 12 2 1 35 -0 25 2 1 30 -0 37 - * 2 1 23 -0 50 : * 1 1 15 -0 62 2 1 05 -o 75 . * 0 0 95 -0 87 1 0 85 -1 00 0 0 74 -1 12 0 0 64 -1 25 0 0 54 -1 37 0 0 45 -1 50 0 0 37 -1 62 0 0 30 -1 75 0 0 24 -1 87 1 0 19 -2 00 * 0 0 15 -2 12 0 0 1 1 -2 25 0 0 08 -2 37 0 0 06 -2 50 1 0 04 -2 62 * 0 0 03 -2 75 0 0 02 -2 87 0 0 02 -3 00 0 0 03 OUT = NORMAL CURVE) CO FILE NONAME (CREATION DATE = 02/06/84) NORMAL PROBABILITY (P-P) PLOT - STANDARDIZED RESIDUAL 1 . 00 + + + * * * 0 B S E . 50 R V E D . 25 25 - + . 5 FILE NONAME (CREATION DATE = 02/06/84) STANDARDIZED SCATTERPLOT ACROSS - *ZPRED DOWN - *ZRESID OUT + + + + + + + - 3 + • 1 + I I -3 + OUT ++- -3 SYMBOLS: MAX N 1 . : 2. 3 OUT 1971 CORRELATION LINC9 AGE L0C1 L0C2 L0C5 LOC LINC9 1 000 -0 693 0 385 0 1 17 -0 074 -0 33 AGE -0 693 1 000 -0 134 0 164 -0 199 0 12 L0C1 0 385 -0 134 1 000 -0 207 -0 139 -0 49 L0C2 0 1 17 0 164 -0 207 1 000 -0 139 -0 49 L0C5 -0 074 -0 199 -0 139 -0 139 1 000 -0 33 L0C6 -0 331 0 125 -0 496 -0 496 -0 334 1 00 FLAR 0 862 -0 613 0 219 0 089 -0 1 13 -0 27 LFAST -0 090 0 012 - o 175 0 104 0 186 -0 27 LNOST 0 464 -0 339 0 395 0 007 0 136 -0 41 LLOT 0 854 -0 666 0 027 0 172 -0 1 10 -0 07 DM2 0 040 0 100 0 093 -0 062 -0 042 0 04 DM3 0 045 -0 072 -0 120 0 216 0 032 -0 18 DM4 0 174 -0 287 0 152 -0 227 0 102 0 02 LGIM 0 173 -0 526 -0 107 -0 087 0 192 0 03 o LGIM LINC9 AGE L0C1 L0C2 L0C5 L0C6 FLAR LF AST LNOST LLOT DM2 DM3 DM4 LGIM 0. 173 -0.526 -0.107 -0.087 0. 192 0.035 0. 302 0.057 0.079 0.312 0.009 O. 178 0.007 1 .000 MULTIPLE R 0.79628 R SQUARE 0.63406 ADJUSTED R SQUARE 0.48159 STANDARD ERROR 0.12099 ANALYSIS OF VARIANCE REGRESSION RESIDUAL DF 10 24 4.15850 VARIABLES IN THE EQUATION VARIABLE B SE B BETA T SIG LINC9 -0.47565 0.11412 - 1 71090 -4 168 0 00 DM2 0.07802 0.07367 0 23077 1 059 0 30 L0C5 0.58540 0.19305 0 98946 3 032 0 00 L0C2 0.61735 0.18945 1 40481 3 259 0 00 L0C1 0.63322 0.20339 1 44092 3 1 13 0 00 DM4 0.06187 0.08531 .0 14943 0 725 0 47 AGE -0.00705 0.00161 -0 91751 -4 376 0 00 DM3 0.15525 0.07762 0 42347 2 000 0 05 FLAR 0.40504E-04 0.1110E-04 1 30687 3 649 0 00 L0C6 0.57993 0.17971 1 74431 3 227 0 00 (CONSTANT) 5.64515 0.94772 5 957 0 00 CASEWISE PLOT OF STANDARDIZED RESIDUAL -3.0 0.0 3.0 SEQNUM 0: : :0 LGIM 1 . * . . 1 .4834 2 . * . 1.9543 3 *. 1.8951 4 * 1.9643 5 . * . 1.9113 6 . * . 1.8254 7 * . 2.0133 8 * . 1.6154 9 * I .7537 10 * . 1.8656 1 1 . * . . 1.9490 12 * 2.2322 13 + . 1.9048 14 * 1 .9720 15 *. . 1.9636 16 * . 1 .9717 17 * . 1.9244 18 * 1.8274 19 * . 1.9720 20 * . 1.7206 21 . + . 1.9438 22 . * 1.8224 23 . * . 1.6513 24 * 1.7217 25 *. 1.8866 26 * . 1.9602 27 * . 1.9113 28 . * . 1.357 1 29 * . 1.5726 30 + . 1 .7837 3 1 . . * 1.8828 32. . * . 1.92 10 33 * 1.7726 34 * . 2.0330 35 * . 1.9643 SEONUM 0: : :0 LGIM -3.0 0.0 3.0 (V) FILE NONAME (CREATION DATE = 02/06/84) * * * * M U L T I P L E DEPENDENT VARIABLE . . LGIM H: $i + ^: ^ ^ rfc RESIDUALS STATISTICS: MIN MAX MEAN STD DEV N *PRED 1 . 5401 2 .0959 1 . 8544 0.1320 35 *ZPRED -2 . 38 17 1 .8300 -0. .0000 1.0000 35 *SEPRED 0 .037 1 0 . 1 189 0 .0585 0.0149 35 *ADJPRED 1 .5122 2 .0623 1 .8556 0. 1361 35 *MAHAL 2 . 3449 33 .0286 7 .7714 5 . 2737 35 *COOK D 0. .0 0. .6118 0 .0408 0.1046 35 TOTAL CASES = 35 DURBIN-WATSON TEST = 2.02195 * * * He * * * * * * * * + OUTLIERS - STANDARDIZED RESIDUAL SEQNUM SUBFILE *ZRESID 1 NONAME -2 . 721'33 28 NONAME -1 .53805 4 NONAME 1 . 34821 24 NONAME 1 .34102 14 NONAME 1 .32322 20 NONAME -1 .25785 26 NONAME 1 .21091 12 NONAME 1 .14536 8 NONAME - 1 .12219 13 NONAME -0. .98796 rv FILE NONAME (CREATION DATE = 02/OG/84) HISTOGRAM - STANDARDIZED RESIDUAL N EXP N ( * = 1 0 0 04 OUT 0 0 02 3 00 0 0 03 2 87 0 0 04 2 75 0 0 06 2 62 0 0 08 2 50 0 0 10 2 37 0 0 14 2 25 0 0 18 2 12 0 0 24 2 00 0 0 30 1 87 0 0 38 1 75 0 0 47 1 62 0 0 57 1 50 3 o 68 1 37 . * * 1 0 80 1 25 1 0 93 1 12 0 1 06 1 00 0 1 19 0 87 2 1 32 0 75 : * 2 1 44 0 62 : * 0 1 54 0 50 5 1 63 0 37 * • * : ( : * 0 1 69 0 25 1 1 73 0 12 * . 4 1 74 0 00 3 1 73 -o 12 4 1 69 -0 25 0 1 63 -0 37 2 1 54 -0 50 * : 2 1 44 -0 62 : * 0 1 32 -0 75 0 1 19 -0 87 1 1 06 -1 00 1 0 93 -1 12 1 0 80 -1 25 0 0 68 -1 37 1 0 57 -1 50 0 0 47 -1 62 0 0 38 -1 75 0 0 30 -1 87 0 0 24 -2 00 0 0 18 -2 12 0 0 14 -2 25 0 0 10 -2 37 0 0 08 -2 50 0 0 06 -2 62 1 0 04 -2 75 0 0 03 -2 87 0 0 02 -3 00 0 0 04 OUT NORMAL CURVE) FILE NONAME (CREATION DATE = 02/06/84) NORMAL PROBABILITY (P-P) PLOT - STANDARDIZED RESIDUAL 1 .00 + + + * * * * * * * * * + ± + * * . 25 FILE NONAME (CREATION DATE = 02/06/84) STANDARDIZED SCATTERPLOT ACROSS - *ZPRED DOWN - *ZRESID OUT ++ + + + + 3 + - + + + SYMBOLS: MAX N 1 . 2 . 3 . -1 -3 + OUT ++- -3 - 1 1 - + ++ 2 3 OUT 1972 CORRELATION LINC9 AGE L0C1 L0C2 L0C5 LOC LINC9 1 000 -0 417 0 554 -0 202 -0 042 -0 24 AGE -0 417 1 000 -0 032 0 160 0 198 - o 20 L0C1 0 554 -0 032 1 000 -0 225 -0 298 -0 43 L0C2 -0 202 0 160 - o 225 1 000 -0 195 -0 28 L0C5 -0 042 0 198 -0 298 -0 195 1 000 -0 24 L0C6 -0 240 -0 200 -0 439 -0 287 -0 248 1 00 FLAR 0 885 -0 247 0 552 -0 210 -0 099 -0 20 LFAST 0 033 0 163 0 080 0 201 0 06 1 - o 24 LNOST 0 664 -0 392 0 454 -0 159 -0 239 - o 06 LLOT 0 808 -0 340 0 356 -0 142 -0 1 16 -0 02 DM2 0 175 -0 009 -0 088 0 262 0 135 -0 02 DM3 -0 200 0 009 -0 107 0 101 -0 036 0 00 DM4 -0 123 -0 003 - o 107 -0 240 -0 036 0 12 LGIM -0 076 -0 294 -0 067 0 161 -0 152 0 21 LGIM LINC9 AGE L0C1 L0C2 L0C5 L0C6 FLAR LF AST LNOST LLOT DM2 DM3 DM4 LGIM -0.076 -0.294 -0.067 0.161 -0.152 0.211 -0.043 0. 182 0. 159 -0.065 0.025 0. 192 -0.251 1 .000 MULTIPLE R 0.61843 R SQUARE 0.38245 ADJUSTED R SQUARE 0.13086 STANDARD ERROR 0.18168 ANALYSIS OF VARIANCE REGRESSION RESIDUAL DF 1 1 27 F = 1.52013 VARIABLES IN THE EQUATION VARIABLE B SE B BETA T SIG LINC9 0 007 15 0 10120 0 02585 0 071 0 94 LF AST 0 29350 0 15226 0 34930 1 928 0 06 DM4 -0 1 1 140 0 08468 - o 26061 - 1 316 0 19 DM2 -0 06060 0 10198 -0 1 1366 -0 594 0 55 L0C6 0 20394 0 10744 0 50859 1 898 0 06 AGE -0 00355 0 00154 -0 43807 -2 303 0 02 DM3 -0 02286 0 08468 -0 05348 -0 270 0 78 L0C2 0 19164 0 14265 0 33306 1 343 0 19 L0C1 0 1 3574 0 13332 0 30813 1 018 0 31 LLOT -0 16461 0 13864 - o 36300 -1 187 0 24 (CONSTANT) 1 4 1488 1 05392 1 342 0 19 CASEWISE PLOT OF STANDARDIZED RESIDUAL -3.0 0.0 3.0 SEQNUM 0: : 0 LGIM 1 * 2.5268 2 * 1.9476 3 * . . 1 .8058 4 * . 1.9210 5 .* . 2.0055 6 + . . 1 . 9 0 5 5 7 * . 1.5619 IV) 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 SEQNUM 0 : . . -3.0 0.0 . . : 0 3.0 2.0381 1.9000 1 .8943 2.0730 2.2933 2 . 0104 1.9119 1.8271 1.9803 1.9699 1.9238 1.9866 1 .792 1 1.7722 2.1669 1.7615 1.9622 1.8933 2.0254 2.0456 1.6797 1.9048 2.0090 1.5968 1.9633 1.9622 1.8874 1.3156 1.9360 2.0480 1.9199 LGIM FILE NONAME (CREATION DATE = 02/06/84) CASEWISE PLOT OF STANDARDIZED RESIDUAL SEQNUM 39 SEQNUM -3.0 0 : . . 0 : . . -3.0 0.0 0.0 3.0 . . : 0 . . : 0 3.0 LGIM 2.0259 LGIM RESIDUALS STATISTICS: MIN MAX MEAN STD DEV N *PRED 1 . 6704 2 .1117 1 . 9269 0.1019 39 *ZPRED -2 .5185 1 .8145 -0 .0000 1.0000 39 *SEPRED 0 .0351 0. .1040 0 .0610 0.0156 39 *ADJPRED I . 7205 2 . 0867 1 . 9277 0.1017 39 *MAHAL 0 . 5406 12 . 36 1 1 3 . . 8974 2.5565 39 *C00K D 0 .0001 0. 6800 0 .0499 0.1332 39 TOTAL CASES 39 DURBIN-WATSON TEST = 1 .7337 1 \ 0 FILE NONAME (CREATION DATE = 02/06/84) OUTLIERS - STANDARDIZED RESIDUAL SEQNUM SUBFILE *ZRESID 1 NONAME 3 .26051 35 NONAME -2 .02022 28 NONAME . - 1 .44804 1 1 NONAME 1 .40058 31 NONAME - 1 .29031 30 NONAME 1 .27666 21 NONAME - 1 .20674 7 NONAME - 1 .20032 27 NONAME 1 .17467 12 NONAME 1 .03389 FILE NONAME (CREATION DATE = 02/06/84) HISTOGRAM - STANDARDIZED RESIDUAL N EXP N ( * 1 0 04 OUT t 0 0 02 3 00 0 0 03 2 87 0 0 04 2 75 0 0 06 2 62 0 0 09 2 50 0 0 12 2 37 0 0 16 .2 25 0 0 20 2 12 0 0 26 2 00 0 0 34 1 87 0 0 42 1 75 0 0 52 1 62 0 0 63 1 50 1 0 76 1 37 1 0 89 1 25 1 1 03 1 12 1 1 18 1 00 2 1 33 0 87 0 1 47 0 75 3 1 60 0 62 0 1 72 0 50 1 1 81 0 37 * 5 1 88 0 25 1 1 93 0 12 * 3 1 94 0 00 2 1 93 -0 12 4 1 88 -0 25 * 1 1 81 -0 37 2 1 72 -0 50 * 2 1 60 -0 62 1 1 47 -0 75 2 1 33 -o 87 0 1 18 -1 00 0 1 03 -1 12 3 0 89 -1 25 0 0 76 -1 37 1 0 63 -1 50 0 0 52 -1 62 0 0 42 -1 75 0 0 34 -1 87 1 0 26 -2 00 * 0 0 20 -2 12 0 0 16 -2 25 0 0 12 -2 37 0 0 09 -2 50 0 0 06 -2 62 0 0 04 -2 75 0 0 03 -2 87 0 0 02 -3 00 0 0 04 OUT NORMAL CURVE) FILE NONAME (CREATION DATE = 02/06/84) NORMAL PROBABILITY (P-P) PLOT - STANDARDIZED RESIDUAL 1 . 00 + + + * * A * + * !̂ .25 .FILE NONAME (CREATION DATE = 02/06/84) STANDARDIZED SCATTERPLOT ACROSS - *ZPRED OUT + + + + - 3 + DOWN - *ZRESID SYMBOLS: MAX N -3 + OUT ++- -3 - - + - - 1 -++ 3 OUT 1973 CORRELATION LINC9 AGE L0C1 L0C2 L0C5 LOC LINC9 1 000 -0 488 0 668 -0 153 -0 060 -0 36 AGE -0 488 1 000 -0 41 1 0 185 0 185 0 27 L0C1 0 668 -0 411 1 000 -0 189 -0 189 - o 58 L0C2 -0 153 0 185 -0 189 1 000 -0 098 -0 30 L0C5 -0 060 0 185 -0 189 -0 098 1 000 0 07 L0C6 -0 369 0 271 -0 584 - o 302 0 074 1 00 FLAR 0 757 -0 418 0 422 -0 120 -0 072 -0 34 LF AST 0 131 0 000 -0 039 0 087 0 100 0 07 LNOST 0 610 -0 387 0 253 0 001 -0 070 -0 17 LLOT 0 213 -0 245 0 229 0 064 -0 444 -0 29 DM2 0 078 -0 005 0 077 0 142 -0 01 1 -0 06 DM3 0 062 -0 101 -0 070 -0 036 -0 181 - o 06 DM4 0 099 - o 007 0 049 0 124 -0 024 -0 02 LGIM 0 073 -0 475 0 144 0 180 -0 063 -0 28 LINC9 LGIM 0.073 AGE LOOM L0C2 L0C5 L0C6 FLAR LF AST LNOST LLOT DM2 DM3 DM4 LGIM -0;. 475 0. 144 0. 180 -0.063 -0.289 0. 201 -0.062 O. 225 0. 222 0.003 -0.093 0. 103 1 .000 MULTIPLE R 0.66608 R SQUARE 0.44367 ADJUSTED R SQUARE 0.32004 STANDARD ERROR O.20867 ANALYSIS OF VARIANCE REGRESSION RESIDUAL DF 10 45 F = 3.58866 VARIABLES IN THE EQUATION VARIABLE B SE B BETA T SIG LINC9 -0 25185 0 09094 -0 64432 -2 769 0 00 L0C5 0 1 1855 0 1 1420 0 13480 1 038 0 30 DM2 -0 00179 0 08536 -0 00293 -0 02 1 0 98 L0C2 .0 22997 0 10554 0 26149 2 179 0 03 DM3 -0 097 14 0 08 158 -0 16773 - 1 191 0 24 LLOT 0 02975 0 02600 0 14790 1 144 0 25 AGE -0 00652 0 00147 -0 58927 -4 440 0 00 DM4 0 06197 0 08534 0 10434 0 726 0 47 L0C1 0 09702 0 09129 0 17133 1 063 0 29 FLAR 0.11436E-04 0.4987E-05 0 43875 2 293 0 02 (CONSTANT) 4 18512 0 88906 4 707 0 00 CASEWISE PLOT OF STANDARDIZED RESIDUAL -3.0 0.0 3.0 SEQNUM 0 : :0 LGIM 1 * 2.1801 2 * 2.1014 3 * . 2.2015 4 .* . 1 . 8 8 8 1 5 * 1.7896 6 * . . 1 .6916 i—* 7 * . . 1.9663 8 * 1.9285 9 * 1 .8540 10 .* . 2.0608 11 * 1.8279 12 . * 2.1469 13 * 1.9671 14 * . 1.7979 15 *. 1.9386 16 .* . 1.7792 17 * . 1.8743 1 8 . . * 2.0609 19 . * 2.1826 20 * . 2.0777 2 1 * 1.8669 22 * . 1.9327 23 * 1.8934 24 * 1.8658 25 . 1.3016 26 * . 2.1292 27 * 1.5306 28 * . 2.0501 29 *. . 1.7042 30 * . 2.0577 31 * 2.0023 32 * 1.8690 33 . * 2.0193 34 * 1.4022 35 * 1.1128 36 * . 1.7336 37 *. 1.6941 38 .* 1.9798 SEONUM 0: :0 LGIM -3.0 0.0 3.0 FILE NONAME (CREATION DATE = 02/06/84) CASEWISE PLOT OF STANDARDIZED RESIDUAL FILE NONAME (CREATION DATE = 02/06/84) * * * * M U L T I P L E DEPENDENT VARIABLE. LGIM RESIDUALS STATISTICS: MIN MAX MEAN STD DEV N *PRED 1 .5744 2 . 2872 1 .8767 0. 1603 56 *ZPRED - 1 .8849 2 . 5604 -0 .0000 1 . 0000 56 *SEPRED 0. .0370 0. . 1268 0 .0633 0. 0229 56 *ADJPRED 1 . . 4902 2 . 3288 1 .8770 0. 1643 56 *MAHAL 0. . 801 1 20. .0048 4 .9107 4 . 6780 56 *COOK D 0. .0000 0. .3179 0 .0244 0. 0575 56 TOTAL CASES = 56 DURBIN-WATSON TEST = 2.01126 OUTLIERS STANDARDIZED RESIDUAL SEQNUM SUBFILE ,:ZRESID 45 NONAME -3 .32209 26 NONAME 2 .70164 35 NONAME -2 .29126 25 NONAME - 1 .84541 14 NONAME - 1 .43436 40 NONAME 1 .35646 6 NONAME - 1 . 34934 46 NONAME -1 .28536 3 NONAME 1 .24258 28 NONAME 1 .07308 FILE NONAME (CREATION DATE = 02/06/84) HISTOGRAM - STANDARDIZED RESIDUAL N EXP N ( * = 1 CASES 0 0 06 OUT 0 0 03 3 .00 0 0 04 2 . 87 1 0 06 2 . 75 * 0 0 09 2 . 62 0 0 12 2 . 50 0 0 17 2 . 37 0 0 22 2 . 25 0 o 29 2.12 0 0 38 2 .00 0 0 48 1 . 87 o 0 60 1 . 75 0 0 75 1 . 62 o 0 91 1 . 50 1 1 09 1 . 37 1 1 28 1 . 25 1 1 48 1.12 2 1 69 1 .00 1 •1 90 0. 87 * 4 2 1 1 0. 75 4 2 30 0. 62 1 2 46 0.50 * 6 2 60 0.37 5 2 7 1 0. 25 4 2 77 0.12 4 2 79 0.00 **:* 2 2 77 -0.12 3 2 71 -0. 25 1 2 60 -0. 37 * 4 2 46 -0. 50 0 2 30 -0.62 3 2 1 1 -0. 75 1 1 90 -0.87 * . 0 1 69 -1 .00 1 1 48 -1.12 1 1 28 - 1 . 25 2 1 09 - 1 . 37 0 0 91 - 1 . 50 0 0 75 - 1 . 62 0 0 60 -1 . 75 1 0 48 -1 .87 0 0 38 -2.00 0 0 29 -2.12 1 0 22 -2 . 25 * 0 0 17 -2 . 37 0 0 12 -2.50 0 0 09 -2 . 62 0 0 06 -2 . 75 0 0 04 -2 . 87 0 0 03 -3.00 1 0 06 OUT * NORMAL CURVE) CO FILE NONAME (CREATION DATE = 02/06/84) NORMAL PROBABILITY (P-P) PLOT - STANDARDIZED RESIDUAL 1 .00 + + + * * :)". * # # 25 5 FILE NONAME (CREATION DATE = 02/06/84) STANDARDIZED SCATTERPLOT ACROSS - *ZPRED DOWN - *ZRESID OUT ++ + + + + 3 + -1 -3 + OUT ++- -3 • - + -2 - + + + SYMBOLS: MAX N 1 . : 2. * 3. - 1 _ + -i + - 0 1 2 + - + + 3 OUT 1974 CORRELATION LINC9 AGE L0C1 L0C2 L0C5 LOC LINC9 1 000 -0 435 0 028 -0 166 -0 234 0 27 AGE -0 435 1 000 -0 034 -0 145 0 123 -0 02 L0C1 0 028 -0 034 1 000 -0 107 -0 160 -0 36 L0C2 -0 166 -0 145 -0 107 1 000 -0 160 -0 36 L0C5 - o 234 0 123 -0 160 -0 160 1 000 -0 54 L0C6 0 277 -0 029 -0 361 -0 361 -0 540 1 00 FLAR 0 906 -0 437 -0 034 -0 202 -0 219 0 34 LFAST -0 275 -0 272 -0 018 0 013 0 262 - o 26 LNOST O 461 -0 446 -0 082 -0 032 -0 438 0 35 LLOT O 760 -0 713 -0 040 -0 1 16 -0 162 0 23 DM2 0 108 -0 107 0 008 -0 226 -0 338 0 34 DM3 - o 1 17 -0 015 -0 107 0 262 0 392 -0 36 DM4 -0 063 -0 008 0 153 0 153 0 007 - o 30 LGIM -0 21 1 -0 541 0 204 0 231 0 063 - o 33 o LGIM LINC9 AGE L0C1 L0C2 L0C5 L0C6 FLAR LFAST LNOST LLOT DM2 DM3 DM4 LGIM -0.211 -0.541 0. 204 0.231 0.063 -0.336 -0.076 O. 552 -0.038 O. 200 -0.016 0. 100 -0.040 1 .000 MULTIPLE R 0.84246 R SQUARE 0.70975 ADJUSTED R SQUARE 0.58535 STANDARD ERROR 0.14058 ANALYSIS OF VARIANCE REGRESSION RESIDUAL DF 9 21 5.70563 VARIABLES IN THE EQUATION VARIABLE B SE B BETA T SIG LINC9 -0 13743 0 05920 -0 35143 -2 32 1 0 03 DM2 -0 C4216 0 06254 -0 09177 -0 674 0 50 DM3 -0 08483 0 10169 -0 1 1678 -0 834 0 41 LFAST 0 19599 0 08475 0 34261 2 313 0 03 DM4 -0 18039 0 08326 -0 30894 -2 167 0 04 L0C5 -0 10506 0 09146 - o 19328 - 1 149 0 26 AGE -0 00569 0 00143 -0 59829 -3 989 0 00 L0C6 -0 14989 0 0831 1 -0 34733 - 1 803 0 08 (CONSTANT) 2 43881 0 977 16 2 496 0 02 CASEWISE PLOT OF STANDARDIZED RESIDUAL -3.0 0.0 3.0 SEQNUM 0: :0 LGIM 1 * 2.2154 2 *. 2.0522 3 * . 2.1143 4 . * . 1.9816 5 * 2.1957 6 * 2.0280 7 . * 1.9432 8 * . 1.8302 9 . * 2 2405 10 * 1 9566 1 1 2 1337 12 1 8681 13 * 2 0610 14 * 1 8704 15 * 2 0129 16 * 1 7265 17 2 3657 18 2 061 1 19 * 1 5293 20 * 2 1876 21 1 591 1 22 * 2 0693 23 * 2 2247 24 * 1 9100 25 * 1 7373 26 2 1317 27 2 0157 28 * 1 4474 29 . * 2 1273 30 2 1 107 31 2 2458 SEONUM 0 : . . . • 0 LGIM -3.0 0.0 3.0 FILE NONAME (CREATION DATE DEPENDENT VARIABLE.. LGIM = 02/06/84) * * * * M U L T I RESIDUALS STATISTICS: MIN MAX MEAN STD DEV N * P R E D 1.5295 2.2355 1.9995 0.1672 31 *ZPRED -2.8113 1.4116 -0.0000 1.0000 31 *SEPRED 0.0318 0.0926 0.0505 0.0170 31 *ADJPRED 1.4898 2.2778 2.0010 0.1672 31 *MAHAL 0.4198 10.7847 2.9032 2.8034 31 *COOK D 0.0000 0.2666 0.0370 0.0554 31 TOTAL CASES = 31 DURBIN-WATSON TEST = 1.51312 $L ft & OUTLIERS - STANDARDIZED RESIDUAL SEONUM SUBFILE *ZRESID 23 NONAME 1.86060 17 NONAME 1.79273 19 NONAME -1.78 101 14 NONAME -1.76555 15 NONAME -1.50444 12 NONAME -1.43084 22 NONAME 1.27155 28 NONAME -1.07586 6 NONAME 1.03803 16 NONAME -1.03069 3 L E FILE NONAME (CREATION DATE = 02/06/84) HISTOGRAM - STANDARDIZED RESIDUAL N EXP N ( * 0 0 03 OUT 0 0 02 3 00 0 0 02 2 87 0 0 04 2 75 0 0 05 2 62 0 0 07 2 50 0 0 09 2 37 0 0 12 2 25 0 0 16 2 12 0 0 21 2 00 1 0 27 1 87 * 1 0 33 1 75 * 0 0 41 1 62 0 0 50 1 50 . 0 0 60 1 37 . 1 0 7 1 1 25 : 0 0 82 1 12 . 1 0 94 1 00 : 0 1 05 0 87 . 4 1 17 0 75 : * 0 1 27 0 62 . 1 1 36 0 50 : 3 1 44 0 37 : * 4 1 50 0 25 : * 0 1 53 0 12 . 2 1 54 0 00 * : 2 1 53 -0 12 * : 1 1 50 -0 25 : 1 1 44 -0 37 : 1 1 36 -0 50 : 1 27 -0 62 . 1 1 17 -0 75 : 1 1 05 -0 87 : 1 0 94 -1 00 : 1 0 82 -1 12 : 0 7 1 -1 25 . 1 0 60 -1 37 : 1 0 50 -1 50 : 0 0 41 -1 62 2 0 33 -1 75 ** 0 0 27 -1 87 0 0 21 -2 00 0 0 16 -2 12 0 0 12 -2 25 O 0 09 -2 37 0 0 07 -2 50 0 0 05 -2 62 0 0 04 -2 75 0 0 02 -2 87 0 0 02 -3 00 0 0 03 OUT NORMAL CURVE) FILE NONAME (CREATION DATE = 02/06/84) NORMAL PROBABILITY (P-P) PLOT - STANDARDIZED RESIDUAL 1 .00 + + + ******** ******* + * * 25 .5 FILE NONAME (CREATION DATE = 02/06/84) STANDARDIZED SCATTERPLOT ACROSS - *ZPRED DOWN OUT ++ + + + - 3 + - 1 -3 + OUT ++- -3 •ZRESID — i + _ - + + + SYMBOLS: MAX N + - 0 - - + + - 1 2 3 OUT 1 . 2 . 1975 CORRELATION LINC9 AGE L0C1 L0C2 L0C5 LOC LINC9 1 000 -0 236 0 247 -0 204 -0 34 1 -0 00 AGE -0 236 1 000 0 271 0 129 -0 009 -0 35 L0C1 0 247 0 27 1 1 000 -0 197 -0 327 -0 54 L0C2 -0 204 0 129 -0 197 1 000 -0 104 -0 17 L0C5 -0 34 1 -0 009 -0 327 -0 104 1 000 -0 1 1 L0C6 - o 003 -0 356 -0 544 -0 173 -0 1 10 1 00 FLAR 0 765 0 087 0 188 -0 162 -0 220 0 01 LFAST -0 070 0 138 0 006 0 024 0 260 -0 20 LNOST 0 524 0 224 0 195 -0 1 1 1 -0 039 -0 14 LLOT 0 775 -0 417 -0 058 -0 130 - o 220 0 20 DM2 -0 208 0 48 1 -0 109 0 1 13 -0 086 -0 03 DM3 0 046 -0 360 -0 370 0 061 0 015 0 23 DM4 0 167 -0 102 0 430 -0 1 16 -0 192 -0 15 LGIM - o 501 -0 330 -0 141 0 312 0 228 0 01 LGIM LINC9 AGE L0C1 L0C2 L0C5 L0C6 FLAR LFAST LNOST LLOT DM2 DM3 DM4 LGIM . 501 . 330 141 .312 . 228 .018 . 366 . 227 -0.411 -0.139 -0.266 0. 222 .045 .000 -O. -0. -0. 0. O. O. -0. 0. 0. 1 10. FLAR MULTIPLE R 0.87239 R SQUARE 0.76106 ADJUSTED R SQUARE 0.65717 STANDARD ERROR 0.10302 ANALYSIS OF VARIANCE REGRESSION RESIDUAL DF 10 23 7.32587 VARIABLE VARIABLES IN THE EQUATION -- B SE B BETA SIG LINC9 -0.37152 0.06511 - 1 . . 38737 -5 . 706 0. .00 DM3 0.00731 0.06512 0. 02049 0 .112 0. .91 L0C2 0. 25810 0.08147 0. 35036 3 . 168 0. .00 L0C5 0.06690 0.06308 0. . 13669 1 .061 0. . 29 DM4 0.05590 0.06638 0. 12293 0 .842 0. .40 AGE -0.00446 0.00118 -0. 58933 -3 . 762 0. 00 L0C1 0.13007 0.05384 0. 36467 2 .4 16 0. 02 DM2 -0.02016 0.06824 -0. 05299 -0. . 295 0. 77 LLOT 0.18458 0.06950 0. 49403 2 .656 0. 01 FLAR 0.88627E-05 0.3688E-05 0. 47139 2 . 403 0. 02 (CONSTANT) 4.19436 0.63355 6 .620 0. 00 CASEWISE PLOT OF STANDARDIZED RESIDUAL SEQNUM 1 2 3 4 -3.0 0 : . . 0.0 3.0 . . : 0 LGIM 2.4785 2.1502 2.1263 1.7989 ^3 5 * 2 .1738 6 * 1.9842 7 . * . 1.7558 8 . * . 1.9280 9 * . 1.9925 10 * . 2.0101 11 . * 2.3183 12 + 1.9821 13 *. 1.9907 14 + . 1.9171 15 * . . 2.2085 16 . * 2.2314 17 * . 2.0329 18 * 2.1449 19 * 1.8819 20 . * 2. 1331 2 1 . * . 2.1320 22 * . 2.1594 23 * 1.9308 24 * . 1.9880 25 .* . 2.0267 26 * 2.4065 27 * . 2.1660 28 * 2.094 1 29 * 2.1308 30 * 1.9891 31 * . 1.6068 32 * 2.0675 33 . * . 2.2595 34 * . 2.1742 SEONUM 0 : :0 LGIM -3.0 0.0 3.0 FILE NONAME (CREATION DATE DEPENDENT VARIABLE . . LGIM = 02/06/84) * * * * M U L T I P L E RESIDUALS STATISTICS: MIN MAX MEAN STD DEV N *PRED 1.6656 2.3677 2.0697 0.1504 34 *ZPRED -2.687 1 1.9812 -0.0000 1.0000 34 *SEPRED 0.0268 0.0821 0.0434 0.0149 34 *ADUPRED 1.7337 2.4074 2.0691 0.1487 34 *MAHAL 1.3624 20.8647 5.8235 5.1259 34 *C00K D 0.0000 0.6438 0.0744 0.1474 34 TOTAL CASES = 34 DURBIN-WATSON TEST = 2.04040 * * + * * * + * * * + * * OUTLIERS - STANDARDIZED RESIDUAL SEONUM SUBFILE *ZRESID 30 NONAME -1.75639 26 NONAME 1.70428 22 NONAME -1.65217 7 NONAME -1.56797 11 NONAME 1.37354 6 NONAME 1.30992 18 NONAME 1.2 1125 28 NONAME -1.09741 1 NONAME 1.09741 19 NONAME - 1 .02437 FILE NONAME (CREATION DATE = 02/06/84) HISTOGRAM - STANDARDIZED RESIDUAL N EXP N ( * 0 0 04 OUT 0 O 02 3 00 0 0 03 2 87 0 0 04 2 75 0 0 05 2 62 0 0 07 2 50 0 0 10 2 37 0 0 14 2 25 0 0 18 2 12 0 0 23 2 00 0 0 29 1 87 1 0 37 1 75 * 0 0 45 1 62 0 0 55 1 50 . 1 0 66 1 37 : 2 0 78 1 25 : * 1 0 90 1 12 : 0 1 03 1 00 . 5 1 16 0 87 : * 0 1 28 0 75 . 0 1 39 0 62 . 0 1 50 0 50 . 1 1 58 0 37 * . 3 1 64 0 25 * : 1 1 68 0 12 * . 5 1 69 0 00 * : 0 1 68 -0 12 1 1 64 -0 25 * . 2 1 58 -0 37 * : 0 1 50 -0 50 . 4 1 39 -0 62 : * 1 1 28 -o 75 : 1 1 16 -0 87 : 1 1 03 -1 00 : 1 0 90 -1 12 : O 0 78 -1 25 . 0 0 66 -1 37 . 0 0 55 -1 50 . 2 0 45 -1 62 ** 1 0 37 -1 75 * 0 0 29 -1 87 0 0 23 -2 00 0 0 18 -2 12 0 0 14 -2 25 0 0 10 -2 37 0 0 07 -2 50 0 0 05 -2 62 0 0 04 -2 75 0 0 03 -2 87 0 0 02 -3 00 0 0 04 OUT NORMAL CURVE) O FILE NONAME (CREATION pATE = 02/06/84) NORMAL PROBABILITY (P-P) PLOT - STANDARDIZED RESIDUAL 1 .00 + + + ***** * 25 5 FILE NONAME (CREATION DATE = 02/06/84) STANDARDIZED SCATTERPLOT ACROSS - OUT ++--- 3 + "ZPRED - - + + - -3 + OUT ++--- -3 DOWN - *ZRESID + + SYMBOLS: MAX N 1 . : 2 . 3 OUT 1976 CORRELATION LINC9 AGE L0C1 L0C2 L0C5 LOC LINC9 1 000 -0 34 1 0 491 -0 148 0 038 -0 44 AGE -0 34 1 1 000 -0 141 -0 246 0 204 0 30 L0C1 0 491 -0 141 1 000 -0 145 -0 24 1 -0 56 L0C2 -0 148 -0 246 -0 145 1 000 -0 094 -0 22 L0C5 0 038 0 204 -0 241 -0 094 1 000 -0 20 L0C6 -o 449 0 304 -0 561 -0 220 -0 206 1 00 FLAR 0 831 -0 258 0 408 -0 169 -0 1 18 -0 43 LF AST 0 173 -0 181 0 118 -0 121 0 084 -0 41 LNOST O 548 -0 235 0 379 -0 044 -0 265 -0 06 LLOT O 798 -0 625 0 282 -0 085 -0 09 1 -0 36 DM2 0 204 -0 226 0 228 0 090 0 064 -0 29 DM3 -0 221 0 331 0 062 -0 105 -0 174 0 18 DM4 0 044 0 187 -0 061 -0 136 0 144 -o 01 LGIM -0 1 17 -0 458 0 139 0 260 -0 058 -0 35 LGIM LINC9 -0.117 AGE -0.458 L0C1 0.139 L0C2 0.260 L0C5 -0.058 L0C6 -0.350 FLAR 0.064 LFAST 0.342 LNOST 0.017 LLOT 0.196 DM2 0.197 DM3 0.021 DM4 -0.402 LGIM 1.000 MULTIPLE R 0.75187 R SQUARE 0.56530 ADJUSTED R SQUARE 0.46038 STANDARD ERROR 0.14574 ANALYSIS OF VARIANCE REGRESSION RESIDUAL DF 7 29 F = 5 . 38760 VARIABLES IN THE EQUATION VARIABLE B SE B BETA T SIG LINC9 -0.22773 0.07250 -0 75092 -3 14 1 0 00 DM4 -0.13210 0.07 127 -0 28960 -1 853 0 07 DM3 0.00329 0.08298 0 00619 0 040 0 96 L0C6 -0.15920 0.05696 -0 4054 1 -2 795 0 00 AGE -0.00370 0.00119 - o 45198 -3 1 13 0 00 DM2 -0.02619 0.06550 -0 06265 -0 400 0 69 FLAR 0.81283E-05 0.4828E-05 0 39095 1 684 0 10 (CONSTANT) 4.47714 0.71372 6 273 0 00 CASEWISE PLOT OF STANDARDIZED RESIDUAL -3.0 0.0 3.0 SEQNUM 0: : : 0 LGIM 1 *. . • 2. 1754 2 * 2.1057 3 * . 1.9478 4 * . 1.9276 5 * . 1.9409 6 * 2.0645 7 * 2.0399 8 9 10 1 1 12 13 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 SEONUM 0 : . . -3.0 0.0 . . :0 3.0 2.1381 1.9774 2 . 2389 2.1036 2.2439 1.6596 1.8933 2.0250 1 . 8644 2 . 1789 1 .8650 1.9122 2 . 2182 1 . 7254 2.057 1 1 .7203 1 .4325 1 .7948 2.1211 1 .9382 2.1029 1.9438 2.1124 2.1519 2.0660 2.0364 1.4161 2.0097 2.0885 2.1594 LGIM FILE NONAME ' (CREATION DATE = 02/08/84) * * * * M U L T I P L E DEPENDENT VARIABLE.. LGIM ft ft ft ft ft ft ft ft ft ft ft ft RESIDUALS STATISTICS: MIN MAX MEAN STD DEV N *PRED 1 . .6000 2 . . 2284 1 .9837 0. 1487 37 *ZPRED -2 . 5791 1 . .6454 -0. .0000 1 . 0000 37 *SEPRED 0. .0376 0. .0939 0. .0558 0. 01 18 37 *ADJPRED 1 . .6521 2 . 2529 1 .9835 0. 1530 37 *MAHA L 1 . . 57 24 14 .8981 4 .8649 2 . 6374 37 *COOK D 0. OOCO 0. . 1736 0 .0408 0. 0534 37 TOTAL CASES = 37 DURBIN-WATSON TEST = 1.80195 ft ft ft. ft ft ft ft ft ft ft ft ft ft OUTLIERS - STANDARDIZED RESIDUAL SEONUM SUBFILE *ZRESID 24 NONAME -2 . 19362 13 NONAME - 1 .67919 12 NONAME 1 .50417 31 NONAME 1 .41750 2 NONAME 1 . 37331 33 NONAME 1 .32154 34 NONAME - 1 .30015 7 NONAME - 1 .27995 1 1 NONAME 1 .24864 23 NONAME - 1 .17994 FILE NONAME (CREATION DATE = 02/08/84) HISTOGRAM - STANDARDIZED RESIDUAL N EXP N ( 0 0 04 OUT 0 0 02 3 00 0 0 03 2 87 0 0 04 2 75 0 0 06 2 62 0 0 08 2 50 0 0 1 1 2 37 0 0 15 2 25 0 0 19 2 12 0 0 25 2 00 0 0 32 1 87 0 0 40 1 75 0 0 49 1 62 1 0 60 1 50 3 0 72 1 37 1 0 85 1 25 2 0 98 1 12 0 1 12 1 00 2 1 26 0 87 0 1 39 0 75 2 1 52 0 62 1 1 63 0 50 1 1 72 0 37 1 1 79 0 25 * 1 1 83 0 12 5 1 84 0 00 * 2 1 83 -0 12 2 1 79 -0 25 0 1 72 -0 37 3 1 63 -0 50 * 2 1 52 -0 62 * 1 1 39 -0 75 2 1 26 -0 87 0 1 12 -1 00 1 0 98 -1 12 2 0 85 -1 25 0 0 72 -1 37 0 0 60 -1 50 1 0 49 -1 62 * 0 0 40 -1 75 0 0 32 -1 87 0 0 25 -2 00 0 0 19 -2 12 1 0 15 -2 25 + 0 0 1 1 -2 37 0 0 08 -2 50 0 0 06 -2 62 o 0 04 -2 75 0 0 03 -2 87 0 0 02 -3 00 0 0 04 OUT NORMAL CURVE) O s FILE NONAME (CREATION DATE = 02/08/84) NORMAL PROBABILITY (P-P) PLOT - STANDARDIZED RESIDUAL 1 .00 + + + * ft. ft ft ft ft ft ft ft ft ft ft ft . 25 FILE NONAME (CREATION DATE = 02/08/84) STANDARDIZED SCATTERPLOT ACROSS - *ZPRED . OUT ++ + + - 3 + -2 -3 + OUT + + - -3 -- + - - 1 DOWN - *ZRESID + SYMBOLS: -++ 3 OUT MAX N 1977 CORRELATION LINC9 AGE L0C1 L0C2 L0C5 LOC LINC9 1 OOO -0 358 0 499 -0 1 15 -0 109 -0 28 AGE -0 358 1 000 -0 187 -0 022 0 149 0 30 L0C1 0 499 -0 187 1 000 -0 257 -0 305 -0 39 L0C2 -0 1 15 -0 022 -0 257 1 000 -0 187 -0 24 L0C5 -0 109 0 149 -0 305 -0 187 1 000 -0 02 L0C6 -0 285 0 307 -0 397 -0 243 -0 024 1 00 FLAR 0 891 -0 242 0 429 -0 200 -0 132 -0 27 LFAST -0 094 -0 137 -0 018 -0 017 0 155 -0 38 LNOST 0 G58 -0 292 0 360 -0 088 -0 098 -0 12 LLOT 0 700 -0 44 1 0 024 0 045 -0 046 -0 17 DM2 0 OOS 0 229 0 051 0 054 -0 024 -0 03 DM3 0 025 0 185 0 017 -0 1 12 0 082 0 16 DM4 -0 073 -0 236 -0 173 0 054 -0 024 -0 14 LGIM -0 185 -0 399 0 074 0 154 0 076 -0 37 LGIM LINC9 -0.185 AGE -0.399 L0C1 0.074 L0C2 0.154 L0C5 0.076 L0C6 -0.371 FLAR -0.158 LFAST 0.309 LNOST -0.109 LLOT 0.012 DM2 -0.128 DM3 -0.019 DM4 -0.116 LGIM 1.000 MULTIPLE R 0.76927 R SQUARE 0.59178 ADJUSTED R SQUARE 0.46807 STANDARD ERROR 0.12720 ANALYSIS OF VARIANCE REGRESSION RESIDUAL DF 10 33 F = 4 . 78382 VARIABLES IN THE EQUATION VARIABLE B SE B BETA T SIG LINC9 -0.28105 0 08095 - 1 02887 -3 472 0 00 DM2 -0.12906 0 06572 -0 33339 -1 964 0 05 L0C5 0.08384 0 06 175 0 18756 1 358 0 18 L0C2 0.08085 0 07436 0 16093 1 087 0 28 DM4 -0.21147 0 06535 -0 54626 -3 236 0 00 L0C6 -0.11286 0 06061 -0 29154 - 1 862 0 07 AGE -0.00502 0 00127 -0 55034 -3 967 0 00 DM3 -0.08481 0 06421 -0 22443 - 1 321 0 19 L0C1 0.05914 0 06613 0 15649 0 894 0 37 FLAR 0.11862E-04 0.607 1E-05 0 55324 1 954 0 05 (CONSTANT) 5.06615 0 79465 6 375 0 00 CASEWISE PLOT OF STANDARDIZED RESIDUAL -3.0 0.0 3.0 SEQNUM 0: : 0 LGIM 1 *. . 2.0558 2 . * . 2.1847 3 * . 1.9060 4 * 2. 1238 5 * 2.2795 6 * 2.0817 G)(?i\r>cviD&0<X)^ca(>>*-o>^~coOci[nincnw^a)Tttr>^tT-Fir>oc!)<3 O m ^ O c o c o c n o o o - ^ c n t ^ o a j c r j O r ^ c N r a ^ - ^ ^ c n r o - ^ c T i c O ' ^ ' ^ o o ) - ! l _ J C J o o o m ' - ' - ' - ' - - ^ ' - ' - ' - ^ c N t M C M P i o i R K N ^ M M n n o n n n c o n n D Z o LU l/J FILE NONAME (CREATION DATE = 02/06/84) CASEWISE PLOT OF STANDARDIZED RESIDUAL SEQNUM 39 40 4 1 42 43 44 SEQNUM LGIM 1 .9728 2.0952 1.8040 1.9063 1.9403 2.0849 LGIM RESIDUALS STATISTICS: MIN MAX MEAN STD DEV N *PRED 1 6364 2 2023 2 0046 0. 1234 44 *ZPRED -2 9842 1 6021 -0 0000 1 . 0000 44 *SEPRED 0 0267 0 0705 0 0423 0. 0108 44 *ADJPRED 1 6225 2 1943 2 0047 0 1231 44 +MAHAL 0 8592 1 1 78 19 3 9091 2 . 5115 44 *COOK D 0 0000 0 5879 0 0357 0. 0934 44 TOTAL CASES 44 DURBIN-WATSON TEST 2.02370 O N FILE NONAME (CREATION DATE = 02/06/84) OUTLIERS - STANDARDIZED RESIDUAL SEQNUM SUBFILE *ZRESID 36 NONAME 2 .52101 33 NONAME -2 .21352 34 NONAME - 1 .79239 24 NONAME - 1 .68049 40 NONAME 1 .67215 19 NONAME - 1 .63291 32 NONAME 1 .41938 31 NONAME 1 .36120 25 NONAME 1 .13806 3 NONAME - 1 .03646 FILE NONAME (CREATION DATE = 02/06/84) HISTOGRAM N EXP N ( * 0 0 05 OUT 0 0 02 3 00 0 0 04 2 87 0 0 05 2 75 0 0 07 2 62 1 0 10 2 50 0 0 13 2 37 0 0 18 2 25 o 0 23 2 12 0 0 30 2 00 0 0 38 1 87 0 0 48 1 75 1 0 59 1 62 0 0 71 1 50 2 0 85 1 37 0 1 00 1 25 1 1 17 1 12 1 1 33 1 00 1 1 50 0 87 0 1 66 0 75 6 1 80 0 62 4 1 94 0 50 * 1 2 04 0 37 * 1 2 13 0 25 2 2 18 0 12 * 3 2 19 0 00 * 1 2 18 -0 12 * 3 2 13 -0 25 1 2 04 -0 37 * 3 1 94 -0 50 * 3 1 80 -o 62 * 2 1 66 -0 75 * 2 1 50 -0 87 1 1 33 -1 00 0 1 17 -1 12 0 1 00 -1 25 0 0 85 -1 37 0 0 71 -1 50 2 0 59 -1 62 1 0 48 -1 75 0 0 38 -1 87 0 0 30 -2 00 0 0 23 -2 12 1 0 18 -2 25 * 0 0 13 -2 37 0 0 10 -2 50 0 0 07 -2 62 0 0 05 -2 75 0 0 04 -2 87 0 0 02 -3 00 0 0 05 OUT STANDARDIZED RESIDUAL * = 1 CASES, = NORMAL CURVE) ON FILE NONAME (CREATION DATE = 02/06/84) NORMAL PROBABILITY (P-P) PLOT - STANDARDIZED RESIDUAL 1 .00 + + + * * * * * * + 25 5 FILE NONAME (CREATION DATE = 02/06/84) STANDARDIZED SCATTERPLOT ACROSS - *ZPRED DOWN - *ZRESID OUT ++ + + + + 3 + -++ + SYMBOLS: MAX N 1 . : 2 . * 3. -3 + OUT ++--- -3 h--2 + - + + 3 OUT 1978 CORRELATION LINC9 AGE L0C1 L0C2 L0C5 LOC LINC9 1 000 -0 297 0 528 -0 1 16 -0 256 -0 16 AGE -0 297 1 000 -0 029 -0 004 0 216 0 00 L0C1 0 528 -0 029 1 000 -0 214 -0 304 -0 41 L0C2 -0 1 16 - o 004 -0 214 1 000 -0 172 -0 23 L0C5 -0 256 0 216 -0 304 -0 172 1 000 -0 12 L0C6 -0 169 0 007 -0 416 -0 235 -0 121 1 00 FLAR 0 572 -0 158 0 258 -0 128 -0 190 0 03 LF AST 0 075 -0 014 0 039 0 051 0 089 -0 20 LNOST 0 658 -0 223 0 528 - o 137 -0 192 -0 14 LLOT 0 281 -0 257 0 120 -0 001 -0 285 -0 14 DM2 -0 1 10 0 046 0 058 0 243 -0 224 -0 04 DM3 0 019 -0 089 -0 109 0 048 0 101 -0 14 DM4 0 103 0 053 0 023 -0 161 0 103 0 15 LGIM -0 275 -0 377 -0 200 0 186 0 245 -0 18 LGIM LINC9 -0.275 AGE -0.377 L0C1 -0.200 L0C2 0.186 L0C5 0.245 L0C6 -0.182 FLAR -0.128 LFAST 0.127 LNOST -0.037 LLOT -0.045 DM2 -0.167 DM3 -0.006 DM4 0.120 LGIM 1.000 MULTIPLE R 0.73732 R SQUARE 0.54363 ADJUSTED R SQUARE 0.49348 STANDARD ERROR 0.10056 ANALYSIS OF VARIANCE REGRESSION RESIDUAL DF 10 91 F = 10.84013 VARIABLES IN THE EQUATION VARIABLE B SE B BETA T SIG LINC9 - o 14632 0.02443 -0 64055 -5 989 0 00 DM3 -0 01471 0.03066 -0 04304 -0 480 0 63 LFAST 0 03072 0.01491 0 15100 2 061 0 04 L0C2 0 12925 0.03524 0 28513 3 667 0 00 AGE -0 00380 0.5300E-03 -0 55465 -7 164 0 00 L0C5 0 09586 0.02894 0 27067 3 3 12 0 00 DM4 0 04962 0.02807 0 16231 1 768 0 08 L0C1 0 03087 , 0.02956 0 09799 1 045 0 29 DM2 -0 06 1 1 5 0.02998 -0 18708 -2 039 0 04 LNOST 0 09775 0.03242 0 30950 3 015 0 00 (CONSTANT) 3 3 1802 0. 251 13 13 2 12 0 00 CASEWISE PLOT OF STANDARDIZED RESIDUAL -3.0 0.0 3.0 SEQNUM 0: : 0 LGIM 1 *. . 1.8635 2 * 2.1455 3 . * . 1.9643 4 * 2.3855 5 * . 1.9184 6 *.. . 2.0580 ON ON ' ' • 2.054 3 8 *• 1.9699 9 *• . 1.8976 1 0 * 1.9574 1 1 •* . 2.0177 1 2 *• . 1.9400 1 3 * • . 1.7267 1 4 •* 2.0719 1 5 * • 1.8842 1 6 * 2.2166 1 7 * 2.0130 1 8 •* . 2.0517 1 9 * . 1.9892 2 0 * . 2.0359 2 1 * • . 2.0537 2 2 • * • 1.8955 2 3 . * . 1.8319 2 4 * 2.0883 2 5 * 1.9216 2 6 * . 1.9492 2 7 * 1.9273 2 8 * . 1.9088 2 9 *• 2.0158 3 0 • * . 1 . 9 5 6 8 3 1 * 1.7751 3 2 *• . 2.1349 3 3 * 1 .8247 3 4 * 2.1577 3 5 * . 2.0065 3 6 * 1.8993 3 7 * • 1 .8057 3 8 * . 1.8089 QNUM 0 : : 0 LGIM - 3 0 0.0 3.0 FILE NONAME (CREATION DATE = 02/06/84) CASEWISE PLOT OF STANDARDIZED RESIDUAL -3."0 0.0 3.0 SEONUM 0: :0 LGIM 39 . * 2.0173 40 * . 2.0371 41 * . 1.9140 42 * . 2.0491 43 * . 1.9633 44 . * . 1 .8597 45 *. 1.8647 46 * 1.8763 47 + . . 2.06 14 43 * 2.1230 49 * 1.7086 50 * . 1.9585 51 * 1.8787 52 + . 2.0130 53 * 2.1799 54 * 2.0505 55 * 1.9719. 56 * . 2.1483 57 .* . 2.0969 58 . + . 2. 1344 59 * . 2.0262 60 * . 2. 1371 6 1 * 1.7253 62 * 2.0335 63 * . 1.6172 64 * . 1.9277 65 .* . 1.9615 66 . * 1.9418 67 . * . 2.0012 68 . + 1 .98 1 7 69 * 1.7537 70 + 1.8756 71 * . 1.9335 72 * 2.0994 73 . * 2.093 1 74 * 2.2290 75 . * . . 1.4500 76 *. 1.7532 77 . * 2.0229 78 * 1.8075 79 * 2 . 1344 80 . * . 1 .9892 SEONUM 0: : 0 LGIM -3.0 0.0 3.0 FILE NONAME (CREATION DATE = 02/06/84) CASEWISE PLOT OF STANDARDIZED RESIDUAL FILE NONAME (CREATION DATE = 02/06/84) * * * * M U L T I P L E DEPENDENT VARIABLE.. LGIM RESIDUALS STATISTICS: MI N MAX MEAN STD DEV N * P R E D 1 . 7381 2 . 2280 1 .991 1 0.1035 102 *ZPRED -2 . 4434 2 . 2881 -0. .0000 1.0000 102 +SEPRED 0 .0167 0. .0939 0 .0284 0.0088 102 •ADJPRED 1 . 5648 2 . 2380 1 . 9374 0.1117 102 *MAHAL 1 .8144 87 . 7467 7 .9216 8.6595 102 *COOK D 0 .0000 1 .4943 0 .0246 0.1481 102 TOTAL CASES = 102 DURB1N-WATS0N TEST = 1.96949 OUTLIERS - STANDARDIZED RESIDUAL SEQNUM SUEFILE *ZRESID 75 NONAME -2 .87535 91 NONAME 2 .30526 13 NONAME -2 .29006 63 NONAME -2 . 28546 4 NONAME 2 .20049 31 NONAME -2 . 12382 74 NONAME 1 .93259 22 NONAME - 1 . .83584 21 NONAME -1 . 64928 23 NONAME - 1 . .63006 -o o FILE NONAME (CREATION DATE = 02/06/84) HISTOGRAM N EXP N 0.11 0.06 0.08 0.12 0. 16 O, 0. O. 0 STANDARDIZED RESIDUAL O 0 0 0 0 O 0 2 0 O 1 0 2 1 2 1 2 5 2 5 2 5 6 6 6 1 1 6 7 4 4 1 2 3 1 3 3 0 2 2 O 1 0 1 2 0 0 0 0 1 0 0 . 22 .30 . 4 1 . 53 O. 69 0.88 1 . 10 1 . 36 1 .65 1 .98 2 . 33 70 .09 .47 . 84 18 4 .49 4 . 74 93 04 08 04 93 74 49 18 84 47 09 70 33 1 .98 1 .65 1 . 36 1 . 10 0.88 0.69 0.53 0.41 0.30 0. 22 0. 16 0.12 0.08 0.06 0.11 ( OUT 00 87 75 62 50 37 25 ** 12 . 00 . 1 . 87 : 1 . 75 . 1 .62 : * 1 .50 *. 1 .37 * : 1 . 25 * . 1.12 * *. 1.00 **:+•* 0. 0. 0.62** . 0.50 * * *: 0.37 * * * * 0.25 * * * * 0.12 * * * * 0.00 * * * * -0.12 **** 25 * * * * 1 CASES, = NORMAL CURVE) .87 ** . .75 ***: 3-7 * * + * 50 ***: 62 * 75 ** 0.87 **: 1.00 * . 12 * * : - 1 • 1 .25 * : 1 -1 . 37 •1.50 *: •1 .62 :* • 1 .75 . •1.87 : 00 . 12 : 25 ** 37 50 62 75 87 * 00 OUT FILE NONAME (CREATION DATE = 02/06/84) NORMAL PROBABILITY (P-P) PLOT - STANDARDIZED RESIDUAL 1 .00 + + + ****** ***** 25 5 FILE NONAME (CREATION DATE = 02/06/84) STANDARDIZED SCATTERPLOT ACROSS - +ZPRED DOWN - *ZRESID OUT + + + + + + 3 + -3 + OUT ++- -3 -2 - - + - - 1 _ + + + - 0 1 2 SYMBOLS: MAX N 1 . : 2. * 4. + -++ 3 OUT 1979 CORRELATION LINC9 AGE L0C1 L0C2 L0C5 LOC LINC9 1 000 -0 44 1 0 434 -0 1 1 1 -0 230 -0 14 AGE -0 441 1 000 0 01 1 0 209 0 159 -0 19 L0C1 0 434 0 01 1 1 000 -0 191 -0 229 -0 42 L0C2 -0 1 1 1 0 209 -0 191 1 000 -0 131 -0 24 L0C5 -0 230 0 159 -0 229 -0 131 1 000 -0 29 L0C6 -0 14 1 -0 193 -0 429 -0 246 -0 295 1 00 FLAR 0 746 -0 314 0 334 -0 170 - o 227 -0 10 LF AST 0 113 0 058 0 016 0 037 -0 078 -0 03 LNOST 0 606 -0 271 0 422 -0 042 -0 128 -0 21 LLOT 0 829 -0 535 0 136 -0 061 -0 227 0 02 DM2 0 034 -0 066 -0 065 0 042 0 101 -0 04 DM3 -0 052 -0 023 -0 049 0 077 -0 051 0 02 DM4 0 010 0 043 0 175 -0 123 -0 1 16 0 03 LGIM -0 170 -0 092 0 1 14 0 086 0 159 -0 23 LGIM LINC9 -0.170 AGE -0.092 L0C1 0.114 L0C2 0.086 L0C5 0.159 L0C6 -0.237 FLAR -0.073 LFAST -0.046 LNOST 0.099 LLOT -0.113 DM2 -0.011 DM3 -0.038 DM4 0.131 LGIM 1.000 MULTIPLE R 0.55209 R SQUARE 0.30481 ADJUSTED R SQUARE 0.24108 STANDARD ERROR 0.12837 ANALYSIS OF VARIANCE DF REGRESSION 11 RESIDUAL 120 F = 4.78305 VARIABLES IN THE EQUATION VARIABLE B SE B BETA T SIG LINC9 -0 19729 0.04249 -0 93869 -4 644 0 00 DM4 0 05256 0.03730 0 15178 1 409 0 16 L0C6 -0 04454 0.03388 -0 14527 - 1 315 0 19 L0C2 0 07398 0.04671 0 15016 1 584 0 1 1 DM3 0 007 14 0.03543 0 02185 0 202 0 84 L0C5 0 08334 0.04239 0 19482 1 966 0 05 AGE -0 00231 0.7581E-03 -0 2941 1 -3 043 0 00 LNOST 0 12975 0.04318 0 32555 3 005 0 00 DM2 -0 00266 0.03473 -0 00849 -0 077 0 93 L0C1 0 10682 0.04110 0 31510 2 599 0 01 LLOT 0 13359 0.05563 0 4 1759 2 401 0 01 (CONSTANT) 2 84831 0.28504 9 993 0 00 CASEWISE PLOT OF STANDARDIZED RESIDUAL -3.0 0.0 3.0 SEQNUM 0: : : 0 LGIM 1 * . 2.1480 2 . + . 2.3284 3 . • * . . 2.0818 4 * . . 2.1619 5 . * 2.3617 6 . * 1.7894 ' • • * • 2.2517 8 * 2.1973 9 •* 1.9663 1 0 * 2.1679 1 1 * 2.0854 1 2 * • 2.0798 1 3 * . 2.0276 1 4 * . 2.1904 1 5 * . 1.9652 1 6 * 1.9719 1 7 • * . 1.9432 1 8 * . 2.0756 1 9 * 2.3657 2 0 * 2.04 10 2 1 * 1.9976 2 2 * • 2.1022 2 3 * 2.0395 2 4 * 2.0707 2 5 •* . 2.1648 2 6 • * 2.0231 2 7 • * . 2.2226 2 8 * . 2.2450 2 9 * 2.3792 3 0 * . 1.8770 3 1 * . 2.1230 3 2 • - * 2.24 53 3 3 * . 1.9131 3 4 • * 2. 1828 3 5 * 2.2338 3 6 * . 1.9811 3 7 * 1.9597 3 8 * 2.3368 SEONUM 0: : : 0 LGIM -3.0 0.0 3.0 FILE NONAME (CREATION DATE = 02/06/84) CASEWISE PLOT OF STANDARDIZED RESIDUAL -3.0 0.0 3.0 SEQNUM 0 : : 0 LGIM 39 + . . 1.9468 40 * . 2.0408 41 .* 2.1084 4 2 * 2.0186 43 * . . 2.5135 44 . * . 1 .8761 45 * 2.0235 46 . * . 2.1936 47 * . . 2.0067 48 * 2.0020 49 * 2.37 12 50 * 2.2017 5 1 . . * 2. 1235 52 * 1.8907 53 * . 2.2150 54 * . 2.0133 55 . * 2.1329 56 * . 2.3508 57 * . 1 .9984 58 * 2. 1581 59 . * 2.0589 60 *. . 2.0823 61 * 2.2454 62 . * 2.0954 63 . . 2. 1941 64 * 1.9753 65 * . 2. 1255 66 • . 2.2566 67 . * . 2.005 1 68 *. 2.0522 69 * . 2. 1645 70 . 2. 1339 7 1 . * . 2.082 1 72 * 1 .8088 73 * . 2.1695 74 . * 2.0148 75 . * . 2.0677 76 . * 2.0888 77 . * . 2.0639 78 * . 2.0669 79 * 1.87 18 80 * . 2.0681 SEONUM 0 : : 0 LGIM -3.0 0.0 • 3.0 FILE NONAME (CREATION DATE = 02/0G/84) CASEWISE PLOT OF STANDARDIZED RESIDUAL -3.0 0.0 3.0 SEONUM 0: : :0 LGIM 81 *. 1 .9356 82 . + . 2.0848 83 * . . 2.0165 84 * 2.0513 85 .+ 1 .9244 86 * 1.8893 87 , *. 1.9463 88 . * 2.228 1 89 * 2.1138 30 . * . 2.2961 91 * . . 2.0670 92 .* 1.9822 93 . + 2.0200 94 * 2.0032 95 . * 2.06 10 96 * . 1.9962 97 * 1.9717 98 * 2.0623 99 . . 2.1229 100 * . 1.9402 101 . * 1.9752 102 . + 2.1567 103 .* 2.0653 104 * 1.8486 105 * 1.9141 106 * 2.0348 107 * 1.8535 108 * . 2 . 1122 109 . *. . 2.0275 110 . * 2.0859 1 1 1 . . * 2.1331 112 * 1.9237 113 * . 2.1662 114 • * . 2.2929 115 * . 1.8907 1 1 6 . *. . 1 .8912 117 * . . 1.7790 118 . + . 1.8527 119 * 2.4389 120 * . 1.9387 121 * 2.2268 122 * . 1.8670 SEONUM 0: : :0 LGIM' -3.0 0.0 3.0 FILE NONAME (CREATION DATE = 02/06/84) CASEWISE PLOT OF STANDARDIZED RESIDUAL LGIM 1.9804 2.0334 1.9784 2.1936 2.0756 2.6285 1.9948 2.1131 2.0503 1.9468 LGIM * * * * * * * * * * * * * RESIDUALS STATISTICS: MIN MAX MEAN STD DEV N *PRED 1 8551 2 2865 2 0785 0 0799 132 *ZPRED -2 7950 2 6025 -0 0000 1 0000 132 *SEPRED O 0176 0 0640 0 032 1 0 0090 132 *ADUPRED 1 8686 2 3162 2 0783 0 0801 132 *MAHAL 1 4813 31 9276 7 9394 4 9515 132 *COOK D O OOOO 0 15 11 0 0095 0 0194 132 TOTAL CASES = 132 DURBIN-WATSON TEST = 2.36166 CO FILE NONAME (CREATION DATE = 02/06/84) OUTLIERS - STANDARDIZED RESIDUAL SEONUM SUBFILE *ZRESID 128 NONAME 3 .96254 1 19 NONAME 2 .44258 5 NONAME 2 .33148 72 NONAME -2 .18144 6 NONAME -2 .03590 43 NONAME 2 .01268 29 NONAME 1 . 84273 49 NONAME 1 .83932 44 NONAME -1 .83622 77 NONAME - 1 . .74171 FILE NONAME (CREATION DATE = 02/06/84) HISTOGRAM - STANDARDIZED RESIDUAL N EXP N ( + = 1 CASE 1 0 14 OUT * 0 0 07 3 00 0 0 1 1 2 87 0 0 15 2 75 0 0 21 2 62 1 0 29 2 50 * 1 0 39 2 37 0 0 53 2 25 0 0 69 2 12 1 0 89 2 00 2 1 14 1 87 • * 0 1 43 1 75 1 1 76 1 62 * 1 2 14 1 50 2 2 56 1 37 * * . 0 3 01 1 25 3 3 50 1 12 * * . 5 3 99 1 00 * * * . * 3 4 49 0 87 * * * _ 4 4 97 0 75 * * * * 8 5 41 0 62 * * * * . * * * 9 5 81 0 50 *****.*** 10 6 13 0 37 ***** . **** 4 6 38 0 25 * * * * 6 6 53 0 12 ****** 7 6 58 0 00 ****** . 6 6 53 -0 12 ****** 6 6 38 -0 25 ***** . 9 6 13 -0 37 ***** . *** 5 5 81 -0 50 ***** 6 5 41 -0 62 ****** 7 4 97 -o 75 ****.** 2 4 49 -0 87 * * 6 3 99 -1 00 * * * . * * 1 3 50 -1 12 * 5 3 01 -1 25 * * • * * 3 2 56 -1 37 * * • 1 2 14 -1 50 * 1 1 76 -1 62 * 2 1 43 -1 75 1 1 14 -1 87 1 0 89 -2 00 1 0 69 -2 12 0 0 53 -2 25 0 0 39 -2 37 0 0 29 -2 50 0 0 21 -2 62 0 0 15 -2 75 0 0 1 1 -2 87 0 0 07 -3 00 0 0 14 OUT = NORMAL CURVE) CO o FILE NONAME (CREATION DATE = 02/06/84) NORMAL PROBABILITY (P-P) PLOT - STANDARDIZED RESIDUAL 1 .00 + + + _ * * * * * * * # * * * + * * * FILE NONAME (CREATION DATE = 02/06/84) STANDARDIZED SCATTERPLOT ACROSS - *ZPRED DOWN OUT + + + + + - 3 + "ZRESID - + + - -3 + OUT ++-- -3 - + + + SYMBOLS: - H ^ + - -2 -1 0 - + + 3 OUT SSIGNOFF 183 APPENDIX C R e t u r n and R i s k S t a t i s t i c s of P r o p e r t i e s R e t u r n on St.Dev Var i a n c e R e t u r n on St.Dev V a r i a n c e C a p i t a l E q u i t y A 1 02 0 .05551 0. 1 4821 0 .02197 0. 03849 0. 20794 0. 04324 A 1 06 0 .05716 0. 1 2962 0 .01680 0. 08064 0. 35527 0. 12621 A 1 07 0 .04951 0. 1 2466 0 .01554 0. 03487 0. 18241 0. 03327 A 1 1 1 0 .05325 0. 1 251 1 0 .01565 0. 09546 0. 421 83 0. 1 7794 A 1 1 2 0 .05526 0. 1 2163 0 .01 479 -0. 33608 1 . 09523 1 . 1 9953 A 1 1 3 0 .05501 0. 1 1 694 0 .01368 0. 07753 0. 42323 0. 1 7912 A 1 15 0 .05902 0. 1 9733 0 .03894 0. 05231 0. 31010 0. 0961 6 A 1 16 0 .05814 0. 1 2309 0 .01515 0. 1 561 5 0. 68062 0. 46325 A 1 19 0 .04638 0. 1 2220 0 .01493 -0. 81 649 1. 66354 2. 76735 A 1 22 0 .04941 0. 1 2225 0 .01494 0. 04461 0. 1 7397 0. 03027 A 1 24 0 .06152 0. 1 2299 0 .01513 0. 04980 0. 1 5049 0. 02265 A 1 25 0 .04237 0. 1 1643 0 .01356 0. 1 3322 0. 94332 0. 88984 A 1 26 0 .04406 0. 1 2668 0 .01605 0. 03357 0. 28978 0. 08397 A 1 27 0 .04407 0. 1 3549 0 .01836 0. 07865 0. 32855 0. 1 0794 A 1 28 0 .04432 0. 1 2776 0 .01632 0. 1 4501 0. 76902 0. 59139 A 1 29 0 .04523 0. 1 1 986 0 .01437 0. 04663 0. 25798 0. 06656 A 1 30 0 .04511 0. 11913 0 .01419 0. 051 1 5 0. 29307 0. 08589 A 1 32 0 .04296 0. 1 1 970 0 .01433 0. 02369 0. 1 6472 0. 0271 3 A 1 33 0 .04480 0. 1 2552 0 .01576 -0. 00062 0. 63334 0. 401 1 2 A 1 34 0 .04642 0. 1 2082 0 .01460 0. 031 49 0. 26341 0. 06938 A 1 37 0 .06216 0. 20285 0 .04115 0. 07694 0. 44796 0. 20067 A 1 38 0 .06975 0. 23581 0 .05561 - o . 04027 0. 56889 0. 32363 A 1 39 0 .05203 0. 1 3547 0 .01835 0. 1 0092 0. 49798 0. 24798 A 1 40 0 .04884 0. 1 3551 0 .01836 0. 06646 0. 3871 9 0. 1 4992 A 141 0 .04576 d. 1 3346 0 .01781 0. 04228 0. 22585 0. 051 01 A 1 42 0 .04642 0. 1 2254 0 .01502 0. 81960 4. 76605 22. 71 527 A 1 43 0 .06675 0. 23380 0 .05466 0. 1 5451 0. 71 622 0. 51 297 A 1 44 0 .05762 0. 1 2753 0 .01626 0. 10532 0. 3661 7 0. 1 3408 A 1 45 0 .05028 0. 11125 0 .01238 0. 05028 0. 11125 0. 01 238 A 1 46 0 .04468 0. 1 351 0 0 .01825 0. 05470 0. 28863 0. 08331 A 1 47 0 .05018 0. 1 6469 0 .02712 0. 08457 0. 37759 0. 1 4258 A 1 49 0 .04756 0. 1 2778 0 .01633 0. 01219 0. 25220 0. 06360 A 1 50 0 .05505 0. 1 2520 0 .01568 0. 1 3267 0. 67089 0. 45009 A 201 0 .04666 0. 1 5997 0 .02559 - o . 03041 0. 42502 0. 1 8064 A 202 0 .05247 0. 20497 0 .04201 0. 1 1 501 0. 55050 0. 30304 A 203 0 .04993 0. 1 4596 0 .02130 0. 09062 0. 54272 0. 29455 A 204 0 .04579 0. 1 5042 0 .02263 - o . 03861 0. 92670 0. 85878 A 205 0 .04794 0. 14289 0 .02042 0. 061 24 0. 31806 0. 10116 A 207 0 .04077 0. 1 3004 0 .01691 0. 04407 0. 33507 0. 1 1 227 A 209 0 .04949 0. 1 231 8 0 .01517 - o . 00373 0. 81690 0. 66732 A 210 0 .06973 0. 25507 0 .06506 0. 31 222 1. 491 31 2. 22402 A 21 1 0 .04566 0. 1 2988 0 .01687 0. 02539 0. 17806 0. 03170 A 212 0 .04697 0. 1 2982 0 .01685 0. 02876 0. 21676 0. 04699 184 A 213 0 .04891 0. 1 2608 0 .01590 0. 07791 0. 3821 2 0. 1 4601 A 214 0 .04844 0. 1 2602 0 .01588 0. 03378 0. 24630 0. 06067 A 215 0 .04417 0. 1 1809 0 .01395 0. 04731 0. 1 3067 0. 01708 A 217 0 .05533 0. 1 3362 0 .01785 0. 1 6905 0. 87696 0. 76906 A 218 0 .04760 0. 1 3587 0 .01846 0. 09667 0. 40352 0. 1 6282 A 219 0 .04850 0. 1 2235 0 .01497 0. 04082 0. 1 8763 0. 03520 A 220 0 .04394 0. 12653 0 .01601 0. 03425 0. 25577 0. 06542 A 221 0 .04931 0. 1 2079 0 .01459 0. 1 4426 1 . 501 77 2. 25533 A 222 0 .04273 0. 1 3358 0 .01784 - o . 061 53 1 . 20741 1 . 45784 A 223 0 .04218 0. 1 91 44 0 .03665 - o . 06481 0. 97603 0. 95263 A 224 0 .04545 0. 1 4368 0 .02064 0. 07223 0. 32116 0. 10314 A 225 0 .06323 0. 25382 0 .06443 0. 08907 1 . 0081 1 1. 01 628 A 226 0 .05967 0. 23678 0 .05607 - o . 60970 2. 62339 6. 88219 A 227 0 .06123 0. 26547 0 .07047 0. 1 5876 0. 6801 0 0. 46253 A 229 0 .04394 0. 14151 0 .02003 -1 . 33565 7. 1 3495 50. 90747 A 231 0 .04175 0. 1 2687 0 .01610 0. 01791 0. 20570 0. 04231 A 232 0 .04974 0. 12507 0 .01564 0. 02563 0. 95044 0. 90334 A 233 0 .05286 0. 1 31 02 0 .01717 0. 01 1 00 0. 45970 0. 21 1 33 A 2 34 0 .06326 0. 26074 0 .06799 0. 7791 2 6. 02329 36. 27998 A 235 0 .04548 0. 1 4021 0 .01966 0. 06898 0. 32574 0. 10611 A 238 0 .04345 0. 1 2685 0 .01609 0. 07933 0. 29256 0. 08559 A 301 0 .05545 0. 1 6547 0 .02738 0. 09962 0. 31 978 0. 1 0226 A 302 0 .04728 0. 17015 0 .02895 0. 1 3623 0. 59096 0. 34923 A 304 0 .05036 0. 1 7844 0 .03184 0. 07402 0. 33352 0. 11124 A 306 0 .05201 0. 17171 0 .02948 0. 06033 0. 27535 0. 07582 A 309 0 .04949 0. 17 125 0 .02933 0. 05073 0. 49119 0. 24127 A 310 0 .05041 0. 1 6455 0 .02708 0. 04642 0. 27061 0. 07323 A 31 1 0 .04935 0. 1 7246 0 .02974 0. 07378 0. 2671 8 0. 071 38 A 312 0 .06209 0. 1 7993 0 .03237 0. 1 6877 0. 61 094 0. 37325 A 313 0 .04825 0. 1 7039 0 .02903 0. 1 301 5 0. 52738 0. 27813 A 314 0 .05945 0. 1 8544 0 .03439 0. 091 57 0. 85754 0. 73537 A 315 0 .05028 0. 1 7363 0 .03015 - o . 00368 0. 48966 0. 23976 A 402 0 .04459 0. 1 21 74 0 .01482 0. 04970 0. 25234 0. 06367 A 403 0 .03839 0. 1 3901 0 .01932 0. 03733 0. 22990 0. 05285 A 404 0 .04802 0. 1 3627 0 .01857 - o . 1 2509 0. 74984 0. 56225 A 405 0 .03944 0. 1 3371 0 .01788 0. 24183 1. 6251 5 2. 641 1 2 A 408 0 .04703 0. 1 3591 0 .01847 0. 55692 2. 07627 4. 31088 A 410 0 .04022 0. 1 3655 0 .01865 0. 0441 5 0. 37377 0. 1 3970 A 413 0 .04329 0. 1 3303 0 .01770 0. 1 4443 1. 03403 1 . 06921 A 414 0 .05083 0. 13459 0 .01812 0. 06038 0. 68330 0. 46689 A 416 0 .04817 0. 1 1 375 0 .01294 0. 051 90 0. 30856 0. 09521 A 417 0 .04702 0. 1 2476 0 .01556 - o . 0271 1 0. 78973 0. 62367 A 418 0 .04486 0. 10897 0 .01187 0. 60778 2. 20052 4. 84229 A 419 0 .04360 0. 1 1 430 0 .01306 0. 03121 0. 21 368 0. 04566 A 420 0 .04140 0. 1 5958 0 .02547 - o . 01 537 0. 37639 0. 14167 A 422 0 .04527 0. 1 1887 0 .01413 0. 06796 0. 32872 0. 1 0806 A 423 0 .03811 0. 1 3769 0 .01896 0. 30973 1. 93694 3. 751 72 A 424 0 .04420 0. 1 2825 0 .01645 0. 03737 0. 21 006 0. 0441 3 A 425 0 .0471 1 0. 14395 0 .02072 0. 1 21 23 0. 78775 0. 62055 A 426 0 .04647 0. 1 0303 0 .01062 0. 06222 0. 76389 0. 58352 A 427 0 .04737 0. 1 2429 0 .01545 0. 07680 0. 331 03 0. 10958 A 428 0 .04175 0. 1 2906 0 .01666 0. 06070 0. 38775 0. 1 5035 185 A 429 0. 04738 0. 1 2474 0. 01 556 0. 04642 0. 31 972 0. 10222 A 430 0. 04264 0. 1 2840 0. 01 649 0. 031 96 0. 21 950 0. 04818 A 431 0. 04894 0. 1 2296 0. 01512 0. 06842 0. 51 979 0. 27018 A 432 0. 04373 0. 1 3069 0. 01 708 0. 07768 0. 40531 0. 1 6428 A 434 0. 04547 0. 1 1 971 0. 01 433 0. 1 2284 0. 57328 0. 32865 A 435 0. 05278 0. 1 4034 0. 01 970 0. 1 2750 0. 48873 0. 23886 A 436 0. 04203 0. 1 1 257 0. 01 267 0. 02880 0. 22108 0. 04888 A 437 0. 04353 0. 1 2731 0. 01 621 0. 40346 1 . 91 763 3. 67731 A 439 0. 04925 0. 1 3367 0. 01787 - o . 02576 0. 75021 0. 56282 A 440 0. 04880 0. 1 5772 0. 02488 0. 061 22 0. 29628 0. 08778 A 441 0. 04204 0. 1 2331 0. 01 521 - o . 85293 6. 34741 40. 28963 A 443 0. 05512 0. 1 2732 0. 01 621 0. 01141 0. 22346 0. 04994 A 444 0. 04731 0. 1 1 994 0. 01439 0. 05600 0. 22799 0. 05198 A 501 0. 06204 0. 22207 0. 04932 0. 1 6736 0. 82018 0. 67270 A 601 0. 06775 0. 21 655 0. 04689 0. 05234 0. 83485 0. 69697 A 602 0. 06227 0. 22341 0. 04991 0. 35340 1. 93324 3. 73742 A 604 0. 06044 0. 1 9867 0. 03947 1 . 57423 9. 63870 92. 90453 A1 004 0. 05825 0. 18538 0. 03437 0. 05806 0. 35743 0. 1 2775 A1 005 0. 05602 0. 1 2568 0. 01 579 - o . 29646 1 . 57856 2. 49185 A1 006 0. 04835 0. 1 3312 0. 01772 0. 1 3234 0. 71 235 0. 50744 A1 009 0. 04094 0. 1 1829 0. 01 399 - o . 00307 0. 51 396 0. 2641 5 Al 01 0 0. 04745 0. 1 2230 0. 01 496 0. 071 22 0. 48672 0. 23690 A1 01 3 0. 04925 0. 1 2641 0. 01 598 0. 09986 0. 34710 0. 1 2048 A1 0 1 5 0. 04456 0. 1 2681 0. 01 608 0. 08445 0. 31 608 0. 09990 A1 0 1 6 0. 04020 0. 1 3579 0. 01 844 0. 06832 0 . 28358 0. 08042 A1018 0. 0 5 1 .1 7 0. 1 2275 0. 01 507 0. 06998 0. 18973 0. 03600 A1 0 1 9 0. 041 96 0. 1 2528 0. 01 569 0. 03428 0. 1 9824 0. 03930 A1 021 0. 05246 0. 1 2696 0. 01612 0. 07867 0. 33700 0. 1 1 357 A1 022 0. 04912 0. 1 2506 0. 01 564 0. 05788 0. 27557 0. 07594 A1 023 0. 04899 0. 1 2562 0. 01 578 1. 1 1 320 6. 81 740 46. 47696 A1 027 0. 05782 0. 23958 0. 05740 0. 09929 0. 53946 0. 291 01 A1 03 1 0. 04641 0. 1 41 97 0. 0201 5 0. 08070 0. 36252 0. 1 31 42 A1 032 0. 04527 0. 1 71 49 0. 02941 0. 06482 0. 21 026 0. 04421 A1 037 0. 0551 3 0. 19110 0. 03652 0. 1 3758 0. 56254 0. 31 645 A1 038 0. 05787 0. 1 9024 0. 0361 9 0. 05787 0. 1 9024 0. 0361 9 A1 039 0. 06607 0. 1 3570 0. 01842 0. 1 7871 1. 0281 3 1 . 05705 A1 041 0. 0591 3 0. 1 21 45 0. 01 475 - o . 49298 2. 74992 7. 56206 A1 042 0. 06403 0. 1 0903 0. 01 189 0. 09331 0. 51 764 0. 26795 Al 044 0. 04320 0. 1 2332 0. 01 521 0. 02093 0. 26282 0. 06907 A1 046 0. 05747 0. 18418 0. 03392 0. 05735 0. 1 8463 0. 03409 A1 049 0. 04547 0. 1 2649 0. 01 600 0. 04544 0. 1 2654 0. 01 601 A1 052 0. 04303 0. 12123 0. 01 470 0. 06672 0. 26267 0. 06900 A1 053 0. 04181 0. 1 2208 0. 01 490 0. 04817 0. 1 5580 0. 02427 A1 054 0. 05897 0. 20387 0. 041 56 -1 . 52234 8. 29928 68. 87808 A1 055 0. 04404 0. 1 2467 0. 01 554 0. 06272 0. 31 683 0. 1 0038 A1 056 0. 04736 0. 1 21 27 0. 01 471 0. 06625 0. 25464 0. 06484 A1 057 0. 04299 0. 1 2493 0. 01 561 0. 0281 6 0. 1 7646 0. 031 14 A1 058 0. 05217 0. 1 2725 0. 01619 0. 06307 0. 23760 0. 05646 A1 061 0. 04980 0. 1 3078 0. 01710 4. 22609 22. 39479501. 52661 A1 066 0. 04491 0. 16894 0. 02854 0. 09873 0. 50293 0. 25293 A1 067 0. 04478 0. 1 2881 0. 01 659 0. 05186 0. 1 5352 0. 02357 A1 069 0. 05929 0. 18942 0. 03588 0. 02794 0. 35878 0. 1 2872 186 A1 072 0. 05950 0. 5369 0 .02362 - o . 46889 1 . 47979 2. 18979 A1 073 0. 0521 4 0. 1 964 0 .01431 0. 48186 2. 66377 7 . 09567 A1 074 0. 04838 0. 0276 0 .01056 0. 04357 0. 1 7703 0. 031 34 A1 075 0. 05296 0. 261 0 0 .01590 0. 1 5405 1 . 43526 2. 05997 A1 076 0. 05842 0. 1307 0 .01278 0. 07068 0. 331 50 0. 10989 A1 077 0. 04601 0. 2327 0 .01519 0. 06853 0. 41 949 0. 17597 A1 078 0. 051 58 0. 2524 0 .01569 -0. 00227 0. 96642 0. 93396 A1 080 0. 05747 0. 1205 0 .01256 0. 05747 0. 1 1 205 0. 01256 A1 082 0. 0651 1 0. 3455 0 .01810 0. 04848 0. 58038 0. 33684 A1 083 0. 04446 0. 1213 0 .01257 0. 04446 0. 11213 0. 01 257 A1 084 0. 05773 0. 7448 0 .03044 - o . 04956 0. 62395 0. 38932 A1 085 0. 05028 0. 1 246 0 .01265 -0. 1 4897 5. 87818 34. 55304 A1 087 0. 06482 0. 11129 0 .01239- 99. 00 0. 00 0. 00 A1 090 0. 05470 0. 1 2673 0 .01606 0. 51 1 08 2. 401 1 4 5. 76548 A1 095 0. 05254 0. 13184 0 .01738 0. 62055 2. 22081 4. 931 98 A1 098 0. 05726 0. 1 1 048 0 .01221 0. 07867 0. 44983 0. 20235 A1 1 00 0. 04841 0. 1 0774 0 .01161 -0. 08290 0. 53355 0. 28468 A1 1 03 0. 04981 0. 1 3298 0 .01768 0. 09380 0. 31 635 0. 1 0008 A1 104 0. 22470 1 . 1 4599 1 .31330 0. 22470 1. 1 4599 1 . 31 330 A1 105 0. 04325 0. 1 1 570 0 .01339 0. 03740 0. 32250 0. 1 0401 A1 1 06 0. 041 78 0. 1 1887 0 .01413 0. 04874 0. 1 5007 0. 02252 A1 107 0. 04528 0. 1 1 469 0 .01315 0. 041 55 0. 1 9944 0. 03978 A1 1 1 4 0. 04241 0. 1 2540 0 .01573 0. 1 21 00 0. 60976 0. 37181 A1 1 1 5 0. 041 60 0. 1 2841 0 .01649 0. 74830 4. 59437 21 . 1 0828 A1 1 1 7 0. 05057 0. 1 1 745 0 .01380 0. 07233 0. 47278 0. 22352 A1 1 1 8 0. 041 76 0. 11731 0 .01376 0. 2281 9 1 . 12128 1. 25727 A1 1 21 0. 04090 0. 1 3493 0 .01821 0. 05594 0. 1 9380 0. 03756 A1 1 22 0. 04507 0. 1 2505 0 .01564 0. 04507 0. 1 2505 0. 01 564 A1 1 24 0. 04405 0. 1 1 251 0 .01266 0. 03937 0. 1 3838 0. 01915 A1 1 27 0. 03770 0. 1 5800 0 .02496 0. 04363 0. 28780 0. 08283 A1 1 29 0. 05431 0. 11617 0 .01350 0. 2321 9 0. 80084 0. 641 35 A1 1 30 0. 04530 0. 10711 0 .01147 0. 01 346 0. 49763 0. 24763 A1 132 0. 06296 0. 1 7090 0 .02921 0. 05262 0. 1 9962 0. 03985 A1 1 33 0. 05638 0. 1 8523 0 .03431 0. 01080 0. 35644 0. 1 2705 A1 1 35 0. 0561 4 0. 1 8976 0 .03601 0. 05614 0. 18976 0. 03601 A2003 0. 05056 0. 1 1896 0 .01415 0. 05056 0. 1 1896 0. 01415 A201 1 0. 04921 0. 1 2793 0 .01637 0. 04573 0. 13916 0. 01937 A201 2 0. 04793 0. 1 2228 0 .01495 0. 05937 0. 1 5430 0. 02381 A201 6 0. 03994 0. 1 2647 0 .01600 0. 1 8757 0. 87602 0. 76741 A201 8 0. 04692 0. 1 1 527 0 .01329 - o . 05630 1. 00587 1. 01 1 78 A 2.01 9 0. 04745 0. 12110 0 .01466 0. 06536 0. 26663 0. 071 09 A2025 0. 0461 1 0. 1 2875 0 .01658 0. 03337 0. 24477 0. 05991 A2027 0. 03649 0. 14125 0 .01995 0. 04840 0. 1 9267 0. 03712 A2033 0. 03561 0. 1 2335 0 .01522 - o . 46232 2. 1 1 797 4. 48578 A2035 0. 05409 0. 1 7026 0 .02899 2. 401 01 10. 93716119. 621 44 A2036 0. 05408 0. 1 6776 0 .02815 0. 45458 2. 18556 4. 77669 A2039 0. 04533 0. 1 3692 0 .01875 0. 03817 0. 63873 0. 40798 A2041 0. 04303 0. 1 1871 0 .01409 0. 04971 0. 24498 0. 06002 A2045 0. 04538 0. 14598 0 .02131 0. 09786 0. 83045 0. 68965 A2047 0. 04755 0. 1 2882 0 .01660 0. 51 689 2. 30664 5. 32061 A2048 0. 04640 0. 1 2591 0 .01585 0. 09310 0. 49861 0. 24861 A2049 0. 04494 0. 1 2461 0 .01553 0. 03304 0. 15173 0. 02302 A2051 A3002 A3003 A3005 A3009 A301 0 A301 2 A3029 A3033 A3035 A3036 A3037 A3039 A3040 A3041 A3042 A3043 A3044 A3046 A3049 A3050 A3051 A3052 A3054 A3055 A3056 A3058 A3059 A3061 A3063 A3064 A3069 A3077 A3080 A3084 A3085 A3088 A3089 A3091 A3093 A3094 A3096 A3097 A3 1 0 1 A3 1 02 A31 04 A31 05 A31 06 A3 1 07 A3! 08 A31 1 0 A31 1 4 0.03870 0.04655 0.04349 0.04736 0.04884 0.04619 0.04477 0.04939 0.04281 0.04624 0.04823 0.04621 0.04512 0.05144 0.04860 0.04248 0.04517 0.04500 0.04575 0.04493 0.061 70 0.04621 0.04077 0.04706 0.04376 0.04511 0.04478 0.05624 0.05046 0.05040 0.05852 0.05798 0.04574 0.04089 0.04458 0.04551 0.04934 0.04581 0.04115 0.04779 0.06107 0.05938 0.05854 0.0558.5 0.05778 0.05994 0.05194 0.05123 0.04636 0.05175 0.04902 0.04500 0. 1 1877 0. 1 1 766 0. 1 1 555 0. 11713 0. 1 1 945 0. 13166 0. 1 231 6 0. 1 2560 0. 12901 0. 1 2829 0. 13033 0. 12869 0. 12745 0. 1 2827 0. 1 2325 0. 12609 0. 1 2673 0. 12731 0. 1 3430 0. 1 2924 0. 25381 0. 13518 0. 1 3928 0. 14122 0. 1 4904 0. 14479 0. 1 4287 0. 10877 0. 1 4064 0. 1 1 844 0. 18056 0. 1 0992 0. 11140 0. 1 3322 0. 12761 0. 12687 0. 13027 0. 10994 0. 11417 0. 1 081 9 0. 18875 0. 1 9496 0. 18420 0. 1 7202 0. 1 9924 0. 20065 0. 1 2731 0. 12104 0. 1 2496 0. 1 3045 0. 12301 0. 12281 0.01411 0.01384 0.01335 0.01372 0.01427 0.01733 0.01517 0.01577 0.01664 0.01646 0.01699 0.01656 0.01624 0.01645 0.01519 0.01590 0.01606 0.01621 0.01804 0.01670 0.06442 0.01827 0.01940 0.01994 0.02221 0.02096 0.02041 0.01183 0.01978 0.01403 0.03260 0.01208 0.01241 0.01775 0.01628 0.01610 0.01697 0.01209 0.01303 0.01171 0.03563 0.03801 0.03393 0.02959 0.03970 0.04026 0.01621 0.01465 0.01561 0.01702 0.01513 0.01508 0.04699 0.08211 -0.27003 0.06580 0.07147 0.05051 0.04211 0.96577 0.05593 -0. 12733 0.09047 0.06197 0. 18708 0.21872 0.06132 0.02781 0.06184 0.21247 0.07757 0.10640 -0.19057 0.04660 0. 10606 0.06394 0.04513 0.23389 0.14060 0.04483 0.76491 0.09248 -0.09990 0.12134 0.02862 0.23016 0.06260 0.05812 0.05943 0.06967 0.11407 0.03347 0.10830 0.07308 0. 16831 0.21329 -0.32231 0.06212 0.04952 0.05880 0.04733 0.02624 0.05031 0.05809 0.16722 0.38635 1.68789 0.40148 0.32977 0.64051 0.20260 5.28492 0.94121 0.69216 0.33302 0.18370 0.97709 2.35339 0.34205 0.74066 0.37031 1.01623 0.21934 0.58995 1.05782 0.24272 0.60677 0.20770 0.22208 1.15224 0.58183 0.18397 6.98121 0.34956 1.71620 0.48542 0.16162 1.58148 0.29280 0.95096 0.30037 0.31310 0.61432 0. 18840 0.46962 0.31786 0.71845 0.73926 2.24813 0.30041 0.31722 0.37096 0.17661 0.20669 0.21960 0.47613 0.02796 0.14926 2.84898 0.16119 0.10875 0.41025 0.04105 27.93034 0.88587 0.47909 0.11090 0.03375 0.95470 5.53842 0.11700 0.54857 0.13713 1 .03273 0.04811 0.34804 1.11899 0.05891 0.36817 0.04314 0.04932 1.32765 0.33852 0.03385 48.73727 0. 12219 2.94535 0.23563 0.02612 2.50107 0.08573 0.90433 0.09022 0.09803 0.37738 0.03549 0.22055 0.10104 0.51617 0.54651 5.05408 0.09025 0.10063 0.13761 0.03119 0.04272 0.04822 0.22670 A3 1 1 5 0.06173 A3 116 0.04623 A3 117 0.04175 A3 118 0.06145 A3123 0.04062 0.19505 0.03804 0.12427 0.01544 0.11901 0.01416 0.21162 0.04478 0.12315 0.01517 0.07764 0.33455 0.11193 0.03552 0.16104 0.02593 0.03838 0.29427 0.08660 2.85418 12.96216168.01765 0.05817 0.68513 0.46941

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