UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Duration and bond returns : empirical tests Freedman, Ruth Janine 1985

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1985_A4_6 F74.pdf [ 3.01MB ]
Metadata
JSON: 831-1.0095926.json
JSON-LD: 831-1.0095926-ld.json
RDF/XML (Pretty): 831-1.0095926-rdf.xml
RDF/JSON: 831-1.0095926-rdf.json
Turtle: 831-1.0095926-turtle.txt
N-Triples: 831-1.0095926-rdf-ntriples.txt
Original Record: 831-1.0095926-source.json
Full Text
831-1.0095926-fulltext.txt
Citation
831-1.0095926.ris

Full Text

DURATION AND BOND RETURNS: EMPIRICAL TESTS by RUTH JANINE FREEDMAN B.Comm., The U n i v e r s i t y of B r i t i s h C o lumbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE (BUSINESS ADMINISTRATION) i THE FACULTY OF F a c u l t y of Commerce and GRADUATE STUDIES B u s i n e s s A d m i n i s t r a t i o n We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF J u l y , © Ruth J a n i n e BRITISH COLUMBIA 1985 Freedman, 1985 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e The U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d , t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . F a c u l t y o f Commerce and B u s i n e s s A d m i n i s t r a t i o n The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e : J u l y , 1985 ABSTRACT The purpose of t h i s t h e s i s i s t o e m p i r i c a l l y i n v e s t i g a t e the r o l e of d u r a t i o n i n e x p l a i n i n g bond p r i c e v o l a t i l i t y caused by i n t e r e s t r a t e movements. S p e c i f i c a l l y , Canadian and American market d a t a a r e used t o t e s t whether M a c a u l a y / F i s h e r and W e i l d u r a t i o n i s an adequate measure of b a s i s r i s k f o r d e f a u l t - f r e e government bonds d u r i n g t h e 20-year sample p e r i o d January 1961 t o December 1980. The most important r e s u l t of the study i s t h a t i n e i t h e r a Canadian or an American c o n t e x t t h e r e i s no s i g n i f i c a n t e v i d e n c e t o suggest t h a t , on a v e r a g e , h i g h e r d u r a t i o n bonds earn h i g h e r r e t u r n s . S p e c i f i c a l l y , t h e r e appears t o be a n e g a t i v e , a l t h o u g h i n s i g n i f i c a n t , r e l a t i o n s h i p on average between bond r e t u r n s and d u r a t i o n . A p o s s i b l e e x p l a n a t i o n f o r t h i s r e s u l t i s t h a t i n t e r e s t r a t e s have t r e n d e d upwards over most of the sample p e r i o d . American r e s u l t s suggest t h a t when changes i n the l e v e l of i n t e r e s t r a t e s have been f i l t e r e d out t h e r e i s a p o s i t i v e , a l t h o u g h i n s i g n i f i c a n t , r e l a t i o n s h i p between bond r e t u r n s and d u r a t i o n . Other r e s u l t s of the s t u d y a r e t h a t coupon r a t e s a r e p o s i t i v e l y r e l a t e d t o bond r e t u r n s (perhaps due t o a t a x e f f e c t ) , and t h a t bond p r i c i n g e r r o r s a r e o f t e n r e l a t e d t o subsequent r e t u r n s . TABLE OF CONTENTS 1 . INTRODUCTION . 1 2. LITERATURE REVIEW 4 2.1 Macaulay's D u r a t i o n 4 2.2 E a r l y Developments 6 2.3 D u r a t i o n as a R i s k Proxy 9 3. TESTABLE IMPLICATIONS 13 3.1 Hypotheses about Expected Returns 13 3.2 A S t o c h a s t i c Model f o r R e t u r n s 14 4. METHODOLOGY 17 4.1 Data 17 4.1.1 American Data 17 4.1.2 Canadian Data 18 4.2 R e g r e s s i o n Model 18 4.3 C o m p u t a t i o n a l D e t a i l s 21 4.3.1 Return Computations 21 4.3.2 D u r a t i o n Computations 22 4.3.3 E r r o r Computations 22 4.4 T e s t s of Hypotheses 26 4.5 L i m i t a t i o n s of Methodology 27 5. RESULTS 29 5.1 American R e s u l t s 29 5.2 Canadian R e s u l t s 50 5.3 L i m i t a t i o n s of A n a l y s i s 55 6. CONCLUSIONS 64 REFERENCES 66 i i i LIST OF TABLES TABLE Page 1 U.S. Bond P r i c e and Y i e l d P r e d i c t i o n s 24 2 Canadian Bond P r i c e and Y i e l d P r e d i c t i o n s 25 3 C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r U.S. Bonds ( H o l d i n g P e r i o d = One Month) 31 4 E s t i m a t e d F i r s t Order A u t o c o r r e l a t i o n s of the Monthly C r o s s - S e c t i o n a l Parameter E s t i m a t e s ( H o l d i n g P e r i o d = One Month) 34 5 E s t i m a t e d C o e f f i c i e n t s and T - s t a t i s t i c s from R e g r e s s i n g 71 on the C o r r e s p o n d i n g Changes i n the Long and S h o r t Rates of I n t e r e s t . . . . 35 6 C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r U.S. Bonds ( H o l d i n g P e r i o d = Three Months) 38 7 E s t i m a t e d F i r s t Order A u t o c o r r e l a t i o n s of the Monthly C r o s s - S e c t i o n a l Parameter E s t i m a t e s ( H o l d i n g P e r i o d = Three Months) 40 8 C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r P o r t f o l i o s of U.S. Bonds ( P o r t f o l i o S e l e c t i o n Method 1; H o l d i n g P e r i o d = One Month) 42 9 C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r P o r t f o l i o s of U.S. Bonds ( P o r t f o l i o S e l e c t i o n Method 2; H o l d i n g P e r i o d = One Month) 44 10 C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r P o r t f o l i o s of U.S. Bonds ( P o r t f o l i o S e l e c t i o n Method 1; H o l d i n g P e r i o d = Three Months) 46 11 C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r P o r t f o l i o s of U.S. Bonds ( P o r t f o l i o S e l e c t i o n Method 2; H o l d i n g P e r i o d = Three Months) 48 12 C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r Canadian Bonds ( H o l d i n g P e r i o d = One Month) 51 13 C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r Canadian Bonds ( H o l d i n g P e r i o d = Three Months) 53 i v TABLE Page 14 C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r U.S. Bonds (AR1 Time S e r i e s Model f o r R e g r e s s i o n Parameters E s t i m a t e d U s i n g G e n e r a l Non-L i n e a r L e a s t Squares Method) 58 15 C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r U.S. Bonds (AR1 Time S e r i e s Model f o r R e g r e s s i o n Parameters E s t i m a t e d U s i n g C o n d i t i o n a l L e a s t Squares Method) 60 16 C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r U.S. Bonds (AR1 Time S e r i e s Model f o r R e g r e s s i o n Parameters E s t i m a t e d U s i n g B a c k c a s t i n g Method) 62 v ACKNOWLEDGEMENTS I w i s h t o thank the members of my d i s s e r t a t i o n committee, P r o f e s s o r s Eduardo Schwartz, Rex Thompson, and M a r t i n Puterman f o r t h e i r guidance and c o n s t r u c t i v e c r i t i c i s m . I am p a r t i c u l a r l y g r a t e f u l t o Dr. S c h w a r t z , who, i n h i s r o l e as chairman, p r o v i d e d generous and i n d i s p e n s a b l e encouragement, a s s i s t a n c e , and a d v i c e . I would a l s o l i k e t o e x p r e s s my thanks t o P r o f e s s o r M i c h a e l Brennan, who o r i g i n a l l y s u g g e sted the t h e s i s t o p i c t o me and who a s s i s t e d me i n f i n d i n g a committee c h a i r m a n , and t o P r o f e s s o r P h e l i m B o y l e , who encouraged me t o embark on t h i s phase of my g r a d u a t e e d u c a t i o n . More than a n y t h i n g e l s e , though, I owe a l a s t i n g debt of g r a t i t u d e t o my mother and t o my good f r i e n d s , Nadine and H e a t h e r , f o r t h e i r w i l l i n g n e s s t o l i s t e n t o me and f o r t h e i r c o n s t a n t s u pport and encouragement d u r i n g the many months t h a t i t has taken me t o complete t h i s p r o j e c t . v i Chapter 1 INTRODUCTION Dramatic i n c r e a s e s i n both the v o l a t i l i t y and the l e v e l of i n t e r e s t r a t e s s i n c e the l a t e 1960s have r e s u l t e d i n a b urgeoning i n t e r e s t i n bond a n a l y s i s , and have m o t i v a t e d r e s e a r c h e r s and p r a c t i t i o n e r s a l i k e t o s e a r c h f o r t o o l s t o a i d i n the measurement and management of i n t e r e s t r a t e r i s k . The quest f o r an a p p r o p r i a t e measure of bond r i s k has l e d t o the r e d i s c o v e r y of a concept o r i g i n a l l y d e v e l o p e d i n 1938 -F r e d e r i c k Macaulay's measure of bond term known as " d u r a t i o n . " The concept of d u r a t i o n has a t t r a c t e d w i d e s p r e a d a t t e n t i o n w i t h i n the investment community, and, as a s i n g l e f a c t o r i n e x p l a i n i n g bond r e t u r n s , has shown s i g n s of becoming as w i d e l y used as b e t a i s f o r e q u i t i e s . In r e c e n t y e a r s , i t has been w i d e l y r e c o g n i z e d t h a t , j u s t as the p r i c e b e h a v i o r of s t o c k s c o n t a i n s a s t r o n g market component, the p r i c e of bonds tends t o be markedly a f f e c t e d by changes i n the l e v e l of i n t e r e s t r a t e s . The i d e a of r i s k adjustment has been w i d e l y a c c e p t e d w i t h r e s p e c t t o s t o c k p o r t f o l i o s : h i g h r i s k ( h i g h beta) p o r t f o l i o s g e n e r a t e the h i g h e s t r e t u r n s i n r i s i n g markets and g e n e r a t e the l o w e s t r e t u r n s i n d e c l i n i n g markets. The i n t e r e s t i n g q u e s t i o n has been whether or not an analogous concept would work f o r the measurement of bond s e n s i t i v i t y . The customary way of computing b e t a c o e f f i c i e n t s f o r e q u i t y i s s u e s i s t o r e g r e s s s e c u r i t y r e t u r n s a g a i n s t index r e t u r n s . However, t h i s approach does not work w e l l f o r bonds 1 2 because bonds change t h e i r s e n s i t i v i t y c h a r a c t e r i s t i c s as time p a s s e s and as the market i t s e l f changes. For example, a bond about t o mature i s f a r l e s s s e n s i t i v e t o i n t e r e s t - r a t e changes than a bond w i t h a l o n g time t o m a t u r i t y . The r e s u l t i s t h a t the r e g r e s s i o n t e c h n i q u e used f o r beta c o m p u t a t ions does not a p p l y v e r y w e l l t o the d e r i v a t i o n of bond r i s k measures. F r e d e r i c k Macaulay's bond s e n s i t i v i t y measure, c a l l e d " d u r a t i o n , " i s f a r s i m p l e r t o compute than the beta c o e f f i c i e n t s f o r s t o c k s . Macaulay and o t h e r s have p o i n t e d out t h a t the most imp o r t a n t s i n g l e source of r i s k f o r d e f a u l t - f r e e bonds i s b a s i s r i s k ; t h a t i s , p r i c e f l u c t u a t i o n s caused by s h i f t s i n i n t e r e s t r a t e s . For a g i v e n s h i f t i n the y i e l d c u r v e , and h o l d i n g o t h e r f a c t o r s unchanged, l o n g e r m a t u r i t y bonds g e n e r a l l y s u f f e r g r e a t e r p r i c e changes than s h o r t e r m a t u r i t y bonds. T h i s c h a r a c t e r i z a t i o n i s not e x a c t because h i g h coupon bonds are l e s s v o l a t i l e than low coupon bonds. " I n t u i t i o n says t h a t t h i s i s t o be e x p e c t e d because, o t h e r t h i n g s b e i n g e q u a l , h i g h coupon bonds have a g r e a t e r p e r c e n t a g e of t h e i r v a l u e due t o t h e i n t e r i m coupons and, hence, have a s h o r t e r ' e f f e c t i v e ' m a t u r i t y . " [16; p.627]. D u r a t i o n may be thought of as an attempt t o q u a n t i f y t h i s q u a l i t a t i v e statement through the use of a s i n g l e , n u m e r i c a l measure. D u r a t i o n , as dev e l o p e d by Macaulay, i s a w e i g h t e d average of the time t o each bond payment, coupon as w e l l as p r i n c i p a l . , where the w e i g h t s a r e the p r e s e n t v a l u e of each 3 of the payments as a p e r c e n t of the t o t a l p r e s e n t v a l u e of a l l the f l o w s . There are d i f f e r e n t d e f i n i t i o n s of d u r a t i o n ; each c o r r e s p o n d s t o a d i f f e r e n t model of t h e y i e l d c u r v e and s h i f t s i n i t . The measure of d u r a t i o n used i n t h i s study i s the M a c a u l a y / F i s h e r and W e i l measure, which computes p r e s e n t v a l u e w e i g h t s u s i n g d i s c o u n t bond y i e l d s from the term s t r u c t u r e . T h i s study e m p i r i c a l l y i n v e s t i g a t e s t h e r o l e of d u r a t i o n i n e x p l a i n i n g average r a t e s of r e t u r n and p r i c e v o l a t i l i t y i n the Canadian and American bond markets over the 20-year p e r i o d January 1961 t o December 1980. Chapter 2 c o n t a i n s a review of some of the i m p o r t a n t l i t e r a t u r e on the s u b j e c t of d u r a t i o n , and Chapter 3 o u t l i n e s t e s t a b l e i m p l i c a t i o n s of the d u r a t i o n model. Chapter 4 d e s c r i b e s the methodology used i n the s t u d y , and Chapter 5 p r e s e n t s the major r e s u l t s . A summary w i t h c o n c l u s i o n s i s p r e s e n t e d i n Chapter 6. Chapter 2 LITERATURE REVIEW 2.1 MACAULAY'S DURATION F r e d e r i c k Macaulay [20] developed the b a s i c concept of duration i n 1938 and p r e s e n t e d i t i n h i s monumental study of i n t e r e s t r a t e s and r a i l r o a d bond p r i c e s . D u r a t i o n was o r i g i n a l l y c o n c e i v e d by Macaulay as a b e t t e r way t o summarize the t i m i n g of bond f l o w s than m a t u r i t y . M a t u r i t y p r o v i d e s i n f o r m a t i o n o n l y about the date of f i n a l payment. However, coupon bonds make r e g u l a r payments b e f o r e the repayment of p r i n c i p a l a t m a t u r i t y . T h e r e f o r e , m a t u r i t y i s an i n c o m p l e t e measure of the l i f e (or l e n g t h ) of a coupon bond. Macaulay c o n c l u d e d t h a t " . . . i t would seem h i g h l y d e s i r a b l e t o have some adequate measure of 'l o n g n e s s . ' L e t us use the word ' d u r a t i o n ' t o s i g n i f y the essence of the time element i n a l o a n . . . . I t i s c l e a r t h a t 'number of y e a r s t o m a t u r i t y ' i s a most inadequate measure of ' d u r a t i o n ' " [ 2 0 ; p . 7 4 ] . Macaulay proposed t h a t d u r a t i o n be computed as a we i g h t e d average of the time p e r i o d s i n which payments a r e made. In Macaulay's d e f i n i t i o n each p e r i o d i s weig h t e d by the p r e s e n t v a l u e of the c o r r e s p o n d i n g payment as a p e r c e n t of the t o t a l p r e s e n t v a l u e of a l l the f l o w s . Macaulay o r i g i n a l l y d e f i n e d d u r a t i o n a s : 4 5 Z ( t , - t 0 ) c ( t . ) q ( t . , t 0 ) i = 1 D, = _ (1) Z c ( t . ) q ( t . , t 0 ) i=1 1 Where: t 0 i s the c u r r e n t d a t e c ( t ^ ) i s the income stream a t time t - (coupon payments and p r i n c i p a l ) 1 q ( t ^ , t 0 ) i s the p r e s e n t v a l u e a t time t 0 of $1 p a y a b l e at time t • , the ' d i s c o u n t f u n c t i o n ' Macaulay, however, proceeded t o s i m p l i f y h i s o r i g i n a l d e f i n i t i o n by u s i n g y i e l d t o m a t u r i t y i n the p r e s e n t v a l u e c a l c u l a t i o n s , r a t h e r than the i n d i v i d u a l p e r i o d d i s c o u n t r a t e s . H i s r e s u l t i n g a l t e r n a t i v e d e f i n i t i o n i s : - y ( t . - t o ) Z ( t . - t 0 ) c ( t . ) e i = 1 D 2 t o = (2) - y ( t . - t o ) Z c ( t . ) e 1 i = l Where: y i s the c o n t i n u o u s l y compounded y i e l d - t o - m a t u r i t y on the bond D e f i n i t i o n s (1) and (2) a r e e q u i v a l e n t i f and o n l y i f the i n t e r e s t r a t e i s known t o be c o n s t a n t over the l i f e of the bond. "When the two measures d i f f e r , i t i s o n l y the o r i g i n a l (1) which i s a p p r o p r i a t e f o r r i s k measurement" [16; p.631]. N e v e r t h e l e s s , as Cox, I n g e r s o l l , and Ross [8] p o i n t o u t , e i t h e r d e f i n i t i o n has i t s advantages. D2 d u r a t i o n , which i s r e f e r r e d t o as " s i m p l e d u r a t i o n , " can be computed 6 knowing o n l y the p r i c e of the bond and the c o n t r a c t payments, and t h e r e f o r e i s most commonly used i n p r a c t i c e . On t h e . o t h e r hand, F i s h e r and W e i l [12] have championed the reemergence of the o r i g i n a l d e f i n i t i o n (1) f o r d u r a t i o n . S i n c e i t i s a more v a l i d r i s k measure, d e f i n i t i o n (1) w i l l be used i n t h i s s t u d y , and i t w i l l h e n c e f o r t h be r e f e r r e d t o as M a c a u l a y / F i s h e r and W e i l d u r a t i o n . D u r a t i o n , as d e f i n e d i n e q u a t i o n s (1) or ( 2 ) , i s measured i n u n i t s of time - e.g., i n months or y e a r s . For z e r o coupon bonds, d u r a t i o n i s e q u a l t o m a t u r i t y . For a l l o t h e r bonds, d u r a t i o n i s s h o r t e r than m a t u r i t y . However, the r e l a t i o n s h i p between d u r a t i o n and m a t u r i t y i s n o n l i n e a r and complex. 2.2 EARLY DEVELOPMENTS Macaulay's d u r a t i o n measure a t t r a c t e d v e r y l i t t l e a t t e n t i o n i n 1938 and was not w i d e l y a c c l a i m e d . In f a c t , the concept of d u r a t i o n remained t o be d i s c o v e r e d , a p p a r e n t l y i n d e p e n d e n t l y , s e v e r a l t i m e s . As B i e r w a g , Kaufman, and Toevs mention, " D u r a t i o n , and p o s s i b l y t h e e n t i r e a r e a of bond p r i c i n g , a p p a r e n t l y had l i t t l e a p p e a l at the t i m e " [2; p. 1 6 ] . In 1939, J.R. H i c k s i n d e p e n d e n t l y d e v e l o p e d e q u a t i o n ( 2 ) , and d i s c u s s e d i t i n h i s major work, V a l u e and C a p i t a l [ 1 4 ] . H i c k s c a l c u l a t e d the e l a s t i c i t y of the v a l u e of a stream of payments w i t h r e s p e c t t o a d i s c o u n t f a c t o r , and r e f e r r e d t o the r e s u l t i n g measure as " e l a s t i c i t y of c a p i t a l 7 v a l u e " and sometimes as "average p e r i o d . " He n o t e d t h a t h i s e l a s t i c i t y measure was not a pure number ( l i k e o t h e r e l a s t i c i t i e s ) , but was i n s t e a d denominated i n u n i t s of t i m e . I t i s i n t e r e s t i n g t h a t H i c k s , who was s e a r c h i n g f o r an e l a s t i c i t y , ended up d e r i v i n g the same measure as Macaulay, who was s e a r c h i n g f o r a measure of t i m e . H i c k s was the f i r s t t o demonstrate the r i s k - p r o x y i n g p r o p e r t y of d u r a t i o n . He showed t h a t f o r a g i v e n ( i n f i n i t e s i m a l ) change i n y i e l d , the per c e n t a g e change i n an a s s e t ' s v a l u e i s p r o p o r t i o n a l t o i t s d u r a t i o n . 1 In 1945, P a u l Samuelson [ 2 4 ] , who was a p p a r e n t l y unaware of H i c k s ' or Macaulay's work, e f f e c t i v e l y d e r i v e d e q u a t i o n ( 2 ) . Samuelson a n a l y z e d the e f f e c t of i n t e r e s t r a t e changes on the c a p i t a l v a l u e s of i n s t i t u t i o n s such as banks, i n s u r a n c e companies, and u n i v e r s i t i e s . He computed the f i r s t d e r i v a t i v e s of the v a l u e s of the i n f l o w s and o u t f l o w s w i t h r e s p e c t t o the y i e l d t o m a t u r i t y and o b t a i n e d a measure, e s s e n t i a l l y e q u i v a l e n t t o d u r a t i o n , which he termed the "weighted average time p e r i o d of payments." He p r o v e d t h a t i f the d u r a t i o n of an i n s t i t u t i o n ' s a s s e t s i s l a r g e r than t h a t of i t s l i a b i l i t i e s , then the i n s t i t u t i o n w i l l l o s e when i n t e r e s t r a t e s r i s e and p r o f i t when i n t e r e s t r a t e s f a l l [24; p. 19]. 1 T h i s r e l a t i o n s h i p i s i m p l i c i t i n Macaulay's a n a l y s i s of s h o r t and l o n g - t e r m i n t e r e s t r a t e s , but i s not r i g o r o u s l y d e v e l o p e d [20; pp.50-51,61-62]. In f a c t , a c l o s e e x a m i n a t i o n of Macaulay's book r e v e a l s t h a t even he was p r i m a r i l y c oncerned w i t h the r i s k - p r o x y i n g p r o p e r t i e s of h i s measure d e s p i t e t h e a s s i g n e d name " d u r a t i o n . " 8 In 1952, a B r i t i s h a c t u a r y by the name of F.M. Redi n g t o n s e t out t o determine what a l l o c a t i o n of a s s e t s and l i a b i l i t i e s would m i t i g a t e the e f f e c t s of i n t e r e s t r a t e changes on l i f e i n s u r a n c e companies' net w o r t h . In the p r o c e s s , R e d i n g t o n [22] computed the f i r s t d e r i v a t i v e s of the v a l u e s of the i n f l o w s and o u t f l o w s w i t h r e s p e c t t o i n t e r e s t r a t e s . These s t a t i s t i c s were e x a c t l y e q u a l t o Macaulay's d u r a t i o n measure. L i k e Samuelson, R e d i n g t o n appeared t o be c o m p l e t e l y unaware of the work of h i s p r e d e c e s s o r s . He c a l l e d h i s s t a t i s t i c the "mean term" and i n t r o d u c e d the term " i m m u n i z a t i o n " t o s i g n i f y the investment of a s s e t s i n such a way t h a t a f i r m would be immune t o a g e n e r a l change i n i n t e r e s t r a t e s . The essence of Re d i n g t o n ' s t h e o r y was t o s e t the d u r a t i o n (or mean term) of the a s s e t s e q u a l t o t h a t of the l i a b i l i t i e s . R e d i n g t o n ' s a r t i c l e was p u b l i s h e d i n an a c t u a r i a l j o u r n a l , and as a r e s u l t was not s u c c e s s f u l i n a t t r a c t i n g the a t t e n t i o n of r e s e a r c h e r s and p r a c t i t i o n e r s i n the f i e l d of f i n a n c i a l economics. In f a c t , the e n t i r e concept of d u r a t i o n was l a r g e l y i g n o r e d u n t i l 1971, when F i s h e r and W e i l [12] p u b l i s h e d t h e i r e x t e n s i v e e m p i r i c a l study e n t i t l e d "Coping w i t h the R i s k of I n t e r e s t - R a t e F l u c t u a t i o n s . " F i s h e r and W e i l showed t h a t d u r a t i o n c o u l d be used t o immunize bond p o r t f o l i o s a g a i n s t i n t e r e s t r a t e r i s k . F i s h e r and W e i l ' s a r t i c l e was w i d e l y read and, as a r e s u l t , d u r a t i o n began t o be r e c o g n i z e d as an i m p o r t a n t t o o l f o r b o t h managing and measuring i n t e r e s t r a t e r i s k . 9 2.3 DURATION AS A RISK PROXY F i s h e r [11] and, s h o r t l y t h e r e a f t e r , Hopewell and " Kaufman [15] d i s c o v e r e d a d i r e c t r e l a t i o n between the d u r a t i o n of a bond and i t s p r i c e s e n s i t i v i t y t o changes i n market i n t e s t r a t e s , and proposed t h a t d u r a t i o n c o u l d t h e r e f o r e s e r v e as a measure of p r i c e r i s k f o r bonds. S p e c i f i c a l l y , Hopewell and Kaufman showed t h a t the d e r i v a t i v e of a bond's p r i c e w i t h r e s p e c t t o the y i e l d t o m a t u r i t y i s p r o p o r t i o n a l t o M a c a u l a y / F i s h e r and W e i l d u r a t i o n . For s m a l l y i e l d changes and c o n t i n u o u s compounding, bond p r i c e v o l a t i l i t y c o u l d be r e l a t e d t o changes i n the y i e l d t o m a t u r i t y by: dP j t = - D , k d r . t (3) Where: P.. i s the i n i t i a l p r i c e of bond j J ^  d P j t i s the p r i c e change of bond j D j t i s the d u r a t i o n of bond j a t time t dr. . i s the change i n y i e l d t o m a t u r i t y E q u a t i o n (3) s t a t e s t h a t a bond's p r i c e changes i n i n v e r s e p r o p o r t i o n t o a change i n the market y i e l d t o m a t u r i t y , where the p r o p o r t i o n a l i t y f a c t o r i s d u r a t i o n . For a g i v e n change i n the y i e l d , the change i n bond p r i c e w i l l be r e l a t i v e l y g r e a t e r the l o n g e r the d u r a t i o n of the bond. As B i e r w a g , Kaufman, and Khang [1] p o i n t o u t , e q u a t i o n s (1) and (3) p e r m i t t h r e e w i d e l y used r u l e s about bond 10 p r i c i n g t o be c o l l a p s e d i n t o one. These r u l e s s t i p u l a t e t h a t f o r a g i v e n change i n i n t e r e s t r a t e s , the p r o p o r t i o n a l change i n the p r i c e of a bond w i l l be g r e a t e r : 1. the lower the coupon r a t e 2. the lower the market y i e l d and 3. the l o n g e r the m a t u r i t y . From e q u a t i o n (1) i t f o l l o w s t h a t the lower the coupon r a t e or the market y i e l d and ( f o r most bonds) the l o n g e r the m a t u r i t y , the l o n g e r the d u r a t i o n . From e q u a t i o n ( 3 ) , "The one complete and a c c u r a t e r u l e of thumb i s t h a t the l o n g e r the d u r a t i o n , the g r e a t e r the p r o p o r t i o n a l p r i c e v o l a t i l i t y f o r a g i v e n change i n y i e l d " [ 2; p.18]. To the e x t e n t t h a t p r i c e v o l a t i l i t y i s viewed as p r i c e r i s k , d u r a t i o n may be viewed as an index of r i s k f o r a g i v e n change i n y i e l d s . However, I n g e r s o l l , S k e l t o n , and W e i l [16] prove t h a t M a c a u l a y / F i s h e r and W e i l d u r a t i o n i s o n l y a v a l i d r i s k measure f o r s h a p e - p r e s e r v i n g s h i f t s i n the e n t i r e y i e l d c u r v e . That i s , t h i s c o m p a r a t i v e s t a t i c p r o p e r t y of d u r a t i o n i s u s e f u l o n l y t o the e x t e n t t h a t y i e l d c u r v e s h i f t s a r e p a r a l l e l over t i m e . Cooper [7] notes t h a t p a r a l l e l s h i f t s a re somewhat u n r e a l i s t i c and a r e a l s o i n c o n s i s t e n t w i t h e q u i l i b r i u m c o n d i t i o n s i n f i n a n c i a l markets s i n c e they imply a r b i t r a g e p r o f i t s . Due t o the h i g h l y r e s t r i c t i v e assumptions needed t o v a l i d a t e the n o t i o n of d u r a t i o n , the u s e f u l n e s s of d u r a t i o n has been i n c r e a s i n g l y q u e s t i o n e d i n r e c e n t y e a r s . In f a c t , d u r a t i o n has f a l l e n from academic f a v o r somewhat as more 11 s o p h i s t i c a t e d e q u i l i b r i u m models of the term s t r u c t u r e have been put f o r t h by Brennan and Schwartz [ 4 ] , Cox, I n g e r s o l l , and Ross [ 8 ] , and o t h e r s . However, i n a r e c e n t paper, Brennan and Schwartz [6] demonstrate q u i t e c l e a r l y t h a t i n p r a c t i c a l p o r t f o l i o a p p l i c a t i o n s , the s i m p l e concept of d u r a t i o n p e r f o r m s as w e l l as a h i g h l y t e c h n i c a l e q u i l i b r i u m bond p r i c i n g model. Brennan and Schwartz c o n c l u d e t h a t "The d u r a t i o n model performs rema r k a b l y w e l l by comparison w i t h the more s o p h i s t i c a t e d e q u i l i b r i u m model, and i n view of the c o m p u t a t i o n a l c o s t of the l a t t e r , the d u r a t i o n model appears t o be more u s e f u l f o r p r a c t i c a l bond p o r t f o l i o management" [ 6; p. 6 ]. In r e c e n t y e a r s , many a u t h o r s have expanded on the M a c a u l a y / F i s h e r and W e i l concept of d u r a t i o n as a s u r r o g a t e f o r r i s k measurement and have d e v e l o p e d a v a r i e t y of new measures of d u r a t i o n . Khang [ 1 8 ] , Cooper [ 7 ] , and B i e r w a g , Kaufman, and Khang [1] have a l l proposed a l t e r n a t i v e measures of d u r a t i o n which they c l a i m a re t h e o r e t i c a l l y s u p e r i o r t o t h e t r a d i t i o n a l measure. G u l t e k i n and R o g a l s k i [13] e m p i r i c a l l y i n v e s t i g a t e seven d i f f e r e n t d u r a t i o n measures and t h e i r r o l e i n e x p l a i n i n g p r i c e v o l a t i l i t y caused by i n t e r e s t r a t e movements. G u l t e k i n and R o g a l s k i ' s r e s u l t s "...do not support the i m p o r t a n t t e s t a b l e i m p l i c a t i o n s of any of the v a r i o u s d u r a t i o n s as measures of b a s i s r i s k " [ 13; p.263].'In f a c t , d e s p i t e the f l o o d of a r t i c l e s c l a i m i n g s u p e r i o r i t y f o r p a r t i c u l a r measures of d u r a t i o n , t h e measures examined were v i r t u a l l y i n d i s t i n g u i s h a b l e e m p i r i c a l l y . S u r p r i s i n g l y , none of them d i d much b e t t e r than m a t u r i t y i n e x p l a i n i n g bond r e t u r n s , and a l l d u r a t i o n measures proved t o be i n f e r i o r t o s i m p l e f a c t o r models. Chapter 3 TESTABLE IMPLICATIONS 3.1 HYPOTHESES ABOUT EXPECTED RETURNS As G u l t e k i n and R o g a l s k i m ention, a c l o s e e x a m i n a t i o n of the d u r a t i o n l i t e r a t u r e y i e l d s s e v e r a l e x p l i c i t h ypotheses about d u r a t i o n ' s a b i l i t y t o serve as a proxy f o r p r i c e v o l a t i l i t y , " . . . t h a t i s , as an index f o r c r o s s -s e c t i o n a l comparisons of bond r i s k " [13; p.246]. The d u r a t i o n l i t e r a t u r e i n d i c a t e s t h a t , under c e r t a i n r e s t r i c t i v e assumptions about y i e l d c u r v e s h i f t s over t i m e , M a c a u l a y / F i s h e r and W e i l d u r a t i o n i s t h e o r e t i c a l l y a good measure of the v a r i a b i l i t y of bond r e t u r n s . To the e x t e n t t h a t p r i c e v o l a t i l i t y i s c o n s i d e r e d t o be p r i c e r i s k , d u r a t i o n may be viewed as a measure of b a s i s r i s k f o r bonds. 2 In a market of r i s k - a v e r s e i n v e s t o r s t h i s r i s k s h o u l d be p r i c e d ; t h a t i s , h i g h e r r i s k s h o u l d be a s s o c i a t e d w i t h h i g h e r e x p e c ted r e t u r n . The d u r a t i o n l i t e r a t u r e a l s o s u g g e s t s s t r o n g l y t h a t d u r a t i o n i s a complete measure of r i s k because i t i n c o r p o r a t e s the e f f e c t s of both couon and m a t u r i t y d i f f e r e n c e s on p r i c e v o l a t i l i t y . D u r a t i o n p e r m i t s the t h r e e r u l e s of bond p r i c i n g n o t e d e a r l i e r t o be c o l l a p s e d i n t o one 2 L a n g s t e i n and Sharpe [19] suggest t h a t measures of d u r a t i o n c a l c u l a t e d ex-ante p r e d i c t the e x p e c t e d r e a c t i o n s of bond p r i c e s t o unexpected changes i n i n t e r e s t r a t e s . T h i s , i n e f f e c t , i m p l i e s t h a t d u r a t i o n i s a r i s k measure s i m i l a r t o bet a f o r common s t o c k s . 13 1 4 and t o s t a n d w i t h o u t e x c e p t i o n . These i m p o r t a n t i m p l i c a t i o n s o b t a i n e d from the d u r a t i o n l i t e r a t u r e g i v e r i s e t o the f o l l o w i n g t h r e e hypotheses about e x p e c t e d bond r e t u r n s : 1. On average, t h e r e i s a p o s i t i v e r e l a t i o n s h i p between bond r e t u r n s and d u r a t i o n . One would expect t h a t , on average, d e f a u l t - f r e e bonds w i t h h i g h e r d u r a t i o n would have h i g h e r r e t u r n s . 2. D u r a t i o n i s a complete measure of the r i s k of bond j . D u r a t i o n i n c o r p o r a t e s the e f f e c t of coupon and m a t u r i t y d i f f e r e n c e s on p r i c e v o l a t i l i t y . 3. The c a p i t a l market f o r bonds i s e f f i c i e n t ; t h a t i s , bond p r i c i n g e r r o r s a r e u n r e l a t e d t o subsequent r e t u r n s . 3.2 A STOCHASTIC MODEL FOR RETURNS The t h r e e hypotheses mentioned e a r l i e r can be t e s t e d w i t h a c t u a l d a t a on government bonds (and p o r t f o l i o s of government bonds) f o r many time p e r i o d s . I t i s impo r t a n t t o choose a model of p e r i o d - b y - p e r i o d r e t u r n s t h a t uses observed average r e t u r n s t o t e s t the t h r e e e x p e c t e d - r e t u r n h y p otheses, but one t h a t i s n e v e r t h e l e s s as g e n e r a l as p o s s i b l e . The f o l l o w i n g s t o c h a s t i c model f o r r e t u r n s i s t h e r e f o r e s u g g e s t e d : s j t - r f t = ; o t + ; i t D j t + ; 2 t c j t + ; 3 t M j t + ^ 4 t E j t + * j t u 1 5 Where: R.. i s the r e a l i z e d h o l d i n g p e r i o d r e t u r n f o r bond j J *-r f t i s the r i s k f r e e r a t e of r e t u r n over the h o l d i n g p e r i o d D. . i s the d u r a t i o n of bond i a t time t C j t i s the coupon on bond j M. i s the time t o m a t u r i t y of bond j , measured at 3 time t E. . i s the p r i c i n g e r r o r f o r bond j a t time t -1 ( a c t u a l p r i c e of bond minus p r e d i c t e d p r i c e ) e.. i s the d i s t u r b a n c e term D ^  E q u a t i o n (4) a l l o w s 70 f c, 7 1 t> ? 2 t ' ^ t ' a n < ^ ^ 4 t t 0 v a r v s t o c h a s t i c a l l y from p e r i o d t o p e r i o d . The d i s t u r b a n c e s (e-,) a r e assumed t o have mean z e r o and t o be independent of D ^  a l l o t h e r v a r i a b l e s i n e q u a t i o n ( 4 ) . The v a r i a b l e s e.,, D t 7<Dt, 7 1 t> 7 2 t ' ^ t ' a n < ^ ^ 4 t a r e a s s u m e < ^ t o f o l l o w a p p r o x i m a t e l y a m u l t i v a r i a t e normal d i s t r i b u t i o n . The f i r s t h y p o t h e s i s , which c o n c e r n s d u r a t i o n ' s a b i l i t y t o s e r v e as a r i s k - p r o x y , i m p l i e s t h a t the e x p e c t e d v a l u e of the r i s k premium 71 i s p o s i t i v e . The second h y p o t h e s i s c o n c e r n s the e x p e c t e d v a l u e s of and 73 f c, the c o e f f i c i e n t s f o r the coupon r a t e and m a t u r i t y . The coupon r a t e ( C j t ) and m a t u r i t y ( M j t ) are i n c l u d e d i n the model as independent v a r i a b l e s i n o r d e r t o t e s t whether d u r a t i o n i s a complete measure of r i s k . Completeness i m p l i e s t h a t E(72 f c)=0 and E(73 t>=0. 1 6 The t h i r d h y p o t h e s i s (market e f f i c i e n c y ) i m p l i e s t h a t bond p r i c i n g e r r o r s are u n r e l a t e d t o subsequent r e t u r n s , so t h a t E(74 f c) e q u a l s z e r o . However, t o the e x t e n t t h a t p r i c i n g e r r o r s a r e a t t r i b u t a b l e t o temporary market d i s e q u i l i b r i a or d e f f i c i e n c i e s i n the d a t a , one s h o u l d expect the c o e f f i c i e n t t o be n e g a t i v e . As y e t , no h y p o t h e s i s has been p r e s e n t e d about th e i n t e r c e p t term, 70 f c, i n ( 5 ) . A r e a s o n a b l e h y p o t h e s i s i s t h a t E ( 7 0 t ) would e q u a l z e r o , p a r t i c u l a r l y because excess r e t u r n (over the r i s k - f r e e r a t e ) i s the dependent v a r i a b l e i n the model. Chapter 4 METHODOLOGY 4.1 DATA 4.1.1 AMERICAN DATA T e s t s a re c a r r i e d out u s i n g monthly p r i c e , coupon, and m a t u r i t y d a t a on U n i t e d S t a t e s T r e a s u r y bonds and notes f o r the twenty - y e a r sample p e r i o d December 1960 t o December 1980. The da t a a re o b t a i n e d from the Cen t e r f o r Research i n S e c u r i t y P r i c e s (CRSP) Government Bond F i l e , which c o n t a i n s month-end data on v i r t u a l l y a l l d i r e c t n e g o t i a b l e o b l i g a t i o n s of the U.S. T r e a s u r y . For each s e c u r i t y , the CRSP Government Bond F i l e c o n t a i n s : 1. B i d p r i c e a t month end 2. Ask p r i c e a t month end 3. Coupon r a t e (per cent per annum) 4. Accrued i n t e r e s t as of month end 5. I n t e r e s t p a y a b l e d u r i n g month 6. M a t u r i t y d a t e a t time of i s s u e The sample i s l i m i t e d t o U.S. T r e a s u r y bonds and note s w i t h m a t u r i t i e s between one and t e n y e a r s . "Flower bonds," c a l l a b l e and deep d i s c o u n t bonds a r e e x c l u d e d from the s t u d y . The d a t a f o r the r i s k - f r e e r a t e a r e the y i e l d s on U n i t e d S t a t e s 91-day T r e a s u r y b i l l s ( t a k en from the Bank of Canada Review, Cansim S e r i e s 2545.1). 1 7 18 4.1.2 CANADIAN DATA The d a t a on Canadian Government s e c u r i t i e s f o r the twenty-year sample p e r i o d December 1960 t o December 1980 are o b t a i n e d from the Wood Gundy M o n t h l y Bond P r i c e Database, which c o n t a i n s monthly ( g e n e r a l l y the l a s t Thursday or F r i d a y ) p r i c e , m a t u r i t y , and coupon i n f o r m a t i o n f o r a s e r i e s of bonds t h a t have been l i s t e d on p u b l i s h e d quote s h e e t s . The sample i n c l u d e s a l l Government of Canada bonds w i t h m a t u r i t i e s between one and t e n y e a r s f o r which p r i c e s a r e a v a i l a b l e i n the Wood Gundy Bond Database and which a r e n e i t h e r c a l l a b l e nor e x c h a n g e a b l e . The d a t a f o r the r i s k - f r e e r a t e of i n t e r e s t a r e the y i e l d s on Government of Canada 91-day T r e a s u r y b i l l s . The i n t e r e s t r a t e s e r i e s a r e average y i e l d s a t Thursday tender f o l l o w i n g the l a s t Wednesday of each month from December 1960 t o December 1980 ( t a k e n from the Bank of Canada Review, Cansim S e r i e s 2560.1). 4.2 REGRESSION MODEL As mentioned e a r l i e r , the l i n e a r i t y , c o m p l e t e n e s s , and e f f i c i e n c y hypotheses can be t e s t e d w i t h a c t u a l Canadian and American market d a t a f o r many time p e r i o d s w i t h the use of government bonds and p o r t f o l i o s of government bonds. More s p e c i f i c a l l y , a r e g r e s s i o n model t h a t uses observed p r i c e changes t o t e s t the t h r e e hypotheses i s e s t i m a t e d monthly over the twenty-year p e r i o d December 1960 t o December 1980. 19 For each month of t h i s p e r i o d , the f o l l o w i n g c r o s s - s e c t i o n a l r e g r e s s i o n — t h e e m p i r i c a l a n a l o g of e q u a t i o n (4) — i s r u n: R. - r f = -YO + -vi D. + v2 C • + v3 M. . Dt t 7 t 7 t ] t 7 t j t 7 t j t Where: + 74 E• + e • (5) 7 t ] t j t v ' R.. i s the r e a l i z e d h o l d i n g p e r i o d r e t u r n f o r bond j -1 (the r e t u r n on bond j from time t t o time t+1) r f f c i s the r i s k - f r e e r a t e of r e t u r n over the h o l d i n g p e r i o d D. i s the M a c a u l a y / F i s h e r and W e i l d u r a t i o n of bond -* j a t time t ( c a l c u l a t e d w i t h e r r o r because the d i s c o u n t f u n c t i o n must be e s t i m a t e d ) C • i s the coupon on bond j j ^  M.. i s the time t o m a t u r i t y of bond j , measured a t J time t E. i s the p r i c i n g e r r o r f o r bond j a t time t -1 ( c a l c u l a t e d as the a c t u a l p r i c e of the bond minus the p r e d i c t e d p r i c e ) The r e s u l t s from ( 5 ) — t h e time s e r i e s of month-by-month v a l u e s of the r e g r e s s i o n c o e f f i c i e n t s 7 n t , 7 l f c / 7 2 t ' 7 ^ t ' and 7 4 t f o r the twenty-year p e r i o d 1961-1980—are the i n p u t s f o r the t e s t s of hypotheses. As y e t , the l e n g t h of the h o l d i n g p e r i o d has not been s p e c i f i e d . I t i s u s e f u l t o examine d i f f e r e n t h o l d i n g p e r i o d s , p a r t l y because the d u r a t i o n l i t e r a t u r e does not g e n e r a l l y s p e c i f y any p a r t i c u l a r h o l d i n g p e r i o d f o r the r e t u r n - d u r a t i o n r e l a t i o n , and a l s o because " . . . i t i s i m p o r t a n t t o determine the r o b u s t n e s s of d u r a t i o n over d i f f e r e n t h o l d i n g p e r i o d s . " [13; p . 2 5 2 ] , For t h e s e two 20 r e a s o n s , a n a l y s i s i s c a r r i e d out f o r h o l d i n g p e r i o d s of one month and t h r e e months. Thus, r e s u l t s a re o b t a i n e d when bond r e t u r n s a re c a l c u l a t e d on a monthly b a s i s and a l s o when r e t u r n s a r e c a l c u l a t e d on a q u a r t e r l y b a s i s . In e i t h e r c a s e , the independent v a r i a b l e s , i n c l u d i n g the p r i c e p r e d i c t i o n e r r o r , a r e c a l c u l a t e d as of the b e g i n n i n g of the o b s e r v a t i o n i n t e r v a l . E q u a t i o n ( 5 ) , which i s e s t i m a t e d month-by-month u s i n g i n d i v i d u a l bond d a t a , can a l s o be e s t i m a t e d u s i n g p o r t f o l i o s of government bonds. To do t h i s , s e c u r i t i e s a r e combined i n t o p o r t f o l i o s each month a c c o r d i n g t o m a t u r i t y . R e t u r n s , d u r a t i o n s , and m a t u r i t i e s f o r the p o r t f o l i o s a r e o b t a i n e d by e q u a l l y w e i g h t i n g the a s s i g n e d s e c u r i t i e s . The (Macaulay/ F i s h e r and W e i l ) d u r a t i o n of a p o r t f o l i o i s , q u i t e c o n v e n i e n t l y , the weighted average of the d u r a t i o n s of the bonds i n the p o r t f o l i o . Two d i f f e r e n t e q u a l l y w e i g h t e d p o r t f o l i o schemes are c o n s i d e r e d : 1. The f i r s t scheme p l a c e s a l l bonds w i t h m a t u r i t i e s between 0 and 1 year i n p o r t f o l i o 1, between 1 and 2 y e a r s i n p o r t f o l i o 2, and so on u n t i l the t e n t h p o r t f o l i o which c o n t a i n s bonds w i t h m a t u r i t i e s between 9 and 10 y e a r s . T h i s scheme i s used by Brennan and Schwartz [4] and [ 6 ] . 2. The second scheme, which t a k e s i n t o account the s c a r c i t y of d a t a i n the 7 t o 10 year m a t u r i t y range, combines the s e t h r e e p r e v i o u s p o r t f o l i o s i n t o a s i n g l e one. 21 Both methods a r e i n c l u d e d i n the study i n o r d e r t o examine the s e n s i t i v i t y of the e s t i m a t e s of the c o e f f i c i e n t s t o the p a r t i c u l a r p o r t f o l i o scheme used. 4.3 COMPUTATIONAL DETAILS 4.3.1 RETURN COMPUTATIONS The r e a l i z e d one-month h o l d i n g p e r i o d r e t u r n f o r bond j i s d e f i n e d t o be the p r i c e change i n the bond p l u s i n t e r e s t d i v i d e d by l a s t month's p r i c e and i s c a l c u l a t e d a s : (P. + A l . ) - (P. + A l . ) + I . R. = Dft+1 ] f t + i j t ] t D,t+1 Dt Where: ( P j t + A I j t ) P j t i s the average of b i d - a s k p r i c e s f o r bond j a t the end of month t A l .. i s the a c c r u e d i n t e r e s t on s e c u r i t y j a t the -1 end of month t I . t + 1 i s the i n t e r e s t p a i d on s e c u r i t y j between 3 ' the end of month t and the end of month t + 1 L i k e w i s e , the r e a l i z e d three-month h o l d i n g p e r i o d r e t u r n f o r bond j i s d e f i n e d t o be the p r i c e change i n the bond over the q u a r t e r p l u s i n t e r e s t d i v i d e d by t h e p r i c e of the bond a t the b e g i n n i n g of the q u a r t e r . The r i s k - f r e e r a t e of r e t u r n i s c a l c u l a t e d as the r e t u r n on government t r e a s u r y b i l l s over the r e l e v a n t h o l d i n g p e r i o d . 22 4.3.2 DURATION COMPUTATIONS The d u r a t i o n of each bond i s c a l c u l a t e d e v e r y month of the sample p e r i o d a c c o r d i n g t o the M a c a u l a y / F i s h e r and W e i l f o r m u l a , e q u a t i o n ( 1 ) , which computes p r e s e n t v a l u e w e i g h t s u s i n g d i s c o u n t bond y i e l d s from the term s t r u c t u r e . A p p l i c a t i o n of the M a c a u l a y / F i s h e r and W e i l d u r a t i o n measure presupposes t h a t the d i s c o u n t f u n c t i o n i s known. In r e a l i t y , s i n c e t h e r e e x i s t no d i s c o u n t government bonds and the s e t of coupon bonds i s not c o m p l e t e , 3 the d i s c o u n t f u n c t i o n f o r each p e r i o d must be e s t i m a t e d . The d i s c o u n t f u n c t i o n i s t h e r e f o r e e s t i m a t e d each month of the sample p e r i o d u s i n g d a t a on a l l t a x a b l e government bonds ( e x c l u d i n g " f l o w e r bonds," c a l l a b l e bonds/ and deep d i s c o u n t bonds). The f u n c t i o n i s a pproximated by a c o n t i n u o u s l y d i f f e r e n t i a b l e p i e c e w i s e q u a d r a t i c f u n c t i o n as suggested by M c C u l l o c h [ 2 1 ] . 4.3.3 ERROR COMPUTATIONS The p r i c i n g e r r o r f o r bond j a t time t , E j t > i s d e f i n e d t o be the a c t u a l p r i c e of the bond minus the p r e d i c t e d p r i c e of the bond. The p r e d i c t e d p r i c e of the bond a t time t i s c a l c u l a t e d by d i s c o u n t i n g the bond's c a s h f l o w s ( i n t e r e s t and p r i n c i p a l ) u s i n g the d i s c o u n t 3A complete s e t of coupon bonds would be s a i d t o e x i s t i f one bond matured each p e r i o d i n the f u t u r e up t o the h o r i z o n . 23 f u n c t i o n e s t i m a t e d at time t . Table 1 conveys i n f o r m a t i o n about the average magnitude of t h e s e p r i c i n g e r r o r s . The f i r s t l i n e of the t a b l e shows the r e s u l t s of the p r e d i c t i o n s f o r each month of the sample p e r i o d f o r a l l U.S. government bonds w i t h m a t u r i t i e s up t o 10 y e a r s . The r o o t mean square e r r o r (RMSE) i s a s t a t i s t i c t h a t p r o v i d e s a measure of the average a b s o l u t e e r r o r ( u n l i k e the mean e r r o r , which a l l o w s p o s i t i v e and n e g a t i v e e r r o r s t o be o f f s e t t i n g ) . . The r o o t mean square p r i c e p r e d i c t i o n e r r o r f o r the whole p e r i o d i s $2.25 per $100 of par v a l u e . A p r e d i c t e d y i e l d t o m a t u r i t y i s a l s o c a l c u l a t e d each p e r i o d based on the p r e d i c t e d bond p r i c e , and the r o o t mean square e r r o r of t h i s p r e d i c t e d y i e l d i s a l s o r e p o r t e d . The r o o t mean square y i e l d p r e d i c t i o n e r r o r f o r the whole p e r i o d i s 1.02 per c e n t . The b a l a n c e of the t a b l e shows the r e s u l t s o b t a i n e d f o r p r e d i c t i o n s i n December of each y e a r . Table 2 c o n t a i n s the p r i c e and y i e l d p r e d i c t i o n e r r o r s f o r Canadian government bonds. The r o o t mean square p r i c e p r e d i c t i o n e r r o r f o r the 20-year sample p e r i o d i s $2.12 per $100 of par v a l u e . The c o r r e s p o n d i n g r o o t mean square y i e l d p r e d i c t i o n e r r o r i s 0.80 per c e n t . 24 TABLE 1: U.S. Bond P r i c e and Y i e l d P r e d i c t i o n s P r i c e s 3 Y i e l d s Number of Mean E r r o r ' 3 O b s e r v a t i o n s ($) RMSE ($) Mean E r r o r (%) RMSE (%) F u l l P e r i o d 0 91 68 -0.08 2.25 0.06 1 .02 I 9 6 1 d 29 -0.04 1 .54 0.02 0.58 1962 32 -0.03 1 .20 0.02 0.48 1 963 32 -0.02 1 .07 0.01 0.37 1 964 31 -0.03 1.10 0.02 0.42 1965 26 -0.03 1 .04 0.02 0.43 1966 25 -0.07 1 .72 0.05 0.66 1 967 26 -0.05 1 .63 0.02 0.69 1968 26 -0.04 1.16 0.02 0.51 1969 28 -0.14 2.80 0.10 1 .27 1 970 31 -0.21 . 3.53 0.16 1 .43 1 971 31 -0.26 4.15 0.17 1 .56 1 972 33 -0.20 3.43 0.11 1 .32 1 973 35 -0.19 3.53 0.14 1.51 1 974 40 -0.22 3.65 0.16 1 .55 1975 44 -0.02 1 .43 0.01 0.70 1976 55 -0.03 1 .50 0.02 0.68 1977 62 -0.02 1 .39 0.02 0.76 1 978 63 -0.03 1 .62 0.02 0.83 1979 66 -0.04 1 .88 0.03 1.15 1980 70 -0.17 3.03 0.13 1 .69 a. Per $100 par v a l u e . b. A c t u a l - P r e d i c t e d . c. January/61-December/80. d. December. TABLE 2: Canadian Bond P r i c e and Y i e l d P r e d i c t i o n s S3 P r i c e s Y i e l d s Number of O b s e r v a t i o n s Mean E r r o r ' 3 (S) RMSE ($) Mean E r r o r (%) RMSE (%) F u l l P e r i o d 0 4109 -0.05 2.12 0.02 0.80 I 9 6 1 d 1 5 -0 .08 1 .79 0.07 0.43 1962 14 -0.05 1.31 0.03 0.34 1963 1 3 -0.04 1 .20 0.05 0.45 1 964 1 4 -0.01 0.92 0.01 0.50 1965 1 5 0.01 0.77 -0.03 0 .39 1 966 13 -0.02 0.77 0.02 0.40 1967 1 4 -0.02 0.86 0.02 0.36 1 968 21 -0.03 1.41 0.02 0.67 1 969 1 7 -0.04 1.71 0.04 0 .80 1 970 22 -0.04 1 .83 0.03 0.68 1 971 27 -0.03 1 .73 -0.01 0.79 1972 22 -0.02 1 .53 0.00 0.62 1 973 1 9 0.05 1 .72 0.05 0.70 1 974 1 5 0.01 1 .95 -0 .09 1 .26 1975 15 -0 .03 1 .32 0.05 0.66 1 976 15 -0 .08 2.28 0.04 . 0.90 1 977 19 -0 .02 2.19 -0 .08 1.11 1978 22 -0.09 2.39 0.05 1 .00 1979 24 -0.15 2.97 0.08 1 .25 1980 20 -0.11 2.76 0.06 1 .47 a. Per $100 par v a l u e . b. A c t u a l - P r e d i c t e d . c. January/61-December/80. d. December. 26 4.4 TESTS OF HYPOTHESES The t h r e e hypotheses are t e s t e d by c o n s t r u c t i n g the means of the time s e r i e s of the c o e f f i c i e n t s of e q u a t i o n (5) and c a l c u l a t i n g the t - s t a t i s t i c s i n the manner f i r s t s u ggested by Fama and MacBeth [10] i n a r e l a t e d c o n t e x t . T h i s method has a l s o been used by Brennan and Schwartz [4] and [ 5 ] , and by G u l t e k i n and R o g a l s k i [13]. E s s e n t i a l l y , a time s e r i e s f o r 7 i s c r e a t e d " . . . i n order t o c o r r e c t f o r any s t a t i s t i c a l b i a s e s i n o b t a i n i n g the p e r i o d - b y - p e r i o d e s t i m a t e s of 7i." [13; p.249]. A s e r i e s of c r o s s - s e c t i o n a l r e g r e s s i o n s a r e r u n , t h e r e b y g e n e r a t i n g f i v e s e r i e s of e s t i m a t e s 7i t i=0,...,4. The sample means are t a k e n as the f i n a l e s t i m a t e s of 7O, 71, 72, 73, and 74, and s t a n d a r d e r r o r s and t - s t a t i s t i c s a r e computed as i f the s e r i e s a r e random samples ( i n d e p e n d e n t l y and i d e n t i c a l l y d i s t r i b u t e d over t i m e ) . The t - s t a t i s t i c s f o r t e s t i n g the h y p o t h e s i s t h a t E(7i)=0 a r e computed by t a k i n g the r a t i o of the average 7i t i m e s the square r o o t of the number of months i n the sample p e r i o d c o n s i d e r e d over the s t a n d a r d d e v i a t i o n of the monthly e s t i m a t e . These t - s t a t i s t i c s a r e : t ( 7 i ) - 7 1 ^ (7) s(7i) Where: n i s the number of months i n the p e r i o d , which i s a l s o the_number of e s t i m a t e s 7i. used t o compute yi and s(7i) 27 yi i s the average of the month-by-month r e g r e s s i o n coef f i c i e n t s s ( 7 i ) i s the s t a n d a r d d e v i a t i o n of the monthly e s t imates I f s u c c e s s i v e v a l u e s of yi^ are i n d e p e n d e n t l y and i d e n t i c a l l y d i s t r i b u t e d normal random v a r i a b l e s , the t - s t a t i s t i c of (7) i s a drawing from the s t u d e n t d i s t r i b u t i o n w i t h n-1 degrees of freedom. S i n c e the s h o r t e s t s u b p e r i o d i n t h i s study i s f i v e y e a r s , n-1 i s always g r e a t e r than 59 and the s t u d e n t d i s t r i b u t i o n i s w e l l a p p r o x i m a t e d by the u n i t normal d i s t r i b u t i o n . 4.5 LIMITATIONS OF METHODOLOGY The r e g r e s s i o n procedure i n t h i s study i s e s s e n t i a l l y the method proposed and used by Fama and MacBeth [10] i n a r e l a t e d c o n t e x t . The Fama-MacBeth p r o c e d u r e , which i n v o l v e s r u n n i n g a s e r i e s of c r o s s - s e c t i o n a l r e g r e s s i o n s , i s e q u i v a l e n t t o j o i n t g e n e r a l i z e d l e a s t squares (GLS) on the whole system i f i n each c r o s s - s e c t i o n a l r e g r e s s i o n : 1. the r e t u r n p r o c e s s e s a r e independent; t h a t i s , t h e r e i s no c r o s s - c o r r e l a t i o n of r e t u r n s , and i f 2. the independent v a r i a b l e s a r e not measured w i t h e r r o r . The f i r s t a s s u m p t i o n , l i k e any a s s u m p t i o n , i s not a c o m p l e t e l y a c c u r a t e d e s c r i p t i o n of the w o r l d . In f a c t , i t i s r a t h e r u n r e a l i s t i c t o assume t h a t the r e t u r n p r o c e s s e s a r e independent, g i v e n t h a t changes i n the y i e l d c u r v e t e n d t o have a somewhat s i m i l a r e f f e c t on a l l bonds. To the e x t e n t t h a t the assumption i s i n a c c u r a t e , the Fama-MacBeth approach 28 w i l l produce e s t i m a t e s of the r e g r e s s i o n parameters which ar e not e f f i c i e n t , but which a r e n e v e r t h e l e s s u n b i a s e d . I f the second assumption i s v i o l a t e d ( t h a t i s , i f any independent v a r i a b l e s a r e measured w i t h e r r o r ) , an " e r r o r s - i n - v a r i a b l e s problem" a r i s e s . In i n t u i t i v e terms, the problem c e n t e r s on the f a c t t h a t i f a proxy e x p l a n a t o r y v a r i a b l e i s used i n a l e a s t squares r e g r e s s i o n , the computed c o e f f i c i e n t s do not have the same p r o p e r t i e s as i f the t r u e e x p l a n a t o r y v a r i a b l e were used. U n f o r t u n a t e l y , two of the independent v a r i a b l e s i n the l i n e a r r e g r e s s i o n model s p e c i f i e d i n e q u a t i o n (5) a r e measured w i t h e r r o r . S i n c e the d i s c o u n t f u n c t i o n must be e s t i m a t e d each month, both the d u r a t i o n c a l c u l a t i o n s and the p r i c i n g e r r o r c a l c u l a t i o n s i n v o l v e measurement e r r o r . As i s w e l l known, the OLS e s t i m a t o r of the r e g r e s s i o n parameters i s b i a s e d under t h e s e c i r c u m s t a n c e s . The two p o s s i b l e s o l u t i o n s t o t h i s problem i n v o l v e u s i n g c o r r e c t i o n f a c t o r s or i t e r a t i v e p r o c e d u r e s , but t h e s e methods w i l l not be employed i n t h i s s t u d y . At any r a t e , the d i s c o u n t f u n c t i o n w i l l i n a l l l i k e l i h o o d be e s t i m a t e d w i t h m i n i m a l e r r o r because, as M c C u l l o c h [21] p o i n t s o u t , the d i s c o u n t f u n c t i o n i s never e x t r a p o l a t e d beyond the m a t u r i t i e s of the s e c u r i t i e s o b s e r v e d . However, any e s t i m a t i o n e r r o r t h a t does e x i s t w i l l t e n d t o i n f l u e n c e a l l the c o e f f i c i e n t s i n the model because the d u r a t i o n measure i s d e r i v e d from the coupon and m a t u r i t y v a l u e s . C h a p t e r 5 RESULTS R e s u l t s are p r e s e n t e d f o r seven p e r i o d s : the o v e r a l l p e r i o d J a n uary 1961 t o December 1980; two 10-year s u b p e r i o d s , January 1961 t o December 1970 and J a n uary 1971 t o December 1980; and f o u r s u b p e r i o d s s t a r t i n g i n January 1961 and c o v e r i n g f i v e y e a r s each. C o n c i s e summary t a b l e s a r e r e p o r t e d , w i t h o t h e r r e s u l t s h i g h l i g h t e d i n the d i s c u s s i o n . R e s u l t s are p r e s e n t e d f o r t h r e e d i f f e r e n t v e r s i o n s of the r e t u r n - d u r a t i o n r e g r e s s i o n e q u a t i o n ( 5 ) : the f i r s t v e r s i o n s u p p r e s s e s t h r e e of the v a r i a b l e s i n ( 5 ) ; the second v e r s i o n s u p p r e s s e s two of t h e v a r i a b l e s i n ( 5 ) ; and the t h i r d v e r s i o n i s based on e q u a t i o n (5) e x a c t l y . 5.1 AMERICAN RESULTS The c r o s s - s e c t i o n a l r e g r e s s i o n r e s u l t s f o r U.S. government bonds a r e p r e s e n t e d i n T a b l e 3. For each p e r i o d and model T a b l e 3 shows the average of the (1-month h o l d i n g p e r i o d ) r e g r e s s i o n c o e f f i c i e n t e s t i m a t e s , yi. The t a b l e a l s o shows t - s t a t i s t i c s f o r t e s t i n g t h e h y p o t h e s i s t h a t E ( 7 i ) = 0 . The r e s u l t s i n T a b l e 3 r e v e a l t h a t over the 20-year p e r i o d , or any s u b p e r i o d , and f o r any model, the v a l u e of t(7l) i s not l a r g e . Thus, t h e r e i s no e v i d e n c e of a s i g n i f i c a n t r e l a t i o n s h i p between bond r e t u r n s (measured on a monthly b a s i s ) and M a c a u l a y / F i s h e r and W e i l d u r a t i o n . I t i s s u r p r i s i n g t h a t a l t h o u g h 71 i s not s i g n i f i c a n t l y d i f f e r e n t 29 30 from z e r o i t i s n e g a t i v e on a v e r a g e . From T a b l e 3 i t can a l s o be observed t h a t the t e s t s t a t i s t i c f o r the m a t u r i t y c o e f f i c i e n t , t ( 7 3 ) , i s g e n e r a l l y v e r y s m a l l . T h e r e f o r e , i t appears t h a t d u r a t i o n i n c o r p o r a t e s the e f f e c t of m a t u r i t y d i f f e r e n c e s on p r i c e v o l a t i l i t y . For most time p e r i o d s , however, the t - v a l u e f o r the coupon c o e f f i c i e n t , t(y2), i s l a r g e and p o s i t i v e . There appears t o be a s t a t i s t i c a l l y o b s e r v a b l e p o s i t i v e r e l a t i o n s h i p between coupon r a t e and bond r e t u r n s . The p o s i t i v e s i g n of t(y2) i s s u r p r i s i n g because i t i s not c o n s i s t e n t w i t h the premise t h a t coupon i s a measure of r i s k . " A p o s s i b l e e x p l a n a t i o n f o r the p o s i t i v e s i g n i s t h a t the coupon r a t e as an independent v a r i a b l e i s e s s e n t i a l l y c a p t u r i n g a t a x e f f e c t . Low coupon bonds have a l a r g e amount of t h e i r r e t u r n coming i n the form of a c a p i t a l g a i n , which i s t a x e d a t a lower r a t e . One would t h e r e f o r e e x pect t h a t the r e t u r n on h i g h coupon bonds would be h i g h e r t o compensate f o r the f a c t t h a t t a x e s would be h i g h e r . The r e s u l t s r e p o r t e d i n T a b l e 3 a l s o r e v e a l t h a t f o r the whole p e r i o d and f o r the f i r s t 10-year s u b p e r i o d a s i g n i f i c a n t r e l a t i o n e x i s t s between the p r i c i n g e r r o r and the r a t e of r e t u r n over the next i n t e r v a l . These r e s u l t s , which t e n d t o suggest t h a t market i n e f f i c i e n c i e s e x i s t , a r e , however, s e n s i t i v e t o the i n c l u s i o n of the coupon "Other t h i n g s b e i n g e q u a l , p r i c e v o l a t i l i t y i s lower f o r h i g h e r coupon bonds, i n d i c a t i n g t h a t i f coupon were a measure of r i s k the c o e f f i c i e n t would be n e g a t i v e 31 TABLE 3: C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r U.S. Bonds ( t a x a b l e bonds w i t h m a t u r i t i e s l e s s than 10 y e a r s ; t - r a t i o s i n p a r e n t h e s e s ) H o l d i n g P e r i o d = One Month I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d yO Al 74 Ja n u a r y 1961 t o December 1980 0.313 (0.80) 0.340 (0.88) -0 (-1 208 14) ( 0.213 •1.16) 0.001 (0.02) -1.958 (-1.78) 0.027 (0.18) 0.797 (2.93) -0.216 (-1.68) -0.237 (-2.24) Ja n u a r y 1961 t o December 1 970 0.507 (2.06) 0.511 (2.06) -0. 170 (-0.90) -0. 169 (-0.89) 0.079 (0.87) -0.645 (-1.44) 0. 1 09 (0.84) 0.629 (3.25) -0.240 (-1.48) -0.168 (-2.29) Ja n u a r y 1971 t o December 1980 0.116 (0.15) 0. 168 (0.23) -0.247 (-0.79) -0.258 (-0.81 ) -0.077 (-0.55) -3.282 (-1.52) -0.055 (-0.20) 0.967 (1.89) -0. 180 (-0.93) -0.306 (-1.54) Note: yO, y\, y2, y3, 74 are the a r i t h m e t i c means of the monthly c r o s s -s e c t i o n a l e s t i m a t e s . The 7i numbers have been m u l t i p l i e d by 1000. The t - v a l u e s (shown i n p a r e n t h e s e s ) t e s t the n u l l h y p o t h e s i s t h a t the mean v a l u e s a r e z e r o . ( c o n t . over) 32 TABLE 3 CONTD. I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d |fj j \ j2 j3 7 4 J a n u a r y 0.155 -0.026 1961 t o (0.44) (-0.21) December 1965 0.113 -0.026 0.050 (0.43) (-0.20) (0.56) -0.478 0.046 0.401 -0.084 -0.105 (-1.16) (0.35) (2.42) (-0.90) (-1.31) January 0.900 -0.314 1966 t o (2.17) (-0.88) December 1970 0.908 -0.313 0.108 (2.18) (-0.87) (0.67) -0.812 0.171 0.857 -0.408 -0.232 (-1.01) (0.77) (2.45) (-1..29) (-1.88) Jan u a r y 0.511 -0.103 1971 t o (0.66) (-0.31) December 1975 0.515 -0.094 -0.022 (0.68) (-0.28) (-0.13) -1.765 0.069 0.771 -0.156 -0.266 (-0.96) (0.20) (1.42) (-0.57) (-1.73) Jan u a r y -0.900 -0.393 1976 t o (-0.22) (-0.73) December 1980 -0.186 -0.425 -0.133 (-0.15) (-0.79) (-0.58) -4.826 -0.182 1.166 -0.204 -0.346 (-1.22) (-0.41) (1.33) (-0.74) (-0.93) Note: t h e 7 i v a l u e s have been m u l t i p l i e d by 1000. 33 and m a t u r i t y v a r i a b l e s . T a ble 4 shows the f i r s t o r d e r a u t o c o r r e l a t i o n s , p ( y i ) , of the v a r i o u s monthly 7it»5 From Table 4 i t can be seen t h a t the s e r i a l c o r r e l a t i o n s pijO), p ( y l ) , P ( J 2 ) , p{y3), and p ( j i ) are o f t e n g r e a t e r than z e r o . The a u t o c o r r e l a t i o n s appear most s i g n i f i c a n t f o r ji and l e a s t s i g n i f i c a n t f o r 71 . As mentioned e a r l i e r , 71 does not appear t o be s i g n i f i c a n t l y p o s i t i v e . One p o s s i b l e e x p l a n a t i o n f o r 71 b e i n g n e g a t i v e on average i s t h a t i n t e r e s t r a t e s have t r e n d e d upwards over most of the sample p e r i o d . In any g i v e n p e r i o d 71 w i l l be i n f l u e n c e d by changes i n i n t e r e s t r a t e s i n the economy, so t h a t i n p e r i o d s of r i s i n g i n t e r e s t r a t e s one -would expect 71 t o be n e g a t i v e on average. H y p o t h e s i s 1 s h o u l d h o l d on average ( t h a t i s , over r e p e a t e d e x p e r i m e n t s ) i f i n t e r e s t r a t e s b o t h r i s e and f a l l but average t o z e r o . In o r d e r t o d e t e r m i n e what p o r t i o n of 71 i s not e x p l a i n e d by changes i n i n t e r e s t r a t e s , the month-by-month v a l u e s of 71 a r e r e g r e s s e d on changes i n the i n t e r e s t r a t e . The i n t e r c e p t term i n such a r e g r e s s i o n can be i n t e r p r e t e d as t h a t p o r t i o n of 71 which i s not d e t e r m i n e d by changes i n the l e v e l of i n t e r e s t r a t e s . One would a n t i c i p a t e t h a t the i n t e r c e p t would have a p o s i t i v e s i g n and t h a t the c o e f f i c i e n t f o r the change i n the l e v e l of i n t e r e s t r a t e s would be e q u a l t o -1.0. 5 A u t o c o r r e l a t i o n s f o r l a g s g r e a t e r than one have been computed but a r e not r e p o r t e d h e r e . 34 TABLE 4: E s t i m a t e d F i r s t Order A u t o c o r r e l a t i o n s of the  Monthly C r o s s - S e c t i o n a l Parameter E s t i m a t e s (U.S. bonds w i t h m a t u r i t i e s l e s s than 10 y e a r s ) H o l d i n g P e r i o d = One Month P e r i o d 3(70) S t a t i s t i c p{y^ ) p(y2) 3(73) p(74) 1/61-12/80 -.097 .042 -.091 .033 -.451* -.132* -.400* -.145* -.394* -.419* 1/61-12/70 .070 .061 -.104 .071 .055 -.093 -.395* 258* -.191* -.210* 1/71-12/80 -.122 -.144 -.477* .031 .025 -.141 -.401* 423* -.095 -.451* 1/61-12/65 -.140 -.137 -.018 .230* .232* .098 -.085 -.079 1 1 1 079 1/66-12/70 .122 . 107 -. 1 36 .038 .021 -.091 -.310* -.217 490*-355* 1/71-12/75 183 187 409* . 1 20 . 1 35 .364* -.246* -.097 296* 321* 1/76-12/80 -.085 .092 -.072 .093 -.498* .009 -.498* -.095 463* 478* • S i g n i f i c a n t a t the 0.05 l e v e l . 35 The f o l l o w i n g r e g r e s s i o n models a r e e s t i m a t e d : 7 l f c = a + b,AYL t (8) and 7 l f c = a + b,AYL t + b 2 A Y S t (9) Where: yl. i s the e s t i m a t e d v a l u e of 71 f o r month t ( o b t a i n e d from the c r o s s - s e c t i o n a l r e g r e s s i o n f o r month t which c o n t a i n s o n l y an i n t e r c e p t term and d u r a t i o n ) AYL i s the c o r r e s p o n d i n g change i n the y i e l d on l o n g -term U.S. Government bonds AYS, i s the c o r r e s p o n d i n g change i n the U.S. T r e a s u r y b i l l r a t e The r e g r e s s i o n c o e f f i c i e n t s and t - s t a t i s t i c s a r e r e p o r t e d below i n Ta b l e 5. TABLE 5: E s t i m a t e d C o e f f i c i e n t s and T - s t a t i s t i c s from  R e g r e s s i n g 71 on the C o r r e s p o n d i n g Changes i n the Long and Short Rates of I n t e r e s t a b, b 2 E s t i m a t e d C o e f f i c i e n t 0.00597 -0.80974 t - s t a t i s t i c (0.41 ) (-12.05) R 2=0.3798 F=145.13 N=239 E s t imated C o e f f i c i e n t 0.00600 -0.99096 0.10668 t - s t a t i s t i c (0.42) (-11.68) (3.38) R 2=0.4085 F=81.488 N=239 36 From the r e s u l t s i n T a b l e 5 i t can be seen t h a t the i n t e r c e p t term i s p o s i t i v e , as p r e d i c t e d , a l t h o u g h i t i s not s i g n i f i c a n t , and t h a t the c o e f f i c i e n t f o r the change i n the l o n g r a t e i s c l o s e t o one and h i g h l y s i g n i f i c a n t . These r e s u l t s suggest t h a t , when changes i n the l e v e l of i n t e r e s t r a t e s have been accounted f o r , t h e r e i s p o s s i b l y a p o s i t i v e r e l a t i o n s h i p between bond r e t u r n s and d u r a t i o n . The a n a l y s i s summarized i n T a b l e 3 has a l s o been c a r r i e d out f o r a h o l d i n g p e r i o d of t h r e e months. S u r p r i s i n g l y , the l e n g t h of the h o l d i n g p e r i o d t u r n s out not t o a f f e c t the b a s i c c o n c l u s i o n s . T a b l e 6 c o n t a i n s the 3-month h o l d i n g p e r i o d r e g r e s s i o n c o e f f i c i e n t e s t i m a t e s , and T a b l e 7 shows the e s t i m a t e d f i r s t o r d e r a u t o c o r r e l a t i o n s of the v a r i o u s monthly yi^ when the h o l d i n g p e r i o d i s t h r e e months. I t s h o u l d be noted a t the o u t s e t t h a t r e t u r n s f o r o v e r l a p p i n g 3-month h o l d i n g p e r i o d s have been used i n the r e g r e s s i o n s . T h i s i m p l i e s t h a t the yi w i l l not be independent over time (as can be seen from the f i r s t - o r d e r a u t o c o r r e l a t i o n s i n Table 7 ) , and hence t h a t the t - s t a t i s t i c s i n T a b l e 6 cannot be i n t e r p r e t e d as b e i n g c o r r e c t . To a c e r t a i n e x t e n t , t h i s problem i s c o r r e c t e d f o r l a t e r when the means and s t a n d a r d e r r o r s of the 3-month h o l d i n g p e r i o d r e g r e s s i o n c o e f f i c i e n t s , yi, are r e c a l c u l a t e d under the a ssumption t h a t each of the yi f o l l o w s an ARIMA p r o c e s s (see T a b l e s 14, 15, and 16). The r e s u l t s c o n t a i n e d i n T a b l e 6 r e v e a l t h a t over most time p e r i o d s , and f o r most models, the v a l u e of 71 i s q u i t e 37 s m a l l . Hence, t h e r e i s l i t t l e e v i d e n c e of a s i g n i f i c a n t ( p o s i t i v e ) r e l a t i o n s h i p between d u r a t i o n and bond r e t u r n s . D u r i n g some time p e r i o d s 71 i s , i n f a c t , n e g a t i v e ( a l t h o u g h i n s i g n i f i c a n t l y s o ) . As mentioned e a r l i e r , a p o s s i b l e e x p l a n a t i o n f o r the n e g a t i v e s i g n i s the upward movement i n i n t e r e s t r a t e s over most of the sample p e r i o d . A comparison of the r e s u l t s i n Tab l e s 3 and 6 a l s o r e v e a l s t h a t t h e r e a re g r e a t s i m i l a r i t i e s between the one-month h o l d i n g p e r i o d and three-month h o l d i n g p e r i o d r e g r e s s i o n c o e f f i c i e n t s f o r coupon, m a t u r i t y , and p r i c i n g e r r o r . T a b l e 6 r e v e a l s t h a t f o r the o v e r a l l p e r i o d 1961-1980, and f o r a l l s u b p e r i o d s , the v a l u e of 72, the coupon c o e f f i c i e n t , i s l a r g e and s y s t e m a t i c a l l y p o s i t i v e f o r a l l models. As i n the one-month h o l d i n g p e r i o d c a s e , the coupon c o e f f i c i e n t i s perhaps c a p t u r i n g a t a x - e f f e c t . T a b l e 6 a l s o shows t h a t , r e g a r d l e s s of the model or the time p e r i o d examined, the v a l u e s of 73 are c o n s i s t e n t l y n e g a t i v e . Thus, i t i s p o s s i b l e t h a t d u r a t i o n does not i n c o r p o r a t e the e f f e c t of m a t u r i t y d i f f e r e n c e s on p r i c e v o l a t i l i t y . F i n a l l y , e v i d e n c e i n T a b l e 6 s u g g e s t s t h a t p r i c i n g e r r o r s a r e r e l a t e d i n a s y s t e m a t i c f a s h i o n t o subsequent bond r e t u r n s . For the t h i r d model and f o r a l l time p e r i o d s the c o e f f i c i e n t f o r the p r i c i n g e r r o r i s l a r g e and n e g a t i v e . From T a b l e 7, i t can be seen t h a t the s e r i a l c o r r e l a t i o n s £(70), £(71), p(y2), p(y3), and £(74) a r e i n th e m a j o r i t y of cases s i g n i f i c a n t l y g r e a t e r than z e r o . T h i s i s u n doubtedly due t o the use of o v e r l a p p i n g d a t a . 38 TABLE 6: C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r U.S. Bonds ( t a x a b l e bonds w i t h m a t u r i t i e s l e s s than 10 y e a r s ; t - r a t i o s i n p a r e n t h e s e s ) H o l d i n g P e r i o d = Three Months I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d yO 71 1 2 73 74 January 1961 t o December 1980 1 .487 (2.66) 1 .480 (2.64) -0.764 (-2.42) -0.758 (-2.40) 0.015 (0.16) -4.947 (-4.92) 0.132 (0.60) 2.171 (8.10) -0.732 (-4.21 ) -0.606 (-6.46) January 1961 t o December 1970 1 .349 (3.28) 1 .335 (3.20) -0.344 (-0.97) -0.337 (-0.96) 0.270 (2.16) -1 .301 (-2.02) 0.248 (1.09) 1 .535 (6.21) -0.540 (-2.42) -0.329 (-3.42) January 1971 t o December 1980 1 .629 (1.55) 1 .629 (1.55) -1.195 (-2.28) -1.190 (-2.26) -0.248 (-1.83) -8.687 (-4.65) 0.013 (0.04) 2.823 (5.97) -0.936 (-3.46) -0.891 (-5.61) Note: the 7 i v a l u e s have been m u l t i p l i e d by 1000. ( c o n t . over) 39 TABLE 6 CONTD. I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d %0 j_\ j2 |3 7 4 January 0.429 -0.142 1961 t o (1.14) (-0.55) December 1965 0.406 -0.136 0.130 (1.07) (-0.53) (1.09) -1.601 0.195 1.330 -0.324 -0.371 (-2.64) (0.99) (5.76) (-2.23) (-2.82) January 2.269 -0.546 1966 t o (3.17) (-0.82) December 1970 2.263 -0.539 0.411 (3.11) (-0.82) (1.87) -1.000 0.301 1.739 -0.756 -0.288 (-0.87) (0.73) (3.97) (-1.78) (-2.03) January 1 .943 -0.581 1971 t o (1.86) (-1.17) December 1975 1.927 -0.556 0.011 (1.85) (-1.10) (0.05) -3.674 0.208 1.914 -0.660 -0.588 (-2.06) (0.56) (3.15) (-1.75) (-3.71) January 1 .300 -1.843 1976 t o (0.69) (-1.96) December 1980 1.316 -1.857 -0.521 (0.70) (-1.98) (-3.03) -13.964 -0.191 3.781 -1.212 -1.209 (-4.34) (-0.28) (5.28) (-3.18) (-4.39) Note: the 7 i v a l u e s have been m u l t i p l i e d by 1000. 40 TABLE 7: E s t i m a t e d F i r s t Order A u t o c o r r e l a t i o n s of the  Monthly C r o s s - S e c t i o n a l Parameter E s t i m a t e s (U.S. bonds w i t h m a t u r i t i e s l e s s than 10 y e a r s ) H o l d i n g P e r i o d = Three Months S t a t i s t i c  P e r i o d p(jO) p ( 7 D P(J2) p(y3) p ( 7 4 ) 1/61-12/80 .558* .635* .560* .631* .298* .253* .384* .159* .509* .159* 1/61-12/70 .611* .685* .610* .679* .329* .453* .605* .217* .535* .409* 1/71-12/80 .551* .605* .554* .601* .219* .176* .306* .111 .480* .015 1/61-12/65 .513* .629* .511* .627* .421* .534* .596* .432* .568* .478* 1/66-12/70 .617* .692* .617* .685* .287* .425* .608* .149 .526* .299* 1/71-12/75 .612* .597* .623* .579* .292* .528* .080 .467* .481* .345* 1/76-12/80 .532* .595* .532* .595* .036 -.004 .371* -.214 .470* -.146 • S i g n i f i c a n t a t the 0.05 l e v e l . 41 The a n a l y s i s i n Table 3 (one-month h o l d i n g p e r i o d ) and Table 6 (three-month h o l d i n g p e r i o d ) has been r e p e a t e d f o r p o r t f o l i o s of T r e a s u r y s e c u r i t i e s . As mentioned e a r l i e r , two a l t e r n a t i v e e q u a l l y - w e i g h t e d p o r t f o l i o schemes a r e c o n s i d e r e d . T a b l e 8 r e p o r t s the r e s u l t s o b t a i n e d when p o r t f o l i o s e l e c t i o n method 1 i s use d , and Ta b l e 9 r e p o r t s the r e s u l t s o b t a i n e d u s i n g p o r t f o l i o s e l e c t i o n method 2. In both T a b l e 8 and Ta b l e 9 bond r e t u r n s a re c a l c u l a t e d on a monthly b a s i s . In T a b l e s 10 and 11 bond r e t u r n s a r e c a l c u l a t e d on a q u a r t e r l y b a s i s . The p o r t f o l i o s e l e c t i o n method appears t o be unimportant when the h o l d i n g p e r i o d i s one month, s i n c e the r e s u l t s i n T a b l e 8 are v e r y s i m i l a r t o those i n T a b l e 9. I t seems t h a t over the 20-year sample p e r i o d t h e r e i s no evidence of a s i g n i f i c a n t r e l a t i o n between one-month p o r t f o l i o r e t u r n s and M a c a u l a y / F i s h e r and W e i l d u r a t i o n . On the p o s i t i v e s i d e , t h e r e i s no e v i d e n c e t o suggest t h a t a s i g n i f i c a n t r e l a t i o n e x i s t s between any of the o t h e r independent v a r i a b l e s and p o r t f o l i o r e t u r n s . The r e s u l t s a re somewhat d i f f e r e n t when the h o l d i n g p e r i o d i s chosen t o be t h r e e months, i n t h a t the p o r t f o l i o s e l e c t i o n method has an apparent impact (see T a b l e s 10 and 11). When p o r t f o l i o s e l e c t i o n method 1 i s used, the average v a l u e of the c o e f f i c i e n t f o r d u r a t i o n , 7 1, i s i n s i g n i f i c a n t , but the c o e f f i c i e n t s f o r the o t h e r independent v a r i a b l e s a r e , on av e r a g e , q u i t e l a r g e ( p a r t i c u l a r l y i n the s u b p e r i o d 1971-1980). However, when the second s e l e c t i o n method i s 42 TABLE 8: C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r P o r t f o l i o s  of U.S. Bonds ( p o r t f o l i o s s e l e c t i o n method 1 used; t - r a t i o s i n p a r e n t h e s e s ) H o l d i n g P e r i o d = One Month I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d 70 J± _2?_ 74 January 1961 t o December 1980 0.265 (0.58) 0. 1 97 (0.44) -0.218 (-1.30) -0.185 (-1.08) 0.343 (1.33) -3.774 (-1.50) 0.038 (0.18) 1.104 (1.47) -0.157 (-0.91) -0.183 (-0.59) January 1961 t o December 1 970 0.476 (1.58) 0.502 (1.75) -0.177 (-0.97) -0.153 (-0.79) 1 . 1 38 (2.60) -0.030 (-0.03) 0.061 (0.23) 0.319 (0.53) -0.157 (-0.65) 0.267 (0.60) January 1971 t o December 1980 0.053 (0.06) -0.111 (-0.13) -7.518 (-1.54) -0.258 (-0.92) -0.218 (-0.76) 0.015 (0.05) 1 .890 (1.37) -0.156 (-0.64) -0.459 (-1.83) -0.633 (-1.44) Note: the 7 i v a l u e s have been m u l t i p l i e d by 1000. ( c o n t . over) 43 TABLE 8 CONTD. Per i o d I n t e r c e p t D u r a t i o n Coupon M a t u r i t y Pr . E r r o r 70 71 72 73 74 January 0.067 -0.023 1961 t o (0.22) (-0.20) December 1 965 0. 129 -0.040 0.131 (0.43) (-0.31 ) (0.47) -0.725 0.035 0.591 -0.089 -0.396 (-0.72) (0.27) (1.03) (-0.89) ( -1.43) January 0.885 -0.332 1966 t o (1.70) (-0.96) December 1 970 0.875 -0.267 2. 145 (1.79) (-0.73) (2.65) 0.676 0.088 0.042 -0.227 0.288 (0.31 ) (0.17) (0.04) (-0.47) (1.12) J a n u a r y 0.429 -0.080 1971 t o (0.52) (-0.22) December 1 975 0.070 0.009 -0.799 (0.08) (0.03) ( -2.07) 1 .842 0.246 -0.464 -0.181 -0.259 (0.48) (0.60) (-0.36) (-0.53) ( -0.57) January -0.329 -0.439 1976 t o (-0.22) (-1.02) December 1980 -0.296 -0.448 -0.114 (-0.20) (-1.00) ( -0.36) -17.035 -0.220 4.284 -0.131 -1.014 (-1.91 ) (-0.43) (1 .77) (-0.37) ( -1.34) Note: the yi v a l u e s have been m u l t i p l i e d by 1000. 44 TABLE 9: C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r P o r t f o l i o s  of U.S. Bonds ( p o r t f o l i o s e l e c t i o n method 2 used; t - r a t i o s i n p a r e n t h e s e s ) H o l d i n g P e r i o d = One Month I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d l l 1 2 11 74 January 1961 t o December 1980 • 0.302 (0.70) 0.203 (0.47) -0.225 (-1 .23) -0.181 (-0.98) 0.372 ( 1 .45) -1.267 (-0.51) -0.131 (-0.41) 0.431 (0.57) -0.029 (-0.14) -0.006 (-0.02) January 1961 t o December 1 970 0.476 (1.55) 0.495 (1.72) 0.131 (0.11) -0. 179 (-0.98) -0.154 (-0.80) 0.024 (0.09) 0.212 (0.34) -0. 122 (-0.47) 1.117 (2.58) 0.274 (0.61) January 0.126 -0.272 1971 t o (0.16) (-0.85) December 1980 -0.092 -0.209 -0.380 (-0.11) (-0.66) (-1.48) -2.665 -0.285 0.651 0.064 -0.287 (-0.55) (-0.50) (0.47) (0.20) (-0.62) Note: t h e 7 i v a l u e s have been m u l t i p l i e d by 1000. ( c o n t . over) 45 TABLE 9 CONTD. I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d ^ XJ l2. lA 74 Jan u a r y 0.068 -0.026 1961 t o (0.21) (-0.22) December 1965 0.116 -0.040 0.088 (0.38) (-0.32) (0.35) -0.404 -0.038 0.378 -0.018 -0.382 (-0.36) (-0.17) (0.58) (-0.09) (-1.24) January 0.885 -0.332 1966 t o (1 .70) (-0.96) December 1970 0.875 -0.267 2.145 (1.79) (-0.73) (2.65) 0.676 0.088' 0.042 -0.019 0.941 (0.31) (0.17) (0.04) (-0.47) (1.12) J a n u a r y 0.553 -0.126 1971 t o (0.66) (-0.34) December 1975 0.179 -0.026 -0.950 (0.21) (-0.07) (-2.20) 3.573 -0.141 -0.951 0.098 -0.286 (0.93) (-0.33) (-0.73) (0.27) (-0.67) Jan u a r y -0.308 -0.420 1976 t o (-0.22) (-0.80) December 1980 -0.369 -0.394 0.200 (-0.26) (-0.76) (0.79) -9.009 -0.432 2.281 0.029 -0.287 (-1.01) (-0.40) (0.92) (0.06) (-0.34) Note: the 5-i v a l u e s have been m u l t i p l i e d by 1000. 46 TABLE 10: C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r P o r t f o l i o s  of U.S. Bonds ( p o r t f o l i o s e l e c t i o n method 1 used; t - r a t i o s i n p a r e n t h e s e s ) H o l d i n g P e r i o d = Three Months I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d yO y2 73 74 January 1961 to December 1980 1 .047 (1 .50) 1 .246 (1.82) -0.690 (-2.38) -0.743 (-2.46) 0.355 (1.15) -5.313 (-2.17) 0.304 (1 .37) 2.286 (3.08) ^0.785 (-3.75) -0.814 (-2.84) January 1961 t o December 1 970 1 . 1 94 (2.38) 1 .061 (2.21) -0.339 (-1.06) -0.291 (-0.84) 0.797 (1.79) -1.504 (-1 .20) 0.237 (0.98) 1 .662 (2.56) -0.457 (-1.65) -0.626 (-1.69) January 1971 t o December 1980 0.896 (0.68) 1 .436 (1.10) -1.051 (-2.16) -1.207 (-2.43) -0.099 (-0.24) -9.254 (-1.93) 0.372 (1.00) 2.931 (2.16) -1.123 (-3.60) -1 .009 (-2.29) Note; the the 7 i v a l u e s have been ( c o n t . over) m u l t i p l i e d by 1000. 47 TABLE 10 CONTD. I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r Per i o d jO yj\ l2. ^3 7 4 January 0.327 -0.159 1961 t o (0.81) (-0.76) December 1965 0.232 -0.129 0.078 (0.56) (-0.53) (0.18) -3.517 0.268 2.689 -0.458 -1.215 (-2.54) (1.76) (3.23) (-2.51) (-3.61) January 2.061 -0.519 1966 t o (2.27) • (-0.85) December 1970 1.891 -0.452 1.517 (2.20) (-0.69) . (1.96) 0.508 0.207 0.635 -0.457 -0.036 (0.25) (0.45) (0.65) (-0.87) (-0.05) January 1.410 -0.419 1971 t o (1.18) (-0.75) December 1975 2.266 -0.667 0.294 (1.82) (-1.21) (0.42) -2.531 0.766 1.613 -0.997 -0.882 (-0.54) (1.55) (1.04) (-2.27) (-1.45) January 0.355 -1.716 1976 t o (0.15) (-2.14) December 1980 0.562 -1.776 -0.513 (-0.24) (-2.12) (-1.10) -16.456 -0.049 4.344 -1.259 -1.143 (-1.93) (-0.09) (1.92) (-2.81) (-1.78) Note: the 7 i v a l u e s have been m u l t i p l i e d by 1000. 48 TABLE 11: C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r P o r t f o l i o s  of U.S. Bonds ( p o r t f o l i o s e l e c t i o n method 2 used; t - r a t i o s i n p a r e n t h e s e s ) H o l d i n g P e r i o d = Three Months I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d yO j r l 1 1 1 1 yi January 1961 t o December 1980 1 .401 (2.29) 1 .524 (2.48) -0.757 (-2.41) -0.788 (-2.47) 0.437 (1.38) 0.077 (0.03) -0.207 (-0.46) 0.518 (0.61) -0.361 (-1.37) -0.237 (-0.72) January 1961 t o December 1970 1 .232 (2.43) 1 .088 (2.25) -0.351 (-1.09) -0.302 (-0.87) 0.741 (1.59) -0.207 (-0.15) -0.129 (-0.43) 0.910 (1 .27) -0.156 (-0.50) -0 (-0 1 70 43) January 1971 t o December 1980 1 .574 (1.40) 1 .971 (1.73) 0.366 (0.07) -1.173 (-2.16) -1.287 (-2.40) -0.287 (-0.33) 0.119 (0.08) -0.569 (-1.34) 0. 1 25 (0.29) -0.304 (-0.58) Note: the yi v a l u e s have been m u l t i p l i e d by 1000. ( c o n t . o v e r ) 49 TABLE 11 CONTD. I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d ^0 l J j2 j3 yi J a n u a r y 0.403 -0.183 1961 t o (0.94) (-0.84) December 1965 0.285 -0.152 -0.035 (0.66) (-0.62) (-0.70) -0.744 -0.507 1.056 0.212 -0.290 (-0.46) (-1.36) (1.02) (0.63) (-0.66) Jan u a r y 2.061 -0.519 1966 t o (2.27) (-0.85) December 1970 1.891 -0.452 1.517 (2.20) (-0.69) (1.96) 0.370 0.255 0.761 -0.531 -0.048 (0.18) (0.55) (0.77) (-1.00) (-0.07) Jan u a r y 1.902 -0.535 1971 t o (1.60) (-0.93) December 1975 2.771 -0.794 0.075 (2.24) (-1.43) (0.10) 6.649 -0.206 -1.408 -0.218 -0.131 (1.32) (-0.35) (-0.82) (-0.42) (-0.20) Ja n u a r y 1.229 -1.845 1976 t o (0.63) (-1.97) December 1980 1.129 -1.805 0.177 (0.58) (-1.93) (0.42) -6.247 -0.372 1.726 -0.938 -0.486 (-0.66) (-0.22) (0.66) (-1.37) (-0.59) Note; the 7 i v a l u e s have been m u l t i p l i e d by 1000. 50 used, the v a l u e s of t ( 7 l ) , t ( 7 2 ) , t { j 3 ) , and t ( y i ) are a l l f a i r l y s m a l l . 5 . 2 CANADIAN RESULTS The c r o s s - s e c t i o n a l r e g r e s s i o n r e s u l t s f o r Canadian government bonds are p r e s e n t e d i n T a b l e s 12 and 13. Table 12 r e p o r t s the r e s u l t s o b t a i n e d when bond r e t u r n s are c a l c u l a t e d on a monthly b a s i s ; Table 13 r e p o r t s the r e s u l t s o b t a i n e d u s i n g q u a r t e r l y r e t u r n s . I t i s i n t e r e s t i n g t o note t h a t t h e r e i s no apparent d i f f e r e n c e between the Canadian and American r e s u l t s over the sample p e r i o d . When r e t u r n s a r e measured oh a monthly b a s i s the v a l u e of t ( 7 l ) i s not l a r g e d u r i n g any s u b p e r i o d f o r any model. T h e r e f o r e , t h e r e i s no s i g n i f i c a n t e v i d e n c e t o suggest t h a t , on a v e r a g e , h i g h e r d u r a t i o n bonds earn h i g h e r r e t u r n s . In a d d i t i o n , f o r most time p e r i o d s the v a l u e of 7 2 i s s i g n i f i c a n t and p o s i t i v e i n s i g n . Once a g a i n , t h i s c o u l d be a t t r i b u t e d t o a t a x e f f e c t . F i n a l l y , the r e s u l t s r e p o r t e d i n T a b l e 12 r e v e a l t h a t f o r many time p e r i o d s a s i g n i f i c a n t r e l a t i o n e x i s t s between the p r i c i n g e r r o r and the r a t e of r e t u r n over the next i n t e r v a l . T h i s r e s u l t i s not s e n s i t i v e t o the i n c l u s i o n of the coupon or m a t u r i t y v a r i a b l e s . When r e t u r n s a r e measured on a q u a r t e r l y b a s i s , the b a s i c r e s u l t s a r e the same. The c o e f f i c i e n t f o r m a t u r i t y i s on average i n s i g n i f i c a n t , and the average v a l u e of the coupon c o e f f i c i e n t , 7 2 , i s once a g a i n p o s i t i v e (and s i g n i f i c a n t l y so i n some s u b p e r i o d s ) . As f a r as H y p o t h e s i s 1 51 TABLE 12: C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r Canadian  Bonds ( t a x a b l e bonds w i t h m a t u r i t i e s l e s s than 10 y e a r s ; t - r a t i o s i n p a r e n t h e s e s ) H o l d i n g P e r i o d = One Month I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d ll j2_ l l 74 Jan u a r y 1961 t o December 1 980 -0. 169 (-0.20) 0. 184 (0.32) 0.087 (0.25) -0.034 (-0.13) -0.333 (-2.57) -4.214 (-2.23) -0.123 (-0.50) 1 .723 (3.07) 0.004 (0.27) -0.555 (-3.77) January 1961 t o December 1 970 0.425 (1.11) 0.415 (1 .08) 0.090 (0.43) 0.087 (0.41) -0.178 (-2.34) -4.648 (-3.91) 0. 1 88 (0.97) 2.302 (4.76) -0.013 (-1 .02) -0.640 (-5.26) Jan u a r y 1971 t o December 1980 -0.767 (-0.46) -0.049 (-0.04) 0.084 (0.13) -0.156 (-0.31) -0.490 (-1.97) -3.775 (-1.05) -0.436 (-0.96) 1 .1 40 (1.12) 0.021 (0.79) -0.468 (-1.74) Note; the 7 i v a l u e s have been m u l t i p l i e d by 1000. ( c o n t . over) 52 TABLE 12 CONTD. I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r Per i o d jO jj y2 j3 7 4 January 0.329 0.129 1961 to (0.66) (0.64) December 1965 0.321 0.124 -0.170 (0.63) (0.60) (-1.64) -3.737 0.150 2.381 -0.013 -0.668 (-2.74) (0.80) (3.36) (-0.96) (-3.94) January 0.521 0.050 1966 t o (0.89) (0.14) December 1970 -0.509 0.050 -0.186 (-0.87) (0.13) (-1.65) -5.560 0.226 2.223 -0.012 -0.613 (-2.85) (0.66) (3.35) (-0.58) (-3.48) Jan u a r y -1.091 0.664 1971 t o (-0.34) (0.55) December 1975 0.281 0.206 -0.911 (0.14) (0.24) (-1.88) -4.477 -0.641 1.715 0.052 -0.708 (-0.64) (-0.80) (0.86) (1.16) (-1.35) January -0.438 -0.507 1976 t o (-0.59) (-1.04) December 1980 -0.384 -0.525 -0.061 (-0.52) (-1.08) (-0.98) -3.062 -0.228 0.555 -0.012 -0.225 (-2.15) (-0.53) (1.46) (-0.48) (-1.98) Note: the yi v a l u e s have been m u l t i p l i e d by 1000. 53 TABLE 13: C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r Canadian  Bonds ( t a x a b l e bonds w i t h m a t u r i t i e s l e s s than 10 y e a r s ; t - r a t i o s i n p a r e n t h e s e s ) H o l d i n g P e r i o d = Three Months I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d 7O 1 1 l i . 1 1 74 January 1961 t o December 1 980 0.365 (0.33) 0.867 (0.95) -0.164 (-0.33) -0.324 (-0.71 ) -0.325 (-2.50) -3.65.1 (-1 .44) -0.077 (-0.26) 1.911 (2.28) -0.019 (-0.96) -0.628 (-3.46) January 1961 t o December 1 970 0.799 (1.00) 0.814 (1.02) 0.353 (0.82) 0.342 (0.79) -0.115 (-1 .15) -2.467 0.787 2.111 -0.429 -0.639 (-0.68) (2.44) (1.66) (-2.16) (-3.55) January 1971 t o December 1980 -0.080 (-0.04) 0.921 (0.56) -0.693 (-0.75) -1.007 (-1.25) -0.542 (-2.24) -4.866 (-1.36) -0.963 (-1.93) 1 .706 (1.56) 0.006 (0.16) -0.617 (-2.34) Note; the 7 i v a l u e s have been m u l t i p l i e d by 1000. ( c o n t . o v e r ) 54 TABLE 13 CONTD. P e r i o d I n t e r c e p t Durat i o n Coupon M a t u r i t y P r . E r r o r y0 71 72 73 74 January 0.926 0.313 1961 t o (0.93) (0.91) December 1 965 0.958 0.291 -0. 164 (0.96) (0.83) (-1.16) -6.139 0.388 3.988 -0.023 -0.982 (-3.27) (1.50) (4.45) (-1.12) (-5.07) January 0.672 0.393 1966 t o (0.54) (0.50) December 1 970 0.670 0.393 -0.065 (0.53) (0.50) (-0.46) 1 .205 1.185 0.235 -0.063 -0.296 (0.17) (2.01) (0.10) (-1.85) (-0.64) January 0.604 0.251 1971 t o (0.15) (0.17) December 1 975 2.257 -0.348 -0.942 (0.83) (-0.28) (-2.05) -5.029 -1.056 2.597 0.044 -0.869 (-0.74) (-1.31) (1.25) (0.85) (-1.74) January -0.800 -1.687 1976 t o (-0.72) (-1.67) December 1980 -0.769 -1.700 -0.120 (-0.70) (-1.69) (-1.31) -4.695 -0.864 0.767 -0.035 -0.351 (-2.48) (-1 .48) (1 .63) (-0.80) (-2.70) Note: the 7 i v a l u e s have been m u l t i p l i e d by 1000. 55 i s c o n c e r n e d , Table 13 i n d i c a t e s t h a t f o r the o v e r a l l p e r i o d 1961-1980 t h e r e i s no s t a t i s t i c a l l y o b s e r v a b l e r e l a t i o n s h i p between d u r a t i o n and bond r e t u r n s . However, f o r the 1 0-year s u b p e r i o d 1961-1970 (model 3 o n l y ) the v a l u e of 71 i s s i g n i f i c a n t and p o s i t i v e , and f o r the f o l l o w i n g 1 0-year s u b p e r i o d 1971-1980 71 i s s i g n i f i c a n t and n e g a t i v e . 5.3 LIMITATIONS OF ANALYSIS B e f o r e drawing any s t r o n g c o n c l u s i o n s from the r e s u l t s p r e s e n t e d i n t h i s c h a p t e r , i t i s impo r t a n t t o e s t a b l i s h whether or not the im p o r t a n t a s sumptions u n d e r l y i n g the r e g r e s s i o n methodology h o l d . To b e g i n w i t h , s e v e r a l a s sumptions have been made about the d i s t u r b a n c e s , e..; i n p a r t i c u l a r , t h a t they a r e n o r m a l l y j d i s t r i b u t e d . A c l o s e e x a m i n a t i o n of the r e s i d u a l s of some of the c r o s s - s e c t i o n a l r e g r e s s i o n s ( 1 0 y e a r s s e l e c t e d a t random) r e v e a l s t h a t t h e r e a r e no s e r i o u s d e v i a t i o n s from n o r m a l i t y . However, p l o t s of s t u d e n t i z e d r e s i d u a l s r e v e a l t h a t i n each c r o s s - s e c t i o n a l r e g r e s s i o n t h e r e a r e one or two o u t l i e r s ( t y p i c a l l y l o n g m a t u r i t y bonds), which c o u l d perhaps be e x e r t i n g an i n f l u e n c e on the r e s u l t s . N ext, the time s e r i e s b e h a v i o r of 7 O , 7 1 , 7 2 , 7 3 , and 7 4 i s examined. I t i s n e c e s s a r y t o determine whether the yi^ a r e u n c o r r e l a t e d through time because the t - s t a t i s t i c s f o r t e s t i n g the h y p o t h e s i s t h a t E ( 7 i ) = 0 have been computed on the assumption t h a t s u c c e s s i v e v a l u e s of yi a r e i n d e p e n d e n t l y and i d e n t i c a l l y d i s t r i b u t e d normal random 56 v a r i a b l e s . An e x a m i n a t i o n of the sample a u t o c o r r e l a t i o n f u n c t i o n s f o r the v a r i o u s yi r e v e a l s t h a t t h i s assumption does not h o l d . When r e t u r n s a r e c a l c u l a t e d on a monthly b a s i s some of the a u t o c o r r e l a t i o n s of the v a r i o u s yi are s i g n i f i c a n t l y d i f f e r e n t from z e r o , and i n the case of the coupon c o e f f i c i e n t , y3), a s t r o n g s e a s o n a l p a t t e r n emerges. When r e t u r n s a r e c a l c u l a t e d on a q u a r t e r l y b a s i s the s e a s o n a l i t y problem d i s a p p e a r s , but almost a l l of the a u t o c o r r e l a t i o n s are s i g n i f i c a n t l y d i f f e r e n t from z e r o f o r s e v e r a l l a g s and they d e c l i n e i n an e x p o n e n t i a l f a s h i o n . T h i s r e s u l t i s und o u b t e d l y due t o the f a c t t h a t r e t u r n s f o r o v e r l a p p i n g 3-month h o l d i n g p e r i o d s a re used i n the r e g r e s s i o n s . An a u t o r e g r e s s i v e model of or d e r one appears t o d e s c r i b e the time s e r i e s of each of the yi v e r y w e l l . T h i s e v i d e n c e i n d i c a t e s t h a t when the h o l d i n g p e r i o d i s t h r e e months the yi a r e s e r i a l l y c o r r e l a t e d and, hence, t h a t the means of thes e c o e f f i c i e n t s s h o u l d be r e c a l c u l a t e d under the a ssumption t h a t each of the yi f o l l o w s an AR(1) p r o c e s s . The a n a l y s i s f o r U.S. bonds i s t h e r e f o r e r e p e a t e d , t a k i n g i n t o account t h i s recommendation. A h o l d i n g p e r i o d of t h r e e months i s chosen, AR(1) models a r e f i t t e d t o the time s e r i e s of the 7 c o e f f i c i e n t s , and the means of the c o e f f i c i e n t s a r e r e e s t i m a t e d . AR(1) models a r e e s t i m a t e d u s i n g t h r e e d i f f e r e n t methods: G e n e r a l N o n l i n e a r L e a s t Squares; C o n d i t i o n a l L e a s t Squares; and B a c k c a s t i n g . The r e s u l t s a r e p r e s e n t e d i n T a b l e s 14, 15, and 16. A l t h o u g h the 57 e s t i m a t e s of the means v a r y from one method t o the n e x t , the b a s i c c o n c l u s i o n s a r e a l l the same: t h e r e i s no e v i d e n c e t o suggest t h a t , on average , h i g h e r d u r a t i o n bonds earn h i g h e r r e t u r n s ; t h e r e i s some e v i d e n c e of a s i g n i f i c a n t p o s i t i v e r e l a t i o n s h i p between bond r e t u r n s and the coupon r a t e ; and f o r some time p e r i o d s a s i g n i f i c a n t l y n e g a t i v e r e l a t i o n s h i p a p p a r e n t l y e x i s t s between the p r i c i n g e r r o r and the r a t e of r e t u r n on the bond over the next i n t e r v a l . 58 TABLE 14: C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r U.S. Bonds  "(Mean V a l u e s of R e g r e s s i o n Parameters C a l c u l a t e d  under Assumption of AR(1) Model) ( t a x a b l e bonds w i t h m a t u r i t i e s l e s s than 10 y e a r s ; AR(1) models e s t i m a t e d u s i n g G e n e r a l N o n l i n e a r L e a s t Squares method) H o l d i n g P e r i o d = Three Months I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d jO l2 7 3 7 4 January 1 .444 -0.702 1961 t o (1.38) (-1.06) December 1980 1.435 -0.698 0.013 (1.37) (-1.06) (0.10) -4.948 0.126 2.170 -0.712 -0.607 (-3.80) (0.38) (6.89) (-2.33) (-5.50) January 1 .283 0.180 1961 t o (1.55) (0.20) December 1970 1.278 0.148 0.260 (1.52) (0.17) (1.47) -1.287 0.514 1.530 -0.507 -0.341 (-1.22) (1.00) (4.95) (-1.26) (-2.29) Jan u a r y 1.499 -1.177 1971 t o (0.77) (-1.13) December 1980 1.470 -1.165 -0.250 (0.76) (-1.12) (-1.47) -8.665 -0.022 2.817 -0.864 -0.891 (-3.88) (-0.04) (5.31) (-1.90) (-5.51) Note: the 7 i v a l u e s have been m u l t i p l i e d by 1000. ( c o n t . over) 59 TABLE 14 CONTD. I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d |o | j j2 |3 7 4 Jan u a r y 0.401 0.096 1961 t o (0.60) (0.18) December 1965 0.375 0.101 0.143 (0.56) (0.19) (0.73) -1.671 0.374 1.341 -0.321 -0.361 (-1.53) (0.96) (3.69) (-1.17) (-1.58) January 2.278 -0.064 1966 t o (1.60) (-0.04) December 1970 2.300 -0.093 0.400 (1.58) (-0.06) (1.35) -0.764 0.544 1.717 -0.695 -0.275 (-0.42) (0.58) (3.35) (-0.93) (-1.42) January 1.609 -0.481 1971 t o (0.78) (-0.50) December 1975 1.489 -0.455 0.001 (0.71) (-0.48) (0.00) -3.511 0.207 1.803 -0.586 -0.585 (-1.12) (0.52) (1.81) (-0.93) (-2.58) Jan u a r y 0.774 -1.453 1976 t o (0.23) (-0.80) December 1980 0.786 -1.466 -0.520 (0.24) (-0.81) (-2.89) -13.963 -0.140 3.733 -1.159 -1.208 (-4.32) (-0.14) (6.48) (-1.61) (-5.03) Note: the yi v a l u e s have been m u l t i p l i e d by 1000. 60 TABLE 15: C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r U.S. Bonds  TMean V a l u e s of R e g r e s s i o n Parameters C a l c u l a t e d  under Assumption of AR(1) Model) ( t a x a b l e bonds w i t h m a t u r i t i e s l e s s than 10 y e a r s ; AR(1) models e s t i m a t e d u s i n g C o n d i t i o n a l L e a s t Squares method) H o l d i n g P e r i o d = Three Months I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d 7 0 1 1 1 1 1 2 . 74 Jan u a r y 1961 t o December 1980 1 .524 (1.44) 1.516 (1.43) -0.837 (-1.24) -0.830 (-1*24) 0.017 (0.13) -4.947 (-3.78) 0.090 (0.27) 2. 1 72 (6.87) -0.726 (-2.35) -0.604 (-5.45) Jan u a r y 1961 t o December 1 970 1 .479 (1 .74) 1 .481 (1.71) -0.221 (-0.24) -0.229 (-0.25) 0.273 (1.53) -1.174 (-1.09) 0.297 (0.57) 1 .526 (4.88) -0.530 (-1.28) -0.324 (-2.15) Jan u a r y 1971 t o December 1980 1 .737 (0.88) 1 .760 (0.88) -1.266 (-1.18) -1.264 (-1.18) -0.239 (-1.39) -8.755 (-3.88) 0.015 (0.03) 2.865 (5.36) -0.947 (-2.06) -0.893 (-5.47) Note; the 7 i v a l u e s have been m u l t i p l i e d by 1000. ( c o n t . over) 61 TABLE 15 CONTD. I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d jO j_\ j2 j3 7 4 J a n u a r y 0.601 -0.345 1961 t o (0.86) (-0.64) December 1965 0.576 -0.338 0.177 (0.83) (-0.63) (0.87) -1.393 0.045 1.314 -0.357 -0.313 (-1.22) (0.12) (3.50) (-1.23) (-1.31) January 2.400 -0.054 1966 t o (1.58) (-0.00) December 1970 2.419 -0.062 0.398 (1.56) (-0.03) (1.31) -1.062 0.573 1.818 -0.720 -0.337 (-0.57) (0.57) (3.53) (-0.92) (-1.74) Jan u a r y 2.251 -0.577 1971 t o (1.03) (-0.57) December 1975 2.297 -0.562 0.036 (1.05) (-0.56) (0.13) -3.656 0.233 2.008 -0.731 -0.582 (-1.11) (0.57) (1.95) (-1.12) (-2.50) Ja n u a r y 1.807 -2.219 1976 t o (0.52) (-1 .17) December 1980 1.823 -2.235 -0.519 (0.53) (-1.18) (-2.83) -13.390 -0.488 3.708 -1.148 -1.195 (-4.13) (-0.47) (6.33) (-1.72) (-4.90) Note: the 7 i v a l u e s have been m u l t i p l i e d by 1000. 62 TABLE 16: C r o s s - S e c t i o n a l R e g r e s s i o n R e s u l t s f o r U.S. Bonds  (Mean V a l u e s of R e g r e s s i o n Parameters C a l c u l a t e d  under Assumption of AR(1) Model) ( t a x a b l e bonds w i t h m a t u r i t i e s l e s s than 10 y e a r s ; AR(1) models e s t i m a t e d u s i n g B a c k c a s t i n g method) H o l d i n g P e r i o d = Three Months I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d |0 | j ll ll 7 4 January 1.468 -0.757 1961 t o (1.40) (-1.13) December 1980 1.461 -0.750 0.014 (1.38) (-1.13) (0.11) -4.947 0.121 2.170 -0.716 -0.607 (-3.79) (0.36) (6.88) (-2.33) (-5.49) January 1.355 -0.011 1961 to (1.61) (-0.01) December 1970 1.352 -0.033 0.261 (1.58) (-0.04) (1.47) -1.263 0.429 1.530 -0.513 -0.338 (-1.19) (0.82) (4.92) (-1.26) (-2.26) January 1.569 -1.210 1971 t o (0.80) (-1.14) December 1980 1.557 -1.201 -0.249 (0.79) (-1.14) (-1.46) -8.669 -0.023 2.817 -0.883 -0.891 (-3.86) (-0.04) (5.30) (-1.93) (-5.48) Note: t h e yi v a l u e s have been m u l t i p l i e d by 1000. ( c o n t . o v e r ) 63 TABLE 16 CONTD. • I n t e r c e p t D u r a t i o n Coupon M a t u r i t y P r . E r r o r P e r i o d 7J) | j j2 yi January 0.455 -0.066 1961 t o (0.66) (-0.12) December 1965 0.429 -0.061 0.149 (0.63) (-0.11) (0.75) -1.594 0.271 1.336 -0.332 -0.349 (-1.43) (0.65) (3.63) (-1.18) (-1.49) January 1966 t o December 1 970 2.322 (1 .58) 2.343 (1.56) -0.814 (-0.44) -0.033 (-0.02) -0.078 (-0.05) 0.556 (0.57) 1.719 (3.36) -0.701 (-0.92) 0.399 (1.33) -0.281 (-1.45) January 1971 t o December 1975 1 .835 (0.86) 1 .782 (0.83) -3.552 (-1.11) -0.513 (-0.52) -0.488 (-0.50) 0.208 (0.51) 1 .845 (1.82) -0.617 (-0.96) 0.071 (0.03) -0.584 (-2.55) January 1976 t o December 1980 1 .042 (0.31) 1 .059 (0.31) -13.960 (-4.36) -1 .708 (-0.91) -1.721 (-0.92) -0.190 (-0.18) 3.770 (6.49) - 1.156 (-1.78) -0.520 (-2.86) -1.208 (-5.00) Note: t h e yi v a l u e s have been m u l t i p l i e d by 1000. Chapter 6 CONCLUSIONS In t h i s study Canadian and American market d a t a have been used t o t e s t whether M a c a u l a y / F i s h e r and W e i l d u r a t i o n i s an adequate measure of b a s i s r i s k f o r d e f a u l t - f r e e government bonds d u r i n g the p e r i o d January 1961 t o December 1980. The most i m p o r t a n t r e s u l t of t h i s study i s t h a t i n e i t h e r an American or a Canadian c o n t e x t t h e r e i s no s i g n i f i c a n t e v i d e n c e t o suggest t h a t , on average, h i g h e r d u r a t i o n bonds earn h i g h e r r e t u r n s . S p e c i f i c a l l y , t h e r e appears t o be on average a n e g a t i v e ( a l t h o u g h i n s i g n i f i c a n t ) r e l a t i o n s h i p between bond r e t u r n s and M a c a u l a y / F i s h e r and W e i l d u r a t i o n . T h i s r e s u l t , which a p p l i e s e q u a l l y t o government bonds and t o p o r t f o l i o s of government s e c u r i t i e s , i s not s e n s i t i v e t o the l e n g t h of the h o l d i n g p e r i o d or t o the p o r t f o l i o s e l e c t i o n method used. One p o s s i b l e e x p l a n a t i o n f o r the d u r a t i o n c o e f f i c i e n t b e i n g n e g a t i v e on average i s t h a t i n t e r e s t r a t e s have t r e n d e d upwards over most of the sample p e r i o d . In any g i v e n p e r i o d the c o e f f i c i e n t f o r d u r a t i o n , 71 , w i l l be i n f l u e n c e d by changes i n the l e v e l of i n t e r e s t r a t e s i n the economy, so t h a t when i n t e r e s t r a t e s are r i s i n g 71 i s l i k e l y t o be n e g a t i v e on average. To d e t e r m i n e what p o r t i o n of 71 i s not e x p l a i n e d by changes i n the l e v e l of i n t e r e s t r a t e s the month-by-month v a l u e s of 71 (American, monthly h o l d i n g p e r i o d v a l u e s o n l y ) 64 65 a r e r e g r e s s e d on changes i n the l e v e l of i n t e r e s t r a t e s . The r e s u l t s suggest t h a t when changes i n the l e v e l of i n t e r e s t r a t e s have been f i l t e r e d out t h e r e i s a p o s i t i v e , a l t h o u g h i n s i g n i f i c a n t , r e l a t i o n s h i p between bond r e t u r n s and d u r a t i o n . T h i s i n d i c a t e s t h a t t h e r e may be c o n s i d e r a b l e v a l u e i n a more d e t a i l e d a n a l y s i s of the r e l a t i o n s h i p between changes i n i n t e r e s t r a t e s and the d u r a t i o n c o e f f i c i e n t , 7 1 . Another r e s u l t of the study i s t h a t d u r a t i o n seems t o i n c o r p o r a t e the e f f e c t of m a t u r i t y d i f f e r e n c e s on p r i c e v o l a t i l i t y , i n t h a t the c o e f f i c i e n t f o r m a t u r i t y i s on average i n s i g n i f i c a n t l y d i f f e r e n t from z e r o . The c o e f f i c i e n t f o r the coupon r a t e i s , however, s i g n i f i c a n t l y p o s i t i v e on average. A p o s s i b l e e x p l a n a t i o n f o r the p o s i t i v e r e l a t i o n s h i p between coupon r a t e and bond r e t u r n s i s t h a t the coupon c o e f f i c i e n t i s c a p t u r i n g a t a x e f f e c t . F i n a l l y , the r e s u l t s i n d i c a t e t h a t i n some p e r i o d s a s i g n i f i c a n t r e l a t i o n e x i s t s between the p r i c i n g e r r o r and the r a t e of r e t u r n over the next i n t e r v a l . REFERENCES [ I ] B i e r w a g , CO., Kaufman, G.G. and Khang, C , " D u r a t i o n and Bond P o r t f o l i o A n a l y s i s : An Overview," 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 13, No.4 (November 1978), 671-681. [2] B i e rwag, CO., Kaufman, G.G. and Toevs, A., " D u r a t i o n : I t s Development and Use i n Bond P o r t f o l i o Management," F i n a n c i a l A n a l y s t s J o u r n a l 39, No.4 ( J u l y - A u g u s t 1983), 15-35. [3] B o q u i s t , J.A., R a c e t t e , C A . and Schlarbaum, G.G., " D u r a t i o n and R i s k Assessment f o r Bonds and Common S t o c k s , " J o u r n a l of F i n a n c e 30, No.5 (December 1975), 1360-1365. [4] Brennan, M.J. and S c h w a r t z , E.S., "An E q u i l i b r i u m Model of Bond P r i c i n g and a Test of Market E f f i c i e n c y , " 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 17, No.3 (September 1982), 301-329. [5] Brennan, M.J. and S c h w a r t z , E.S., "Bond P r i c i n g and Market E f f i c i e n c y , " F i n a n c i a l A n a l y s t s J o u r n a l 38, No.5 (September-October 1982), 49-56. [6] Brennan, M.J. and S c h wartz, E.S., " D u r a t i o n , Bond P r i c i n g and P o r t f o l i o Management" i n G. Kaufman, G. Bierwag and A. Toevs, eds., I n n o v a t i o n s i n  P o r t f o l i o Management: D u r a t i o n A n a l y s i s and  I m munization, Greenwich, CT: JAI P r e s s (1983). [7] Cooper, I.A., "Asset V a l u e s , I n t e r e s t - R a t e Changes, and D u r a t i o n , " 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 12, No.4 (December 1977), 701-724. [8] Cox, J.C., I n g e r s o l l , J.E. and Ross, S.A., " D u r a t i o n and the Measurement of B a s i s R i s k , " j o u r n a l of  B u s i n e s s 52, No.1 (January 1979), 51-62. [9] Fama, E.F., F o u n d a t i o n s of F i n a n c e , New York: B a s i c Books Inc~. (1 976). [10] Fama, E.F. and MacBeth, J.D., " R i s k , R e t u r n , and E q u i l i b r i u m : E m p i r i c a l T e s t s , " J o u r n a l of  P o l i t i c a l Economy 81, No.3 (May-June 1973), 607-636. [ I I ] F i s h e r , L., "An A l g o r i t h m f o r F i n d i n g Exact Rates of R e t u r n , " J o u r n a l of B u s i n e s s 39, No.1 ( J a n u a r y 1966), 111-118. 66 67 [12] F i s h e r , L. and W e i l R.L., "Coping With the R i s k of I n t e r e s t Rate F l u c t u a t i o n s : R e t u r n s t o Bondholders From Naive and O p t i m a l S t r a t e g i e s , " J o u r n a l of B u s i n e s s 44, No.4 (October 1971), 408-431. [13] G u l t e k i n , N.B. and R o g a l s k i , R.J., " A l t e r n a t i v e D u r a t i o n S p e c i f i c a t i o n s and the Measurement of B a s i s R i s k : E m p i r i c a l T e s t s , " J o u r n a l of B u s i n e s s 57, No.2 ( A p r i l 1984), 241-264. [14] H i c k s , J.R., V a l u e and C a p i t a l , O x f o r d : C l a r e n d o n P r e s s (1939) . [15] H o p e w e l l , M.H. and Kaufman, G.G., "Bond P r i c e V o l a t i l i t y and Term t o M a t u r i t y : A G e n e r a l i z e d R e s p e c i f i c a t i o n , " American Economic Review 63, No.4 (September 1973), 749-753. [16] I n g e r s o l l , J.E., S k e l t o n , J . and W e i l , R.L., " D u r a t i o n F o r t y Years L a t e r , " 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 13, No.4 (November 1978), 627-650. [17] Kaufman, G.G., "Measuring R i s k and Re t u r n f o r Bonds: A New Approach," J o u r n a l of Bank Research 9, (Summer 1978), 82-90. [18] Khang, C , "Bond Immunization When Short-Term Rates F l u c t u a t e More than Long-Term R a t e s , " 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 14, No.5 (December 1979), 1085-1090. [19] L a n g s t e i n , R. and Sharpe, W.F., " D u r a t i o n and S e c u r i t y R i s k , " 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 13, No.4 (November 1978), 653-668. [20] Macaulay, F.R., Some T h e o r e t i c a l Problems Suggested by  the Movements of I n t e r e s t R a t e s , Bond Y i e l d s , and  Stock P r i c e s i n the U.S. S i n c e 1856, New York: N a t i o n a l Bureau of Economic Research (1938). [21] M c C u l l o c h , J.H., "Measuring the Term S t r u c t u r e of I n t e r e s t R a t e s , " J o u r n a l of B u s i n e s s 44, No.1 (January 1971), 19-31. [22] R e d i n g t o n , F.M., "Review of the P r i n c i p l e s of L i f e - O f f i c e V a l u a t i o n s , " J o u r n a l of the I n s t i t u t e  of A c t u a r i e s 78, No.3 (1952), 286-340. [23] R e i l l y , F.K. and S i d h u , R.S., "The Many Uses of Bond D u r a t i o n , " F i n a n c i a l A n a l y s t s J o u r n a l 36, No.4 ( J u l y - A u g u s t 1980), 58-72. 68 [24] Samuelson, P.A., "The E f f e c t s of I n t e r e s t Rate I n c r e a s e s on the Banking System," American  Economic Review 35, No.1 (March 1945), 16-27. [25] S c h a e f e r , S.M., "Immunization and D u r a t i o n : A Review of Theory, Performance and A p p l i c a t i o n s , " M i d l a n d  C o r p o r a t e F i n a n c e J o u r n a l 2, No.3 ( F a l l 1984), 41-58. [26] W e i l , R.L., "Macaulay's D u r a t i o n : An A p p r e c i a t i o n , " J o u r n a l of B u s i n e s s 46, No.4 (October 1973), 589-592. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0095926/manifest

Comment

Related Items