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Risk-aversion in open-end investment companies White, Robert Wayne 1968

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RISK-AVERSION IN OPEN-END INVESTMENT COMPANIES by .ROBERT WAYNE WHITE Sc. (Honours), U n i v e r s i t y of B r i t i s h Columbia, 1 A THESIS SUBMITTED IN PARTIAL FULFILMENT. OF THE .REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION fn the Department o f Commerce and Business A d m i n i s t r a t i o n We accept t h i s t h e s i s as conforming to the requ i red standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1968 In presenting t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the Li 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s represen-t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department nf Commerce and Business 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 of B r i t i s h Columbia Vancouver 8, Canada Date A p r i l , 1968. ABSTRACT OF THESIS The purpose of t h i s study i s to examine the three hypotheses: (a) expected returns and r i s k as measured by the d i s p e r s i o n of returns a s s o c i a t ed w i t h r i s k c l a s s e s of s e c u r i t i e s are p o s i t i v e l y c o r r e l a t e d , (b) tha t the va r i ance of returns i s the app rop r i a te measure of r i s k and (c) Canadian i nve s to r s in open-end investment companies do not e x h i b i t g r ea te r r i s k a ve r s i on t han t h e i r American coun te rpa r t s . The method of study employed i s the a p p l i c a t i o n of reg re s s i on ana l y s i s to samples of Canadian and American open-end. investment companies. The conc lu s ions drawn from the study are expected returns and r i s k are p o s i t i v e l y c o r r e l a t e d , r i s k c l a s s e s of s e c u r i t i e s are s ca led accord ing to expected re tu rns and va r i ance o f r e tu rn s , and Canadian i nves to r s in open-end investment companies do not e x h i b i t more nor les s r i s k ave r s i on than t h e i r American coun te rpa r t s . • • . TABLE OF CONTENTS CHAPTER - . PAGE I. THE PROBLEM AND DEFINITIONS OF TERMS USED .'. . '"1 The Problem • ••• 3 The Statement of the problem 3 Importance of the study . . . 3 D e f i n i t i o n s of Terms Used 6 Investmeht . 6 Open-end investment companies (mutual funds) 7 Rate o f re tu rn 9 Expected re tu rn 9 Risk 10 Pe r f e c t markets 1 16 Ri sk -avers ion . . . 17 R i s k - f r e e i n t e r e s t rate 18 Methodology 19 Assumptions 21 Assumptions regard ing i nve s to r s 21 Market assumptions 21 Organ i za t i on of the t h e s i s • • 22 I I . A FRAMEWORK OF I NVESTMENT 23 Investment Ob jec t i ve s . 2k The Hypothesis of i n c rea s i n g returns 25 T h e o r e t i c a l a n a l y s i s 26 V CHAPTER PAGE Summary J1 I I I . THEORY OF INVESTMENT 32 Markowitz Theory of D i v e r s i f i c a t i o n 33 E f f i c i e n c y F r o n t i e r :. 33 Investors Preferences 35 The opt imal p o r t f o l i o 37 Investment Opportun i ty Loci 38 C a p i t a l Market Curve 42 Standard d e v i a t i o n model . . . . 42 Var iance model 43 C o e f f i c i e n t of v a r i a t i o n model .. 46 The Treynor Index 46 Limi t a t ions 49 Summary . 50 IV. EMPIRICAL ANALYSIS 52 Input data : . 52 Resul ts and analys i s . . . . . . . . . . . . . . . . . . 55 C a p i t a l market curve 56 Risk premiums 66 Summary '. 67 V. SUMMARY AND CONCLUSIONS 68 Recommendations f o r f u r t h e r research 72 VI CHAPTER f-AGE BIBLIOGRAPHY . . 75 APPENDIX A. Regress ion Ana l y s i s 78 APPENDIX B. Treynor Index 86 LIST OF TABLES TABLE PAGE I. Growth Rate of Canadian Open-End Investment Companies k I I. Segment of Canadian Investment D o l l a r Owned by Open-End Investment Companies k I I I . Trend of Canadian Investment in American Stock 5 IV. Rates of Return and V a r i a b i l i t y : Canadian Mutual Funds . . . . 59 V. Rates o f Return and V a r i a b i l i t y : American Mutual Funds . . . . 60 VI. Regress ion A n a l y s i s : Standard Dev i a t i on Model 61 V l l . Regress ion A n a l y s i s : Var iance Model 62 V I I I . Regress ion A n a l y s i s : C o e f f i c i e n t of V a r i a t i o n Mode l . - ' . - . . . 63 IX. C l a s s i f i c a t i o n o f Mutual Funds by Investment P o l i c y 6k X. Summary of Regress i on .Ana l y s i s 80 XI. Average Annual Rates of Return on Sample Canadian Mutual Funds, American Mutual Funds and the Dow-Jones I n d u s t r i a l Average 88 XI I . V o l a t i l i t y (W) of Canadian Mutual Funds . . . . . 88 X I I I . V o l a t i l i t y (W) of U.S. Mutual Funds . 89 XIV. Treynor Index f o r Sample Canadian and U.S. Mutual Funds . . . . . 89 LIST OF FIGURES FIGURE PAGE 1.1. Expected Return--D i spers ion of R e t u r n s . . . . . . : 12 2.1. C a p i t a l Market Curve • . .'. 27 2.2. T yp i c a l C a p i t a l Market Curve f o r Mutual Funds 30 3.1. E f f i c i e n c y F r o n t i e r \ 35 3.2. Fami ly o f I nd i f f e rence Curves 36 3.3. Optimal P o r t f o l i o . . . 38 _3.4. Market Opportun i ty L ine 41 3.-5- C h a r a c t e r i s t i c L ine 47 3.6. P o r t f o l i o P o s s i b i 1 i t y L ine 48 4.1. Canadian and U.S. 'Bus iness C y c l e s ' 54 A. 1. Regress ion A n a l y s i s : Canadian Mutual Funds — Standard Dev i a t i on Model 81 A.2. Regress ion A n a l y s i s : American Mutual Funds — Standard Dev i a t i on Model 82 A.3. Regress ion A n a l y s i s : Canadian Plus American Mutual Funds—Standard Dev i a t i on Model 83 A.4. Regress ion A n a l y s i s : Canadian Mutual Funds--Varfance Mode l . . . 84 A.5- Regress ion A n a l y s i s : American Mutual Funds—Var iance Mode l . . . 85 . A. 6. Regress ion A n a l y s i s : Canadian Plus American Mutual Funds — "Var iance Model 86 B. I. C h a r a c t e r i s t i c L ines .....90 B.2. P o r t f o l i o P o s s i b i l i t y L ines 91 CHAPTER 1 THE PROBLEM AND DEFINITIONS OF TERMS USED In a wor ld c h a r a c t e r i z e d by pe r f e c t c a p i t a l markets and c e r t a i n t y there would be no d i f f e r e n c e whatsoever in the y i e l d s on var ious . s e c u r i t i e s . In p a r t i c u l a r there would be no d i f f e r e n c e between debt and e q u i t y . Investors would know p r e c i s e l y what present and f u t u r e cash f lows to expect from any investment. Where unce r t a i n t y p r e v a i l s , equ i t y investment is u s u a l l y cons idered more r i s k y than investment in bonds o f the same company. R i sk ave r s i on has s t rong i n t r o s p e c t i v e appea l . S t a r t i n g w i th the observat ions that insurance i s a w ide l y purchased commodity and. that insurance companies make money. The l a t t e r ob se rva t i on i nd i c a te s that the expected re tu rn on investment in insurance, on the average, must be negat i ve . Thus, in purchas ing insurance an i n d i v i d u a l i s accept ing a c e r t a i n loss of a smal l amount (the insurance premium) in preference to the combinat ion of a smal l chance of a much l a r ge r loss and a large chance of no lo s s .^ There fo re , the premise i s drawn that i nves to r s have an ave r s i on to r i s k and are w i l l i n g to pay something in o rder to s h i f t i t . It i s l o g i c a l to po s tu l a te that in order to induce M. Friedman and L . J . Savage, "The U t i l i t y Ana l y s i s of Choices Invo lv ing R i s k , " The Journal of P o l i t i c a l Economy. LVI , (August, 1948), pp. 279—304. 2 an i n d i v i d u a l to bear a d d i t i o n a l r i s k , he must be compensated by h igher r e t u r n s . The g rea te r the r i s k the g rea te r should be the expected . e q u i l i b r i u m rate of r e t u r n . Given that the r a t i o n a l i n ve s to r i s averse to bus iness r i s k s , what i s the trade o f f f u n c t i o n between r i s k and expected re tu rn ? 2 The ques t i on has both a normative and a p o s i t i v e s i d e . Norma-t i v e l y we are concerned w i th the f o l l o w i n g : g iven an expected re tu rn -how should the i n d i v i d u a l eva luate the d i s p e r s i o n of returns about the aj^ithemetic mean? P o s i t i v e l y we wish^to a s c e r t a i n whether r i s k premiums a r i s e in o p e r a t i v e c a p i t a l markets and, i f they do, to determine t h e i r r e l a t i v e magnitude. The major complex i t y a r i s e s w i th respect to the measurement of i n v e s t o r s ' e x p e c t a t i o n s . ^ A s i m p l i f y i n g technique is to s i de s tep the problem by assuming that i nves to r s e x h i b i t pe r f e c t f o r e s i g h t . The consequence of which i s that, ex post returns are i d e n t i c a l to ex ante re tuns . Th is means that the i nves to r s frequency d i s t r i b u t i o n of re tu rns f o r a s e c u r i t y can be c a l c u l a t e d from h i s t o r i c data as a proxy f o r f u t u r e behav io r . By measuring the h i s t o r i c a l market y i e l d s f o r ..various r i s k c l a s s e s of s e c u r i t i e s the r i s k premiums can be determined f o r a p a r t i c u l a r segment of the c a p i t a l market. Jack H i r s h l e i f e r , " R i s k the Discount Rate, and Investment D e c i s i o n , "American Economic Review, L I , No. 2, (May, 1961), p. 112. ^E. Bruce F r e d r i c k s o n , Front i e r s of Investment Analys? s, (Scranton: I n t e r na t i ona l Textbook Company, 1966), p.7. 3 1. THE PROBLEM Statement of the problem. In i n v e s t i g a t i n g one segment of the c a p i t a l market, namely open-end investment companies, three hypothes is were exp lo red . S p e c i f i c a l l y that (a) expected returns are an i n c rea s i ng f u n c t i o n of r i s k ; ^ (b) the va r i ance i s the appropr i a te measure of r i s k ; (c) the Canadian i n ve s to r in open-end investment companies does not e x h i b i t g reater r i s k ave r s i on than h i s American coun te rpa r t . Importance of the study. Investment trends in the l a s t decade and a h a l f has r e s u l t ed in the increased prominence of open-end investment companies. In 1950 these funds counted assets of approximately $42 m i l l i o n s ; by 1962 assets of $1,056.3 m i l l i o n s . Table 1 presents the r e l a t i v e growth rates o f open-end investment companies and Gross Nat iona l Product. Thus, ' mutual funds grew approx imately 6,402 per cent over the per iod 1950-1966, or at an annual rate of 400 per cen t , compared to a growth in Gross Nat iona l Product f o r the same per iod of on l y 320 per cent , or an average annual rate of 20 per cent . As o f f i s c a l year end 1966 mutual funds owned approx imately 3 per cent of the va lue of a l l s e c u r i t i e s W.F. Sharpe, "R i s k Avers ion in the Stock Market: Some Emp i r i ca l Ev idence, "The Journal of F inance, XX, No. 3» (September, 1965), p. 416-422. 4 l i s t e d or- the Montreal and Canadian Stock E x c h a n g e s . T a b l e 11 i l l u s t -rates the crowing segment of the Canadian investment d o l l a r owned by TABLE 1 GROWT;' RATE OF CANADIAN O P E N - E N D INVESTMENT COMPANIES Open-end Investment Companies G.N. P. Per iod 1950-1966 6,402% 370% Average annual ra te over -per iod 1950-1966 400% 20% Sources—Bank of Canada, S t a t i s t i c a l Summary, 1965 and 1966 and F i n a n c i a l Post, Survey o f Investment Funds, 1967. open-end investment companies. The importance of and the concern over the growth and s i z e of mutual funds in Canada i s r e f l e c t e d in the es tab l i shment o f a s p e c i a l committee, Canadian Committee on Mutual Funds TABLE 11 SEGMENT OF CANADIAN INVESTMENT DOLLAR OWNED BY OPEN l-END INVESTMENT COMPANIES Value of L i s t i n g s Tota l Net Assets on Montreal and of Open-end Canadian Stock Per Cent of L i s t i n g s Investment Companies Exchange Held by Open-end Year ($ Mi 11 ion) ($ M i l l i o n ) Investment Companies 1966 2,677.9 91,740 2.93% 1965 2,482.8 94,150 2.61 1964 1,811.7 .69,700 2.59 1963 1,305.0 56,220 2.32 1962 1,056.3 48,580 2.17 1950 42 8,970 0.47 • Source--Bank of Canada, S t a t i s t i c a l Summary, I965 and I966. F i n a n c i a l Post , Survey of Investment Funds, 1967. • 5 and Investment Con t r ac t s ? , in 1967 tc i n v e s t i g a t e t h e i r ope r a t i on s . Recent a r t i c l e s in the F i n a n c i a l Post have created concern along another 1 i ne—Canad ian investment funds ore sw i t ch ing to American s e c u r i t i e s . Table III i n d i c a te s the recent t rend to investment by Canadians and mutual funds in American s e c u r i t i e s . Over the per iod of 1963 to 1967 the market va lue of f o r e i g n s tock held by Canadian mutual TABLE III TREND OF CANADIAN INVESTMENT IN AMERICAN STOCK Canadian Net Market Value of Purchase of Foreign Stocks American Held by Mutual Stocks Funds (IV Quarter) Year ($ M i l l i o n s ) . ($ M i l l i o n s ) . 1967 ( I I I ) Quarter) 967 1966 314.7 608 1965 • 48.8 419 1964 -38.5 236 1963 -35.1 183 Source—Dominion Bureau of S t a t i s t i c s , Sales and Purchase of  Secur ? t ies Between Canada and Other Count r i e s , Catalogue No. 67—002. funds has grown at an annual rate of 525% compared to an annual growth ra te in mutual fund assets f o r the same per iod of on ly 153%. As of f i s -c a l year end I967 the funds had 45% of t h e i r p o r t f o l i o s in U.S. s tocks compared to 30% at year end 1966. If c a r r i e d to the extreme and the mutual funds p laced 100% of t h e i r assets in f o r e i g n markets, such a d r a i n of c a p i t a l , c on s i de r i n g the r e l a t i v e magnitude of t h e i r ho ld ings and the growth rate of the i ndu s t r y , cou ld have f a r reaching consequences on the Canadian economy. Two po s s i b l e i m p l i c a t i o n s can be drawn from 6 such a t r e n d : (1) that g rea te r investment o p p o r t u n i t i e s e x i s t in the American market, that i s , h igher l e v e l s of expected re tu rn (and co r re spond ing l y h igher l e v e l s of r i s k ) or (2) at any g iven l e ve l of r i s k American s e c u r i t i e s o f f e r g reate r expected returns than Canadian. " " I t i s c la imed that Canadians are ' l e s s w i l l i n g ' to take r i s k s than Amer i cans . " ^ If t h i s i s t rue then the second i m p l i c a t i o n drawn from the investment trends i s c o n t r a d i c t e d . For any g iven l e v e l of r i s k Americans r equ i re less expected returns than Canadians. Based on the hypothes i s that r i s k and expected returns are p o s i t i v e l y c o r r e l a t e d the r i s k a ve r s i on c h a r a c t e r i s t i c of the r e spec t i ve markets can be c a l c u l a t e d . By c o n t r a s t i n g the r i s k aver s ion e x h i b i t e d in each of the market segments the quest ion of whether Canadians are more con se r va t i ve r e l a t i v e to Americans or whether "Canadians are sw i t ch i ng to U.S. s e c u r i t i e s in o rder to p a r t i c i p a t e in ' g lamour ' a r e a s . . . . c a n be a s c e r t a i n e d . I I . DEFINITIONS OF TERMS USED Investment. The term ?nvestment i s i n t e r p r e t e d as f i n a n c i a l investment, which F i n a n c i a l Post , A p r i l 15,'1967, p. 1. I b i d . 7 r e f e r s to the buying and s e l l i n g of assets in the form of s e c u r i t i e s that w i l l produce a p r o f i t f o r the i n ve s t o r . Open-End Investment Companies (Mutua1 Funds). An investment company i s de f ined as "an i n s t i t u t i o n that i nves t s i t s c a p i t a l in s e c u r i t i e s such c a p i t a l being obta ined by the sa le s of i t s own s e c u r i t i e s . The income from i t s investments i s used to def ray «»8 i t s ope ra t i n g expenses and issued as p r o f i t to the shareho lder s . W i th in t h i s broad d e f i n i t i o n investment companies may be c l a s s i f i e d by ba s i c investment p o l i c y . Thus, there are common stock funds, s p e c i a l i z e d common stock funds, balanced funds, and bond funds. These terms r e f e r to the type of s e c u r i t i e s in which a fund normal ly i nves t s most or a l l o f i t s a s se t s . Balanced Funds seek to prov ide long-term growth of c a p i t a l and income. To o f f s e t the v o l a t i l e nature o f common s t ock s , some bonds and debentures are a l s o h e l d . Emphasis on p r e se r va t i on of c a p i t a l o f t en means these funds tend to concent rate on r e l a t i v e l y ' s a f e ' shares w i th backgrounds of s t a b l e earn ings and d i v i dend s . The purpose of balanced funds i s to minimize r i s k so f a r as p o s s i b l e wi thout unduly s a c r i f i c i n g the p o s s i b i l i t i e s f o r long-term growth which common stocks prov i de. F u l l y Managed Funds o f f e r some income but the emphasis i s more on 8 H.S. Sloan and A . J . Zucher, A D i c t i o n a r y of Economics, 4th E d i t i o n (Barnes & Noble Inc., 1961), p. 184. 8 growth s e c u r i t i e s than in the Balanced Fund. They a l s o inves t in a l l types of s e c u r i t i e s . The p ropo r t i on invested in each however, v a r i e s w i t h the cu r ren t market c o n d i t i o n s . A category u s u a l l y cons idered separate from the F u l l y Managed and Balanced Funds are the Common Stock Funds. As t h e i r name suggests, these fund concent rate t h e i r investments in e q u i t i e s . Just as some i n v e s t o r s , w i th at l e a s t part o f t h e i r c a p i t a l , seek maximum p r o f i t s from stock p r i c e movements, r ecogn i z i ng the g rea te r r i s k s i n vo l ved , so do c e r t a i n mutual funds p lace t h e i r p r i n c i p a l or e n t i r e emphasis on c a p i t a l a p p r e c i a t i o n . The S p e c i a l t y Funds are so named as they concentrate t h e i r investment in p a r t i c u l a r geographica l areas or i n d u s t r i e s and do not e x e r c i s e f u l l d i v e r s i f i c a t i o n . They hope to ga in more from s p e c i a l i -z a t i o n than they lose from l i m i t e d d i v e r s i f i c a t i o n . The c l a s s i f i c a t i o n of Bond Funds app l i e s to the l i m i t e d number of mutual funds which con f i ne t h e i r investments to bonds, p re fe r r ed s t o ck s , debentures or some combinat ion of the th ree . Ob jec t i ve s of Bond Funds vary in terms of emphasis p laced upon cu r ren t income and r e l a t i v e s t a b i l i t y . Mutual Fund shares can be bought and redeemed v i r t u a l l y on demand at the net asset va lue per share. The net asset va lue i s the market va lue of the investments and o the r assets of the fund les s the l i a b i l i t i e s . It v a r i e s as va lue o f the s e c u r i t i e s in the p o r t f o l i o change. If s tock p r i c e s drop, f o r i n s tance , the net asset va lue drops too. If they r i s e net asset va lue r i s e s . Rate of Return. Frequency d i s t r i b u t i o n of returns were c a l c u l a t e d from the h i s t o r i c a l annual rates of returns f o r each mutual fund f o r the nine year pe r i od of 1957 to 1966. A measure o f c e n t r a l tendency, the mean, was used as an e s t imate of the expected r e t u r n . To o b t a i n the approximate percentage y i e l d , or es t imated r e t u r n , f o r an investment fund the sum of the change in net asset va lue per share and d iv idends d i s t r i b u t e d was d i v i d e d by the net asset va lue per share at the beg inn ing of the p e r i o d : x = (N .A .V . ) t - (N.A.V.) t _ 1 + D t (N.A.V.) t _ 1 where (N.A.V.) i s the net asset va lue per share and i s the d iv idends d i s t r i b u t e d dur ing time per iod t . This assumes that a d o l l a r of r e a l i z e d o r u n r ea l i z ed c a p i t a l gains i s e x a c t l y equ i va l en t to a d o l l a r 9 of d i v i dend s , no b e t t e r or no worse. The annual rate of r e t u r n , as de f i ned above, i s a measure of net per formance—gross y i e l d less the expenses of management and admi n ? s t r a t i o n — b u t , is not .net of the a c q u i s i t i o n charge. Expected Return. The average va lue of a past s e r i e s o f numbers i s de f ined as the C a p i t a l gains d i s t r i b u t i o n s f o r American mutual funds are t r e a t ed as increases in the number o f shares ou t s tand ing . 10 sum of the s e r i e s d i v i d e d by the number of items in the s e r i e s . An average r e tu rn i s a number about which vhe past returns f l u c t u a t e d . The expected re tu rn of a fund i s the a r i t h m e t i c mean of the ac tua l r e t u r n s . Expressed mathemat i ca l l y : r x. where E(X) » expected re tu rn of the fund, N = number of years X. = a c tua l annual rates o f r e t u r n , i R i sk . The term r i s k r e f e r s to a v a r i e t y of f a c t o r s which a f f e c t s the r e tu rn on investment. When an i n ve s to r buys a s e c u r i t y the expected r e tu rn i s comprised of two e lements, the i n t e r e s t or d i v i dend and the c a p i t a l ga in or loss a s soc i a ted w i th p r i c e movements. He may s e l l the s e c u r i t y at a p r o f i t and thus increase the s i z e of re tu rn on h i s investment; but of course, the p r i c e may go down and he may s e l l at a l o s s . The s i z e o f gains o r losses from investment in a s e c u r i t y are a f f e c t e d by developments u n t i l the moment the s e c u r i t y i s so ld and are t h e r e f o r e , u n c e r t a i n up to that f i n a l moment. Th is ob se r va t i on about unce r t a i n returns leads to the assumption that i n ve s to r s form s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s of returns and an important c o r o l l a r y to the general, p r o p o s i t i o n that the expected returns and r i s k are p o s i t i v e l y c o r r e l a t e d . The c o r o l l a r y i s that the s i z e of u l t i m a t e returns becomes more unce r t a i n as the degree of r i s k i nc rea se s . The d i s p e r s i o n of the s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n 11 may be thought of as va r y i ng i n v e r s l y w i t h the conf idence an i n ve s to r has in h i s e s t ima te . F igure 1.1 i l l u s t r a t e s g r a p h i c a l l y the degree ov r i s k represented by the d i s p e r s i o n of returns f o r three broad ly de f i ned r i s k c l a s s e s o f s e c u r i t i e s . The g rea te r the d i s p e r s i o n the g rea te r the r i s k . The s t a t i s t i c a l measures of d ? spers i o n — t h e standard d e v i a t i o n , va r i ance and c o e f f i c i e n t o f v a r i a t i o n — a r e used as measures of r i s k . The app rop r i a te measure o f r i s k is a c o n t r o v e r s i a l i s sue . J . H i r s h l e i f e r ^ and W.F. Sha rpe^ employ the standard d e v i a t i o n o f the 12 p r o b a b i l i t y d i s t r i b u t i o n as a measure of r i s k . Archer and D'AmbrOsio suggest that the standard d e v i a t i o n i s . no t a s u f f i c i e n t measure but the c o e f f i c i e n t o f v a r i a t i o n i s . John L i n t n e r ^ argues that the v a r i ance i s the app rop r i a te measure. R i sk , as de f ined above, can be c l a s s i f i e d in a number of ways depending, f o r example, on whether i t is the r e s u l t of vagar ie s in the market p lace o r i n t r i n s i c s t o c h a s t i c p r o p e r t i e s of the p a r t i c u l a r f i r m . Cohen and Z inbarg have summarized the major types of r i s k that an ^ J . H i r s h l e i f e r , op. c i t . . ^W.F . Sharpe, op. c i t . . 12 S.H. Archer and C A . D 'Ambrosio, Business F inance: Theory and  Management, (New York:MacMi11 an, 1966);. 13 J . L i n t n e r , "The Va l ua t i on of R i sk Assets and the S e l e c t i o n of R i sky Investments i n Stock P o r t f o l i o s and C a p i t a l Budgets , " The Review  o f Economics and S t a t i s t i c s , k7 (February, 1965), p. 13—37. 12 FIGURE 1.1 EXPECTED RETURN--DISPERSI ON OF RETURNS P robab i1 i ty 1 .0 • 0.9-0.5 J . A Rate o f Return (a) Government Bonds rr~r/o) i.o-0.5- (c) Specu l a t i ve Issue P robab i1 i ty — . j — : _ • • -10 0 5 Rate o f Return TO 15 (%) 13 14 i n ve s to r i s sub jec t t o . These i n c l u d e : (1) Business R i sk , which i s r i s k inherent in the phy s i ca l ope ra t i on s of the f i r m ; i t a r i s e s s imply from the i n a b i l i t y to ensure a b s o l u t e l y s t ab l e s a l e s , c o s t s , and p r o f i t s . (2) Market R i sk , ( r e s u l t of market psychology) p r i c e s change d a i l y and c y c l i c a l l y , but noth ing r e a l l y has changed about the investment c h a r a c t e r i s t i c s of the s e c u r i t y but the market p r i c e has changed. (3) Purchas ing Power R i sk , which i s r i s k of or gains losses due to the f l u c t u a t i o n s in the purchas ing power of the d o l l a r . (4) I n te re s t Rate R isk. V a r i a t i o n s in the l e ve l of i n t e r e s t ra tes cause f l u c t u a t i o n s in the p r i c e of marketable bonds. (5) P o l i t i c a l R i sk. . Disturbances of a pure ly random and temporary cha rac te r r e s u l t i n g from such f a c t o r s as sudden i n t e r n a t i o n a l c r i s e s , pr ice-wage c o n t r o l s , p r e s i d e n t i a l heart a t t a c k s , change in subs id ie s and tax inc reases . Common stocks are most vu l ne rab le to (1), (2) and (4). Bonds are most v u l ne r ab l e to (1), (3) and (4). C l e a r l y no s e c u r i t i e s are f ree from a l l r i s k s . Var iance o f re tu rn of a_ secu r i t y . Var iance i s the average of squared d e v i a t i o n s from the mean. Var iance i s in e f f e c t a measure of J.B. Cohen and E.D. Z inbarg , Investment Ana l y s i s and P o r t f o l i o  Management, (Richard D. I rw in, Inc., 1967), p. 47--48. 14 d i s p e r s i o n from the average va lue of re tu rn s . Mathemat ica l l y the v a r i 2 ance 2 _ . (X. - X ) 2 6 .- ' i - i N-1 where X. _ a c t u a ] annual rates of r e t u r n , N = number of years and X = mean annual r e t u r n . Standard d e v i a t i o n of re turn of a_ secur i t y . The standard d e v i -a t i o n ( Cf ) i s de f ined as the p o s i t i v e square root of the v a r i ance . C o e f f i c i e n t o f v a r i a t i o n . The actua l v a r i a t i o n or d i s p e r s i o n as determined from the standard d e v i a t i o n or va r iance Is c a l l e d the abso lu te d i s p e r s i o n . However, a v a r i a t i o n or d i s p e r s i o n of one hundred d o l l a r s in measuring a re tu rn of one m i l l i o n d o l l a r s i s qu i te d i f f e r e n t in e f f e c t from the same v a r i a t i o n of one hundred d o l l a r s in a r e tu rn o f one thousand d o l l a r s . A measure of t h i s e f f e c t i s supp l i ed by the re 1 a t i v e d i s p e r s i o n def ined by ' , .. ». . Absolute D i sper s i on R e l a t i v e D i spe r s i on = c  Average If the abso lu te d i s p e r s i o n i s the standard d e v i a t i o n , , and the average i s the mean, X, the r e l a t i v e d i s p e r s i o n i s c a l l e d the c o e f f i c i e n t of v a r i a t i o n g iven by C o e f f i c i e n t of V a r i a t i o n = C = ^ * — For example, cons ider the f o l l o w i n g two investments. Investment (b) has an expected re tu rn o f 8 per cent and standard d e v i a t i o n of approx imate ly 5 per cent . Investment (c) has an expected re turn of 15 11 per c=nt w i th a standard d e v i a t i o n o f re tu rn o f 5 per cent . Using an abso lute measure of d i s p e r s i o n , . t h e standard d e v i a t i o n , the r i s k s are i d e n t i c a l . Using a r e l a t i v e measure of d i s p e r s i o n the r i s k s asso-c i a t e d w i th each investment are 0.625 and 0.455, r e s p e c t i v e l y . By t h i s measure the r i s k of investment (c) i s l e s s . Note that the c o e f f i c i e n t of v a r i a t i o n i s independent of un i t s used. For t h i s reason i t i s u se fu l in comparing d i s t r i b u t i o n s where un i t s may be d i f f e r e n t — a disadvantage of the c o e f f i c i e n t of v a r i a t i o n i s that i t f a i l s to be usefu l when X i s c l o se to zero . Standard e r r o r of es t imate. If the est imated expected r e t u r n , E(X) e s t j m a t e c j ' °^ t n e a c t u a l expected r e t u r n , E (X), f o r a g iven l e ve l o f r i s k , R, i s g iven by the l e a s t squares reg re s s i on equat ion E (X) = A + B R a measure of the s c a t t e r about the reg re s s i on l i n e of E (X) on R i s s upp l i ed by the quan t i t y /goo - coo e s t )2 g E ( x ) - R - \ y - v / i r which is c a l l e d the standard e r r o r e s t imate of E(X) on R, or the standard e r r o r of the mean. The standard e r r o r of e s t imate has p r ope r t i e s analogous to those of the standard d e v i a t i o n . C o r r e l a t i o n C o e f f i c i e n t . The r a t i o of the exp la i ned v a r i a t i o n to the t o t a l v a r i a t i o n i s c a l l e d the c o e f f i c i e n t of de te rminat ion 16 2 ( /O ). If there i s zero exp l a i ned ' a r i a t i o n , that i s , the t o t a l v a r i a t i o n i s a l l unexp la ined, t h i s ra t i " ) is zero . If there i s zero unexpla ined v a r i a t i o n , that i s , the t o t a l v a r i a t i o n i s a l l e x p l a i n e d , the r a t i o is one. In other cases the r a t i c l i e s between zero and one. The quan t i t y yd , c a l l e d the c o e f f i c i e n t of corre1 at ion is g iven by /O = + / Expla ined V a r i a t i o n Tota l V a r i a t i o n and v a r i e s between -1 and +1. The + are used f o r p o s i t i v e l i n e a r c o r r e l a t i o n and negat ive l i n e a r c o r r e l a t i o n r e s p e c t i v e l y . Covar iance. If x . and x. are random v a r i a b l e s ( return on j , s e c u r i t i e s ) w i t h standard dev i a t i o n s C*5" and <5 r e s p e c t i v e l y , and X • X • I J c o e f f i c i e n t o f c o r r e l a t i o n y&; ;, of x. and x . , the covar iance i s . 1 J i j denoted by the symbol cov ( x. , x^.) and de f ined by cov (x. , x . ) = E (x. , x.) - E (x.) E (xj) or X . . = /O. . &x. ^ x . ' J ' . ' J ' J Two s e c u r i t i e s which f l u c t u a t e in the same d i r e c t i o n over time w i l l show a p o s i t i v e cova r i ance . S i m i l a r l y , two s e c u r i t i e s which tend to move in oppos i te d i r e c t i o n s w i l l have a negat ive cova r i ance . If x. and X j are independent, t h e i r covar iance w i l l be ze ro , s?nee E (x . , x^) i s equal to E ( x . ) E ( X j ) . Pe r f ec t Markets. Merton H. M i l l e r and Franco Mod i g l i an i have summarized the requirements as f o l l o w s 17 15 In ' p e r f e c t c a p i t a l market s ' no buyer or s e l l e r (or i s suer ) of s e c u r i t i e s i s la rge enough f o r h i s t r an sac t i on s to have an app rec i ab l e impact on the then r u l i n g p r i c e . A l l t r ader s have equal and c o s t l e s s access to in fo rmat ion about the r u l i n g p r i c e and about a l l o ther re levent c h a r a c t e r i s t i c s of shares . . . . . . . No brokerage f ee s , t r a n s f e r taxes or t r a n s a c t i o n cos t s are i ncu r red when s e c u r i t i e s are bought, s o l d , or i s sued. The supply o f funds i s p e r f e c t l y e l a s t i c . Consequently, there would be no l i m i t to the amount of funds which an i n d i v i d u a l can r a i s e at the p r e v a i l i n g r i s k - f r e e i n t e r e s t r a t e . R isk avers i on . An i n d i v i d u a l i s aversed to r i s k in a g iven s i t u a t i o n i f (a) g iven the cho i ce between two investments w i th the same expected re tu rn s , he choses the a l t e r n a t i v e w i th the less r i s k or (b) g iven the cho ice between two investments w i th same r i s k , he choses the a l t e r n a t i v e w i th the l a r ge s t expected r e t u r n . R isk being d e f i n e d , as some f u n c t i o n of the s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n of expected r e t u r n . These preferences are imp l ied by max imizat ion of the expected va lue of a von Neumann-Morgenstern u t i l i t y f u n c t i o n i f the i n v e s t o r ' s u t i l i t y f u n c t i o n i s concave w i th respect to expected r e t u r n . ^ 5 M.H. M i l l e r and F. M o d i g l i a n i , "D i v idend P o l i c y , Growth and the Va l ua t i on of Shares , " Journal of Bus iness, XXXIV, No. k, (1961), p. 412. ^ F o r a d i s c u s s i o n of the nature of such u t i l i t y f unc t i on s see: M. Friedman and L . J . Savage, op_. c i t . ; D.E. F a r r a r , The Investment  Dec i s i on Under Unce r t a i n t y , (Englewood C l i f f s : P r e n t i c e - H a l l , Inc., 1962) John L i n t n e r , op_. c i t . ; and J . Tob in , " L i q u i d i t y Preference as Behavior Towards R i sk , "The Review of Economic S tud ie s , XXVI, (February, 1958). 18 Risk f r e e ?nteres t r a t e . The - i s k f r e e i n t e r e s t rate i s the p r i c e o f time or the ra te of -exchange between p re sen t -and - fu tu re -investments . The ques t ion of what determines the magnitude of the r i s k f r e e ^ i n t e r e s t ra te can be answered by r e s o r t i n g to economic theory . It i s determined by the i n t e r s e c t i o n of the supply and demand curves f o r investment funds in a r i s k l e s s economy. The nature of the supply curve ( i n d i c a t i n g the terms on which investment funds w i l l be supp l i ed ) depends on the i n d i v i d u a l s ' time p re fe rences , expec ta t i on s concern ing the f u t u r e , and present l e ve l o f wea l th . The demand curve, f o r the present d i s c u s -s i o n , i s the marginal e f f i c i e n c y of c a p i t a l (MEC) ( i n d i c a t i n g what can be done w i th investment funds ) . The MEC curve depends on two aspects of investment o p p o r t u n i t i e s : t h e i r rate of r e tu rn and the amount of the investment. I n o t h e r words, the r i s k f t e e - i n t e r e s t ra te in a r i s k -les s economy w i l l depend on peop l e ' s d e s i r e s - t o i nves t and the a v a i l -a b i l i t y of investment o p p o r t u n i t i e s . So._f.ar .the d i s c u s s i o n has .u t i LLzed the._s imp 1 if.y? ng . assumption of a r i s k l e s s economy. In the rea l wo r l d , there are no investments which have t h i s c h a r a c t e r i s t i c and a r e a l i s t i c a p p r a i s a l of the investment process must, e x p l i c i t l y , take i n to account the element of r i s k . In p r a c t i c e , the nearest t h i ng to the r i s k f r e e i n t e r e s t r a te among marketable s e c u r i t i e s i s the re tu rn on 30 day Treasury B i l l s . Investments r i s k , d i scussed p r e v i o u s l y , i n f l uence Treasury B i l l the l e a s t o f a l l s e c u r i t i e s . They are sub ject on l y to a smal l extent by 19 market and i n t e r e s t rate r i s k s . For both Canada and United S ta te s , the average y i e l d on 30 day Treasury. B i l l s over the per iod 1957 to 1966 was approx imately 4 per cent . I I I . METHODOLOGY The recent extens ions of J . Tob in s ' work on l i q u i d i t y ' ^ by W.F. 18 19 20 Sharpe J .L . Treynor and J . L i n t ne r have prov ided the methodology necessary to make such an e v a l u a t i o n . The a r t i c l e on the 'The Theory of C a p i t a l Asset P r i ce s Under Cond i t ions of R i s k ' by W.F. Sharpe desc r ibes the manner by which the p r i c e of r i s k r e s u l t s from the ba s i c i n f l uences of i nve s to r performance, and the phy s i c a l a t t r i b u t e s by two parameters, the expected re tu rn and the p red i c t ed v a r i a b i l i t y of re tu rn expressed as the standard d e v i a t i o n . The f u n c t i o n a l r e l a t i o n s h i p i s determined by l e a s t squares 17 . T L. J . Tob ins , op. c i t . 18 W.F. Sharpe, " C a p i t a l Asset P r i c e s : A Theory of Market Equ i1 i b r i urn Under Cond i t ions of R i s k , " The Journa1 of Fi nance, XIX, (September, 1964), p. 425—427. 19 J , Treynor, "How to Rate Management o f Investment Funds, " Harvard Business Review, 43, ( January—February , 1965), p. 63—79. 20 J . L i n t n e r , op. c i t . 20 r eg re s s i on a n a l y s i s . J. L i n t n e r has po s tu l a ted that the expected re tu rn i s l i n e a r l y r e l a t e d to the va r i ance not the standard d e v i a t i o n . Risk c l a s se s o f s e c u r i t i e s should be sca led in terms of va r i ance of returns ra ther than standard d e v i a t i o n . His model is b a s i c a l l y the same as the one developed by Sharpe. Regress ion a n a l y s i s was employed to a s c e r t a i n the app rop r i a te measure of r i s k — t h e v a r i a n c e , standard d e v i a t i o n , or c o e f f i c i e n t of v a r i a t i o n — a n d the r e l a t i o n s h i p between expected re tu rn and r i s k by t e s t s of s i g n i f i c a n c e . The reg re s s i on c o e f f i c i e n t i s a measure of the r i s k - a v e r s i o n e x h i b i t e d by the mutual fund i n v e s t o r s . T reyno r ' s model measures r i s k in terms of v o l a t i l i t y of expected returns ra ther than v a r i a b i l i t y . V o l a t i l i t y i s de f ined as the per cent change in rate of r e tu rn r e l a t i v e to a one per cent change in a standard of comparison, say the Dow-Jones I n d u s t r i a l Average. The c h a r a c t e r i s t i c l i n e i s the l e a s t squares reg re s s i on l i n e of fund rate of r e tu rn over time and market ra te of r e t u r n . The reg re s s i on c o e f f i c i e n t i s a measure of the f und ' s v o l a t i l i t y . A graph of the expected re tu rn and v o l a t i l i t y ( r i s k ) y i e l d the p o r t f o l i o p o s s i b i l i t y l i n e — a s t r a i g h t l i n e pass ing through the fund (expected r e t u r n , v o l a t i l i t y ) i n t e r s e c t i n g the v e r t i c a l a x i s at the r i s k f r e e i n t e r e s t r a t e . The s lope of the p o r t f o l i o p o s s i b i l i t y l i n e f o r a p a r t i c u l a r i n ve s to r i s a measure of the i nve s to r s r i sk -avers i on . Return is made up of two components. The f i r s t i s the i n t e r e s t 21 or d i v idends pa id to the s e c u r i t y ho lde r , and the second i s the c a p i t a l ga in or loss to the ho lder caused by marr.et p r i c e changes. Risk was based on the v a r i a t i o n s from year to y e a r " i n h i s t o r i c a l rates of r e tu rn s . IV. ASSUMPTIONS Assumption regard i ng i nves to r s . It was assumed (1) the i n v e s t o r ' s e s t imate of rate of re turn w i l l be equal to the sum of cash d iv idends rece ived plus the change in net asset va lue expressed as a percentage of the net asset va lue at 21 the beg inn ing of the pe r i od . They e x h i b i t pe r f e c t f o r e s i g h t and the ex post re turns are equal to the ex ante expected re tu rn s . (2) In-ve s to r s have i d e n t i c a l sets of p r o b a b i l i t y d i s t r i b u t i o n s f o r each mutual fund. Th is means that the same percentage ho ld ings w i l l be opt imal f o r each i n ve s t o r . (3) The inves to r s are r i s k avo ide r s . (4) I n ve s to r ' s t a s t e s and a t t i t u d e s change r e l a t i v e l y s l ow ly over t ime, such that they can be assumed to remain constant over the per iod 1957 to 1966. Market assumpt i on . (1) S e c u r i t i e s are traded in a s i n g l e pe r f e c t market. (2) the p r i c e s of s e c u r i t i e s are g i ven , (3) each i n ve s to r can borrow or lend funds at a common p o s i t i v e r i s k f r e e i n t e r e s t rate and (4) an i nve s to r can inves t any p o r t i o n of h i s c a p i t a l in r i s k investments. W.F. Sharpe, op_. c i t . and J . L i n t n e r , op_. c i t . 22 V. ORGANIZATION OF THE THESIS The study i s devided i n to f i v e chapte r s . Chapter 1 i s an i n t r o d u c t o r y chapter o u t l i n i n g the problems and i t s importance, the methodology and the assumptions r equ i r ed . Chapter II prov ides a framework o f investment and 'a p r i o r i ' reasoning f o r the models developed in Chapter I I I . The normative models used to de sc r i be the c a p i t a l markets are developed in Chapter I I I . The data and a n a l y s i s of r e s u l t s are presented in Chapter IV. The f i n a l chapter conta in s a summary and conc lu s i on s of the study. CHAPTER I I A FRAMEWORK OF I Nv'ESTMENT This chapter presents some of the elements of a theory of the market process by which market premiums on investments are determined. R i sk premiums are de f ined as the d i f f e r e n t i a l between the r i s k - f r e e i n t e r e s t ra te and the market y i e l d on r i s k y investments. The o b j e c t i v e s o f the i n ve s to r and the hypothes i s of i n c rea s i n g returns are combined in a t h e o r e t i c a l a n a l y s i s of the investment process. In b r i e f , there are three ba s i c concepts in the framework of thought: i nve s to r s g e n e r a l l y p r e f e r l a r ge r to sma l l e r returns on s e c u r i t i e s ; i n ve s to r s d e s i r e to min imize r i s k ; and r i s k and returns are p o s i t i v e l y c o r r e l a t e d . In general an investment invo lves an exchange. Such a t r a n s -s a c t i o n can be desc r ibed by two elements. The f i r s t i s s h i f t i n g cu r ren t funds to the f u t u r e . This v i o l a t e s the g e n e r a l l y accepted concept that " a b i r d in the hand i s worth two in the bu sh "—a c h a r a c t e r i s t i c c a l l e d time p re fe rence . The second element i s the exchange o f secure and c e r t a i n funds f o r something less than c e r t a i n . This a l s o v i o l a t e s a g e n e r a l l y accepted p r i n c i p l e — t h a t i n d i v i d u a l s are r i s k averse. Obv ious l y , such a t r a n s a c t i o n w i l l occur i f and on ly i f the i n ve s to r i s H. Sauv Ain, Investment Management, Th i rd E d i t i o n , (Englewood C l i f f s : P r e n t i c Hal 1 , 1967) and S.H. Archer and C A . D 'Ambrosio, Business F inance: Theory and Management, (New York: MacM i l l an , 1966). 2k adequately compensated f o r assuming the a d d i t i o n a l r i s k and w a i t i n g . The magnitude of the market premium depends on the a t t i t u d e s of i n ve s to r s toward r i s k , the time to r e a l i z e the r e tu rn and the degree o f r i s k . I INVESTMENT OBJECTIVES The purpose of investment i s pecuniary ga i n . This i s what mot ivates people to take t h e i r money out of bank accounts , s a f e t y depos i t boxes and other forms fo sav ing to buy r i s k y a s se t s . Through investments i t i s assumed that the r a t i o n a l i n d i v i d u a l attempts to maximize h i s wea l th . The d e c i s i o n r u l e o f max imizat ion o f r e tu rn s , w i thout i n co rpo r a t i n g the element of r i s k , leads to the conc l u s i on that the e n t i r e p o r t f o l i o would be invested in a s i n g l e s e c u r i t y , the one y i e l d i n g the g rea te s t r e tu rn . This r u l e i s r e j ec ted on the grounds that d i v e r s i f i c a t i o n i s a w ide l y p r a c t i c e d p r i n c i p l e . The aim of d i v e r s i f i c a t i o n i s to achieve a c e r t a i n o v e r - a l l r e tu rn w i th a minimum of r i s k . Thus, the i n ve s to r attempts to m in im i ze , r i s k f o r a g iven expected r e t u r n . Since r i s k and re tu rn are assumed to be p o s i t i v e l y c o r r e l a t e d , i n c r ea s i n g returns may be viewed as a p o s i t i v e f o r ce whereas, i n c rea s i n g r i s k is a negat ive or opposing f o r c e . The i n d i v i d u a l at tempt ing to maximize h i s f u t u r e wealth must a t t a i n a balance between the two fo rce s compat ib le w i th h i s a t t i t u d e s toward r i s k and r e t u r n . In t h e i r investment i n d i v i d u a l s s t r i v e to achieve severa l o b j e c t i v e s . The f o l l o w i n g are o b j e c t i v e s f r e q u e n t l y of importance to 25 i n v e s t o r s : ( i ) The p s ycho l og i ca l e f f e c t , of owning a p o r t f o l i o of s e c u r: -t i e s . There is a c e r t a i n a i r of a f f l u e n c e to being ab le to ' p l a y the market. ' There i s a c e r t a i n element of enjoyment f o r some in t r y -ing to out-guess s e c u r i t y p r i c e movements. ( i i i ) The d i s t r i b u t i o n of returns between d iv idends and c a p i t a l ga ins . ( i v ) The t im ing o f returns in the f u t u r e . (v) The a c q u i s i t i o n of r e tu rn s , (v i ) The avoidence of r i s k . It i s assumed that the f i r s t f ou r o b j e c t i v e s are subord inate to the l a s t two, to maximize returns and minimize r i s k a s soc i a ted w i th any g i ven expected r e t u r n , which governs the i n d i v i d u a l ' s investment p o l i c y . II THE HYPOTHESIS OF INCREASING RETURNS It i s g e n e r a l l y be l i e ved tha t the rates of net re tu rn by a l l s e c u r i t i e s over long per iods are p o s i t i v e l y c o r r e l a t e d w i th r i s k i n e s s of s e c u r i t i e s . Th is g ives r i s e to the concept of r i s k c l a s s e s : group-ing o f s e c u r i t i e s accord ing to rate o f net re tu rn and l e ve l of r i s k . In o rder to grasp the f u l l meaning o f the model i t is necessary to d e l i n e -ate i t s unde r l y i ng assumptions. The model views the the market as a w h o l e - - a l l s e c u r i t i e s . The market can be s u b - c l a s s i f i e d i n to r i s k c l a s se s o f s e c u r i t i e s accord ing 26 to t h e i r expected re tu rn and r i s k c h a r a c t e r i s t i c s . The ra te of r e tu rn i s f o r a large number of s e c u r i t i e s . The performance i s a net f i g u r e^ h o p e f u l l y the gains w i l l more than o f f s e t the l o s se s . It does not r e f e r to a s i n g l e investment nor a s i n g l e i n v e s t o r . The phrase 'over long p e r i o d s ' i s important. Even aggregate net returns may vary from year to year as the economy passes through the peaks and troughs of the succes s i ve bus iness c y c l e s and as i n v e s t o r s ' conf idence in the i n v e s t -ment ou t look f l u c t u a t e . A long pe r iod is one long enough to average out c y c l i c a l and e r r a t i c v a r i a t i o n s in investment re tu rn s . I l l THEORETICAL ANALYSIS The concept that the rate o f r e tu rn is p o s i t i v e l y c o r r e l a t e d w i t h r i s k f o l l o w s d i r e c t l y from the a t t i t u d e s o f i n ve s to r towards assuming r i s k and f o l l o w s from the premise that i n ve s to r s are r i s k avo ide r s . Owners o f r i s k y s e c u r i t i e s must r e a l i z e l a r ge r returns in the aggregate than ho lders of r i s k l e s s investments. The ob se rva t i on that they cont inue to do so means that they are compensated by l a r ge r r e tu rn s . What i s o f i n t e r e s t to the i n ve s to r i s not the 'degree of c e r t a i n t y ' o f returns on a s i n g l e s e c u r i t y but the re tu rn on a h i g h l y d i v e r s i f i e d p o r t f o l i o . The re turn on t h i s p o r t f o l i o may be thought as having a p r o b a b i l i t y d i s t r i b u t i o n which i s the j o i n t p r o b a b i l i t y d i s t r i b u t i o n of the component investments weighted by the amount invested in each. By changing the we ight s , making marginal adjustments between one s e c u r i t y and another, the p r o b a b i l i t y d i s t r i b u t i o n of the 27 whole may be v a r i e d . The r e s u l t , in the l i m i t , i s a whole spectrum of combinat ions of s e c u r i t i e s d i s t i n g u i s h e d by t h e i r expected re tu rn and r i s k parameters. Assuming a p e r f e c t market, the c a p i t a l market curve (CMC) represents the sum of such combinat ions f o r a l l i n v e s t o r s . A l l investments l y i n g on the c a p i t a l market curve (CMC), i l l u s -t r a t e d in F igure 2.1, represents the two parameters of t h e i r p r o b a b i l i t y d i s t r i b u t i o n : expected re tu rn and r i s k . The expected re turn i s that re tu rn which i s necessary to compensate the i n ve s to r f o r the sum of the r i s k f r e e i n t e r e s t rate (p r i ce of time) and the r i s k premium (p r i ce of r i s k ) . Any combinat ion of investments below the CMC represents an investment overva1ued by i n v e s t o r s . Anything above CMC i s undervalued combinat ion; that i s , the returns are more than s u f f i c i e n t to induce i n d i v i d u a l s to undertake them. FIGURE 2.1 CAPITAL MARKET CURVE Expected Return r f Risk 28 A necessary c o n d i t i o n f o r the c a p i t a l market to be in a s t a t e of e q u i l i b r i u m i s that no two s e c u r i t i e s of equal ' a t t r a c t i v e n e s s ' can e x i s t at d i f f e r e n t p r i c e s . Two s e c u r i t i e s are e q u a l l y a t t r a c t i v e i f and on l y i f , they both l i e a long the CMC. This i s the e q u i l i b r i u m p o s i t i o n f o r the c a p i t a l market. If the c a p i t a l market was p e r f e c t l y c o m p e t i t i v e , a l l investments would l i e a long the c a p i t a l market curve in e q u i l i b r i u m . Any d e v i a t i o n from the l i n e would tend to set in motion an a r b i t r a g e process which would r e tu rn the system to e q u i l i b r i u m . Suppose that three combinat ions of s e c u r i t i e s X, Y and Z e x i s t and t h e i r r i s k - r e t u r n parameters r e l a t i v e to the c a p i t a l market curve are as represented in F igure 2.1. The attempts by i nve s to r s to purchase the s e c u r i t i e s in combinat ion X and lack of i n t e r e s t in ho l d i n g a s ses t s not in combinat ion X would lead to a r e v i s i o n of p r i c e s . The combinat ion X i s p r e f e r r ed to Y, f o r example, s i nce i t o f f e r s a g rea te r expected re tu rn f o r the same l e ve l of r i s k . From the assumption made of i nve s to r s being r i s k averse the combinat ion X would be d e s i r e d . The p r i c e s of s e c u r i t i e s in X w i l l r i s e and, s ince a s e c u r i t i e s ' expected re tu rn r e l a t e s f u t u r e income to present p r i c e s , t h e i r expected re tu rn w i l l f a l l . This w i l l reduce the a t t r a c t i v e n e s s o f the combinat ions which inc lude such s e c u r i t i e s ; thus po in t X wi11 move downward from i t s i n i t i a l p o s i t i o n . On the o ther hand, the p r i c e s o f s e c u r i t i e s not in X wi11 f a l l caus ing an increase in t h e i r expected returns and an upward movement of po in t s rep re sent ing combinat ions 29 which include them. Such price changes will lead to a revision of investors actions. Some new combination or combinations will become attractive, leading to different demands and thus to further revisions to prices. As the process continue, the capital market curve will tend to become more linear, with points such as X moving down and formerly inefficient points (such as Y and Z) moving upward. Security prices continue to change until a set of prices is attained for which every asset enters at least one combination lying on the capital market curve. The CMC is the indifference curve for the market as a whole. Indifference curves present a graphic analysis of consumer tastes and preferences. The indifference curve of a single investor for risk and expected return preferences is obtained by con-fronting the investor with a whole range of choices among various possible combinations of the two parameters. The crucial assumption made is that the investor is able to tell which combinations yield equivalent satisfaction and which ones yield greater or less satisfaction A single indifference curve shows the various combinations of risk and expected return that yield equal satisfaction to the investor or among which the investor is indifferent. The CMC represents the combination of indifference curves of all investors in the capital market. In practice there are a number of reasons why the relationship between expected return and risk will only be approximately a curve such as the CMC. Imperfections in the market (trademarks, patents, and image), imperfect knowledge on the part of the investor and government 30 •regulations are among the important reasons. However, the CMC i s the 'Jbest f i t " toward which the investment o p p o r t u n i t i e s in the market tend -to g rav i t a t e . The i n t e r s e c t i o n of the CMC w i th the v e r t i c a l a x i s g ives the r i s k - f r e e i n t e r e s t r a te . The s lope of CMC i nd i c a t e s the a d d i t i o n a l expected re tu rn rece ived per un i t of a d d i t i o n a l r i s k . The expected r e t u r n from any g iven investment w i l l thus be made up o f the r i s k - f r e e i n t e r e s t ra te (p r i ce o f time) plus an a d d i t i o n a l amount, the r i s k  prem?urn (p r i ce o f r i s k ) . The a p p l i c a t i o n of t h i s t h e o r e t i c a l model to a segment of the c a p i t a l market i s s t r a i g h t forward. A t y p i c a l CMC f o r mutual funds i s i l l u s t r a t e d in F igure 2.2. • FIGURE 2.2 TYPICAL CAPITAL MARKET CURVE FOR MUTUAL FUNDS Expected Return R i sk 31 Much of the v a r i a t i o n s among mutual funds i s e xp l a i ned by v a r i a t i o n s in the r i s k i n e s s o f t h e i r p o r t f o l i o s . By a l t e r i n g the mixture of s e c u r i t i e s in i t s p o r t f o l i o a mutual fund can a t t a i n any po in t on the CMC. The management of the mutual fund s e l e c t s an a t t i t u d e toward expected re turn and r i s k and then i nvi tes.. i nvestors w i t h s i m i l a r preferences to purchase shares in the fund. At one extreme, the fund might attempt to de sc r i be an e n t i r e pa t t e rn o f r e l a t i v e p re ference f o r expected re tu rn v i s - a - v i s r i s k . a much more l i k e l y method, one that seems to be fo l l owed in p r a c t i c e , invo lves merely a d e s c r i p t i o n of the general degree of r i s k planned f o r the f und ' s p o r t f o l i o ; the fund then s imply attempts to s e l e c t the e f f i c i e n t p o r t -f o l i o f o r that degree of r i s k . For example, the, ' I n ve s to r s Group' manage f i v e complementary open-end investment companies. IV SUMMARY This chapter developed an ove r - v iew of the investment process i n c o r p o r a t i n g the three concepts : (1) i nves to r s p r e f e r l a r ge r to sma l l e r r e t u r n s ; (2) i nves to r s attempt to minimize r i s k a s soc i a ted w i t h any p a r t i c u l a r expected r e t u r n ; and (3) r i s k and expected re tu rn are p o s i t i v e l y c o r r e l a t e d . The concept o f the c a p i t a l market curve (CMC) was i n t roduced. W i th i n t h i s framework, Chapter III developes a theory of m i c r o -economics to e x p l a i n how market p r i c e s are e s t a b l i s h e d from inves to r s per ferences and c h a r a c t e r i s t i c s of c a p i t a l a s se t s . CHAPTER I I I THEORY OF INVESTMENT Recent a r t i c l e s on investment behavior have emphasized the element of r i s k , as measured by year to year v a r i a t i o n s in r e t u r n , in a d d i t i o n to expected r e tu rn . Normative and p o s i t i v e models have been fo rmulated to e x p l a i n the ope ra t i on of c a p i t a l markets under c o n d i t i o n s of r i s k . Based on the premise that i n ve s to r s are r i s k aver se , Markowitz fo rmulated a technique f o r the s e l e c t i o n of e f f i c i e n t p o r t f o l i o s . ^ The e f f i c i e n t combinat ions of assets form an " e f f i c i e n c y f r o n t i e r . " A p o r t f o l i o i s e f f i c i e n t i f , f o r a g iven va r i ance no o ther p o r t f o l i o e x i s t s w i t h a g rea te r expected r e t u r n , o r (a p o r t f o i o ) w i th the same expected re tu rn but a sma l l e r va r i ance (of r e t u r n ) . An important 2 ex ten s i on of the Markowitz theory was made by J . Tobin. He showed t h a t , assuming i nves to r s can borrow or lend funds at the r i s k - f r e e i n t e r e s t r a t e , there e x i s t s a p r e f e r r ed e f f i c i e n t combinat ion of r i s k y a s s e t s ; the i n ve s to r need on l y l eve r h imse l f up or down the ' o ppo r t un i t y l o c u s ' to a t t a i n any combinat ion o f expected re tu rn and r i s k c o n s i s t e n t w i t h h i s p a r t i c u l a r a t t i t u d e towards r i s k v i s - a - v i s expected r e t u r n . Harry Markowitz, " P o r t f o l i o S e l e c t i o n , " The Journal of F inance, V l l , (March 1952), p. 7 7 - 9 1 . 2 J . Tob in , " L i q u i d i t y Preference as Behavior Towards R i s k , " The Review of Economic S tud ie s , XXVI, (February, 1958), p. 65—86. W.F. Sharpe and J . L i n t ne r extended Tob in ' s . theory to e x p l a i n the fo rmat ion of r i s k premiums in c a p i t a l markets. The remaining sec t i on s of t h i s chapter summarize the pos tu l a te s of the authors mentioned above. I.. MARKOWITZ THEORY OF DIVERSIFICATION Markowitz suggested that the process of p o r t f o l i o s e l e c t i o n con s i s t ed of three s tages : ( l ) o b t a i n i n g p r o b a b i l i s t i c e s t imates f o r the f u t u r e performance of s e c u r i t i e s , (2) d e r i v i n g an e f f i c i e n t set o f combinat ions of s e c u r i t i e s and (3) s e l e c t i n g the p o r t f o l i o which i s c o n s i s t e n t w i th i n v e s t o r ' s p re fe rences . r E f f i c i e n c y f r o n t i e r . Markowitz presented a r a t i o n a l e f o r d i v e r s i f i c a t i o n based on the hypothes i s of i n c rea s i n g returns and inves to r s r i s k a ve r s i on . A frequency d i s t r i b u t i o n of returns can be generated from the rates of re tu rn o f a s e c u r i t y over a number of year s . From t h i s , an a p p r o x i -mation f o r the expected re tu rn and r i s k f o r the s e c u r i t y can be ob ta i ned . Investors are assumed, to eva luate s e c u r i t i e s on the bas i s of two parameters, the mean and v a r i a n c e , of t h e i r p r o b a b i l i t y d i s t r i b u t i o n of annual r e tu rn s . The mean i s i d e n t i c a l to the expected re tu rn and the va r i ance i s used as a measure of r i s k . The expected re tu rn on a p o r t -W.F. Sharpe, " C a p i t a l Asset P r i c e s : A Theory of Market E q u i l i -brium Under Cond i t ions of R i s k , " The Journal of F inance, XIX, (September, 1964), p. 425-442. 4 J . L i n t n e r , "The Va lua t i on of Risk Assets and the S e l e c t i o n of R i sky Investments in Stock P o r t f o l i o s and C a p i t a l Budgets, " The Review  of Economics and S t a t i s t i c s , XLVLL, (February, ^^6S), p. 13—37. 34 f o l i o Is the weighted sum of the expected returns on each s e c u r i t y . Stated mathemat i ca l l y , E ( X ) =9 X X where % . i s the f r a c t i o n of the p o r t f o l i o invested in the i t h s e c u r i t y and X. i s the mean re tu rn on the i t h s e c u r i t y . The va r i ance o f re tu rn on a p o r t f o l i o i s . V(X) = 1 £ < i # 3 $ c where O .. i s the covar iance between the i t h s e c u r i t y and the j t h IJ s e c u r i t y . The covar iance is a s t a t i s t i c a l measure o f c o r r e l a t i o n , in t h i s case between the returns on two s e c u r i t i e s . Stated mathemat i ca l l y , where . . i s the c o e f f i c i e n t o f c o r r e l a t i o n between the re tu rn of the ' J i t h s e c u r i t y and the j t h s e c u r i t y and is the standard d e v i a t i o n of the re tu rn of the i t h s e c u r i t y . Accord ing to t h i s model, i f the i n ve s to r acts in a r a t i o n a l manner, he w i l l s e l e c t the combinat ion of s e c u r i t i e s which y i e l d s the maximum expected re tu rn f o r any g iven l e ve l of r i s k . A s e r i e s of a t t a i n -ab le e f f i c i e n t combinat ions of (E,V) can be c a l c u l a t e d from a l i s t of M s e c u r i t i e s and p l o t t e d on expected r e tu rn - va r i ance axes to form a l i n e . ' ' P o r t f o l i o S e l c t i o n : E f f i c i e n t D i v e r s i f i c a t i o n York: John Wi ley and Sons 1959), Chapter V I I I . H. Markowitz, of Investments. (New 35 This l i n e i l l u s t r a t e d in F igure 3.1 i s known as the ' e f f i c i e n c y f r o n t i e r . ' ^ FIGURE 3.1 EFFICIENCY FRONTIER Expected Return Risk A combinat ion which l i e s below t h i s l i n e i s i n e f f i c i e n t ; there e x i s t s , at l e a s t one p o r t f o l i o w i t h a g rea te r expected re tu rn f o r the same l e ve l of r i s k . Any po in t above the l i n e is u n a t t a i n a b l e . I I INVESTORS PREFERENCES A convenient dev ice f o r summarizing an i n d i v i d u a l ' s a t t i t u d e s toward r i s k and expected re turn i s a f a m i l y of i n d i f f e r e n c e curves such In order to those formulated by ma inta in cons i s tency the axes have been reversed to Markowitz. 36 as shown i'n F igure 3.2.^ Each po in t on an i n d i f f e r e n c e curve represents a conce i vab le p o r t f o i i o c h a r a c t e r i z e d by a p a r t i c u l a r combinat ion of ^variance and expected r e t u r n . A s i n g l e i n d i f f e r e n c e curve shows the d i f f e r e n t combinat ions of expected re tu rn and va r i ance that y i e l d equal s a t i s f a c t i o n to the i n v e s t o r , or which are e q u a l l y p r e f e r r e d . FIGURE 3.2 ^FAMILY OF INDIFFERENCE CURVES Risk A system of r i s k expected re turn i n d i f f e r e n c e curves e x h i b i t four ba s i c c h a r a c t e r i s t i c s : ( i ) They are concave upwards, the marginal rate of sub-s i t u t i o n of expected re tu rn f o r r i s k is i n c r e a s i n g . An increase in expected re tu rn i s requ i red to induce an . i n ve s to r to bear more r i s k and the more r i s k which i s ^R ichard H. L e f t w i c h , The P r i c e System and Resource A l l o c a t i o n , Th i rd Ed i t i on , - (New York: H o i t , R inehar t , -and Winston, Inc.,1966), Chapter. 5. 37 i n c u r r e d , the g rea te r w i l l be the requ i red increase in expected r e t u r n , ( i i ) The p o r t f o l i o s represented by h igher i n d i f f e r e n c e curves ••epresent g reate r u t i l i t y , s ince f o r any l e ve l of r i s k the i nves to r would p r e f e r maximum expected r e t u r n , ( i i i ) The g rea te r the degree o f r i s k ave r s i on that an i n ve s to r has the steeper w i l l be h i s i n d i f f e r e n c e curves . S i m i l a r l y the f l a t t e r the d i f f e r e n c e curves the more w i l l i n g the i n ve s to r i s to accept r i s k , ( i v ) I nd i f f e rence curves are n o n - i n t e r s e c t i n g . This can be seen by. r e f e r r i n g to the f am i l y of i n d i f f e r e n c e curves in F igure 3.2. A l l combinat ions of r i s k and expected re tu rn on curve 2 are p r e f e r r ed to combinations on curve 1, but a l l combinations on curve 1 are e q u i v a l e n t . There fo re , b y d e f i n i t i o n two i n d i f f e r e n c e curves cannot i n t e r s e c t . The opt imal p o r t f o l i o . The f a m i l y of i n d i f f e r e n c e curves are superimposed on the e f f i c i e n c y f r o n t i e r in F igure 3-3. The i n ve s to r maximizes h i s u t i l i t y at the po in t of tangency, E, between the h ighest i n d i f f e r e n c e curve and the e f f i c i e n c y f r o n t i e r . The steeper the i n d i f f e r e n c e curves the f a r t h e r to the l e f t w i l l be the po in t of tangency. S i m i l a r l y , the sha l lower the i n d i f f e r e n c e curves the f a r t h e r to the r i g h t w i l l the opt imal p o r t f o l i o l i e . I l l INVESTMENT OPPORTUNITY LOCI Tobin showed that the p o r t f o l i o s e l e c t i o n problem can be simpl f i e d by break ing i t down in to two s t a g e s — a l l o c a t i o n f i r s t among and g then w i t h i n asset ea t ago r i e s . Assume that the i n ve s to r i s t r a d i n g in p e r f e c t market. He can borrow or lend any de s i r ed amount at the r i s k f r e e i n t e r e s t r a t e , r^ . and i nves t in a p o r t f o l i o w i th p r e d i c t e d pe r -formance (E(X), & ). Let x be the re tu rn on e q u i t i e s and dx be the standard d e v i a t i o n of r e tu rn s . The t o t a l net re tu rn on h i s t o t a l c a p i t a l i s : FIGURED. 3 • OPTIMAL PORTFOLIO Expected Return Risk J . Tob in , o_p_. c? t . , p. 85. 39 (;) XA = (1 - A 2 ) A r f + A A 2 x ; 0 < A < GD where h" the f r a c t i o n of i n ve s to r s c a p i t a l A he ld in the p o r t f o l i o of equ i t i es. (2) E(X) = r f + A 2 ( x - r f ) . The r i s k attached to a p o r t f o l i o i s g iven by the standard d e v i a t i o n of r e t u r n , X, & . High va lues of <^ represent s p e c u l a t i v e investments and presents the p o s s i b i l i t y of large r e t u rn s , but i t might produce on l y modest r e t u rn s , and might r e s u l t in the r e a l i z a t i o n of a s u b s t a n t i a l l o s s . It i s j u s t as t rue to say that the s i z e and the p ro -b a b i l i t y of p o t e n t i a l losses increases w i t h increases in r i s k as i t i s to say that the s i z e and the p r o b a b i l i t y of p o t e n t i a l gains increases w i t h increases in r i s k . A low c_5"~x p o r t f o l i o " p r o t e c t s " the i n ve s to r from c a p i t a l loss and l i k e w i s e g ives him l i t t l e prospect of unusual gai hs. The standard d e v i a t i o n of X depends on the standard d e v i a t i o n of x , ^ and on the investment in e q u i t i e s , A„. X ( 3 ) ^ X = A2 « ^ x . Thus, the p r opo r t i on the i n ve s to r holds i n e q u i t i e s determines both h i s expected r e t u r n , E (X) , and r i s k The terms on which the i n ve s to r can a t t a i n g rea te r expected re tu rn at the expense of assuming more r i s k can be d e r i v e d . E l i m i n a t i n g A^ from equat ions (2) and (3) E(X) = r f + ( 0 ) 4o where 0 = i^ _____L>) • This the equat ion of the " investment oppo r tun i t y l o c u s " w i th i n t e r c e p t , r f , and s lope (x - r . ) . The i n ve s to r by a l l o c a t i n g h i s funds ^ x between the p o r t f o l i o and borrowing or lend ing can a t t a i n any po in t on such a 1i ne. Any p o r t f o l i o w i l l g i ve r i s e to a l i n e a r boundary of E (X), & A combinat ions . If a l l p o s s i b l e p o r t f o l i o s are cons idered a whole f a m i l y o f investment oppo r tun i t y l o c i , as shown in F igure 3.4, w i l l be generated. Since any expected re tu rn can be a t t a i n e d from any p o r t f o l i o , he w i l l min imize h i s r i s k a s soc i a ted w i th any p a r t i c u l a r expected re turn by -c o n f i n i n g a l l h i s investments in e q u i t i e s to the combinat ion y i e l d i n g the l a r ge s t va lue of 0. This corresponds to that investment p lan where the oppo r t un i t y locus i s tangent to the e f f i c i e n c y f r o n t i e r . The r e s u l t i n g p o r t f o l i o has the necessary c h a r a c t e r i s t i c s minimum r i s k f o r any l e ve l of expected r e t u r n . By means of leverage the i nve s to r can reach any r i s k - e xpec ted re turn combinat ion. The unique opt imal p o r t -f o l i o i s independent of the i n v e s t o r ' s de s i r ed expected re tu rn and A^, the p r opo r t i on invested in e q u i t i e s . The market oppo r tun i t y l i n e i s i l l u s t r a t e d g r a p h i c a l l y in F igure 3.4 and corresponds to the investment oppo r tun i t y locus which maximizes 0. The i n d i f f e r e n c e curves f o r each i n d i v i d u a l i nve s to r are represented by I. , . where i r e f e r s to the i t h i n ve s to r and j to the 41 . FIGURE 3.4 MARKET OPPORTUNITY LINE Lend Risk (Standard Dev ia t ion ) (%) p a r t i c u l a r i n d i f f e r e n c e curve. An i nve s to r represented by i n d i f f e r e n c e curves 1^, . would lend a p o r t i o n of h i s wealth at the r i s k - f r e e rate r^ . and i nves t the remainder in the combinat ion represented by the po int (E) on the e f f i c i e n c y f r o n t i e r , to a t t a i n an o v e r a l l p o s i t i o n of F. An J . L i n t n e r , "The Va lua t i on o f Ri sk A s s e t s o p _ . c i t . , pp. 679—681 . In t h i s a r t i c l e i t i s argued that the assumption of quad ra t i c u t i 1 i t y ' f u n c t i o n s by Markowitz and Tobin i s unduly r e s t r i c t i v e . The r e s u l t s obta ined are v a l i d whenever the p r o b a b i l i t y d i s t r i b u t i o n s are mul t i - v a r i ate normal, the u t i l i t y f u n c t i o n maybe polynomial .exponent ia l or any g iven form. This overcomes the apparent ab su rd i t y that the s lope of the demand curve f o r r i s k y investments becomes negat ive and conse-quently, i n f e r r i n g that r i s k y investments are " i n f e r i o r goods. " i n ve s t o r w i t h preferences i n d i c a t e d by . w i l l i nves t a l l h i s fur.ds in the combinat ion E, wh i l e the i nve s to r w i th i n d i f f e r e n c e curves l _ , . 3 J w i l l borrow funds at the r i s k - f r e e r a t e , r^, and i nves t the e n t i r e sum in combinat ion of r i s k y assets E to a t t a i n an o v e r a l l p o s i t i o n of G. On the bas i s of the i n v e s t o r ' s u t i l i t y f unc t i on s he decides how much of h i s weal th to inves t in r i s k assets and consequent ly what p o r t i o n to i nve s t in r i s k l e s s investments. Since parameters of the market oppor-t u n i t y l i n e are expected re tu rn and standard d e v i a t i o n , the i n ve s to r bas i s h i s d e c i s i o n on how much to inves t in r i s k y as set s on the expected r e tu rn and standard d e v i a t i o n of returns on the opt imal p o r t f o l i o . IV CAPITAL MARKET CURVE In e q u i l i b r i u m , p r i c e s of i n d i v i d u a l s e c u r i t i e s ad jus t so that the expected re tu rn of a s e c u r i t y can be expressed as a l i n e a r f u n c t i o n o f some measure of r i s k . Th is f u n c t i o n a l r e l a t i o n s h i p i s expressed in the c a p i t a l market curve. The area o f c o n f l i c t i s the app rop r i a te measure of r i s k . Sharpe ' s model imp l i e s that the standard d e v i a t i o n i s the c o r r e c t measure. J . L i n t n e r po s tu l a te s ' that r i s k c l a s s e s of s e c u r i t i e s should be s ca led by va r i ance o f r e t u r n . The t h i r d v iew expressed by Archer and D'Ambrasio i s that the app rop r i a te measure of r i s k i s the c o e f f i c i e n t of v a r i a t i o n . Standard d e v i a t i o n mode 1. On the assumption that the p o r t f o l i o s of r i s k y investments on Markowi t z ' s e f f i c i e n c y f r o n t i e r are p e r f e c t l y c o r r e l a t e d , Sharpe pos tu -l a t ed an e q u i l i b r i u m model f o r c a p i t a l asset p r i c e s . He argued that by 43 an a r b i t r a g e process the c u r v i l i n e a r (E, ) boundary would become a s t r a i g h t 1 ine. The i n v e s t o r ' s de s i r e s to hold the r i s k y assets in combinat ion E w i l l cause t h e i r p r i c e s to r i s e and consequent ly , t h e i r returns to f a l l . S i m i l a r l y , i n ve s to r s are not i n t e re s t ed in s e c u r i t i e s which are not inc luded in combinat ion E. Consequently, t h e i r p r i c e s w i l l f a l l and expected returns w i l l i nc rease . The o v e r a l l e f f e c t i s that combin-a t i o n E s h i f t downward whereas, combinations adjacent to E wi11 s h i f t upward. Th is process would cont inue u n t i l e q u i l i b r i u m i s reached and the e f f i c i e n c y f r o n t i e r becomes c o i n c i d e n t w i th the market oppo r tun i t y l i n e . A continum of e f f i c i e n t p o r t f o l i o s are now opt imal depending on the p a r t i c u l a r i n v e s t o r ' s preferences f o r r e tu rn and r i s k . Based on t h i s reasoning the conc l u s i on was drawn that c a p i t a l assets were sca led in r i s k c l a s s e s accord ing to expected returns and standard d e v i a t i o n of r e t u r n . Var i ance mode 1. L i n t n e r ' s reasoning proceeds from the premise that on l y one po in t on the e f f i c i e n c y f r o n t i e r needs to be con s ide red , namely the one which y i e l d s the maximum va lue o f In o rder to a r r i v e at the c o r r e c t rank ing procedure i t i s necessary to de r i ve an expres s i on f o r the com-W.F. Sharpe, op_. c i t . J . L i n t n e r , op_. c i t . kk p o s i t i o n ->f the opt imal p o r t f o l i o . Suppose that the i n ve s to r has Incorporated s e c u r i t y i in h i s p o r t f o l i o w i th the parameters E ( x . ) , ^ . -^rand .. ,--the-corvar- iance-between-the 4-th s e c u r i t y - a n d the j t h s e c u r i t y . If X . i s the p ropo r t i on of the t o t a l p o r t f o l i o invested in the i t h - s e cu r i t y , then the expected re tu rn on the p o r t f o l i o of e q u i t i e s i s i and the expected r i s k premium i s "(2) p = x - r f =M_pT<x j - r<l The standard d e v i a t i o n of the p o r t f o l i o ' s f u l l ra te of re turn and of i t s r i s k premium w i l l be the same. Stated mathemat i ca l l y , I (3) Jr= d^HZ S + zz x{ % 4 p ~ I S u b s t i t u t i n g the express ions f o r r i s k premium and va r i ance of t o t a l r e t u rn i n t o equat ion f o r 0 y i e l d s , (4) 4 = -In-order to o b t a i n the -va lues of % [ which maximize 0, d i f f e r e n t i a t e the f u n c t i o n 0 p a r t i a l l y w i t h respect to )f . sub jec t to the c o n s t r a i n t -that (5) H X: = a 2 f' ~ . _ < where K = " ^ 2 Since . for the maximum a t t a i n a b l e 0, a l l the £ f . must have been ad -j u s t e d up or down u n t i l the va lue of i s zero f o r them s imu l taneous l y the set of va lues of X . must s a t i s f y the f o l l o w i n g system of equa t i on s : z.4_. + 2*4* 4 2 ^ 3 4 •• • 4-*,<*2Pa = ?2 45 where 2j = K ^ The nature of the i n d i f f e r e n c e curves between expected r i s k p re -miums and va r i ance can be determined from the above set of s imultaneous equat ions . By t o t a l l y d i f f e r e n t i a t i n g the e q u i l i b r i u m cond i t i o n s when a l l covar iances are constant or i n v a r i a n t i t i s shown that there i s a 46 l i n e a r i n d i f f e r e n c e contour between the expected re tu rn and the va r i ance 12 of the i n d i v i d u a l s e c u r i t y . Subject to the assumption that the covar iances are i n v a r i a n t , i t f o l l o w s that r i s k c l a s s e s of s e c u r i t i e s should be s ca led in terms of va r i ance of returns ra ther than standard d e v i a t i o n s . . C o e f f i c i e n t of v a r i a t i o n mode 1. On h e u r i s t i c and informal bases Archer and D'Ambrasio propose that the c o e f f i c i e n t of v a r i a t i o n i s a b e t t e r measure of r i s k than the standard d e v i a t i o n . ^ The standard d e v i a t i o n does not r e f l e c t the magni-tude of the expected r e t u r n . The c o e f f i c i e n t o f v a r i a t i o n , ' a r e l a t i v e measure of d i s p e r s i o n , enables a meaningful comparison of r i s k e x i s t e n t in d i f f e r i n g investments. V THE TREYNOR INDEX The concept of the p o r t f o l i o p o s s i b i l i t y l i n e developed by Treynor summarizes the performance of a fund In terms of the imp l ied t r a d e - o f f between r i s k and expected r e t u r n . The model" uses v o l a t i l i t y of returns of a fund as a measure of i t s r i s k ra ther than v a r i a b i l i t y of 14 r e tu rn s . S ince the returns on a l l d i v e r s i f i e d p o r t f o l i o s tend to move 12 J. L i n t n e r , op. c i t . , pp. 35—36. 13 Stephen H. Archer and Char les A. D 'Ambrasio, Bus i ness ,Fi nance: Theory and Management, (The MacMi l lan Company, New York^ 1966), p. 71. 14 Jack L. Treynor, "How to Rate Management of Investment Funds, " Harvard Business Review, 43, ( January—February , 1965). pp. 53 — 79. hi wi th the market the extent to which changes in a f und ' s rate of re turn v a r i e s w i t h the market can stand as a good measure of the t o t a l v a r i -a b i l i t y of the f und ' s re tu rn over t ime. By observ ing t h i s r e l a t i o n s h i p over some past p e r i o d , a reasonably good es t imate of vo l a t i 1 i t y - - t h e change in the ra te of re tu rn on a fund a s soc i a ted w i th a one per cent change in the ra te of re tu rn on, say, the Dow-Jones I n d u s t r i a l A v e r a g e -can be ob ta i ned . The s lope of the c h a r a c t e r i s t i c l i n e , which r e l a t e s the expected re tu rn of a fund to the market rate of r e t u r n , measures the r i s k or v o l a t i l i t y , W., of the fund. FIGURE 3.5 CHARACTERISTIC LINE Market of Return (%) The v o l a t i l i t y and expected re tu rn of the fund are combined in the p o r t f o l i o p o s s i b i l i t y l i n e to measure the degree of r i s k ave r s i on e x h i b i t e d by the fund. The s lope o f the l i n e i s a measure of fund performance which transcends d i f f e r e n c e s in the i n v e s t o r ' s a t t i t u d e s U8 .toward r i . i k . A p o r t f o l i o p o s s i b i l i t y l i n e i s i l l u s t r a t e d in F igure 3.6. FIGURE 3.6 PORTFOLIO POSSIBILITY LINE R i s k ( % ) In a p e r f e c t c a p i t a l market the locus of p o s s i b l e combinations o f investment a v a i l a b l e to the i nve s to r i s a s t r a i g h t l i n e . Th is corresponds to the market oppo r tun i t y l i n e developed e a r l i e r . The p r e f e r r ed f und - - the po in t o f tangency between the p o r t f o l i o p o s s i b i l i t y line and the i n v e s t o r ' s i n d i f f e r e n c e c u r v e — w i l l d i f f e r f o r each i n ve s to r depending on the shape and p o s i t i o n o f h i s i n d i f f e r e n c e curves . A r i s k averse i n ve s to r would p r e f e r fund G to fund C, s i nce f o r the same l e v e l of r i s k fund G o f f e r s a g rea te r expected r e t u r n . The p o r t f o l i o p o s s i b i l i t y l i n e i s a ray j o i n i n g the r i s k f r e e i n t e r e s t rate, r^., and the fund; The steepness o f the l i n e i s thus, a d i r e c t measure of the d e s i r a b i 1 i t y of the fund to the r i s k averse i n ve s t o r . The s lope of the l i n e i s r e f e r r e d to as the Treynor Index and i s g iven by T. I . = Tangent <X = X i ^ r f , i where X. i s the expected re tu rn f o r i t h fund, i s the r i s k f r e e rate of i n t e r e s t and W. is the f und ' s v o l a t i l i t y — a measure of r i s k , i VI LIMITATIONS Several h i g h l y r e s t r i c t i v e assumptions have been employed in deve lop ing the theory. The assumption of pe r f e c t c a p i t a l market, which forms a keystone in the theory , stands seve re l y c r i t i c i z e d . . The i n ve s to r i s assumed to be ab le to borrow or lend as much money as he wants at a s i n g l e r a te . Th i s , however, i s not as r e s t r i c t i v e as i t , at f i r s t , appears to be. Most i nve s to r s w i l l be s u f f i c i e n t l y r i s k averse that t h e i r e q u i l i b r i u m p o s i t i o n w i l l i n vo l ve ho ld ing both r i s k l e s s assets and stocks w i thout borrowing. Others who.are w i l l i n g to accept g reate r degrees of r i s k but w i l l not borrow beyond the amounts a v a i l a b l e wi thout i n c rea s i n g the r a t e . For a l l these i n v e s t o r s , the assumption has no bear ing on the r e s u l t s of the a n a l y s i s . The remaining i nves to r s who d e s i r e to use ex ten s i ve leverage, the i n t e r e s t ra te would be expected to i nc rease . The e f f e c t of which would cause the curve to bend down-wards at high l e v e l s of r i s k . - • Investment markets are a l s o imper fec t , in the sense that a l l i nve s to r s do not possess pe r f e c t knowledge, l ega l and i n s t i t u t i o n a l r e s t r i c t i o n s e x i s t , and some i nve s to r s are s u f f i c i e n t l y large to be ab le to i n f l uence the market. The g rea te r the market imper fec t ions the g rea te r w i l l be the s c a t t e r about the c a p i t a l market curve. The arguments put f o r t h by Sharpe, Archer and D'Ambrasio in t h e i r r e spec t i ve c a p i t a l market models lack mathematical r i g o r . They are h e u r i s t i c and informal r a the r than formal and a n a l y t i c a l . However, L i n t n e r ' s model c e r t a i n l y , meets the t e s t s of mathematical r i g o r . C l e a r l y , in us ing ex post performance of s e c u r i t i e s as surrogates f o r ex ante performance c reates the danger that the a n a l y s i s w i l l not prov ide adequately f o r the tendency of the p r i c e mechanism to ad jus t over time (and perhaps to over ad jus t ) to whatever r e l evan t i n fo rmat ion i s conta ined in past trends and f l u c t u a t i o n s in average re tu rn s . Investor preferences and investment o p p o r t u n i t i e s presumably change over t ime. This poses a major problem: i f the e q u i l i b r i u m va lues of r i s k f r e e i n t e r e s t rate and r i s k premium change from year to year , i t may be dangerous to use data from severa l years to es t imate t h e i r va lue s . VI I SUMMARY Based on the premise that i nves to r s are r i s k averse, the concept of the e f f i c i e n c y f r o n t i e r was developed. I nd i f f e rence cu rves , which summarize i n v e s t o r ' s preferences f o r r i s k and r e t u r n , were combined w i t h the e f f i c i e n c y f r o n t i e r to a s c e r t a i n the opt imal p o r t f o l i o f o r an i n ve s t o r . The ex ten s i on of Markowi tz ' s theory made by Tob in , who cas t i t in a p o s i t i v e r o l e , shows that the market oppo r tun i t y l i n e i s a f u n c t i o n of the p o r t f o l i o ' s expected re tu rn and standard d e v i a t i o n of r e tu rn s . Combining the i n v e s t o r ' s i n d i f f e r e n c e curves and the market oppo r tun i t y l i n e enabled the quest ion of how much of h i s assets an i n ve s to r would 51 i nves t in a s tock p o r t f o l i o . The next s e c t i o n presented the theory of the capi t a l market cu rve : the f u n c t i o n a l r e l a t i o n s h i p between expected re tu rn on i n d i v i d u a l s e c u r i t i e s and t h e i r r i s k . The three models, standard d e v i a t i o n , v a r i -ance, and c o e f f i c i e n t o f v a r i a t i o n f o r s c a l i n g r i s k c l a s se s of s e c u r i t i e s were presented. The f i n a l s e c t i o n o u t l i n e d the more s t r i n g e n t l i m i t a t i o n s of the theory . CHAPTER IV EMPIRICAL ANALYSIS This chapter presents the e m p i r i c a l t e s t s of the models f o r the s c a l i n g of r i s k c l a s s e s of s e c u r i t i e s and the de te rm ina t i on of r i s k premiums f o r Canadian and American open-end investment companies. The t e s t samples of twenty -e ight open-end investment companies were prepared us ing annual p r i c e and y i e l d data f o r the pe r iod 1957 to 1966. Before p re sen t i ng a d e s c r i p t i o n of the e m p i r i c a l f i n d i n g s ; some of the c on s i de r a t i on s in generat ing the input data are presented. I INPUT DATA Annual s e c u r i t y data f o r the per iod 1957 to I966 were used to develop inputs f o r each of the models. The sources of t h i s i n fo rmat ion were the F i n a n c i a l P o s t ' s Survey o f Investment Companies^ f o r Canadian 2 and Ar thur Weisenberger ' s Investment Companies f o r American mutual funds. Although t h i s inc luded over 100 Canadian and approx imate ly 200 American mutual funds, on l y 28 Canadian funds o f f e r e d shares to the p u b l i c over the e n t i r e pe r iod 1957 to 1966 i n c l u s i v e . Consequently, the t e s t samples were r e s t r i c t e d to 28 comapnies. F i n a n c i a l Post , Survey o f Investment Companies, (MacLean-Hunter, 1967). 2 Ar thur Weisenberger, Investment Companies 1967, (Arthur Weisenberger and Company, 1967). 53 .A - t rade-of f was necessary between the number o f comapnies conta ined in the samples and the per iod over which data were c o l l e c t e d . The time span must be long enough to smooth out market trends and to o b t a i n a meaninful f requency d i s t r i b u t i o n o* r e t u rn s , yet short enough to o b t a i n a sample s i z e large enough to make the reg re s s i on a n a l y s i s , mean ing fu l . A nine year pe r iod was cons idered to be adequate to generate a meaningfu l f requency d i s t r i b u t i o n of r e tu rn s , s ince the span conta ins at l e a s t three ' bus iness c y c l e s . 1 The time span was decreased from ten — y e a r s - t o nine years in o rder - to- increase the number of Canadian funds from 19 to 28. In c o l l e c t i n g data f o r two economies over t ime, po s s i b l e d i f f e r e n c e s in bus iness c y c l e s r a i s e s the quest ion of whether the data f o r two p a r t i c u l a r economies over time i s comparable. In comparing the U.S. and Canadian economies over the per iod 1957 to 1966 both conta ined three bus iness c y c l e s . The main d i sc repency being in the ampl itude of the swings as shown in F igure 4 . 1 . On t h i s bas i s the data were assumed to be comparable. To measure the y i e l d f o r a fund in any year , both the market a p p r e c i a t i o n and d i v idends were cons ide red . Annual y i e l d s were c a l c u -l a t ed in the f o l l o w i n g manner: - - ( X . ) L = ( N . A . V . . ) t - (N.A.V.,) t _ -| + x 100 (N.A.V..) t _ ] —where ( X j ) t i s the y i e l d of the i t h fund in year t . C l e a r l y , t h i s i s on l y an approximate measure of the rate o f r e t u r n . It neg lect s such s u b t l i t i e s as the t im ing w i t h i n the per iod of d i v i dend d i s t r i b u t i o n , the 210 (100) 190 (90) \ Index 170 - (80) \ 150 (70) FIGURE k.1 CANADIAN AND U.S.'BUS I NESS CYCLES' Standard and 59 1960 61 62 63 64 65 Year Source—Bank of Canada, S t a t i s t i c a l Summary, I965 and 1966. r e l a t i v e a f t e r - t a x va lue to the shareholder of market a p p r e c i a t i o n , the d i v i dend i n t e r e s t income, and whether the value of funds a v a i l a b l e f o r investment f l u c t u a t e s more o r les s con t i nuous l y throughout the year. Annual y i e l d s were computed f o r each of the funds f o r the per iod 1957 to 1966. These annual rates of re tu rn were than used to generate the expected re tu rn and the s t a t i s t i c a l measures o f d i s p e r s i o n f o r each fund The American mutual funds were s e l e c ted from Table 18 of Ar thur Weisenberger ' s Investment Companies. The sample was s e l e c t ed by the use Ar thur Weisenberger, op_. c i t . p. 1 \k. 55 of a ranc'om number t ab l e from the funds l i s t e d , except f o r s p e c i a l i z e d funds—Canodian and I n te rna t i ona l i s s ue s - - which e x i s t e d p r i o r to 1957. II RESULTS AND ANALYSIS If the hypothes i s of i n c rea s i n g returns i s v a l i d , then the expected re tu rn should be p o s i t i v e l y c o r r a l a t e d w i th the r e spec t i ve measure of d i s p e r s i o n : funds expe r i enc i ng g rea te r v a r i a b i l i t y should prov ide g reate r expected r e tu rn . By means of l e a s t squares reg re s s i on a n a l y s i s , the pos tu la ted l i n e a r r e l a t i o n s h i p , the c a p i t a l market cu rve , between expected returns and the three measures of d i s p e r s i o n was examined. Thus, the hypothes i s to be te s ted by s imple c o r r e l a t i o n and k reg re s s i on a n a l y s i s can be formulated as f o l l o w s : EXPECTED RETURN = r f + B RISK where r^ . i s the r i s k f r e e rate of i n t e r e s t , RISK being measured by the d i s p e r s i o n o f r e tu rn s , and B i s the r i s k premium. The constant r^ . i s the y i n t e r cep t of the leas t - squares e s t i m a t i n g equat ion . The r i s k premium, B, i s the s lope of. the reg re s s i on l i n e . Tables IV and V show the r e t u r n - - r i s k parameters f o r the Canadian and American mutual fund samples r e s p e c t i v e l y . Since both the expected re tu rn and measures of r i s k are c a l c u l a t e d from the frequency d i s t r i b u -t i o n of annual rates o f r e t u r n , any change in investment re tu rn w i l l The regress ions f o r t h i s study were done by a l i b r a r y program, TRIP at the U n i v e r s i t y of B r i t i s h Co 1umbia 1 s Computing Centre. 56 i n tu rn a l t e r the va lues of both expected re tu rn and r i s k . Since both expected re tu rn and r i s k are random v a r i a b l e s , ne i t he r can be unequi -v o c a l l y des ignated as the independent v a r i a b l e . Consequently, the app rop r i a te l e a s t squares reg re s s i on l i n e presumably l i e s between the reg re s s i on l i n e s formed by the reg re s s i on of expected returns on r i s k and r i s k on expected re tu rn s . Such a l i n e i s s p e c i f i e d by the average o f the s lopes of the two reg re s s i on l i n e s and passes through t h e i r i n t e r s e c t i o n . The reader i s r e f e r r e d to Table X, in Appendix A, f o r a summary of the s p e c i f i c s t a t i s t i c s . Cap?tal Market Curve. The parameters f o r the reg re s s i on and intermediate l i n e s cor respond ing to each of the measures of d i s p e r s i o n f o r the Canad ian, 1 U.S., and combined samples are shown in Tables IV and V I I I . Standard Dev i a t i on Model. Table III shows the r e s u l t s f o r the three samples us ing the standard d e v i a t i o n as a measure o f r i s k . F igures A . I , A.2 and A.3 i l l u s t r a t e the reg re s s i on l i n e s and s c a t t e r -grams g r a p h i c a l l y . The c o e f f i c i e n t s of c o r r e l a t i o n between expected r e tu rn and standard d e v i a t i o n of returns are +0.725, +0.816 and +0.849, which are h i g h l y s i g n i f i c a n t and c o n s i s t e n t w i th the assumption of r i s k a v e r s i o n . Moreover, the F - r a t i o f o r these l ea s t squares reg re s s i on equat ions are s i g n i f i c a n t at the 1 per cent l e v e l , meaning that the p r o b a b i l i t y i s less than 0.01 that a f i t as good as those observed cou ld have happened by chance. The reg re s s i on l i n e s e x p l a i n more than 52 per cen t , 66 per cent and 72 per cent r e s p e c t i v e l y of the t o t a l v a r i a b i l i t y in the Canadian, American, and combined samples. The p re -57 d i e t e d f u n c t i o n a l r e l a t i o n s h i p i s c l e a r l y p r e s e n t . However, the m o d s l , f o r t h e s e samples o f d a t a , f a i l s t o y i e l d m e a n i n g f u l i n t e r c e p t s . 'The r i s k f r e e i n t e r e s t r a t e s p r e d i c t e d a r e 0 . 0 779 , 0.161 and - 1.750 per c e n t f o r the t h r e e s a m p l e s , w h i c h d i f f e r s u b s t a n t i a l l y f r o m the o b s e r v e d r a t e o f a p p r o x i m a t e l y 4 to 5 per c e n t . V a r i a n c e Mode 1. The s t a t i s t i c a l t e s t s f o r t h i s model y i e l d n e a r l y i d e n t i c a l r e s u l t s as f o r the s t a n d a r d d e v i a t i o n model. The s t a t i s t i c s a r e shown i n T a b l e VII f o r the t h r e e s a m p l e s : and a r e i l l u s t r a t e d g r a p h i c a l l y i n F i g u r e s A.4, A.5 and A.6. The c o e f f i c i e n t s o f c o r r e l a t i o n a r e l a r g e and p o s i t i v e . The p e r c e n t a g e s o f t o t a l v a r i a b i l i t y e x p l a i n e d b y t h e r e g r e s s i o n l i n e s were 49 per c e n t , 65 per c e n t and 73 per c e n t . The computed F - r a t i o s g r e a t l y e xceed the t h e o r e t i c a l v a l u e o f F a t the one per c e n t l e v e l o f s i g n i f i c a n c e , t h e r e f o r e , t h e p r o b a b i l i t y i s much s m a l l e r t h a n 0.01 t h a t the o b s e r v e d r e l a t i o n s h i p between E and c o u l d have o c c u r e d by chance. The i m p o r t a n t d i f f e r e n c e i n the e m p i r i c a l r e s u l t s between the v a r i a n c e and s t a n d a r d d e v i a t i o n models i s t h a t the f o r m e r y i e l d s m e a n i n g f u l i n t e r c e p t s , w h e r e a s - t h e l a t t e r f a i l e d t o do so. The r i s k - f r e e i n t e r e s t r a t e , r ^ . , — w h i c h c o r r e s p o n d t o the i n t e r c e p t s o f t h e i n t e r m e d i a t e r e g r e s s i o n l i n e s f o r the t h r e e s a m p l e s — o f 3.95. 5 -99 and 3 .92 per c e n t a r e c o n s i s t e n t w i t h t h o s e o b s e r v e d i n p r a c t i c e ( y i e l d on 30 day t r e a s u r y b i l l s ) . C o e f f i c i e n t o f V a r i a t i o n Model. T h i s model i s not s u p p o r t e d by the e m p i r i c a l r e s u l t s , shown i n T a b l e V I M . The c o r r e l a t i o n c o e f f i c i e n t s a r e s m a l l and n e g a t i v e : - 0.122, -0.261 and - 0.412 f o r C a n a d i a n , U.S. and 58 and combined samples, r e s p e c t i v e l y . Tnly 1.5%, 7% and 16% of the t o t a l v a r i a b i l i t y in the three samples were exp la i ned by the reg re s s i on l i n e s . Table IX shows the grouping of Car.adian and American mutual funds , forming the samples, accord ing to t h e i r s ta ted investment p o l i c i e s . Several deduct ions can be drawn from Tables I V , V , and IX. F i r s t , the funds which are c l a s s i f i e d as h i g h l y v o l a t i l e in general produce g rea te r expected returns and co r re spond ing l y , exper ience g rea te r v a r i -a t i o n o f returns and v i c e ve r sa . Second, an exp l ana t i on f o r the r i s k -f r e e i n t e r e s t ra te p r ed i c t ed by the va r i ance model f o r the U.S. sample i s h igher than expected, iThe sample was dominated by common stock furids--21 out of the 28 funds forming the sample. As a r e s u l t of t h i s heavy c oncen t r a t i on of high r e t u r n — h i g h r i s k funds, the reg re s s i on l i n e was r o ta ted c l o ckw i se caus ing the i n t e r c e p t to inc rease . This reasoning i s f u r t h e r s ub s t an t i a t ed by the reg re s s i on a n a l y s i s of the Canadian and American funds combined: the r i s k - f r e e i n t e r e s t ra te f o r the combined markets was 3-92 per cent . T h i r d , that the Canadian and U.S. markets are one and the same—th i s i s a l s o shown by the f a c t that the c o e f f i c i e n t o r c o r r e l a t i o n f o r the combined markets increased to 0.852 from 0.808 and 0.690 f o r the separate U.S. and Canadian samples. F i n a l l y , the reason why the mean return f o r the sample of American mutual funds i s s i g n i f i c a n t l y g rea te r than f o r the sample of Canadian funds, 12.68 per cent compared to 7.928 per cent r e s p e c t i v e l y , the Canadian sample was concentrated in high r i s k - h i g h re tu rn funds. From these data two conc lu s ions may be drawn, s u b j e c t , of course, TABLE IV 59 RATES OF RETURN AND VARIABILITY: CANADIAN MUTUAL FUNDS Mean of Standard Annaul Dev i a t i on Var i ance Rates of of Rates of Rates Return of Return of Return 1957-1966 1957-1966 1957-1966 Mutual Fund ( E , ) (<5V ( <f.2) 1. A l l Canadian Compound Fund 8.990 12.446 159.902 2. A l l Canadian Div idend Fund 9.397 13.665 186.732 3. American Growth Fund L td . 11.531 17.448 304.432 4. As soc i a te Investors L td . 7.990 12.336 156.177 5. Beaubran Corporat ion 7.171 9.958 99.162 6. Canadian Investment Fund L td . 7.417 10.732 115.176 7. Canafund Company L td . IO.656 14.155 200.364 8. Champion Mutual Fund of Canada L td . 6.346 11.658 135.909 9. Commonwealth I n t ' l L t d . 7.301 11.376 129.413 10. Commonwealth I n t ' l Leverage Ltd . 8.994 15.979 249.545 11. Corporate Investors L td . 7. 122 9.426 88.849 12. Dominion Equ i ty Investments Ltd . 8.521 9.581 91.746 .13. Fonds C o l l e c t i f " A " 6.170 8.915 79.477 14. Grouped Income Shares L td . 6.175 13.869 192.349 15. Investors Growth Fund of Canada L td . 7.428 10.575 111.831 16. Investors Mutual of Canada L t d . 7.340 9.763 95.316 17. Mar i t ime Equ i ty Fund 10.874 17.989 323.604 18. Mutual Accumulat ing Fund 8.203 10.757 : 115.713 19. Mutual Bond Fund 4.255 4.090 16.728 20. Mutual Income Fund 7.898 10.548 111.260 21. Savings and Investments Corpor-a t i o n Mutual Fund of Canada Ltd . 6.432 9.749 95.043 22. Timed Investment Fund L td . 7.030 10.929 119.443 23 . United Accumulat ive Fund 8.767 12.935 167.314 24. D i v e r s i f i e d income Shares Ser ies "A ' 1 8.893 13.959 194.854 25. D i v e r s i f i e d Income Shares Ser ies " B " 7.960 13.516 182.682 26. Trans-Canada Shares Ser ies " C " 6.817 14.061 197.712 27 . Trans-Canada Shares Ser ies " A " 6.718 15.477 239.538 28. Trans-Canada Shares Ser ies " B " 6.801 13.866 192.266 TABLE V 60 -RATES IF RETURN AND VARIABILITY: AMERICAN MUTUAL FUNDS Mean of Standard Annual Dev i a t i on Var i ance 'Rates of of Rates of Rates Return o f Return of Return 1957-1966 1957-1966 1957-1966 Mutual Fund (E,) ( <f.) ( ^ 2 ) 1. ""Delaware Fund 15.744 19.255 370.755 2. Morton (B.C.) Growth Ser ies 16.156 19.058 363.207 23. Investment Trust of Boston 11.044 17.181 295.187 4. Kn ickerbocker Growth Fund 16.944 22.987 528.402 5. T e l e v i s i o n - E l e c t r o n i c s Fund 15.222 20.387 415.630 6. W i n f i e l d Growth Fund 17.222 20.692 428.159 7. United Accumulat ive Fund 12.867 15.403, 237.252 8: "Commonwealth Investment Co. 8.700 11.391 129.755 9. Putman Investors Fund 9.767 18.171 330.195 .40. - Insurance Investors Fund 15.222 20.897 .' 436.185 11. S e c u r i t y Investment Fund M. 7.711 11.061 122.346 12. .Commonwealth Stock Fund 10.956 15.747 247.968 13. Fundamental Investors 12.589 17.543 307.757 14. Putman Income Fund 10.733 14.011 196.308 15. Axe Science Corporat ion 12.767 16.163 261.242 16. Eaton and Howard Balanced Fund 7.877 10.026 100.521 17. American Mutual Fund 13.156 14.325 205.206 18. de Vegh Mutual Fund 13.322 16.713 ,279.324 19. Washington Mutual Investors Fund 13.756 17.412 393.178 20. Corporate Leaders T r u s t — Ser ie s " B " 11.178 16.495 272.085 21. Keystone (K - l ) Income Fund 10.211 10.980 . 120.560 22. Fund of America (c) 13.178 20.408 416.486 23. F i n a n c i a l I n d u s t r i a l Fund 12.667 . 17.746' 314.921 24. Texas Fund 12.422 15.799 249.608 ' 25. D i v e r s i f i e d Growth Stock Fund 16.044 23.874 569.968 26. I s t e l Fund 13.456 11.880 139.240 27. Lex ington Income Trust 9.056 12.898 166.358 28. Massachusetts Investors "Growth Stock Fund 15.133 18.219 331.932 61 TABLE V! REGRESSION ANALYSIS; STANDARD DEVIATION MODtZL REGRESSION LINE F -Rat io CANADI AN E on <_T <J»on E Intermed i ate AMERICAN E o n t f ^ o n E I ntermed i ate CANADIAN PLUS AMERICAN E on*5" d ' on E I ntermed i ate 2.966 0.403 •1.490 O.766 0 .779 0 .584 2.672 0.601 -2.349 0. 161 0.204 •3.670 •1.750 0 .900 0.751 0 .698 0 .969 0 .834 0 .725 0.725 0 .816 0 .816 0.849 0.849 0.526 28;862 0 .526 28.862 0 .667 O.667 0.721 0.721 51 .826 51 .826 139.043 139.043 Level of S i g n i f i c a n c e 0.01 0.01 0.001 0.001 Degrees of Freedom ( 1 , 27) ( 1 , 55) ( 1 , 27) ( 1 , 55) F Value 7.68 7.12 13.5 12.2 62 TABLE VI I REGRESSION ANALYSIS: VARIANCE MODEL REGRESSION LINE r f B R R2 F - R a t i o " CANAD1 AN E on C ^ 2 5.397 0.016 0.690 0.476 23.613 <52 on E 2.739 0.033 0.690 0.476 23.613 1ntermed i ate 3.950 0.025 AMERICAN E on <5~ 2 7.502 0.018 0.808 0.653 48.955 t_r2 on E 4 . 750 0.027 0.808 0.653 48.955 1ntermed i ate 5.990 0.223 CANADIAN PLUS AMERICAN E o n 2 <=5"2 5.057 0.023 O.852 0.726 142.439 cr5^ on E 3.0860 0.032 0.852 0.726 142.439 Intermediate 3.919 0.028 Level of S i g n i f i c a n c e Degrees of Freedom F Value 0.01 (1 , 27) 7.68 0.01 ( 1 , 55) 7.12 0.001 (1, 27) 13.5 0.001 ( 1 , 55) 12.2 63 TABLE VIII REGRESSION ANALYSIS: COEFFICIENT OF VARIATION MODEL r 2 Regress ion L ine f B R R F -Ra t i o ' CANADIAN : . E o n <5*/E 8.793 - 0 . 6 20 -0.122 0 .015 0 .374 <5" /E on E 72.708 -41.667 -0.122 0 .015 0 .374 AMERICAN E on <57E 17.498 - 3 . 653 -0.161 0 .066 1.908 & /E onE 81.890 - 5 2 . 667 -0 .161 0 .066 1.908 CANADIAN PLUS AMERICAN E o n <^/E 17.301 - 4 . 8 8 5 -0.412 0 .170 11.045 CS /E on E 51.400 -28.571 -0.412 0 .170 11.045 * L e v e l of S i g n i f i c a n c e Degrees of Freedom F Value 0 .05 ( 1 , 27) 4.21 0.05 ( 1 , 55) 4 . 02 0.01 ( I , 27) 7.68 0.01 ( ] , 55) 7.12 64 TABLE IX CLASSIFICATION OF MUTUAL FUND BY INVESTMENT POLICY Canad ian American .1. 2. I. 2. 3. 4. 5. 6. 7. 8. 9. 10. S p e c i a l t y American Growth Fund Canafund L t d . Common Stock Investors Growth Fund o f Canada L t d . F u l l y Managed A l l Canadian Compound Fund A l l Canadian Div idend Fund A s soc i a te Investors L td . Beaubran Corpora t ion Canadian Investment Fund L td . Dominion Equ i ty Investment L td . Grouped Income Shares L t d . Mar i t ime Equ i ty Fund Mutual Accumulat ing Fund United Accumulat ive Fund 1. ' 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 2 1 . 1. 2. 3. S p e c i a l t y Insurance Investors Funds Common Stock Delaware Morton (B.C.) Growth Se r ie s Investment Trust of Boston Kn ickerbocker Growth Fund T e l e v i s i o n E l e c t r o n i c s Fund W i n f i e l d Growth Fund United Accumulat ive Putman Investors Funds Commonwealth Stock Fund Fundamental Investors Axe Science Corpora t ion American Mutual de Vegh Mutual Fund Washington Investors Fund Corporate Leaders T r u s t - - S e r i e s I I Q I I Fund of American (c) F i n a n c i a l I n d u s t r i a l Fund Texas Fund D i v e r s i f i e d Growth Stock Fund Iste ' l Fund Massachusetts Investors Growth Stock Fund F l e x i b l e S e cu r i t y Investment Fund M Putman Income Fund Keystone (K - l ) Income Fund 6 5 . TABLE IX (continued) Canadian American Balanced Fund 1. Champion Mutual Fund of Canada 1. L t d . 2. 2. Commonwealth I n t ' l L t d . 3. 3- Commonwealth I n t ' l Leverage L t d . k. Corporate Investors L td . 5. Fonds C o l l e c t i f " A " 6. Investors Mutual of Canada L td . 7. Savings and Investments Corpor-a t i o n Mutual Fund of Canada L td . 8. Timed Investment Fund L td . 9. D i v e r s i f i e d Income Shares Se r ie s " A " 10. D i v e r s i f i e d Income Shares Ser ie s " B " 11. Trans-Canada Ser ie s " A " 12. Trans-Canada Ser ie s " B " . 13. Trans-Canada Ser ie s " C " Bonds 1. Mutual Bond Fund 2. Mutual Bond Income Fund Balanced Fund Commonwealth Investment Company Eaton and Howard Balanced Fund Lex ington Income Trust * F ixed t r u s t funds are grouped w i th balanced fund. 66 to the l i m i t e d s i z e and nature of the samples. F i r s t , the expected returns and r i s k are p o s i t i v e l y c o r r e l a t e d . This means that funds which exper ience g rea te r v a r i a t i o n s in re tu rn y i e l d h igher rates of r e t u r n . Second, the' c a p i t a l market curve i s a 1inear f u n c t i o n of expected returns and va r i ance o f r e t u rn s , ra ther than the standard d e v i a t i o n or the c o e f f i c i e n t of v a r i a t i o n . R i sk premi urns. In the prev ious s e c t i o n the conc l u s i on drawn was that expected re tu rns were p o s i t i v e l y c o r r e l a t e d to va r i ance of returns or a l t e r n a -t i v e l y , that r i s k c l a s s e s of s e c u r i t i e s are sca led by expected returns and va r i ance of r e tu rn s . The quest ion which remains to be answered i nvo l ve s the de te rm ina t i on of the r i s k premiums demanded by the i nves to r s f o r assuming t h i s a d d i t i o n a l r i s k . The r i s k premiums demanded and rece ived by Canadian and American mutual fund i n ve s to r s are g iven by the s lopes of the in termed iate reg re s s i on l i n e s ( c a p i t a l market curve w i th r i s k measured by va r i ance o f r e t u r n s ) . To take on an a d d i t i o n a l un i t of r i s k the Canadian i nves to r requ i red (and rece ived) an increase in expected re turn o f 0.025 per cent . The American i n ve s t o r requ i red (and rece ived) an a d d i t i o n a l 0.023 per cent in expected r e tu rn f o r each a d d i t i o n a l u n i t of r i s k assumed. The ques t ion remaining to be answered i s whether a r i s k premium of 0.025 per cent i s s i g n i f i c a n t l y g rea te r than a r i s k premium of 0.023 per cent? S ince the sample s i z e s are less than 30, i t i s necessary to invoke smal l sampling theo ry : " S t uden t s " t d i s t r i b u t i o n . The c o e f f i c i e n t s 67 B-.,s,P ana 8,, a r e , in f a c t , random v a r i a b l e s w i th means and standard CDN U.S. dev i a t i o n s of 0 . 0 2 5 , 0 . 0 2 3 , 0 .0169 and 0 .032 r e s p e c t i v e l y . Examine the n u l l hypothes i s that the reg re s s i on c o e f f i c i e n t s are the same. H 0 : = B U S ' a n c ' t n e r e ' s n o e s s e n t i a l d i f f e r e n c e between ' ' the two c o e f f i c i e n t s . H, : B.,.., ^ B,, _ , and there i s a s i g n i f i c a n c e d i f f e r e n c e between 1 CDN U.S. ., „ . . , the two c o e f f i c i e n t s . On the bas i s of a two t a i l e d t e s t at a 0.01 l e v e l of s i g n i f i c a n c e H n would be r e j e c t ed i f the c a l c u l a t e d t va lue were ou t s i de the range - t g to t-j which f o r 54 degrees of freedom is the range - 2 . 6 8 to +2.68. The computed va lue of t i s 1 - BCDN " B U.S. where NCDN N U.S. 2 2 & " 1 CDN S CDN + N U.S. S U.S. NCDN + N U.S. " 2 The d i s t r i b u t i o n of t i s S tudent ' s d i s t r i b u t i o n w i th N--.. + N.. _ - 2 CDN U.S. degrees of freedom. 6 = 0 . 0 1 5 5 t = 0.483 Thus, the n u l l hypothes i s cannot be r e j e c t ed at the 0.01 l e ve l of s i g n i f i c a n c e . The r i s k premiums requ i red by Canadian i nve s to r s in open-end investment companies is not s i g n i f i c a n t l y d i f f e r e n t from that requ i red by t h e i r American coun te rpa r t s . -Summary. This chapter has s t a t i s t i c a l l y examined the three c a p i t a l market 68 curves developed in the proceding chapter by means of reg re s s i on a n a l y s i s . On the bas i s of these samples f o r the per iod cons idered the va r i ance model of the c a p i t a l market was .Nhown to be the most c o n s i s t e n t w i t h the e m p i r i c a l ev idence. It was consequently shown that expected returns and v a r i a t i o n s of returns were p o s i t i v e l y c o r r e l a t e d . The r i s k f r e e i n t e r e s t r a te was determined to be approx imately four per cent . On the bas i s o f conc lu s i on s drawn w i th respect to the c a p i t a l market curve the e m p i r i c a l r e s u l t s i nd i c a t ed that Canadian inves to r s in open-end investment companies are ne i t he r more nor less r i s k averse than t h e i r American coun te rpa r t s . CHAPTER V SUMMARY AND CONCLUSIONS The purpose of t h i s study has been to examine s t a t i s t i c a l l y a segment of the c a p i t a l market namely, mutual funds. Three hypotheses were exp l o red . S p e c i f i c a l l y , that (a) expected returns are an i n c r e a s -ing f u n c t i o n of r i s k (b) the va r i ance i s the app rop r i a te measure of r i s k and (c) the Canadian i nves to r s in open-end investment companies do not e x h i b i t g rea te r r i s k ave r s i on than t h e i r American coun te rpa r t s . The technique employed to t e s t the hypotheses was l e a s t squares reg re s s i on a n a l y s i s o f samples of Canadian and American open-end invest ment companies. The input data f o r the study has placed r e s t r i c t i o n s on the i n t e r p r e t a t i o n of r e s u l t s and the conc lu s i on s reached. The data was inadequate w i th respect to sample s i z e s and the concen t r a t i on of the funds in s p e c i f i c r i s k c l a s s e s . Chapter II presented a t h e o r e t i c a l framework of the investment process . Chapter III developed the concept of the c a p i t a l market curve The f i r s t s e c t i o n presented an exp l ana t i on of Markowi tz ' s e f f i c i e n c y f r o n t i e r . The next s e c t i o n seveloped the l i n e a r r e l a t i o n s h i p between expected returns and standard d e v i a t i o n of r e tu rn s . Th is l i n e was des ignated the market oppo r tun i t y l i n e and represents the t r a d e - o f f between a r i s k l e s s asset and a s i n g l e r i s k y s e c u r i t y (or a g iven mix of r i s k assets to be he ld in f i x e d p r o p o r t i o n s ) . Any p o r t f o l i o w i l l g ive r i s e to a l i n e a r boundary of E, ctT comb i na t i on s . By i n v e s t i n g in the i t h p o r t f o l i o and borrowing o r lend ing at the r i s k l e s s r a t e , r f , an 70 i n ve s to r -.an a t t a i n any po int a long t h i s l i n e . The f i n a l s e c t i o n presents th= concept of the c a p i t a l market curve, the f u n c t i o n a l r e l a t i o n s h i p between expected returns and r i s k , and the reasoning behind the three measures of r i s k proposed. The e m p i r i c a l r e s u l t s demonstrated that expected returns and r i s k are p o s i t i v e l y c o r r a l a t e d . The c o e f f i -c i e n t s of c o r r e l a t i o n were +0.849 and +0.852 r e s p e c t i v e l y f o r combined Canadian and U.S. samples us ing the standard d e v i a t i o n and va r i ance of returns as measures o f r i s k . The economic i n t e r p r e t a t i o n i s that us ing the standard d e v i a t i o n as a measure of r i s k the reg re s s i on l i n e exp la i ned 72 per cent of the t o t a l v a r i a b i l i t y and us ing the va r i ance as a measure of r i s k i t exp la i ned 73 per cent of the t o t a l v a r i a b i l i t y . The p o s i t i v e s igns i n d i c a t e that high values of expected re turn are a s soc i a ted w i th high values r i s k and low returns are a s soc i a ted w i th low r i s k . Thus, the hypothes i s that expected returns and r i s k are p o s i t i v e l y c o r r e l a t e d was accepted. The reg re s s i on a n a l y s i s of the standard d e v i a t i o n model of the c a p i t a l market curve y i e l d e d s t a t i s t i c a l l y s i g n i f i c a n t r e s u l t s at the 0.01 l e ve l of s i g n i f i c a n c e . However, i t d i d not y i e l d a meaningful i n t e r c e p t . On t h i s b a s i s , the model was r e j e c t e d . The c o e f f i c i e n t of v a r i a t i o n was r e j ec ted as a measure of r i s k on the grounds that the c o e f f i c i e n t s of c o r r e l a t i o n were small and negat ive f o r a l l three samples. If a negat ive c o e f f i c i e n t of c o r r e l a t i o n represented the true r e l a t i o n s h i p , then the degree of r i s k assumed would decrease w i th i n c r ea s i n g expected r e tu rn s , which i s con t r a r y to what i s observed i n p r a c t i c e . The va r i ance model produced r e s u l t s c o n s i s t e n t w i th the hypothes i s of i n c r ea s i n g returns and a a d d i t i o n y i e l d e d meaningful i n t e r c e p t s p r e d i c t e d by the theory. Thus, the second hypothes i s to the e f f e c t that r i s k c l a s s e s of s e c u r i t i e s are sca led accord ing to expected returns and va r i ance of returns or a l t e r n a t i v e l y , the c a p i t a l market curve i s a l i n e a r f u n c t i o n of expected returns and va r i ance of r e t u rn s , was accepted Based on the conc lu s i on s drawn w i t h respect to the c a p i t a l market cu rve , the r i s k premiums requ i red by Canadian and American i nves to r s in open-end investment companies were c a l c u l a t e d . Canadian i nve s to r s requ i red (and rece ived) a 0.025 per cent increase in expected returns f o r each u n i t inc rease in r i s k . The i r American counte rpar t s requ i red (and rece ived) a 0.023 per cent increase in expected re tu rn f o r each u n i t increase in va r i ance of r e tu rn s . A s t a t i s t i c a l t e s t at the 0.01 l e v e l of s i g n i f i c a n c e showed that there was no s i g n i f i c a n t d i f f e r e n c e in the r i s k premiums demanded by the i nves to r s in the two markets. There fo re , the t h i r d hypothes i s was re jec ted -and the a l t e r n a t e hypothes i s that Canadian i nve s to r s in open-end investment companies e x h i b i t n e i t he r g r ea te r nor less r i s k ave r s i on than t h e i r American counte rpa r t s was accepted. On the bas i s of the s t a t i s t i c a l r e s u l t s f o r these samples over the time span 1957 to 1966, the conc l u s i on was drawn that the recent investment trends are a r e s u l t o f a lack o f investment o p p o r t u n i t i e s in Canada. The r e l a t i v e l y heavy concen t r a t i on of the Canadian mutual funds 72 i n l o w - r i s k - l o w re tu rn s e c u r i t i e s and the recent trend to investment i n U.S. s e c u r i t i e s cou ld be due to a w ios r cho ice of investments open to i nve s to r s in the U.S. c a p i t a l market and may a l s o stem from less r i gorous d i s c l o s u r e requirements in Canada which requ i re les s informed r i s k - t a k i n g by Canadian i n ve s to r s . ^ In s ho r t , the investment funds p o s s i b l y have s imply out-grown the s i z e and f a c i l i t i e s of the Canadian market p l a ce . Recommendations f o r f u t u r e research. Data from d i f f e r e n t segments of the c a p i t a l market might y i e l d d i f f e r e n t r e s u l t s . Further research i n co rpo r a t i n g severa l segments of the c a p i t a l market over severa l time per iods would produce more un i v e r s a l r e s u l t s . Such a m u l t i - i n d e x model cou ld be of b e n e f i t to both p r i v a t e i nve s to r s and p u b l i c agenc ies . I f , f o r example, p r i v a t e c a p i t a l i s t i m i d in a p a r t i c u l a r segment of economy e x h i b i t i n g s t rong r i s k - a v e r s i o n through custom or because large par t s of domestic savings are channeled through f i n a n c i a l i n te rmed ia r i e s having con se rva t i ve p o r t f o l i o p o l i c i e s , p u b l i c agencies cou ld i n i t i a t e f i s c a l and monetary p o l i c i e s to o f f s e t such a t r end . Such a market e v a l u a t i o n would enable the p u b l i c agencies to a s c e r t a i n where p u b l i c investment cou ld best supplement p r i v a t e c a p i t a l f o r the o v e r - a l l economic development of the Royal Commission on Banking and F inance, (Queens P r i n t e r , Ottawa, 1963), p. 30. 73 count ry . 11 would, ass i s t the i n d i v i d u a l i n ve s to r in such dec i s i on s as whether to inves t in a f o r e i g n s u b s i d i a r y or in the parent company and the investment o p p o r t u n i t i e s which o f f e r the g rea te s t r e tu rn per d o l l a r r i s k e d . Past performance is a good guide in making d e c i s i o n s , but i s on l y a gu ide. There i s no necessary reason f o r past performance to be repeated in the f u t u r e . This study has incorporated the s i m p l i f y i n g technique of us ing ex post data as surrogates f o r ex ante. The e r r o r s i ncu r red by doing so may be neg l i g ab l e or on the other hand, they may be s u b s t a n t i a l . A study ana l y s i ng the d i f f e r e n c e s between ex ante p r e d i c t i o n s and ex post performance would a s s i s t in v a l i d a t i n g the use of ex post data as surrogates f o r ex ante da ta . Th i s . s tudy has requ i red the homogeneity of p r o b a b i l i t y d i s t r i b u -t i o n s of returns on i n d i v i d u a l s e c u r i t i e s . It i s f e l t that g e n e r a l i z i n g the r e s u l t s would not r e fu te the model or i t s r e s u l t s but would merely 2 comp l i ca te i t s o p e r a t i o n . 3 The reward to v a r i a b i l i t y r a t i o de f ined as the rate of the r i s k premium to v a r i a b i l i t y of returns on a p a r t i c u l a r s e c u r i t y or c l a s s of John L i n t n e r , " S e c u r i t y P r i c e s , R i sk , on Maximum Gains from D i v e r s i f i c a t i o n , " The Journal of F inance, XX, No. k (December, 1965), p. 600. . . . . . . 3 W i l l i a m F. Sharpe, "Mutual Fund Per formance, " The Journal of  Bus iness, XXXIX, No. 1, Part I I , (January, 1966), p. 119—137. 74 s e c u r i t i e s , can be developed as a l o g i c a l step from the present ana l y -s i s . This would a f f o r d a method of ranr ing s e c u r i t i e s accord ing to t h e i r " r e l a t i v e e f f i c i e n c y . " Coupled w i th a m u l t i - i n d e x model, suggested e a r l i e r , not on l y cou ld the most e f f i c i e n t ' n d u s t r i e s be sought out but the more e f f i c i e n t s e c u r i t i e s as w e l l . However, i t must be kept in mind that t h i s model on l y incorporates two parameters, the expected re tu rn and va r i ance of r e tu rn s . An a d d i t i o n a l f i e l d of study would be to determine the i m p l i c a t i o n s of combining other i nd i ce s represent ing the va r ious f a c t o r s of importance in e v a l u a t i n g p ro spec t i ve investment. F i n a l l y , an i n v e s t i g a t i o n i n to the r e l a t i o n s h i p between the degree of r i s k ave r s i on e x h i b i t e d by a country and i t s stage of economic development cou ld be a f r u i t f u l area f o r research. One po s s i b l e avenue would be to use the l e ve l of d i s c r e t i o n a r y income as a surrogate f o r the stage of economic development. It is f e l t that the degree of r i s k a ve r s i on and the stage of economic development are i n v e r s e l y r e l a t e d . As the s o c i e t y becomes more a f f l u e n t i t demands a sma l le r r i s k premium per u n i t of r i s k assumed. BIBLIOGRAPHY BOOKS A r c h e r , S.H., and D'Ambrosio, C.A. B u s i n e s s F i n a n c e : Theory and  Management. New Yor k : The M a c M i l l a n Company, 1966. C l e l l a n d , R i c h a r d A., and o t h e r s . B a s i c S t a t i s t i c s w i t h B u s i n e s s  App 1i c a t i o n s . New Y o r k : John W i l e y & Sons, I n c . , 1966. Cohen, Jerome B., and Z i n b a r g , Edward D. Investment A n a l y s i s and P o r t -f o l i o Management. Homewood: R i c h a r d D. I r w i n , I n c . , 1967. Cohen, Kalman J . , and Hammer, F r e d e r i c k . A n a l y t i c a l Methods i n B a n k i n g . Homewood: R i c h a r d D. I r w i n , I n c . , 1966. F a r r a r , D.E., The Investment D e c i s i o n Under U n c e r t a i n t y . Englewood C l i f f s : P r e n t i c e - H a l l , I n c . , 1962. L e f t w i c h , R i c h a r d H., The P r i c e System and Resource A l l o c a t i o n . T h i r d E d i t i o n . New Yor k : H o l t , R i n e h a r t , and W i n s t o n , I n c . , 1966. M a r k o w i t z , H a r r y . Po r t f o 1 i o Se 1ect i o n : E f f i c i e n t D i v e r s i f i c a t i o n o f Investment. (Cowles F o u n d a t i o n Monograph No. 16 ) . N e w York: John W i l e y and Sons, 1959. S a u v a i n , H a r r y . Investment Management. T h i r d E d i t i o n . Englewood C l i f f s P r e n t i c e - H a l l , I n c . , 1967. R o b i c h e k , A l e x a n d e r A., and Myers, S t e w a r t C. Op t i m a l F i n a n c i n g D e c i s i o n s Englewood C l i f f s : P r e n t i c e - H a 11, I n c . , 1965. PUBLICATIONS OF THE GOVERNMENT Royal Commission on Banking and F i n a n c e . Ottawa: Queen's P r i n t e r , 1964. PERIODICALS A r d i t t i , F r e d D. , " R i s k and R e q u i r e d R e t u r n on E q u i t y , " The J o u r n a l o f  F i n a n c e , X X I I , (March 1967). p. 1 9 — 3 6 . B i e r w a g , G.O., and Grove, M.A., "On C a p i t a l A s s e t P r i c e s : Comment, 'The  J o u r n a l o f F i n a n c e , XX (March, I 9 6 5 ) , p. 8 9 — 9 3 . 76 Fama, Eugene F., " R i s k , R e t u r n , and E q u i l i b r i u m : Some C l a r i f y i n g Comments," The J o u r n a l o f F i n a n c e , X X I I I , (March, 1968), pp. 29- - k o . F i n a n c i a l P o s t , Survey o f Investment Companies, 1967. T o r o n t o : Mac-Lean-Hunter, 1967. F r i e n d , I r w i n , and V i c k e r s , D o u g l a s , " P o r t f o l i o S e l e c t i o n and I n v e s t -ment P e r f o r m a n c e , " The J o u r n a l o f F i n a n c e , XX, (September, I 965) , pp. 391—415. F r i e d m a n , M. and Savage, L . J . , "The U t i l i t y A n a l y s i s o f C h o i c e s I n v o l -v i n g R i s k , " The J o u r n a l o f P o l i t i c a l Economy, L V I , (August, 1948), p. 2 7 9 - - 3 0 4 . H i r s h l e i f e r , J a c k , " R i s k The D i s c o u n t R a t e , and The Investment D e c i s i o n , " A m e r i c a n Economic Review, L I , (May, 1961), p. 1 1 2 — 1 2 0 . H o r o w i t z , I r a , "The 'Reward t o V a r i a b i l i t y 1 R a t i o and Mutual Fund P e r f o r m a n c e , " Jou r na1 o f Bus i ne s s, XXIX, ( O c t o b e r , I 966) , p. 485—488. L i n t n e r , John, "The V a l u a t i o n o f R i s k A s s e t s and The S e l e c t i o n o f R i s k y Investments i n S t o c k P o r t f o l i o s and C a p i t a l B u d g e t s , " Review o f  Economics and S t a t i s t i c s , 47, ( F e b r u a r y , I965), p. 13 — 37-, " S e c u r i t y P r i c e s , R i s k , and Maximum Gains from D i v e r s i f i -c a t i o n ," The J ojjr_n_aJ_ o f HjTajTce, XX, (December, 1965), p. 5 8 7 — 6 1 5 . M a r k o w i t z , H., " P o r t f o l i o S e l e c t i o n , " The J o u r n a l o f F i n a n c e , V l l , (March, 1952), p. 7 7 - 9 1 . M i l l e r , M.H. and M o d i g l i a n i , F., " D i v i d e n d P o l i c y , Growth and the V a l u a t i o n o f S h a r e s , " J o u r n a l o f B u s i n e s s , XXXIV, ( O c t o b e r , 1961), p. 411—433. Sharpe, W.F., " C a p i t a l A s s e t P r i c e s : A Theory o f Market E q u i l i b r i u m Under C o n d i t i o n s o f R i s k , " The J o u r n a l o f F i n a n c e , XIX, (September, 1964), p. 425—442. , • . , " R i s k - A v e r s i o n i n the S t o c k M a r k e t : Some E m p i r i c a l E v i d e n c e , " The J o u r n a l o f F i n a n c e , XX, (September, I965), p. 416—422. , "Mutual Fund P e r f o r m a n c e , " J o u r n a l o f B u s i n e s s , XXXIX, P a r t I I , ( J a n u a r y , 1966). p. 119—138. 77 , " S e c u r i t y P r i c e s , R i s k , and Maximal Gains From D i v e r s i f i -c a t i o n : R e p l y , " X X I , (December, 1966), p. - ] k l ~ - l h k . T o b i n , J . , " L i q u i d i t y Preference as Behavior Toward R i s k , " The Review o f Economic S t u d i e s , XXVI , (February, 1958), p. 65—86. T r e y n o r , Jack L . , "How to Rate Management o f Investment F u n d s , " Harvard  Business Review, 43, ( J a n u a r y - F e b r u a r y , 1965), p. • 63 — 79. Weisenberger , A r t h u r , Investment Companies, New York: A r t h u r Weisen-berger and Company, 19&7. APPENDIX A REGRESSION ANALYSIS OF THE CAPITAL MARKET CURVE Appendix A c o n s i s t s of the data used in the reg re s s i on a n a l y s i s , the r e s u l t s of the reg re s s i on a n a l y s i s , the scattergrams and g r aph i ca l p r e sen t a t i on of the reg re s s i on l i n e s . TABLE X j SUMMARY OF REGRESSION ANALYSIS Model Data Equat ion Std. Er r Dep. Var a b Std. Er r b F -Rat io Coef. of Corr . (R) Standard Dev i a t i on Canadian Y = a + bX 1.1048 2.9656 0.4029 0.0750 28.8616 0.7253 X = a + bY 1.9889 1.8478 1.3058 0.2431 28.8616 0.7253 American Y' = a + bX 1.5882 2.6721 0.6005 0.0834 51.8260 0.8160 X = a + bY, 2.1582 2.6055 1.IO89 0.1540 51.8260 0.8160 Canadian p lus American Y = a + bX 1.7528 0.2036 O.698I 0.0592 139.0428 0.8487 X = a + bY 2.1308 3.8176 1.0317 0.0875 139.0428 0.8487 Var iance Canad i an Y = a + bX 1.1617 5.3967 0.0157 0.0032 23.6129 0.8899 X = a + bY 51.1271 • -82.5560 30.3615 6.2481 23.6129 0.6899 Canad i an Plus American Y = a + bX 1.7376 5.0571 0.0233 0.0020 142.4287 0.8515 X = a + bY 63.4536 • •95.9672 31.0954 2.6055 142.4287 0.8515 C o e f f i c i e n t o f V a r i a t i o n Canadian Y = a + bX 1.5928 8.7928 -0.6198 0.9873 0.3741 -0.1222 • X = a + bY 0.3140 1.7446 -0.0241 0.0384 0.3941 -0.1222 American Y = a + bX 2.6523 17.4984 -3.6532 2.6450 1.9075 -0.2614 •<• X = a + bY 0.1898 1.5557 -0.0187 0.0135 1.9075 -0.2614 Canadi an Plus -Ameri can Y = a + bX 3.0196 17.3014 -4.8845 1.4697 11.0450 -0.4121 X = a + bY 012547 1.7991 -0.0238 0.0104 11.0450 -0.4.21 83 1« 84 o 85 o APPEND .'X B TREYNOR INDEX The Treynor Index y i e l d s the investment oppo r tun i t y locus , assuming r i s k averse i n ve s t o r , f o r a p a r t i c u l a r segment o f the c a p i t a l market. The funds v o l a t i l i t y w i th respect to a market index is used as a measure o f r i s k . By aggregat ing the annual returns f o r the 28 funds forming the samples, the average annual rates of re tu rn were computed f o r the Canadian and American open-end investment companies. The v o l a t i l i t y f o r each market was measured by means of reg re s s i on a n a l y s i s of average annual aggregate rates of r e tu rn f o r the samples and the annual rates of r e tu rn f o r the Dow-Jones I n d u s t r i a l s . The average annual ra tes of r e tu rn f o r the Dow-Jones I n d u s t r i a l s were c a l c u l a t e d by the f o rmu la : M t = P t - Pt_~ + D t x 100 where P ^ i s the p r i c e of the p o r t f o l i o at the s t a r t of the per iod ( c l o s i n g p r i c e of the prev ious p e r i o d ) , P i s the c l o s i n g p r i c e at the end of pe r i od t , D^ are the es t imated d iv idends f o r per iod t , and M i s the p o r t f o l i o r a te of r e tu rn f o r per iod t . ^ Table XI shows the average annual returns f o r the American and Canadian samples and the Dow-Jones I n d u s t r i a l s . The r e s u l t s of the The data was obta ined from Arthur Weisenberger and Co., Investment  Companies, 1967, p. 339. 87 reg re s s i on a n a l y s i s are shown in Tables XII and XIII and are i l l u s t r a t e d g r a p h i c a l l y in F i gu re B . i . . The p o r t f o l i o p o s s i b i l i t y curves f o r the two markets are shown g r a p h i c a l l y in F i gu re B.2. The economic i n t e r p r e t a t i o n of these l i n e s i f t ha t f o r the samples of open-end investment companies over the pe r i od 1957 to 1966 the American market prov ided a g rea te r investment oppo r t un i t y f o r the r i s k averse i nve s to r than the Canadian market. To the ex tent that these samples are r ep re sen ta t i ve of the r e spec t i ve c a p i t a l markets the American market prov ides g reater investment o p p o r t u n i t i e s than the Canadian market. 88 TABLE XI . AVERAGE ANNUAL RATES OF RETURN ON SAMPLE CANADIAN MUTUAL FUNDS, U.S. MUTUAL FUNDS AND THE DOW-JONES INDUSTRIALS Canadian U.S. Dow-Jones Year Mutual Funds Mutual Funds 1 ndus t r i a 1s A„ A, M t t t .1966 - 9 . 3 7 2 % -3 .411% -15.649% 1965 7 .947% 2 0 . 7 7 1 % 14.135% 1964 15 .890% 12.468% 18.667% 1963 13 .545% 16 .125% 20 . 588% 1962 -5 .004% - 11 . 925% - 1 0 . 8 1 0 % 1961 19.967% 23.864% 22.400% I960 - 1 . 1 9 2 % 3 . 257% - 6 . 1 9 8 % 1959 6 . 329% 13 .089% .19.952% 1958 22.346% 3 9 . 9 0 0 % 28 . 777% <A> CON - ( A )u.s. - <">DJ= 7 .828% 12 .660% 10 . 21% TABLE XI I VOLATILITY(W) OF CANADIAN MUTUAL FUNDS C o r r e l a t i o n W C o e f f i c i e n t F -Rat io Regress ion L ine : A on M 0 .655 0 .9476 61.533 Regress ion L ine : M on A 0 .717 O.9476 61.533 Intermediate L i n e : 89 TABLE XIII VOLATILITY(W) OF U.S. MUTUAL FUND W Corre1 at ion C o e f f i c i e n t F -Rat io Regress ion Lino : A on M 0 .832 Regress ion L ine : M on A 1.054 Intermediate L ine : 0.9^3 0.888 0.888 26-225 26.225 TABLE XIV TREYNOR INDEX FOR SAMPLE CANADIAN AND U.S. MUTUAL FUNDS T. 1 . Canadian Mutual Funds 5.621 U.S. Mutual Funds 9.183 1 PORTFOLIO POSSZQXCTTV t-zveS 

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