FORMULA TIMING PMRS AND? THE BETA C‘DEFFECI MEN? AR EMPIRICAL STUDY Thesis“ nor the Degree of Ph. D MICHIGAN STATE UNIVERSITY RICHARD EDGAR WILLIAMS 1 9 75 - L IE R A R Y Michigan 5‘ 1m University This is to certify that the thesis entitled FORMULA TIMING PLANS AND THE BETA COEFFICIENT: AN EMPIRICAL STUDY presented by Richard E. Williams has been accepted towards fulfillment of the requirements for Ph.D. degree in Business - Finance QZ/la/L/ (J. WW-m/ Major professor / DateLWL‘x/V} 92/973 0-7 639 '= mam-«sour- " BUUK BINDERY IND. LIBRARY BINDERSI II F— __——— mecpogzn HICIIISA] I l ’2' x ' ‘ H’ ‘V This t femala p1; Periods of The in t0 deveIOp ally Emplo and twenty the two 10 ABSTRACT FORMULA TIMING PLANS AND THE BETA COEFFICIENT: AN EMPIRICAL STUDY By Richard Edgar Williams This thesis examined the investment performance of various formula plan strategies versus the buy-and-hold strategy during periods of cyclical stock price movements. The beta coefficient was utilized both as a measure of risk and to develop the high— and low-risk portfolios that have been tradition- ally employed in formula plan operations. Twenty high-beta securities and twenty low-beta securities were selected from the two highest and the two lowest beta deciles respectively as compiled by Sharpe & 1 for the 1957-1961 period. As an alternative to low-beta Cooper stock, Treasury bills were also utilized in calculating formula plan performance. Two types of formula plans were employed. One plan was of the constant-ratio variety in which the proportions invested in the high- and lowhrisk portfolios were rebalanced at quarterly intervals to 50%- SOZ. The other type of formula plan tested was of the variable-ratio 1Sharpe, William F. and Cooper, Gary M. "Risk-Return Classes of New York Stock Exchange Common Stocks, 1931-1967," Financial Analysts Journal 28 (March-April, 1972): 46—54, 81, 95-101. :pe vith the pi as monthly depex Stazdard 6 Poor ratio, the larg ad vice-versa. Performanc trestaent of S. dictate the nu: me Percent cor far transactior Pcrtfolios was ad 1962-1971 . lated for each 39311 for each tins invested The audit] 91m were the PattfoliOS th Witfolio’ a I “11.11 a'molll'ltsI The reSu‘ \ ‘ Q “named super Richard Edgar Williams type with the proportions invested in each portfolio changed as often as monthly depending on the level of the price-earnings ratio of the Standard & Poor's 500 Stock Index. The higher the price-earnings ratio, the larger was the proportion invested in the low-risk portfolio and vice-versa. Performance was simulated in two ways, first by assuming an initial investment of $100 in each security and letting subsequent market action dictate the number of shares of each security that would be held. A one percent commission on all sales and purchases was used as a proxy for transaction costs. The ending value of the high- and lowhrisk portfolios was then calculated at the end of the 1962—1966, 1967-1971 and 1962-1971 periods. Secondly, monthly price relatives were calcu- lated for each security. From these monthly price relatives a geometric mean for each plan in each of the three periods was derived using propor— tions invested in the lowhrisk and high-risk portfolios as weights. The ending dollar and geometric mean returns for the formula plans were then compared with similar measures for four buy-and-hold portfolios which consisted of the lowhbeta portfolio, the high-beta portfolio, a portfolio combining the high— and lowbbeta portfolios in equal amounts, and the Standard & Poor's 500 Stock Index. The results of the study indicate that the formula plans generally produced superior investment returns compared to the buy-and-hold portfolios during the three periods studied. When the results were adjusted for risk by calculating the Treynor Index, the conclusion that the formula plans generated superior returns relative to the buy— and-hold plans was further reinforced. Using the t fxrdaplans d' knead-hold pl. "sad as the per reflects primar fizzle plans. rally had a h The effici sense of reduci :orrelation (:05 is that fomul; 5“ buY‘aIld-hc the for—ma Pl. Performance di The use 0 "carted magma Fity betas mov ms did Hot ”Mono theo Rhone d. The reSul asset Such as Produced bette TI (A u leveraging Rants. Richard Edgar Williams Using the ending dollar value as a measure of performance, the formula plans did not produce as good a performance relative to the buyband-hold plans after commissions as when the geometric mean was used as the performance measure. The relative decline in performance reflects primarily the substantial impact of commissions on the formula plans. However, even after commissions, the formula plans usually had a higher ending dollar value than the S&P 500 Index. The efficiency of the formula plans, in the portfolio theory sense of reducing unsystematic risk, was not particularly good with correlation coefficients ranging from .703 to .779. The implication is that formula plan results are less predictable than results obtained from buy-and-hold portfolios. However, this result was expected since the formula plan portfolios used in the study were intended to produce performance different from the performance of the market as a whole. The use of a sixty-month beta coefficient as the measure of risk worked reasonably well in that only one-fourth of the individual secu- rity betas moved out of their original risk class and most of these ‘moves did not occur until the end of the ten-year period. Thus, the portfolio theory assumption of a stable portfolio beta over time is supported. The results of this study also show that the use of a risk—free asset such as Treasury bills with a high-risk portfolio of stocks produced better results than when high- and lowbbeta stock portfolios were used. This finding suggests that using an extreme beta portfolio and leveraging with a risk-free asset can produce superior investment results. In sum, t plan as an im the risk-retui Richard Edgar Williams In sum, this study has demonstrated the usefulness of the formula plan as an investment timing device when proper attention is given to the risk-return characteristics of the portfolio securities. FORMULA TIMING PLANS AND THE BETA COEFFICIENT: AN EMPIRICAL STUDY BY Richard Edgar Williams A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTORAL OF PHILOSOPHY Department of Accounting and Financial Administration 1975 Ivish to . @‘Conaor and Dr ideas which mad the original. Caaiman of my RY? beginning Of course wife, Sharon , ‘fiiCh this pro ACKNOWLEDGMENTS I wish to acknowledge my sincere appreciation to Dr. Melvin O'Connor and Dr. Robert Dolphin for their assistance and many helpful ideas which made the final version of this thesis much better than the original. Special thanks is owed to Dr. Alden Olson who, as Chairman of my thesis committee, worked closely with me from the very beginning and whose guidance and advice are greatly appreciated. Of course, more appreciation than words can express is due my wife, Sharon, for her support, both moral and financial, without which this project would surely not have been completed. ii LIST OF TABLES. . 1251' OF FIGURES . Chapter I. INTRODU 11. RESEARC State Resea Techn 111- mum TABLE OF CONTENTS Page LIST OF TABLES. O O O O O O O O O O O O O O O O O O O O O O O 0 iv LIST OF FIGURES O O O O O O O O O O O O O O O O O O O O O O 0 0 Vi Chapter I 0 INTRODUCTION 0 O O O O O O O O O O O O O O O O O O O O 1 II. RESEARCH DESIGN AND TECHNIQUE. . . . . . . . . . . . . 14 Statement of Hypothesis. . . . . . . . . . . . . . . 14 ResearCh DeSign. I O O O O O O O O O O O O O O O O O 14 TeCMiqueo o o o o o o o o o o o o o o o o o o o o o 25 III. ANALYSIS OF RESULTS. . . . . . . . . . . . . . . . . . 35 Summary. . . . . . . . . . . . . . . . . . . . . . . 35 Constant-Ratio Plan Results. . . . . . . . . . . . . 46 1962-1966 . . . . . . . . . . . . . . . 46 1967-1971 . . . . . . . . . . . . . . . . . . . 49 1962-1971 . . . . . . . . . . . . . . . . . . . . 52 Variable-Ratio Plan Results. . . . . . . . . . . . . 56 1962-1966 . . . . . . . . . . . . . . . . . . . . 56 1967-197100000000000000000... 59 1962-1971 . . . . . . . . . . . . . . . . . . . . 62 General Observations . . . . . . . . . . . . . . . . 65 IV. RESERVATIONS, IMPLICATIONS AND CONCLUSIONS . . . . . . 69 Reservations 0 o o o o o o o o o o o o o o o o o o o 70 Impl icat ions 0 o o o o o o o o o o o o o o o o o o o 7 9 Conclusions . . . . . . . . . . . . . . . . . . . . 84 APPmDIXo O I O O O O O O O O O O O O O O O O O O O O O O O O O 89 REFERWCES. O O 0 O O O O O O O O O O O O O O O O O O O O C O O 9 3 iii Chapter 11"]. Fri: Var: 11-2 Ret' 11-3 Re: III-la COT: III-lb Con I-IHC Cor: Co: Pla End 21-3 Cor Bet am I214. C01 Th St: En III-5 Co 19 111-5 Co 19 31.7 Pr in 19 TII‘S CC 19 “‘9 Er 1? ‘ZI‘IO P] Chapter 11-1 11-2 11-3 III-la III-lb III-1c III-2 III-3 III-4 III-5 III-6 III-7 III-8 III-9 III-10 LIST OF TABLES Price-Earnings Ratio Ranges Used in variable-RatioplanSoo o so... .00. o o 0 Return Matrix I for Buy-and-Hold Calculations. . . Return Matrix II for Formula Plan Calculations . . Comparative Results, 1962-1966 . . . . . . . . . . Comparative Results, 1967-1971 . . . . . . . . . . Comparative Results, 1962-1971 . . . . . . . . . . Comparisons Between Constant & Variable-Ratio Plan Results, Annualized Geometric Mean and Ending Total Dollar Value. . . . . . . . . . . . . Comparisons Between Unadjusted-Beta and Adjusted- Beta Portfolio Results, Annualized Geometric Mean and Ending Total Dollar Value. . . . . . . . . . . Comparison of Results Between Formula Plans Using Treasury Bills and Formula Plans Using Lobieta Stock Portfolio, Annualized Geometric Mean and hiding Dallar value. 0 O O O O O O O O O I O O O 0 Comparative Results - Constant-Ratio Plans, 1962-196600000000000000000000. Comparative Results - Constant-Ratio Plans, 1967-197100.000000000000000... Price Change Comparisons Between Individual Stocks in Lobieta High-Beta Portfolios, 1962-1966 V8 1967-1971 c o o o o o o o o o o o o o Comparative-Results - Constant-Ratio Plans, 1961-1971.00000000000000.0000o Ending Value of Unadjusted—Beta Portfolios, 1962-1971000000000000000coo... Proportions Invested in High- and Low-Beta Portfolios, 1962-1966. . . . . . . . . . . . . . . iv Page 19 27 3O 4O 41 42 43 44 45 47 50 51 54 55 57 hapter III-ll III-l4 III-15 Ccr 3:1 iia Ctr Sea lfi Chapter III-11 III-12 III-13 III-14 III-15 IV-l Appendix Arl A-2 Comparative Results - Variable-Ratio Plans, 1962-1966..coco-00000.00...o. Proportions Invested in High- and Lobieta Portfolios, 1967-1971. . . . . . . . . . . . . . Comparative Results - Variable-Ratio Plans, 1967-197100000000000000000... Comparative Results - Variable-Ratio Plans, 1962-19710000000000.0.0000... Comparative Rankings Between Ending Dollar Value and Annualized Geometric Mean for Variable- Ratio Plans, 1962-1971 0 o o o o o o o o o o o 0 Comparison of Low-Beta Portfolio Value With and Without Utility Stocks . . . . . . . . . . . Securities Used In Study . . . . . . . . . . . . Timing of Security Replacements. . . . . . . . . Page 58 6O 61 63 67 75 9O 91 LIST OF FIGURES Chapter Page I-l Standard & Poor's 500 Stock Index, Monthly Prices, 1946-1971. C C O O C . C O O O C O O C 6 vi Formula iisastrous l of 1929-1932 theory that 0f continuec $3317. pOpI of My 111V! i3"es'iments Chases was held fol. a Stock Mrke idEa that 1 18:21 at w? 3“ the rat Nate CYCl the Pl‘oble atteution n s *4. ties as \61{ CHAPTER I INTRODUCTION Formula-timing plans originated in the 1940's as a reaction to the disastrous losses experienced by investors in the stock market collapse of 1929-1932. Prior to 1929 there had been wide acceptance of the theory that common stock prices would increase over time as the result of continued profitable reinvestment of earnings by corporations. This theory, popularized by Edgar Lawrence Smith (42), confirmed the belief of many investors that common stocks could be purchased as long-term investments at any level of the market. Careful timing of stock pur- chases was unnecessary, it was thought, as long as the securities were held for a sufficiently long period of time. However, the precipitous stock market crash which began in September, 1929 quickly dispelled the idea that timing was unimportant. Instead it was recognized that the level at which stocks are purchased and sold can have a major impact on the return earned by investors, especially when stock prices fluc- tuate cyclically around a secularly rising trend. Since the 1930's the problem of timing purchases and sales has received considerable attention in the popular literature of investments as evidenced by such titles as The Profit Magic of Stock Transaction Timing (25), How Charts Can Help You In the Stock Market (26), A Strategy of Daily Stock Market Timing for Maximum Profit (23), or Fundamentals for Profit in Under- valued Stocks (35), all of which, in one way or another, purport to l shes investo Formula naL-ze the tim glam or fort: sipals for these trend investor se] "defensive" Senerated b: ieclines bej are gradual. 011th "de the Princip with minimu PP. 128-29) 2 show investors how to make selection and/or timing decisions. Formula-timing plans are investment strategies which attempt to make the timing decision automatically for the investor. That is, a plan or formula is devised which provides automatic buy or sell signals for the investor to follow. The general procedure is to make certain assumptions about the future trend of the market. Then, using these trend assumptions, a set of rules is constructed which has the investor sell stock as the market rises and invest the proceeds in a "defensive" portfolio of securities thus preserving the capital gains generated by the beginning portfolio. When the market turns down and declines below some pre-established level, the defensive securities are gradually sold and "aggressive" securities are purchased. In this context "defensive" is taken to mean the preservation of the value of the principal amount invested. That is, defensive securities are those with minimum financial risk which has been defined by Sauvain (43, pp. 128-29) as: ". . . . the uncertainty of a series of promised or expected cash receipts by owners of securities due to changes in the financial abilities of issuers to make payments to them, or due to changes in in- vestors' estimates of their financial abilities." Formula plan literature of the 1940's defined the defensive portfolio as consisting of spaced-maturity bonds with the implicit assumption being that bonds would not fluctuate in price because of changes in financial risk. By the same reasoning, common stock was considered an "aggressive" security in that purchase of common stock involved the risk that changes in the financial abilities of the issuing corpora- tions or expectations about their abilities would cause fluctuations in the price of their stock. Cottle anc goint out that abcut the futur afressive and systematic purc these systemat: tjces of plans :xstant-ratio Plans a fixed j tions of the f as stock price 1° remain cons Biases, Afte Sane BiVEn per stocks are so] meinitial p1 stock Prices ‘ Variable. fluctuatiOns total fund C0 9’59“ timin :‘39 mrket C)‘ EMS being tiring, the 1 the market c. The SPQI Plan S are to. 3 Cottle and Whitman (8) in their classic study of formula plans point out that all plans are characterized by (l) certain assumptions about the future; (2) the division of total portfolio assets into aggressive and defensive portions; and (3) the use of rules for systematic purchases and sales. It is the construction and use of these systematic rules which differentiate formula plans. Two general types of plans are usually discussed in investment textbooks, the constant-ratio plan and the variable-ratio plan. With constant-ratio plans a fixed percentage relationship between the bond and stock por- tions of the fund is maintained by periodic rebalancing. Consequently, as stock prices rise from the initial level (bond prices are presumed to remain constant) the percentage of stock value to bond value in- creases. After some predetermined time period has passed, or after some given percentage increase in a market index has taken place, stocks are sold and bonds purchased in sufficient quantity to restore the initial proportions. The same procedure, in reverse, is used when stock prices are falling. Variable-ratio plans are more aggressive with respect to market fluctuations in that, as stock prices rise, the proportion of the total fund committed to stocks is continuously reduced until, assuming perfect timing, the fund is mostly invested in bonds at the peak of the market cycle. As stock prices fall the process is reversed, with stocks being purchased and bonds sold until, again assuming perfect timing, the fund is mostly invested in common stock at the bottom of the market cycle. The specific trading rules used by constant and variable-ratio Plans are too numerous to discuss here except to point out that one frequent mod trend line a allowing for of the mine} Tmlinson (I Unfort' reverent of east plans since there :unity to I able-ratio invested 11 a regult’ and little Toe leadin Binge who, 25.1218 a VE : “Rated ‘7) r under- C.) I '1 the Pt aka: ‘1 e thc 4 frequent modification of variable-ratio plans is to include a rising trend line around which stock prices are expected to fluctuate, thus allowing for a secular upward trend in stock prices. For a cataloging of the numerous possible types of trading rules see Persons (34) or Tomlinson (44). Unfortunately for the users of most formula plans, the upward movement of stock prices in the 1950's was so strong and sustained that most plans performed poorly when compared with a buy-and-hold strategy, since there were few cyclical declines which could be used as an oppor- tunity to rebalance the fund by purchasing stock. Indeed, most vari- able-ratio type plans found themselves completely or almost completely invested in a bond portfolio while the market continued to rise.1 As a result, formula plans fell out of favor in the investment community and little has been written about them in the literature of the 1960's. The leading academic proponent of formula plans during the 1960's was Dince who, in a series of journal articles (9), (10), (11), suggested using a variable ratio plan in which the bond-stock ratio at any point in time was tied to a regression equation relating Gross National Product to the Dow Jones Industrial Average. The hypothetical DJIA computed from the regression equation was compared to the actual DJIA at quarterly intervals to see whether the market as a whole was over- or under-valued. This comparison determined the bond-stock proportions of the portfolio. Thus, if the actual DJIA was 141 percent or more above the regression formula value, the bond-stock proportions were 1A8 Sauvain (43) p. 466, says: "The long bull market simply ran away from plans that assumed some cyclical ceiling, and investors who adhered to them were left in their minimum stock position." set at 801-207.. :he beginning 0 his amunted t 5.66 percent as Stack Average c credence to the by a buy-and-hc relating currer Tw devel< further Sugges1 discarded. Fi: SiCCk prices 0‘ °5 Mb greate Period. Vlth t 3.3.36ar that th 1950's Came tc 3mmher 1971. Stock Index 37 Cyclical an early 197 fitmula Plans behave in thi Ede" flde a ber '53? ch worked far ' Example ’ \ 2 - Dince 5 set at 80Z-20%.2 Dince's results indicated that $100,000 invested at the beginning of 1930 would have grown to $431,000 by the end of 1962. This amounted to a compounded annual return (including dividends) of 8.66 percent as compared with a buy-and-hold return on Moody's 125 Stock Average of 6.6 percent over the same period. Dince's work adds credence to the idea that formula plans need not always be outperformed by a buy-and-hold strategy given a reasonably accurate method for relating current stock prices to some type of "intrinsic" value. Two developments in the investment environment of the later 1960's further suggest that the formula-timing idea should not be too quickly discarded. First, an examination of Figure I-l shows that average stock prices over the 1960's went through three major cyclical declines of much greater severity than anything experienced during the 1946-1961 period, with the exception of the 1957 decline. Furthermore, it would appear that the strongly rising secular trend of the 1950's and early 1960's came to a halt during the sixryear period from January 1966 to December 1971. By the end of this period the Standard and Poor's 500 Stock Index stood at virtually the same level as at the beginning. Cyclical behavior of the kind experienced in the latter 1960's and early 1970's provides the sort of market environment in which formula plans should perform best. If the stock market continues to behave in this cyclical fashion in future years, formula plans may provide a better performance than the simple buy-and-hold strategy which worked so well in the 1953-1965 period. 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I I- .II I 7. $52... fizhmge stc vmdd have 1 3 xmmflly. 366-1971 pl :ion ad a Second theoretical traces its 1932 which lished in 1 3&5 not rec talicatiox Pricing of folio of S Essen :511 ‘ ...-IOUlng (1) 7 Exchange stocks bought at the end of 1953 and sold at the end of 1965 would have produced a tax-exempt return of 16.2 percent compounded annually.3 0n the other hand, holding the S&P 500 stocks during the 1966-1971 period would have resulted in virtually zero price apprecia— tion and a dividend yield of only about 3.3 percent. Secondly, the 1960's witnessed the deve10pment of a body of theoretical literature known as portfolio theory. This body of theory traces its origin back to a seminal article by Harry Markowitz (32) in 1952 which was subsequently refined and developed in a monograph pub- lished in 1959 (33). The full importance of Markowitz's work, however, was not recognized by the investment community until a series of articles by Sharpe (37), (38), (39), Lintner (31), and Treynor (45) revealed the implications of the Markowitz "efficient" diversification idea for the pricing of common stock and measurement of the performance of a port- folio of securities. Essentially, the Markowitz-Sharpe-Treynor thesis contains the following propositions: (1) Investors are, as a group, risk-averse and expect there- fore to earn a higher return on a risky security than on a riskless one. Risk in this context is defined as inter— period variability of price plus dividends and measured as the variance or standard deviation around the expected return over some time period. The objective of investment management is taken to be the maximizing of expected return within the constraint of a risk-class specified by the 3Fisher and Lorie (17), p. 7, Table l. (2) Harko; invest combir cient. ....— -_— — of th: both ' combi temat be re Syste Simil such inter all 5 ChOO: atte with mess SYs1 0f for Fe: CQu r1: to‘ Do (2) (3) (4) (5) 8 investor's risk-preference function. Markowitz or "efficient" diversification consists of combining securities which have low correlation coeffi- cients in such a way as to maximize the expected return of the portfolio for some given level of risk. The total risk of an individual security is made up of both "systematic" and "unsystematic" components. When combined in an efficient portfolio, however, the unsys- tematic risk component of the total portfolio risk can be reduced so as to approximate zero, leaving only the systematic risk component. Systematic risk cannot be similarly eliminated since it is the risk produced by such factors as changes in general economic conditions, interest rate levels or purchasing power which affect all securities to some extent. The investor, then, must choose some level of systematic risk within which he attempts to maximize return. Within a diversified portfolio of securities, the riski- ness of an individual security can be measured by its systematic risk component (i.e. the variability of return of the security as compared with the variability of return for the market as a whole). Performance of a portfolio can be measured relatively by comparing excess return (i.e. portfolio return minus the risk-free interest rate) per unit of either systematic or total risk with that of other portfolios. The optimal portfolio is the one having the highest excess return per unit An implici :3;er the assu efficient. Per elicit a securit nation is rapic security. Sim raid-om fashion, of each other :ion is that p' 0f little valu. the i2i‘Plicatio TEdUCEd to an matted retur 'Pit'n the highe 3??”ng a qua 33 using Sharp (II EC”rity anal} TElEVant to ti PriCEd aCCOrd: Portfolio of : C'bQEHSUI-ate ‘ tside inform. fell("Wing com Exp. Cla Vili fUn 9 unit of systematic or total risk. An implicit assumption of portfolio theory should be pointed out; namely the assumption that securities markets are perfect or at least efficient. Perfection in this context means that new information about a security is available to all investors and that such new infor- mation is rapidly if not instantaneously reflected in the price of the security. Since such information is presumed to enter the market in random fashion, security price changes will be random and independent of each other (14). A major conclusion to be drawn from this assump- tion is that prediction of future prices based on past information is of little value to the investor. When coupled with portfolio theory the implication for investors is that portfolio management can be reduced to an almost mechanical operation. Given estimates of the expected return and variability of a group of securities, the portfolio with the highest return for a given level of risk can be constructed by applying a quadratic programming model developed by Markowitz (33) or by using Sharpe's simpler diagonal model (37). The use of traditional security analysis techniques to discover undervalued issues is not relevant to this framework since all securities are already correctly priced according to their risk level. Buying an efficiently diversified portfolio of securities, given some level of risk, and earning a return commensurate with this risk level is the best that an investor without inside information can hope for. Francis (19), for example makes the following comments about the role of security analysis: Expert fundamental analysts who discover new finan- cial information and quickly interpret it correctly will earn higher-than-average returns, but most fundamental analysts will not earn a return above I: a simi] VESEOY! DEr; 10 what could be achieved with a naive buy-and-hold strategy. In a similar vein, Francis comments on the role of the average in- vestor: Selecting the most efficient portfolio in the preferred risk-class will enable the investor to attain his highest indifference curve in risk-return space. This investment may or may not earn an above-average rate of return - this depends on the risk—class the investor selects and when he liquidates his investment. But such analysis will maximize the investor's expected utility.5 Needless to say, the framework outlined above has not been com- pletely accepted by practicing security analysts and portfolio managers trained in the intrinsic-value analysis methods of Graham and Dodd (22). There is, however, a growing body of literature, summarized by Fama in (15), which concludes that security markets are efficient enough to make the search for undervalued securities a fruitless occupation except for those individuals with exceptional insight or those having the time, data and computing equipment necessary for the type of highly sophisticated analysis which might give a temporary advantage over less knowledgeable investors. One aspect of portfolio management which, as yet, has received relatively little attention in the portfolio theory literature is the question of timing. The Markowitz-Sharpe models are static, one-period equilibrium models which show how to obtain an effi- cient portfolio at a point in time. The problem of transition from one period to the next as prices and investor expectations change and the initial portfolio becomes less and less efficient has not yet been 4Francis (19), p. 547. 51bid., p. 548. ll satisfactorily worked out in theory. Rather, most of the literature has concentrated on measuring returns over some holding period assuming a naive buy-and-hold policy in which the portfolio is left unchanged during the period. Recently, however, Evans (13) and Cheng and Deets (7), have shown that a policy of periodic rebalancing so as to maintain equal dollar amounts in each portfolio security produces better returns than the buy-and—hold policy. These studies along with those of Fisher and Lorie (17) and Brigham and Pappas (6), indicate that the timing of purchases and sales is in fact a major determinant of multi-period portfolio performance given cyclically fluctuating markets. Formula plans represent a timing device which make periodic timing decisions automatically and thus fit in well with the mechanistic approach to portfolio selection proposed by portfolio theory. In order for formula plans to be successful it is necessary to forecast the trend and amplitude of stock prices with some accuracy. The failure of formula plans to perform well in the post-World War II era was a direct result of the failure by formula plan users to forecast the sustained bull market of that period. There is some question, how— ever, whether future bull markets will be of the long and sustained nature experienced during the 1950's and early 1960's. Seligman (36), for example, has argued that inflation and the ending of several types of special market forces may usher in a "Bad New Era" for common stocks. According to Seligman, the relatively high levels of inflation experi- enced since 1965 have had the effect of both reducing corporate profit margins and raising interest rates. This has meant that the prices of commmn stocks declined for two reasons; first, by reducing expectations abOLrt future earnings; and secondly, by raising the discount rate that 12 investors use to determine the present value of those future earnings. Seligman argues that the rate of inflation and interest rates will continue at relatively high levels throughout the 1970's, and therefore that the growth of common stock prices will continue to be less than the rate experienced during the 1953-1965 period. Furthermore, he argues that the relationship between the supply of and demand for com- mon stock will be less favorable than in the past. On the supply side, corporations will continue the trend begun in 1969 of issuing large amounts of new stock. On the demand side both institutions and indi— vidual investors will reduce their purchases of common stock as high interest rates pull investment funds into the bond markets and as the institutions reach optimal bond/stock proportions in their portfolios. Similarly, Bernstein (1) has argued that increased supply and reduced demand for common stock will make high rates of return less likely in the 1970's than in the 1960's. Grunewald and Klemkosky (24) have also argued that large stock market gains may be a thing of the past. Citing the special conditions which resulted in large earnings per share growth and price-earnings ratio increases during the 1950- 1965 period as well as reduced liquidity in the stock market and the impact of wage and price controls on price-earnings ratios, they con- clude that the stock market is likely to be more volatile in the future with a much lower secular growth rate. They point out that in such a market environment correct timing of purchases and sales becomes the key to successful investing and that formula plans are one type of timing device which could be used by investors to improve their investment returns. In sum, the implication of the articles cited above is that stock 13 price trends in the future may be more amenable to the formula plan approach. Stock prices may continue to increase but the percentage increases may be much smaller than in the past and, more importantly for this thesis, price trends may be much more volatile as changes in the rate of inflation, interest rates, wage-price controls and corpo- rate earnings all combine to cause waves of optimism and pessimism in the marketplace. Given this additional volatility and the corresponding increase in the necessity for better timing of purchases and sales, formula plans might once again come into widespread use by the invest— ment community. v. .3 F.‘ u re‘v'l' are a: a: xix. CHAPTER II RESEARCH DESIGN AND TECHNIQUE Statement of Hypothesis This thesis proposes to re-examine the formula plan concept in light of perceived changes in the market environment and using some of the newer analytical techniques developed in the literature as reviewed in Chapter I. Specifically, it is hypothesized that formula plans can earn a larger return than a buy-and-hold strategy without any additional increase in risk in a market environment characterized by cyclical fluctuations around a slowly rising trend line. This hypothesis will be tested by comparing the returns that would have been earned by a buy-and-hold strategy with the returns that would have been earned by several types of formula plans during the 1962- 1971 period. Research Design Performance of the formula plans and the buy-and-hold policies will be simulated for the two non-overlapping five-year periods 1962- 1966 and 1967-1971 as well as for the entire ten-year period 1962- 1971. This time period was chosen for a number of reasons. First, during this ten-year period the stock market went through three major cycles. Standard & Poor's 500 Stock Index reached cyclical troughs in June 1962, October 1966 and June 1970. Cyclical peaks occurred in 14 L. . u. .-' Jars: .’ {St E E L I] 1 q. a J. . «D L . .. y a: s1 ..t .3 mi - v H ‘ .§ .... c4 . s ...t. k. ... :nw MW E J g Q l. .3 . c _ a S 15 March 1962, January 1966 and December 1968. The cyclical nature of the stock market during this period provides the type of environment necessary for the successful operation of a formula plan. Second, formula plans have long time horizons and the five- and ten-year intervals being examined seem the minimum appropriate for testing the usefulness of formula plans as a portfolio management device. Third, the increase in the S&P 500 Stock Index from January 1962 until December 1971 averaged about 3.5 percent per year, the kind of increase that, it was argued in Chapter I, might be a reasonable expectation in future years. Two types of formula plans will be tested: the constant-ratio type and the variable-ratio type. The specific trading rules employed by each were chosen on the basis of simplicity. The goal was to test formula plans that involved as few complex rules as possible in order to make application of the plans straightforward and to make the results as general as possible. The constant-ratio plan is the simplest of all the formula plans in that no prediction about future market levels is necessary (except, of course, the assumption of cyclical movement in stock prices). Action points (i.e. points in time when the portfolio composition will be altered) in a constant-ratio plan can be based on some percentage change in a market average or they can be based on some time period. An example of the former would call for rebalancing the aggressive and defensive portfolios every time the market average rises or falls by say, ten percent. The latter type of constant-ratio plan and the type that will be used in this study is even simpler to operate, in that it calls for rebalancing the two portfolios after a fixed period of time ¢ H...— I ....A CJr CE ...-“ y‘an ssJ- A DflCE . C a V ht r: a C 16 without reference to the level of the market. Variable-ratio plans, by contrast, are more difficult to develop since a market forecast is required. As a means of avoiding a predic- tion of the absolute level of the market, this study will use a vari- able-ratio plan in which action points are dictated by a change in the price-earnings ratio of a market average, the Standard & Poor's 500 Stock Index. By incorporating the price-earnings ratio as a decision criterion, any secular increase in corporate earnings is taken into account by relating these earnings to the current market price index. An explicit forecast of market levels is not necessary. Rather, the assumption is made that the price-earnings ratio of the index will remain within a certain range and that action points in the formula plan can be con- structed within that range. Two versions of the constant-ratio plan and two versions of the variable-ratio plan will be tested. One constant-ratio plan will use a bond portfolio and a stock portfolio. The two portfolios will be rebalanced at quarterly intervals in order to maintain a market value ratio of 50 percent in each. In order to hold the risk of capital losses in the bond portfolio to zero, ninety-one day Treasury bills are assumed to be purchased at the beginning of the quarter and held until they mature. Treasury bills are used in place of the traditional portfolio of spaced maturity bonds in order to remove a major weakness of previous formula plans--name1y, the assumption that bond prices would be stable over the life of the plan. Earlier plans did not recognize the problem of interest rate risk and the corresponding necessity of forecasting bond prices or investing in only short—term 17 maturities in order to reduce this type of risk. The stock portfolio will consist of New York Stock Exchange secu- rities chosen from those in the two highest deciles of market sensi- tivity coefficients as of January 1, 1962 as compiled by Sharpe and Cooper (41). In this study, Sharpe and Cooper computed market sensi- tivity coefficients (i.e. the slope coefficient relating the capital gains returns on individual stocks to the capital gains returns on the market as a whole) for each year from 1931 to 1967 for all listed stocks on the New York Stock Exchange. The slope coefficients were based on monthly returns for the preceding five years. All the secu- rities were then grouped into ten deciles according to the size of their coefficients during the preceding five-year period. This lepe coefficient is analogous to the beta coefficient used in portfolio theory as a measure of the riskiness of an individual security in a portfolio. It differs only in that the beta coefficient relates the total return for each period, including dividends, to the total return on the market as a whole while the Sharpe-Cooper measure excludes dividends. Thus, a ranking of securities according to their market sensitivity coefficients is also a ranking of the securities according to their level of systematic risk.6 Using securities from the two highest deciles of the Sharpe-Cooper study implies that highly volatile securities are being used in the aggressive portfolio of the formula plan. Consequently, the aggressive component of the total fund used 6Because the market sensitivity coefficient and the beta coeffi- cient are so similar, the high- and low-risk market sensitivity coefficient portfolios used in the formula plans will be referred to henceforth as the "high-beta" and "low-beta" portfolios respectively. Mum .1 a u ‘l‘ ad C“ c flu. ‘ rt .0 .l .1 a T. .. .l .. l p 5 U U — C a n! . J y . «v. 2.. .-. E . 1 .. : u . t a .1 .1 .4- e C x. S .M . 3 3 ,. .. . . 4 C mi u a «u .C .1 L . ... . C l L.“ P» p . p: -..; .1 $« . « . u in» .m vL a f P . a .J. a a. VHJ r >1.“ 2w 51L 0 if“. t .4.» PH 5 r s 0 >1. .»h e . : l 18 in the plan assumes more risk than the stock portfolio usually seen in discussions of formula plans. Cottle and Whitman, Dince and others typically use the returns from a market index in computing formula plan results. When coupled with the defensive or bond portfolio, the total variability of the entire fund will be less than the market averages. It is not surprising then, that the return from formula plans is often found to be less than the return from a buy-and-hold policy, since, from a portfolio theory standpoint, less total risk is being assumed in the formula plan. By including only high-risk secu- rities in the stock portfolio of this constant-ratio plan it is hoped that total variability will be comparable to the market as a whole and thus make comparisons with market averages more meaningful. The second type of constant-ratio plan to be examined will con- tinue to use the quarterly rebalancing technique. However, instead of using Treasury bills in the defensive portfolio, a portfolio of low-beta stocks will be utilized. This portfolio will be chosen from the two lowest deciles of the Sharpe-Cooper study. These securities, according to portfolio theory, are defensive in that their price changes over time are less than those of the market as a whole. Nevertheless, this plan is more aggressive with respect to financial risk than the other constant-ratio plan in that losses in the defen- sive portion of the total portfolio are now possible. The purpose of substituting stocks for bonds in the defensive portfolio is simply to examine the returns earned by an all-common stock plan as compared with the returns earned by the more traditional bond-stock plan given the same trading rules. Presumably, since the all-common stock plan is riskier, it should earn a higher return than the bond-stock plan. 19 The third type of plan to be tested is a variable-ratio plan in which the ratio of the dollar investment in bonds to dollar investment in stock is dependent upon the level of the price-earnings ratio for the Standard & Poor's 500 Stock Index. Portfolio revision will occur only when the monthly price-earnings ratio moves from one level to another as shown in Table II-l. TABLE II-l PRICE-EARNINGS RATIO RANGES USED IN VARIABLE-RATIO PLANS P-E of S&P Index1 Defensive-Aggressive Ratio 14.0 or less 10%-90% 1400-1505 3004-702 15.5-17.5 50%-50% 17.5-19.0 7OZ-3OZ 19.0 or more 90%-10Z l Yearly earnings for the most recent four quarters divided by the value of the Index at the end of the month. The price-earnings ratio range of 14-19 was established after studying the range of the S&P 500 price-earnings ratio over the 1955- 1961 period. During this period, the price-earnings ratio ranged from extremes of 11.78 in December 1957 to 22.18 in August 1961. Since both of these extremes occurred prior to the beginning of the time period used in this study, the range established in Table II—l would have been a valid one to use during the period beginning January 1962. The median level is taken to be the range from 15.5-17.5. In actuality, the mean price earnings ratio for the S&P 500 Index during the 1962-1971 period was 17.2 with a high of 20.79 in February 1962 rati LL (1) r -O| m Vari 9;an Port the 20 and a low of 13.58 in June 1970. Thus the range shown in Table II—l seems reasonable and is likely to be similar to the range in any future bull or bear markets unless there is a significant reappraisal by investors of the value of a dollar's worth of earnings. As in the first constant-ratio plan, this plan will utilize a defensive portfolio consisting of Treasury bills and the aggressive portfolio will consist of the same high-beta securities. However, since portfolio readjustments can occur every month under this plan, thirty-day Treasury bills assumed to be purchased at the beginning of the month will be used instead of the 91-day bills of the constant- ratio plan in order to minimize the possibility of losses in the defensive portfolio. The fourth and last formula plan to be tested is again of the variable-ratio type. Its operation is identical to that of the other variable-ratio plan described above except that the low-beta stock portfolio used in the second constant-ratio plan is substituted for the 30-day Treasury bills. This plan is the most aggressive of the four in that it keeps the fund fully invested in common stock at all times and relies on average price-earnings ratios to dictate the per- centage of the fund invested in high-and low-risk securities. Unlike the traditional formula plan which utilized a defensive fund made up of bonds, this plan should lessen the risk of an improper market fore- cast since the fund will still be fully invested in common stock, albeit mostly lowerisk, low-volatility stocks when the market is near a peak and in high-risk, high-volatility stocks when the market is near a cyclical low. The possibility of being entirely, or mostly, invested in bonds when the stock market was continuing to rise to new . 5"“ I- .n , .:. 2‘ fl vv‘ H L. 1 “on“ 3 J5 21 highs plagued earlier variable ratio plans and is avoided with this plan. In addition, the substitution of a low-volatility portfolio of stock for a bond portfolio allows a test of the portfolio theory tenet that historically-derived low-beta stocks will provide less volatility than historically-derived high-beta stocks during cyclical swings in the market. If this proposition is true then the high-low beta portfolios should perform better than a policy of buying and holding a portfolio of stock with average price volatility since, during upswings of the market, low-beta stocks will be progressively substituted for high-beta stocks and vice-versa during downswings. Thus the large gains in price generated by the high-beta stocks as the market rises are preserved by transfering these gains into the low- beta portfolio. As the market moves down the more stable low-beta stocks are gradually replaced with high-beta stocks which have fallen more than the lOWbbECa stocks and thus can be bought, on balance, at lower prices than the prices at which they were sold. The above scenario will prove accurate only if the betas of high- beta stocks, whose coefficients have been calculated from historical data, remain relatively stable during the time period in which the formula plan is in Operation. Instability of the beta coefficients during a market cycle would imply that high-beta stocks might become lowhbeta stocks during a market downturn. As a result, the formula plan results would be, at best, no better than the results to be attained by buying and holding an average-beta portfolio throughout the market cycle. Fortunately, there is an accumulating body of empirical evidence .' YR: ~ ' ' J ‘.I ' I (“(112, 22 which supports the stationarity property of the beta coefficient. At the level of the individual security, Blume (4) examined the station- arity of the beta coefficient for 251 securities over the 1927-1960 period. Individual security betas for six 7-year non-overlapping subperiods were regressed against the corresponding betas calculated for the subsequent subperiod. The average correlation coefficient was found to be about .72 which Blume interpreted in favor of the stationarity thesis. Levy (30) calculated a 52-week beta coefficient for each of 500 New York Stock Exchange securities for each of the ten years 1961-1970. He compared each individual beta with the corresponding beta of the following year to test the ability of the current beta to predict the beta for the next period. Levy was thus able to correlate successive pairs of betas for each security for nine different periods. The quadratic mean of the nine correlation coefficients for individual securities was found to be .485.7 However, when individual securities were combined into portfolios of 5, 10, 25 and 50 securities, correla- tion coefficients increased to .713, .815, .914 and .985 respectively.8 Levy concludes: Evidence indicates that this risk measure is remarkably stationary for larger portfolios, less stationary for smaller portfolios and unpredictable for individual securities. Predictability improves materially as the forecast period lengthens. . . .(30, p. 62) In a similar vein, Blume, in another study (5) found that 84-month individual beta coefficients calculated for portfolios of 20 and 50 7Levy (30), p. 58. 81bid. ’ p. 62. 23 securities exhibited considerable stationarity over time with product moment correlation coefficients of .92 and .98 respectively when the July 1954 to June 1961 period was compared with the July 1961 to June 1968 period. The conclusion to be drawn from these studies can be summarized by saying that the stationarity property of the beta coefficient apparently increases as a function of both the time period over which beta is computed and the number of securities in the portfolio. For purposes of this study it will be assumed, based on the studies cited above, that a portfolio of high-beta securities and a portfolio of low- beta securities can be expected to remain high and low respectively over the 1962-1971 period under examination. However, as a precaution against the possibility that the beta coefficient of the securities used in this study might become unstable, the coefficients will be recalculated each year using the monthly returns for the preceding five years. That is, the securities used in the high- and low-beta portfolios at the beginning of the study are selected from the two highest and two lowest deciles of the Sharpe-Cooper study based on monthly price changes during the 1957-1961 period. This study will continually update this original data which ends with the coefficients for the 1962-1966 period so that, for example, new coefficients will be calculated in 1968 based on monthly returns from the preceding five-year period, 1963-1967. Thus a set of five market sensitivity coefficients will be calculated for each security for the overlapping five-year periods 1962-1966, 1963-1967, 1964-1968, 1965-1969, 1966- 1970. This set of coefficients will indicate whether or not individual securities are maintaining their volatility level relative to other .C a!— 24 securities in the portfolio. Formula plan results will be calculated using the original securities throughout the ten-year test period. However, an attempt will also be made to keep the average beta of each portfolio at roughly its beginning level by substituting other high (low) volatility securities for those securities whose coefficient decreases (increases) be enough to move them out of their previous risk class. Thus, if a security's market sensitivity coefficient for the most recent sixty months should fall below the average of the Sharpe-Cooper risk-class nine (1.20), the security is assumed to be sold at the beginning of the year following the 60—month period for which the coefficient was calculated and replaced with another secu- rity whose coefficient has remained above 1.20. Similarly for the lowbbeta portfolio, substitution will occur when a security's market sensitivity coefficient rises above .75. Formula plan results will be calculated using this substitution method although it is expected that there will be little difference in the results since most of the securities in each beginning portfolio are expected to remain in their original risk class. The high- and lowbbeta portfolios will each consist of twenty securities. Securities for each portfolio will be randomly selected from those available in the two upper and two lower Sharpe-Cooper risk-class deciles. In the process of selecting portfolio securities, only those securities which have shown previous risk-class stability-— defined as having remained within two deciles of their 1962 decile level over the preceding ten years--will actually be used. Securities which are randomly selected but do not meet the test of stability will be discarded. Random selection will be continued in this fashion M A a 25 until a total of twenty-five securities for each portfolio is selected. Five of these twenty-five (the last five chosen) will be used as substitute securities when any of the remaining twenty move out of their risk class. The use of twenty securities in each portfolio was based partly on the beta-stationarity studies previously referenced, which indi- cated that a portfolio of twenty or more securities was likely to experience significant stability in the average beta over time. In addition, other studies have shown that a portfolio of this size is large enough to provide adequate diversification from the standpoint of eliminating unsystematic risk. Thus, Evans and Archer (12), Fisher and Lorie (18), and Latane and Young (29), have all concluded that 90-95 percent of all unsystematic variability of return can be eliminated with a portfolio of eight to fifteen randomly selected securities. Consequently, it is assumed that the twenty securities being used in each portfolio for this study will provide both a high degree of stability in the portfolio beta as well as providing enough diversification to eliminate most of the unsystematic risk component of the individual securities contained in the portfolios. Technigue The performance of the four formula plans will be compared with the performance of four portfolios assumed to be purchased on either January 15, 1962 or January 15, 1967 and held until either December 15, 1966 or December 15, 1971. One of these buy-and-hold portfolios will be the Standard & Poor's 500 Stock Index, a widely used surrogate for the "market" in general. The other three buy-and-hold portfolios will no .. AU “1'. {u r 9)» a \. Jt SECL 26 consist of (l) the twenty high-beta stocks; (2) the twenty low-beta stocks; and (3) a portfolio of the high- and low-beta stock portfolios combined in equal amounts. One reason for using the high-beta and lowbbeta portfolios is to test the portfolio theory idea that a high- risk portfolio should earn a higher return than the market portfolio over a long period of time. Similarly, the low—risk portfolio should be outperformed by both the market average and the high-risk portfolio. Performance of the Standard & Poor's 500 Stock Index will be measured by computing a monthly geometric mean where the monthly price used in the price relative calculation is the monthly average price of the index. The geometric mean is defined as the nth root of the product of n monthly price relatives: n /P /“ n 1 [i=1 (P11 1—1,1 + D11)] (2.1) R1 where: i = l—---60 observations for the five-year periods 1962-1966 and 1967-1971 and i = l----120 observations for the 1962-1971 period. P = Value of the index for month i 11 = Value of the index for month i—l P i-1,l D11 = Dividend paid on the index during the period i-l to 1, expressed as a percentage of the index. Performance of the high-beta, low-beta and equallydweighted buy- and-hold portfolios will be measured using a technique developed by Evans (13). Monthly prices for each security are taken as the closing price on the fifteenth of each month as reported in the ISL Daily Stock §£i£g_books and adjusted for stock splits and stock dividends. These prices are then used to calculate monthly price relatives for each security: -J C71. 27 + 2.2) R1 = Pi} Di] 3 1914’j where: i - 1 60 or 120 observations j = l 20 or 40 securities Rij = Price relative for security j over period i—l to i Pij = Price of security j at beginning of period 1 P1_1 j - Price of security j at beginning of period i—l 9 Dij = Dividend paid on security j over period i-l to i. For the buy-and-hold plans (except the S&P 500 Index) the equal dollar weighted compound return for each security will be calculated as: m The portfolio geometric mean return is then given by: n 1/m (2.4) RBH = (l/njg1 Rj) where: RBH 3 average geometric mean return for a portfolio of n securities over m monthly periods. For example, assume a three-security portfolio with the price relatives shown in Table II-2. TABLE II-2 RETURN MATRIX 1 FOR BUY-AND-HOLD CALCULATIONS Period Price Relatives Security 1 2 3 4 5 1.01 1.02 1.03 .99 1.01 .98 1.01 1.02 1.03 1.02 3 1.04 .96 1.05 .98 1.03 .1 is Y6 1: f3 28 If equal dollar amounts are invested in each security then, from equation 2.3: R1 = (1.01 x 1.02 x 1.03 x .99 x 1.01) = 1.0610 R2 = ( .98 x 1.01 x 1.02 x 1.03 x 1.02) = 1.0607 R3 = (1.04 x .96 x 1.05 x .98 x 1.03) = 1.0581 From equation 2.4: RBH RBH RBH Thus, the monthly geometric mean for this five-month period is [(1.061 + 1.0607 + 1.058l)/3]l/5 (1.0599)1/5 1.0117 1.17 percent. Under the formula plan strategies, returns must be calculated for subperiods which vary from as little as one month to periods of several months in length. For the constant-ratio plans this subperiod is defined as a fixed interval of three months. Thus, over the ten- year period 1962-1971, there will be a total of 40 subperiods. For the variable-ratio plans, the length of the subperiods varies since the percentage composition of the aggressive and defensive portfolios will be changing according to the price-earnings ratio of the S&P 500 Index. Within each subperiod two returns will be calculated, one for the lowbrisk portfolio (RE) and one for the high-risk portfolio (RE). For each portfolio the monthly return is calculated as: n . = l 2 (2 5) R1 /n j=1 Rj where: R = average return in month i for a portfolio of n securities. 29 For the constant-ratio plan where the subperiods are fixed intervals of three months and the proportion invested in each port- folio is maintained at SOZ-SOZ, the subperiod return (Rk) is the simple average of the monthly returns in each three-month subperiod for each portfolio. 3 3 H L 1 z + 2 /3 (i=1 R1 i=1 R1) (2.6) Pm = 2 where: R. = Combined return in subperiod k for the low risk portfolio and the high-risk portfolio. The portfolio geometric mean (Rcr) is then calculated as the nth root of the product of the p subperiod portfolio returns. P (2.7) Rcr = (r21 Rk)l/p For the variable-ratio plans the subperiods are variable in length and the proportions invested in the lowerisk and high-risk portfolios change according to the price-earnings ratio of the S&P 500 Index. Thus the procedure for calculating the geometric mean is somewhat different than for the constant-ratio plans. The monthly return for each port- folio will still be calculated as in equation 2.5. However, the sub- period return-defined as the interval during which the proportions invested in the two portfolios remain unchanged-dwill be calculated as: 2 L 2 H _ P R + P R 2 where: z = number of months in subperiod k. Pr = proportion of the total invested in the low~risk portfolio. P3 = proportion of the total invested in the high—risk portfolio. 100% and Pr + PS n) (I) m “1‘ in Pe‘ the 30 Since the subperiods are of different length the geometric mean return for the entire five— or ten-year period must be weighted by the length of each subperiod. q (2.9) R = ( g l/zz where: 22 = sum of the z subperiods of length q. As an example of the variable-ratio calculation procedure, first assume the same returns matrix as shown in Table II-2 and, in addition, assume a second portfolio of three securities with period price rela- tives as shown in Table II-3. TABLE II-3 RETURN MATRIX II FOR FORMULA PLAN CALCULATIONS Period Price Relatives Security 1 2 3 4 5 1.03 1.01 .98 1.02 1.00 1.01 1.03 .94 .98 1.01 1.05 1.04 .98 1.00 1.02 Secondly, assume that during periods one and two the price- earnings ratio of the S&P 500 calls for 70 percent of the total to be invested in the three securities comprising Return Matrix I and 30 percent in the three securities of Return Matrix II. With these assumptions the price relative for the two months of the first subperiod would be calculated as: L 1.01 + .98 + 1.04 1.02 + 1.01 + .96 R1 = 3 + 3 2.007 31 . . + . . + . + . R: = 1 03 + 1 01 l 05 + l 01 l 03 1 04 = 2.057 3 3 R = .70(2.007) + .30(2.057) z 2 _ 1.049 + .6171 R— Z 2 R = 1.0110 Now assume that in period 3 a change in the price-earnings ratio of the S&P 500 Index results in a change in the percentage composition of the two portfolios so that the total dollar investment is split--50 percent in each portfolio. A new subperiod thus begins and the weighted arithmetic return for subperiod 2 must be calculated. L g 1.03 + 1.02 + 1.05 + .99 + 1.034'.98 + 1.01 + 1.02 + 1.03 R2 3 3 3 = 3.053 a}; — .98 + .94 + .98 1.02 + .98 + 1.00 1.00 + 1.01 + 1.02 + + 3 3 3 = 2.978 .5(3.053) + .50(2.978) R = z 3 _ 1.5265 + 1.4890 R _ z 3 R = 1.0052 The geometric mean return for the formula plan during these five months (two subperiods) is then calculated by: .. 1/5 er - (2Rz x 3R2) 21o 1.011 + 310 1.0052 log R.vr = ‘E 5 g R = 1.007515 vr 32 Thus the monthly geometric mean for the five-month period is 0.7515 percent. Annualized over a lZ-month period the return would be 9.40 percent (i.e. 1.00751512). Terminal dollar values for the four buy-and-hold plans and the four formula plans will also be calculated and reported with the geometric mean returns. It is recognized that the geometric mean return is a much better measure of a portfolio's performance over time than the ending dollar value of the portfolio. For one thing ending dollar value figures fail to take into account dividends received on the portfolio securities during the period studied. Secondly, the ending dollar value is a point in time value reflecting only the value of the portfolio as of the terminal date of the study; as a result, it says nothing about the value of or returns earned on the portfolio at intervening points in time. Nevertheless, the ending dollar value of a portfolio has the advantage of being easily under- stood. The ending value can be compared with the beginning value to see the absolute dollar magnitude of the capital gains realized during the period studied. Transactions costs will be taken into account by assuming that a one percent commission would have been paid on all purchases and sales. The one percent figure was arrived at by examining a commis- sion schedule for 1971 and noting that the commission paid on the sale or purchase of a round lot of a $50 stock would approximate one per- cent of the stock's value. No attempt is made to calculate the exact commission on buying and selling stocks of varying prices since the fifty stocks used in this study vary in price from under $10 per share to over $100 per share and thus exact calculation of commissions 33 would be very cumbersome. Rather, the commission on a $50 stock is used to obtain a rough estimate of the magnitude of commissions that would actually be generated in a real world investment situation. The total commissions generated during each subperiod from rebalancing the aggressive and defensive portfolios are deducted from the total market value of the two portfolios and this reduced amount then becomes the beginning value for the next subperiod. Terminal values for each formula plan using this method of accounting for transaction costs will be reported along with the terminal values before considering transaction costs. In this way the effect of transaction costs on the formula plan results can be readily compared with the transaction costs associated with the buy-and-hold plans. The question of whether the various formula plans and buy-and— hold portfolios attained their return because of differences in the amount of risk assumed will be analyzed by using the Treynor Index (45). This index computes the excess return on the portfolio in question per unit of systematic risk. The formulation is: R - R T=_;L_—f— BiP where: R. = return on portfolio p during period 1 expressed as an annualized monthly geometric mean. Rf = risk-free rate of interest over the portfolio time horizon. The proxy for This rate will be the average interest rate on intermediate—term government bonds at the beginning of 1962 and 1967 and the average interest on long-term government bonds at the beginning of 1962. Bi = weighted average beta of portfolio p p during period 1. 34 Sharpe in (40) has shown that a weighted average of individual betas multiplied by the variance of the market index provides a good measure of the riskiness of an efficiently diversified portfolio. That is: m . = X .b Blp j=l ’5 j 2 _ 2 and op - Bip OI Since the market variance is a constant for any security or portfolio, the weighted average of the individual betas may be used to measure the ex—post risk of a diversified portfolio. CHAPTER III ANALYSIS OF RESULTS Summary This chapter reports the results obtained from simulating the performance of the four types of formula timing plans described in Chapter II. As an aid to the reader a summary of the chapter is provided before the detailed results are reported. It should be pointed out that for each type of formula plan two sets of results for three time periods were calculated. For each formula plan in each of the three time periods, results were obtained using port- folios of common stock in which the component securities were left unchanged throughout the period. For clarity of exposition these results will be referred to henceforth as "unadjusted beta" portfolio results, referring to the fact that the securities were selected based on their volatility at the beginning of 1962 and no substitution of other securities was made to adjust for changes in their beta coefficients over time. Results were also obtained for each formula plan using portfolios in which some substitution of securities was made to allow for the fact that some securities in the lowh and high- beta portfolios did shift out of their relative risk-class during the time period studied. This substitution procedure is explained in more detail in Appendix A. Formula plans in which substitution took place will be referred to as "adjusted beta" portfolio results. 35 36 A major conclusion to be drawn from the results is that the formula plans tested demonstrated superior performance relative to buying and holding the "market" as represented by the Standard & Poor's 500 Stock Index. As shown in Tables III-1a, III-1b, and III-1c all of the formula plan portfolios produced higher ending dollar values after allowing for a one percent commission on all purchases and sales than the S&P 500 in the 1962-1966 and 1962-1971 periods. However, in the 1967-1971 period the S&P 500 produced higher ending dollar values than any of the unadjusted beta portfolios. For all time periods covered, the monthly geometric means (calculated before commissions) of the formula plans were substantially higher than the geometric mean of the S&P 500. Such comparisons, of course, do not allow for differences in volatility between the market port- folio and the formula plan portfolios. Volatility differences are taken into account in calculating the Treynor Index which measures the excess return on the portfolio per unit of market risk assumed. As shown in Tables III-1a, III-lb, and III-1c the Treynor Index was, ‘with one exception, always higher for the formula plan portfolios than for the S&P 500 when the annualized monthly geometric mean is used as the basis for measuring return on the portfolios. In addition to the S&P 500, results were also simulated for three other buy-and-hold portfolios. These portfolios consisted of the lowbbeta portfolio used in the unadjusted beta formula plans, the high-beta portfolio from the same plans and a portfolio made up of both the high- and lowbbeta portfolios with equal investment in each. This plan is denoted as the "Equally-weighted" buy-and—hold plan in the tables. Comparison of the results from these buy-and-hold portfolz picture As and-hol: sent for and 1961 be.a f0} adjuster compari: hold p1; the Trey cate, a risk. '1 consiste formula were mix hi8h“bet the high Fin Vith the generate formula I the form: hold Plat but Smal; beta buy. SidEring 37 portfolios with the formula plan results presents a somewhat mixed picture. As might be expected from portfolio theory, the high-beta buy- and-hold portfolio produces higher ending dollar values after adjust- ment for commissions than any of the formula plans in the 1962-1966 and 1962-1971 periods. However, in the 1967-1971 period, the adjusted- beta formula plans produced higher ending dollar values while the un- adjusted-beta portfolios performed about the same. Again, however, comparison of the formula plan results with the high-beta buy-and- hold plan does not allow for differences in portfolio volatility. As the Treynor Index numbers in Tables III-la, III-lb, and III-1c, indi- cate, the formula plans produced higher returns per unit of market risk. This was true for all periods in cases where the formula plans consisted of high-beta stock portfolios and Treasury bills. When the formula plan used high- and low-beta stock portfolios, the results were mixed with the variable-ratio plans always outperforming the high-beta buy-and-hold and constant-ratio plans usually outperforming the high-beta portfolio. Finally, comparing the equallydweighted buy—and-hold portfolio with the formula plans leads to conclusions somewhere between those generated by comparing the high-beta buy-and-hold portfolio with the formula plans and those produced when the S&P 500 is compared with the formula plan results. That is, the equallydweighted buy-and- hold plan generated higher dollar returns than the S&P 500 portfolio but smaller dollar returns and geometric mean returns than the high- beta buy-and-hold plan. This result would have been expected con- sidering that the average beta level of the equally-weighted portfolio 38 lies between that of the high-beta portfolio and the S&P 500 portfolio. Within the formula plans themselves, three conclusions stand out. First, it is apparent from Table III-2 that the variable-ratio plans usually did not perform better than the comparable constant-ratio plans. This conclusion is somewhat unexpected from the nature of the two types of plans since more high-beta stock should be bought at low prices and sold at high prices with the variable-ratio plan than with the constant-ratio plan. Secondly, Table III-3 indicates that adjusting beta levels, at least in the limited manner employed in this study, did not improve on average, and, in fact, worsened the perform- ance of the formula plans. In eight out of twelve comparisons the geometric mean of the constant-ratio plans was higher than that of the comparable variable-ratio plans. Third, it was clear from the results that the variable-ratio plans using Treasury bills were superior to the variable-ratio plans which used lowbbeta stock. Table III-4 shows that the formula plans using Treasury bills as the lowerisk portfolio consistently performed better than the formula plans using a low-beta stock portfolio. This result may be biased by the particular lowhbeta sample chosen and the time period studied. Nevertheless, on a risk-adjusted basis, performance of the lowebeta portfolio would have to be significantly better than the results reported here in order to make up for the increase in the average beta of the portfolio which occurred when low-beta stocks were substituted for Treasury bills. As a general observation it should be pointed out that the geometric mean return provides a better measure for ranking perform- ance than ending dollar value because it measures total return, 39 including both dividends and capital gains, over the entire period whereas the dollar value figures measure only capital gains at a point in time. Thus the validity of the conclusions drawn in this chapter and the next should be based primarily on the geometric mean returns reported. The ending dollar value results are presented only as a secondary guage of the performance of the portfolios studied and to give the reader some idea of the magnitude of the dollar capital gains that would have been earned during the periods studied on the various types of formula and buy-and-hold plans. The remainder of this chapter presents more detailed results for the constant-ratio plan during the 1962-1966, 1967-1971 and 1962-1971 periods followed by the results for the variable-ratio plans over the same three periods. 40 TABLE III-la COMPARATIVE RESULTS, 1962-1966 Ending Dollar Annual- Value* ized Tgizn°f Before After MOnthly Geo. Ave. Treynor** Comm. Comm. Geo. Mean Mean' Beta Index Constant-Ratio Unadjusted Betas $5519 $5438 1.0149 19.42% .918 .1820 W/ Treasury Bills Constant-Ratio Adjusted Betas $5395 $5311 1.0108 13.76% .955 .1366 W/ Treasury Bills Constant-Ratio Unadjusted Betas $5336 $5244 1.0094 11.88% 1.229 .0949 W/ Lotheta Stock Constant-Ratio Adjusted Betas $5288 $5189 1.0067 8.34% 1.235 .0877 W/ Lotheta Stock Variable-Ratio Unadjusted Betas $6142 $5810 1.0124 15.94% .679 .2927 W/ Treasury Bills Variable-Ratio Adjusted Betas $6147 $5810 1.0101 12.82% .707 .2383 W/ Treasury Bills Variable-Ratio Unadjusted Betas $5952 $5328 1.0108 13.76% 1.071 .1269 W/ LOWbBeta Stock Variable-Ratio Adjusted Betas $6013 $5377 1.0101 12.82% 1.059 .1753 W/ Lotheta Stock Buy 8 Hold Lobieta Stock $3848 $3808 1.0037 4.53% .622 .0347 Buy & Hold High-Beta Stock $7632 $7592 1.0124 15.94% 1.836 .0941 Buy & Hold Equally weighted $5740 $5700 1.0080 10.03% 1.229 .0938 High & Low Beta Buy & H°1d $4710 $4670 1.0060 7.44% 1.000 .0331 S&P 500 Index *Beginning Value for each plan was $4000. **The risk-free rate used in computing the Treynor Index for this period was 4.13%. Con: Una: '—'/ '. Cons A311 Con: Una: 111/] Cons Adjt U/I Varj Unac W/ 1 Vari Adjl W1 Varj Unac 41 TABLE III-lb COMPARATIVE RESULTS, 1967-1971 Ending Dollar Annual- T e of Value* ized Plan Before After Menthly Geo. Ave. Treynor** Comm. Comm. Geo. Mean Mean Beta Index Constant-Ratio Unadjusted Betas $4543 $4468 1.0100 12.68% .945 .0463 W/ Treasury Bills Constant-Ratio Adjusted Betas $5173 $5069 1.0114 14.57% 1.002 .0698 W/ Treasury Bills Constant-Ratio ‘ Unadjusted Betas $4481 $4402 1.0069 8.47% 1.280 .0291 W/ Lobieta Stock Constant-Ratio Adjusted Betas $5098 $4980 1.0070 8.73% 1.277 .0414 W/ Low~Beta Stock Variable-Ratio Unadjusted Betas $4846 $4578 1.0076 9.51% .894 .0664 W/ Treasury Bills Variable-Ratio Adjusted Betas $5528 $5209 1.0083 10.43% .948 .1064 W/ Treasury Bills Variable-Ratio Unadjusted Betas $4526 $4042 1.0053 6.55% 1.247 .0340 W/ LowaBeta Stock Variable-Ratio ‘ Adjusted Betas $5302 $4706 1.0076 9.51% 1.238 .0478 W/ Low—Beta Stock Buy & Hold . Lotheta Stock $3962 $3922 1.0049 6.04% .671 .0117 Buy & Hold High-Beta Stock $4506 $4466 1.0039 4.78% 1.889 .0321 Buy & Hold . ' Equally weighted $4234 $4194 1.0044 5.41% 1.280 .0150 High & Low Beta Buy & H°1d $4697 $4657 1.0054 6.68% 1.000 .0206 S&P 500 Index *Beginning Value for each plan was $4000. **The risk-free rate used in computing the Treynor Index for this period was 4.61%. 42 TABLE III-1c COMPARATIVE RESULTS, 1962-1971 Ending Dollar Annual- Value* ized Tgpznof Before After Monthly Geo. Ave. Treynor** Comm. Comm. Geo. Mean Mean Beta Index Constant-Ratio Unadjusted Betas $6638 $6464 1.0124 15.94% .931 .0964 W/ Treasury Bills Constant-Ratio Adjusted Betas $7494 $7251 1.0111 14.16% .979 .0807 W/ Treasury Bills Constant-Ratio Unadjusted Betas $6461 $6298 1.0081 10.16% 1.255 .0482 W/ Low-Beta Stock Constant-Ratio Adjusted Betas $7389 $7171 1.0069 8.60% 1.260 .0543 W/ Lotheta Stock Variable-Ratio Unadjusted Betas $8030 $7173 1.0109 13.89% .785 .1612 WI Treasury Bills Variable-Ratio Adjusted Betas $9319 $8296 1.0092 11.62% .825 .1669 W/ Treasury Bills Variable-Ratio Unadjusted Betas $7306 $5909 1.0076 8.51% 1.159 .0891 W/ Low-Beta Stock Variable-Ratio Adjusted Betas $8861 $7061 1.0089 11.22% 1.146 .0877 W/ Lotheta Stock Buy & Hold Lotheta Stock $3900 $3860 1.0043 5.28% .647 .0200 Buy & Hold High-Beta Stock $9301 $9261 1.0082 10.30% 1.862 .0511 Buy & Hold Equally Weighted $6600 $6560 1.0062 7.70% 1.255 .0545 Buy & Hold S&P 500 Index $5743 $5703 1.0057 7.06% 1.000 .0299 *Beginning Value for each plan was $4000. **The risk-free rate used in computing the Treynor Index for this period was 4.10%. 1 111 1 1| I! 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Hooum mummuaoH \z mqao mNN H awn HH «moo H qqmmm ommma mmumm emumsflwmap . . o . . wHHHm anammmus \2 ONmH me “Ne 0H qu0 H wmqmm mlfimmw mmumm Humuwnofiwmcp xmucH wumm awmz .ooo 5mm: .omw .8800 .5800 Hoammuh .m>¢ mmuwamsas< hasucoz umum< muommm swam mo make .msHm> umHHoo wcham oomalwoma .mz¢qm OHHHH mcHaanmma xmvcH com mam cowo. ooo.H Nwm.m «moo.H amoqa acme“ eHomuaqmusam mmumm 30H a swam omHo. omN.H NHq.m «coo.H quqm qmmqm wouanwz mHHmovm nHomImcmlksm . . o . . xUOum wuomlanm Hmmo mwm H awn c amoo H ooqqm oomqm mHomlmsmlzom . . . . . xooum mummlzog NHHo Hmo see c mqoo H mummw momma vHomlmswlmsm . . . . . xuoum wommlaog \3 HHHO aam H ama w choc H ommsa wmoma mmumm emumsns< . . . . . mHHHm huommmua \3 wmoo moo H mum «H «HHo H moomw mnHmm mmuwm moumSmm< o o o o o Umuoum mummlgq \3 Homo owm H ans m mcoo H Nose» quqm mmumm umumsmema: . . . . . mHHHm >u=mmmuH \3 mesa was awe NH OOHo H moses mmmqa mmumm umumamemcs somaH ouom cmoz .060 com: .omu .eaoo .8800 uoahoua .m>< mouHHmsoa< anuaoz umum< ouommm GMHm mo mama QIHHH mamHH waHocmem« NowaH 00m mam ammo. coo.H Nmo.a amoo.H moamm mqamm eHomuecmuasm mmumm sou m eme memo. mm~.H Noa.a Nmoo.H ommmm oommm emuemHmz sHHmsam wHomI00MI>=m 000 mom I w m HHmo. 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