ESTEW‘TEGN OF BETA RISK WNNENTS' FGR REEUCTEON 8F PREQICTIGN ERROR IN PfiRTFQLfiO MflDELS Thasis for the Dem of Ph: 3‘ Evil-Cmfifili‘é SWTE rUE-‘ii‘JERSiTY‘ R. CGRWKN GRBB‘E E974 ....... LIBRARY _ Michigmsmtc O 'r‘ Umvcrsxy r This is to certify that the thesis entitled ESTIMATION OF BETA RISK COMPONENTS FOR REDUCTION OF PREDICTION ERROR IN PORTFOLIO MODELS presented by R. Corwin Grube has been accepted towards fulfillment of the requirements for Ph.D Business Adm., Finance degree in Major professor Date 4/74 0-7639 ‘ no 1MA-' ' 800K BINDERY INC. 95511193333. ABSTRACT ESTIMATION OF BETA RISK COMPONENTS FOR REDUCTION OF PREDICTION ERROR IN PORTFOLIO MODELS By R. Corwin Grube Two primary uses have been made of systematic risk as it applies to common stocks: (1) the evaluation of historical performance of common stock portfolios (2) the construction of common stock portfolios according to the risk-return (RR) criteria. This research focuses on the construction of common stock portfolios according to the RR criteria. In order to estimate the worth of a common stock by the RR criteria, some ex ante estimate of the systematic risk of the stock is required. Typically it has been assumed that systematic risk is stationary hence a computation of ex post systematic risk can serve very well as an esti- mate of ex ante systematic risk. This assumption, however, still leaves the question of how to compute ex post systematic risk e.g., should the historical estimate be computed over three years, five years, ten years or some other time period and should observations of return within this period (used to compute the systematic risk) be taken weekly, monthly, quarterly, etc. Furthermore, should the relationship of risk and return be assumed to be R = a i i + Bi Rm + e or should some alternative market model be used? Two market models were considered in this paper: (1) the ordinary R. Corwin Grube least squares equation presented above and (2) the market model pre- sented by Lawrence Fisher and Jules Kamin at the Midwest Finance Meetings in March 1972 which depicts the relationship of risk and return as R1 3 Bi Rm. Twenty-eight estimates of ex post systematic risk were com- puted for each of the thirty-five securities in the sample in each of two different (but overlapping) test periods. These twenty-eight esti- mates were then compared to the volatility (Ri/Rm) of the individual securities in each of twelve different holding periods in test period I and each of eight different holding periods in test period II. These differences were defined as prediction error and the mean absolute value of prediction errors was computed for all securities for each measurement period, observation interval and holding period. Prediction errors were computed for both market models and these results also compared. The results indicated that there is not a great deal of consistency between securities i.e., the estimate of ex post systematic risk which minimizes prediction error for security i will not necessarily minimize prediction error for security j. The results also indicated that the minimum mean prediction error is most decidedly a function of the time period examined. In test period I the minimum mean prediction error arose from 69 observations of monthly return used to estimate volatility in a fifteen month holding period while in test period II the minimum mean prediction error arose from three observations of quarterly returns used to estimate volatility in a three month holding period. In general and on average, it was noted that prediction error was positively corre- lated with holding period length and that between 75 and 100 observations R. Corwin Grube of return provided the minimum mean prediction error. It was also noted that the ordinary least squares procedure of computing ex post systematic risk provided a smaller mean prediction error than the Fisher-Kamin procedure in the majority of instances. The results further indicated that highly accurate estimates of ex ante systematic risk were difficult to obtain e.g., the mean abso- lute prediction error averaged across all holding periods and measure- ment periods ranged between 0.45 (quarterly observation interval) and 1.20 (weekly observation interval) in test period I and between 0.37 (weekly observation interval) and 0.48 (quarterly observation interval) in test period II. Given the magnitude of these errors and the inability of a particular measurement period or observation interval to consis- tently provide superior prediction results, it would appear that without improved techniques to estimate ex ante systematic risk, the Risk-Return programming model is of very limited use in the real world. ESTIMATION OF BETA RISK COMPONENTS FOR REDUCTION OF PREDICTION ERROR IN PORTFOLIO MODELS By R. Corwin Grube A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Accounting and Financial Administration 1974 Copyright by R. CORWIN GRUBE 1974 ACKNOWLEDGMENTS Many faculty members at Michigan State have been supportive and helpful throughout my doctoral program. Dr. Hugo Nurnberg deserves credit for my first publication and for providing an example of hard work and discipline. Dr. Gardner Jones and Dr. Ronald Marshall deserve credit for agreeing to serve on the dissertation committee. Dr. Myles Delano deserves special credit. He served as academic advisor during my coursework and as committee chairman during my dis- sertation. In both these roles he displayed an unconventional wisdom. As academic advisor he permitted me to study extensively outside the College of Business. As committee chairman he permitted me to experi- ence rather than learn research. My colleagues at Michigan State provided invaluable assistance in many ways throughout the doctoral program. In particular I would like to acknowledge Tom Ulrich for his intellect and wit and Dick Williams for his maturity and wisdom. Dr. Marvin DeVries deserves a special note of thanks. His continued interest and encouragement both during and after my work at Grand Valley College will always be appreciated. Perhaps the ultimate debt is due to my family: to my parents for their concern; to my wife for her patience and understanding; to my two baby girls for their lack of understanding. iii TABLE OF CONTENTS Chapter I INTRODUCTION. . . . . . . . . Background and Objectives . Problem Identification. . . Market Model . . . . . . Model Specification. . . Portfolio Analysis Using Volatility as a Measure of Risk . . . Model Limitations. . . . Justification for Research. II PAST STUDIES. . . . . . . . . Appropriate Time Interval Considered by Investors . . . . . . . . . . . . . . . . . The Gonedes Paper . . . . . The Observation Interval Problem . . . . . . The Confidence Interval Problem. . . . . . . Observation Periods up to Three Years. . . . smaryoooooooooo III RESEARCH DESIGN . . . o . . . General Approach. . . . . . The Evaluation Model. . . . Computational Procedures. . Alternative Market Mbdel. . The Sample. . . . . . . . . Sample Size. . . . . . . Selection Procedures . . IV ANALYSIS OF RESULTS . . . . . Ordinary Least Squares (OLS) Beta Coefficients Computed from Weekly Observations of Return. . Ordinary Least Squares (OLS) Beta Coefficients Computed from Mbnthly Observations of Return . Ordinary Least Squares (OLS) Beta Coefficients Computed from Quarterly Observations of Return A Comparison of Weekly, Mbnthly, and Quarterly Observation Intervals. . . . . . . . Implications for Portfolio Construction . . . . iv 15 23 25 28 31 34 35 37 37 38 39 48 49 49 51 53 S3 59 61 62 66 Chapter The Effect of Holding Period Length on Minimum MAPE . . . . . . . . . The Fisher-Kamin Beta . . . . . . . . . . A Comparison of F—K and Ordinary Least Squares (OLS) Beta Coefficients. . . . . V SUMMARY AND CONCLUSIONS . . . . . . . . . . Specification of the Market Model . . . . Holding Period Length . . . . . . . . . . Measurement Period and Observation Interval Limitations and Suggestions for Further Research. Appendices Appendix A—l . . . . . . . . . . . . . . . . . . . Appendix A-2 . . . . . . . . . . . . . . . . . . . Appendix B-l . . . . . . . . . . . . . . . . . . . Appendix B-Z . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY O I O O O O O O O O O O O O O O O O O O O Page 67 68 68 8O 8O 83 84 86 89 100 107 120 129 CHAPTER I INTRODUCTION Bagkggound and Objectives In finance assets are frequently evaluated using quantitative measures for the expected return and anticipated risk associated with the asset.1 The relationship of these two parameters is frequently written in the following form, R - a1 + b (I—l) it t itRmt where Rit - the historical return during period t on an individual asset i a a l/n 2 R it t it - b/n E Rmt b = n 2 R R - Z R 2 R = the systematic risk it t mt it t it t mt component of asset i Rmt - the historical return on a portfolio of assets similar to asset 1 during period t n - the number of historical observations of Rit and Rmt In this expression the systematic risk component, b describes the it’ total risk of the asset so long as the asset is held in a portfolio of 1See for example, Chris A. Welles, "The Beta Revolution: Learning to Live with Risk", The Institutional Investor, Vol. V, No. 9 (September 1971), pp. 21-26 and following. See also, Frank E. Block, "Beta Evaluation", Wall Street Transcript, July 3, 1972, p. 29,056. 2 similar assets numbering at least twenty.2 The financial community uses the two parameter model in the follow- ing two ways: (1) for the evaluation of historical investment perform— ance and (2) for the prediction of the future relationship between the return on asset 1 (R1) and the portfolio of similar assets (Rm). Since it is easier to predict the behavior of a portfolio of assets than an individual asset (errors tend to cancel), and since the relationship is assumed to be stationary over time, the model can be used to predict the behavior of an individual asset with historical information and some estimate of the behavior of the portfolio. The historical information used consists of the n observations of R and Rmt made during time it interval t. The purpose of this research is to examine how the parameter, bit’ can best be estimated to provide the model user with the minimum pre- diction error when predicting systematic risk in some future time interval. Problem Identification There are several factors which will influence the predictability of future systematic risk. Equation I-2 below provides a model which measures prediction error as predicted less actual return of the indi— vidual security and shows the several factors which influence the computation of bit'3 In equation I-2, the size of the prediction error 2J. L. Evans and Stephen H. Archer, "Diversification and the Reduc- tion of Dispersion: An Empirical Analysis", Journal of Finance, December 1968, pp. 761-769. 3A. M. Mood and F. M. Graybill, Introduction to the Theory of Statistics, McGrawbHill, 1963, pp. 335-343. 3 will be a function of the ability to correctly estimate b it' - 2 EP1 (Rp - Ra) — oei l + 1/n + (Rmk - Rmt) (I-2) - 2 2(R - R ) where t mt mt R k = the observed return on the portfolio of assets in time period k Oei = i—{E (Rit - air _ bitRmt)2} n-2 and n, ait’ and bit are as described in equation (I-l). The problem of estimating bit breaks down into two major com- ponents: (l) establishing the Optimal market model and (2) specifying the parameters of the market model selected. Market Model There exist an infinite number of potential market models. The market model specifies the computational equation of the historical beta coefficient, bit’ and correction factors of any sort can be in— cluded in the computation methodology. Thus there would exist, in theory, as many market models as correction factors. Until recently however, there was basic agreement that the prOper (or at least suit— able) relationship between the return on an individual security and the return on the market could be expressed as in equation (I-l). At the Midwestern Finance Association Meetings in 1972, Professor Lawrence Fisher and Jules H. Kamin found that unbiased estimates of b t deve10ped in equation (I-l) provided predictions of future systematic 1 risk, bik’ which were inferior to those deve10ped from equation (I—3) below.4 b = X R R t E mt This develOpment of the historical beta coefficient assumes an alternative relationship between risk and return viz, F Rit bit Rmt (1-4) Equation (1-3) is an unbiased estimator of beta i£_the functional form exhibited in equation (I—4) is assumed.6 What is at issue here is whether the relationship of risk and return for an individual security is best represented as it ‘ bit Rmt or R1: = air + bit Rmt Under the functional form assumed by Fisher, F bit Rit/Rmt whereas under the normal linear form, /R b = (R' - ait)/Rmt = Rit/Rmt - ait mt it it Clearly, if ai = 0.0, both estimates of systematic risk will be the t 4Lawrence Fisher and Jules H. Kamin, "Good Betas and Bad Betas; How to Tell the Difference", A Presentation at the Midwest Finance Association Meeting, St. Louis, Missouri, April, 1972. 5A11 statiStics generated by Fisher and Kamin will be denoted by superscript F. 6Harold J. Larson, Introduction to Probability Theory and Statis— tical Inference, John Wiley and Sons, New York, 1969, p. 320. 5 same. While ait typically approaches zero, it is seldom identically zero.7 This suggests that the two systematic risk measures will vary /R Which of the two forms is more correct will by a factor of ait mt' depend on which is the more useful i.e. which produces the least pre- diction error. Model Specification Once the market model has been selected, the parameters of the model must still be Specified. The prediction error associated with either equation (I-l) or (I—4) will depend on (1) the measurement period over which returns are taken, t, (2) the number of observations of return within the measurement period, n, and (3) the period over which the systematic risk to be predicted is computed. The importance of parameter specification to the selection of the market model is shown below. The prediction error model of equation (I-2) can be applied to the Fisher market model as follows:8 F F 2 - 2 EP1 a Oei {I + [(Rmk) /E(Rmt - Rmt) ]} (I 6) It can also be shown that9 7 Nancy L. F. Jacob, "Theoretical and Empirical Aspects of the Measurement of Systematic Risk for Securities and Portfolios", Unpub— lished Ph.D. dissertation, University of California - Irvine, 1970, p. 74. 8 F _ 2 * c ‘ ‘ 2 EP1 oe1 + Rm: 0 b + ZRkE [(a - a)(b - b)] + o a B ~ . c “t E[(a-a) (b-b)]-oa=0.0 F-“ 2.2:. 2 2 — 2 Hence EP1 Oei + Rm: ob oei + Rmt Oei [l/E (R1 - Rmt) ] Q.E.D. 9H. F. Larson, Op. cit., p. 339. 6 "F 2 1 _ F 2 * (Oei) ‘n-l Emit bit Rut) (I 6) This means that the squared prediction error associated with the market model of equation (I-l) will be less than the prediction error asso— ciated with the Fisher formulation of equation (I-4) for any observa- tion of R when mk _ 2 2 . (R - R ) (R ) o l + 1-+ mk mt < (oF )2 l + mk (I-7) ei n X - 2 ei Z - 2 t(Rmt - Rmt) t(Rmt - Rmt) . .2 F 2 Breaking the inequality into factors it can be shown that Oei < (Oei) when - R (b + bF) b R > mt it it t m a " T R1. “‘7“ it Further, - 2 2 1 + l + (Rink ' Rmt) (Rink) n 2 _ 2 < l + Z _ 2 (I-7B) t Rmt‘bit + bit) bit R and R > Bil E(Rmt - Rmt) mt R - T it -— mt 2n it Rmk It is not obvious when these conditions will prevail. It can be agreed, however, that the quantity (Rmt - fimt)2 will depend in part upon the 7 observation interval selected. A riori, a larger absolute percentage could be expected if daily rather than annual returns were used be- cause of the greater variability of returns associated with shorter time periods.10 Similarly, the variability of bit (ob) will depend upon the length of the observation interval. (The observation interval being that length of time between price observations used to compute returns). If bit and bIt are independent of the observation interval, an assumption used in some applications, then Rmt (bit + bit) will be positively correlated with the observation interval. Since the obser- vation interval is determined in part by the total period over which observations are taken (the measurement period), a seven year period, for example, would virtually eliminate annual observations of return since the significance of the statistical parameters would hold only for very large confidence intervals. The prediction results obtained by Fisher and Kamin may be due to the measurement period and/or observation interval employed rather than to any difference in the functional form of risk and return exhibited in equations (I-1) and (1—4). The anticipated holding period, discussed later, can also influence these results. Portfolio Analysis Using4Volatility_as a Measure of Risk The application of the two parameter model was originally presented by Harry Markowitz in his 1952 article.11 Because Markowitz utilized 10Lawrence Fisher and James C. Lorie, "Some Studies of the Vari- ability of Returns on Investments in Common Stock", Journal of Business, April 1970, p. 110. 11Harry Markowitz, "Portfolio Selection", Journal of Finance, March 1952, pp. 77-91. 8 variance of return as a measure of risk, it has frequently been argued that the attendant co—variance matrix required to evaluate combinations of securities is both too costly and too time consuming to be useful 12,13 in the selection of portfolios. The substitution of the beta coefficient for variance of return to measure risk was introduced by Professor Sharpe in 1967.14 This substitution reduced the information required to obtain estimates of risk for combinations of securities and provided a low cost, viable alternative to the Markowitz mean-variance approach. The Sharpe model is described here as it applies to the evaluation of individual securities. The Sharpe model requires two inputs: (1) expected return for each security = E i (2) expected systematic risk of each security 8 b1 By plotting ex ante estimates of E1 and b1 for each security and com- bination of securities, a feasible region of combinations would exist similar to Figure I—l below. Efficient combinations (maximum return for any given level of risk) are indicated by the darker (upper left hand) border. The goal is to determine this set of efficient portfolios given the input requirements mentioned above. 12William F. Sharpe, Portfolio Theory and Capital Markets, McGraw- Hill, 1970, p. 118. 13John Clark Francis and Stephen H. Archer, Portfolio Analysis Prentice-Hall, Englewood Cliffs, New Jersey, 1971, pp. 95-96. lAWilliam F. Sharpe, "A Linear Programming Algorithm for Mutual Fund Portfolio Selection", Management Science, March 1967, pp. 499- 510. FIGURE I-l Linear programming can be used to accomplish this goal. Since the objective is to maximize portfolio return for any given level of risk, the following notation is in order: n Ep = iélxifii n b = 2 x b p i=1 i i where n - the total number of securities under consideration Xi - the portion of the total portfolio commited to security i The objective function can be stated as Maximize Z = l - E - p ( q) p qbp subject to N v _ 0 for all i L 5 xi 5 U1 10 and q = the relative importance of risk vs. return The constraints can be interpreted as follows: X1 > 0 dis-investment is not allowed 1" IA N IA C..‘ at least L prOportion and not more than U proportion of total funds must be held in security i XX = l 1002 Of the available money shall be invested 0 5 q 5 1 not more than 100% risk nor less than 0.0% risk can be assumed. For each arbitrary q selected (0 f q 5 l), the objective function will determine an optimal combination of risky securities i;g,, an Optimum portfolio. If q = 0, indicating risk (bp) is to be totally disregarded, then maximizing Z1) is equivalent to maximizing Ep. Figure I—2 shows that the linear programming algorithm would determine A as the optimum port- folio for q - 0. 0.0 FIGURE 1-2 P For q = l, the Optimum portfolio would be that portfolio which minimizes risk regardless of return. In this case, maximizing Zp is equivalent to minimizing bp. Portfolio B in Figure I—2 depicts this ll situation. For 0 > q > 1, say q a .5, the appropriate Zp line lepes upward at .5 and denotes C as the Optimum portfolio in Figure I-2. When q is chosen such that the slope of Zp is parallel to a linear segment of the efficient border, then any linear combination of the corner portfolios is equally acceptable and Optimal. In Figure I—3 this situation is shown. If q - .4 and is parallel to line segment CD as in Figure I-3, then any linear combination of portfolios C and D which satisfies the expression X D + XZD = l is equally optimal. l 0.0 FIGURE I-3 Model Limitations Two inputs are required to use the model, E and b1. From a i conceptual standpoint, E1 earlier, historical estimates of b1 are assumed to be the best esti- mates of future b1. The reasoning proceeds as follows. By arguing that bp is relatively stable over time, many authors have assumed that ex post estimates of b1 will provide the best estimate of and b1 are ex ante measures. As was pointed out 12 15 . future b1. The generally accepted methodology of determining his- torical b1 has been to use the linear regression analysis associated with the market model of equation (I-l). This reasoning suffers from two important limitations. First, stability Of portfolio volatility, bp, is no assurance of the stability of individual security volatility, b Also, the stability of the beta 1' coefficient is not necessarily equivalent to the predictability of the beta coefficient if, indeed, the slope of the least squares regression line is what should be predicted. Finally, no specifications have been set down as to the measurement period or observation interval which the model user should consider in determining the historical bi.l6 Justification for Research Incorrectly estimating the systematic risk of an individual security may cause the model user to incorrectly rank securities hence include in the portfolio securities which are less desirable than others which have been excluded. An examination of the linear programming model will demonstrate this argument. Just as the value of a particular combination of securities was expressed as Zp - (l - q)Ep - qbp, the value of any particular security can be expressed as zi = (l - q)Ei - qb1 (I-9) 15See for example, William F. Sharpe, Portfolio Theory and Capital Markets, MbGraw-Hill, 1970, p. 142. 16Nicholas J. Gonedes, "Evidence on the Information Content of Accounting Numbers: Accounting-Based and Market-Based Estimates of Systematic Risk", Journal of Financial and Quantitative Analysis, 13 For any given value of q = qo, 2 becomes a function of E and bi' i i For example, let qo - 1/2 and E1 = 0.10, then 21 assumes different values depending upon the estimate of b Table I-l provides an i. illustration. 21 3 (l-q)Ei — qbi qo - 1/2 E1 = 0.10 2.1-. :4. .1 0.00 .3 -O.10 .7 —0.30 1.0 -0.45 1.5 -0.70 TABLE I-l From Table I-l, it is apparent that 21 is a decreasing function of hi. This is reasonable since the relationship between value and risk is negative by assumption i.e., individuals are assumed to be risk averters. Figure I—4 plots the data in Table I-l. 1 0'0 0.3 14.0 b 1 -o.45 FIGURE I~4 14 Figure I-S below demonstrates a simple four security universe. Each security is assumed to have a different expected value but the same systematic risk. To simplify the example, it is assumed that each security's bi has been estimated at 1.00. A ranking of the four securities according to the 2 criteria would indicate security 1 as i zi 0'0 0.5 - 1:0 b 1 1 2 4 3 FIGURE 1—5 the first choice, security 2 as the second choice, security 4 as the third choice, and security 3 as the fourth choice. Satisfying the constraint that two securities must be held would indicate holding securities 1 and 2 in equal prOportions in the portfolio. If however, the correct values of bi are 0.5 not 1.0, then the prOper ranking of the securities would change. Now the highest ranking securities are 2 and 4, but not 1. In fact, security 1 now ranks last rather than first. With a large sample of securities and a prOportionately smaller number of securities held, the change in rankings could be even more significant. CHAPTER II PAST STUDIES The literature on portfolio theory is abundant. For the most part this literature has focused on the evaluation of historical performance of securities and portfolios or on implications of the theory itself. Little attention has been given to the problem of Specification of the risk-return model parameters when the model is to be used for predictive purposes. Michael C. Jensen's work with the portfolio model can be gener- ally cast into that category which attempts to measure the ex post performance of portfolios.l Specifically, he set out to determine whether the returns achieved by mutual funds were consistent with their systematic risk exposure. He did, however, partially test his input data by comparing one year return with two year return (for use in computing his beta co-efficients) over twenty years for each port- folio. Regressing the betas so derived, Jensen found a correlation co-efficient of 0.89 which he interpreted to mean that use of one year observations or two year Observations was largely immaterial in com— puting returns for use in estimating beta co-efficients. Marshall E. Blume, II, recognizing that the results obtained by 1Michael C. Jensen, "Risk, Capital Asset Prices and Evaluation of Portfolios", Journal of Business, April 1969, pp. 167-245. 15 16 Jensen need not apply to individual securities, tested 251 securities listed continuously on the NYSE over the period 1927-1960, for station- arity.2 He divided the period into four equal sub-periods, computed a beta co-efficient for each security in each sub-period, and regressed the 251 individual betas from each sub-period against the respective beta in each subsequent sub-period. His average correlation coeffi— cient was approximately 0.72. Blume interpreted this to mean that individual security betas were stationary over time. Both Jensen and Blume tested the stationarity prOperty of the beta co-efficient; Blume for individual securities, and Jensen for groups of securities. These tests are not really a specification of how the beta co-efficient should be computed. Blume for example, arbitrarily selected one month as the observation interval and seven years as the measurement period. Jensen did compare beta co-efficients deve10ped from one and two year observation intervals but he did not attempt to reconcile differences nor determine which of the methods provided the best results.4 As part of her 1970 dissertation entitled Theoretical and Empiri— cal Aspects Of the Measurement of Systematic Risk for Securities and 2Marshall E. Blume II, "The Assessment of Portfolio Performance: An Application of Portfolio Theory", Unpublished Ph.D. dissertation, University of Chicago, 1968. 3The stationarity property of the beta co-efficient and the pre- dictability of the beta co-efficient are similar but not identical. If the beta co-efficient is stationary then using historical informa— tion to estimate the future will produce small prediction errors. On the other hand, a particular beta co-efficient could be predictable but not stationary. 4Jensen showed that in the absence of measurement error the period over which these returns is computed is immaterial. 17 Portfolios,5 Professor Nancy L. Jacob examined various prOperties of the beta co-efficient for 593 securities on the NYSE listed continu— ously between 1945-1965. To determine the predictive ability of beta, she compared adjacent time periods of identical length, utilized the same observation interval for measurement of return, and regressed the individual security's betas so derived. She used three distinct time period lengths: one year, five years and ten years and three observation intervals: monthly, quarterly and annually to compute beta for the regression test. This procedure was employed for indi- vidual securities and for random and sorted portfolios of various size. Using the co-efficient of determination (R2) as her measure of predictive ability, she found that it was extremely difficult to pre— dict betas for one year periods (average R2 of approximately .100).6 As the period lengthened, however, from one to five years, the pre- dictive ability substantially improved (R2 of approximately .370). Dr. Jacob did not test the predictive ability of beta between one and five years. As a result of these findings, Dr. Jacob argued "... an investor cannot use the past average volatility... of a portfolio as a best "8 guess of its future value.... In other words, the beta co—efficient 5Nancy L. F. Jacob, Theoretical and Empirical Aspects of the Measurement of Systematic Risk for Securities and Portfolios, Unpublished Ph.D. dissertation, University of California - Irvine, 1970. 6The figures cited for one year were reported for monthly obser— vations of return only. 7Nancy L. F. Jacob, Op. cit., p. 91. 81bid., p. 93. 18 of an individual security or portfolio is not a particularly reliable estimate of either volatility or expected return of the same security or portfolio in future time periods. In the design of her experiment, Dr. Jacob specified the computa— tion of beta for individual securities as I: - .- t:(mri,t - mri)(mruht - mmmbt) (II-1) 2 .. t (Mrm,t ' mm) where mr1 and mrm are excess returns on the individual security and the market respectively. The market index used was unique in that it was comprised of the 593 securities which she studied. In addition, returns on the index were computed under the assumption that funds were re-allocated equally to each of the S93 securities in the market portfolio at each obser- vation. This technique of utilizing precisely the 593 securities in the sample as the market index may cause the residual risk of any security, e1, to be correlated with the market index.9 Portfolio theory typically assumes that E(ei) = 0. The effect on Jacob's results is uncertain. Because of the large number of securities, it may be insig— nificant. Professor Jacob found some results which do have implications for the specification of the market model of equation I—l. First, she found that the standard error of beta, 0 b’ is inversely related to the 9Michael C. Jensen, "The Performance of Mutual Funds in the Period 1945-1964", Journal Of Finance, May 1968, p. 392. 19 length of the period over which beta is measured.10 For example, systematic risk exhibited less dispersion when measured with annual returns over five years than when measured with annual returns over two years. Secondly, she found 8b was directly related to the length of the observational interval.11 That is, greater dispersion of the beta coefficient was evident for annual and quarterly observations of return than for monthly observations of return for any given period. She did not, however, reach any conclusion as to which observation interval provided the best estimates of beta (minimum Ob) nor over how long a period returns should be observed. One additional observation made by Professor Jacob is worth noting here, Capital market theory Specifies a perfect linear relationship between R1 and Rm. She found this relationship less than perfect but most in evidence when Rm was significantly different from zero. Another 1970 dissertation, this by F. B. Campanella Of Harvard University, addressed itself to three specific questions concerning the measurement of beta.12 First, Campanella asked, "Do computed values of beta vary with the length of the time period considered, Stflha five years vs. ten years; or are the values of beta sensitive to the time interval over which security returns are calculated, e.g., 3 n monthly vs. quarterly returns?"l He also asked, ...wou1d beta 1oNancy L. F. Jacob, Op. cit., p. 61. 11Nancy L. F. Jacob, op. cit., p. 61. 12F. B. Campanella, Measurement and Use Of Portfolio Systematic Risk, Unpublished DBA dissertation, Harvard University, 1970. 131b1d., p. 29. 20 computed for the time interval t=1951—60 be a good estimate of the appropriate beta for time interval t=1961—70?"l4 To test the stationarity of beta, Campanella regressed (bi)t where t= 1956-66 against (bi)t where t=1946~56 and found a correlation co—efficient Of .513. For the identical regression but t=l946-56 and 1936-46, he found a correlation co-efficient of .707. And for 1936—46 vs. 1926-36, he found a correlation co-efficient of .655. To make these results similar to Jacob's, the corresponding R2 has been com- puted and is shown below in Table II—l. Campanella's conclusion was that a relatively high correlation exists between the betas computed in different time periods and there is stationarity over time. In testing the effects of varying the observation intervals and time period lengths, Campanella used five, ten and twenty year periods and computed beta from both monthly and quarterly data. He used 1956-66 betas (computed from monthly observation intervals) as a standard and correlated all other betas with these. He found that betas computed over ten years are highly correlated with betas computed over either five years or twenty years for both monthly and quarterly TABLE II-l 2 Regression Study R 1936-46 vs. 1926-36 .429 1946—56 vs. 1936-46 .500 1956-66 vs. 1946-56 .263 14loc cit. 21 observations. His conclusion was that security betas are not sensi- tive to either the length of the time period used or the observational interval used.15 While Campanella argues that betas are largely stationary over time, he calls for more work to be done in this area. He specifically says, "... (we) need more research in the area of measuring the systematic risk of individual securities. The need is to test that stationarity in individual security betas over the most recent decade and further to test for within period stationarity."l6 One recent study which did examine the short term stationarity prOperty of beta over time is described below. Robert A. Levy developed a beta co-efficient for each of 500 NYSE securities in each of the ten years 1961-70.17 He then compared the betas so deve10ped in adjacent one year intervals to determine the predictability of (hi) using (bi)t via regression analysis. t+l The quadratic mean of the nine correlation co—efficients so obtained was .486. Levy went on to test the predictability of both thirteen week and twenty-six week betas using the immediately preceding 52 weeks as a measurement period for (bi)t' His correlation co-efficient was lower in each case. It should be noted that Levy used one week differencing intervals in his computation of the beta co-efficients. An interesting result is obtained if the Jacob and Levy studies 15F. B. Campanella, op. cit., p. 5. 16Ibid., p. 39. 17 Robert A. Levy, "On the Short Term Stationarity of the Beta Co-efficient", Financial Analysts Journal, November/December 1971, pp. 55-62. 22 are compared. Dr. Jacob found that for each of the three years 1963, 1964, and 1965, she was able to obtain an R2 of about .10.18 This R2 was Obtained, recall, by regressing (hi) on (bi)t for about 600 t+1 securities where (bi)t was computed over the immediately preceding one year time period using monthly data. Levy was able to obtain an R2 of more than .20 or twice that Of Jacob by using the measurement period of one year immediately preceding the year he was attempting to predict but using weekly rather than monthly observations of return.19 This suggests that Professor Jacob's contention regarding the inability of historical betas to predict future betas may have been premature. If predictive results can be improved 100% (.20/.10) by altering the specifications by which beta is computed, then perhaps further modifi— cation would provide even better results. Additionally, this result stands in direct contradiction to Campanella's claim that the obser— vation interval is irrelevant to the stationarity prOperty of beta. It would appear that additional empirical testing is desirable in order to resolve some of these controversies. One additional point should be made here since it applies to all of the studies cited above. The experimental design utilized in all cases involved regressing the beta co-efficient of one time period against the beta co-efficient Of an adjacent time period. As pointed out earlier, this test of stationarity is not identical to the test of predictability. Conceptually, beta is an indicator of a security's systematic risk i.e., its volatility vis a vis the market. For any 18Nancy L. F. Jacob, Op. cit., p. 91. 19Robert A. Levy, op. cit., p. 57. 23 given period of time, the real variable of interest (in a prediction sense) is the volatility of the security with the market: Ri/Rm' Examining the predictability of beta as constructed through regression analysis in adjacent time periods is equivalent to examining the pre- dictability of the predictor itself. Hence the tests have examined the predictability of the next predictor, not the predictability of volatility. Utilizing the beta co-efficient as the variable to be predicted requires an estimate of the quantity Ri - ai _ 6i. If volatility or E; is the variable of interest, then predicting a beta co-efficient will provide a prediction error which includes both a1 and e1 and thus differs from the variable of interest by a factor of 48“ + e1) Since R1 "(31“? __ Ii _., _-(ai+ei) . It may be argued R... Rm Rm Rm that this will not be of great concern since the expected value of the error term is zero, i.e., E(ei) - 0.0 and a will typically be near 1 zero. However, e will typically not be zero as evidenced by the i need to combine securities to eliminate residual risk. This suggests that to measure the predictive ability of beta an improved experi- mental design is required which utilizes volatility Or Ri/Rm as the variable to be predicted. Appropriate Time Interval Considered by Investors The agreement reached by Blume and Jensen would seem to indicate that specification of the beta co-efficient is not of critical impor— tance so long as the use Of beta is limited to longer time periods and the historical evaluation of performance. For shorter time periods, predictive purposes, and individual securities, the evidence is contra— dictory and the results of previous studies are inconclusive. Recent 24 evidence which indicates both the normative and positive apprOpriate- ness of these shorter time periods is discussed below. Friend, Blume, and Crockett have found that institutional turn- over of securities within portfolios has been increasing steadily over the last twenty years.20 In the case of mutual funds they found turn— over approximated 472 in 1968 and nearly 56% in the second quarter of 1969.21 This turnover rate implies an expected average holding period for any security in the portfolio of less than one year; since 100% turnover would occur in (l/.56) 1.79 years, the expected holding period for any security would be 1.79/2 or .895 years (assuming a uniform dis- tribution of the anticipated holding periods). This positive evidence is further augmented in a recent article by Ralph A. Bing. Bing found that nearly 802 of the institutional investors whom he surveyed were concerned with time horizons of three years or less; 602 were concerned with time horizons of two years or less; and 402 were concerned with less than one year.22 His sample included not only mutual funds, but insurance companies and in-house pension funds as well. From a normative standpoint, the work done by Evans indicates that this orientation toward shorter holding periods may well have theoretical justification.23 Evans suggests that by re-allocating 20Irwin Friend, Marshall Blume and Jean Crockett, Mutual Funds and Other Institutional Investors; A New Perspective, New York, MccrawbHill, 1970. 211bid., p. 9. 22Ralph A. Bing, "A Survey of Fractioners' Stock Evaluation Methods," Financial Analysts Journal, May/June 1971, pp. 55—60. 23John L. Evans, "An Analysis of Portfolio Maintenance Strategies", Journal of Finance, June 1970. 25 portfolio dollars periodically, one could eXpect a higher return on the portfolio than by simply buying and holding. Presumably the ran- dom nature of stock price changes is responsible for this phenomenon, but for this paper that is unimportant. What is important is that from both a normative and a positive standpoint, investors consider substantially shorter holding periods than the five to ten year periods for which studies of beta stationarity (predictive) prOperties are conclusive. The Gonedes Paper Before introducing the Gonedes paper it will be instructive to point out the relationship between the prediction error associated with equation 1-2 and the number of observations of R and Rm used to i compute beta. From equation I-2 recall, - 2 (R - R ) (EP )2 = E(Rp —R. )2 = o 1 + -1—+ m’k "’4' i i,k i,k e n z - 2 i (R - R ) Notice that the prediction error is a decreasing function of n since 6e is a decreasing function of n. This inverse relationship between i the prediction error and the number of observations suggests that to minimize the prediction error it is desirable to Observe the relation— ship between return on the security and return on the market for as long a period as possible e.g., since the inception of the company. Alternatively, for some given length of time, the observations of return should be for as short a differencing interval as possible e.g., hourly. It seems reasonable to eXpect that companies will change their 26 management, marketing, and/or financial policies over time thus providing investors with different information upon which to base expectations regarding the company. If this is the case, then it seems likely that the relationship between the return on the security and the return on the market will change over time. It would also seem reasonable that returns computed from very short observational intervals would be less than meaningful and subject to random dis- turbances of a large magnitude.24 Professor Nicholas Gonedes has suggested that "structural changes" will occur to the extent necessary to offset any benefits from increasing the number of observations or R and Rm after about 1 seven years.25 The observation interval used by Gonedes was one month, hence 84 observations provided the best estimates of a and hi i for use in predicting R given Rm. 1 Gonedes first reserved the initial six monthly returns of each of 99 securities in 1960 and 1968 as returns to be predicted. He then constructed a set of beta co-efficients for each firm by ordinary least squares procedures over the period 1946-67. The 99 firms were selected according to the following criteria: (1) monthly price- relative data were available for the firm from the CRSP tape for the period January 1946 to June 1968 and that selected annual data were available from the COMPUSTAT tapes for the same period; (2) the firm 2('Jack A. Treynor, W. W. Priest, L. Fisher, and C. A. Higgins, "Using Portfolio Composition to Estimate Risk," Financial Analysts Journal, September/October 1968, pp. 93-100. 25Nicholas J. Gonedes, "Evidence on the Information Content of Accounting Numbers: Accounting Based and Market Based Estimates of Systematic Risk", Journal of Finance and Quantitative Analysis (forthcoming). 27 was a member of a two—digit S&P industry grouping with at least fifteen member firms. The set of beta co—efficients for each firm was created by breaking down the total twenty—one year period into sub-periods. Gonedes selected the sub—periods just prior to the years of the reserved returns and divided the sub-periods into three, five, seven, ten, and twenty-one years of length. Figure II-l shows the schematic approach utilized by Gonedes for establishing sub-periods and reserved returns. FIGURE II—l b(l957-59) b(l965—67) b(l955-59 b(l963—67) b(l953-59) ResefzggoReturn b(l961-67) ReseiyggsReturn b(l950-59) b(l958—67) b(l946-67) b(l946—67) The prediction error was computed as EPi = (jo - R13)2 where R33 is the predicted one period rate of return for the ith firm at time j and R1:. is the actual rate of return. The predicted return jo was computed by first Observing the mar- ket return at time j, then using this to estimate the return on the individual security using one of the available historical betas and the relationship jo = a1+b1ij. Since there are six reserved returns in each of two years to be predicted and five betas to be used for predictive purposes, there are (6x2x5) 30 prediction errors to be recorded for each of the ninety-nine firms in the sample. 28 Cross sectional summary statistics indicated, as mentioned earlier, that the seven year period with 84 monthly Observations used to estimate beta provided the minimum prediction error. Gonedes concluded that "... for the aggregate and 'on average' the seven year observation period provides a (relatively) better set 26 of estimates". The Observation Interval Problem Using monthly observations of R t and Rmt to construct the Optimum 1 beta co-efficient as Gonedes has done implicitly assumes that the one month Observation interval is Optimal. Jacob, however, suggests that the variability of beta, 3% will be a function of the observation interval.27 Note that A (R r R ) EP = a: 1 + 1114 “’1‘ Fit 2 (II—2a) E(Rmt - Rmt) — 2 g . (n+1) ~2 (R - R ) 06 n + 02 mk tilt 2 (II-2b) £(Rmt - Rmt) *2 - 2 “2 but, 06 / E 0 implies mt Rit > O and Rmt < 0 implies Rit < 0. If, however, measurement error exists, this relationship will not hold. Professor Jacob has pointed out that the observation interval will influence this measurement error.31 This suggests that the interval of time used to observe R and Rmt will also affect the prediction error through the determina- it tion of b hence Us. A priori, one might expect that returns measured it’ over a short interval of time would be more vunerable to random effects than returns measured over a longer time period. The expected return for say, a one week Observation interval will be very near zero. Ignoring discounting, an upper bound for this weekly return might be the Fisher and Lorie average annual return for securities of about 0From equation I—l nZR R — ZR ZR bit = t it mt t mt t it 2 2 nERmt — (ERmt) 31Nancy L. F. Jacob, Op. cit., p. 63. 31 .09 divided by 52 or .00167. Very little random disturbance could cause R1t > 0 and Rmt < 0 or vice versa. If this occurs then the computation of bit would be biased downward (due to the monotonically increasing nature of ERit) relative to b computed for longer obser- it vation intervals. It may be that monthly or quarterly observations are less vunerable to small random fluctuations and in addition, would provide more Opportunity for random disturbances to cancel out. But longer Observation intervals imply fewer total Observations for any given period of time. Some trade off function is suggested for evaluating the effect on the prediction error of increasing the length of the Observation interval to reduce the random 'noise' versus the effect of reducing the number of observations. The Confidence Interval Problem In linear bivariate regression analysis the standard error of an estimated value of the dependent variable, x1 given some value of the independent variable, x2 is given by the expression32 A A l — O (xllxz) ‘ 012 l + n + (x2 "' X2) (II-6) 2 2 2x2 - (2x2) 11 For any given value of x2 it is possible to compute the standard error of the estimated value of x1 where x1 is given by x1 - a + bx2 + e. Mbre importantly, the set of confidence limits established for x1, given x , are non-linear.33 Figure II—2 on the following page provides 2 32A. M; Mbod and Franklin A. Graybill, Introduction to the Theory of Statistics, New York, McGraw-Hill, 1963, p. 336. 33Ib1d., p. 337. 32 an illustration of a typical set of confidence limits for x: < x2 _ where x; = a lower bound for the observed values of x2 Kg = an upper bound for the observed values of x2 FIGURE II-2 Note that equation II—6 is precisely the square root of the Gonedes prediction error model. What Professor Gonedes has done then is establish confidence intervals for the predicted values of return on individual securities given some return on the market. The market returns observed by Gonedes are presented in Table II—2.34 In rela- tion to Figure II-2 above, these returns correspond for the most part to values of x2 near either x: or Kg. Averaging the prediction error (confidence intervals) across all Observed values of RInk for each number of observations, n will pro— 1): duce a mean prediction error for each n gig, EPID) =‘Tf t EPk(n) 34The returns reported here are those for the Dow Jones 65 Com- posite Average. These returns would be similar to the returns ob— served by Gonedes because of the high correlation between returns on indicies. See Sharpe, Op. cit., p. 114. NC 33 TABLE II-2 Annualized Monthly MOnth Observations of Return 1960 1968 January -.716 —.413 February .089 -.302 March -.216 -.082 April -.228 .871 May .258 .222 June .380 .216 where EPk(n) = the total prediction error for all companies given beta co-efficients computed from n observations of return and market observa- tion, Rmk' For a larger n say n* = n+c, it would be expected that EP(n*){EP(n) since a larger number of observations of return will, ceterusgparibus, reduce the prediction error. If the differences in individual values of EPk(n) and EPk(n*) arise primarily in conjunction with Observations of very large and very small values of Rmk then the difference in the mean values EP(n) and EP(n*) need not hold for values of Rmk near the expected value of Rmk' Based upon historical evidence, the expected value of an annualized observation of Rmk would approximate 0.09.35 The one observation by Gonedes of annualized R.Ink 1960) would not seem sufficient to justify his conclusion that the near 0.09 (February seven year measurement period (n=84) is Optimal for values of Rmk SLawrence Fisher and James H. Lorie, "Rates of Return on Investments in Common Stocks", Journal of Business, January 1964, pp 0 1-17 0 34 near 0.09. An illustration follows: suppose for n = 24 that RInk is observed at .5, .1, and ~.6 and the corresponding EPk(n) values are .8, .l, and 1.2. These observations provide a mean prediction error of O.7=(1/3(.8+.l+l.2)=0.7). For n*=n+36, let the observations of R.Ink provide EPk(n*) values of .7, .12, and 1.10. The mean prediction error would be 0.64=(l/3(.7+.12+l.l)=0.64). This illustration demon- strates that while fame) < EM), EPk(n) < EPk(n*) for Rmk = .10. Observation Periods up to Three Years In selecting sub-periods for the determination of mean predic- tion error, Gonedes arbitrarily selected three, five, seven, ten, and twenty-one year periods for testing purposes. In the analysis of time series data a frequently utilized technique is that of weighting recent data more heavily than earlier data.36 This often gives better prediction results than simply considering all data equally. This suggests that future results may be more closely related to recent events than to earlier events. The presence Of weighting devices would seem to lend validity to the idea of testing sub-periods up to three years since in the prediction of monthly reserved returns (annualized), it may well be that a sub-period within the most recent three years provided the minimum prediction error. Figure II-4 on the following page shows the Gonedes assump- tion (line A) while lines B and C show potential alternative possi— bilities. 36Charles T. Clark and Lawrence L. Schkade, Statistical Methods for Business Decisions, South—Western Publishing Co., 1969, pp. 702-711. 35 Before any judgement can be exercised with respect to the appropriateness of line segments A, B, or C, further empirical work must be done in order to observe points between zero and three years. FIGURE II—4 Gonedes' Results Prediction ‘5 Error \\ ./ B \. ./ / \ ""' "7' 1c Measurement Period Length (in years) Summary The number of observations of Rit and Rmt’ the length of the period over which these observations are taken, and the variability of the returns will all affect the predictive ability of a security's future volatility. These items comprise the essentials of the speci- fication problem for the risk return model. Previous studies have been shown deficient in methodology and contradictory in results when attempting to predict the volatility of securities' returns in short time periods. It has also been shown that the market model itself ipgp, the ordinary least squares regres— sion procedures for computing beta co-efficients has also recently come under attack. What is required is a simultaneous attention to the number of 36 observations of R and Rm it the length of the period over which t9 these observations are taken, and the variability of these returns as they affect both the market models introduced in order to consider the influences on the prediction error of the various trade—off functions which exist. Only by dropping the ceterus paribus assump- tion can the combined effect of all these factors be ascertained and consequently the optimal market model and specification of parameters of the market model be determined. CHAPTER III RESEARCH DESIGN General Approach The purpose of this research is to examine the two market models presented in Chapter I utilizing alternative parameter specifications to determine which model and which set of specifi- cations provide the minimum prediction error when estimating the future volatility Of a security based upon its historical beta co-efficient. Unfortunately, a mathematical solution to the prediction error problem is not possible because the relationships (parameters) of the prediction error model are subject to change. Ideally, one could establish for an equation in several variables the set of partial derivatives at zero and solve the resulting series of equa- tions simultaneously for the minimum value of the dependent variable. This minimum value of the dependent variable would then depict the Optimum relationship of the independent variables. In the predic- tion error model, however, the question is not of establishing the optimal relationship of the independent variables, rather it is to specify the parameters of the model 143., what are the relationships which subsequently can be optimized? Accordingly, an empirical approach will be used in order to estimate the parameters of the model and the relationship of the independent variables. 37 38 The Evaluation Model This paper utilizes prediction error defined as in equation III-l. III-1 where Rik the return on security i in time period k R = the return on the market surrogate in mk . time period k. The quantity Rik/Rmk is the volatility of security i, will be defined as the systematic risk of security i, and is the measure of concern to investors. This is the relationship which will reflect the vol- atility of an investor's portfolio, not the regression equation of + a b tRmt + e it i The regression equation is the estimating device it' 1In the prediction model utilized by Gonedes (see equation II-2a) the predicted return, Rit will be a function of the historical beta used in the market model. The observed return, Rik will be condi- tional upon the market return, Rmk and the volatility of the security at that time (k). Implicitly this measure incorporates the slope of the regression equation at time k and as such represents an attempt to predict the future predictor rather than systematic risk only. Rewriting the equation will illustrate this point: *2 1 (Rink " 2:192 2.2 n+1 ‘2 (3e 1 + I; + 2(R - a )2 = (Rmk - Rmt) Ob + (T)Oe (III-2) '1 mt mt 2‘2 n+1 .2 _ . . 2 but (Rink ' Rmt) °b + ( n )Oei " 1“ [(ait ' “119 + (b1: ' bik)Rmk " 61k] (III-3) From equation III-3 the prediction error will be a function of the difference in the expected value of the p_parameter from time t to time k, and the residual error at time k as well as the change in the beta parameter from time t to time k. If the objective is to determine the best predictor for a firm's systematic risk behavior, then the error measurement should include only the systematic risk component of the model as in equation III-1. 39 for volatility; Rik/Rmk is the measure of interest. The term Rik/Rmk will be referred to as the sensitivity of the security throughout the remainder of this paper. The quantity bit is, Of course the historical beta co-efficient computed from the least squares procedures of linear bivariate regression analysis. It has been shown that average prediction error, when computed from returns to be predicted which are very different from their expected value, will not assure the model user that the beta value is Optimal because the average could conceal information to the contrary. A more representative sample of returns would be generated if longer holding periods were considered and the observed returns adjusted to a basis consistent with those used in the computation of the historical beta co-efficient, b The model used in this paper allows this it' flexibility since the quantity Rik/Rmk can be observed over a period of any length desired. In addition, Rik/R need not be adjusted to mk a shorter time basis (weekly, monthly, quarterly) since it is a ratio and the absolute difference in return is not measured. This procedure allows a greater range of returns to be observed without the additional time and expense for adjustment to a basis consistent with the returns used in the computation of the historical beta co-efficient while at the same time providing returns nearer those an investor might expect on an ex ante basis i.e., nearer the expected value of the market return distribution. Computational Procedures Two test periods were selected to observe the sensitivity measure. The first was the three year period 1/1/67 — 12/31/69, the second was 40 the two year period l/1/70 - 12/31/71. Allowing the first holding period to correspond to the subperiod l/l/67 — 4/1/67, the second holding period to 1/1/67 - 7/1/67 etc., Table III—1A shows the twelve holding periods corresponding to test period one (TPl) and Table III-lB shows the eight holding periods corresponding to test period two (TP2). A value of Rik/Rmk was observed for each holding period, for each security. Thus there were (20x35) 700 sensitivities computed for the sample.1 These sensitivities were computed as in equation III-4 below. Pit-l-lDit Pmt+l ik “'37-"1 T‘— '1 *9... it mt mk III~4 50 Pit = the price of security i at time t Pit+l = the price of security i at time t+l D = the dividends declared on security i it during time t Pmt = the value of the S&P 500 index at time t P = the value of the S&P 500 index at time t+l mt+l Dmt = dividends (in per cent) on the S&P 500 index during time t 1See subsequent section on Sample Size, p. 43 TABLE III-1 Time Time Designation Period Designation Period HP(l) 1/1/67 — 3/31/67 HP(13) 1/1/70 - 3/31/70 HP(Z) 1/1/67 - 6/30/67 HP(14) 1/1/70 - 6/30/70 HP(3) 1/1/67 - 9/30/67 HP(lS) 1/1/70 - 9/30/70 HP(4) 1/1/67 - 12/31/67 HP(16) 1/1/70 - 12/31/70 HP(S) l/l/67 - 3/31/68 HP(17) 1/1/70 - 3/31/71 HP(6) 1/1/67 - 6/30/68 HP(18) 1/1/70 — 6/30/71 HP(7) 1/1/67 - 9/30/68 HP(19) 1/1/70 — 9/30/71 HP(8) l/l/67 - 12/31/68 HP(ZO) 1/1/70 - 12/31/71 HP(9) 1/1/67 - 3/31/69 HP(IO) 1/1/67 - 6/30/69 HP(ll) 1/1/67 - 9/30/69 HP(12) 1/1/67 - 12/31/69 The importance of using dividends (and the benefits of using longer holding periods) in the computation of sensitivity is evidenced here by examination of Table III-2 which shows the S&P 500 return for each of the twelve holding periods in TPl. Table III-2 indicates that over longer periods the dividends act as a smoothing device for returns. In addition, since dividend yield differs substantially among individual securities, ignoring dividends will bias upward the variability Of returns (relative to the market) of those companies following a stable, high-payout dividend policy. 42 TABLE III-2 Holding Return + Annualized Period Return Dividends Return + Dividend HP(I) 13.2% 14.0% 56.0% HP(Z) 14.5 16.1 32.2 HP(3) 20.7 23.1 30.8 HP(4) 18.5 21.7 21.7 HP(S) 10.1 14.1 11.3 HP(6) 25.3 30.1 20.1 HP(7) 27.4 33.0 18.9 HP(8) 30.2 36.5 18.3 HP(9) 28.3 35.3 15.7 HP(lO) 20.4 28.2 11.3 HP(ll) 17.2 25.8 9.4 HP(12) 2.9 12.4 4.1 In summary, since the holding period is important for obtaining estimates of return consistent with investors' expectations and since typical holding periods may be thought of as one-three years or longer, sensitivity was computed for periods up to three years and included the return provided by dividends. An additional advantage to this method of testing was that it provided insight into the length of the holding period required before returns were sufficiently stable for reliable estimates of sensitivity and prediction error to be obtained. Returns used in estimating bit for both TPl and TP2 were computed over twenty-eight periods of different length utilizing weekly, monthly and quarterly Observation intervals. The observation periods extended 43 from three months to eighty-four months and each period represented a three month increment over the previous observation period. Thus there were twenty-eight observation periods for each test period ranging from three months to seven years. Designating the first observation period corresponding to TPl as OP(ll) and the first observation period corresponding to TP2 as OP(21), Table III-3 indi- cates the observation periods and corresponding time interval for each test period. A regression equation was computed for each OP(ij) using weekly, monthly, and quarterly observations of return. The particular obser- vational interval being used was represented by its first letter. Thus OPw(ij) corresponded to weekly observations of return taken in TP(i) in observation period j, used in estimating the historical beta coefficient. Weekly price observations used to compute weekly returns were taken as the closing Friday prices. When holidays or other events caused the New York Exchange to be closed on Friday, the nearest preceding day's closing price was used. Mbnthly price observations were taken as every fourth weekly price. Thus in this paper there are thirteen observations of monthly return per year and 30 measurement periods over the total seven year period. The effect of obtaining thirteen rather than twelve monthly observations per year was not considered important since the focus of this paper is on the effect of lengthened observation intervals not the specific interval of month end prices for use in the computa- tion of the historical beta co-efficients. 44 TABLE III-3 TPl TP2 Time Time Designation Period Designation Period OP(1l) 10/1/66 12/31/66 OP(21) 10/1/69 12/31/69 OP(12) 7/1/66 12/31/66 OP(22) 7/1/69 12/31/69 OP(13) 4/1/66 12/31/66 OP(23) 4/1/69 12/31/69 OP(l4) 1/1/66 12/31/66 OP(24) 1/1/69 12/31/69 OP(15) 10/1/65 12/31/66 OP(25) 10/1/68 12/31/69 OP(16) 7/1/65 12/31/66 OP(26) 7/1/68 12/31/69 OP(17) 4/1/65 12/31/66 OP(27) 4/1/68 12/31/69 OP(l8) 1/1/65 12/31/66 OP(28) 1/1/68 12/31/69 OP(19) 10/1/64 12/31/66 OP(29) 10/1/67 12/31/69 OP(llO) 7/1/64 12/31/66 OP(210) 7/1/67 12/31/69 OP(lll) 4/1/64 12/31/66 OP(211) 4/1/67 12/31/69 OP(112) 1/1/64 12/31/66 OP(212) l/l/67 12/31/69 OP(ll3) 10/1/63 12/31/66 OP(213) 10/1/66 12/31/69 OP(114) 7/1/63 12/31/66 OP(214) 7/1/66 12/31/69 OP(115) 4/1/63 12/31/66 OP(215) 4/1/66 12/31/69 OP(ll6) 1/1/63 12/31/66 OP(216) 1/1/66 12/31/69 OP(117) 10/1/62 12/31/66 OP(217) 10/1/65 12/31/69 OP(118) 7/1/62 12/31/66 OP(218) 7/1/65 12/31/69 OP(119) 4/1/62 12/31/66 OP(219) 4/1/65 12/31/69 OP(120) 1/1/62 12/31/66 OP(220) 1/1/65 12/31/69 OP(121) 10/1/61 12/31/66 OP(221) 10/1/64 12/31/69 OP(122) 7/1/61 12/31/66 OP(222) 7/1/64 12/31/69 OP(123) 4/1/61 12/31/66 OP(223) 4/1/64 12/31/69 OP(124) l/l/6l 12/31/66 OP(224) 1/1/64 12/31/69 OP(125) 10/1/61 12/31/66 OP(225) 10/1/63 12/31/69 OP(126) 7/1/61 12/31/66 OP(226) 7/1/63 12/31/69 OP(127) 4/1/61 12/31/66 OP(227) 4/1/63 12/31/69 OP(128) 1/1/61 12/31/66 OP(228) 1/1/63 12/31/69 45 Quarterly price observations were taken as every thirteenth weekly price observation hence, the observation interval was thirteen times the weekly interval. Again, for purposes of this study, this was considered acceptable since quarterly prices (or price changes) were not directly under study. Returns were computed for individual securities and the market index according to equation III-4. Continuously compounded returns were not utilized for the computation of historical betas because returns for periods up to thirteen weeks were typically very small. For small returns the difference between return as computed by equa- tion III-4 and the natural log function of the price relative is also very small (near zero). For example, a return of 3% computed by equation III-4 would provide a continuously compounded return of 2.956%, a difference of .044% or .00044. Recalling that the expected weekly return using the Fisher—Lorie results2 would be (8.7%/52) = .167% = .00167, then all differences on weekly returns would be expected to be on the order of magnitude of (.00167/.03 = .0556) .0556 x .00044 = .00002446. Differences of this magnitude (or even thirteen times this magnitude in the case of quarterly price observa- tions) were considered insignificant for purposes of this study. Returns used to compute sensitivities were also taken as abso— lute rather than continuously compounded returns. Absolute returns were used because using continuously compounded returns for computing sensitivities while using absolute returns for computing historical 2Lawrence Fisher and James H. Lorie, "Rates of Return on Investments in Common Stocks", Journal of Business, January 1964, pp. 1-17. 46 beta co-efficients seemed a priori unreasonable. Secondly, since sensitivities are ratios, sensitivity computed from continuously com- pounded returns was very similar to sensitivity computed from absolute returns. As an example, if R1k = 0.15 and Rmk = 0.20 on an absolute basis, sensitivity would be Rik/Rmk = .15/.20 = 0.75. On a continuously compounded basis, sensitivity would be ln(R1k + 1.0) 1n(1.15) .14 =————-—=——=.77. 1n(RInk + 1.0) 1n(l.20) .18 Differences of the order of magnitude of 0.02 (.77 - .75), for observa- tions of R1k and RInk at 0.15 and 0.20, respectively, were not considered significant for purposes of this study. Finally, the lower sensitivi- ties computed from absolute returns are consistently lower than those computed with continuously compounded returns. While the absolute value of the prediction error might be different, the relative differ- ence (rankings) between individual prediction errors would be equal. Thus the same ranking for alternative specifications of the historical beta co-efficient should occur regardless of how sensitivities were computed. Quarterly dividends for individual companies were added to price in the week, month, or quarter in which the stock went ex-dividend. For the S&P 500, annual percentage rates were observed quarterly and divided by fifty-two to estimate weekly dividend return, thirteen to estimate monthly dividend return, and four to estimate quarterly divi- dend return. The S&P 500 Index was selected as the surrogate for the market. 47 While all indicies are highly correlated,3 it is argued that the S&P 500 is more representative of the entire market than say the Dow Jones 65 Composite Average.4 The S&P 500 was preferred to Moody's 300 Aver- age because of the availability of data and to the NYSE Index because of its (NYSE's) shorter history. An absolute prediction error defined as the absolute value of the difference between the beta coefficient and sensitivity corresponding to every combination of observation period, observation interval, and holding period was computed for each security using the above tech- niques. In TPl, the twenty-eight observation periods, twelve holding periods and three different observation intervals provided (3x12x28) 1008 absolute prediction errors for each company in the sample. In TP2 the three observation intervals, twenty-eight observation periods and eight holding periods provided 672 absolute prediction errors for each security in the sample. The total number of absolute prediction errors computed was ((672+1008)35)=58,800.5 The absolute prediction errors were averaged across all securities in the sample by observation period, by observational interval and by length of holding period. Designating the absolute prediction error W 1,2 absolute prediction error across all 35 securities for observation period across all 35 securities as 3, then (81) would correspond to the mean one and holding period two computed from weekly returns in TPl. Similarly, 3William F. Sharpe, Portfolio Theory and Capital Markets, McGraw- Hill, 1970, p. 148. 4Jerome Cohen and Edward Zinbarg, Investment Analysis and Port— folio Management, Richard D. Irwin, p. 631. 5See the subsequent section on Sample Size. 48 (32): 10 would correspond to the mean absolute prediction error across 9 all 35 securities for observation period eight and holding period ten computed from quarterly returns in TP2. Table III—4 provides a schematic plan for the methodology. TABLE II I-‘l _w —W “W "'W (‘11)11 (d1)1,12 (d2)11 (d2)1,8 _m -111 (d1)11................ .(dl)l’12 (d2)11 ............. (d2)1 8 -q °. -q -q '3 -q (d1)11 -. (d1)1,12 (d2)11 g (d2)1,8 é ...('d1)isj i : .°..(d'2') 19:] g 3 a 3 3 1.. E 5 (d9) ' j (gq) : I . 1 j . 2 i,j -€1 '- —w 3 “W. .‘ -w ‘ (d1)28,1 33(d1)28,12 (‘12)28'1 3(d2)28, 8 _m (d1)28 1. ................(c11)28’12 (d2)28 1........ ..(d2)28 8 -q -q -q -q (d1)28,l (d1)28,12 (d2)28,1 (d2)28,8 Alternative Market MOdel In Chapter I it was pointed out that Fisher and Kamin have argued for an alternate form of the market model: Rit - bitRmt' They have not however, specified which data should be used in the computation of the historical beta co-efficient to obtain optimal results when considering the model in a predictive sense.7 7Lawrence Fisher and Jules H. Kamin, "Good Betas and Bad Betas; How to Tell the Difference", a handout to accompany the presentation at The Meeting of the Midwest Finance Association, St. Louis, Missouri, 1972. 49 The same tests and methodology specified in Chapter III will be utilized in evaluating alternative specifications of the "Fisher beta". Finally, absolute prediction errors will be compared utilizing the "Fisher beta" and the "ordinary beta" to determine which provides the minimum absolute prediction error for all holding periods on average and for specific holding periods. The Sample Sample Size The problem of determining sample size was essentially a statisti- cal problem. The population was considered to be all stocks listed continuously on the New York Stock Exchange from 1960-1972. The objec- tive was to draw from this population a random sample of sufficient size in order to make inferences about the population. It was known that according to the index model approach to port— folio theory, any representative sample of securities will have a beta of 1.0, where bp = f Xib1 and X1 = 1/n. Hence a sample of securities which would allow inferences to be made about the population should also, when combined into a portfolio, have a beta of 1.0. As a first approximation, sixteen securities were selected at random from the population.8 Betas were constructed for each security. A portfolio beta, bp’ was then computed by the technique shown above. 8See following section on Selection Procedures. 9The betas constructed for testing purposes were computed according to the optimal Gonedes results i.e., one month observation interval and a seven year observation period. 50 The standard deviation of the sample betas was also computed and after adjusting for small sample methods, taken as the population standard deviation. At this point, available information consisted of the following: = population mean = 1.0 R = sample mean = 1.1304 0; = the sample standard deviation = .2404 S = the estimated population standard deviation = o—IE = .015 x n = the number of elements in the sample = 16 The objective was to determine if u and R were significantly different or alternatively, if the quantity (i - u) was significantly different from zero. The hypothesis to be tested was taken as Ho: (i - u) = 0.0. Again, standard statistics texts demonstrate that the two-tailed T-test will provide a confidence interval for accepting this hypothesis. The confidence interval selected was .01 143;, the results can be accepted with 99% confidence. The results indicated that the sample of sixteen was not sufficiently large to accept with 99% probability the hypothesis under consideration. The next step was to determine how large a sample was necessary. Letting then - u t S '- .99 log/ya I and for t 99 = 2.947 a value of n can be determined which will satisfy the criteria. Here, n = 29.512. To ensure results, an additional 19 companies were selected making the total sample 35 companies. 51 Selection Procedures The number of securities which appeared in the Wall Street Journal under the New York Exchange heading were counted as of the first trading day in 1960. The total number which appeared on that date was 1472. Each security was then assigned a number between 1 and 1472. Random numbers (250) between zero and one were then generated from the Michigan State University - CDC 6500 random number generator program. Multiplying each random number generated by the total number of securities available provided an adjusted random number corresponding to the number of one of the securities in the population. Each security was then reviewed for preferred status. All pre- ferred stocks were eliminated from consideration. The remaining common stocks were then compared to a list of the common stocks appearing in the Wall Street Journal on the first trading day of 1972. Any stocks which did not appear in this edition of the Journal were also excluded from further consideration. The remaining securities were then con— sidered to conform to the criteria of common stocks, continuously listed and the first thirty-five chosen as the sample. Table 111—5 lists the securities selected for the study. 52 TABLE III-5 Admiral Corporation Alleghany Ludlum Steel American Electric Power American Metal Climax Boeing Company Borden Continental Can Emerson Electric Co. Falstaff Brewing Co. Ferro Corp. General Cigar General Tire and Rubber Getty Oil Granby Mining Harsco Corp. International Mining Joy Manufacturing Co. S. S. Kresge Co. Kroehler Mfg. Lone Star Gas McAndrews & Forbes McCraw-Edison National Can Owens-Illinois Phelps-Dodge Public Service Electric & Gas Scott Paper J. P. Stevens Sunshine Mining Tri—Continental TRW Inc. U. S. Tobacco Upjohn Walgreen Inc. Washington Water and Power CHAPTER IV ANALYSIS OF RESULTS Ordinary Least Squares (OLS) Beta Coefficients Computed from Weekly Observations of Return Beta coefficients computed from weekly observations of return exhibited nearly without exception a hump-backed shape over time 2:2;9 as more observations of return were included in the computation of the beta coefficient, the beta value first increased, then decreased. This would not be surprising if the value of the beta coefficient approached 1.0 as the measurement period increased for it would tend to indicate that periods of instability had cancelled over time and with enough observations of return (a sufficiently long measurement period) the security moved very much like the market. This hump- backed nature of the beta values over time, however, held even in cases where the beta value of MP(1) was near 1.0 and thus approached 1/2 or 1/4 as the measurement period lengthened. This characteristic of the beta values over time held for both test period one (TPl) and test period two (TP2). Typically, the maxi- mum beta value occurred between MP(5) and MP(12) indicating that the value taken on by the beta coefficient computed from weekly returns is generally increasing between zero and three years and declining thereafter. The actual number of observations of weekly returns during this interval ranged from sixty five to 156. As evidenced in column 53 54 one of Table IV-lA and IV-lB, the maximum values of the beta coefficient observed between MP(S) and MP(12) typically provided the minimum mean absolute prediction error (MAPE) across securities.1 TABLE IV-lA MAPE (minimum values) TPl Ordinary least squares beta co-efficients Weekly SHEI. Monthly (Mgl. Quarterly (MP). ‘HP .50891 (6) .03320 (9) .01874* (3) 1 .93445 (6) .45874 (9) .13840* (8) 2 .99483 (6) .51911 (9) .19877* (8) 3 1.16025 (6) .68454 (9) .36420* (8) 4 1.51196 (6) 1.04394 (9) .72360* (8) 5 1.39540 (6) .91968 (9) .59934* (8) 6 1.25708 (6) .78137 (9) .46103* (8) 7 1.53797 (6) 1.06225 (9) .74191* (8) 8 1.28177 (6) .80605 (9) .48571* (8) 9 1.05982 (6) .58411 (9) .26377* (8) 10 1.31285 (6) .83714 (9) .51680* (8) 11 1.43040 (6) .95468 (9) .63434* (8) 12 *Minimum row entry. 1Results shown in Tables IV-lA and IV-lB are computed from OLS beta coefficients using weekly observations of return (column 1), monthly observations of return (column 2) and quarterly observations of return (column 3). Each value corresponds to the minimum MAPE value from all twenty-eight measurement periods (1) for the holding period shown (j). Thus in column 2 for TPl a value of .03320 was the minimum MAPE found from all 28 measurement periods when attempting to predict systematic risk in HPl (the first 3 months subsequent to the computation of the beta coefficient) and occurred in MP(9) i.e., with 27 observations of monthly return used in the historical estimate. The average MAPE values for both TPl and TP2 are shown in Table IV-ll at the end of Chapter IV. MAPE values for each measurement period are shown in Appendix A. 55 TABLE IV—lB MAPE (minimum values) TP2 Ordinary least squares beta co-efficients Weekly (MP) Monthly (ME) Quarterly (MP) .33 .34211* (21) .90994 (30) .98726 (16) 1 .08094 (1) .06154* (30) .13886 (16) 2 .12355 (1) .00377 (14) .00123* (25) 3 .62614* (21) 1.19397 (30) 1.27128 (16) 4 .04049 (1) .00086* (23) .01743 (16) 5 .01109* (7) .21877 (30) .29609 (16) 6 .00811* (14) .46891 (30) .54623 (16) 7 .01651* (14) .44429 (30) .52160 (16) 8 *Minimum row entry. In TPl, seventy-eight observations of return (MP(6)) provided beta coefficients which minimized MAPE for every one of twelve holding periods considered. In TP2 the results were not so consistent, but of the eight holding periods considered, an optimum beta fell within thirteen to 182 observations of return in seven instances. Thus in nineteen of the twenty holding periods considered in both TPl and TP2, the minimum MAPE was produced by beta coefficients computed over a measurement period of less than three and one-half years. Sijf 56 The observations made by Professor Gonedes that it is advantageous to increase the number of observations to eighty-four (seven years for monthly observations of return) seems to hold here even though eighty- four observations of weekly returns covers a period of only about 1.6 years and hence could not encompass sufficient time for structural changes to occur as he suggests. A potential explanation of the declining value of the weekly beta coefficient after about MP(12) and the similarity of the Gonedes optimum number of observations to the weekly optimum number of observa- tions reported here lies in a closer examination of the formula for computing beta coefficients. The regression coefficient for a particular security i can be written for computational purposes as in Equation IV-l below. As n increases the term nXR R. in the t mt 1t nZR R. - ZR. ZR t mt it t 1tt mt 2 2 nERmt - (ERmt) it- and Rit = historical return during period t for security i Rmt = historical return during period t for the S&P 500 Index n = the number of historical observations of Rit and Rmt' runnerator becomes very large relative to (ER The same is true t mtERit)' ix: the denominator, nZR:t becomes large relative to (ZRmt)2. After a t t snafficient number of observations, the value of the beta coefficient 57 is largely a function of the expression: nZRmtRit/nXR2 Now since t mt' 2 2 Rmt is always positive for any Rmt value, ERmt is a monotonically increasing function. The product (RitRmt) will not necessarily always be positive and in fact will be negative whenever Rmt and R t are of i opposite sign. Thus, the function ZRmtR is not necessarily monotoni- it cally increasing. It seems reasonable to assume that Rmt and Rit will most often be of opposite sign when index return, R.In is near zero. If so, small t’ returns on the market index may introduce a downward bias into the regression equation viz, the monotonically increasing nature of the denominator (nERit) ys, the fluctuating positive and negative behavior of the numerator (nZRmtRit).3 To determine whether this potential bias had entered the computa— tion of beta values computed from weekly observations of return, a runs test was performed on weekly individual security returns given both positive and negative returns on the market index for the 364 week period: 1960-1967. A runs test is designed to determine whether the number of obser- vations of a particular statistic are sufficiently different (in a two alternative situation) to be considered random. The statistic under consideration here was the sign of the return for an individual security given the sign of the return of the market index. If the sign of the return on the index is positive, then so should be the sign of the 2 a 2 In the limit as n+w, beta XRmtRit/ZRmt. 3This assumes, of course, that n is sufficiently large to effec- tively eliminate the influence of the other term in both numerator and denominator. 58 return on an individual security (assuming a positive relationship). If, however, a positive return on the market index brings a sufficient number of negative returns for a particular security, then the rela— tionship between the market returns and individual security returns is not strongly positive and the test would indicate that the return on the security is randomly positive and negative when the return on the index is positive. The runs test was performed twice, once for all observations of weekly returns on the index such that 0.0 H mqm¢a 64 HHO0M.H wmoom. mammq. omwmm. womno.n.n w 00m~q.H 00mmm. 00mmq. oqqno. oqqnm. n mm0m¢.a mmomm. muomq. nmmqo. nmmqm. 0 0000N.H 00005. 0000N. Hommm. Hmmmm. m meamm. mqamq. nmwmo. mmmnm. nmw~0.a q mmmma.m wwmn0.H wNmNH.H mmmno. mmmma. m nmwmw. mmwmm. moaoa. moaoo. NOHOH.H N mmw¢0.a mwmqm. ommqo. «Hume. «Hmmm. H mmamw.a mNHmm.H owamm. omamm. ammoa. mm 0.N u muom m.H n oumm 0.H u muom m.0 u ouom 0.0 I ouom NmH zH mMDA<> H mqm<8 65 to the minimum row values in Tables IV—lA and IV-lB. The weekly, monthly and quarterly MAPE results were averaged sepa- rately across all holding periods and these figures are shown in Table IV-S. Comparing the minimum MAPE across all holding periods with the MAPE from arbitrarily setting the beta coefficient equal to 1.0 pro- duces some surprising results. In TPl setting beta equal to 1.0 pro- vides an overall MAPE for all holding periods of .76103 while the minimum MAPE for beta coefficients computed from weekly returns was 1.19945, from monthly returns was .72373, and quarterly returns was .45125. In TP2, an arbitrary beta value of 1.0 produced an overall MAPE of .42819 while the minimum MAPE for weekly returns was .36745, for monthly returns was .43753, and for quarterly returns was .48055. The results of combining TPl and TP2 are also shown in Table IV-5 on the fOllowing page. The results indicate that if an investor could ex ante determine which of the differencing intervals and measurement periods would provide the minimum MAPE he could obtain superior results relative to arbitrarily selecting a beta coefficient of 1.0. If he is unable to ex ante determine which combination of differencing interval and measurement period will provide the minimum MAPE consistently using a particular observational interval may provide inferior results to an arbitrary selection of the beta value as 1.0. 66 TABLE IV-S A COMPARISON OF OLS BETA COEFFICIENT VALUES AND AN ARBITRARY BETA VALUE OF 1.0 FOR ALL HOLDING PERIODS TPl TP2 (TPl+TP2)/2 W 1.19945 .36745 .78798 M .72373 .43753 .64274 O .45125 .48055 .54529 Beta 8 1.0 .76103 .42819 .59461 Implications for Portfolio Construction Portfolio theory suggests that the investor should purchase the market (or some reasonable facsimile thereof) and borrow or lend to obtain his desired risk exposure. If the investor draws a random selection of securities from the market sufficiently large (about twenty securities) the results of the preceding section do not apply. In practice, however, investors still appear to evaluate and rank securities in an attempt to out-perform the market.5 For individuals who are attempting to construct a portfolio of securities which exhibits a beta value near 1.0 by the inclusion of some high and low beta secu- rities, the results may be important. They suggest that the ranking of securities might most efficiently be carried out (in a cost-benefit sense) by simply assuming the systematic risk component of each security to be 1.0, hence, ranking securities on an expected return basis. 5Chris A. Welles, "The Beta Revolution: Learning to Live with Risk", The Institutional Investor, Vol. V, No. 9. 67 For individuals who desire a portfolio which exhibits a beta value dissimilar to 1.0, the results, again, need not apply. The sample under consideration here was random and hence has no implications for groups of securities which all exhibit a high or low historical beta value. These portfolios may exhibit a beta value which is sytematically higher or lower than predicted, whereas the sample here consisted of a random selection of securities and presumably errors which were randomly posi- tive and negative. The effect of Holding Period length on Minimum MAPE Increasing the holding period length tended to decrease the minimum MAPE in TP2 and increase the minimum MAPE in TPl. Combining the minimum MAPE for TPl and TP2 by holding period indicates that overall the error tends to increase as the holding period increases. All three observa- tion intervals provided essentially the same results, however, weekly observations of return were more consistent in exhibiting a higher pre- diction error as the holding period lengthened than prediction error from monthly observations of return which in turn were more consistent than prediction errors from quarterly observations of return. Apparently, in general, the longer one allows the period to be predicted extend, the more difficult it becomes to Obtain good predic- tions. The idea that the security's behavior will be easier to predict if a sufficiently long term interval is examined i.e., an opportunity for "normal" behavior to occur is allowed, does not appear valid when predicting an individual security's responsiveness with the market for periods up to three years. For periods longer than three years this statement may not be valid. 68 The conclusions must be qualified by the fact that in the two test periods examined only one indicated a higher prediction error corre- sponding to increased holding period length. The higher prediction error associated with increased holding period length experienced in TPl was, however, sufficiently large to influence the combined results of TPl and TP2. The Fisher-Kamin Beta The characteristics of the Fisher-Kamin (F-K) beta coefficient are nearly identical to the ordinary least squares beta coefficients and results of the prediction tests are listed in Table IV—6A and IV-OB. A Comparison of F-K and Ordinary Least Squares (OLS) beta Coefficients For each test period and holding period, the minimum MAPE was taken for both OLS and F-K beta coefficients. The summary of results are shown in Tables IV-7 through IV-9. The tables show that in four of the six separate cases (two test periods and three observation intervals) OLS beta coefficients provided a smaller MAPE than the F-K beta coefficients. Only the F-K beta coefficients computed from quar- terly observations of return in TPl provided a smaller MAPE (see Table IV-9B). MAPE computed from OLS beta coefficients using monthly observa- tions of return were consistently lower in both TPl and TP2 than the F-K beta coefficients. This appears to be in direct contradiction to the Fisher-Kamin results cited in Chapter I. There are two poten- tial explanations for this event. First, Fisher-Kamin used a five year measurement period to compute beta coefficients. This measurement 69 TABLE IV-6A MAPE (minimum values) TPl F-K beta coefficients Weekly .IHEI Monthly (MP) Quarterly (MP)_ HP .50891 (6) .08169 (4) .00926* (3) 1 .93445 (6) .50723 (4) .10029* (10) 2 .99483 (6) .56760 (4) .16066* (10) 3 1.16025 (6) .73302 (4) .32609* (10) 4 1.51965 (6) 1.09242 (4) .68548* (10) 5 1.39540 (6) .96817 (4) .56123* (10) 6 1.25708 (6) .82985 (4) .42291* (10) 7 1.53797 (6) 1.11074 (4) .70380* (10) 8 1.28177 (6) 085454 (4) .44760* (10) 9 1.05982 (6) .63260 (4) .22566* (10) 10 1.31285 (6) .88563 (4) .47868* (10) 11 1.43040 (6) 1.00317 (4) .59623* (10) 12 *Minimum row entry. The results for each measurement period appear in Appendix B. 70 TABLE IV-OB MAPE (minimum values) TP2 F-K beta coefficients Weekly (MP) Monthly (MP) Quarterly (MP) ‘HP .34534* (21) .92800 (30) .99120 (16) 1 .06343* (1) .07960 (30) .14280 (16) 2 .14106 (1) .00748 (16) .00440 (18) 3 .62937* (21) 1.21202 (30) 1.27523 (16) 4 .05800 (1) .00151* (24) .02137 (16) 5 .00703* (12) .23683 (30) .30003 (16) 6 .00971* (14) .48697 (30) .55017 (16) 7 .01491* (14) .46234 (30) .52554 (16) 8 *Minimum row entry. The results for each measurement period appear in Appendix B. 71 TABLE IV-7A MAPE (weekly observations) TP1** F-K and OLS beta coefficients compared Ordinary least squares 12E2_ Fisher 3MP) .HP .50891 (6) .50891 (6) 1 .93445 (6) .93445 (6) 2 .99483 (6) .99483 (6) 3 1.16025 (6) 1.16025 (6) 4 1.51965 (6) 1.51965 (6) 5 1.39540 (6) 1.39540 (6) 6 1.25708 (6) 1.25708 (6) 7 1.53797 (6) 1.53797 (6) 8 1.28177 (6) 1.28177 (6) 9 1.05982 (6) 1.05982 (6) 10 1.31285 (6) 1.31285 (6) 11 1.43040 (6) 1.43040 (6) 12 **No minimum in TPl; both ordinary least squares and Fisher betas are equal. 72 TABLE IV-7B MAPE (weekly TP2 observations) F-K and OLS beta coefficients compared Ordinary least squares .34211* .08094 .12355* .62614* .04049* .01109 .00811* .01651 £3§Q_ Fisher (21) .34534 (1) .06343* (1) .14106 (21) .62937 (1) .05800 (7) .00703* (14) .00971 (14) .01491* (MP) (21> <1) <1) <21) <1) (12> <14) (14) IE (”\l05U'IJ-‘UJNH *Minimum row entry. 73 TABLE IV-8A MAPE (monthly observations) TPl F—K and OLS beta coefficients compared Ordinary least squares 52§Q_ Fisher (M21. ‘HP .03320* (9) .08169 (5) 1 .45874* (9) .50723 (4) 2 .51911* (9) .56760 (4) 3 .68454* (9) .73302 (4) 4 1.04394* (9) 1.09242 (4) 5 .91968* (9) .96817 (4) 6 .78137* (9) .82985 (4) 7 1.06225* (9) 1.11074 (4) 8 .80605* (9) .85454 (4) 9 .58411* (9) .63260 (4) 10 .83714* (9) .88563 (4) 11 .95468* (9) 1.00317 (4) 12 74 TABLE IV-SB MAPE (monthly observations) TP2 FdK and OLS beta coefficients compared Ordinary least Squares .90994* .06154* .00377 1.19397* .00086* .21877* .46891* .44429* IHZI. Fisher (30) .92800 (30) .07960 (14) .01691 (30) 1.21202 (23) .00151 (30) .23683 (30 .48697 (30) .46234 5.14.11 (30) (30) (14) (30) (24) (30 (30) (30) :1: GNOUIwal-‘l'd 75 TABLE IV-9A MAPE (quarterly observations) TPl F-K and OLS beta coefficients compared Ordinary least squares (MP) Fisher (MP) .EE .01874 (3) .00926* (3) 1 .13840 (8) .10029* (10) 2 .19877 (8) .16066* (10) 3 .36420 (8) .32609* (10) 4 .72360 (8) .68548* (10) 5 .59934 (8) .56123* (10) 6 .46103 (8) .42291* (10) 7 .74191 (8) .70380* (10) 8 .48571 (8) .44760* (10) 9 .26377 (8) .22566* (10) 10 .51680 (8) .47868* (10) 11 .63434 (8) .59623* (10) 12 76 TABLE IV-9B MAPE (Quarterly observations) TP2 F-K and OLS beta coefficients compared Ordinary least squares .98726* .13886 .00123* 1.27128* .01743* .29609* .54623* .52160* (MP) (l6) (16) (25) (l6) (l6) (l6) (16) (16) Fisher .99120 .14280 .00440 1.27523 .02137 .30003 .55017 .52554 (MP) (l6) (16) (18) (16) (16) (l6) (16) (16) [E CDVO‘Ulbb-JNH 77 period and the sixty monthly observations of return could have been optimal for the F—K beta but not the OLS beta; this would provide superior prediction results for the F-K beta. A second possibility is that neither the F—K nor the OLS beta were Optimal with sixty monthly observations. Under this assumption, the difference between the five year measurement period beta and the Optimal beta value is important. But this is equivalent to examining the variance between the estimates of the beta coefficients computed from the twenty-eight different measurement periods. If less variability exists between estimates of the beta coefficient computed according to the Fisher-Kamin criteria then it is possible that the F-K nonoptimum beta estimate utilizing sixty observations of return would be closer to the optimum estimate than the OLS beta estimate even though the OLS beta is a better esti— mate of the security's behavior with the market. Accordingly, the variance of the different beta estimates was computed for each of the holding periods and averaged together for all measurement periods for both Fisher-Kamin and OLS beta coefficients. The results are shown in Table IVelO. In both TPl and TP2, the F-K TABLE IV-lO VARIANCE OF BETA ESTIMATES TPl TP2 OLS F-K OLS F-K W .09843 .06730 .07331 .07308 .07620 .07620 .15643 .14480 Q .34900 .32700 7.32551 .36143 78 beta provided a minimum mean error greater than the corresponding OLS beta but with less variance between estimates. This suggests a situ- tion in which the relationship R1 = a1 + biRm describes more accurately a security's behavior with the market than R1 = biRm but produces worse estimates of the security's future behavior. Alternatively, it may be said that OLS beta coefficients provide better estimates of a security's behavior with the market (smaller minimum prediction error) but F-K beta coefficients are more useful in a prediction sense because there exists less variance in the estimates of the beta coefficients and from the results of the previous section, it is extremely difficult, a priori, to select that measurement which will provide the optimum estimate of the beta coefficient for prediction purposes. 79 TABLE IV-ll AVERAGE MAPE FOR TPl AND TP2 BY HOLDING PERIOD* OLS BETA COEFFICIENTS w M Q HP .42551 .47517 .50300 1 .50720 .26014 .13863 2 .55919 .26144 .10000 3 .89319 .93925 .81774 4 .71794 .46027 .37052 5 .63408 .56922 .44771 6 .63259 .62514 .50363 7 .77724 .75327 .63176 8 1.28177 .80605 .48571 9 1.05982 .58411 .26377 10 1.31285 .83714 .51680 11 1.43040 .95468 .63434 12 *Minimum prediction errors for holding periods 9 through 12 are TPl only. CHAPTER V SUMMARY AND CONCLUSIONS The purpose of the present research has been to accurately specify the market model for use in making predictions about an individual security's systematic risk behavior. The specification problem was considered in two areas: the market model, and the parameters of the market model. Specification of the market model was limited to the choice of two available alternatives, (1) R1 = a + b R t which is t it it m called the ordinary least squares procedure and (2) R = b R t which it it m is called the Fisher-Kamin procedure. The parameters of the market model to be specified included the measurement period, the observation interval, and the length of the holding period. The proper specifica- tion of the market model and the parameters of the model are necessary if investors are to accurately rank securities by means of the pro- gramming model introduced by Professor Sharpe and presented in Chapter 1. Specification of the Market Model As was pointed out in Chapter I, the OLS beta coefficient will generate less prediction error than the Fisher beta coefficient for any given combination of measurement period, holding period, and observation interval when: F R > Rmt(bit+bit) _ bitRit v-1 mt Rit n 80 and 2 - n+1 (Rmt'Rmt) Rmt > 53:. R V-2 mk In general these two conditions appear to have been met since the OLS beta coefficients produced lower mean absolute prediction error (MAPE) values for most measurement periods and observation intervals. In only one instance, MAPE computed with quarterly observations of return in test period two (TP2), did Fisher beta coefficients consistently provide lower MAPE values than the OLS beta coefficients. Table V-l indicates that the returns observed on the S&P 500 Index were sufficiently similar in TPl and TP2 to be unable to explain why Fisher beta coefficients produced less prediction error in TP2 but not TPl. That the Fisher beta coefficient provided less prediction error using quarterly observations of return rather than monthly or weekly observations is not surprising. Condition two (equation V-2) is influ- enced to a large degree by the number of observations of return, (n) hence fewer returns associated with the quarterly observation interval tends to prohibit condition (2) from being satisfied. When n81, the expression ((n+1)/2n) is maximized with a value of 1.0. As n increases, ((n+1)/2n) approaches 1/2, its minimum value. Fewer observations of return would quite naturally tend to maintain ((n+1)/2n) near its maxi- mum value, thereby decreasing the probability that condition (2) will be satisfied. A second factor which will influence the prediction errors asso- ciated with the two models is the sensitivity of both the average historical return of the market index and the individual security beta values to changes in the observation interval. One might expect average 82 TABLE V-l RETURNS FOR S & P 500 CORRESPONDING TO MEASUREMENT PERIODS IN TPl & TP2 MP TPl TP2 1 .05600 -.00500 2 -.O99OO —.O49OO 3 -.O407O -.O4SOO 4 -.01760 .02300 5 .03640 .02500 6 .06340 .04400 7 -.00840 -.12400 8 .03570 -.07000 9 .00690 -.00100 10 .04120 .08700 11 .03600 .01300 12 .07110 .13000 13 .03940 .05600 14 .04730 -.09900 15 .04980 -.04100 16 .06530 -.01800 17 .12700 .03600 18 .03630 .06300 19 -.20350 -.00800 20 -.02110 .03600 21 .08390 .00700 22 .03910 .04100 23 -.00110 .03600 24 .12810 .07100 25 .11120 .03900 26 -.06660 .04700 27 .03720 .05000 28 -.05810 .06500 83 historical returns of the market to become prOportionately smaller as the observation interval is reduced. Weekly returns would be expected to be about one-thirteenth of quarterly returns and one-fourth of monthly returns. Beta coefficients although biased downward by the use of weekly data, are not so sensitive to changes in the observation interval because beta coefficients are ratios and ratios express one variable in terms of another. As a result, decreasing the length of the observation interval reduces nearly proportionately the left hand side of the inequality, fimt of condition (1). As fimt decreases, bit and bit decrease to a lesser extent thus the probability that fimt will F exceed the quantity Rmt (bit+bit) - bit R't also decreases. Rit n The ability of the OLS beta coefficient to produce in most instances smaller MAPE values does not mean that the OLS formulation is most useful in a prediction sense. The wider variability of OLS beta estimates among the various measurement periods coupled with the diffi- culty in a priori selecting the optimal measurement period suggests that the Fisher beta may be more useful in making predictions. Holding Period Length The empirical evidence indicated that for periods ranging from three months to three years, the prediction error increased as the holding period length increased (Table IV—ll). Contrast these results with the results of Blume who, when comparing beta coefficients in adjacent periods of seven years in length, found a high degree of predictability. These results suggest that prediction error may, over a lengthening holding period, exhibit a hump-backed shape i.e., first 84 increasing, then decreasing. In order for the results of this study and Blume's study to be consistent, it appears that sometime after three years and before seven years, the prediction error turns down- ward and the effect of lengthening the holding period on prediction error becomes favorable. Unfortunately, investors seem to anticipate a holding period of one to three years in length, in most instances. This holding period is precisely that interval when the effect of the holding period on prediction error is most unfavorable. It would appear that in order to minimize the prediction error arising from the use of beta coefficients, one must either (a) pick those securities which can comfortably be held for at least three years or (b) continu— ously revise and update the historical beta coefficients for all secu- rities under observation. Measurement Period and Observation Interval There existed no one best combination of measurement period length and observation interval which consistently provided maximum MAPE (Tables IV-lA and IV-lB), rather, the optimum combiantion seemed to change between the two test periods. The difficulty in a priori selecting a combination which will provide an acceptable prediction error may suggest the use of an arbitrary value of 1.0 assigned to each security for the investor who wishes to assemble a portfolio such that the portfolio beta approaches 1.0. It would appear from the results obtained that both weekly and monthly observations of return are acceptable if no more than 75 - 100 observations of return are collected for each beta estimate. The absence of benefits from collecting weekly returns past 100 suggests 85 that the "structural changes" hypothesized by Professor Gonedes are in reality a tendency of the individual security to exhibit random posi— tive and negative price movement for small changes in the value of the market index. Thus the lack of a sufficiently strong relationship which exists for most securities when the market exhibits a low abso- lute value return is sufficient to offset any benefits of increasing the number of observations used to compute the beta coefficient. Quarterly observations of return used to compute beta coefficients did not provide, in all instances, a sufficient number of observations over seven years to consistently insure the statistical significance of the beta estimate. It may be possible that over a longer measurement period a sufficient number of observations could be obtained which com- bined with the tendency of quarterly returns of individual securities to exhibit more reliable co-movement with the market (less random fluc- tuation) would provide a beta coefficient which produces consistently less prediction error than beta coefficients computed from either weekly or monthly observations of return. This would appear to be a promising area for future research. At this stage it seems fair to say that the beta coefficient is an average relationship between the behavior of an individual security and the market of all securities. An average necessarily aggregates behavior and thus may hide important deviations viz, the potential dual response of a security in response to upward and downward movements in the index. The size of the prediction errors obtained suggests that making the assumption that a particular historical relationship between an indi- vidual security and the market will continue to hold in the future is 86 a risky business. The size of the prediction errors obtained also suggests that highly accurate estimates of a security's systematic risk behavior are extremely difficult to obtain. Given this difficulty it would appear that the programming model proposed by Sharpe which utilizes as an input the future systematic risk behavior of individual securities is of only limited usefulness for ranking securities in the real world. Perhaps, if ranking must be done, fundamental analysis should be utilized. Both techniques require estimates to be made; the programming technique requires estimates of return and systematic risk; fundamental analysis requires estimates of earnings, dividends, etc. If more reliable estimates of fundamental factors can be made than can be made for return and systematic risk than there may be justification in utilizing a fundamental rather than a programming approach. Alternatively it appears that some combination of the programming approach and the fundamental approach may be useful. If those funda- mental factors whiCh determine systematic risk can be identified, then it may be possible to predict systematic risk more accurately from estimates of better understood fundamental factors. Some research has already appeared with this thrust.1 Limitations and Suggestions for Further Research Perhaps the single largest and most important limitation to any empirical study is that data is gathered from a specific time interval and any results generated are limited in applicability to that time 1See for example, William J. Breen and Eugene M. Lerner, "Corporate Financial Strategies and Market Measures of Risk and Return," Journal of Finance, May 1973, pp. 339-351. 87 interval. It may be useful, however, to determine empirically pre- cisely that fact: results are in fact limited to a specific time interval and will not hold in general. Witness for example the results in TPl of this research vs. the results of TP2. An optimum measurement period was found for each observation interval to be between two and three years in TPl while in TP2 the results were more variable but nevertheless tended toward a longer measurement period. This incon- sistency seems important for individuals attempting to use estimates of systematic risk generated from historical information to construct portfolios. A second limitation is the blue chip bias inherent in requiring firms in the sample to be continuously listed on the NYSE during the period 1960-1972. Many firms have joined the NYSE since 1960 but were not considered in this study because data for these firms was not readily available or because of the impact on the market model which listing itself may contain. Finally, the portfolio systematic risk component of all thirty- five securities in combination was very near 1.0. This does not allow generalization of the results to securities which in combination exhibit a systematic risk component very much different from 1.0. As mentioned earlier, these securities may exhibit a systematic predic- tion error which invalidates the results found here. This attribute of the sample does allow future research to be conducted which uses as a sample securities that exhibit either a very high or a very low historic beta value when combined. Other suggestions for future research might include Runs Test for observation intervals other than one week for values of the S&P 500 88 index different from [0.005 . A repeated testing procedure might provide the researcher with information as to the size of the return of the index required to provide a reliable response in the price of the individual security. This might suggest an alteration of the methodology associated with the computation of the historic beta coefficient. In particular, it might suggest eliminating all obser- vations of return for an individual security given that the return on the market index was less than some predetermined absolute value. APPENDICES APPENDIX A—l Mean Absolute Prediction Error for each Measurement Period Com- puted Using Ordinary Least Squares Beta Coefficients in Test Period 1. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 1 Measurement Period \OmNO‘Ui-fiwNi-i *Minimum column element. 89 ABSOLUTE VALUE OF Weekly 0.62288 0.59546 0.54220 0.53057 0.51583 0.50891* 0.52217 0.51528 0.67168 0.67986 0.70445 0.69937 0.72148 0.72417 0.72654 0.72002 0.68968 0.67917 0.70165 0.70154 0.69454 0.70740 0.70728 0.70317 0.70228 0.69425 0.70768 0.70651 Observation Interval Monthly 0.47700 0.23797 0.16597 0.12900 0.08311 0.07609 0.03674 0.04346 0.03320* 0.04371 0.07831 0.09603 0.11660 0.12891 0.14066 0.13640 0.14860. 0.15857 0.14809 0.14857 0.10943 0.13671 0.12494 0.16380 0.15943 0.14014 0.15657 0.15037 0.16957 0.16923 This value appears in Table IV-lA. Quarterly 0.01874* 0.02751 0.12720 0.21317 0.23511 0.28714 0.28034 0.27297 0.25177 0.20603 0.16171 0.11100 0.08160 0.03614 0.13720 0.14243 0.13277 0.13709 0.14460 0.14563 0.14811 0.11086 0.16011 0.17060 0.17277 0.17214 90 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 2 Observation Interval Measurement Period Weekly Monthly anrterly 1 1.04843 0.90254 2 1.02100 0.66351 3 0.96774 0.59151 0.44429 4 0.95611 0.55454 0.39803 5 0.94137 0.50866 0.29834 6 0.93445* 0.50163 0.21237 7 0.94771 0.46229 0.19043 8 0.94083 0.46900 0.13840* 9 1.09722 0.45874* 0.14520 10 1.10540 0.46926 0.15257 11 1.13000 0.50386 0.17377 12 1.12491 0.52157 0.21951 13 1.14702 0.54214 0.26383 14 1.14971 0.55446 0.31454 15 1.15208 0.56620 0.34394 16 1.14557 0.56194 0.38940 17 1.11522 0.57414 0.56274 18 1.10471 0.58411 0.56797 19 1.12720 0.57363 0.55831 20 1.12708 0.57411 0.56263 21 1.12008 0.53497 0.57014 22 1.13294 0.56226 0.57117 23 1.13282 0.55048 0.57365 24 1.12871 0.58934 0.53640 25 1.12782 0.58497 0.58565 26 1.11980 0.56568 0.59614 27 1.13322 0.58211 0.59831 28 1.13205 0.57591 0.59768 29 0.59511 30 0.59477 *Minimum column element. This value appears in Table IV-lA. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 3 Measurement Period \DmNO‘UIbWNI-J *Minimum column element. 91 ABSOLUTE VALUE OF Weekly 1.10880 1.08137 1.02811 1.01648 1.00174 0.99483* 1.00808 1.00120 1.15760 1.16577 1.19037 1.18528 1.20739 1.21008 1.21245 1.20594 1.17559 1.16508 1.18757 1.18745 1.18045 1.19331 1.19319 1.18908 1.18820 1.18017 1.19359 1.19242 Observation Interval Monthly 0.96291 0.72388 0.65188 0.61491 0.56903 0.56200 0.52266 0.52937 0.51911* 0.52963 0.56423 0.58194 0.60251 0.61483 0.62657 0.62231 0.63451 0.64448 0.63400 0.63448 0.59534 0.62263 0.61086 0.64971 0.64534 0.62605 0.64248 0.63628 0.65548 0.65514 This value appears in Table IV-lA. Quarterly 0.50466 0.45840 0.35871 0.27274 0.25080 0.19877* 0.20557 0.21294 0.23414 0.27989 0.32420 0.37491 0.40431 0.44977 0.62311 0.62834 0.61868 0.62300 0.63051 0.63154 0.63403 0.59677 0.64603 0.65651 0.65868 0.65806 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 4 Measurement Period \OmVOM-DUONH *Minimum column element. 92 ABSOLUTE VALUE OF Observation Interval Weekly 1.27422 1.24680 1.19354 1.18191 1.16717 1.16025* 1.17351 1.16663 1.32302 1.33120 1.35580 1.35071 1.37282 1.37551 1.37788 1.37137 1.34102 1.33051 1.35299 1.35288 1.34588 1.35874 1.35862 1.35451 1.35362 1.34560 1.35902 1.35785 Monthly 1.12834 0.88931 0.81731 0.78034 0.73446 0.72743 0.68808 0.69480 0.68454* 0.69506 0.72966 0.74737 0.76794 0.78025 0.79200 0.78774 0.79994 0.80991 0.79942 0.79991 0.76077 0.78805 0.77628 0.81514 0.81077 0.79148 0.80791 0.80171 0.82091 0.82057 This value appears in Table IV-lA. Quarterly 0.67008 0.62383 0.52414 0.43817 0.41623 0.36420* 0.37100 0.37837 0.39957 0.44531 0.48963 0.54034 0.56974 0.61520 0.78854 0.79377 0.78411 0.78843 0.79594 0.79697 0.79945 0.76220 0.81145 0.82194 0.82411 0.82348 93 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 5 Observation Interval Measurement Period Weekly Monthly anrterly 1 1.63362 1.48774 2 1.60620 1.24871 3 1.55294 1.17671 1.02948 4 1.54131 1.13974 0.98322 5 1.52657 1.09385 0.88354 6 1.51965* 1.08683 0.79757 7 1.53291 1.04748 0.77563 8 1.52602 1.05420 0.72360* 9 1.68242 1.04394* 0.73040 10 1.69060 1.05445 0.73777 11 1.71520 1.08905 0.75897 12 1.71011 1.10677 0.80471 13 1.73222 1.12734 0.84903 14 1.73491 1.13965 0.89974 15 1.73728 1.15139 0.92914 16 1.73077 1.14714 0.97460 17 1.70043 1.15934 1.14794 18 1.68991 1.16931 1.15317 19 1.71239 1.15883 1.14351 20 1.71228 1.15931 1.14783 21 1.70528 1.12017 1.15534 22 1.71814 1.14745 1.15637 23 1.71802 1.13568 1.15885 24 1.71391 1.17454 1.2160 25 1.71302 1.17017 1.17085 26 1.70500 1.15088 1.18134 27 1.71842 1.16731 1.18351 28 1.71725 1.16111 1.18288 29 1.18031 30 1.17997 *Minimum column element. This value appears in Table IV-lA. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 6 Measurement Period \OCDNO‘UIL‘LDNH *Minimum column element. 94 ABSOLUTE VALUE OF Obseryation Interval Weekly 1.50937 1.48194 1.42868 1.41705 1.40231 1.39540* 1.40865 1.40177 1.55817 1.56634 1.59094 1.58585 1.60797 1.61065 1.61302 1.60651 1.57617 1.56565 1.58814 1.58802 1.58102 1.59388 1.59377 1.58965 1.58877 1.58074 1.59417 1.59299 Monthly, 1.36348 1.12445 1.05245 1.01548 0.96960 0.96257 0.92322 0.92994 0.91968* 0.93020 0.96480 0.98251 1.00308 1.01540 1.02714 1.02288 1.03508 1.04505 1.03457 1.03505 0.99591 1.02320 1.01143 1.05028 1.04591 1.02662 1.04305 1.03685 1.05605 1.05571 This value appears in Table IV-lA. Quarterly 0.90522 0.85897 0.75928 0.67331 0.65137 ‘ 0.59934* 0.60614 0.61351 0.63471 0.68045 0.72477 0.77548 0.80488 0.85034 1.02368 1.02891 1.01925 1.02357 1.03108 1.03211 1.03460 0.99734 1.04660 1.05708 1.05925 1.05862 95 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 7 Observation Interval Measurement Period Weekly Monthly Quarterly 1 1.37105 1.22517 2 1.34362 0.98614 3 1.29037 0.91414 0.76691 4 1.27874 0.87717 0.72065 5 1.26400 0.83128 0.62097 6 1.25708* 0.82425 0.53500 7 1.27034 0.78491 0.51306 8 1.26345 0.79163 0.46103* 9 1.41985 0.78137* 0.46783 10 1.42802 0.79188 0.47520 11 1.45262 0.82648 0.49640 12 1.44754 0.84420 0.54214 13 1.46965 0.86477 0.58646 14 1.47234 0.87708 0.63717 15 1.47471 0.88882 0.66657 16 1.46820 0.88457 0.71203 17 1.43785 0.89677 0.88537 18 1.42734 0.90674 0.89060 19 1.44982 0.89625 0.88094 20 1.44971 0.89674 0.88525 21 1.44271 0.85760 0.89277 22 ' 1.45557 0.88488 0.89380 23 1.45545 0.87311 0.89628 24 1.45134 0.91197 0.85903 25 1.45045 0.90760 0.90828 26 1.44242 0.88831 0.91877 27 1.45585 0.90474 0.92094 28 1.45468 0.89854 0.92031 29 0.91774 30 0.91740 *Minimum column element. This value appears in Table IV-lA. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 8 Measurement Period @mNO‘MbWNl-J *Minimum column element. 96 ABSOLUTE VALUE OF Weekly 1.65194 1.62451 1.57125 1.55962 1.54488 1.53797* 1.55122 1.54434 1.70074 1.70891 1.73351 1.72842 1.75054 1.75322 1.75559 1.74908 1.71874 1.70822 1.73071 1.73059 1.72360 1.73645 1.73634 1.73222 1.73134 1.72331 1.73674 1.73557 Observation Interval Monthly 1.50605 1.26702 1.19502 1.15805 1.11217 1.10514 1.06580 1.07251 1.06225* 1.07277 1.10737 1.12508 1.14565 1.15797 1.16971 1.16545 1.17765 1.18762 1.17714 1.17762 1.13848 1.16577 1.15400 1.19285 1.18848 1.16920 1.18562 1.17942 1.19862 1.19828 This value appears in Table IV-lA. Quarterly 1.04780 1.00154 0.90185 0.81588 0.79394 0.74191* 0.74871 0.75608 0.77728 0.82303 0.86734 0.91805 0.94745 0.99291 1.16625 1.17148 1.16182 1.16614 1.17365 1.17468 1.17717 1.13991 1.18917 1.19965 1.20182 1.20120 97 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 9 Observation Interval Measurement Period Weekly Monthly anrterly 1 1.39574 1.24985 2 1.36831 1.01083 3 1.31505 0.93882 0.79160 4 1.30342 0.90185 0.74534 5 1.28868 0.85597 0.64565 6 1.28177* 0.84894 0.55968 7 1.29502 0.80960 0.53774 8 1.28814 0.81631 0.48571* 9 1.44454 0.80605* 0.49251 10 1.45271 0.81657 0.49988 11 1.47731 0.85117 0.52108 12 1.47222 0.86888 0.56683 13 1.49434 0.88945 0.61114 14 1.49702 0.90177 0.66185 15 1.49940 0.91351 0.69125 16 1.49288 0.90925 0.73671 17 1.46254 0.92145 0.91005 18 1.45202 0.93142 0.91528 19 1.47451 0.92094 0.90563 20 1.47440 0.92143 0.90994 21 1.46739 0.88228 0.91745 22 1.48025 0.90957 0.91848 23 1.48014 0.89780 0.92097 24 1.47602 0.93665 0.88371 25 1.47514 0.93228 0.93297 26 1.46711 0.91300 0.94345 27 1.48054 0.92943 0.94563 28 1.47937 0.92322 0.94500 29 0.94242 30 0.94208 *Minimum column element. This value appears in Table IV-lA. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 10 Measurement Period \OCDNO‘LDJ-‘UQNH *Minimum column element. 98 ABSOLUTE VALUE OF Observation Interval Weekly 1.17380 1.14637 1.09311 1.08148 1.06674 1.05982* 1.07308 1.06620 1.22260 1.23077 1.25537 1.25028 1.27240 1.27508 1.27745 1.27094 1.24059 1.23008 1.25257 1.25245 1.24545 1.25831 1.25819 1.25408 1.25320 1.24517 1.25860 1.25742 Monthly 1.02791 0.88888 0.71688 0.67991 0.63403 0.62700 0.58765 0.59437 0.58411* 0.59463 0.62923 0.64694 0.66751 0.67983 0.69157 0.68731 0.69951 0.70948 0.69900 0.69948 0.66034 0.68762 0.67585 0.71471 0.71034 0.69106 0.70748 0.70128 0.72048 0.72014 This value appears in Table IV-lA. Quarterly 0.56965 0.52340 0.42371 0.33774 0.31580 0.26377* 0.27057 0.27794 0.29914 0.34489 0.38920 0.43991 0.46931 0.51477 0.68811 0.69334 0.68368 0.68800 0.69551 0.69654 0.69902 0.66177 0.71103 0.72151 0.72368 0.72305 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 11 Measurement Period \DQNOUkUJNO-d *Minimum column element. 99 ABSOLUTE VALUE OF Weekly 1.42682 1.39940 1.34614 1.33451 1.31977 1.31285* 1.32611 1.31922 1.47563 1.48380 1.50840 1.50331 1.52542 1.52811 1.53048 1.52397 1.49362 1.48311 1.50560 1.50548 1.49848 1.51134 1.51122 1.50711 1.50622 1.49820 1.51162 1.51045 Observation Interval Monthly 1.28094 1.04191 0.96991 0.93294 0.88705 0.88003 0.84068 0.84740 0.83714* 0.84766 0.88225 0.89997 0.92054 0.93285 0.94460 0.94034 0.95254 0.96251 0.95202 0.95251 0.91337 0.94065 0.92888 0.96774 0.96337 0.94408 0.96051 0.95431 0.97351 0.97317 This value appears in Table IV—lA. Quarterly 0.82268 0.77643 0.67674 0.59077 0.56883 0.51680* 0.52360 0.53097 0.55217 0.59791 0.64223 0.72234 0.72234 0.76780 0.94114 0.94637 0.93671 0.94103 0.94854 0.94957 0.95205 0.91480 0.96405 0.97454 0.97671 0.97608 100 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 12 Observation Interval Measurement Period Weekly Monthly, anrterly 1 1.54437 1.39848 2 1.51694 1.15945 3 1.46368 1.08745 0.94023 4 1.45205 1.05048 0.89397 5 1.43731 1.00460 0.79428 6 1.43040* 0.99757 0.70831 7 1.44365 0.95822 0.68637 8 1.43677 0.96494 0.63434* 9 1.59317 0.95468* 0.64114 10 1.60134 0.96520 0.64851 11 1.62594 0.99980 0.66971 12 1.62085 1.01751 0.71545 13 1.64297 1.03808 0.75977 14 1.64565 1.05040 0.81048 15 1.64803 1.06214 0.83988 16 1.64151 1.05788 0.88534 17 1.61117 1.07008 1.05868 18 1.60065 1.08005 1.06391 19 1.62314 1.06957 1.05425 20 1.62302 1.07005 1.05857 21 1.61602 1.03091 1.06608 22 1.62888 1.05819 1.06711 23 1.62877 1.04642 1.06960 24 1.62465 1.08528 1.03234 25 1.62377 1.08091 1.08160 26 1.61574 1.06162 1.09208 27 1.62917 1.07805 1.09425 28 1.62800 1.07185 1.09362 29 1.09105 30 1.09071 *Minimum column element. This value appears in Table IV-lA. APPENDIX A-2 Mean Absolute Prediction Error for each Measurement Period Com- puted Using Ordinary Least Squares Beta Coefficients in Test Period 2 101 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 1 Observation Interval Measurement Period Weekly Monthly anrterly 1 0.92934 1.60065 2 0.73048 1.68071 3 0.74851 1.16848 4.68282 4 0.71886 1.16680 4.17877 5 0.74086 1.15491 2.64202 6 0.72414 1.15394 1.86882 7 0.68008 1.15097 1.61511 8 0.66205 1.10574 1.46637 9 0.65968 1.08580 1.46731 10 0.67563 1.08711 1.49594 11 0.66185 1.10011 1.49925 12 0.67826 1.09105 1.30168 13 0.53611 1.07743 1.12248 14 0.44914 1.05665 1.00325 15 0.42314 1.03577 0.99900 16 0.41848 1.05920 0.98726* 17 0.41766 1.03414 1.00128 18 0.40863 1.03520 1.03465 19 0.39160 1.03162 1.04811 20 0.38877 1.02648 1.06202 21 0.34211* 1.02525 1.05897 ‘ 22 0.34511 1.00791 1.06445 23 0.34831 0.96897 1.06294 24 0.35020 0.96005 1.06768 25 0.34974 0.96177 1.05411 26 0.35197 0.96223 1.04400 27 0.35180 0.95568 1.04360 28 0.34743 0.93877 1.00614 29 0.93420 30 0.90994* *Minimum column element. This value appears in Table IV-lB. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 2 Measurement Period \OQNGU‘IL‘WNH *Minimum column element. 102 ABSOLUTE VALUE OF Weekly 0.08094* 0.11791 0.09989 0.12954 0.10754 0.12426 0.16831 0.18634 0.18871 0.17277 0.18654 0.17014 0.31229 0.39926 0.42526 0.42991 0.43074 0.43977 0.45680 0.45963 0.50628 0.50328 0.50008 0.49820 0.49866 0.49643 0.49660 0.50097 Observation Interval Monthly 0.75225 0.83231 0.32009 0.31840 0.30651 0.30554 0.30257 0.25734 0.23740 0.23871 0.25171 0.24266 0.22903 0.20826 0.18737 0.21080 0.18574 0.18680 0.18323 0.17809 0.17686 0.15951 0.12057 0.11166 0.11337 0.11383 0.10729 0.09037 0.08580 0.06154* This value appears in Table IV-lB. Quarterly 3.83442 3.33037 1.79362 1.02042 0.76671 0.61797 0.61891 0.64754 0.65086 0.45328 0.27409 0.15486 0.15060 0.13886* 0.15289 0.18626 0.19971 0.21363 0.21057 0.21606 0.21454 0.21929 0.20571 0.19560 0.19520 0.15774 103 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 3 Observation Interval Measurement Period Weekly Monthly anrterly 1 0.12355* 0.54776 2 0.32240 0.62782 3 0.30437 0.11560 3.62994 4 0.33403 0.11391 3.12588 5 0.31203 0.10203 1.58914 6 0.32874 0.10203 1.58914 7 0.37280 0.09808 0.56222 8 0.39083 0.05285 0.41348 9 0.39320 0.03291 0.41442 10 0.37726 0.03423 0.44305 11 0.39103 0.04723 0.44637 12 0.37463 0.03817 0.24880 13 0.51677 0.02454 0.06960 14 0.60374 0.00377* 0.04963 15 0.62974 0.01712 0.05389 16 0.63440 0.00631 0.06563 17 0.63523 0.01875 0.05160 18 0.64426 0.01769 0.01823 19 0.66129 0.02126 0.00477 20 0.66411 0.02640 0.00914 21 0.71077 0.02763 0.00608 22 0.70777 0.04497 0.01157 23 0.70457 0.08392 0.91005 24 0.70268 0.09283 0.01480 25 0.70314 0.09112 0.00123* 26 0.70091 0.09112 0.00123* 27 0.70109 0.09066 0.00889 28 0.70546 0.11412 0.04675 29 0.11412 30 0.11868 *Minimum column element. This value appears in Table IV—lB. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 4 Measurement Period \DQNChU'Iwal-J *Minimum column element. 104 ABSOLUTE VALUE OF Observation Interval Weekly 1.21337 1.01451 1.03254 1.00288 1.02488 1.00817 0.96411 0.94608 0.94371 0.95965 0.94588 0.96228 0.82014 0.73317 0.70717 0.70251 0.70169 0.69266 0.67563 0.67280 0.62614* 0.62914 0.63234 0.63423 0.63377 0.63600 0.63583 0.63146 my. 1.88468 1.96474 1.45251 1.45083 1.43894 1.43797 1.43500 1.38977 1.36983 1.37114 1.38414 1.37508 1.36145 1.34068 1.31980 1.34322 1.31817 1.31922 1.31565 1.31051 1.30928 1.29194 1.25300 1.24408 1.24580 1.24625 1.23971 1.22280 1.21823 1.19397* This value appears in Table IV-lB. Quarterly 4.96685 4.46279 2.92605 2.15285 1.89914 1.75040 1.75134 1.77997 1.78328 1.58571 1.40651 1.28728 1.28302 1.27128* 1.28531 1.31868 1.33214 1.34605 1.34300 1.34848 1.34697 1.35171 1.33814 1.32802 1.32763 1.29017 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 5 Measurement Period xooouo‘uale-J *Minimum column element. 105 ABSOLUTE VALUE OF Weekly 0.04049* 0.23934 0.22131 0.25097 0.22897 0.24569 0.28974 0.30777 0.31014 0.29420 0.30797 0.29157 0.43371 0.52068 0.54668 0.55134 0.55217 0.56120 0.57823 0.58105 0.62771 0.62471 0.62151 0.61962 0.62008 0.61785 0.61803 0.62240 Observation Interval Monthly 0.63083 0.71088 0.19866 0.19697 0.18509 0.18411 0.18114 0.13591 0.11597 0.11729 0.13029 0.12123 0.10760 0.08683 0.06594 0.08937 0.06431 0.06537 0.06180 0.05666 0.05543 0.03809 0.00086* 0.00977 0.00806 0.00760 0.01414 0.03106 0.03563 0.05989 This value appears in Table IV-lB. Quarterly 3.71299 3.20894 1.67220 0.89900 0.64528 0.49654 0.49748 0.52611 0.52943 0.33186 0.15266 0.03343 0.02917 0.01743* 0.03146 0.06483 0.07829 0.09220 0.08914 0.09463 0.09311 0.09786 0.08429 0.07417 0.07377 0.03631 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 6 Measurement Period \DmNGUIbU-DNH *Minimum column element. 106 ABSOLUTE VALUE OF Observation Interval Weekly 0.23817 0.03931 0.05734 0.02769 0.04969 0.03297 0.01109* 0.02911 0.03149 0.01554 0.02931 0.01291 0.15506 0.24203 0.26803 0.27269 0.27351 0.28254 0.29957 0.30240 0.34906 0.34606 0.34286 0.34097 0.34143 0.33920 0.33937 0.34374 Monthly 0.90948 0.98954 0.47731 0.47563 0.46374 0.46277 0.45980 0.41457 0.39463 0.39594 0.40894 0.39989 0.38626 0.36449 0.34460 0.36803 0.34297 0.34403 0.34046 0.33531 0.33409 0.31674 0.27780 0.26889 0.27060 0.27106 0.26451 0.24760 0.24303 0.21877* This value appears in Table IV-lB. Quarterly 3.99165 3.48760 1.95085 1.17765 0.92394 0.77520 0.77614 0.80477 0.80808 0.61051 0.43131 0.31209 0.30783 0.29609* 0.31011 0.34349 0.35694 0.37086 0.36780 0.37329 0.37177 0.37651 0.36294 0.35283 0.35243 0.31497 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 7 Measurement Period \OGDVO‘UikLANi-J *Minimum column element. 107 ABSOLUTE VALUE OF Observation Interval Weekly 0.48831 0.28946 0.30749 0.27783 0.29983 0.28311 0.23906 0.22103 0.21866 0.23460 0.22083 0.23723 0.09509 0.00811* 0.01789 0.02254 0.02337 0.03240 0.04943 0.05226 0.09891 0.09591 0.09271 0.09083 0.09129 0.08906 0.08923 0.09360 Monthly 1.15903 1.23968 0.72746 0.72577 0.71388 0.71291 0.70994 0.66471 0.64477 0.64608 0.65908 0.65003 0.63640 0.61563 0.59474 0.61817 0.59311 0.59417 0.59060 0.58546 '0.58423 0.56688 0.52794 0.51903 0.52074 0.52120 0.51466 0.49774 0.49317 0.46891* This value appears in Table IV-lB. Quarterly 4.24180 3.73774 2.20099 1.42780 1.17408 1.02534 1.02628 1.05491 1.05823 0.86065 0.68146 0.56223 0.55797 0.54623* 0.56026 0.59363 0.60708 0.62100 0.61794 0.62343 0.62191 0.62666 0.61308 0.60297 0.60257 0.56511 108 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 8 Observation Interval Measurement Period Weekly Monthly anrterly 1 0.46369 1.13500 2 0.26483 1.21505 3 0.28286 0.70283 4.21716 4 0.25320 0.70114 3.71311 5 0.27520 0.68926 2.17636 6 0.25849 0.68828 1.40317 7 0.21443 0.68531 1.14945 8 0.19640 0.64009 1.00071 9 0.19403 0.62014 1.00166 10 0.20997 0.62146 1.03028 11 0.19620 0.63446 1.03360 12 0.21260 0.62540 0.83603 13 0.07046 0.61177 0.65683 14 0.01651* 0.59100 0.53760 15 0.04251 0.57011 0.53334 16 0.04717 0.59354 0.52160* 17 0.04800 0.56848 0.53563 18 0.05703 0.56954 0.56900 19 0.07406 0.56597 0.58246 20 0.07689 0.56083 0.59637 21 0.12354 0.55960 0.59331 22 0.12054 0.54226 0.59880 23 0.11734 0.50331 0.59728 24 0.11546 0.49440 0.60203 25 0.11591 0.49611 0.58846 26 0.11369 0.49657 0.57834 27 0.11386 0.49003 0.57794 28 0.11823 0.47311 0.54048 29 0.46854 30 0.44429* *Minimum column element. This value appears in Table IV—lB. APPENDIX B-l Mean Absolute Prediction Error for each Measurement Period Com- puted Using Fisher-Kamin Beta Coefficients in Test Period 1. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 1 Measurement Period OflNO‘UIJ-‘UJNH *Minimum column element. 109 ABSOLUTE VALUE OF Weekly 0.60463 0.58823 0.53451 0.52948 0.51860 0.50891* 0.52237 0.51337 0.66994 0.67574 0.69834 0.68863 0.71051 0.71231 0.71285 0.70405 0.67188 0.66317 0.69674 0.69731 0.68877 0.70111 0.70111 0.69320 0.69125 0.68534 0.69854 0.69922 Observation Interval Monthly 0.19451 0.12200 0.08777 0.08169* 0.08323 0.08451 0.28391 0.14840 0.11409 0.11829 0.12883 0.12731 0.09817 0.11709 0.10449 0.10389 0.10740 0.11709 0.12563 0.12580 0.11446 0.12749 0.12843 0.15151 0.14577 0.12134 0.12223 0.11989 0.12660 0.13720 This value appears in Table IV-6A. Quarterly 0.00926* 0.07289 0.03803 0.21114 0.22651 0.31331 0.30600 0.32526 0.31854 0.29031 0.23520 0.17263 0.13769 0.08657 0.08137 0.10131 0.11911 0.12229 0.12883 0.12937 0.12949 0.08634 0.14060 0.15017 0.15451 0.15603 110 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 2 Observation Interval Measurement Period Weekly Monthly anrterly 1 1.03017 0.62006 2 1.01377 0.54754 3 0.96005 0.51331 0.41629 4 0.95503 0.50723* 0.49843 5 0.94414 0.50877 0.38751 6 0.93445* 0.51006 0.21440 7 0.94791 0.70945 0.19903 8 0.93891 0.57394 0.11223 9 1.09548 0.53963 0.11954 10 1.10128 0.54383 0.10029* 11 1.12388 0.55437 0.10700 12 1.11417 0.55286 0.13523 13 1.13605 0.52371 0.19034 14 1.13785 0.54263 0.25291 15 1.13840 0.53003 0.28786 16 1.12959 0.52943 0.33897 17 1.09742 0.53294 0.50691 18 1.08871 0.54263 0.52686 19 1.12228 0.55117 0.54466 20 1.12285 0.55134 0.54783 21 1.11431 0.54000 0.55437 22 1.12665 0.55303 0.55491 23 1.12665 0.55397 0.55503 24 1.11874 0.57706 0.51188 25 1.11679 0.57131 0.56614 26 1.11088 0.54689 0.57571 27 1.12408 0.54777 0.58006 28 1.12477 0.54543 0.58157 29 ' 0.55214 30 0.56274 *Minimum column element. This value appears in Table IV-6A. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 3 Measurement Period WQNO‘M-FUNH *Minimum column element. 111 ABSOLUTE VALUE OF Weekly 1.09054 1.07414 1.02042 1.01540 1.00451 0.99483* 1.00828 0.99928 1.15585 1.16165 1.18425 1.17454 1.19642 1.19822 1.19877 1.18997 1.15779 1.14908 1.18265 1.18322 1.17468 1.18702 1.18702 1.17911 1.17717 1.17125 1.18445 1.18514 Observation Interval Monthly, 0.68043 0.60791 0.57368 0.56760* 0.56914 0.57043 0.76983 0.63431 0.60000 0.60420 0.61474 0.61323 0.58408 0.60300 0.59040 0.58980 0.59331 0.60300 061154 0.61171 0.60037 0.61340 0.61434 0.63742 0.63168 0.60726 0.60814 0.60580 0.61251 0.62311 This value appears in Table IV-6A. Quarterly 0.47666 0.55880 0.44789 0.27477 0.25940 0.17260 0.17991 0.16066* 0.16737 0.19560 0.25071 0.31329 0.34823 0.39934 0.56728 0.58723 0.60503 0.60820 0.61474 0.61528 0.61540 0.57226 0.62651 0.63608 0.64043 0.64194 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 4 Measurement Period \DmNOUI-L‘UJNH *Minimum column element. 112 ABSOLUTE VALUE OF Weekly 1.25597 1.23957 1.18585 1.18083 1.16994 1.16025* 1.17371 1.16471 1.32128 1.32708 1.34968 1.33997 1.36185 1.36365 1.36419 1.35539 1.32322 1.31451 1.34808 1.34865 1.34011 1.35245 1.35245 1.34454 1.34260 1.33668 1.34988 1.35057 Observation Interval Monthly 0.84585 0.77334 0.73911 0.73302* 0.73457 0.73586 0.93525 0.79974 0.76543 0.76962 0.78017 0.77865 0.74951 0.76843 0.75582 0.75523 0.75874 0.76843 0.77697 0.77714 0.76580 0.77883 0.77977 0.80285 0.79711 0.77268 0.77357 0.77123 0.77794 0.78854 This value appears in Table IV-6A. Quarterly 0.64208 0.72423 0.61331 0.44020 0.42483 0.42483 0.34534 0.32609* 0.33280 0.36103 0.41614 0.47871 0.51366 0.56477 0.73271 0.75265 0.75265 0.77362 0.78017 0.78071 0.78082 0.73768 0.79194 0.80151 0.80585 0.80737 TEST PERIOD 1 TEST PERIOD 5 Measurement Period \DQNO\U‘<§WNH *Minimum column element. 113 ABSOLUTE VALUE OF Weekly 1.61537 1.59897 1.54525 1.54023 1.52934 1.51965* 1.53311 1.52411 1.68068 1.68648 1.70908 1.69937 1.72125 1.72305 1.72359 1.71480 1.68262 1.67391 1.70748 1.70805 1.69951 1.71185 1.71185 1.70394 1.70200 1.69608 1.70928 1.70997 AVERAGE PREDICTION ERROR FOR 35 COMPANIES Observation Interval Monthly, 1.20525 1.3274 1.09851 1.09242* 1.09397 1.09525 1.29465 1.15914 1.12482 1.12902 1.13957 1.13805 1.10891 1.12782 1.11522 1.11462 1.11814 1.12782 1.13637 1.13654 1.12520 1.13822 1.13917 1.16225 1.15651 1.13208 1.13297 1.13062 1.13734 1.14794 This value appears in Table IV-6A. Quarterly 1.00148 1.08362 0.97271 0.79960 0.78423 0.69743 0.70474 0.68548* 0.69220 0.72043 0.77554 0.83811 0.87305 9.92417 1.09211 1.11205 1.12985 1.13303 1.13957 1.14011 1.09708 1.14022 1.15134 1.16091 1.16525 1.16677 114 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 6 Observation Interval Measurement Period Weekly Monthly anrterly 1 1.49111 1.08100 2 1.47471 1.00848 3 1.42100 0.97425 0.87723 4 1.41597 0.96817* 0.95937 5 1.40508 0.96971 0.84845 6 1.39540* 0.97100 0.67534 7 1.40885 1.17040 0.65997 8 1.39985 1.03488 0.57317 9 1.55642 1.00057 0.58048 10 1.56223 1.00477 0.56123* 11 1.58483 1.01531 0.56794 12 1.57511 1.01380 0.59617 13 1.59700 0.98465 0.65128 14 1.59879 1.00357 0.71385 15 1.59934 0.99097 0.74880 16 1.59054 0.99037 0.79991 17 1.55837 0.99388 0.96785 18 1.54965 1.00357 0.98780 19 1.58322 1.01211 1.00560 20 1.58379 1.01228 1.00877 21 1.57525 1.00094 1.01531 22 1.58759 1.01397 1.01585 23 1.58759 1.01491 1.01597 24 1.57968 1.03800 0.97283 25 1.57774 1.03225 1.02708 26 1.57182 1.00782 1.03665 27 1.58502 1.00871 1.04100 28 1.58571 1.00637 1.04251 29 1.01308 30 1.02368 *Minimum column element. This value appears in Table IV-6A. 115 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 7 Observation Interval Measurement Period Weekly Monthly Quarterly 1 1.35280 0.94268 2 1.33640 0.87017 3 1.28268 0.83594 0.73891 4 1.27765 0.82985* 0.82105 5 1.26677 0.83140 0.71014 6 1.25708* 0.83268 0.53703 7 1.27054 1.03208 0.52166 8 1.26154 0.89657 0.43486 9 1.41811 0.86225 0.44217 10 1.42391 0.86645 0.42291* 11 1.44651 0.87700 0.42963 12 1.43680 0.87548 0.45786 13 1.45868 0.84634 0.51297 14 1.46048 0.86525 0.57554 15 1.46102 0.85265 0.61048 16 1.45222 0.85205 0.66160 17 1.42005 0.85557 0.82954 18 1.41134 0.86525 0.84948 19 1.44491 0.87380 0.86728 20 1.44548 0.87397 0.87045 21 1.43694 0.86263 0.87700 22 1.44928 0.87565 0.87754 23 1.44928 0.87660 0.87765 24 1.44137 0.89968 0.83451 25 1.43942 0.89394 9.88877 26 1.43351 0.86951 0.89834 27 1.44671 0.87040 0.90268 28 1.44739 0.86805 0.90420 29 0.87477 30 9.88537 *Minimum column element. This value appears in Table IV-6A. r-—' 116 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 8 Measurement Observation.Interva1 Period Weekly Monthly Quarterly 1 1.63368 1.22357 2 1.61728 1.15105 3 1.56357 1.11682 1.01980 4 1.55854 1.11074* 1.10194 5 1.54765 1.11228 0.99102 6 1.53797* 1.11357 0.81791 7 1.55142 1.31297 0.80254 8 1.54243 1.17745 0.71574 9 1.69900 1.14314 0.72305 10 1.70480 1.14734 0.70380* 11 1.72740 1.15788 0.71051 12 1.71768 1.15637 0.73874 13 1.73957 1.12722 0.79385 14 1.74137 1.14614 0.85643 15 1.74191 1.13354 0.89137 16 1.73311 1.13294 0.94248 17 1.70094 1.13645 1.11042 18 1.69222 1.14614 1.13037 19 1.72579 1.15468 1.14817 20 1.72637 1.15485 1.15134 21 1.71782 1.14351 1.15788 22 1.73017 1.15654 1.15842 23 1.73017 1.15748 1.15854 24 1.72225 1.18057 1.11540 25 1.72031 1.17482 1.16965 26 1.71440 1.15040 1.17922 27 1.72760 1.15128 1.18357 28 1.72828 1.14894 1.18508 29 1.15565 30 1.16625 *Minimum column element. This value appears in Table IV-6A. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 9 Measurement Period omflOUIbUNt-I' *Minimum column element. 117 ABSOLUTE VALUE OF Observation Interval Weekly 1.37748 1.36108 1.30737 1.30234 1.29145 1.28177* 1.29522 1.28622 1.44280 1.44860 1.47120 1.46148 1.48337 1.48517 1.48571 1.47691 1.44474 1.43602 1.46960 1.47017 1.46162 1.47397 1.47397 1.46605 1.46411 1.45820 1.47140 1.47208 Monthly 0.96737 0.89485 0.86062 0.85454* 0.85608 0.85737 1.05677 0.92125 0.88694 0.89114 0.90168 0.90017 0.87103 0.88994 0.87734 0.87674 0.88025 0.88994 0.89848 0.89865 0.88731 0.90034 0.90128 0.92437 0.91863 0.89420 0.89508 0.89274 0.89945 0.91005 This value appears in Table IV-6A. Quarterly 0.76360 0.84574 0.73482 0.56171 0.54634 0.45954 0.46686 0.44760* 0.45431 0.48254 0.53765 0.60023 0.63517 0.68628 0.85423 0.87417 0.89197 0.89514 0.90168 0.90222 0.90234 0.91345 0.91345 0.92302 0.92737 0.92888 118 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 10 Observation.lnterva1 Measurement Period Weekly Monthly anrterly 1 1.15554 0.74543 2 1.13914 0.67291 3 1.08543 0.63868 0.54165 4 1.08040 0.63260* 0.62380 5 1.06951 0.63414 0.51288 6 1.05982* 0.63543 0.33977 7 1.07328 0.83483 0.32440 8 1.06428 0.69931 0.23760 9 1.22085 0.66500 0.24491 10 1.22665 0.66920 0.22566* 11 1.24925 0.67974 0.23237 12 1.23954 0.67823 0.26060 13 1.26143 0.64908 0.31571 14 1.26322 0.66800 0.37829 15 1.26377 0.65540 0.41323 16 1.25497 0.65480 0.46434 17 1.22280 0.65831 0.63228 18 1.21408 0.66800 0.65223 19 1.24765 0.67654 0.67003 20 1.24822 0.67671 0.67320 21 1.23968 0.66537 0.67974 22 1.25202 0.67840 0.68028 23 1.25202 0.67934 0.68040 24 1.24411 0.70243 0.63725 25 1.24217 0.69668 0.69151 26 1.23625 0.67226 0.70108 27 1.24945 0.67314 0.70543 28 1.25014 0.67080 0.70694 29 0.67751 30 0.68811 *Minimum column element. This value appears in Table IV-6A. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 11 Measurement Period \DmNO‘U'l-L‘wNH *Minimum column element. 119 ABSOLUTE VALUE OF Weeklz 1.40857 1.39217 1.33845 1.33342 1.32254 1.31285* 1.32631 1.31731 1.47388 1.47968 1.50228 1.49257 1.51445 1.51625 1.51680 1.50800 1.47582 1.46711 1.50068 1.50125 1.49271 1.50505 1.50505 1.49714 1.49520 1.48928 1.50248 1.50317 Observation Interval Monthly 0.99845 0.92594 0.89171 0.88563* 0.88717 0.88845 1.08785 0.95234 0.91802 0.92223 0.93277 0.93125 0.90211 0.92103 0.90843 0.90783 0.91134 0.92103 0.92957 0.92974 0.91840 0.93143 0.93237 0.95545 0.94971 0.92528 0.92617 0.92382 0.93054 0.94114 This value appears in Table IV-6A. Quarterlz 0.79468 0.97683 0.76591 0.59280 0.57743 0.49063 0.49794 0.47868* 0.48540 0.51363 0.56874 0.63131 0.66625 0.71737 0.88531 0.90525 0.92305 0.92623 0.93277 0.93331 0.93343 0.89028 0.94454 0.95411 0.95845 0.95997 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 1 HOLDING PERIOD 12 Measurement Period \DGJVO‘UJ-‘UONH *Minimum column element. 120 ABSOLUTE VALUE OF Observation Interval Weeklx 1.52611 1.50971 1.45600 1.45097 1.44008 1.43040* 1.44385 1.43485 1.59142 1.59722 1.61982 1.61011 1.63200 1.63380 1.63434 1.62554 1.59337 1.58465 1.61822 1.61880 1.61025 1.62260 1.62259 1.61468 1.61274 1.60682 1.62002 1.62071 (11 Monthly 1.11600 1.04348 1.00925 1.00317* 1.00471 1.00600 1.20540 1.06988 1.03557 1.03977 1.05031 1.04880 1.01965 1.03857 1.02597 1.02537 1.02888 1.03857 1.04711 1.04728 1.03594 1.04897 1.04991 1.07300 1.06725 1.04282 1.04371 1.04137 1.04808 1.05868 This value appears in Table IV-6A. Quarterly 0.91222 0.99437 0.88345 0.71034 0.69497 0.60817 0.61548 0.59623* 0.60294 0.63117 0.68628 0.74885 0.78380 0.83491 1.00285 1.02280 1.04060 1.04377 1.05031 1.05085 1.05097 1.00782 1.06208 1.07165 1.07600 1.07751 APPENDIX B-2 Mean Absolute Prediction Error for each Measurement Period Computed Using Fisher-Kamin Beta Coefficients in Test Period 2. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 1 Measurement Period \OCDNO‘LDJ-‘UJNH *Minimum column element. 121 ABSOLUTE VALUE OF Weeklx 0.91182 0.73151 0.75985 0.73388 0.74414 0.72565 0.68040 0.66160 0.65931 0.67648 0.66326 0.68414 0.54297 0.45074 0.42374 0.41883 0.41877 0.41174 0.39389 0.39211 0.34534* 0.34903 0.35323 0.35683 0.35663 0.35966 0.36006 0.35717 Observation Interval Monthlz 1.58605 1.62485 1.19140 1.22314 1.16805 1.15708 1.15231 1.10511 1.08554 1.08714 1.10114 1.09382 1.08685 1.06980 1.03837 1.06037 1.03500 1.03714 1.03825 1.03125 1.03122 1.01337 0.97768 0.97448 0.97448 0.97837 0.97297 0.95677 0.95203 0.92800* This value appears in Table IV-6B. Quarterlx 1.43597 1.78842 2.01319 1.80794 1.61422 1.45685 1.45568 1.50308 1.51208 1.32908 1.13645 1.01160 1.00311 0.99120* 1.01057 1.05728 1.06383 1.08602 1.08234 1.09400 1.09800 1.11305 - 1.10008 1.09108 1.09425 1.05923 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 2 Measurement Period cmuomwar-d *Minimum column element. 122 ABSOLUTE VALUE OF Weeklz 0.06343* 0.11689 0.08854 0.11451 0.10426 0.12274 0.16800 0.18680 0.18909 0.17191 0.18514 0.16426 0.30543 0.39766 0.42466 0.42957 0.42963 0.43666 0.45451 0.45629 0.50306 0.49937 0.49517 0.49157 0.49177 0.48874 0.48834 0.49123 Observation Interval Monthlx 0.73765 0.77645 0.34300 0.37474 0.31966 0.30869 0.30391 0.25671 0.23714 0.23874 0.25274 0.24543 0.23846 0.22140 0.18997 0.21197 0.18660 0.18874 0.18986 0.18286 0.18283 0.16497 0.12929 0.12294 0.12609 0.12997 0.12457 0.10837 0.10363 0.07960* This value appears in Table IV-6B. Quarterlx 0.58757 0.94002 0.16479 0.95954 0.76583 0.60845 0.69728 0.65468 0.66368 0.48068 0.28806 0.16320 0.15471 0.14280* 0.16217 0.20889 0.21543 0.23763 0.23394 0.24560 0.24960 0.26466 0.25169 0.24269 0.24586 0.21083 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 3 Measurement Period \OCDNO‘Uil-‘LJNH *Minimum column element. 123 ABSOLUTE VALUE OF Ueeklx 0.14106* 0.32137 0.29303 0.31900 0.30874 0.32723 0.37249 0.39129 0.39357 0.37640 0.38963 0.36874 0.50991 0.60214 0.62914 0.63406 0.63411 0.64114 0.65900 0.66077 0.70754 0.70386 0.69966 0.69606 0.69626 0.69323 0.69283 0.69571 Observation Interval Monthlz 0.53317 0.57197 0.13851 0.17025 0.11517 0.10420 0.09942 0.05222 0.03265 0.03425 0.04825 0.04094 0.03397 0.01691 0.01452 0.00748* 0.01789 0.01575 0.01463 0.02163 0.02166 0.03952 0.07520 0.08155 0.07840 0.07452 0.07992 0.09612 0.10086 0.12489 This value appears in Table IV-6B. Quarterlz 0.38308 0.73554 0.96031 0.75505 0.56134 0.40397 0.40279 0.45020 0.45919 0.27620 0.08357 0.04129 0.04977 0.06169 0.04232 0.00440* 0.01094 0.03314 0.02945 0.04111 0.04511 0.06017 0.04720 0.03820 0.04137 0.00634 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 4 Measurement Period \Dmfla‘mwaH *Minimum column element. 124 ABSOLUTE VALUE OF Weeklz 1.19585 1.01554 1.04388 1.01791 1.02817 1.00968 0.96443 0.94563 0.94334 0.96051 0.94728 0.96817 0.82700 0.73477 0.70777 0.70286 0.70280 0.69577 0.67791 0.67614 0.62937* 0.63306 0.63726 0.64086 0.64066 0.64369 0.64408 0.64120 Observation Interval 21929111 1.87008 1.90888 1.47542 1.50717 1.45208 1.44111 1.42634 1.38914 1.36957 1.37117 1.38517 1.37785 1.37088 1.35383 1.32240 1.34440 1.31903 1.32117 1.32228 1.31528 1.31525 1.29740 1.26171 1.25537 1.25851 1.26240 1.25700 1.24079 1.23605 1.21202* This value appears in Table IV-6B. Quarterlz 1.72000 2.07245 2.29722 2.09197 1.89825 1.74088 1.73971 1.78711 1.79611 1.61311 1.42048 1.29562 1.28714 1.27523* 1.29460 1.34131 1.34785 1.37005 1.36637 1.37802 1.38202 1.39708 1.38411 1.37511 1.37828 1.34325 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 5 Measurement Period \DCDNOUIbWNI-J *Minimum column element. 125 ABSOLUTE VALUE OF Weekly 0.05800* 0.23831 0.20997 0.23594 0.22569 0.24417 0.28943 0.30823 0.31051 0.29334 0.30657 0.28569 0.42686 0.51908 0.54608 0.55100 0.55105 0.55808 0.57594 0.57771 0.62448 0.62080 0.61660 0.61300 0.61320 0.61017 0.60977 0.61266 Observation Interval Monthly 0.61623 0.65503 0.22157 0.25331 0.19823 0.18726 0.18249 0.13529 0.11571 0.11731 0.13131 0.12400 0.11703 0.09997 0.06854 0.09054 0.06517 0.06731 0.06843 0.06143 0.06140 0.04354 0.00786 0.00151* 0.00466 0.00854 0.00314 0.01306 0.01780 0.04183 This value appears in Table IV-6B. Quarterly 0.46614 0.81860 1.04337 0.83811 0.64440 0.48703 0.48586 0.53326 0.54226 0.35926 0.16663 0.04177 0.03329 0.02137* 0.04074 0.08746 0.09400 0.11620 0.11251 0.12417 0.12817 0.14323 0.13026 0.12126 0.12443 0.12443 126 ABSOLUTE VALUE OF AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 6 Observation Interval Measurement Period Weekly Monthly Quarterly 1 0.22066 0.89488 2 0.04034 0.93368 3 0.06869 0.50023 0.74480 4 0.04371 0.53197 1.09725 5 0.05297 0.47689 1.32203 6 0.03449 0.46591 1.11677 7 0.01077 0.46114 0.92306 8 0.02957 0.41394 0.76568 9 0.03186 0.39437 0.76451 10 0.01469 0.39597 0.81191 11 0.02791 0.40997 0.82091 12 0.00703* 0.40266 0.63791 13 0.14820 0.39569 0.44529 14 0.24043 0.37863 0.32043 15 0.26743 0.34720 0.31194 16 0.27234 0.36920 0.30003* 17 0.27240 0.34383 0.31940 18 0.27943 0.34597 0.36611 19 0.29729 0.34709 0.37266 20 0.29906 0.34009 0.39486 21 0.34583 0.34006 0.39117 22 0.34214 0.32220 0.40283 23 0.33794 0.28651 0.40683 24 0.33434 0.28017 0.42189 25 0.33454 0.28331 0.40891 26 0.33151 0.28720 0.39991 27 0.33111 0.28180 0.40309 28 0.33400 0.26560 0.36806 29 0.26086 30 0.23683* *Minimum column element. This value appears in Table IV-6B. AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 7 Measurement Period xooowa‘mwaH *Minimum column element. 127 ABSOLUTE VALUE OF Weekly 0.47080 0.29049 0.31883 0.29286 0.30311 0.28463 0.23937 0.22057 0.21829 0.23546 0.22223 0.24311 0.10194 0.00971* 0.01729 0.02220 0.02226 0.02929 0.04714 0.04891 0.09569 0.09200 0.08780 0.08420 0.08440 0.08137 0.08097 0.08386 Observation Interval Monthly 1.14502 1.18382 0.75037 0.78211 0.72703 0.71606 0.71128 0.66408 0.64451 0.64611 0.66011 0.65280 0.64583 0.62877 0.59734 0.61934 0.59397 0.59611 0.59723 0.59023 0.59020 0.57234 0.53666 0.53031 0.53346 0.53734 0.53194 0.51574 0.51100 0.48697* This value appears in Table IV-6B. Quarterly 1.34740 1.57217 1.36691 1.17320 1.01583 1.01465 1.06205 1.07106 0.88805 0.69543 0.57057 0.56208 0.55017* 0.56954 0.61626 0.62280 0.64500 0.64131 0.65297 0.65697 0.67203 0.65906 0.65006 0.65323 0.61820 AVERAGE PREDICTION ERROR FOR 35 COMPANIES TEST PERIOD 2 HOLDING PERIOD 8 Measurement Period \OCDNO‘UIJ-‘MNH *Minimum column element. 128 ABSOLUTE VALUE OF Weekly 0.44617 0.26586 0.29420 0.26823 0.27849 0.26000 0.21474 0.19594 0.19366 0.21083 0.19760 0.21849 0.07731 0.01491* 0.04191 0.04683 0.04689 0.05391 0.07177 0.07354 0.12031 0.11663 0.11243 0.10883 0.10903 0.10600 0.10560 0.10849 Observation Interval Monthly 1.12040 1.15920 0.72574 0.75748 0.70240 0.69143 0.68666 0.63946 0.61988 0.62148 0.63548 0.62817 0.62120 0.60414 0.57271 0.59471 0.56934 0.57148 0.57260 0.56560 0.56557 0.54771 0.51203 0.50568 0.50883 0.51271 0.50731 0.49111 0.48637 0.46234* This value appears in Table IV-6B. Quarterly 0.97031 1.32277 1.54754 1.34228 1.14857 0.99120 0.99003 1.03743 1.04643 0.86343 0.67080 0.54594 0.53746 0.52554* 0.54491 0.59163 0.59817 0.62037 0.62834 0.62834 0.63234 0.64740 0.63443 0.62543 0.62860 0.59357 BIBLIOGRAPHY Bing, Ralph A., "Survey of Practitioners' Stock Evaluation Methods," Financial Analysts Journal, May/June 1971, 55-61. I? Block, Frank E., "Elements of Portfolio Construction," Financial Analysts Journal, May/June 1969, 123-130. Blume, Marshall E., "Portfolio Theory: A Step Towards Its Practical u Application," Journal of Business, June 1970, 602-612. ‘ , "The Assessment of Portfolio Performance: An Application of Portfolio Theory," Unpublished Ph.D. dissertation, Graduate School of Business, University of Chicago, 1968. Breen, W. and Lerner, E. M., "Corporate Financial Strategies and Market Measures of Risk and Return," Journal of Finance, June 1973, pp. 339-351. 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