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A statistical investigation of the returns on closed-end investment companies Ellis, Denise Taylor 1977

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A STATISTICAL INVESTIGATION OF THE RETURNS ON CLOSED-END INVESTMENT COMPANIES by DENISE TAYLOR ELLIS A.B., University of C a l i f o r n i a , Berkeley, 1968 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUSINESS ADMINISTRATION i n THE FACULTY OF GRADUATE STUDIES (The Faculty of Commerce and Business Administration) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1978 Denise Taylor E l l i s , 1978 In presenting th i s thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree l y ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho lar l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f i nanc ia l gain sha l l not be allowed without my wri t ten permission. Department of Commerce and Business Administration The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 6 Abstract The common shares of closed-end funds, unlike mutual funds, trade on the stock exchanges. A market determined value of the assets of closed-end funds (net asset value) i s published weekly for those funds l i s t e d on the New York Stock Exchange. A discrepancy exists between the market p r i c e of the common share of the closed-end fund and the net asset value per common share of the fund. The size of these discrepancies, premiums and discounts, has never been adequately explained within the context of f i n a n c i a l theory. Furthermore, estimates of r i s k c o e f f i -cients (betas) are such that the common equity appears les ri s k y than the closed-end fund i t s e l f . An inve s t i g a t i o n was undertaken of the s t a t i s t i c a l properties associated with both weekly and monthly market value and net asset value return series for twelve closed-end funds l i s t e d on the New York Stock Exchange from 1965 to the end of 1972. These twelve funds account for approx imately f i f t y percent of a l l funds by asset size l i s t e d during that period. Non-parametric tests demonstrated a lack of inde-pendence i n contiguous observations and some additional support was given by a measure of s e r i a l c o r r e l a t i o n . Goodness-of-fit tests were performed for the normal d i s t r i b u t i o n and i t was rejected as representative of the data. The d i s t r i b u t i o n of the return series, as v e r i f i e d by the sample moments, i s leptokurtic and shows properties consistent with a stable d i s t r i b u t i o n . The lack of independence and normality i n the data causes serious v i o l a t i o n s of the assumptions necessary to f i t the market model i n order to estimate the betas of the closed-end funds. The vi o l a t i o n s are such that the market return betas are l i k e l y to be seriously underestimated and therefore cause the common equity of closed-end funds to appear less r i s k y than the funds themselves. Some support for theory which indicates that the common equity should be r i s k i e r i s given by the results of the lagged market model. i v Table of Contents i n t r o d u c t i o n 1 Sample Data 5 Study Design 10 S i n g l e Sample Runs T e s t 12 Kolmogorov-Smirnov Two-Sample Goo d n e s s - o f - F i t 18 S e r i a l C o r r e l a t i o n 25 Sample Moments 32 Good n e s s - o f - F i t T e s t s 37 Chi-Square T e s t 39 Kolmogorov-Smirnov One-Sample T e s t 46 Cramer-von Mises Go o d n e s s - o f - F i t T e s t 50 Anderson-Darling T e s t o f Goodness-of-Fit 54 Re g r e s s i o n A n a l y s i s 58 C o n c l u s i o n 68 Appendix 1: B r i e f H i s t o r y o f Closed-End Investment Companies 71 Appendix 2: Overview o f C a u s a l i t y o f Discounts/Premiums 77 Appendix 3: Runs Test, D e t a i l e d R e s u l t s 81 Appendix 4: T r a d i n g Volumes 83 Appendix 5: Sample Moments 84 Appendix 6: Goo d n e s s - o f - F i t T e s t s — Comparative R e s u l t s 89 Appendix 7: Simple Re g r e s s i o n — D e t a i l e d R e s u l t s 93 V Appendix 8: Lagged Re g r e s s i o n Model — D e t a i l e d R e s u l t s 97 Footnotes 98 B i b l i o g r a p h y 102 v i L i s t o f Tables I Companies i n Sample 9 II Runs T e s t — Weekly Data 15 III Runs T e s t — Monthly Data 15 IV Kolmogorov-Smirnov Two Sample T e s t — Weekly Data 22 V Kolmogorov-Smirnov Two Sample T e s t — Monthly Data 22 VI S e r i a l C o r r e l a t i o n — Weekly Data 27 VII S e r i a l C o r r e l a t i o n — Monthly Data 28 V I I I S e r i a l C o r r e l a t i o n — M a r k e t Index — Weekly Data 31 IX Groups and Range S p e c i f i c a t i o n — Chi-Square T e s t 43 X Chi-Square T e s t o f G o odness-of-Fit where the D i s t r i b u t i o n o f I n t e r e s t i s the Normal D i s t r i b u t i o n 44 XI Kolmogorov-Smirnov One Sample T e s t 48 XII Crame'r-von Mises T e s t of G o odness-of-Fit 52 XIII Anderson-Darling G o o d n e s s - o f - F i t T e s t 56 XIV E s t i m a t e d Betas 62 XV C o r r e l a t i o n - Market Returns and Index — Weekly Data 65 XVI Lagged Market Model 66 XVII C i t a t i o n o f Reasons f o r Disounts/Premiums 80 XVIII Runs T e s t 81 XIX Sample Moments — Market Returns 85 XX Sample Moments — Net A s s e t Value Returns 87 XXI Comparative R e s u l t s G o o d n e s s - o f - F i t T e s t s — Market Returns 89 v i i XXII Comparative R e s u l t s o f G o o d n e s s - o f - f i t T e s t s — Net A s s e t Value Returns 91 XXIII R e g r e s s i o n R e s u l t s — Market Returns — Weekly 93 XXIV Reg r e s s i o n R e s u l t s - Net A s s e t Value Returns — Weekly 94 XXV R e g r e s s i o n R e s u l t s - Market Returns — Monthly 95 XXVI Re g r e s s i o n R e s u l t s — Net A s s e t Value Returns -- Monthly 96 XXVII R e g r e s s i o n R e s u l t s — Lagged Market Model 97 V l l l L i s t of I l l u s t r a t i o n s Chernoff and Lehman's r e s u l t i l l u s t r a t e d i x Acknowledgment The committee has been invaluable i n encouraging me forward and providing patient assistance. Dr. Robert White provided much of the raw data and I am very thankful to avoid that thankless task. For emotional and f i n a n c i a l support, I am t r u l y i n -debted to B i l l y C. E l l i s , without whom t h i s opportunity would never have existed. 1 Introduction A closed-end investment company, as defined by the Investment Company Act of 1940 (U.S.) i s a company en-gaged i n the business of investing, reinvesting, or trading i t s f i n a n c i a l assets. More then fo r t y percent of i t s assets must be held i n invested s e c u r i t i e s , excluding government s e c u r i t i e s and those of major owned subsidiaries which are not investment companies. The company does not engage i n the continuous sale and redemption of shares i n the company, and i t i s t h i s aspect which d i f f e r e n t i a t e s closed-end invest-ment companies from open-end mutual funds. The lack of con-tinuous redemption re s u l t s i n the number of shares of the company outstanding remaining r e l a t i v e l y constant over time. Closed-end funds are not new, having existed i n one form or another since 1868''". They tend to be ignored i n the finance l i t e r a t u r e because l i t t l e adequate explanation can be given for the existence of premiums and discounts on the market value of the common stock of the fund when compared to the market determined value of the fund's net assets. This work investigates the s t a t i s t i c a l properties of returns on closed-end fund assets and common equity. I t becomes apparent that closed-end funds during the time period 1965-1972 do not always exhi b i t c h a r a c t e r i s t i c s believed 2 t y p i c a l of common stock. In fact, they exh i b i t character-i s t i c s which create serious estimation problems when one attempts to apply s t a t i s t i c a l methodology based upon the assumption of normality. It i s believed that one source of deviation i n the s t a t i s t i c a l behaviour of closed-end funds arises from i n -freguent trading, r e l a t i v e to the stock market as a whole, and while t h i s matter i s not d i r e c t l y investigated, there i s some evidence that returns on closed-end funds lag returns on the market Index. The existence of discounts and premiums i s often treated as an example of i n e f f i c i e n t markets by f i n a n c i a l writers. The argument i s that i f the market value of the common equity of a closed-end fund r e f l e c t s the net present value of future earnings to a shareholder, and the market determined net asset value of the fund r e f l e c t s the net present value of future earnings of the fund, then the market value should be the same. This argument neglects the fac t that future mana-g e r i a l fees must be paid out of future earnings, and they may be s u f f i c i e n t to create a discount. Furthermore, some of the funds have had preferred shares outstanding i n the past (Tri-Continental s t i l l does) and the p o s s i b i l i t y of such p r i o r claimants to earnings may e x i s t i n the future. Whether the si z e of discounts observed i s reasonable, i s another question. 3 Discount i s defined as (MV. . - NAV. .) / NAV. . where MV. . i s the market value per share of common equity of company i at time t and NAV i s the corresponding per share net asset value of the underlying s e c u r i t i e s i n the firm's p o r t f o l i o . The size of the discount varies greatly across companies. Rising markets are associated with increasing discounts and f a l l i n g markets i n decreasing discounts. That i s , the net asset value of the firm's p o r t f o l i o increases at a faster rate i n a r i s i n g stock market than the market value of the shares of the closed-end fund held by the public. (A d i s -count of -17% i s larger than a discount of -10%). Research i n t h i s area has been e s s e n t i a l l y h e u r i s t i c although more recent work by Ingersoll and Malkiel attempts 2 to provide some t h e o r e t i c a l underpinnings. Their economic arguments are not borne out by consistent empirical t e s t s . A f u l l l i s t of the reasons posited for discounts and premiums and t h e i r source i s contained i n Appendix 2. One reason frequently put forth for the existence of discounts i s that ownership of stock i n a closed-end fund i s viewed as r i s k i e r than d i r e c t ownership of the under-l y i n g p o r t f o l i o . I f t h i s i s so, then the pri c e behaviour of shares i n closed-end funds seems at variance with the be-haviour of common stocks i n general. If the r i s k i n owning 4 common shares i s related to the v a r i a b i l i t y of expected future returns and t h i s i s somehow measured by the v a r i a -b i l i t y of past returns, then those stocks which are usually perceived as " r i s k i e r " are those which increase i n value at a faster rate than the stock market i n general during r i s i n g markets and f a l l i n value at a faster rate than the stock market average during declining markets. An inves t i g a t i o n was undertaken to see i f the s t a t i s -t i c a l c h a r a c t e r i s t i c s exhibited by closed-end funds on weekly and monthly (four-week period) return series was con-s i s t e n t with e a r l i e r work on common stocks. The seminal 3 work m t h i s area was Fama who worked with 30 stocks com-p r i s i n g the Dow Jones Industrial Average, using d a i l y data. His findings, b r i e f l y , were that the natural logarithms of pric e r e l a t i v e s , ( l o g e w e r e independent, did not exhi b i t s i g n i f i c a n t s e r i a l c o r r e l a t i o n , and could be modelled well by a stable Paretian d i s t r i b u t i o n (stable d i s t r i b u t i o n ) with an alpha exponent s l i g h t l y less than two. Fama's res u l t s made use of s t a t i s t i c a l techniques based on the normal d i s t r i b u t i o n questionable, for other than point estimators, as the stable d i s t r i b u t i o n s are character-ized by i n f i n i t e variance. Estimates of variance for returns are understated and significance l e v e l s for results obtained using these techniques are unknown as the estimates of beta 5 4 are baised upwards. Later work by others has lead to the generally accepted b e l i e f that stock returns may be well represented by a symmetric stable, i . e . , skewness i s essen-5 t i a l l y zero, although Gonedes suggests that a Student t d i s t r i b u t i o n (also symmetric about zero) may have greater descriptive v a l i d i t y , since aggregating the data (forming k-sums) res u l t s i n "more normal" d i s t r i b u t i o n s which i s inconsistent with the stable d i s t r i b u t i o n hypothesis. One would l i k e to be able to use tests based on the normal d i s t r i b u t i o n because i t allows one to have some con-fidence i n estimates made using the simple market model, R i , t = a + b Rm,t + e t which i s used to estimate the relationship that exists between returns on closed-end funds and returns on the Index. This model i s used to determine how much of the behaviour of a stock or p o r t f o l i o i s due to market factors and how much i s due to other factors. Sample Data The sample data consist of 377 weekly observations on market values and net asset values for 14 closed-end funds from 15 October 1965 to 29 December 1972, from which 376 weekly returns were computed. The observations 6 were taken from the Monday e d i t i o n of The Wall Street  Journal which reports the information as at the close of the New York Stock Exchange on Friday. Capital and income dividends were taken from Moody's Annual Dividend Record for the appropriate year and are as of the week the stock went ex-dividend. Net asset values which were not published i n The  Wall Street Journal were obtained through correspondence with the companies i n New York which disseminate t h i s information. Every e f f o r t was made to insure the accuracy of the information. Extreme observations were checked for typo-graphical errors against those printed i n Barron's and the discount r a t i o recomputed. Undoubtedly errors p e r s i s t but the data f a i r l y represent information available to an informed observer of that time period. The Index used was taken from the Center for Research i n Security Prices (CRSP) tapes and i s a weekly value c a l -culated from the d a i l y CRSP equally-weighted index corres-ponding to the same time period as the data. As an i n i t i a l rough check on the accuracy of the data, returns were computed as 7 - P . +C. + D . i , t + l f i , t Li,t+1 u i , t + l i , t = R i , t where P. . i s the p r i c e of security i at time t, C i s the x, c. c a p i t a l dividend and D i s the income dividend. These values were regressed against the market index, The residuals were then pl o t t e d and examined for extreme values. The corresponding raw data for the large o u t l i e r s was again checked and corrected where errors could be located. When apparent errors i n the raw data could not be v e r i f i e d , the values were l e f t unchanged, which resulted i n increased variance but also approximated the information on which the public would act. This i n i t i a l check on data accuracy spot-lighted prob-lems with Japan Fund Inc. and American South A f r i c a n Investment. Both funds were subject to the 15 percent Interest Equalization Tax (U.S.) enacted i n 1964 and res-cinded i n 1974. This was a tax on s e c u r i t i e s purchased by U.S. residents from foreigners. Both of these investment companies were foreign companies for the purposes of the Act and while U.S. c i t i z e n s could buy and s e l l these shares among themselves, transactions with non-residents resulted i n double l i s t i n g s on the New York Stock Exchange. 8 I t was apparent t h a t when the market v a l u e s o f these companies had been c o l l e c t e d , t h e r e had been some c o n f u s i o n as t o which market v a l u e should be used. I t would be expected t h a t the r e l a t i o n s h i p between the t a x a b l e ( P t ) and non-taxable ( P n t ) shares would be l - i 5 p _ = P n t b u t t h i s W a S not always the case. A t times the p r i c e s o f the two shares were almost equal and a t other times the r a t i o was g r e a t e r than 1.15:1. The n et a s s e t v a l u e o f these two s e c u r i t i e s d u r i n g t h i s time p e r i o d were a l s o d i r e c t l y a f f e c t e d by exchange r a t e s . The Japanese yen was on a f l o a t i n g b a s i s and the South A f r i c a n rand f o l l o w e d the B r i t i s h pound d e v a l u a t i o n v i s - a - v i s the U.S. d o l l a r . T h i s s u b j e c t e d the data on these stocks to d i r e c t exchange f l u c t u a t i o n s as the b u l k o f t h e i r a s s e t s were i n the r e s p e c t i v e c o u n t r i e s , whereas other closed-end funds i n the sample were s u b j e c t e d t o exchange r a t e changes o n l y through an i n d i r e c t p r o c e s s . Due t o the data c o l l e c t i o n e r r o r and the f l u c t u a t i n g exchange r a t e s , American South A f r i c a n and Japan Fund were dropped from the sample, l e a v i n g a reduced sample o f 12 companies which r e p r e s e n t e d approximately 46 p e r c e n t o f the p o p u l a t i o n based upon a s s e t s i z e . 9 TABLE I Companies i n Sample T o t a l Net As s e t s ( m i l l i o n s ) 31 Dec. 65 31 Dec. 72 The Adams Express Co. $ 123 .1 $ 210 .0 C a r r i e r s & General Corp. 21 .9 25 .0 The Dominick Fund Inc. 55 .9 68 .8 I n t e r n a t i o n a l Holdings Corp. 94 .2 124 .0 The Lehman Corp. 441 .0 641 .8 Madison Fund Inc. 210 .8 336 .8 N a t i o n a l A v i a t i o n Fund 74 .1 119 .2 Niagara Share Corp. 96 .3 131 .7 Petroleum Corp. o f America 49 .2 96 .0 T r i - C o n t i n e n t a l Corp. 563 .6 849 .3 The U n i t e d Corp. 162 .4 160 .8 US & F o r e i g n S e c u r i t i e s Corp • 144 .1 172 .4 $ 2,037 .0 $2,935 .8 T o t a l A s s e t s - a l l funds 4,024 .4* 6,416 .0 Percentage t h i s sample/ A l l funds 50 .6 45 .8 * 46 companies ** 56 companies 10 Study Design This work had two objectives. F i r s t was to look at the return series computed for closed-end funds and see i f these series were sim i l a r to returns on common stocks. Common stock returns are usually treated as i f they came from normal d i s t r i b u t i o n , although there i s evidence that other d i s t r i b u t i o n s may have better descriptive v a l i d i t y . The data was investigated, primarily through a series of non-parametric t e s t s . F i r s t the data was checked for s e r i a l independence by the use of the single sample runs t e s t because the one assumption that generally must be made for the application of other non-parametric tests i s that of s e r i a l independence. Next the data was examined by the Kolmogorov-Smirnov Two Sample Goodness-of-Fit t e s t to see i f the series exhibited roughly the same d i s t r i b u t i o n i n two d i f f e r e n t time periods, since i n most tests there i s the i m p l i c i t assumption that the sample observations are contemporaneous and here they are time dependent. The conventional c o r r e l a t i o n c o e f f i c i e n t was calcu-lated as i t provides some idea of the strength and d i r e c t i o n of dependency when lack of independence i s present. 11 Sample moments were estimated to provide a description of the data's d i s t r i b u t i o n . They suggested that the samples were non-normal. A series of goodness-of-fit tests, which compared the sample d i s t r i b u t i o n against a normal d i s t r i b u t i o n , were applied and indicated where the discrepancies between the two were greatest. Overall these tests gave some idea of how "normal" the data appeared to be. The second objective was to apply the simple market model to the data to determine what proportion of a closed-end fund's returns could be explained by market factors and what was due to other factors. This was done by the use of ordinary l e a s t squares estimation. Extensive t e s t i n g of the residuals was not necessary as the previous tests had indicated v i o l a t i o n of c e r t a i n assumptions. Since the non-parametric tests had suggested a lack of independence i n the observations, c o r r e l a t i o n was checked between return on the Index and pr i c e r e l a t i v e s for lagged periods, with the res u l t s suggesting the application of a lagged simple market model as appropriate. 12 Single Sample Runs Test The f i r s t t e s t performed was a single sample runs t e s t to determine i f the data were s e r i a l l y independent. This was done f i r s t because common stock returns are believed to be random independent observations. Since the observations are time dependent, i t was es p e c i a l l y desirable to know i f adjacent observations are independent, as the assumption of independent random observations i s necessary for many of the other tests to be performed. The single sample runs t e s t checks changes i n a dicho-tomous variable to determine the number of runs observed, where runs are defined as a s t r i n g of observations with the same value the varia b l e . I f the observations are independent, there should be neither too few nor too many runs. When the sample size i s greater than 20, the t e s t s t a t i s t i c , ^2n 1 n 2(2n 1 n 2 - i i 1 - n 2 ) ? % (n 1+n 2) 2(n 1+n 2-l) where r i s the number of runs, n-^  i s the number of zeroes observed and n 2 i s the number of ones. Z i s approximately Q normally d i s t r i b u t e d . A two-tailed t e s t of the proba-b i l i t y i s performed. Z = r- 2n n n 1"2 +1 n 1 + n 2 13 When the o r i g i n a l variables are ratio-scaled, assign-ment to the dummy variable i s usually based upon values greater than the median or less than the median. A d i f f i -c u l t y arises when values equal the median. One treatment of t i e s i s to assign the observation the same value as the preceeding observation. This treatment of t i e s w i l l under-state the number of runs and thus make a more conservative t e s t of the n u l l hypothesis that the observations are 9 independent. The t e s t was performed on the natural logarithm of p r i c e r e l a t i v e s for market value of the common equity and net asset value, where: log (R. . ) = log ,' P i , t + 1 + C i , t + 1 + D i , t + 1 e i , t ' 3 e ( p ) t Further reference to returns w i l l always be to the natural logarithm of the p r i c e r e l a t i v e . The weekly and monthly ( i . e . 4-week) returns were sorted according to size and the median calculated as the average value of the two middle observations. The unsorted data i n t h e i r o r i g i n a l sequence were converted to a zero or one, depending on whether they were less than the median or greater than the median. Ties were broken i n the most con-servative way. I f an observation was equal to the median, 14 i t was assigned the value of the previous observation. I f the i n i t i a l value of the series was equal to the median, the following values were searched and the i n i t i a l value assigned the dummy value taken by the f i r s t following ob-servation not equal to the median. The p r o b a b i l i t i e s i n Tables II and III (which follow) are the p r o b a b i l i t y of observing a Z-score as large or larger i n absolute value than the value observed i f the data were independent. The re s u l t s were surprising. A negative Z-score indicates fewer runs than might be expected under inde-pendence. I n i t i a l l y the method of t i e breaking was suspect but for the net asset values only Dominick had any observa-tions equal to the median which would r e s u l t i n fewer runs. With the market returns a l l three companies with s i g n i f i c a n t scores had observations equal to the median which may have resulted i n s i g n i f i c a n t scores. However, the Index i s i n d i c a t i v e of the f a c t that the market i t s e l f displayed far fewer runs than expected. (See Appendix 3 for a breakdown of the d e t a i l s . ) 15 TABLE II Runs t e s t - Weekly Data Market Returns Net A s s e t Value Z Prob. Z Prob. Adams -1. .74 .0820 -2. .48 .0132* C a r r i e r s -1, .75 .0802 -3. .31 .0010* Dominick -1, .24 .2150 -4, .04 .0001* I n t e r n a t i o n a l -2. .96 .0030* -2. .79 .0052* Lehman 1. .34 .1802 -3. .61 .0032* Madison -2. .70 .0070* -4, .23 .0000* N a t i o n a l -1. .45 .1470 -3, .92 .0001* Niagara 1. .12 .2628 -3, .30 .0010* Petroleum -1, .37 .1670 -2. .27 .0232* T r i - C o n t i n e n t a l -1, .71 .0872 -3, .30 .0010* U n i t e d -2, .47 .0136* -1. .65 .0990* US & F o r e i g n -0, .10 .9204 -2. .27 .0232* Index -4, .13 .0001* * S i g n i f i c a n t a t the 5% l e v e l TABLE I I I Runs T e s t - Monthly Data Market Returns Net A s s e t Value Prob. Prob, Adams 0. .830 .4066 -0. .207 .8336 C a r r i e r s -1, .182 .2380 1. .037 .2984 Dominick -0. .415 .6744 -1. .437 .1498 I n t e r n a t i o n a l -1. .659 .0970 -1. .244 .2150 Lehman -1. .659 .0970 -0. .830 .4066 Madison -1, .182 .2380 -1, .452 .1470 N a t i o n a l -1. .341 .2150 -1, .659 .0990 Niagara 1. .249 .2112 -1. .452 .1470 Petroleum -0. .618 .5352 -0. .622 .5352 T r i - C o n t i n e n t a l -0. .622 .5353 1. .037 .2984 U n i t e d -4, .056 .0001* -0. .207 .8336 US & F o r e i g n 0. .112 .9124 0. .207 .8336 Index -1, .244 .2150 * S i g n i f i c a n t a t 5% l e v e l . The monthly data provide a s t a r t l i n g contrast to the weekly. The tendency to have fewer than expected runs p e r s i s t s but only i n one case, United Fund, would one r e j e c t the hypothesis that the observations are independent. As a cross check as to whether acceptance or r e j e c t i o n of independence was a function of the method by which t i e s were broken, the t h e o r e t i c a l number of runs which would cause one to r e j e c t the hypotheses that the observations were independent at the f i v e percent l e v e l was calculated. This assumed an equal number of observations each side of the median. No changes i n acceptance would occur for either the weekly or monthly sample. In fact, the method of t i e breaking seemed to have resulted i n a f a i r l y equitable s p l i t of the data. The apparent lack of independence i n the weekly data i s s urprising. Fama"*"0 demonstrated a small tendency for stock changes to p e r s i s t on a d a i l y basis but one week and three week changes were e s s e n t i a l l y independent. (The methodology he used was s l i g h t l y d i f f e r e n t from t h i s . ) The lack of independence i n net asset values may be caused by a number of factors. The net asset value published i s an unaudited figure and the p o r t f o l i o managers might not re-calculate the true value each week. This i s u n l i k e l y since 17 valuations are available d a i l y by telephoning or writing the head o f f i c e of the firm. Furthermore, publication was undertaken at the request of the funds as they f e l t the information would reduce discounts. A more l i k e l y cause may be that the p o r t f o l i o s of the company contain l e t t e r stock or shares of infrequently traded companies. One reason for the formation of closed-end funds was to obtain c o n t r o l l i n g i n t e r e s t . If the p o r t f o l i o stocks trade infrequently, the prices used may be old r e l a -t i v e to current market values. I f the p o r t f o l i o contains l e t t e r stock, i t i s impossible to assign an accurate market value, since by d e f i n i t i o n i t does not p u b l i c l y trade, and the values used again may not be current. This could not be investigated without obtaining further data. The best source would probably be the quarterly and annual reports f i l e d with the S.E.C. Some evidence for t h i s may e x i s t i n M a l k i e l ' s 1 1 r e s u l t s which showed a s i g n i f i c a n t r elationship between the discount on closed-end funds and the r a t i o of l e t t e r stock to the t o t a l p o r t f o l i o . Another cause of lack of independence i n the data would be i f time trends had not been eliminated. The time period under analysis had b u l l , bear and f l a t markets. The logarithmic form of returns should have eliminated exponen-t i a l growth factors over time. 18 Both the weekly and monthly series had fewer runs than expected. For the weekly series, there are so few runs one would r e j e c t the hypothesis of independence, but not for the monthly. This suggests that some exogenous variable affects the return series, changing slowly i n terms of weeks, but quickly i n terms of months. Since i t also appears to a f f e c t the Index, one would hesitate to say i t was the le t t e r - s t o c k factor. The results also indicate that while information i s l o s t i n aggregation, through comparison of the two l e v e l s one may become aware of the presence of factors that might otherwise be obscured. Kolmogorov-Smirnov Two-Sample Goodness-of-Fit Test This non-parametric t e s t i s generally used to determine i f two independent samples have been drawn from the same population or from populations with d i f f e r e n t d i s t r i b u t i o n s . Here i t was used as a rough and ready t e s t of whether the d i s t r i b u t i o n of the returns remained the same or changed over time. Since the observations are time dependent, one e s s e n t i a l l y samples from a d i s t r i b u t i o n at each point i n time. There i s no reason for the d i s t r i b u t i o n s from which each point has been drawn to remain constant across time and 19 t h e r e f o r e the r e s u l t i n g sample d i s t r i b u t i o n may be the r e s u l t o f data from many d i f f e r e n t d i s t r i b u t i o n s . T h i s t e s t g i v e s one some rough i d e a i f the d i s t r i b u t i o n o f the sample appears t o be changing over time. The Kolmogorov-Smirnov t e s t compares the cumulative d i s t r i b u t i o n f u n c t i o n o f two samples. The f i r s t step o f one sample i s compared w i t h the f i r s t s tep o f the oth e r and the v e r t i c a l d i f f e r e n c e s i n the h e i g h t h o f the two f u n c t i o n s c a l c u l a t e d . T h i s i s the d i f f e r e n c e i n cumulative p r o b a b i l i t y f o r each step o f the f u n c t i o n . For a t w o - t a i l e d t e s t , which i s a p p r o p r i a t e when one i s concerned w i t h the d i f f e r e n c e i n the f u n c t i o n s r e g a r d l e s s o f d i r e c t i o n , the l a r g e s t d i f f e r -ence i n the h e i g h t h o f the two f u n c t i o n s D, i s found. For l a r g e samples, one computes the Kolmogorov-Smirnov z-sc o r e : Z = D / ( n 1 n 2 > where n-^  and n 2 are the number o f o b s e r v a t i o n s i n each sample. Acceptance or r e j e c t i o n o f the n u l l h y p o t h e s i s t h a t the data come from the same d i s t r i b u t i o n a t a p r e -determined l e v e l o f s i g n i f i c a n c e i s then made by comparing the Kolmogorov-Smirnov Z-score w i t h the va l u e s i n the 12 t a b l e s c a l c u l a t e d by Smirnov. 20 For small samples (40 or less) with an equal number of observations i n each group, the largest absolute d i f -ference i s found and m u l t i p l i e d by the number of observa-tions i n either group. The r e s u l t i n g "D" score i s com-pared with values associated with a two-tailed t e s t at a s p e c i f i e d l e v e l of s i g n i f i c a n c e . The small sample values 13 are usually taken from Massey tables. In performing t h i s test, one assumes that the obser-vations i n each sample are independent. The r e s u l t s of the runs t e s t indicate t h i s assumption may be v i o l a t e d . The lack of independence i n the weekly data as demon-strated by too few runs, indicated that values below the median were followed by values below the median and vice versa for values above the median. This indicates that the data would c l u s t e r together. Clustering may cause the step function to increase i n large jumps, rather than step upwards smoothly. This could cause the difference between the two functions to be larger than i t would have been i f the observations were independent. A larger difference w i l l cause the corresponding Z or D score to be greater and the true p r o b a b i l i t y would be larger than the values reported. The size of the bias i s unknown. 21 Weekly and monthly returns series were each s p l i t into three segments. The central segment was dropped to eliminate continuity and the sample d i s t r i b u t i o n of the returns i n th f i r s t and t h i r d segments were compared. For monthly data, 14 observations were dropped and the f i r s t and l a s t 40 compared. For weekly data, 76 observa-tions were dropped and the f i r s t and l a s t 150 compared. The data i n each h a l f of the sample were sorted i n ascending order and two cumulative step functions computed. The t e s t was then run. 22 Table IV Kolmogorov-Smirnov Two Sample T e s t Weekly Data Market Returns Net A s s e t Value Max D Z Prob. Max D Z Prob. Adams .1600 -1.386 .043* -.0667 -.577 .893 C a r r i e r s .1333 1.155 .139 .1067 .924 .361 Dominick •.1667 -1.443 .031* -.1200 -1.039 .230 I n t e r n a t i o n a l •.0867 - .751 .626 .0467 .404 .997 Lehman .0933 .808 .531 .0800 .693 .723 Madison -.1867 -1.617 .011* -.1200 -1.039 .230 N a t i o n a l -.1533 -1.328 .059 -.1267 -1.097 .180 Niagara -.1067 - .924 .361 -.0533 - .462 .983 Petroleum -.1067 - .924 .361 .0867 .751 .626 T r i -C o n t i n e n t a l .1000 .866 .441 .0800 .693 .723 U n i t e d -.1933 -1.674 .007* .0800 .693 .723 US & F o r e i g n .0800 .693 .723 .1467 1.270 .079 Index .0867 • .750 .626 * S i g n i g i c a n t a t 5% l e v e l . T able V Kolmogorov-Smirnov Two Sample T e s t Monthly Data Market Returns Net A s s e t Value Max|D| D-score Max |D1 D -score Adams .175 7 .250 10 C a r r i e r s .125 5 .225 9 Dominick .175 7 .100 9 I n t e r n a t i o n a l .200 8 .125 5 Lehman .100 4 .150 6 Madison .375 15* .225 9 N a t i o n a l .275 11 .200 8 Niagara .175 7 .200 8 Petroleum .150 5 .175 7 T r i - C o n t i n e n t a l .200 8 .200 8 U n i t e d .300 12 .100 4 US & F o r e i g n .175 7 .225 9 Index .275 11 * S i g n i f i c a n t a t 5% l e v e l , D-score >^ 13. For the net a s s e t r e t u r n s e r i e s on both a weekly and monthly b a s i s , i t appears s a f e t o assume t h a t the d i s t r i -b u t i o n s are the same. With the weekly market r e t u r n s e r i e s , f o u r firms have d i s t r i b u t i o n s i n one sample which appear q u i t e d i f f e r e n t from the second sample. One o f these f i r m s , Madison, a l s o e x h i b i t s t h i s c h a r a c t e r i s t i c on a monthly b a s i s . R e j e c t i o n of o n l y Madison on a weekly and monthly b a s i s f o r the market r e t u r n s i s not s u r p r i s i n g . T h i s 14 company has g i v e n other r e s e a r c h e r s d i f f i c u l t y , as i t s behaviour i s o f t e n d i f f e r e n t from other funds, such as s e l l i n g a t a premium when other funds are a t a d i s c o u n t . The weekly and monthly r e s u l t s are e s s e n t i a l l y i n concurrence. That three f i r m s appear to l a c k s t a t i o n a r i t y i n t h e i r market r e t u r n s e r i e s weekly but not monthly i n d i c a t e s t h a t extreme v a l u e s may be p r e s e n t i n the s e r i e s which c o u l d e f f e c t the r e s u l t s o f the t e s t on weekly ob-s e r v a t i o n s but these v a l u e s are masked i n the aggregation p r o c e s s . Extreme v a l u e s c o u l d r e s u l t from an e x t e r n a l shock to the market such as o c c u r r e d i n the week o f 10 May 1970. P l o t s o f the data show l a r g e n e g a t i v e r e t u r n s f o r most 24 companies i n the sample. T h i s was the week Nixon ordered bombing i n Cambodia, Kent State students r i o t e d , and American c o n f i d e n c e plummeted. I f the closed-end fund was s e n s i t i v e and demonstrated a much lower r e t u r n i n t h i s time p e r i o d , and th e r e was not a corresponding nega-t i v e r e t u r n i n the other h a l f o f the sample, one c o u l d r e j e c t the sameness of the d i s t r i b u t i o n s under t h i s t e s t . There does not appear t o be a d i r e c t r e l a t i o n s h i p between l a c k o f independence and a r e j e c t i o n o f sameness. The net a s s e t v a l u e samples appear q u i t e s i m i l a r and were not independent. The market r e t u r n s were r e l a t i v e l y independent, y e t some companies appear v e r y d i f f e r e n t . Madison and U n i t e d were both q u i t e dependent, and appear d i s s i m i l a r on a weekly b a s i s , but the Index which showed the l e a s t l i k e l i h o o d o f b e i n g independent appears t o have remained the same. I f i n v e s t o r s had non-constant e x p e c t a t i o n s o f r e t u r n s f o r t h e i r h o l d i n g s of shares i n closed-end funds, t h i s c o u l d cause the mean o f the s e r i e s t o change over time. A non-constant mean c o u l d r e s u l t i n a r e j e c t i o n o f the hypothesis t h a t the s p l i t s e r i e s were drawn from the same d i s t r i b u t i o n . 25 S e r i a l Correlation Correlation i s a scaled l i n e a r measure of the r e l a t i o n -ship between two variables. The runs t e s t indicated a possible lack of independence between observations. A t e s t for s e r i a l c o r r e l a t i o n may give some in d i c a t i o n of how strong and what type of dependency e x i s t s . The term auto or s e r i a l c o r r e l a t i o n i s used to describe the relationship between ordered observations on the same variab l e . The autocovariance i s the expected product of the deviation of two observations from the mean of the ser i e s . Autocorrelation i s the scaled autocovariance, where the sc a l i n g factor i s the variance of the serie s . If the autocorrelation i s po s i t i v e , i t would indicate that large returns i n one period are followed by large returns i n the same d i r e c t i o n i n the next period. I f the auto-c o r r e l a t i o n i s negative, i t i s in d i c a t i v e of large returns i n one d i r e c t i o n followed by large returns i n the opposite d i r e c t i o n i n the next time period. The c o r r e l a t i o n co-e f f i c i e n t ranges from -1 to 1, with absolute values close to one in d i c a t i n g a strong relationship and values close to zero i n d i c a t i n g l i t t l e or no relat i o n s h i p . 26 One problem with s e r i a l c o r r e l a t i o n i s that i f the population i s stable, s e r i a l c o r r e l a t i o n i s undefined since the variance i s undefined. The one exception i s the normal d i s t r i b u t i o n which has a f i n i t e variance, but the assumption of normality may be unwarranted. Auto-c o r r e l a t i o n can be computed for the sample data and the c o e f f i c i e n t s may provide some information as to the nature of the dependency observed i n the runs t e s t . I t i s not possible to assess significance without assuming normality. Under the assumption of normality, a f i v e percent l e v e l of significance corresponds to two-standard deviations for a two-tailed t e s t . While only the f i r s t four s e r i a l c o r r e l a t i o n co-e f f i c i e n t s are presented here, the f i r s t forty-eight were calculated. The relationship was such that the c o e f f i c i e n t s quickly approached values close to zero. Given the d i f f i c u l t i e s i n int e r p r e t a t i o n of auto-co r r e l a t i o n , the results presented here should be interpreted with great care. 27 Table VI S e r i a l C o r r e l a t i o n - Weekly Data Market Returns / t , t - l „/t,t-2 / t , t - 3 / t , t - 4 Adams -.07 .00 .01 -.06 C a r r i e r s -.02 .04 .01 .01 Dominick -.20* -.03 .08 -.03 I n t e r n a t i o n a l -.10* -.03 .03 -.04 Lehman .14* .02 .00 -.08 Madison -.15* .01 -.03 .06 N a t i o n a l -.01 -.08 -.05 -.01 Niagara -.11* .00 -.03 -.04 Petroleum -.22* .10* -.09 .02 T r i - C o n t i n e n t a l -. 14* .02 -.07 .01 U n i t e d -.19* .00 -.01 .05 US & F o r e i g n -.17* .05 .02 .03 Index .20* .02 -.07 .00 Net A s s e t Values Adams -.05 .02 .01 .01 C a r r i e r s .17* -.01 -.08 -.02 Dominick .11* -.05 -.05 -.01 I n t e r n a t i o n a l .14* .08 -.17 .01 Lehman .13* -.02 -.06 .00 Madison .05 .07 -.03 .05 N a t i o n a l .25* .03 -.09 .01 Niagara .24* -.03 -.08 .03 Petroleum .11* .09 -.08 -.05 T r i - C o n t i n e n t a l .12* -.02 -.09 .02 U n i t e d .12* .09 .02 -.01 US & F o r e i g n .07 -.02 .06 -.01 *Under the assumption o f n o r m a l i t y , s i g n i f i c a n t a t 5% l e v e l . 28 Table VII S e r i a l C o r r e l a t i o n - Monthly Data Market Returns y ^ t - i jPt,t-2 V°t,t-3 Adams -.22 .10 -.06 -.05 C a r r i e r s -.01 .12 .08 -.05 Dominick -.25* .08 -.04 .12 I n t e r n a t i o n a l -.13 .07 .21* -.09 Lehman -.07 -.41* .29* .04 Madison .04 .00 -.09 .03 N a t i o n a l -.03 -.09 .08 .12 Niagara -.14 .10 .11 .07 Petroleum -.05 -.08 -.09 .13 T r i - C o n t i n e n t a l -.16 -.18 .16 -.01 U n i t e d .01 .10 .00 .05 US & F o r e i g n -.08 -.02 .00 -.11 Index -.02 .00 .08 .07 Net A s s e t Values Adams -.03 -.16 .15 .07 C a r r i e r s -.06 -.10 .08 .11 Dominick -.02 .02 .13 .05 I n t e r n a t i o n a l -.04 -.03 .10 .00 Lehman -.06 .00 .07 .05 Madison .15 .02 .08 .06 N a t i o n a l .03 .02 .15 .03 Niagara .03 .10 .06 .00 Petroleum -.01 .13 .04 .24* T r i - C o n t i n e n t a l -.06 -.04 .10 .03 U n i t e d .15 -.09 -.06 -.04 US & F o r e i g n -.06 .00 .06 .09 *Under assumption o f n o r m a l i t y , s i g n i f i c a n t a t 5% l e v e l . As was expected, weekly n et a s s e t v a l u e s showed p o s i t i v e s e r i a l c o r r e l a t i o n , w h i l e f o r the monthly net a s s e t v a l u e s the s i g n s were mixed. T h i s r e s u l t would be c o n s i s t e n t w i t h a tendency f o r r e t u r n s t o p e r s i s t f o r a 29 s h o r t p e r i o d o f time. The weekly net a s s e t v a l u e s f o r t,t-2 are not s i g n i f i c a n t l y d i f f e r e n t from zero. A more s u r p r i s i n g r e s u l t was the presence o f n e g a t i v e s e r i a l c o r r e l a t i o n i n weekly market r e t u r n s . T h i s r e s u l t appears i n c o n s i s t e n t w i t h the r e s u l t s o f the runs t e s t . Instead o f showing too few runs, t h e r e would have been too many. A s s e s s i n g the s i g n i f i c a n c e o f these r e s u l t s i s d i f f i c u l t . I f the r e t u r n s were g e n e r a l l y s m a l l and p o s i t i v e a few l a r g e n e g a t i v e r e t u r n s w i t h no l a r g e o f f s e t t i n g p o s i t i v e r e t u r n s c o u l d cause these r e s u l t s . On a weekly b a s i s , the Index c o n s t r u c t e d shows d e f i n i t e s i g n i f i c a n t p o s i t i v e s e r i a l c o r r e l a t i o n , i n d i c a t i n g a tendency f o r market r e t u r n s t o p e r s i s t i n a c e r t a i n d i r e c t i o n f o r a v e r y s h o r t p e r i o d o f time. Ingersoll"'"^ found p o s i t i v e s e r i a l c o r r e l a t i o n i n h i s weekly v a l u e r e t u r n s e r i e s which he f e l t was a r e s u l t o f the manner i n which the s e r i e s had been c o n s t r u c t e d . H i s market s e r i e s was f r e e o f s i g n i f i c a n t a u t o c o r r e l a t i o n . Negative s e r i a l c o r r e l a t i o n i n s h o r t run market r e t u r n s , when the Index i t s e l f i s showing p o s i t i v e auto-c o r r e l a t i o n , i n d i c a t e s t h a t some f a c t o r , other than the " u s u a l " market f a c t o r s , may be working on the market p r i c e s 30 of closed-end investment companies. I t i s not known what these other factors might be but one p o s s i b i l i t y might be infrequent trading of the closed-end funds common stock. This could cause market values to be adjusted less frequently and i n larger r e l a t i v e jumps than the majority of stocks i n the market place and t h i s trading pattern might cause negative s e r i a l c o r r e l a t i o n . As a very rough, informal check on frequency of trading, the actual trading volumes for three funds were co l l e c t e d over three d i f f e r e n t four-week periods from the ISL Daily Stock Price Index. The d a i l y average volume over each time period for each fund was compared with the d a i l y average per company of the Standard and Poor 500 Index. The shares of closed-end funds traded far less frequently. (See Appendix 4.) An even more desirable check would be i f the frequencies of trading could be compared. One further check was made on the Index. The sample was s p l i t into two 180 week observation periods, the middle 16 observations were dropped, and the corre-l a t i o n c o e f f i c i e n t s computed for the two time periods. The re s u l t s were consistent with those found for the 31 continuous weekly sample which suggests the p o s i t i v e s e r i a l c o r r e l a t i o n was not a f u n c t i o n of a p a r t i c u l a r time p e r i o d . Table V I I I S e r i a l C o r r e l a t i o n - Index - Weekly Data / t , t - l ft, t-2 ft, t-3 / t , t - 4 Index - 1 s t h a l f .23* .07 -.19* .04 Index - 2nd h a l f .17* -.03 .03 -.04 *Under assumption o f n o r m a l i t y , s i g n i f i c a n t a t 5% l e v e l . The presence o f n e g a t i v e s e r i a l c o r r e l a t i o n would seem t o c o n t r a d i c t Fama's"^ e a r l i e r work but most of h i s work was done on f r e q u e n t l y t r a d e d l a r g e i n d u s t r i a l s and he i n d i c a t e d t h a t h i s r e s u l t s may be a p p l i c a b l e to o n l y 17 those s t o c k s . In more r e c e n t work, Scholes and W i l l i a m s observe t h a t the use o f d a i l y data may make the problem o f measurement e r r o r more c r i t i c a l and f o r the Index cause the autocovariance o f l a g one to be o v e r s t a t e d . I t may be t h a t t h i s problem p e r s i s t s w i t h closed-end funds on a weekly b a s i s , s i n c e the s i z e o f the e r r o r i s a f u n c t i o n o f t r a d i n g frequency. 18 Scholes and W i l l i a m s a l s o found f o r i n d i v i d u a l s t o c k r e t u r n s , measured autocovariances of lag-one appear n e g a t i v e ; w i t h other remaining c o v a r i a n c e s o f v a r i o u s l a g s 32 vanish. The results found e a r l i e r are consistent with t h i s . Due to errors i n measurement of the variable be-cause of frequency of trading, Scholes and Williams demonstrated that the d i r e c t i o n of the bias introduced i n the use of l e a s t square estimators when applying the Sharpe-Lintner model i s a function of the frequency of trading. I f a security trades less or more frequently than the "average" stock i n the index, p o s i t i v e betas are asymptotically biased downwards. If on average a security trades as frequently, p o s i t i v e betas are biased upwards. The ordinary l e a s t square estimators of alpha w i l l be biased i n the opposite d i r e c t i o n . This r e s u l t arises from errors i n the measure of the variance. When trading i s less frequent than "average", errors i n measure-ment of the variance of the stock returns w i l l dominate errors i n the measurement of the Index. However, i f trading i s as frequent, errors i n measurement of the Index 19 variance w i l l dominate. Sample Moments The apparent inconsistency observed between the re s u l t s of the runs t e s t and s e r i a l c o r r e l a t i o n for market returns made i t desirable to investigate the sample moments of the return series, as a check on the accuracy of the 33 r e s u l t s . Sample moments are descriptive summary s t a t i s t i c s which provide one with some idea of the appearance of the sample d i s t r i b u t i o n s . The sample moments were calculated according to the formulae l i s t e d i n Appendix 5. The results are l i s t e d i n d e t a i l there. The estimators used are unbiased estimators of the population s t a t i s t i c ( i f i t e x i s t s ) . The estimators of the standard deviations assume popula-20 t i o n normality. Under the assumptions of independent, normally d i s t r i b u t e d observations, the expected value for skewness and kurtosis i s zero. The standard devia-tions of the skewness and kurtosis s t a t i s t i c s can be used as a rough guide as to whether the sample under observation demnostrated a s i g n i f i c a n t amount of t h i s property. The presence of p o s i t i v e skewness would cause one to expect to observe more frequently a negative return but large returns would most often be p o s i t i v e . Persistent leptokurtosis i n comparison to a normal d i s t r i b u t i o n , which i s consistent with e a r l i e r work on stock returns, would seem to indicate more returns i n the t a i l s . If, add i t i o n a l l y , more returns clustered 34 c l o s e l y about the mean t h i s c o u l d be i n d i c a t i v e o f a s t a b l e p o p u l a t i o n d i s t r i b u t i o n . For the market r e t u r n s e r i e s , a l l companies on both a weekly and monthly b a s i s had s l i g h t l y p o s i t i v e means but the means were not s i g n i f i c a n t l y d i f f e r e n t from zero . Market r e t u r n medians were u s u a l l y zero but Lehman Corp. had p o s i t i v e medians on both a weekly and monthly b a s i s . The weekly and monthly r e s u l t s were c o n s i s t e n t , w i t h the monthly v a l u e s somewhat l a r g e r (but s t i l l not s i g n i f i c a n t ) as might be expected i f the u n d e r l y i n g r e t u r n s e r i e s was l a r g e r i n s i z e due to aggregation. On a weekly b a s i s , n et a s s e t v a l u e medians were p o s i t i v e and j u s t as l i k e l y to be g r e a t e r than the mean as l e s s . On a monthly b a s i s , the medians o f net a s s e t v a l u e were d e f i n i t e l y s m a l l e r than the means. The Index showed a p o s i t i v e median g r e a t e r than the mean on both a weekly and monthly b a s i s . I f these sample d i s t r i b u t i o n s were symmetric, one would expect the mean and the median to be equal. The means were not s i g n i f i c a n t l y d i f f e r e n t from zero . In most cases, the median was zero or c l o s e r t o zero than the mean. 35 Weekly market returns showed s i g n i f i c a n t skewness i n ten out of 13 cases, including the Index. Eight cases were p o s i t i v e . The Index and Niagara were the two s i g n i -f i c a n t negative cases. The monthly market returns were skewed i n eight cases, seven were negative and National Aviation was p o s i t i v e . Consistency i n skewness was demon-strated by Carriers (0), National Aviation (+), Niagara (-), and the Index (-). A l l others changed signs. Possi-bly the apparent inconsistency demonstrated by the weekly and monthly return series i s due to the presence of a few large o u t l i e r s . The e f f e c t of these o u t l i e r s could be moderated by the aggregation process or aggravated. A very large negative return i n the weekly series could cause the d i s t r i b u t i o n to appear negatively skewed. When the data are aggregated for the monthly return, the impact of that single event may be diminished and the series looks p o s i t i v e l y skewed. Weekly market returns showed s i g n i f i c a n t leptokur-t o s i s i n a l l cases ( i . e . , a tendency to have f a t t e r t a i l s than a normal d i s t r i b u t i o n ) . With monthly values, a l l but Carriers and United again demonstrated t h i s property. This i s consistent with Gonedes' findings. 36 Net asset values also showed a tendency towards skewness but the results were mixed. Weekly, eight out of twelve cases demonstrate p o s i t i v e skewness but monthly ten out of twelve demonstrate negative skewness with no company remaining consistent. Weekly net values were leptokurtic but on a monthly baiss only seven cases demonstrated s i g n i f i c a n t lepto-kurtosis. The variance of the market returns was i n a l l cases higher than the variance of the net asset values. In size, the variance of net asset values was very close to the variance of the Index. The higher variance on the market returns may be i n d i c a t i v e of some element external to the market increasing the r i s k on the stock. That the Index shows negative skewness and the market returns of closed-end funds p o s i t i v e skewness on a weekly but not a monthly basis may indicate i n the very short run the funds are subject to other than market factors, but over the longer horizon they tent to follow the market. This tendency to go i n opposite directions i s s i m i l a r to the behaviour of the signs of the auto-c o r r e l a t i o n c o e f f i c i e n t s . However, since there i s evidence 37 that the assumptions of independence and normality may well be violated, one cannot i n fact conclude the d i s -t r i b u t i o n s are skewed. In general, these data for closed-end fund returns seem to have properties s i m i l a r to other stock returns. Goodness of F i t Tests The sample moments indicate that the data series do not appear to come from a normal d i s t r i b u t i o n . Application of the market model to determine the relationship between closed-end fund returns and market returns requires caution i f the data i s non-normal since the assumptions of the model are vi o l a t e d . I t i s desirable to know how badly the data series v i o l a t e s normality as t h i s would give some idea i f the application of the model would provide any information at a l l . Goodness of f i t tests are one way of approaching t h i s . They can provide confirmation of e a r l i e r findings and some information as to how d i f f e r e n t the data series are from a normal d i s t r i b u t i o n . Goodness of f i t tests are generally non-parametric tests used to ascertain how well sample data approximates the t h e o r e t i c a l d i s t r i b u t i o n of i n t e r e s t . For the purpose of the four tests following, the sample data w i l l be 38 compared against the normal d i s t r i b u t i o n . The assumption one must usually make to apply these tests i s that the data are independent. The r e s u l t s of the runs tests and the presence of s e r i a l c o r r e l a t i o n indicate that, i n fact, t h i s assumption would be v i o l a t e d . An alternative factor that may be a f f e c t i n g the observations may be the presence of a non-constant mean for the data ser i e s . This i s most l i k e l y to be a problem with the market return series, since the Kolmogorov-Smirnov Two Sample te s t found one t h i r d of the sample to have d i f f e r e n t d i s -t r i b u t i o n s i n d i f f e r e n t time periods. This could arise as investors adjusted t h e i r expectation of return on the shares of the closed-end funds. I n t u i t i v e l y the presence of p o s i t i v e s e r i a l corre-l a t i o n could cause a clumping of data. This could cause a non-normal sample to appear normal and a normal sample to appear non-normal. I t i s more d i f f i c u l t to discern the e f f e c t of negative s e r i a l c o r r e l a t i o n . One might expect i t to increase the dispersion but t h i s could have the same e f f e c t as p o s i t i v e c o r r e l a t i o n i n that i t may cause things appear to be what they are not. The c o r r e l a t i o n present i n the data, while s i g n i f i c a n t , i s not large and probably becomes c r i t i c a l only for those 39 cases where acceptance or r e j e c t i o n of the hypotheses i s marginal. The Chi-Square Test The chi-square t e s t i s probably the best known goodness of f i t t e s t and since i t uses grouped data i s p a r t i c u l a r l y well suited for nominally measured variables. A disadvantage encounted when data are measured on a r e l a t i v e l y ' continuous scale i s the loss of information through the grouping procedure. To perform t h i s t e s t the data must be grouped and a t h e o r e t i c a l d i s t r i b u t i o n s p e c i f i e d . I t i s t h e o r e t i c a l l y desirable that no more than twenty percent of the groups should have an expected frequency of f i v e or less and that the expected frequency of any group should not be less than one. What biases are incurred by misspecification of grouping i s unknown, but the results are sensitive to group frequencies. If the parameters of the t h e o r e t i c a l d i s t r i b u t i o n can be spec i f i e d , a p r i o r i , from p r i o r work or theory, then comparison i s made between the expected t h e o r e t i c a l f r e -quency of each group and the actual number of observations i n each group. 40 A Q - s t a t i s t i c i s calculated where: 4r 0 . - E . _ i = l ( E T > i and i s the observed frequency of group i , and E^ i s the 21 expected frequency. The number of groups i s n. Q approx-imately follows a chi-square d i s t r i b u t i o n with n-1 degrees of freedom. Frequently a researcher does not know i n advance what the parameters of the th e o r e t i c a l d i s t r i b u t i o n should be. If t h i s i s the case, then the parameters may be estimated from the sample data, with a r e s u l t i n g loss of degrees of freedom, losing one degree for each parameter estimated. The r e s u l t i n g Q - s t a t i s t i c has n-k-1 degrees of freedom where k i s the number of estimated parameters. If the th e o r e t i c a l d i s t r i b u t i o n of i n t e r e s t i s the normal d i s t r i b u t i o n and " . . . i f the o r i g i n a l data i s un-grouped and maximum l i k e l i h o o d estimators are based on the l i k e l i h o o d functions of a l l observations, the l i m i t i n g d i s t r i b u t i o n of the s t a t i s t i c computed, Q, i s not the 22 chi-square" . That i s , i f the researcher s p e c i f i e s the parameters of the th e o r e t i c a l normal d i s t r i b u t i o n as the values obtained by the use of maximum l i k e l i h o o d estimators 41 on the ungrouped sample data, the r e s u l t i n g Q - s t a t i s t i c does not follow a chi-square d i s t r i b u t i o n of n-k-1 degrees of freedom. The sample mean and variance are maximum l i k e l i h o o d 2' estimators for a normal d i s t r i b u t i o n . Chernoff and Lehman have found that the e f f e c t of the bias introduced i n the Q - s t a t i s t i c through the use of the maximum l i k e l i h o o d estimators i s to make the chi-square t e s t anti-conservative for the normal d i s t r i b u t i o n . One w i l l r e j e c t the n u l l hypothesis that the sample i s drawn from the th e o r e t i c a l population too frequently, i . e . , at a lower l e v e l of signi f i c a n c e than intended. The severity of underestima-t i o n of the chi-square p r o b a b i l i t y i s unknown but they prove that the true p r o b a b i l i t y of observing a sample such as the one being tested, given that the true population i s a normal d i s t r i b u t i o n , follows a d i s t r i b u t i o n that l i e s 2 2 between "X . , and ' X -, • This r e s u l t allows one n-k-1 n-1 to obtain an upper l i m i t for the error r e s u l t i n g from the bias introduced by the estimation procedure used. This r e s u l t can be i l l u s t r a t e d with the monthly market value Q - s t a t i s t i c for Dominick (see table X). The proba-b i l i t y of Q=22.268 i f a "Xg , given H Q i s true i s .01 >p> .005. However, using Chernoff and Lehman's res u l t , i f 2 Q was a "X n , the p r o b a b i l i t y i s .025 ^  p ^ . 0 1 . T h i s r e s u l t allows one to d e f i n i t e l y conclude t h a t i f the assumption o f independence i s met, a t a pre-determined f i v e p e r c e n t l e v e l o f s i g n i f i c a n c e , Dominick's monthly market v a l u e does not f o l l o w a normal d i s t r i b u t i o n . T h i s r e s u l t i s p r e s e n t e d g r a p h i c a l l y i n i l l u s t r a t i o n 1. I l l u s t r a t i o n 1. P r o b a b i l i t y 1.00 0 22.3 Q - s t a t i s t i c (Not to scale) The sample data were ungrouped. The program used (UBC FREQ) has a f u n c t i o n a l l i m i t a t i o n o f twenty groups and the data were a r b i t r a r i l y grouped as s p e c i f i e d i n Table IX. The sample moments i n the pr e c e e d i n g s e c t i o n , which would have been o f use i n s p e c i f y i n g the groups, were p a r t o f the output o f t h i s program. 43 Table IX Group and Range S p e c i f i c a t i o n - Chi-square T e s t Number o f Groups Range Weekly Monthly 20 16 -.05 to .05 -.16 to .16 I f an o b s e r v a t i o n f a l l s o u t s i d e the range s p e c i f i e d i t i s i n c l u d e d i n the group to which i t i s c l o s e s t . I f the expected frequency i n a group i s l e s s than f i v e , the program aggregates groups u n t i l i t a r r i v e s a t a group w i t h an expected frequency o f f i v e . U n f o r t u n a t e l y the program used computes the e s t i -mates of the t h e o r e t i c a l d i s t r i b u t i o n from the ungrouped data and t h i s i n t r o d u c e s the b i a s mentioned above i n t o the r e s u l t s . 44 Table X Chi-square T e s t o f Goodness o f F i t where the D i s t r i b u t i o n o f I n t e r e s t i s the Normal D i s t r i b u t i o n Weekly Data Net A s s e t Value Market Returns prob. V? prob. Adams 50 .70 __ 16 0 .000* 88 .35 __ 17 0 .000* C a r r i e r s 19 .93 14 0 .132 54 .38 17 0 .000* Dominick 48 .25 17 0 .000* 115 .52 17 0 .000* I n t e r n a t i o n a l 20 .33 17 0 .258 104 .43 17 0 .000* Lehman 20 .98 17 0 .227 43 .76 17 0 .000* Madison 35 .81 17 0 .005* 110 .60 17 0 .000* N a t i o n a l 25 .42 16 0 .063 45 .40 17 0 .000* Niagara 21 .64 16 0 .155 67 .97 17 0 .000* Petroleum 25 .08 16 0 .112 109 .68 17 0 .000* T r i -C o n t i n e n t a l 28 .28 17 0 .042** 32 .37 17 0 .014* U n i t e d 17 .99 13 0 .158 280 .59 17 0 .000* US & F o r e i g n 24 .40 17 0 .111 58 .52 17 0 .000* Index 27 .88 16 0 .033*** Monthly Data Adams 7 .07 6 0 .315 15 .85 9 0 .070 C a r r i e r s 5 .91 5 0 .315 1 .28 6 0 .973 Dominick 6 .50 7 0 .483 22 .27 9 0 .008* I n t e r n a t i o n a l 9 .88 6 0 .130 7 .86 9 0 .549 Lehman 8 .54 8 0 .383 6 .09 10 0 .808 Madison 9 .21 7 0 .238 13 .29 9 0 .150 N a t i o n a l 2 .19 6 0 .901 12 .99 13 0 .449 Niagara 8 .68 6 0 .192 15 .58 11 0 .157 Petroleum 8 .28 6 0 .219 6 .06 9 0 .734 T r i -C o n t i n e n t a l 6 .27 6 0 .394 9 .22 9 0 .417 U n i t e d 7 .78 4 0 .100 11 .62 8 0 .169 US & F o r e i g n 7 .58 6 0 .270 9 .42 8 0 .308 Index 7 .41 6 0 .285 * R e j e c t H J o at 5% l e v e l i of s i g n i f i c a n c e . . 10>p^>.05, t h e r e f o r e i n d e t e r m i n a t e . *** % . 10>p>.05, t h e r e f o r e i n d e t e r m i n a t e . 45 Using Chernoff and Lehman's r e s u l t , a t the f i v e per-cent l e v e l one would accept the a l t e r n a t i v e h y p o t h e s i s , i . e . : t h a t the data are not from a normal d i s t r i b u t i o n , o n l y i n one case w i t h the monthly net a s s e t v a l u e s and market r e t u r n s . S t r i k i n g l y d i f f e r e n t r e s u l t s are o b t a i n e d w i t h the weekly data. A l l market r e t u r n s , except the Index are non-normal. The weekly Index f a l l s i n t o a grey zone w i t h a p r o b a b i l i t y o f coming from a normal d i s t r i b u t i o n between .10 and .03. With the weekly net a s s e t v a l u e s , the r e s u l t s are mixed. In t h r e e cases one would conclude non-normality, and one case i s in d e t e r m i n a t e . Given these r e s u l t s , i t c o u l d be s a f e l y concluded t h a t the monthly data approximated a normal d i s t r i b u t i o n , i n view o f the runs t e s t and c o r r e l a t i o n c o e f f i c i e n t s . One would a l s o conclude the weekly market r e t u r n s were not normal and net a s s e t v a l u e s p r o b a b l y were but the runs t e s t s i n d i c a t e d n et a s s e t v a l u e s weren't independent and market r e t u r n s g e n e r a l l y were. The r e s u l t s o f t h i s t e s t i n d i c a t e t h a t the net a s s e t v a l u e s e r i e s appears to come from a normal d i s t r i b u t i o n on both a weekly and monthly b a s i s . While the monthly market 46 return series appear normal, the weekly series does not. This t e s t i s sensitive to the s p e c i f i c a t i o n of the groups and miss p e c i f i c a t i o n of the groups could cause t h i s r e s u l t . Another p o s s i b i l i t y , again, i s that the aggrega-t i o n into monthly data has possibly eliminated the e f f e c t of extreme values which are more prevalent i n the market return series (as indicated by higher variances) than i n the net asset value ser i e s . Since i t i s not known how the results may have been biased by the group selection, i t i s necessary to use further d i s t r i b u t i o n a l tests before a r r i v i n g at a conclusion. Kolmogorov-Smirnov One-Sample Test This t e s t i s s i m i l a r to the Kolmogorov-Smirnov Two-Sample Test except i t determines the largest d i s c r e -pancy between a sample and a t h e o r e t i c a l cumulative p r o b a b i l i t y d i s t r i b u t i o n , whereas the two sample t e s t compared two samples. Because the Kolmogorov-Smirnov t e s t looks at each observation i t i s more powerful than the chi-square t e s t . The r e s u l t i n g D - s t a t i s t i c i s d i s t r i b u t i o n - f r e e and c r i t i c a l values have been tabled i f one could f u l l y specify a p r i o r i the parameters of the th e o r e t i c a l d i s t r i b u t i o n . It i s known that i f the 47 t h e o r e t i c a l parameters are estimated from the sample data a bias i s introduced which causes one to accept the n u l l hypothesis that the sample was drawn from a population with the t h e o r e t i c a l d i s t r i b u t i o n too frequently. Large scale Monte Carlo simulations have been con-ducted to table the appropriate asymptotic s i g n i f i c a n t point values for D when the t h e o r e t i c a l population i s normal and parameters are estimated from the sample data and there seems to be a consensus on the r e s u l t s . Recent 24 work by Stevens has estimated the adjustments necessary for smaller samples. The largest absolute deviation, | D | , i s adjusted for f i n i t e sample size as follows: D a d j = J D | (n* - 0.01 + 0.85/n5 ) and the adjusted score compared against significance 25 leve l s published by Stevens This and the two following goodness of f i t tests use data converted to standardized scores and sorted into as-cending order. The r e s u l t i n g values were compared with the cumulative standard normal values. The maximum abso-lute D was then calculated i n the same manner as for the Kolmogorov-Smirnov two sample t e s t . The r e s u l t s were somewhat at variance with those of the chi-square t e s t . (Appendix 6 compares the results of each company on a l l goodness of f i t tests performed.) 48 Table XI Kolmogorov-Smirnov One Sample T e s t Market Returns Weekly _ Monthly Max. IDI D Ad j * Max IDI D Ad]•** Adams .091 1.767 .084 .819 C a r r i e r s .073 1.425 .110 1.077 Dominick .090 1.752 .073 .713 I n t e r n a t i o n a l .091 1.774 .078 .765 Lehman .074 1.432 .068 .661 Madison .090 1.742 .086 .843 N a t i o n a l .075 1.460 .110 1.076 Niagara .067 1.308 .074 .720 Petroleum .050 1.649 .110 1.076 T r i - C o n t i n e n t a l .064 1.249 .106 1.033 U n i t e d .126 2.451 .113 1.099 US & F o r e i g n .047 .910 .087 .848 Index .084 1.635 .087 .855 Net A s s e t Values Adams .063 1 .214 .092 .903 C a r r i e r s .075 1 .458 .119 1. .163 Dominick .096 1 .862 .103 1. .002 I n t e r n a t i o n a l .081 1 .565 .152 1. .488 Lehman .067 1 .298 .107 1. .050 Madison .059 1 .140 .099 .969 N a t i o n a l .069 1 .337 .115 1, .128 Niagara .069 1 .342 .090 .881 Petroleum .071 1 .370 .116 1, .131 T r i - C o n t i n e n t a l .074 1 .443 .102 .999 U n i t e d .068 1 .320 .108 1. 059 US & F o r e i g n .073 1 .410 .111 1. .081 * Adjustment f a c t o r = 19.4246, n=376. ** Adjustment f a c t o r = 9.773, n=94. • • 27 S i g n i f i c a n c e l e v e l s : 15.0 0.775 2.5 0.955 10.0 0.819 1.0 1.035 5.0 0.895 49 T h i s t e s t f r e q u e n t l y r e j e c t e d the n u l l h y p othesis o f the sample data coming from a normal d i s t r i b u t i o n and u s u a l l y a t a h i g h e r s i g n i f i c a n c e l e v e l . The r e s u l t i s p a r t i c u l a r l y s t r i k i n g f o r the weekly net a s s e t v a l u e s . Under the ch i - s q u a r e t e s t , o n l y t h r e e cases would have been r e j e c t e d as non-normal, but w i t h the Kolmogorov-Smirnov a l l cases are r e j e c t e d a t the one p e r c e n t l e v e l or g r e a t e r . One would a l s o r e j e c t n o r m a l i t y f o r the monthly net a s s e t v a l u e s , except f o r one case, a t the f i v e p e r c e n t l e v e l or g r e a t e r . T h i s r e s u l t i s not b e l i e v e d t o be a f u n c t i o n o f l a c k o f independence. Monthly net a s s e t v a l u e s were not s i g n i f i c a n t l y a u t o c o r r e l a t e d and t h e r e f o r e i t i s not b e l i e v e d t h a t a b i a s was i n t r o d u c e d i n t o the t e s t . The r e s u l t i s more l i k e l y a comment on the l a c k o f s e n s i t i v i t y o f the ch i - s q u a r e t e s t , p r o b a b l y because o f i n f o r m a t i o n l o s s through grouping. For the market r e t u r n s e r i e s , one r e j e c t s the hypo t h e s i s o f n o r m a l i t y f o r the weekly data. The weekly s e r i e s were r e j e c t e d a t the one p e r c e n t l e v e l o f s i g n i f i -cance or g r e a t e r f o r a l l but one company (US & For e i g n ) which was r e j e c t e d a t the f i v e p e r c e n t l e v e l . The monthly market r e t u r n s e r i e s s t i l l show a g r e a t e r tendency t o appear normal. F i v e companies would be r e j e c t e d as normal, but o n l y one (Dominick) was r e j e c t e d by the ch i - s q u a r e t e s t . 50 Cramer-von Mises Goodness of F i t Test The Kolmogorov-Smirnov t e s t i s based on the largest discrepancy observed. The Cramer-von Mises t e s t looks at a l l the discrepancies between the observed data and the t h e o r e t i c a l function and tests the o v e r a l l f i t of the data to the t h e o r e t i c a l d i s t r i b u t i o n . I t may be that no single deviation i s large enough to cause one to r e j e c t the f i t between the sample and the t h e o r e t i c a l d i s t r i b u t i o n , but o v e r a l l there may be s u f f i c i e n t discrepancies that might cause one to r e j e c t the n u l l hypothesis that the sample i s drawn from a normal d i s t r i b u t i o n . Like the Kolmogorov-Smirnov one sample t e s t and the Anderson-Darling t e s t (which follows), the Cramer-von Mises t e s t has only recently become useful for empirical work when the t h e o r e t i c a l parameters of the population d i s t r i b u t i o n must be estimated from the sample data. When the d i s t r i b u t i o n parameters are estimated from the sample data, these three tests are s t i l l only well documented for tests of a normal or exponential d i s t r i b u t i o n . To calculate the Cramer-von Mises s t a t i s t i c , W , l e t w^  be the standardized observations i n ascending order. 51 F(X^) i s the cumulative p r o b a b i l i t y of a standard normal d i s t r i b u t i o n and n i s the number of observations i n the sample. w 2 = - j - J - + f -12n „d_ i = l The derivation of the computational formula i s i n 2 6 2 7 Lindgren . Stevens has estimated the small sample 0 5 bias and provides an adjustment factor of (1 + ———) when estimating both the mean and the variance of the the o r e t i c a l d i s t r i b u t i o n . Asymptotic upper t a i l percen-tage points have been found by Stevens by the use of curve f i t t i n g techniques to the cumulative of the 2 8 asymptotic d i s t r i b u t i o n , and i t i s these values rather than those i n his e a r l i e r work that are used to test for significance. 2 i - l 2n - F(x i) 52 Table XII Crame'r-Von Mises T e s t o f Goodness of F i t Market Returns Weekly Monthly W 2 2* Ad: • w W Ad:, v r Adams .482 .483 .152 .153 C a r r i e r s .666 .667 .341 .343 Dominick .581 .582 .115 .115 I n t e r n a t i o n a l .572 .573 .122 .122 Lehman .578 .579 .104 .104 Madison .218 .219 .119 .120 N a t i o n a l .480 .481 .152 .153 Niagara .419 .419 .125 .125 Petroleum .235 .235 .181 .182 T r i - C o n t i n e n t a l .507 .508 .370 .372 U n i t e d .825 .826 .304 .306 US & F o r e i g n .192 .193 .170 .170 Index .815 .816 .224 .225 Net A s s e t Values Adams .428 .429 .198 .199 C a r r i e r s .767 .768 .295 .296 Dominick .784 .785 .295 .296 I n t e r n a t i o n a l .837 .839 .513 .516 Lehman .722 .723 .250 .251 Madison .396 .397 .313 .315 N a t i o n a l .536 .537 .383 .385 Niagara .680 .681 .171 .172 Petroleum .585 .586 .323 .325 T r i - C o n t i n e n t a l .833 .834 .205 .206 U n i t e d .515 .515 .211 .214 US & F o r e i g n .782 .783 .293 .295 V Adjustment f a c t o r = W2 (1.0013). ** Adjustment f a c t o r = Wz (1.0053). S i g n i f i c a n c e l e v e l s : 2 9 10.0 .104 2.5 .148 5.0 .126 1.0 .178 53 It was expected that the Cramer-von Mises t e s t would r e j e c t the hypothesis of normality for a l l cases where i t was rejected under the Kolmogorov-Smirnov t e s t and at the same or smaller l e v e l of si g n i f i c a n c e . In general t h i s was true but two exceptions were found, the monthly market returns for Madison and National Aviation. At the f i v e percent l e v e l one would have rejected both as normal under the Kolmogorov-Smirnov, but only National Aviation under the Cramer-von Mises. With the Crame'r-von Mises s t a t i s t i c , one would r e j e c t normality for a l l weekly data, for monthly net asset values and for eight out of 13 monthly market returns. I t was expected that the Crame'r-von Mises t e s t would r e j e c t where the Kolmogorov-Smirnov t e s t did because i t was believed that i f the difference cumulative p r o b a b i l i t y at any single point was large enough to cause reject i o n , surely the o v e r a l l differences would be s i g n i f i c a n t . It was hoped i t would be more powerful, however, where no single difference was large enough to cause reject i o n , but consistent small differences were present. This appears to be what occurred. The monthly market returns are s t i l l the most normal appearing. In general the results are consistent with the chi-square and Kolmogorov-Smirnov t e s t . 54 Anderson-Darling Test of Goodness of F i t Unlike the preceeding goodness-of-fit tests, the Anderson-Darling t e s t s a c r i f i c e s s e n s i t i v i t y to deviations around the median and places much more weight upon devia-tions i n the t a i l of a d i s t r i b u t i o n . This seems to make th i s a most appropriate t e s t for stock prices which are generally believed to be l e a s t l i k e a normal d i s t r i b u t i o n i n the t a i l s . While the sample s t a t i s t i c s indicated s i g n i -f i c a n t leptokurtosis, there were s t i l l monthly market values that would have been accepted as normal at the f i v e percent l e v e l under the preceeding goodness-of-fit t e s t s . The Anderson-Darling t e s t i s a variant of the Crame'r-von Mises but instead of uniformly weighting the deviations, i t weights more heavily the deviations i n the t a i l s , and puts smaller weights to deviations around the median. The weighting factor used here i s the r e c i p r o c a l of the variance of the sample cumulative d i s t r i b u t i o n function under the n u l l hypothesis. Let F(X^) be the cumulative pro-b a b i l i t y of a standard normal d i s t r i b u t i o n . The weighting factor i s then -F ( x i ) ( l - F ( x i ) ) 55 which has the property of being large when F(X^) i s near zero and one. The computational formula used i s n ' l o g e F ( X i ) + l o g e ( l - F ( X n _ i + 1 ) ) A 2 = -n - ( 2 i - l ) i = l 30 and i s derived m Anderson and Darling . The upper t a i l asymptotic points when the th e o r e t i c a l d i s t r i b u t i o n parameters are estimated from the sample parameters are 31 tabled i n Stevens and the small sample correction factor, 4 25 32 1 + — - —2 , i s from his e a r l i e r work n 56 Table XIII Anderson-Darling Goodness of F i t Test Market Returns Weekly Monthly A 2 A 2 adj.* A 2 A 2 Adj. Adams 7 .091 7.984 2.099 2.182 Carriers 11 .020 11.135 4.307 4.478 Dominick 8 .715 8.806 2.027 2.108 International 6 .935 7.008 1.875 1.949 Lehman 9 .103 9.198 1.907 1.983 Madison 3 .570 3.608 1.924 2.001 National 9 .740 9.842 2.823 2.935 Niagara 5 .145 5.199 1.888 1.963 Petroleum 4 .344 4.389 2.641 2.746 T r i -Continental 6 .859 6.931 3.258 3.387 United 9 .650 9.751 4.399 4.573 US & Foreign 4 .414 4.460 2.290 2.381 Index 10 .303 10.410 2.868 2.982 Net Asset Values Adams 6 .071 6.124 2.725 2.833 Carriers 10 .165 10.271 3.836 3.988 Dominick 10 .090 10.195 3.015 3.135 International 10 .589 10.700 4.196 4.363 Lehman 8 .376 8.464 2.885 2.999 Madison 4 .375 4.421 3.681 3.827 National 10 .281 10.389 4.428 4.603 Niagara 9 .649 9.750 2.427 2.523 Petroleum 8 .563 8,652 4.326 4.498 T r i -Continental 9 .654 9.755 2.974 3.093 United 6 .371 6.438 3.297 3.428 US & Foreign 8 .722 8.813 3.436 3.573 * Adjustment factor = 1.0105. ** Adjustment factor = 1.0397. Significance l e v e l s 0 : 10.0 .632 2.5 .870 5.0 .751 1.0 1.029 57 Because of the presence of leptokurtosis, i t was expected that most samples would f a i l t h i s t e s t but not that every single sample would be rejected as non-normal at the one percent l e v e l or greater, as occurred. The r e s u l t s of t h i s t e s t coupled with sample moments ind i c a t i n g s i g n i f i c a n t kurtosis would seem to indicate that closed-end fund market returns exhibit a tendency not to follow a normal d i s t r i b u t i o n and t h i s i s d i s t i n c t l y evidenced i n the t a i l s of the d i s t r i b u t i o n . It i s a well recognized f a c t that stock market returns are shocked or jarred from time to time by unanticipated economic or p o l i t i c a l events and these i n turn may cause the extreme o u t l i e r s which cause the d i s t r i b u t i o n of the returns to appear non-normal. A d d i t i o n a l l y expectation of returns i s not a constant and w i l l change over time. A s h i f t i n expected returns may cause price changes and actual returns w i l l adjust accordingly. However, i t i s generally believed that logarithmic returns seem to approximate a normal d i s t r i b u t i o n and few researchers investigate t h i s property of t h e i r data and instead proceed as i f i t were true. 58 Taking the r e s u l t s o f a l l the goodness o f f i t t e s t s t o gether, one would conclude t h a t f o r weekly market r e t u r n s and n et a s s e t v a l u e s o f closed-end funds from 1965 t o 1972, are not w e l l - d e s c r i b e d by a normal d i s t r i b u t i o n . The monthly r e t u r n s e r i e s appear t o be normal except f o r e x c e s s i v e r e t u r n s i n the t a i l s o f the data d i s t r i b u t i o n . R e g r e s s i o n A n a l y s i s R e g r e s s i o n a n a l y s i s i s used t o f i t the market model. The l e a s t s t r i n g e n t s e t of assumptions one must make i n order t o make i n f e r e n c e from the r e s u l t s , i s u s u a l l y s t a t e d w i t h r e f e r e n c e t o the e r r o r term o f the model. That i s , the e r r o r term i s normally d i s t r i b u t e d w i t h a zero mean and con s t a n t v a r i a n c e . A d d i t i o n a l l y the c o r r e -l a t i o n between e r r o r terms i s zero (/ 5(e^,e^. +^)=0). A common procedure when working w i t h t h i s s e t o f assumptions i s t o develop the model, estimate the c o e f f i c i e n t s and then t e s t the e r r o r term t o see i f i t meets the assumptions. With the presence o f s e r i a l c o r r e l a t i o n , the model used t o estimate the r e l a t i o n s h i p o f r e t u r n s on a stock or 59 p o r t f o l i o to returns on the market Index, R' += o(. + A . R . + 6 . . , i f used for closed-end funds, v i o l a t e s the assumption of independent error terms. This causes the standard deviation of the c o e f f i c i e n t s to be underestimated and the corresponding t - s t a t i s t i c s w i l l be too high. Hypothesis t e s t i n g i s not possible with ordinary l e a s t squares. The beta c o e f f i c i e n t i s an estimate of the r e l a t i v e r i s k of the stock. If the c o e f f i c i e n t has a value of approximately one, then i t s r i s k i s e s s e n t i a l l y the same as i f one held a market p o r t f o l i o . If the beta c o e f f i -c i e n t i s greater than one, the stock has greater r i s k than a market p o r t f o l i o and vice versa i f less than one. A stock with a beta c o e f f i c i e n t of zero would be one for which the returns were independent of returns i n the stock market. The beta c o e f f i c i e n t i s then a measure of the vola-t i l i t y of a security's rate of return r e l a t i v e to changes i n the Index rate of return. How much the returns on the stock co-vary with a l l the other s e c u r i t i e s which comprise the Index has yet to be determined. 60 Betas were estimated on both market returns and net asset values of the closed-end funds. If the assets of closed-end funds consisted wholly of publicly-traded s e c u r i t i e s which were contained i n the Index, the beta of the net asset value would provide a measure of r i s k for the firm r e l a t i v e to the r i s k of the stock market. While t h i s would be desirable for a l l firms, estimation problems arise as i t i s impossible to obtain a market determined value for many assets. The assets of closed-end funds contain primarily p u b l i c l y traded s e c u r i t i e s and the closed-end funds, themselves, publish t h i s information weekly. Unfortunately, information i s not r e a d i l y available as to the actual composition of 34 t h e i r p o r t f o l i o s on a weekly or monthly basis. Malkiel has shown a s i g n i f i c a n t r elationship between the premiums and discounts of the firm's stock and the proportion of l e t t e r stock i n the p o r t f o l i o on an annual cross-sectional basis. If the porportion was s i g n i f i c a n t i t would cause d i s t o r t i o n i n the net asset value. I t i s d i f f i c u l t to assess what the e f f e c t s would be but i f the non-traded assets were revalued infrequently, t h i s might reduce the relationship between returns on a net asset value and returns on market value. A d d i t i o n a l l y i t could cause increases or decreases i n variance. 61 Returns on net asset value are measured as returns to shareholders and are therefore net of managerial fees and expenses. This has the e f f e c t of biasing the returns downward. While managerial fees do not bear a constant 35 relationship to assets across firms, Malkiel found that on an annual basis they were never s i g n i f i c a n t i n r e l a t i o n -ship to discounts and premiums. Ingersoll has argued that management fees w i l l cause discounts and one observes premiums due to market i n e f f i c i e n c i e s . Given Malkiel's re s u l t s i t appears other factors must a f f e c t discounts and premiums more strongly than managerial fees, so t h e i r omission i n measuring return may not be serious. I t i s believed that the r i s k associated with the common equity of a firm i s greater than that of the firm i t s e l f . This i s because the common shareholder i s e n t i t l e d only to the residual value of a firm after a l l p r i o r claims have been s a t i s f i e d . P r i o r claimants to a firm's assets may be secured or unsecured creditors and preferred share-holders. In the extreme event of bankruptcy, owners of common shares may f i n d t h e i r i n t e r e s t i n the firm worthless. The assets of closed-end funds are t h e i r p o r t f o l i o s of stocks and as the returns earned must f i r s t be used to s a t i s f y managerial fees and other p r i o r claimants, i t was expected that the estimated betas of the market returns 62 to the shareholder would be g r e a t e r than the betas o f the r e t u r n s on net a s s e t v a l u e t o the f i r m . Table XIV Estimated Betas Weekly Monthly Q u a r t e r l y * NAV MV NAV MV NAV MV Adams .869 .606 .883 .764 .92 .67 C a r r i e r s .827 .517 .840 .488 .87 .78 Dominick 1.113 .756 1.053 .876 1.06 .82 I n t e r n a t i o n a l .882 .685 .985 .894 .97 1.16 Lehman 1.062 .824 1.076 .909 1.09 1.15 Madison 1.021 .825 1.016 .887 1.18 1.08 N a t i o n a l .721 1.325 .715 1.100 .97 1.39 Niagara .987 .703 1.011 .699 1.12 .92 Petroleum .846 .796 .808 .746 .74 .69 T r i - C o n t i n e n t a l .987 .968 .974 1.038 1.04 .97 U n i t e d .445 .360 .548 .669 .69 .90 US & F o r e i g n .897 .751 .951 .816 1.01 1.23 *1966-73, a f t e r Sharpe and S o s i n . Not s t r i c t l y comparable as estimated from r e t u r n s a d j u s t e d f o r r i s k f r e e r a t e . Regressions were run on both the weekly and monthly s e r i e s . The estimated betas were s i g n i f i c a n t but the su r -p r i s e was t h a t the betas o f the market r e t u r n s were lower than n et a s s e t v a l u e r e t u r n s i n 11 out o f 12 weekly cases and 9 out o f 12 monthly cases. Under the assumption o f no r m a l i t y , the d i f f e r e n c e s i n v a l u e s were s i g n i f i c a n t a t l e a s t 50 p e r c e n t o f the time f o r weekly data. ( F u l l d e t a i l s o f the r e g r e s s i o n s are i n Appendix 7.) Some evidence o f t h i s problem appears i n an e a r l y v e r s i o n o f a paper by Sharpe and S o s i n . W a l l i n g f o r d and Z a v a n e l l i noted a s i m i l a r r e s u l t over a s h o r t e r time p e r i o d . T h i s r e s u l t i s c o n s i s t e n t w i t h the observed behaviour o f d i s c o u n t s , which i n c r e a s e i n r i s i n g markets and decrease i n f a l l i n g markets. I t would be tempting t o say t h a t these r e s u l t s a r i s e because o f s e r i a l l y c o r r e l a t e d data, p a r t i c u l a r l y s i n c e N a t i o n a l A v i a t i o n i s the o n l y f i r m w i t h a market v a l u e b eta g r e a t e r than a net a s s e t v a l u e b e t a and o f a l l the equations estimated i t showed the l e a s t s e r i a l l y c o r r e l a t e d r e s i d u a l s . I t d i d show s i g n i f i c a n t a u t o c o r r e l a t i o n i n i t s n e t a s s e t v a l u e s e r i e s on a weekly b a s i s . The a u t o c o r r e l a t i o n i s not the whole e x p l a n a t i o n . In the monthly r e g r e s s i o n s , Madison showed no s i g n i f i c a n t a u t o c o r r e l a t i o n i n the e r r o r term o f e i t h e r r e g r e s s i o n but the market b e t a remains much lower than the net a s s e t v a l u e b e t a . A d d i t i o n a l l y t h e r e was no s i g n i f i c a n t a u t o c o r r e l a t i o n i n the monthly data f o r t h i s company. 2 On s t r i k i n g f a c t o r i s t h a t the R v a l u e s o f the market r e t u r n r e g r e s s i o n s are much lower than those o f the net a s s e t v a l u e s — o f t e n by a f a c t o r o f two. On a weekly 2 b a s i s the market r e t u r n R va l u e s range from 0.03 to 0.29, 64 whereas the net a s s e t v a l u e v a l u e s range from 0.28 to 0.84. With r e s u l t s as low as 0.03 f o r a s t o c k l i s t e d on the New York Stock Exchange, i t i s an i n d i c a t i o n o f almost complete independence between r e t u r n s on t h a t closed-end fund share and the Index based on the New York Stock Exchange i n the same time p e r i o d . Since the net a s s e t v a l u e r e g r e s s i o n f o r t h a t f i r m ( U n i t e d Fund) had an 2 . . . R v a l u e of 0.27, one suspects i t s p o r t f o l i o d i d not c o n t a i n many shares which were i n the Index. Returns on net a s s e t v a l u e s are a v a i l a b l e a t the c l o s e of the market on F r i d a y and g e n e r a l l y d i s t r i b u t e d i n the p r e s s on Monday. Because o f the time l a g t h a t e x i s t s w i t h t h i s i n f o r m a t i o n , c o r r e l a t i o n s were computed between weekly market v a l u e r e t u r n s and the market index f o r a p e r i o d o f up to 12 l a g s . S i g n i f i c a n t p o s i t i v e c o r r e l a t i o n was found between the r e t u r n on the s t o c k and the Index lagged one p e r i o d f o r s i x out o f 12 companies. Those companies t h a t showed s i g n i f i c a n t c o r r e l a t i o n w i t h the Index lagged one p e r i o d u s u a l l y had lower c o r r e l a t i o n w i t h the contemporaneous Index, than those companies t h a t d i d not. 65 Table XV C o r r e l a t i o n - Market Returns and Index - Weekly Data T T-1 Adams .411 .131 C a r r i e r s .398 .249* Dominick .399 .193* I n t e r n a t i o n a l .399 .266* Lehman .478 .149* Madison .401 .125 N a t i o n a l .543 .095 Niagara .370 .153* Petroleum .413 .102 T r i - C o n t i n e n t a l .575 .069 U n i t e d .204 .173* US & F o r e i g n .422 .181* * S i g n i f i c a n t a t 5% l e v e l , t > 2.92. 40 . • Ibbotson , working w i t h unseasoned new i s s u e s i n p o r t f o l i o s , found t h a t due to i n f r e q u e n t t r a d i n g p r i c e quotes a t the time o f o b s e r v a t i o n were outdated and the s t o c k s ' a c t u a l r e t u r n was a combination o f the two p e r i o d r e t u r n . He t h e r e f o r e suggested and estimated a two market p e r i o d model, i n c o r p o r a t i n g a one-period l a g . Given the p o s i t i v e c o r r e l a t i o n between r e t u r n on the s t o c k a t t and the Index a t t-1, i t was decided t o estimate R i , t =°i± +ARm,t + A - l V t - l + * i , t 41 • • • f o r the weekly market v a l u e s . The estimated c o e f f i c i e n t s are shown i i i T a ble XVI and d e t a i l e d r e s u l t s are i n Appendix 9. 66 Table XVI Lagged Market Model A Adj . A A A + A R 2 A t-- s t a t A-i t-• s t a t Adams .16 .591 8 .31 .076 1. .06 .666 C a r r i e r s .18 .471 7 .69 .227 3. .70 .698 Dominick .16 .712 7 .82 .221 2. .43 .969 I n t e r n a t i o n a l .19 .618 7 .57 .333 4. .07 .951 Lehman .23 .805 10 .07 .096 1. .19 .901 Madison .16 .806 8 .09 .096 .96 .902 N a t i o n a l .29 1.33 12 .27 -.035 .32 1.296 Niagara .14 .672 7 .23 .154 1. .66 .826 Petroleum .16 .790 8 .47 .038 .41 .828 T r i - C o n t i n e n t a l .33 .985 13 .62 -.082 -1. .14 .902 U n i t e d .05 .311 3 .44 .243 2 .69 .544 US & F o r e i g n .20 .718 8 .97 .163 2 .04 .881 The ^ t c o e f f i c i e n t a s s o c i a t e d w i t h the contemporaneous Index was not s i g n i f i c a n t l y d i f f e r e n t from the beta co-e f i f c i e n t e s t imated f o r the non-lagged model. T n e / ^ t - l c o e f f i c i e n t was p o s i t i v e i n t e n out o f 12 cases, and s i g n i f i c a n t i n f i v e cases (assuming t h a t none o f the model's assumptions were v i o l a t e d , which i s i n c o r r e c t ) . In two cases, ( C a r r i e r s and I n t e r n a t i o n a l ) , the t - s t a t i s t i c s were l a r g e enough to accept the c o e f f i c i e n t s as v e r y s i g -n i f i c a n t . C a r r i e r s was one fund t h a t had a much s m a l l e r volume o f shares t r a d e d than those stocks i n the Standard 2 and Poors 500 (Appendix 4 ) . The a d j u s t e d R v a l u e s f o r the lagged e q u a t i o n were about the same as f o r the unlagged model, but the two companies w i t h s i g n i f i c a n t c o e f f i c i e n t s d i d have l a r g e r R v a l u e s , i n d i c a t i n g a l a r g e r p o r t i o n o f the v a r i a n c e was e x p l a i n e d by the a d d i t i o n o f the lagged 2 f a c t o r . O v e r - a l l , the R v a l u e s remained low, r a n g i n g 67 from 0.05 to 0.33. When the and c o e f f i c i e n t s were added t o -gether, the r e s u l t i n g v a l u e was l a r g e r than the be t a estimated f o r the non-lagged model i n t e n out o f 12 cases. The two exceptions were N a t i o n a l and T r i - C o n t i n e n t a l which had n e g a t i v e c o e f f i c i e n t s . These two companies a l s o had the s m a l l e s t c o r r e l a t i o n c o e f f i c i e n t s between market r e t u r n s a t time t and the Index a t t - 1 . For th r e e companies, I n t e r n a t i o n a l , N a t i o n a l and U n i t e d the summed be t a v a l u e was g r e a t e r than the be t a c o e f f i c i e n t a s s o c i a t e d w i t h the net a s s e t v a l u e r e t u r n , whereas i n the non-lagged model, o n l y N a t i o n a l demonstrated t h i s . The r e s u l t s o f t h i s r e g r e s s i o n suggest t h a t t h e r e i s some r e l a t i o n s h i p between r e t u r n on the common e q u i t y o f closed-end funds and r e t u r n s on the Index i n the week p r i o r . While many reasons may be hypo t h e s i z e d as t o why t h i s might be, t h e r e i s no e x p l a n a t i o n g i v e n by the r e s u l t s . The f a c t t h a t the summed estimated betas are g r e a t e r than the b e t a estimated f o r the s i n g l e Index model a l s o suggest t h a t the " t r u e " b e t a o f r e t u r n s on market v a l u e o f closed-end fund may be g r e a t e r than the net a s s e t v a l u e b e t a and econometric problems prevent us from a c c u r a t e l y determining i t . 68 Conclusions Returns for closed-end funds do not appear to follow a normal d i s t r i b u t i o n . If one was w i l l i n g to assume a constant mean for the series (which implies a constant expected return), one could conclude that the properties of the series were such that they were from a lon g - t a i l e d population. However, there i s some evidence i n the Kolmogorov-Smirnov Two Sample t e s t to suggest that the assumption of a constant mean for the series i s dubious. 42 Scholes and Williams have shown that infrequent trading r e l a t i v e to the market "average", i f instan-taneous returns are normally dis t r i b u t e d , w i l l cause observed returns to appear leptokurtic and reported variances to be overstated, while contemporaneous co-variances (R.,R ) are understated. Furthermore, auto-1 m covariances of lag one appear negative. The runs test, Kolmogorov-Smirnov two sample t e s t and t e s t of s e r i a l c o r r e l a t i o n gave res u l t s consistent with t h i s . Scholes and Williams further demonstrated that because instantaneous trading does not occur, i f trading i s less frequent the estimated betas on market returns w i l l be understated and the alphas overstated. The 69 estimated betas are too low due to measurement error which causes the variance of the market return of the fund to be overstated and dominate the measurement error of the Index. This also causes the estimated contempor-aneous covariance to be understated. This may p a r t i a l l y explain why the betas computed on market returns were less than those on net asset value returns. I f the true beta of the market return series of a closed-end fund i s greater than the beta of the net asset value return, and some evidence of t h i s i s given i n the results of the lagged model, then the common shares of closed end funds are r i s k i e r than the underlying p o r t f o l i o . This r e s u l t i s i n keeping with the b e l i e f that the common equity of a firm i s r i s k i e r because of the existence of p r i o r claims. 43 Sharpe argues that the existence of discounts/ premiums makes closed-end funds r i s k i e r than mutual funds as the opportunity for greater losses and returns e x i s t s . If the ownership of common shares of closed-end funds i s r i s k i e r , and one was able to hold a proportionate share of the underlying assets at less r i s k , then a greater return would be necessary to induce i n d i -viduals to hold shares of the fund. A discount on the market price of the fund i n excess of managerial fees, 70 could r a i s e the return such that one became i n d i f f e r e n t between the two alternatives. This would seem to imply that no advantage accrued to the shareholder by the fund's a b i l i t y to take large positions i n s e c u r i t i e s , acquire control of firms, or undertake managerial r e s p o n s i b i l i t i e s of p o r t f o l i o s . I t seems that one of the most f r u i t f u l areas of further i n v e s t i g a t i o n i n closed-end funds would be to look at the relationship between the frequency of trading and discounts. The return series show evidence consistent with infrequently traded stock. 71 Appendix 1 A B r i e f History of Closed-End Investment Companies The h i s t o r i c a l predecessors of closed-end funds are investment t r u s t s . I t appears that The Foreign and Colonial Government Trust, formed i n London i n 1868 was the f i r s t 44 one. Its p o l i c y was to raise a li m i t e d amount of c a p i t a l from investors. The proceeds were invested i n government bonds, d i v e r s i f i e d as to geographical location. Investors were afforded the opportunity to have an i n t e r e s t i n a number of s e c u r i t i e s , thus reducing the default r i s k , an opportunity that did not e x i s t for investors of moderate c a p i t a l before the formation of Foreign and Colonial. The introduction of l i m i t e d l i a b i l i t y laws made the formation of such a t r u s t f e a s i b l e . This f i r s t t r u s t was rapidl y followed by many others i n England and Scotland. The nature and form of investment trusts i n B r i t a i n today were established during the nine-teenth century. The emphasis was on security of income. Capital gains were not paid out but set aside i n a reserve fund to enable the maintenance of a constant rate of income. Capital gains treated that way were not subject to U.K. income tax. C a p i t a l i z a t i o n was primarily through preferred and common shares. Preferred shares, when available, sold at premiums. The p r i n c i p l e of d i v e r s i f i c a t i o n was geo-graphical and these funds survived the world-wide depression of the 1930's through the use of t h e i r reserve funds. U n t i l the 1930's, a l l B r i t i s h trusts were closed-end funds. North American trusts were f i r s t formed i n the United States i n the 1880's under state laws. The e a r l i e r ones appear to have been primarily holding companies for the assets of extended families. Following the B r i t i s h example, and contrast to present policy, the e a r l i e r investment trusts were leveraged. Rapid growth of investment companies occurred during the 1920's. They were primarily holding companies, and pyramiding occurred frequently. The popularity of the funds i s r e f l e c t e d i n the fac t that over 700 companies were formed 45 between 1927 and 1929 with 265 of them i n 1929 alone. One company raised c a p i t a l of over $300 m i l l i o n i n less than one year. Shares of investment trusts frequently sold at large 46 premiums over net asset value. The crash of 1929 hurt many U.S. investment companies as they had issued debentures and preferred shares but were prim a r i l y invested i n common stock. Unlike t h e i r B r i t i s h counterparts, the conservative practice of reserve funds had not been adopted and with the cessation of the common d i v i -73 dends, the closed-end funds were unable to meet t h e i r debt requirements. Furthermore, most investors did not know the s e c u r i t i e s held by the investment funds. Frequently they were con-t r o l l i n g interests of other investment funds. I t was after the crash, that discounts f i r s t appeared, allowing some funds to p r o f i t by s t r i p p i n g other funds. Discounts have prevailed since that time. Yet there were funds formed during t h i s time period with sound management. Eleven of the fourteen companies i n the i n i t i a l sample were i n existence i n 1929 and the twelfth i s an amalgamation of a company i n existence at that time and a l a t e r company. The crash of 1929 marked the end of investor enchantment with investment t r u s t s . I t has been estimated that the aggregate market value of shares was 35 percent less than • • 47 the net asset value of the underlying s e c u r i t i e s . The following decade was a period of consolidation. Some companies repurchased t h e i r shares on the open market at a 25 to 50 percent discount. Others bought c o n t r o l l i n g interests i n smaller investment companies which were liquidated. The s e c u r i t i e s of the l i q u i d a t e d company were added to the c o n t r o l l i n g company's p o r t f o l i o and minority shareholders 74 o f f e r e d s e c u r i t i e s i n the c o n t r o l l i n g company or a propor-t i o n a l share o f the a s s e t s o f the company b e i n g s t r i p p e d . D uring 1930-1931, the formation o f f i x e d t r u s t s was p o p u l a r . In f i x e d t r u s t s , the c a p i t a l r a i s e d i s i n v e s t e d i n a pre-determined p o r t f o l i o and v i r t u a l l y no changes are made. In v e s t o r s a t l e a s t knew what they were buying. In 1932, mutual funds were e s t a b l i s h e d and s i n c e t h a t time they have become the ascendent form o f investment t r u s t p r e f e r r e d by the p u b l i c . However, the closed-end form has remained v i a b l e , p a r t i c u l a r l y f o r s p e c i a l i n t e r e s t investments and has had r e c e n t waves o f p o p u l a r i t y as r e a l e s t a t e investment t r u s t s (REITs), Dual-Purpose Funds, and Bond Funds. The Investment Company A c t o f 1940 (U.S.) and subsequent r e v i s i o n s are r e s p o n s i b l e f o r the s t r u c t u r e o f closed-end investment companies as they are today. I t f o l l o w e d two years o f p u b l i c h e a r i n g s t h a t arose out o f the abuses o f the 1920's. To q u a l i f y as a r e g i s t e r e d investment company, the r e must be more than 100 s h a r e h o l d e r s and c a p i t a l i n excess o f $100,000. A company must f i l e a statement w i t h the S e c u r i t i e s and Exchange Commission (SEC) which d e c l a r e s i t s c l a s s i f i c a t i o n , borrowing p o l i c i e s , the i s s u a n c e o f s e n i o r s e c u r i t i e s , i n d u s t r y c o n c e n t r a t i o n , e t c . , none o f which may be changed without s h a r e h o l d e r s ' p e r m i s s i o n . Most companies i n the U.S. are r e g i s t e r e d . Without r e g i s t r a t i o n 75 they cannot use the U.S. mail or engage i n interstate commerce. The Act sets a number of r e s t r i c t i o n s on company a c t i -v i t i e s with the objective of preventing the excesses of the 1920's and providing shareholders with a modicum of protection. Some regulations probably do r e s t r i c t the companies from p r o f i t a b l e and le g a l a c t i v i t i e s . One such r e s t r i c t i o n i s the p r o h i b i t i o n of the purchase of s e c u r i t i e s from an under-writing or s e l l i n g syndicate i f any director, o f f i c e r , or employee of the investment company i s d i r e c t l y or i n d i r e c t l y connected with the underwriting or s e l l i n g group, unless the investment company i s the underwriter. If a t r i p l e A security was issued, the investment company could only purchase i n the after market i f one of i t s directors happened to be a member of a brokerage company i n the s e l l i n g group. After market purchases of well-priced long-term bonds can be more expensive due to brokerage fees. Exceptions may be made with SEC approval but obtaining SEC approval may be lengthy and new issues can s e l l out rapidly. The other side of the story i s that many a c t i v i t i e s may be taken only with shareholder approval. Few companies have much d i f f i c u l t y i n obtaining approval as shareholders are notoriously lax i n exercising t h e i r voting rights, or merely accede to management wishes and return t h e i r proxies without s p e c i f i c a t i o n as to t h e i r desires. In some cases, c o n f l i c t 76 o f i n t e r e s t might w e l l a r i s e between management who n a t u r a l l y wishes to see a s s e t s remain s t a b l e or i n c r e a s e as managerial fee s are based on a s s e t s and i n v e s t o r s whose i n t e r e s t s might be maximized by c o n v e r t i n g to an open-end fund or winding-up and t a k i n g the u n d e r l y i n g s e c u r i t i e s i n exchange f o r t h e i r shares. Companies t h a t have long been at a d i s c o u n t might adopt t h i s s t r a t e g y as maximizing t h e i r s h a r e h o l d e r s ' wealth, y e t o n l y once i n r e c e n t years has t h i s happened w i t h the d i s s o l u t i o n o f the M.A. Hanna Company i n 1965-1966. The c a p i t a l i z a t i o n o f closed-end funds i s r e s t r i c t e d by the Investment Company Act. The company may i s s u e debt s e c u r i t i e s or borrow from a bank, i f such a c t i v i t i e s are p e r m i t t e d by t h e i r r e g i s t r a t i o n , o n l y i f a s s e t coverage i s g r e a t e r than 300 p e r c e n t o f the amount r a i s e d . E q u i t y c a p i t a l cannot be i s s u e d f o r l e s s than net a s s e t v a l u e . Since most o f the companies have s o l d a t d i s c o u n t s most of the time s i n c e the 1930's, r a i s i n g a d d i t i o n a l e q u i t y c a p i t a l has been d i f f i c u l t , although many companies now o f f e r r e i n v e s t -ment programs f o r t h e i r s t o c k h o l d e r s . 77 Appendix 2 Overview of Causality of Discounts/Premiums There are eleven main reasons that reappear i n the l i t e r a t u r e on closed-end funds for the existence of premiums and discounts. Since most of the arguments pro and con are not based on any underlying theory but rather upon "street knowledge", they are not discussed at length here. I t was decided to b r i e f l y l i s t them and the table accompanying t h i s appendix shows the sources i n which the arguments appear. Management-shareholder c o n f l i c t . Management interests l i e i n maximizing t h e i r fee over the l i f e of the fund. The existence of management fees creates a discount because the f u l l earnings of the assets aren't passed onto the share-holders. The value of the fund to the shareholders would be maximized by winding-up and d i s t r i b u t i n g the underlying assets. Turnover. The holdings of the funds are churned or traded unnecessarily to maximize fees. Or the turnover i n the p o r t f o l i o i s excessive for maintenance of the r i s k class desired. 78 Performance. The funds do not perform as well as the market index. P o r t f o l i o composition. Either the p o r t f o l i o i s not well d i v e r s i f i e d or holds l e t t e r stock which i s inaccurately valued. The l e t t e r stock issue may be of some importance. The d i v e r s i f i c a t i o n of funds that s e l l as d i v e r s i f i e d funds i s f a i r l y broad (usually more than 100 d i f f e r e n t holdings). Capital structure. This i s regulated by law and d i f f i -c u l t to change. Opportunities for leverage are l i m i t e d as well as the i m p o s s i b i l i t y of r a i s i n g new c a p i t a l at less than net asset value. D i s t r i b u t i o n p o l i c y . To a c e r t a i n extent t h i s i s determined by tax l e g i s l a t i o n . Ninety percent of the income must be d i s t r i b u t e d or the fund i s subject to taxation. The shareholder has no control over timing or size of income or c a p i t a l d i s t r i b u t i o n s . Lack of merchandising. The public i s n ' t aware of closed-end funds. Security salesmen don't s e l l them as i t ' s a one-shot commission and the commission i s higher on a mutual fund. 79 Thin markets. The stocks don't trade very frequently. Some evidence i s presented i n t h i s paper that suggests t h i s might be a factor. Potential tax on unrealized c a p i t a l d i s t r i b u t i o n s . The purchaser acquires a poten t i a l tax l i a b i l i t y for c a p i t a l gains at time of purchase. There i s no corresponding pot e n t i a l for o f f s e t t i n g c a p i t a l loss. Risk. As common shareholders, purchaser has only the residual value. I r r a t i o n a l i t y and/or market i n e f f i c i e n c y . This i s the ca t c h a l l condition. Table XVII Reason Source Boudreaux Close Edwards Fishbein Homer In g e r s o l l M a l k i e l P r a t t Roenfeldt & T u t t l e S c h i f f Sharpe & Sosin S t o l l Vives Citations of Reasons for Premiums/Discounts C o n f l i c t Turnover Perfor- Port- C a p i t a l D i s t r i - Lack of Thin P o t e n t i a l Risk I n e f f i mance f o l i o Structure bution S e l l i n g Markets Tax c i e n t Comp. P o l i c y E f f o r t Market X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X x X X X X X X Wallingford & Z a v a n e l l i X X TABLE XVIII Runs Test Weekly Data Market Returns Net Asset Value Returns Number of Observations Number of Observations Company <median > median = median #runs <median > median = median #runs Adams 183 193 56 172 188 188 0 165 C a r r i e r s 173 203 38 171 188 188 0 157 Dominick 186 190 61 177 162 214 39 147 International 179 197 60 160 188 188 0 162 Lehman 188 188 0 202 188 188 0 154 Madison 174 202 48 162 188 188 0 148 National 188 188 0 175 188 188 0 151 Niagara 170 206 33 198 188 188 0 157 Petroleum 176 200 49 175 188 188 0 167 T r i -Continental 178 198 27 172 188 188 0 157 United 192 184 78 165 188 188 0 173 US & Foreign 188 188 0 188 188 188 0 167 Index 188 188 0 149 - - - -> va a CD ui 3 OO Ch 1-3 H-CD X Ul cf LO The number of observations less than the median and greater than the median contain those observations which were equal to the median and assigned a value such that they f e l l i nto that group. I f no values had been equal to the median, there would have been 188 observations i n each group. Based upon t h i s s p l i t , i f le s s than 171 runs or greater than 206 runs occurred i n a sample, one would conclude at the f i v e percent l e v e l that the data were not independent. TABLE XVIII (continued) Runs Test Monthly Data Market Returns Net Asset Value Returns Number of Observations Number of Observations Company <median > median = median #runs < median > median = median #runs Adams 47 47 0 52 47 47 0 47 C a r r i e r s 43 51 10 42 47 47 0 53 Dominick 47 47 13 46 45 49 10 41 International 47 47 12 40 47 - 47 0 42 Lehman 47 47 0 40 47 47 0 44 Madison 51 43 11 42 47 47 0 41 National 48 46 5 42 47 47 0 40 Niagara 46 48 7 54 47 47 0 41 Petroleum 46 48 9 45 47 47 0 45 T r i - C o n t i n e n t a l 47 47 8 45 47 47 0 53 United 41 53 21 28 47 47 0 47 US & Foreign 42 52 10 48 47 47 0 49 Index 47 47 0 42 - - - -The number of observations less than the median and greater than the median contain those observations which were equal to the median and assigned a value such that they f e l l into that group. I f no values had been equal to the median, there would have been 47 observations i n each group. Based upon t h i s s p l i t , i f l e s s than 40 or greater than 56 runs occurred, one would conclude at the f i v e percent l e v e l of s i g n i f i c a n c e that the data were not inde-pendent . 83 Appendix 4 Trading Volume Three funds, Carriers & General, Lehman Corp., and Petro-leum Corp., were a r b i t r a r i l y chosen and the number of shares traded were c o l l e c t e d for three d i f f e r e n t time periods. The average number of shares traded per day during each time period was calculated and compared with the average traded for a stock i n the Standard & Poors 500 Index. Far fewer shares of closed-end funds traded. Fund 15 January 1968 to 9 February 1968 48 Total Shares Traded Carriers Lehman Petroleum Standard & Poors 500 5,800 141,800 11,900 215,750,000 9 November 1970 to 4 December 1970 Fund Carriers Lehman Petroleum Standard & Poors 500 Total Shares Traded 3,200 106,200 20,400 250,700,000 49 Daily Average 290 7,090 595 21,575 Daily Average 168 5,589 1,074 26,389 5 June 1972 to 30 June 1972 Fund Carriers Lehman Petroleum Standard & Poors 500 Total Shares Traded 10,600 132,200 42,900 283,870,000 50 Daily Average 530 6,610 2,145 28,387 84 Appendix 5 Sample Moments Below are the formulae used i n the estimation of the sample moments of the data on closed-end funds. Let be the sample observations, i=l,2,...,n and n i s the number of observations. The median i s computed on ordered data, from smallest to largest. MED = MEAN Xr\ + Xn+| 5 =wl X; 1*1 Let M^ be the kth moment about the mean /I K = TT i ( x - ; - x j STANDARD DEVIATION = S = — Q— M (n-l) N* SKEWNESS = (n-l)(n-a) JL 2 STANDARD DEVIATION OF SKEWNESS 6n(n-l) KURTOSIS = rr STANDARD DEVIATION OF KURTOSIS = (n-2)(n-H)(n+3) TABLE XIX Sample Moments Market Returns Weekly r--Company Median Mean Standard 2 Deviation Skewness Kurtosis Adams 0 .0018 .025 -.192 2.471 Carriers 0 .0014 .022 .140 .639 Dominick 0 .0013 .033 .427* 1.832* International 0 .0019 .030 .331* 3.677* Lehman .0007 .0021 .030 .231 1.461 Madison 0 .0012 .035 .636* 12.386* National -.0002 .0013 .042 .552* 1.315* Niagara 0 .0019 .032 -1.081* 8.247* Petroleum 0 .0024 .033 .620* 10.097* Tri-Continental 0 .0020 .029 .995* 6.733* United 0 .0018 .030 .390* 2.099* US & Foreign .0002 .0025 .029 .606* 6.562* Index .0028 .0014 .017 .271* 1.114* 1. Standard Deviation (under assumption of normality) = .126 2. Standard Deviation (under assumption of normality) = .251 Si g n i f i c a n t at 5% l e v e l TABLE XIX (Continued) Market Returns Monthly Company Median Mean Standard Deviation Skewness"^ Kurtosis Adams .0041 .0058 .054 -1.459* 5. .023* Carriers 0 .0048 .039 - .326 .088 Dominick 0 .0030 .056 - .113 2. .451* International 0 .0057 .056 - .746* 3. .800* Lehman .0041 .0066 .056 - .616* 2. .652* Madison 0 .0022 .057 - .365 2. .445* National 0 .0015 .075 .569* 1. .736* Niagara 0 .0054 .064 - .882* 2. .986* Petroleum 0 .0074 .055 - .650* 1. .648* Tri-Continental 0 .0082 .054 -1.098* 3. .979* United 0 .0054 .052 - .058 -, .061 US & Foreign 0 .0081 .051 - .472 1, .995* Index .0101 .0056 .040 - .729* 1 .525* 4 3. Standard Deviation (under assumption of normality) = .249 4. Standard Deviation (under assumption of normality) = .493 * S i g n i f i c a n t at 5% l e v e l . TABLE XX Sample Moments Net Asset Value Returns Weekly Company Median Mean Standard , Deviation Skewness Kurtosis Adams .0013 .0016 .018 .296* 4.453* Carriers .0013 .0015 .016 - .133 1.094* Dominick 0 .0012 .021 - .239 1.437* International .0022 .0015 .019 - .125 1.100* Lehman .0034 .0020 .020 .485* 2.562* Madison .0034 .0017 .022 .498* 14.461* National .0008 .0014 .018 .241* 1.006* Niagara .0024 .0021 .018 - .069 1.381* Petroleum .0017 .0022 .017 .155 1.714* Tri-Continental .0028 .0019 .019 .116 1.563* United .0009 .0016 .015 1.211* 11.326* US & Foreign .0009 .0013 .019 .287* 2.647* 1. Standard Deviation (under assumption of normality) = .126 2. Standard Deviation (under assumption of normality) = .251 Si g n i f i c a n t at 5% l e v e l TABLE XX (Continued) Net Asset Value Returns Monthly Company Median Mean Standard 3 • 4 Deviation Skewness Kurtosis Adams -.0017 .0060 .038 - .845* 1.912* Carriers -.0004 .0054 .035 - .550* .586 Dominick 0 .0038 .046 - .561* 1.196* International .0027 .0053 .041 - .917* 1.610* Lehman .0008 .0073 .046 - .808* 1.753* Madison .0003 .0058 .044 - .753* .897 National .0001 .0052 .041 .287 -.133* Niagara .0006 .0078 .043 - .537* 1.812* Petroleum -.0005 .0080 .038 - .789* .812 Tri-Continental .0005 .0068 .041 - .558* 1.263* United .0008 .0059 .032 - .260 .790 US & Foreign -.0010 .0040 .040 - .542* .661 3. Standard Deviation (under assumption of normality) = .249 4. Standard Deviation (under assumption of normality) = .493 Si g n i f i c a n t at 5% l e v e l TABLE XXI Comparative Results — Goodness-of-Fit Tests Market Returns Weekly Level of Level of Kolmog- Level of Crame'r- Level of Level of Runs S i g n i f i - Chi S i g n i f i - orov S i g n i f i - von S i g n i f i - Anderson S i g n i f i -Company Test cance Square cance Smirnov cance Mises cance Dar l i n g cance Adams -1 .74 10 88. 35 1 1. .767 1 .483 1 7 .984 1 Car r i e r s -1 .75 10 54. 38 1 1. .425 1 .667 1 11 .135 1 cn 0 Dominick -1 .24 115. 53 1 1. .752 1 .582 1 8 .806 1 o P J International -2 .96 1 104. 43 1 1. .774 1 .573 1 7 .008 1 CD Lehman 1 .34 43. 76 1 1, .432 1 .579 1 9 .198 1 05 cn 3 OO Madison -2 .70 1 110. 60 1 1. .742 1 .219 1 3 .608 1 1 0 H -X National -1 .45 45. 50 1 1. .460 1 .481 1 9 .842 1 H i 1 as Niagara 1 .12 67. 97 1 1. .308 1 .419 1 5 .199 1 Petroleum T r i -Continental -1 .37 109. 68 1 1 .649 1 .235 1 4 .389 1 -1 .71 10 32. 37 5 1 .249 1 .508 1 6 .931 1 United -2 .47 2.5 280. 59 1 2 .451 1 .826 1 9 .751 1 US & Foreign - .10 58. 52 1 .910 5 .193 1 4 .460 1 Index -4 .13 1 27. 88 10 1 .635 1 .816 1 10 .410 1 1. S i g n i f i c a n t at 1% l e v e l . 2.5 S i g n i f i c a n t at 2.5% l e v e l . 5 S i g n i f i c a n t at 5% l e v e l . 10 S i g n i f i c a n t at 10% l e v e l . TABLE XXI (Continued) Comparative Results -- Goodness-of-Fit Tests Market Returns Monthly Level of Level of Kolmog- Level of Crame'r- Level of Level Runs S i g n i f i - Chi S i g n i f i - orov S i g n i f i - von S i g n i f i - Anderson Signi: Company Test cance Square cance Smirnov cance Mises cance Darling cance Adams .83 15. 85 10 .819 10 .153 2.5 2 .182 1 Car r i e r s -1 .18 1. 28 1.076 1 .343 1 4 .478 1 Dominick - .42 22. 27 10 .713 .115 10 2 .108 1 International -1 .66 10 7. 86 .765 .122 10 1 .940 1 Lehman -1 .66 10 6. 09 .661 .104 1 .983 1 Madison -1 .18 13. 29 .843 5 .120 10 2 .001 1 National -1 .24 12. 99 1.076 1 .153 2.5 2 .935 1 Niagara 1 .25 15. 58 .720 .125 10 1 .963 1 Petroleum T r i -Continental - .62 6. 06 1.076 1 .182 1 2 .746 1 .62 9. 22 1.033 2.5 .372 1 3 .387 1 United -4 .06 1 11. 62 1.099 1 .306 1 4 .573 1 US & Foreign .11 9. 43 .848 10 .170 2.5 2 .381 1 Index -1 .24 7. 41 .855 10 .225 1 2 .982 1 1 S i g n i f i c a n t at 1% l e v e l . 2.5 S i g n i f i c a n t at 2.5% l e v e l 5 S i g n i f i c a n t at 5% l e v e l . 10 S i g n i f i c a n t at 10% l e v e l . TABLE XXII Comparative Results — Goodness-of-Fit Tests Net Asset Value Returns Weekly Level of Level of Kolmog- Level of Cramer- Level of Level Runs S i g n i f i - Chi- S i g n i f i - orov S i g n i f i - von S i g n i f i - Anderson Si g n i i Company Test cance Square cance Smirnov cance Mises cance Darling cance Adams -2.48 2.5 50.70 1 1.214 1 .429 1 6.135 1 Ca r r i e r s -3.31 1 19.93 1.458 1 .768 1 10.271 1 Dominick -4.04 1 48.25 1 1.862 1 .785 1 10.195 1 International -2.79 1 20.33 1.565 1 .838 1 10.700 1 Lehman -3.61 20.98 1.298 1 .723 1 8.464 1 Madison -4.23 1 35.81 2.5 1.140 1 .397 1 4.421 1 National -3.92 1 25.42 10 1.337 1 .537 1 10.389 1 Niagara -3.30 1 21.64 1.342 1 .681 1 9.750 1 Petroleum -2.27 2.5 25.08 1.370 1 .586 1 8.652 1 T r i -Continental -3.30 1 28.28 10 1.443 1 .834 1 9.755 1 United -1.65 10 17.99 1.320 1 .515 1 6.438 1 US & Foreign -2.27 2.5 24.34 1.410 1 .783 1 8.813 1 1. S i g n i f i c a n t at 1 % l e v e l . 2.5 S i g n i f i c a n t at 2.5 % l e v e l . 5 S i g n i f i c a n t at 5 % l e v e l . 10 S i g n i f i c a n t at 10 % l e v e l . TABLE XXII (Continued) Comparative Results -- Goodness-of-Fit Tests Net Asset Value Returns Monthly Level of Level of Kolmog- Level of Crame'r- Level of Level of Runs S i g n i f i - Chi- S i g n i f i - orov S i g n i f i - von S i g n i f i - Anderson S i g n i f i -Company Test cance Square cance Smirnov cance Mises cance Darlmg cance Adams - .21 7.07 .903 5 .199 1 2.833 1 Ca r r i e r s 1.04 5.91 1.163 1 .296 1 3.988 1 Dominick -1.44 6.50 1.002 2.5 .296 1 3.135 1 International -1.24 9.88 1.488 1 .516 1 4.363 1 Lehman - .83 8.54 1.050 1 .251 1 2.999 1 Madison -1.45 9.21 .969 2.5 .315 1 3.827 1 National -1.66 10 2.19 1.128 1 .385 1 4.603 1 Niagara -1.45 8.68 .881 10 .172 2.5 2.523 1 Petroleum - .62 8.28 1.131 1 .325 1 4.498 1 T r i -Continental 1.04 6.27 .999 2.5 .206 1 3.093 1 United - .21 7.78 10 1.059 1 .212 1 3.428 1 US & Foreign .21 7.58 1.081 1 .295 1 3.573 1 1 S i g n i f i c a n t at 1 % l e v e l . 2.5 S i g n i f i c a n t at 2.5 % l e v e l . 5 S i g n i f i c a n t at 5 % l e v e l . 10 S i g n i f i c a n t at 10 % l e v e l , : TABLE XXIII Regression Results — Markets Returns Weekly Standard Durbin Von Neumann A Standard t- A t - R Company CX Error Stat. A Error Stat. Adj . Watson Z Adams .0006 .0012 .522 .6057 .0696 8.703 .1643 2.304 2.95 Ca r r i e r s .0006 .0010 .580 .5169 .0611 8.465 .1566 2.252 2.45 Reg] Dominick -.0003 .0015 -.165 .7562 .0897 8.427 .1555 2.687 6.67 Reg] I n t ernational .0004 .0014 .304 .6854 .0816 8.399 .1545 2.484 4.70 cess: App< Lehman .0005 .0013 .364 .8245 .0784 10.523 .2248 2.466 4.53 cess: App< Madison -.0007 .0017 -.393 .8254 .0976 8.459 .1564 2.443 4.31 r" 0 ID 3 VD (h OJ H-X National -.0015 .0018 -.817 1.3245 .1062 12.473 .2905 2.204 1.98 13 Niagara .0004 .0016 .264 .7030 .0912 7.706 .1327 2.328 3.18 n> Petroleum .0007 .0016 .454 .7975 .0912 8.746 .1658 2.562 5.46 w C —1 Tri-Continental .0008 .0012 .654 .9681 .0709 13.663 .3299 2.510 4.95 h-1 rt United .0009 .0015 .551 .3596 .0892 4.034 .0367 2.498 4.84 w US & Foreign .0010 .0014 .725 .7507 .0787 9.532 .1916 2.521 5.06 TABLE XXIV Regression Results — Net Asset Value Returns Weekly Company A o( Standard Error t-Stat. A A Standard Error t-Stat. R 2 Adj . Durbin Watson Von 1 Z Adams .0002 .0005 .519 .8688 .0298 29.111 .6928 2.7551 7.33 Ca r r i e r s .0002 .0004 .480 .8271 .0217 38.080 .7943 2.1540 1.50 Dominick -.0006 .0005 •1.229 1.1133 .0288 38.650 .7991 2.0460 .45 International .0001 .0006 .175 .8822 .0360 24.470 .6141 2.7320 7.11 Lehman .0004 .0004 .895 1.0624 .0239 44.390 .8399 2.4119 4.00 Madison .0000 .0007 .038 1.0210 .0422 23.307 .6090 2.6268 6.09 National .0003 .0007 .414 .7215 .0389 18.560 .4774 1.6631 -3.27 Niagara .0006 .0003 1.671 .9866 .0197 50.110 .8702 2.1046 .14 Petroleum .0008 .0005 1.630 .8464 .0291 29.067 .6921 1.9406 -.58 Tri - C o n t i n e n t a l .0003 .0004 .746 .9874 .0244 40.522 .8139 2.3975 3.86 United .0009 .0006 1.370 .4453 .0370 12.048 .2762 2.1010 .98 US & Foreign -.0002 .0005 -.311 .8972 .0313 28.640 .6857 2.6457 6.27 TABLE XXV Regression Results -- Net Asset Value Returns Monthly Company Adams Ca r r i e r s Dominick International Lehman Madison National Niagara Petroleum T r i - C o n t i n e n t a l United US & Foreign Standard t-CX Error Stat .0010 .0014 .757 .0007 .0012 .549 .0020 .0019 -1.098 .0002 .0014 .157 .0013 .0017 .744 .0001 .0018 .747 .0011 .0031 .366 .0022 .0015 1.470 .0035 .0021 1.701 .0014 .0014 1.000 .0029 .0024 1.216 .0010 .0014 -.678 A A Standard Error t-Stat. .8826 .0340 25.955 .8396 .0303 27.669 1.0536 .0479 22.002 .9852 .0344 28.610 1.0760 .0436 24.666 1.0160 .0451 22.530 .7154 .0783 9.135 1.0110 .9367 27.580 .8076 .0514 15.702 .9744 .0345 28.252 .5480 .0596 9.197 .9506 .0360 26.424 R 2 Durbin Von 1 Adj . Watson Z .8785 2.66 3.23 .8915 1.54 -2.24 .8386 1.90 - .47 .8979 1.89 - .52 .8672 2.24 1.18 .8449 2.02 .12 .4469 2.01 .05 .8909 1.83 - .83 .7252 1.62 -1.86 .8956 2.14 .69 .4733 1.86 - .69 .8823 2.02 .07 TABLE XXVI Regression Results - Market Returns Monthly Company A <* Standard Error t-Stat. A /* Standard Error t -Stat. R 2 Adj. Durbin Watson Von I Z Adams .0015 .0047 .327 .7638 .1168 6.539 .3099 2.73 3.57 Ca r r i e r s .0021 .0036 .579 .4881 .0892 5.474 .2375 2.27 1.30 Dominick -.0019 .0046 -.418 .8756 .1137 7.700 .3853 2.80 3.92 International .0007 .0045 .164 .8944 .1111 8.048 .4068 2.62 3.01 Lehman .0015 .0045 .328 .9088 .1127 8.063 .4077 2.45 2.18 Madison -.0028 .0047 -.589 .8867 .1178 7.525 .3743 2.12 1.0.2 National -.0046 .0063 -.735 1.1000 .1573 6.994 .3401 2.29 1.42 Niagara .0015 .0061 .248 .6690 .1519 4.602 .1783 2.51 2.50 Petroleum .0033 .0048 .682 .7461 .1195 6.242 .2899 2.25 1.22 Tri-C o n t i n e n t a l .0024 .0036 .653 1.0376 .0898 11.557 .5877 2.45 2.20 United .0016 .0046 .351 .6691 .1151 5.812 .2606 2.13 .66 US & Foreign .0036 .0042 .864 .8160 .1030 7.920 .3991 2.64 3.15 Appendix 8 Lagged Market Model cr> i> r- r>- C M 0 0 co 1 0 o> to o C M c r > o - r ~ o u n n o N ' t co oo C M C M C M uo ^ N m m m H o> o H r » m H r> o c r i ro C M r> i n ^ * (M ro i n m <3< m C M C M C M C M C M C M C M C M C M C M C M C M LO \£> ( O U O (Ji i> ^ O C M 00 V J D O O U O oo C M m oo ro I D toin m H H H i — I C M H C M H i— I C O O H ro i n oo m ^ m m H o uo cn oo U3CJ1CM r> CT> lO CM H COOO CO O U 3 < * O H CP fO iS H iD O H ro CM H I H i—irM CM H H c-» o r> r- o co H H H H H O d C O rO C O C M O O r - v O C X i O O O O C T i O C J i C T i r> 0 1 0 0 O O O O O O r H O O o o o u o o o c o o m c M c o m m C M O m uo H ro m i n >* oo C M co ro r~ C M C M ro rx> cn co m ro 0 0 uo O C M C M r O O O O r H O O C M H I I H uO O r> C O O tO N ^ C— C M uD i—I 0 0 CM U0 r> CT> uD C O O H U D ro v O 0 0 i n O O C M CM 1 0 > * O~> oo r> r- > o co N M D C M C O O O O C O O lO Oi vD m Oi C M C O C O H i—I i—I r H H CJ> Cf> 0 0 C N C O C M O o r ^ u o c n oo r- cn o rx> cn c-~ cn cn O O O O O O r H O O o o o i n C M r> <j ro o J ) o to i> i> oo O H H oo m io H C M cn o r~ cn r- r H H o o c o r - o o O O H H m r P r - i o co oo ro I D r- cn ro r-ro H i—i c n i > c o > * r > v D i n C M m m c M ^ to m ( T i n C M cs ioo ro ro o c M " * r - H < * t - ro c M o m r~ oo uo io C M m >* H H H r-i r-i r-i r-i r-i r-i H H r-i o o o o o o o o o o o o o o o o o o o o o o o o m «3< i n H ^ f o o m m o cn uo oo O O O O O O H C M O O O O O O O O O O O O O o o o o o o o o o o o o o o o CO G U U fd flrBfOO) <U c O J - H l C C O C S - H C T ) W - H G I J O C S M O C O O - H O ) - H S U -H (D -H £ -H -H Ol S-i I +J +J eg CD r O W g J - i - M ^ T i - P n j + J - H C - H U T 3 ( t l O f i r t 4 i n ! t l ! H l l J l . O C t n O 98 Footnotes 1. See Appendix 1 f o r a b r i e f review of the e v o l u t i o n of closed-end funds. 2. J.E. I n g e r s o l l , J r . , " A n a l y s i s o f dual purpose funds," J o u r n a l o f F i n a n c i a l Economics, V o l . 3, (1976), 83-123. Burton G. M a l k i e l , "The V a l u a t i o n o f Closed-end Investment Company Shares," The J o u r n a l of Finance, V o l . 32, No. 3., (June 1977), 847-859. They both argue management fees should c r e a t e a d i s c o u n t , but M a l k i e l f i n d s i t i n s i g n i f i c a n t e m p i r i c a l l y . 3. Eugene F. Fama, "The Behavior of Stock Market P r i c e s , " J o u r n a l o f Business, V o l . 38, (1965), 34-105. 4. Eugene F. Fama, Foundations o f Finance, (New York, 1976), Chap. 1, c i t e s the r e l e v a n t work. 5. N i c h o l a s J . Gonedes, "A Comparison o f the S t a b l e and Student D i s t r i b u t i o n s as S t a t i s t i c a l Models f o r Stock P r i c e s , " J o u r n a l o f Business, V o l . 47, (1974), 244-80. 6. Merton H. M i l l e r and Myron G. Scholes, "Rates o f Return i n R e l a t i o n t o R i s k : A re-examination o f some r e c e n t f i n d i n g s , " S t u d i e s i n the Theory o f C a p i t a l Markets, (ed. M i c h a e l C. Jensen), (New York, 1972), 54-58. 7. Wiesenberger S e r v i c e s Inc., Investment Companies Mutual  Funds & Other Types, V o l 26 (1966), and V o l . 33 (1973), (New Y o r k ) . 8. Robert L. Winkler and W i l l i a m L. Hays, S t a t i s t i c s :  P r o b a b i l i t y , Inference and D e c i s i o n , (New York, 1975), 852. 9. Sidney S i e g e l , Non-parametric S t a t i s t i c s f o r the  B e h a v i o r a l Sciences, (New York, 1956), 57. 10. Fama, "Stock Market P r i c e s " , 34-105. 11. M a l k i e l , " V a l u a t i o n o f closed-end...shares", 854. 12. N. Smirnov, "Table f o r E s t i m a t i n g the Goodness of F i t o f E m p i r i c a l D i s t r i b u t i o n s , " Annals o f  Mathematical S t a t i s t i c s , V o l . 19, (1948), 279-81. 99 13. F . J . Massey, J r . , "The D i s t r i b u t i o n o f the Maximum D e v i a t i o n Between Two Sample Cumulative Step F u n c t i o n s , " Annals o f Mathematical S t a t i s t i c s , V o l . 22, (1951), 126-7. 14. M a l k i e l , " V a l u a t i o n o f closed-end...shares", 855. 15. I n g e r s o l l , " A n a l y s i s " , 99. 16. Fama, "Stock Market P r i c e s " , 69-74. 17. Myron Scholes and Joseph W i l l i a m s , " E s t i m a t i n g Betas from D a i l y Data," Graduate School o f Business, ( U n i v e r s i t y o f Chicago, 1976), 2. 18. Scholes and W i l l i a m s , 2. 19. Scholes and W i l l i a m s , 17-20. 20. J.F. Kenney and E.S. Keeping, Mathematical S t a t i s t i c s ,  P a r t One, ( P r i n c e t o n , 1954), 3rd ed., 100. A l s o E.S. Keeping, I n t r o d u c t i o n t o S t a t i s t i c a l Inference, ( P r i n c e t o n , 1962), 197. 21. S i e g e l , Non-parametric S t a t i s t i c s , 42-47. 22. Jean Dickenson Gibbons, Nonparametric S t a t i s t i c a l  Inference, (New York, 1971), 73. 23. Herman Chernoff and E.L. Lehman "The Use o f Maximum L i k e l i h o o d Estimates i n X z T e s t s f o r Goodness o f F i t , " Annals o f Mathematical S t a t i s t i c s , V o l . 25, (1954), 579-86. 24. M.A. Stevens, "EDF S t a t i s t i c s f o r Goodness o f F i t and Some Comparisons," J o u r n a l o f the American  S t a t i s t i c a l A s s o c i a t i o n , V o l . 69, No. 343, (Sept. 1974), 730-737. 25. Stevens, 732. 26. B.W. Lindgren, S t a t i s t i c a l Theory, (New York, 1968), second ed., 333. 27. Stevens, "EDF S t a t i s t i c s , " 732. 100 28. M.A. Stevens, "Asymptotic R e s u l t s f o r Goodness-o f - F i t S t a t i s t i c s with Unknown Parameters," Annals  of S t a t i s t i c s , V o l . 4, (1976), No. 2, 357-69. 29. Stevens, 367. 30. T.W. Anderson and D.A. D a r l i n g , "A Te s t of Goodness of F i t , " American S t a t i s t i c a l A s s o c i a t i o n J o u r n a l , V o l . 66, ( D e c , 1954), 765-9. 31. Stevens, "Asymptotic R e s u l t s , " 365. 32. Stevens, "EDF S t a t i s t i c s , " 732. 33. Stevens, "Asymptotic R e s u l t s , " 365. 34. M a l k i e l , " V a l u a t i o n of closed-end shares," 854. 35. M a l k i e l , 855-856. 36. I n g e r s o l l , "Dual Purpose Funds," 92. 37. W i l l i a m F. Sharpe and Howard S o s i n , "Closed-end Investment Companies: F a c t and F i c t i o n : Risk and Return," unpublished paper, (Stanford U n i v e r s i t y , 1974), 35 and 51. 38. Sharpe and S o s i n , 35, 51. 39. Buckner A. W a l l i n g f o r d I I and Max Z a v a n e l l i , "A Comparison of In v e s t o r and P o r t f o l i o Risks and Returns i n Closed-End Investment Companies," un-p u b l i s h e d paper, (Columbia U n i v e r s i t y , 1974) , 14-15. 40. Roger G. Ibbotson, " P r i c e Performance of Common Stock New Issues," J o u r n a l of F i n a n c i a l Economics, V o l . 2, (1975), 242. 41. In a d d i t i o n to v i o l a t i o n of the assumption of ind e -pendent e r r o r terms, the i n t r o d u c t i o n of the one-week lagged Index as an independent v a r i a b l e i n the equation i n t r o d u c e s m u l t i c o l l i n e a r i t y i n t o the e s t i -mation s i n c e there was s i g n i f i c a n t a u t o c o r r e l a t i o n i n the Index. T h i s has the e f f e c t of causing the estimates of fa and to be i n a c c u r a t e . 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