THE PROBABILITY DISTRIBUTION OF SECURITY RETURNS.. . S A N K A U N

The Probability Distribution of Security Returns: Canadian Evidence from the Toronto Stock Exchange

Swaminathan Sankaran Faculty of Administration University of Regina

Abstract This article examines the appropriate probability

distribution for security returns traded on the Toronto Stock Exchange. The sample consists of 185 stocks traded continuously over a 19 year post war period from 1963 to 1981 in the Toronto Stock Exchange. The lognormality model of wealth relative is compared with the normality models ofprice change and rate of return. I t is shown that there is better support for the lognormality model than for the other two.

Rdsumd Cet article examine la rkpartition approprik des

probabilitis du rendement des titres transigbs d la Bourse de Toronto. L’echantillon comprend 180 actions transigbes sans air& durant une piriode de 19 ons, m e 1963 et 1981, h la Bourse de Toronto. Le mod& de lognormaliti de la richesse relative est comparbe QYOC

les modgles de normaliti de la fluctuation ak prir et de taux de rendement. I1 a btk dkmontrb que le m d l e de lorgnormolitb est favorisb.

Modern theoretical and empirical research in the portfolio choice area centres around the Capital Asset Pricing Model or CAPM. The bulk of this research is based on the assumptions that the decision maker’s utility function for wealth is quadratic and /o r that the probability distribution of security returns is Gaussian. Either of these assumptions leads to the conclusion that means and variances are sufficient statistics for the portfolio choice decision. Other assumptions about externalities such as transaction costs and taxes, homogeneity of investor expectations, the existence of a riskless asset,et cetera, have been relaxed and modified, and yet it has been shown that the general edifice of the CAPM still stands (Fama, 1976).

However, serious objections and criticisms have been raised against the quadratic utility assumption (Arrow, 1971; Mossin, 1973) and against the Gaussian probability distribution assumption (Cootner, 1962; Lintner, 1976; Rosenberg, 1972). Any attempt to drop the quadratic utility function and replace it with a ‘more reasonable’ one like the logarithmic utility function as suggested by Arrow (1971) must first come to grips with the probability distribution of security returns. Lintner on U.S. data has argued that the lognormal distribution

is the appropriate candidate and provided empirical support based on a study of 168 U.S. stocks over the 25 year period 1946-1971. Later theoretical and practical extensions have followed from this evidence (kyle , 1976;1985; Bawa and Chakrin, 1979). Somewhat less extensive studies of the distribution of security rcbrrns elsewhere also exist (Teichmoeller, 1971; Roux and Gilbertson, 1978).

The work presented here examines the empirical evidence for the Canadian scene based on 185 socks listed on the Toronto Stock Exchange (TSE) over the 19 years 1963-1981. As such, it should provideint-ing comparison with the Lintner study covering the U .S. economy and a different time period. This set of Canadian data also includes figures from the turbulent seventies which fall outside the Lintner time fram

Section I1 provides a brief summary of. the myirical research on the probability distribution of security returns leading upto Lintner’s 1976 work. S e c t h 111 gives a brief description of the Canadian data bsrsr used in this study. Section IV presents the results ofthisstudy and Section V concludes with a summary and discmion.

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EMPIRICAL EVIDENCE ON THE PROBABILITY DISTRIBUTION OF SECURtTY RETURNS

Cootner’s book (1964) is an excellent survey of the research till then on the behavior of stock prices. Later surveys can be found in Ball et al. (1980) and Lintner (1976). Almost all the articles in Cootner’s book assume that share transactions occur over fairly uniform time intervals and that price changes between successive transactions are i.i.d. random variables with a finite variance. Further assuming that a large number of transactions occur over a chosen time interval (day, month, or year) and then invoking the central limit theorem leads one to conclude that price changes across time intervals will tend to have the normal probability density function (Fama, 1965).

Mandelbrot (1963) questioned the correctness and validity of dropping or trimming numerous outliers from economic time series data such as stock prices without any causal justification. He was probably the first to argue that it would be unwise to ignore the observed fat tails and leptokurtosis. He hypothesized that the p.d.f.’s of the random variables in speculative prices were not Gaussian, but stable Paretian, i.e., distributions with finite means but whose variances may be infinite. The stable Paretians were still sufficiently useful for analysis as limiting distributions of sums of i.i.d. random variables and because of their invariance under addition.

Fama (1965) studied daily stock price data for each of the thirty stocks included in the Dow Jones Industrial Average (DJIA) from about the end of 1957 to September 26, 1962. He concluded that a symmetric stable Paretian distribution with its characteristic exponent alpha less than 2 seemed to fit the data better than the normal or a mixture of normals.

Clark (1973) introduced an important change dropping the assumption of uniform time intervals between successive transactions and, instead, treated the operating time as a random variable, this directing process itself being subordinate to the data generating process guiding the price differences between clock time intervals. Based on this subordinate stochastic process model, he examined the data on daily price, transactions and volume for cotton futures for the years 1945 to 1958. He concluded that the limit distribution of price changes is subordinate to the finite variance normal distribution and the distribution of operating time between transactions is lognormal.

Rosenberg (1972) examined a 100 year series of monthly changes in Standard and Poor’s composite index, using a two-parameter model to predict fluctuations in the variance. He concluded that log price changes obey a normal distribution with predictably fluctuating variances. He suggested that the study of the p.d.f. of security prices be conducted in two parts: (i) modelling the fluctuation in the series variance; and, (ii) analyzing the p.d.f. of the series after modifying it for the predictable variance. He called for much additional research in the area.

Lintner (1976) carried out an extensive study of postwar US. securities which, among other issues, examined the p.d.f. of security returns in great detail. He correctly pointed out that most of the work cited earlier studied daily or weekly data whereas most investors are likely to have longer planning horizons. He also argued that one should be studying the distribution of returns rather than that of prices or price changes, as the market moves from its initial position to the market-clearing equilibrium vector of current prices. In addition, the sample sizes in most of the earlier studies were relatively small; many of them also included data from earlier periods thereby ignoring the structural and institutional changes in the world economy since World War 11. He studied the p.d.f. of the wealth relative, i.e., (1+ the rate of return), of 569 stocks on the CRSP tape, of which 168 stocks traded continuously from I946 to 1971. He examined their p.d.f.’s, skewness and kurtosis over four differencing intervals, viz., monthly, quarterly, semi-annual and annual.

Lintner tested the lognormal model against the Gaussian. Based on Kolmogorov-Smirnov tests, he found the lognormal model to be the better one, even for the longer intervals, regardless of the level of significance used. He found that the logarithmic wealth relatives, on the average, had essentially no significant skewness and that the skewness statistic itself seemed to be symmetrically distributed over the population. The evidence from the kurtosis statistic was somewhat mixed, although even here he found the deviation from the model r5pidly narrowing as one moved to longer differencing intervals or planning horizons. He concluded that the lognormal model was superior, and hypothesized that either some averaging process akin to the central limit theorem was “working on a random sequence of proportional shocks whose sum tends to approximate normality in the logs ......” or that, reflecting barriers as earlier suggested by Cootner ( I 962) were at work.

Given Clark’s earlier results and the generality of the assumption of non-uniformity of time intervals between successive transactions, the rigour and tractability of portfolio choice models based on lognormality (Merton, 1971; Bawa and Chakrin, 1979), and the extensiveness of Lintner’s inquiry, it is likely to remain the standard work of comparison for the future. Further, combining the well known properties of the lognormal with the desirable risk aversion properties of the logarithmic utility (Arrow, 1971), and also the optimality of myopic policies under such a utility function (Mossin, 1968), Lintner’s findings point to promising developments in the area of multiperiod decision models. It would be helpful to find out if his results are general enough to be applicable over other industrial economies. This article therefore concentrates on the test of lognormality of wealth relatives of common stocks derived from a sample comparable to Lintner’s in size drawn from the Toronto Stock Exchange.

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One must however hasten to point out that there are Canadian studies which deal with many of the issues addressed in this article. This study differs from the earlier ones in terms of sample size, length, differencing intervals, and therefore the scope. Rorke, Wills, Hagerman and Richmond (1976) studied the random walk hypothesis in the Canadian equity market based on monthly returns of 133 companies listed on the TSE and MSE from January 1958 to December 1967. Their data base (Rorke and Love, 1974) was a forerunner of the Laval data base used in this study. They concluded from serial correlation tests that there was support for the weak form of the efficient market hypothesis, for the hypothesis that the distributions underlying their data were non-normal, and that, based on runs tests similar to the ones reported here, there was “nothing here that can be interpreted as contradicting the random walk hypothesis.”

Singh and Rahman (1986) used a sample of 100 randomly selected stocks from the same Laval tape for the ten-year period 1972-198 1. They were concerned with the variance of security returns. They suggested that monthly variances converge and, hence, are finite. Further, they concluded that, irrespective of industry classification and thinness, for 70% of their sample, the variances followed a white noise with constant non-zero drift.

Hatch and White (1983) use a different data base located at the University of Western Ontario, which will probably be made available to other researchers in the near future. Theirs is a wide ranging study of returns from treasury bills, bonds and equities and also of the inflation rate. For a weighted index of monthly equity returns they found patterns consistent with the normal distribution although with some modest skewness to the left and slight kurtosis. They found no serial correlation in that series. However, for a random sample of ten individual securities from the same 1950-1983 period they found that the hypothesis of normality of security returns was rejected and that they were generally more peaked. The focus of their study is however different and it is likely to become a standard reference source of historical return and variability measures of several asset classes in the Canadian capital markets.

THE CANADIAN DATA BASE

The data used in this study are from the Laval TSE Master File 1981, created, maintained and updated by Universitt Laval, in QuCbec, Canada. The tape consists of monthly data from January 1963 through December 1981 on the common stocks of firms incorporated in Canada and listed on the Toronto Stock Exchange. All industrial firms are included. Any mining or oil firm whose common stock traded at $5 per share or over is included from the time it first traded at such a price. Once included, a stock is never cast out even if its price fell below $5 subsequently. There were 1012 such stocks

on the tape as of December 1981, of which 185 bad complete data over the entire 228 months.

The Laval Master File data consist of monthly opening and closing stock prices, and monthly split-adjusted returns, dividends, splits, rights, distributions of capital, ticker symbols, names, name changes and number of shares outstanding at yearend. The file development, testing and evaluation are described in Morgan m d Turgeon (1978).

Using this file, the 185 stocks for which dl the information is provided over the 19 years were e x t r a . For each of these stocks, then, price changes, @it- adjusted capital gains, dividend and total returns m e calculated for monthly, quarterly, semi-annual and annual intervals. The following distributional fits were investigated over each of the four aforementioned t h e intervals: normality for price changes, and total rcturns, and lognormality for the wealth relative, ie, ( I + tutal returns). The stochastic ‘independence’ of the cross- sectional observations of each of these variables was also analyzed using the non-parametric runs-up- d- down test (Gibbons, 1976). The results of these tests and some of the comparisons are reported in the next section.

RESULTS

As mentioned earlier, most of the empirical evidence about the distribution of security returns comes from non-Canadian data, and/ or is based on a smaller sample size and on daily or weekly horizons which are much shorter than investors’ actual horizons. This stetion provides the evidence for Canadian stock market data over longer holding periods - monthly, quarterly, semi- annual and annual. It compares the normality models of price change and rate of return in natural units with the lognormality model of wealth relatives.

Table 1 presents the results first on the randomness of the data - price changes and logarithm of the wealth relative, based on the non-parametric runs up-and-down test. Since the logarithm of the wealth relative is a monotone transformation of the rate of return no separate test results are produced for that variable. The non-parametric test was used in preference to the traditional zero serial correlation test for two reasons: (a) a correlation of zero is only a necessary but not sufficient condition for independence for all random variables; (b) it is not a distribution-free test, whereas it is advisable to establish randomness of the variable first before making assumptions about its probability density function.

The results of the test indicate that it‘ is reasonable to assume randomness for all the variables under consideration for all time intervals, with muginally better support as the length of the differencing inFerva1 increases. At the monthly interval, less than 3% of the sample are rejected at the 1% level and less than 7% at the 5% level when testing for randomness ofthewealth

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TABLE 1: p - values from the RUNS UP and DOWN TEST.

H,: The series is random. HA: The series is random. Let V be the total number of runs whether up or down, and N be the number of observations in the sample. The mean of V is (2N-i)/3 and its standard deviation ‘s’ is the square root of (16N-29)/90. Then, with a continuity correction of 0.5 we get the two standardized variables:

V + 0.5 - (2N - 1)/3 2, = S

and 2, = V + 0.5 - (2N - 1)/3

S Then,

Is( 2, otherwise

is a standardized normal variable (Gibbons, 1976).

-2, if V § (2N - 1)/3

~~ ~ ~~

log(l+r) I Price Change p- Value i- M I O I S I A I M I O I S I A

I 0.010-0.025 I 41 21 31 01 41 61 11 21 C I

I I

I 0.150 - 1.000 1159116011551171 ~162~164115711701 I I

relative. At the annual interval 0% is rejected for the same variable at both levels of significance. For the price change variable, the corresponding figures are less than 2% at the 1% level for all intervals, and less than 7% at !he 5% level for monthly data and less than 3% for the annual data. Similar results hold for the quarterly and semi-annual data.

Table 2 provides the results of the Kolmogorov- Smirnov goodness of fit test for normality of the logarithm of the wealth relative and normality of the price change and rate of return variables in natural units over all four intervals. From the p-values it can be seen that there is stronger support for the lognormality of the wealth relative than for the normality of either of the other variables over all four differencing intervals. This is in accord with Lintner’s results for the U.S. data. Further, as the length of the differencing interval increases, the evidence supporting hnormality also grows stronger for all three variables. This is also in agreement with the results obtained for the U.S. data. And, one is led to conclude with Lintner that as the length of the differencing interval increases, so also do the number of transactions taking place, and hence it is reasonable to postulate that a process akin to the law of large numbers may be at work bringing in the effects of the central limit theorem.

TABLE 2: p - values from the KOLMOGOROV - SMIRNOV TEST.

H,,: The cumulative distribution function of the observed random variable is that of a Normal random variable. HA: It is different from that of a Normal random variable. The test statistic is the well-knows Kolrnogorov-Smirnov D statistics.

10.150 - I.OOO1 881147116311831 661121114111511 341 11(1461168]

As for lognormality of the wealth relative, 31% of the monthly sample, 7% of the quarterly, 2.2% of the semi-annual and 0% of the annual sample are rejected at the 1% level of significance. The comparative figures for the price change variable are 38%, 20%, 16% and 2.2% respectively; and for the rate of return variable the corresponding figures are 58%, la%, 5.4% and 0.5% respectively. At the 5% level of significance, the rejection percentages are slightly higher for all three variables over all four differencing intervals. Once again, on this evidence, the lognormality of the wealth relative is a more plausible assumption than the normality of price change or rate of return variable. Further, even for this variable at the monthly interval, normality is a tenuous assumption at best.

Many scholars, notably Cootner (1962), Mandelbrot (1963), and Fama (I965), have drawn attention to the significance of higher moments of the distribution of security returns and especially to their skewness as measured by the third moment. Thus, while accepting the normality of the three variables, viz., the logarithm of the wealth relative, and the price change and rate of return variables in natural units, it is useful to consider the degree of support for the observed skewness and kurtosis coefficients. These tests are based on Fisher (1928) (see also Kendall(1943), Johnson (1949) ).

Table 3 provides the evidence for the skewness coefficient, which measures the symmetry of the distribution around the mean. The null hypothesis here is that the observed skewness is the same as one would expect for a normal random variable, against the alternative that it is not. Table 4 provides the evidence for the kurtosis coefficient which measures the “peakedness”of the density function. The null hypothesis here is that the observed kurtosis is the same as one would expect if the random variable under consideration is normal, versus the alternative that it is not.

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TABLE 3

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TABLE 4 p-Values from the NORMALITY TEST

for SKEWNESS

Let the standardized skewness statistic be called SK, the sample size N and the sample mean X. Then assuming that the underlying random variable is Normal, we have the following test:

H,: SK = 0

HA: SK # 0

The test statistic is ( N / ( N - 1) (N - 2)] 2 (X, - k;y/sJ. It follows a standardized ‘t’ distribution, but for large samples is asymptotically normal. (Johnson, 1949).

N

FI

From both tables, we can conclude first that there is more evidence in support of the null hypothesis for the logarithm of the wealth relative than for the other two, and marginally better support for the price change variable than for the rate of return variable. Further, for any given variable there is better support as the length of the differencing interval increases, the lone exception being the price change as we move from monthly to quarterly data. It is also found that all variables are more peaked than they are skewed as can be seen by comparing the p-values for skewness and kurtosis for any given variable under any of the four intervals.

However, except for the annual logarithm of wealth relatives data, there is only patchy support at best for either of the two null hypotheses. But for this exception, the standardized values of the skewness and kurtosis coefficients are significantly different from what one would expect under normality for all the other eleven situations. Even for the annual logarithm of wealth

p-Values from the NORMALITY TEST for KURTOSIS

In addition to the notation used in the skewness test, let KU be the standardized kurtosis. Then, assuming that the underlying random variable is normally distributed the hypothesis test for testing that KU does not differ from that of a standardized normal random variable is as follows:

H& K U = O

HA: KU # 0

The test statistic is:

[N(N+I)/(N-I)(N-2)(N-3)] c!,, (Xi - k)4/s4 - 3(N-1)2/(N-2)(N-3). It follows a standardized ‘t’ distribution, but is asymptotically Normal for large samples (Johnson, 1949). The tests for skewness and kurtosis are stable only when the sample size is large or if the distribution of the random variable is well behaved.

relatives data, 12.5% of the skewness values and 16% of the kurtosis values are rejected at the 1% level of significance, and 33.2% and 20% respectively are rejected at the 5% level of significance. Thus, although the lognormal model is better supported than the other two, it still exhibits higher skewness and kurtosis than one would expect under that model.

SUMMARY AND DISCUSSION

In this article, the empirical evidence on the probability distribution of Canadian security returns was studied, based on a 19 year post war sample of 185 stocks on the Toronto Stock Exchange. The lognormality model of the wealth relative was compared against normality models of the price change and the rate of return over four differencing intervals. These results for the Canadian data were also compared with Lintner’s earlier results for the U.S. data.

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It was found that the lognormality of the wealth relative was a much better model in describing the behavior of security returns than the normality models. This is in accordance with Lintner’s earlier results for U.S. data. However, all three distributions exhibited greater asymmetry and even greater peakedness than one would expect under normality. Finally, as the length of the differencing interval increases asymptotically better fits are obtained under all three models.

This study, however, did not deal with the problems posed by thinness of trading. The important work on this aspect in Canada is by Fowler, Rorke and Jog (1980). They found approximately 58% of the stocks traded in the TSE were thinly traded at one time or another during the ten year period 1970 - 1979. They concluded that thinness leads to predictable biases in estimation, but cautioned however: “Overall, there does not yet exist a technique that seems to have general applicability and effectiveness in reducing thin trading induced bias......”. In the absence of such a technique they advised that there may very well be downward bias in the return estimates; however, there would be a simultaneous increase in the variance of the estimator if any of the corrective techniques suggested in the literature were applied. This would tend to suggest that the tests of hypotheses reported in the previous section are somewhat conservative.

Only nominal returns and their distributions are addressed here. It may be possible to consider real returns. But, the choice of the appropriate deflator does pose a problem. If a “risk free” rate were to be applied then probably the treasury bill rate of the same differencing interval could be used. If one were to assume that investors are ultimately concerned with the prices of goods they expect to purchase in the current period as Hatch and White have done, then, the CPI may be used. Whichever measure is used, the problems posed by unanticipated inflation and the consequent variation in the real rate still remain. Statistically, an intractability problem would arise if the deflator and the security’s wealth relative are correlated and if they do not both follow the lognormal distribution. In addition, the randomness or otherwise of these proposed deflators, the p.d.f. of their underlying data generating processes, and the joint p.d.f. of any of these deflators and the return on a given security are all insufficiently studied phenomena, Apart from these difficulties, only nominal returns were considered to facilitate comparisons with the results of other studies quoted earlier.

The results here point in two directions: (1) As the length of the differencing interval increases and becomes a year or longer, the error from using the “incorrect” model becomes less and less. Therefore, in analyzing the decisions of investors with such long time horizons or holding periods, tractability, availabilty of readily developed inference techniques and computational convenience may dictate the choice of the probability distribution to be used. This would then argue in favour of the normality of rates of return in natural units and

the mean-variance approach to portfolio choice. (2) Insufficient attention has been given so far to the kurtosis characteristics of return distributions than to their skewness. Available evidence suggests that these distributions are more peaked than they are skewed. Further research needs to be done to study its implications for the observed behavior of stock market participants and for market efficiency.

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Bawa, V.S., and L.M. Chakrin, (1979) ’bptimal Portfolio Choice and Equilibrium in a Lognormal Securities Market,“ in Elton and Gruber. 47-62.

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Fowler, D.J., C.H.Rorke and V.M.Jog, (1980) “Thin Trading and Beta Estimation Problems on the Toronto Stock Exchange,” Journal of Business Administration, 12, No. I, 77-90.

Gibbons, J.D.. (1976) Nonparametric Methods for Quantirative Analysis. New York:Holt. Rinehart and Winston.

Hatch, J.E., and R.W.White, (1985) Canadiun Stocks, Bonds. Bills, and Inflation: 1950-1983, Charlottsville,VA: The Financial Analysts Research Foundation.

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Lintner. J., (1976) “The Lognormality of Security Returns, Portfolio Selection and Market Equilibrium,” Working Paper No.235 (Revised), Harvard Institute of Economic Research, Boston,Mass.

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Merton. R.C., (1971) “Optimum Consumption and Portfolio Rules in a Continuous-Time Model,” Journal of Economic Theory. 3.373-413.

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Mossin, J., (1973) Theoryof Financial Markets, Englewood Cliffs: Prcnticc- Hall, Inc.

Rorke, C.H., and C.E. Love, (1974) “A Note Concerning the Availability of the Study: Some Indices and Rates of Return to Canadian Equity Investment for the Period January 1951 to December 1967.” Journal of Business Administration, 6, No. I , Fall 64-66.

199-238.

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Rorke. C.H., 1. Wills, R.L. Hagennan and R.D. Richmond, (1976), "The Random Walk Hypothesis in the Canadian Equity Market," Journa/ of Business Administration. 8. No. I , Fall 2341.

Rosenberg, B., (1972) "The Behavior of Random Variables with Nonrtationary Variance and the Distribution of Security Prices," Working Paper No. 11. Rescorch Program in Finance, Insfiture of Businea Research. Berkeley. Uniwrsify of California, 42 pages.

Roux, F.J.P.. and B.P. Gilbcrtron. (1978) 'The khaviour of Share P r i m on the Johannesburg Stock Exchange," Journal of Business Finance ond Accounting, 5,223-232.

Singh, B.. .nd A.H. Rahman. (1986) "An Econometric Analysis of the Variability of Security Returns." Canadian Journal of Adrninirrariw Sciences. 3, No. I, (June) 65-80.

Tcichmoeller, J.. (1971) "Distribution of Stock Price Changes," Journal of the American Starirfical Association, 66,282-284.

Cct article a CtC dlcct iond par le professeur Ron Burke. / This article was selected by Professor Ron Burke.

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