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Testing the random walk hypothesis on Swedish stock prices: 1919–1990
Journal
of Banking
and Finance
17 (1993)
175~191. North-Holland
Testing the random walk hypothesis Swedish stock prices: 19 19-l 990 Per Frennberg Department
and Bjiirn Hansson*
of Economics
Received October
on
and Institute of Economic
1991. final version
received
March
Research,
University
of Lund, Lund, Sweden
1992
This paper tests the random walk hypothesis on a new set of monthly data for the Swedish stock market, 1919-1990. We use both the variance ratio test and the test of autoregressions of multiperiod returns. Our results suggest that Swedish stock prices have not followed a random walk in the past 72 years. For short investment horizons, one to twelve months, we find strong evidence of positively autocorrelated returns. For longer horizons, two years or more, we find indications of negative autocorrelation, so called ‘mean reversion’. Our results are in line with recent research on the U.S. stock market and may have several implications for the practical investor. They point in particular towards a larger proportion of stocks in the portfolio for long term investors.
1. Introduction Several studies have reported empirical evidence that stock returns, contrary to the random walk hypothesis, contain relatively large predictable components [e.g. Fama and French (1988), Poterba and Summers (1988) and Lo and MacKinlay (1988 and 1989)]. The evidence found suggests that nominal and real stock returns up to one year may be positively autocorrelated while real stock returns for longer investment horizons may be negatively autocorrelated. But the results are still controversial. On the one hand, they could be interpreted as a violation of the efficient market hypothesis, though their implications for portfolio choice and option pricing are more interesting. On the other hand, the statistical significance of the results has been questioned: Kim et al. (1988) argue that the mean-reversion phenomenon is concentrated in the pre-war period and that it is also very sensitive to the choice of market index and the type of return measure.
Correspondence to: Dr. Bjiirn Hansson, Department of Economics, University of Lund, Box 7082, 220 07 Lund, Sweden. *We like to thank two anonymous referees for constructive comments. We are also grateful to seminar participants at the Swedish School of Economics and Business Administration in Helsinki and at the Department of Economics at the University of Lund. Support from the Institute of Economic Research at the University of Lund is gratefully acknowledged.
03784266/93/$06.00
&‘J
1993-Elsevier
Science Publishers
B.V. All rights reserved
176
P. Fwnnher~
nnd B. Hansson.
Swedish .s~ock prices
Richardson and Stock (1989) show that the results provided by Fama and French (1988) are less significant than reported. Unfortunately, most previous research has focused on the U.S. stock market.’ This is probably due to the fact that the issue of autocorrelated stock returns, especially multiperiod returns, requires long time series of high quality data. The purpose of this paper is to test the random walk hypothesis for the Swedish stock market using a new set of data for the period 1919P1990.2 Our data are the monthly returns, including reinvested dividends, on a valueweighted market index of the Swedish stock market, the so called ‘Affarsvarldens Generalindex’. To make our results comparable with those reported earlier for the U.S. data, we use two standard techniques: the variance ratio and multiperiod autocorrelated tests. The analysis considers investment horizons from one to forty years. Following the American studies we focus on the inflation adjusted returns, but nominal and excess returns are also examined. Real returns are believed to be the most relevant variable for an in particular for investment horizons spanning several years. investor, However, due to the quality of the consumer price data for short periods we analyse nominal returns for investment horizons below one year. Another measure of interest is excess returns, i.e. the difference between the return on stocks and the return on a risk-free short term security, since many dynamic asset allocation strategies focus on the return differential between stocks and short term money market instruments. Unfortunately, the short term money market in Sweden has only existed since the early 1980s. Therefore, the results of the excess return analysis are less reliable than the ones from the real return analysis. The paper is organized in the following way. Section 2 gives a brief theoretical background to the question under consideration and a presentation of the testing methodologies used in the paper. Section 3 provides empirical results as well as tests for robustness. Section 4 is the conclusion.
2. Theory and methodology In this section we first recapitulate the random walk model and the testable null hypothesis derived from it. Then follows a presentation of the two testing methodologies: the variance ratio test and the test of autoregressions on multiperiod returns.
‘Poterba and Summers (198X) examined a number of non U.S. stock markets. among them the Swedish one, using the IFS data from the IMF. However, these data cover only the period 1956-1986 and do not include dividends. ‘In an early study of the Swedish and the Norwegian stock market Jennergren and Korsvold (1974) studied daily prices of individual stocks for the period 1967-1971.
P. I‘rennhrrg
2.~.
The random
and B. ffansson.
Swedish
stock
177
prices
walk hypothesis
Let P, denote the stock price at time t and define p1-In process given by the recursive relation: pt=p+pr_,
P, as a stochastic
+e,~r(t-ll,t)-p,-pp,~,=C(+r,,
(1)
where p is a constant drift parameter, e, is the randon disturbance and r(t - 1, t) is the actual return for the period t - 1 to t. The one-period return is denoted r( 1) and the q-period return, which is the sum of q one-period returns, is denoted r(q). The e,s are assumed to be strict ‘white noise’, that is to say independently and identically distributed (i.i.d.) and having a normal distribution. Since there is strong evidence of time-varying volatilities and deviation from normality [e.g. Hsu (1984) and Schwert (1989 and 1990)] the strict random walk hypothesis can easily be rejected. However, the essence of the random walk hypothesis is that the disturbances r, are serially uncorrelated, i.e. it is not possible to forecast future returns from past returns. Consequently the interesting testable null hypothesis is a formulation which focuses on the serial independence of the error term e, and takes heteroscedasticity and non-normality into consideration: H,: For all t, E[e,]=O,
E[e,e,._i]=O
for any i#O.
Several alternatives to the random walk model have been proposed in the light of new empirical evidence. One of the more cited, the so-called meanreversion model, was proposed by Poterba and Summers (1988). In their model the observed stock price, at any time t, is the sum of the ‘true’ stock price, following a random walk, and a stationary ‘mispricing’ or ‘noise’ term, following an AR1 process. The mispricing term causes long term stock returns to be negatively autocorrelated. In terms of the random walk model it implies that E[erel_i] ~0.
2.h. The vuriunce This methodology walk in GNP. It character of stock logarithm of the random walk, then
rutio methodology
was first used by Cochrane (1988) for a test of random has since been frequently used to examine the random returns.3 The variance ratio test uses the fact that if the stock price, including reinvested dividends, follows a the return variance should be proportional to the return
“Poterha and Summers (1988), Lo and MacKinlay al. (1991) to mention a few.
(198X), Kim et al. (1988) and Berglund
et
178
P. Frennherg
and B. Hmsson.
horizon. In this study the variance ment horizon - VR(q) - is calculated
Swedish stock prices
ratio test-statistic for a q-period in the following way:
VR(q)=Var[r(q)]/qVar[r(l)].
invest-
(2)
The expected value of VR(q) is one for any investment horizon q under the assumption of a random walk. Values below one are consistent with stock price models implying negatively autocorrelated returns, such as the meanreversion model. Values above one are consistent with positively autocorrelated returns. There exists no analytically derived distribution for jinite samples of the variance ratio. Poterba and Summers (1988) as well as Lo and MacKinlay (1989) have therefore estimated the standard error of the variance ratio using Monte-Carlo simulations. To assess the significance of test statistics without making any assumptions concerning the distribution of stock returns, Kim et al. (1988) have used randomization (or shuffling) methods. Lo and MacKinlay (1989) have derived the asymptotic distribution of finite samples for heteroscedastic disturbances. We denote this test statistic z(q). Lo and MacKinlay have also performed extensive Monte-Carlo simulations to test the performance of z(q) in finite samples under H,. Their results indicate that the asymptotic distribution of z(q) performs reasonably well in finite samples. We use z(q) as test statistic in section 3.
2.~. The methodologll
of uutoregressions
on multiperiod
returns
Another method for testing the random walk hypothesis, which has been used by Fama and French (1988), is to regress the cumulative return from t to t + q on the return from t-q to t. The estimaton equation is: r(r,r+4)=m%(q)+B(q)r(r-qr)+e(r,r+q).
(3)
Under H,, serial independence, the theoretical value of /l(q) is zero for all q. as Fuma and French (1988), the When using overlapping observations, interpretation of both the point estimates and the usual t-statistics becomes complicated due to bias in the OLS estimates as well as heavy residual autocorrelation. However, Richardson and Smith (1989) have derived the asymptotic approximation to the distribution of o(q) for the overlapping case. This distribution is used below in table 4 for the inference. Fama and French (1988) used instead a Monte-Carlo simulated distribution for p(q), but only from q = 24 up to q = 120. To assess /3(q) one can also use the following exact connection between the variance ratio and p for the theoretical q-period return distribution. The variance for the 2q-period return can always be written:
P. Frennherg
and B. Hansson,
179
Swedish stock prices
Var [r( t - 2q, t)] = Var [r( t - 2q, t - q)] + Var [r( t - q, t)] =2Var[r(q)]
+2Cov[r(t-2q,t-q),r(t-q,
t)].
(4)
Note that r(t -2q, t -q) and r(t-q, t) belong to the same theoretical distribution, but this is only asymptotically true for the sampling distribution. The following result holds:
VRGWIWq) = VarCr(WPVarCr(q)l = 1 + Cov [r(t - 2q, t-q),
r(t - q, t)l/Var [r(q)]
= 1 +/I(q)=/?(q)zOoVR(2q)z2VR(q) This result is used in the subsequent section to compare with the so-called implied beta, which is denoted:
Pimp(4) = VarCrCQ)llVar CrMl.
for all q.
(5)
the estimated
/l(q)
(6)
3. The evidence 3.a. Data We use the monthly returns of the value-weighted Swedish stock index presented in Frennberg and Hansson (1992). This index includes all stocks listed on the main list of the Stockholm stock exchange (SSE). The stock prices are the bid-prices registrated at the end of the last trading day in each month. Stock dividends are included in the return series and are assumed to be reinvested at end of each month. The index is free from so-called survivorship bias, i.e. it takes proper account of the effect when stocks are delisted due to mergers, bankruptcy or other reasons. The number of companies included in the index range from just below 5G at the beginning of 1919 to just above 130 at end of 1990. The total market value of the index was $92 billion at the end of 1990. Foreign stocks traded on the SSE (which are very few) are not included in the index. Most of the 20 largest companies, to which more than 60 percent of both trading volume and market value were concentrated in 1990, also have their stocks listed in London and New York. The series for price changes is the consumer price index calculated by the Central Bureau of Statistics for the period 1949-1990 and before that the cost-of-living index of the Swedish National Board of Health and Welfare, but somewhat modified for the period 1919-1930 [see Frennberg and
Table
Summary a~~tisticsfor
monthly
Mean
Standard deviation
Nominal
0.754
0.0446
Real
0.467
0.0446
Excess
0.295
0.0445
I
nominal return\ on the value-weighted 1YlY:l lYYO:l2.
Min. -0.316 (1932:3) -0.319 (1932:3) 0.321 (193X)
Max. 0. I77 (19X3:2) 0.173 (1921:7) 0. I78 (iY31:7)
Skewness -0.788
(0.00) ~ 0.823 ( 0.00) 0.875 (0.00)
Swedish
Excess kurtosis 5.634
(0.00) 5.754 (0.00) 5.780 (0.00)
stock
index
T* 3.09 I (
Monthly returns. R,, are defined as In P,-In P, I. where P, is the value-weighted stock index (adjusted for inflation or the risk-free rate of return) at the end of period r. including dividends payed in period r. The T*-statistic ib the test of H,,: constant variance, against HI: at least one shift m variance. proposed by Hsu (1979). Prob. values are giben in parentheses and refer to the null hypotheses of normal distribution (for the skewness and excess kurtosis statistics) and constant variance (for the T*-test).
Hansson (1992)]. The consumer price figures are all monthly averages. To measure price changes as monthly averages rather than end of period measures, as for nominal stock returns, and the possibility of straightfoward measurement errors make them a bit unsuitable for calculating the real returns of stocks for a particular month. But for longer periods, spanning several years, these kind of errors should not play any significant role. Finally, we consider the so-called excess returns. They are calculated by subtracting the risk-free rate of return from the nominal stock return. The risk-free rate is the monthly return on short term money market instruments. Unfortunately, the short term money market in Sweden was born as late as in the early 1980s. Before 1980 there existed no market determined risk-free rate in Sweden. As a substitute the official discount rate of the Swedish central bank is used. Some basic properties of the data are worth noting before proceeding with the test procedure (see table 1). Our stock return data do neither match the normal distribution assumption ~ the skewness and kurtosis statistics ~ nor the identical distribution assumption - the T*-test ~ required by the strict random walk hypothesis. Thus the null hypothesis allowing non-normality and time varying volatility seems to be the adequate. It may be noted that the strong departure from normality indicated by the skewness and kurtosis statistics is due to the presence of a few ‘abnormal’ returns. Excluding 12 out of 864 months is sufficient for the sample to pass the normality test.4
‘However, these ‘abnormal’ returns are not due to data errors. They are related to well known events: March 1932 (-27.1”,,) when lvar Kreuger shot himself in Pals. February 1983 (+ 19.4”,,) when business conditions recovered in Sweden after a huge devaluation of the krona in October 1982, the October 1987 crash (-20.5”,,), or September 1990 ( - 21.6”,,)to mention a few.
P. Frrnnhvrg
and B. Hansson,
181
Swedish stock prices
Table 2 Actual
versus
theoretically
implied holding-period variances test. real returns, 1919~1990.
Holding-period
Investment horizon, q (months)
Actual
2 3 4 6 12 24 36 4x 60 72 84 96 108 120 132 144 180 216 240 300 360 480
0.0020 0.0047 0.0073 0.0101 0.0154 0.0368 0.0765 0.1068 0.1425 0.1671 0.1860 0.2008 0.2093 0.2221 0.2327 0.2380 0.2477 0.23 I8 0.2107 0.2174 0.1785 0.2242 0.4849
variance
and
Variance
ratio test
Implied
Variance
ratio
0.0040 0.0060 0.0080 0.0120 0.0241 0.0482 0.0723 0.0963 0.1204 0.1445 0.1686 0.1927 0.2 168 0.2408 0.2649 0.2890 0.3613 0.4335 0.48 17 0.602 1 0.7225 0.9634
1.171 1.215 I.253 I.278
I .x28 1.588 1.478 1.479 1.387 1.287 1.190 I.086 I.024 0.966 0.898 0.X57 0.64 1 0.486 0.45 1 0.296 0.310 0.485
variance
ratio
Prob. value 0.00 I 0.003 0.004 0.014 0.001 0.010 0.082 0.121 0.254 0.435 0.624 0.834 0.952 0.936 0.826 0.772 0.503 0.374 0.361 0.280 0.322 0.522
The first column shows the actual holding-period variance. The second column gives the implied holding-period variance. if the return process had followed a random walk with drift. i.e. with the drift and the variance equal to the actual mean and variance of the monthly returns (i.e. Var(y),,,=qVar( 1)). The third column is the variance ratio VR(q). The prob. values in the fourth column refer to a two-sided variance ratio test. Under H, the test statistic z(q) = I(gS’nt/J (VR(q) ~
t)l/IJ?tq)l
is asymptotically distributed N[O, I] where O(q) is the estimated [see Lo and MacKinlay (1989) for further details].
3.h. Act&
holding-period
variance
for VR(q)
variances jbr real returns
In table 2 the first column shows the actual holding-period variances of the real returns for different investment horizons. If monthly (one-period) stock returns are independent then the variance for the q-month horizon is q times the variance of monthly returns, i.e. q. 0.0020. As expected, the total actual variance first increases with q, though not perfectly proportional to q, but after q = 144 (12 years) the variance falls and reaches a minimum at q= 300 (25 years). Column 2 shows the implied holding period variance under the assumption of independent returns. If stock returns where random, i.e. if H, is true, then implied and actual variance should be roughly the same. If the q-period returns where positively autocorrelated then actual 2q-
182
P. Frennherg
and B. Hansson,
Swedish stock prices
period variance should be above the implied variance, and vice versa for negatively autocorrelated returns. As can be seen in table 2 the actual variance is generally above the implied variance for holding periods up to nine years (q =96), but for longer return horizons the relation is reversed. This result indicates positive autocorrelation for short horizons and negative autocorrelation for longer horizons. 3.~7. Variance ratio tests for real returns The results of the variance ratio test for real returns are shown in the last two columns in table 2. The variance ratio increases up to q=24 months and then it decreases monotonically, with one small exception, to reach a minimum at q= 300 months (25 years). The variance ratios are significantly above one at the 95’7” level for q =2 up to q= 24. Thus, H, is rejected. Since the variance ratios are above one we have indications of positive autocorrelation. The variance ratios below one, for qz 120, are never statistically significant, However, the declining variance ratio from q=48 to q=300 is a strong indication of negative autocorrelation for large qs. Our results are similar to those reported for the U.S. data by Poterba and Summers (1988). They found positive autocorrelation for holding periods shorter than one year and negative autocorrelation for investment horizons of two years and above. They reported a minimum variance ratio for an investment horizon of seven years while ours is reached at a horizon of 25 years. However, their study only covered holding periods up to eight years. 3.d.
Autoregressions
of multiperiod
real returns
Table 3a and 3b summarize the result of the autoregressions of the multiperiod real returns. We find that /l(q) is positive for investment horizons up to twelve months, with some significant values, but the explained variance or RZ is small. For longer investment horizons there are several significant values of /l(q) and R2 is above 300/,. The z-values are calculated following Richardson and Smith’s (1989) derivation of the distribution of J(q) under He. The point estimates of Fama and French (1988), from one to ten years, show no particular similarity with our evidence. Our beta estimates show no U-shaped pattern since there is significant positive autocorrelation for short horizons. However, starting with q= 12 there is a U-shaped pattern as in the Fama and French study, but the minimum is at eleven years instead of three. It has been noted by several authors that OLS estimates of p(q) tend to be biased downwards when using overlapping observations. In table 4 we compare the estimated p(q) and the implied Pimp(q) using eq. (6). The difference is small for qs up to 96 months and then negative. This matches
P. Frennherg and B. Hanson,
183
Swedish stock prices
Table 3 Investment horizon, q (months)
Real prob. value
8(q)
Rz-adj. (%) ~____
a. Multiperiod autoregressions of real and nominal 1919~1990: r(r,r+q)=x(q)+/I(q)r(-q,t)+e(t,f+q). 1 2 3 4 5 6 I 8 9 10 11 12 b. Multiperiod 24 36 48 60 72 84 96 108 120 132 144 180 216 240
0.168 0.068 0.073 0.119 0.159 0.213 0.228 0.211 0.188 0.151 0.096 0.033 autoregressions - 0.094 -0.175 -0.324 -0.338 -0.344 - 0.465 -0.562 - 0.805 - 0.850 -0.883 -0.871 ~ 0.798 - 0.622 - 0.386
0.000 0.047 0.129 0.014 0.007 0.002 0.001 0.006 0.026 0.074 0.292 0.737
8(q) stock returns
2.1 0.3 0.4 1.2 2.2 4.2 4.9 4.2 3.3 2.1 0.8 0.0
0.8 2.9 9.7 9.4 8.3 11.3 13.2 25.3 30.7 36.6 43.6 35.0 19.6 9.8
for q
R2-adj. (%)
for q up to 12 months,
0.182 0.076 0.09 1 0.150 0.197 0.256 0.280 0.271 0.257 0.232 0.192 0.144
of real and excess returns 0.504 0.315 0.112 0.145 0.183 0.101 0.068 0.016 0.017 0.021 0.033 0.101 0.183 0.549
Nominal prob. value
0.000 0.026 0.060 0.002 0.001 0.000 0.000 0.000 0.002 0.006 0.035 0.140
up to 240 months,
- 0.043 -0.176 -0.317 -0.235 - 0.048 0.024 0.037 - 0.097 -0.134 - 0.249 -0.326 -0.344 -0.398 -0.410
0.758 0.311 0.120 0.311 0.854 0.933 0.904 0.772 0.708 0.515 0.424 0.480 0.490 0.525
3.2 0.5 0.7 2.0 3.5 6.2 7.5 7.1 6.4 5.2 3.5 1.9 1919-1990. 0.1 3.7 11.9 6.0 0.1 0.0 0.0 0.6 1.6 8.1 18.6 22.1 30.9 40.7
OLS slopes of q-period returns regressed on lirst q-period lag, using a sample of overlapping qperiod returns starting in January 1919. The prob.values are calculated using Richardson and Smith’s (1989) finding under H, is distributed
that [&n - 2q + 1)1/I(q) (n = total N[0,(2q2+ 1)/3q].
number
of one-period-returns
= 864)
the Monte-Carlo simulations by Fama and French. However, b(q) is still negative. Finally we study the implied R2 statistics in table 4, which is the square of the implied beta. For investment horizons below one year, where we have found significant positive autocorrelation, R2 is never above 5%, i.e. of little practical significance for the investor. For longer horizons R2 increases and reaches a maximum above SOo/, at a horizon of twelve years. This is a very large number, which implies possibilities of timing for long term investors [see for example Samuelson (1989)]. Campbell (1991) has calculated the
184
Implied Investment horizon (4) (months)
/j(q)
compared
to actual
Table
4.
OLS
slopes
Estimated
Implied
P(4)
PC(/) VR(Zq).VR(q)
OLS
I
0.I68
2 3 4 6 I2 24 36 48 60 72 84 96 IOX I 20 I32 I44 I80 216 240
0.068 0.073 0.1 IX 0.2 I3 0.033 ~ 0.094 PO.175 ~ 0.324 PO.338 ~ 0.344 ~ 0.466 -0.562 - 0.805 ~O.XSO ~~ o.xx2 -0.x71 -0.79x ~ 0.622 Po.3x6
Sourw.~: Actual OLS I +/l(q)= VR(2q),VR(q).
and
implied
Implied
0.16X 0.067 0.04x 0.075 0.187 0.025 m~o.095 ~ 0. I67 ~0.309 PO.356 -0.395 ~ 0.479 -0.516 PO.594 PO.608 ~ 0.649 ~0.710 0.643 0.46X .- 0.314
slopes see Table 3. The and implied R’ is the square
R’. Real
implied of /j,,,Jy).
returns.
bias
lmphed R’
(1)i2)
(“J
~ 0.0003 0.0004 0.0248 0.0434 0.0259 0.0080 0.00 I8 ~ 0.0077 ~0.0154 0.0177 0.05 IO 0.0134 ~ 0.0458 PO.21 I5 ~ 0.2422 -0.2334 PO.1606 PO.1555 PO.1542 ~ 0.0722
2.8 0.4 0.2 0.6 3.5 0. I 0.9 2.8 9.5 12.7 15.6 22.9 26.6 35.3 37.0 42. I 50.4 41.3 21.9 9.9
/j(q) comes Estimation
from period
the relation is 1919P1990.
implied R2 using the bias adjusted beta-estimates from the Fama and French study (1988). The shape of the curve is similar to the Swedish one, but the American data peak already at a horizon of four to five years and the maximum R2 is only about 7?,,.
The whole test procedure for real returns has been applied to both nominal and excess returns. The results are shown in figs. 1 and 2 and in tables 3a and 3b. Nominal returns are a good alternative to real returns when analysing shorter investment horizons. As the investment horizon gets shorter the inflation measure becomes less accurate for reasons mentioned above in the data section. It is also true that the variance of short horizon inflation is very small compared to the variance in short horizon stock returns. As can be seen the estimated fi(q)s for nominal returns are all greater than the p(q)s for the real returns. Thus, the results of positive autocorrelation for short horizon stock returns can not be due to measurement errors in the inflation measure. Positive first-order autocorrelation for short horizon nominal stock returns has also been found in the U.S. data by
Nominal
Fig. I. 0t.s estimates of /j(q) for overlapping monthly nominal, real and excess return data for the Swedish stock market, 1919P1990. Solid lines represent a two-sided 95”,, confidence interval for /j(q) under H,, according to Richardson and Smith (1989). See also Kim et al. (198X) for a compartson wtth U.S. data.
Lo and MacKinlay (1988). Using weekly data for the period 1962:9-1985:12 and the variance ratio methodology, they found evidence of positive autocorrelation for several shorter investment horizons, though the magnitude of the correlation was lower than in our study of the Swedish stock market. For investors involved in long term asset allocation strategies, excess returns may be more interesting that real returns. We have already mentioned that our figures for excess returns are marred by the fact that the risk free rate of return has only been market determined for the last ten years. The results for excess returns (see figs. 1 and 2 and table 3b) are almost identical to the results for nominal and real returns for short investment horizons. Again the reason is that the variance in the risk-free rate is negligible compared to the variance in the stock returns. For longer investment horizons the similarities with real returns are weaker. There are still signs of negative autocorrelation but the statistical significance is very low. To summarize: we conclude that the evidence of positive autocorrelation for shorter return horizons is statistically highly significant and holds for both nominal and excess returns. The dependence emerges from the stock market itself. The signs of mean-reversion in the longer horizons found in the
P. Frennherg
186
and B. Hansson, Swedish stock prices
2 I.0 -1.6 -1.4 -1.2 -3$
I-0.6 -0.6 -0.4 -0.2 --
Fig. 2. Variance ratio 1919-1990. Estimated
of nominal, real and excess return from overlapping monthly data MacKinlay (1989).
data for the Swedish stock market, using the definitions of Lo and
real return data are considerably less pronounced in the excess return data. Does this indicate that the mean reversion is in the inflation data? The answer is no: the evidence shows that the inflation data are itself highly positively autocorrelated for all horizons q.’ 3.f.
Checking
,for robustness
To check for robustness we have results are the effects of the following and/or extreme subperiods. 3:f.I.
Nonsynchronous
in particular phenomena:
looked at whether our nonsynchronous trading
trading
The most common explanation of the phenomenon of positive first order autocorrelation in stock index return data is to consider it to be a statistical artifact, caused by infrequent or nonsynchronous trading [e.g. Scholes and Williams (1977), Atchinson et al. (1987) or Lo and MacKinlay (1988)]. The idea is that small capitalization stocks trade less frequently than larger stocks, which means that new information affects large firms’ stocks before small firms’ stocks. This lag leads to a positive serial correlation in stock 5Results available
from authors.
P. Frennherg
and B. Hansson,
Swedish stock prices
187
index returns. Since the trading volume on the Swedish stock market has been rather thin for long periods, especially during World War II and the following twenty years, could this perhaps explain the high positive autocorrelation observed for the one-month horizon returns in this study? We will argue that is not the case. First, autocorrelation caused by infrequent trading was originally put forward as an explanation of positive autocorrelation in daily data. We are using monthly data. Furthermore, we can also rely on the evidence from the following model presented by Lo and MacKinlay (1988): the autocorrelation due to nonsychronous trading is determined by the average probability of non-trading for a stock during a trading day. In their model all stocks are assumed to have equal probability of non-trading and the index under consideration is assumed to be equally weighted. The first-order nonsynchronous trading induced autocorrelation coefficient, p(l), is then simply the probability of non-trading [see Lo and MacKinlay (1988)]. Assuming that a month has on average 20 trading days, the first-order monthly autocorrelation coefficient induced by the nonsynchronous trading, p”(l), is given by:
(20+38p(l)+36~(2)+34~(33+~~~+2p(19)}, where p(j) by: /J(j)
is the jth-order
= P!
autocorrelation
coefficient
(7) for daily returns
given
(8)
where p is the (equal) probability for every individual stock of non-trading during a trading day. Our estimates of p”(1) is around 0.17 (see table 3a). By using (7) we have to presume an implied probability of non-trading of nearly 90% to explain our pm(l)-estimate with nonsynchronous trading. The median non-trading probability per trading day on the Stockholm stock exchange was 0.21 for the period 197771989. This leads to an induced first-order monthly autocorrelation of only 0.01. However, as mentioned in the data section, we use a value-weighted index, which is dominated by some 2C-30 large companies. This weakens the case for thin trading effects even further. A second and more serious objection to the nonsynchronous trading hypothesis is the fact that our index is based on bid prices, not the latest transaction data. As long as bid prices are adjusted to reflect current information there will be no nonsynchronous trading effect at all in our stock index. A third argument against the nonsynchronous trading hypothesis is given by studying a moving ten-year period estimation of B(l) (see fig. 3). The
JBF
G
188
-0.2 -Lower bound
End of sample period
Fig. 3. OLS estimates of /j(I) for monthly nominal returns. estimated over a muking ten year sample period starting with 1919:2~1929:1 and ending with 1981:1-199O:lZ. Solid lines represent a two-sided 95”,, confidence interval for /j(q) under H,, according to Richardson and Smith
(19X9).
highest /j( 1) coefficient is estimated for the period 1981-1990: a period when trading volume on the Swedish stock market rose far above its historical levels. In contrast, the lowest [I( 1) coefficient was estimated for the period 194G-1949: a period with very low trading volume. These results are in conflict with the nonsynchronous trading hypothesis.
A further check for robustness is to divide the data into two subperiods. This follows the approach by Kim et al. (1988), who argue that the evidence of mean reversion for the U.S. stock market is concentrated in the pre-war period. The data have been divided over two periods of equal size, 1919-1954 and 19.55-1990. The results for the multiperiod autoregressions arc presented in table 5. The signs of positive autocorrelation for short investment horizons are of the same size in both periods. The sign of mean reversion for longer investment horizons are also similar except that there is not such a distinct picture of mean reversion for the later period. The same procedure has also been performed with the variance ratio test with very similar results.” As a ‘Results
are available
on requests from the authors
P. Frennherg
and B. Hansson,
Table Multiperiod Investment horizon, q (months) 1
2 3 4 6 12 24 36 48 60 72 84 96 108 120 OLS
195412
Subperiod P(Y) 0.137 0.027 0.000 0.030 0.252 0.191 ~ 0.087 - 0.447 - 0.767 -0.722 ~ 0.497 -0.259 -0.237 -0.321 -0.505
autoregressions
1919-1954 Subperiod
R’-adj. (“A)
Prob. value 0.004 0.570 0.996 0.658 0.010 0.173 0.67 1 0.083 0.013 0.044 0.223 0.573 0.646 0.578 0.434
5
of real returns
1 (1919-1954)
1.7 0.0 0.0 0.0 6.0 3.3 0.5 20.7 61.5 59.4 36.2 12.4 15.4 19.0 27.7
189
Swedish stock prices
and
1955-1990.
2 (1955-1990)
B(Y) 0.193 0.099 0.131 0.192 0.184 -0.096 -0.088 0.131 0.120 0.047 -0.258 -0.780 - I.194 - 1.693 - 1.836
Prob.
R’-adj.
VdW
("0)
0.000 0.040 0.056 0.005 0.059 0.494 0.666 0.613 0.697 0.895 0.527 0.090 0.02 I 0.003 0.004
3.5 0.7 1.3 3.0 2.9 0.6 0.5 1.3 1.0 0.0 2.7 15.1 26.1 49.5 61.4
slopes of y-period real returns regressed on lirst q-period lag for two subsamples: 1919:1-and 1955:1~1990:12. The prob.values are calculated using Richardson and Smith’s (1989)
finding that [,/(rt - 2q + I )]/Qq) distributed N[0.(2$+ 1)/3q].
(n = total
number
of one-period-returns
=432)
under
H,
is
further check of the robustness of the results for different time periods we can again use the ten year moving estimates of p(1). As can be noted in fig. 3 [I(l) is definitely varying over time. It is positive for most of the sample periods and significantly different from zero for roughly half of the overlapping ten-year periods. These significant values are estimated for such different time periods as the 1930s the 1950s the 1960s and the 1980s. Thus, the phenomenon of positive autocorrelation for short horizon returns is not concentrated to any particular time period with very specific conditions, like the 1930s. To sum up: it seems that the basic findings - rejection of a random walk hypothesis in favour of a hypothesis suggesting positive autocorrelation for short return horizons and negative for long horizons - are reasonably robust.
4. Summary and concluding notes The results in this study suggest that Swedish stock returns are positively autocorrelated over short return horizons, i.e. one to twelve months, and negatively autocorrelated over longer return horizons, i.e. five years or more. Thus, the random walk hypothesis, interpreted as serial independence, is rejected. The basis for our conclusion are two common tests performed on a newly constructed set of monthly value-weighted Swedish stock return data
190
P. Frennherg
and B. Hansson,
Swedish stock prices
for the period 1919-1990. Our results are in line with recent research on the U.S. stock market. The results of positive short horizon autocorrelation holds for real as well as for nominal and excess return data. We have found that our results can not be explained by the nonsynchronous trading hypothesis, and the choice of sample period was only of marginal importance. Predictable short term returns could be a violation of the efficient market hypothesis [e.g. Fama (1970)]. However the low R2 statistics, less than five percent, indicates that the possibilities for economic profits are very limited. A more important implication of the short term autocorrelation is that estimation of stock volatility, a necessary input in option pricing, using the assumption of serial independence may lead to an underestimation of the true volatility. The conclusion that stock returns over longer periods show negative autocorrelation, also known as mean reversion, are less robust. For the period 1919-1990 both variance ratio tests and OLS estimates suggest negative autocorrelation, but the significance levels are low, especially for the low-powered variance ratio test. Divided into subperiods the negative autocorrelation for horizons from three to six years only show up in the early sample. However, for very long return horizons, ten years or more, and for real returns, the evidence seems to be in favour of a mean-reversion hypothesis. The presence of mean reversion in long period stock returns implies that the portfolio choice is not invariant to the investment horizon. Stocks are relatively less risky for long investment horizons and therefore more attractive to long term investors than for short term investors.’ Thus, the rule of thumb that investors with short investment horizon should have a smaller proportion of stocks than investors with long investment horizons, given the same level of risk aversion, might have an empirical explanation. ‘See for example
Samuelson
( 1989). Lofthouse
(1990) and Butler and Domian
(1991)
References Atchison, M.D., K.C. Butler and R.R. Simonds, 1987, Nonsynchronous security trading and market index autocorrelation. Journal of Finance 42, Ill&l 18. Berglund, T., K. Hedvall and E. Liljeblom, 1991. Predicting volatility of stock indexes for option pricing on a small security market, Working paper no. 34 (Stockholm School of Economics). Butler, K.C. and D.L. Domian, 1991, Risk, diversification and the investment horizon, Journal of Portfolio Management. Spring, 41-47. Campbell. J.Y.. 1991. A variance decomposition for stock returns, The Economic Journal 101, 1577179. Cochrane, J.Y., 1988, How bid is the random walk in GNP?. Journal of Political Economy 96, 8933920. Fama, E.F., 1970, Eflicient capital markets: A review of theory and empirical work. Journal of Finance 25, 3833417. Fama, E.F. and K.R. French, 1988, Permanent and temporary components of stock prices, Journal of Political Economy 96, 246273.
P. Frennberg nnd B. Hnnsson, Swedish stock prices
191
Frennberg, P. and B. Hansson, 1992, Computation of a monthly index for Swedish stock returns: 1919-1989, Scandinavian Economic History Review, forthcoming. Hsu, D.A., 1979. Detecting shifts of parameter in gamma sequences with applications to stock price and air traflic flow analysis, Journal of American Statistical Association 74, 31-40. Hsu, D.A., 1984, The behaviour of stock returns: Is it stationary or evolutionary?, Journal of Financial and Quantitative Analysis 19, 1 l-28. Jennergren, P. and P. Korsvold, 1974, Price formation in the Norwegian and Swedish stock markets Some random walk tests, Swedish Journal of Economics 76, 171~185. Kim, M.J.K. and C.R. Nelson and R. Startz. 1988, Mean reversion in stock prices‘? A reappraisal of the empirical evidence, National Bureau of Economic Research, Working Paper no. 2795. Lo. A.W. and A.C. MacKinlay, 1988, Stock market prices do not follow random walks: Evidence from a simple specification test, The Review of Financial Studies 1, 41-66. Lo, A.W. and A.C. MacKinlay, 1989, The size and power of the variance ratio test in finite samples, Journal of Econometrics 40, 2033238. Lofthouse, S., 1990, Sovereign portfolio management, Journal of International Security Markets, Summer, 133-137. Poterba, J.M. and L.H. Summers, 1988, Mean reversion in stock prices. Evidence and implications. Journal of Financial Economics 25, 3233348. Richardson, M. and T. Smith, 1989, Tests of financial models in the presence of overlapping observations, Review of Financial Studies 2. Richardson, M. and J.H. Stock, 1989, Drawing inference from statistics based on multiyear asset returns, Journal of Financial Economics 25, 3233348. Samuelson, P., 1989, The judgement of economic science on rational portfolio management: Indexing, timing, and long-horizon effects, The Journal of Portfolio Management 16, Fall, 412. Scholes, M. and J. Williams, 1977, Estimating betas from nonsynchronous data, Journal of Financial Economics 5. 309-328. Schwert, G.W., 1989, Why does stock market volatility change over time?, Journal of Finance 44, 1115~1153. Schwert, G.W., 1990, Indexes of U.S. stock prices from 1802 to 1987, Journal of Business 63, 399-426.
of Banking
and Finance
17 (1993)
175~191. North-Holland
Testing the random walk hypothesis Swedish stock prices: 19 19-l 990 Per Frennberg Department
and Bjiirn Hansson*
of Economics
Received October
on
and Institute of Economic
1991. final version
received
March
Research,
University
of Lund, Lund, Sweden
1992
This paper tests the random walk hypothesis on a new set of monthly data for the Swedish stock market, 1919-1990. We use both the variance ratio test and the test of autoregressions of multiperiod returns. Our results suggest that Swedish stock prices have not followed a random walk in the past 72 years. For short investment horizons, one to twelve months, we find strong evidence of positively autocorrelated returns. For longer horizons, two years or more, we find indications of negative autocorrelation, so called ‘mean reversion’. Our results are in line with recent research on the U.S. stock market and may have several implications for the practical investor. They point in particular towards a larger proportion of stocks in the portfolio for long term investors.
1. Introduction Several studies have reported empirical evidence that stock returns, contrary to the random walk hypothesis, contain relatively large predictable components [e.g. Fama and French (1988), Poterba and Summers (1988) and Lo and MacKinlay (1988 and 1989)]. The evidence found suggests that nominal and real stock returns up to one year may be positively autocorrelated while real stock returns for longer investment horizons may be negatively autocorrelated. But the results are still controversial. On the one hand, they could be interpreted as a violation of the efficient market hypothesis, though their implications for portfolio choice and option pricing are more interesting. On the other hand, the statistical significance of the results has been questioned: Kim et al. (1988) argue that the mean-reversion phenomenon is concentrated in the pre-war period and that it is also very sensitive to the choice of market index and the type of return measure.
Correspondence to: Dr. Bjiirn Hansson, Department of Economics, University of Lund, Box 7082, 220 07 Lund, Sweden. *We like to thank two anonymous referees for constructive comments. We are also grateful to seminar participants at the Swedish School of Economics and Business Administration in Helsinki and at the Department of Economics at the University of Lund. Support from the Institute of Economic Research at the University of Lund is gratefully acknowledged.
03784266/93/$06.00
&‘J
1993-Elsevier
Science Publishers
B.V. All rights reserved
176
P. Fwnnher~
nnd B. Hansson.
Swedish .s~ock prices
Richardson and Stock (1989) show that the results provided by Fama and French (1988) are less significant than reported. Unfortunately, most previous research has focused on the U.S. stock market.’ This is probably due to the fact that the issue of autocorrelated stock returns, especially multiperiod returns, requires long time series of high quality data. The purpose of this paper is to test the random walk hypothesis for the Swedish stock market using a new set of data for the period 1919P1990.2 Our data are the monthly returns, including reinvested dividends, on a valueweighted market index of the Swedish stock market, the so called ‘Affarsvarldens Generalindex’. To make our results comparable with those reported earlier for the U.S. data, we use two standard techniques: the variance ratio and multiperiod autocorrelated tests. The analysis considers investment horizons from one to forty years. Following the American studies we focus on the inflation adjusted returns, but nominal and excess returns are also examined. Real returns are believed to be the most relevant variable for an in particular for investment horizons spanning several years. investor, However, due to the quality of the consumer price data for short periods we analyse nominal returns for investment horizons below one year. Another measure of interest is excess returns, i.e. the difference between the return on stocks and the return on a risk-free short term security, since many dynamic asset allocation strategies focus on the return differential between stocks and short term money market instruments. Unfortunately, the short term money market in Sweden has only existed since the early 1980s. Therefore, the results of the excess return analysis are less reliable than the ones from the real return analysis. The paper is organized in the following way. Section 2 gives a brief theoretical background to the question under consideration and a presentation of the testing methodologies used in the paper. Section 3 provides empirical results as well as tests for robustness. Section 4 is the conclusion.
2. Theory and methodology In this section we first recapitulate the random walk model and the testable null hypothesis derived from it. Then follows a presentation of the two testing methodologies: the variance ratio test and the test of autoregressions on multiperiod returns.
‘Poterba and Summers (198X) examined a number of non U.S. stock markets. among them the Swedish one, using the IFS data from the IMF. However, these data cover only the period 1956-1986 and do not include dividends. ‘In an early study of the Swedish and the Norwegian stock market Jennergren and Korsvold (1974) studied daily prices of individual stocks for the period 1967-1971.
P. I‘rennhrrg
2.~.
The random
and B. ffansson.
Swedish
stock
177
prices
walk hypothesis
Let P, denote the stock price at time t and define p1-In process given by the recursive relation: pt=p+pr_,
P, as a stochastic
+e,~r(t-ll,t)-p,-pp,~,=C(+r,,
(1)
where p is a constant drift parameter, e, is the randon disturbance and r(t - 1, t) is the actual return for the period t - 1 to t. The one-period return is denoted r( 1) and the q-period return, which is the sum of q one-period returns, is denoted r(q). The e,s are assumed to be strict ‘white noise’, that is to say independently and identically distributed (i.i.d.) and having a normal distribution. Since there is strong evidence of time-varying volatilities and deviation from normality [e.g. Hsu (1984) and Schwert (1989 and 1990)] the strict random walk hypothesis can easily be rejected. However, the essence of the random walk hypothesis is that the disturbances r, are serially uncorrelated, i.e. it is not possible to forecast future returns from past returns. Consequently the interesting testable null hypothesis is a formulation which focuses on the serial independence of the error term e, and takes heteroscedasticity and non-normality into consideration: H,: For all t, E[e,]=O,
E[e,e,._i]=O
for any i#O.
Several alternatives to the random walk model have been proposed in the light of new empirical evidence. One of the more cited, the so-called meanreversion model, was proposed by Poterba and Summers (1988). In their model the observed stock price, at any time t, is the sum of the ‘true’ stock price, following a random walk, and a stationary ‘mispricing’ or ‘noise’ term, following an AR1 process. The mispricing term causes long term stock returns to be negatively autocorrelated. In terms of the random walk model it implies that E[erel_i] ~0.
2.h. The vuriunce This methodology walk in GNP. It character of stock logarithm of the random walk, then
rutio methodology
was first used by Cochrane (1988) for a test of random has since been frequently used to examine the random returns.3 The variance ratio test uses the fact that if the stock price, including reinvested dividends, follows a the return variance should be proportional to the return
“Poterha and Summers (1988), Lo and MacKinlay al. (1991) to mention a few.
(198X), Kim et al. (1988) and Berglund
et
178
P. Frennherg
and B. Hmsson.
horizon. In this study the variance ment horizon - VR(q) - is calculated
Swedish stock prices
ratio test-statistic for a q-period in the following way:
VR(q)=Var[r(q)]/qVar[r(l)].
invest-
(2)
The expected value of VR(q) is one for any investment horizon q under the assumption of a random walk. Values below one are consistent with stock price models implying negatively autocorrelated returns, such as the meanreversion model. Values above one are consistent with positively autocorrelated returns. There exists no analytically derived distribution for jinite samples of the variance ratio. Poterba and Summers (1988) as well as Lo and MacKinlay (1989) have therefore estimated the standard error of the variance ratio using Monte-Carlo simulations. To assess the significance of test statistics without making any assumptions concerning the distribution of stock returns, Kim et al. (1988) have used randomization (or shuffling) methods. Lo and MacKinlay (1989) have derived the asymptotic distribution of finite samples for heteroscedastic disturbances. We denote this test statistic z(q). Lo and MacKinlay have also performed extensive Monte-Carlo simulations to test the performance of z(q) in finite samples under H,. Their results indicate that the asymptotic distribution of z(q) performs reasonably well in finite samples. We use z(q) as test statistic in section 3.
2.~. The methodologll
of uutoregressions
on multiperiod
returns
Another method for testing the random walk hypothesis, which has been used by Fama and French (1988), is to regress the cumulative return from t to t + q on the return from t-q to t. The estimaton equation is: r(r,r+4)=m%(q)+B(q)r(r-qr)+e(r,r+q).
(3)
Under H,, serial independence, the theoretical value of /l(q) is zero for all q. as Fuma and French (1988), the When using overlapping observations, interpretation of both the point estimates and the usual t-statistics becomes complicated due to bias in the OLS estimates as well as heavy residual autocorrelation. However, Richardson and Smith (1989) have derived the asymptotic approximation to the distribution of o(q) for the overlapping case. This distribution is used below in table 4 for the inference. Fama and French (1988) used instead a Monte-Carlo simulated distribution for p(q), but only from q = 24 up to q = 120. To assess /3(q) one can also use the following exact connection between the variance ratio and p for the theoretical q-period return distribution. The variance for the 2q-period return can always be written:
P. Frennherg
and B. Hansson,
179
Swedish stock prices
Var [r( t - 2q, t)] = Var [r( t - 2q, t - q)] + Var [r( t - q, t)] =2Var[r(q)]
+2Cov[r(t-2q,t-q),r(t-q,
t)].
(4)
Note that r(t -2q, t -q) and r(t-q, t) belong to the same theoretical distribution, but this is only asymptotically true for the sampling distribution. The following result holds:
VRGWIWq) = VarCr(WPVarCr(q)l = 1 + Cov [r(t - 2q, t-q),
r(t - q, t)l/Var [r(q)]
= 1 +/I(q)=/?(q)zOoVR(2q)z2VR(q) This result is used in the subsequent section to compare with the so-called implied beta, which is denoted:
Pimp(4) = VarCrCQ)llVar CrMl.
for all q.
(5)
the estimated
/l(q)
(6)
3. The evidence 3.a. Data We use the monthly returns of the value-weighted Swedish stock index presented in Frennberg and Hansson (1992). This index includes all stocks listed on the main list of the Stockholm stock exchange (SSE). The stock prices are the bid-prices registrated at the end of the last trading day in each month. Stock dividends are included in the return series and are assumed to be reinvested at end of each month. The index is free from so-called survivorship bias, i.e. it takes proper account of the effect when stocks are delisted due to mergers, bankruptcy or other reasons. The number of companies included in the index range from just below 5G at the beginning of 1919 to just above 130 at end of 1990. The total market value of the index was $92 billion at the end of 1990. Foreign stocks traded on the SSE (which are very few) are not included in the index. Most of the 20 largest companies, to which more than 60 percent of both trading volume and market value were concentrated in 1990, also have their stocks listed in London and New York. The series for price changes is the consumer price index calculated by the Central Bureau of Statistics for the period 1949-1990 and before that the cost-of-living index of the Swedish National Board of Health and Welfare, but somewhat modified for the period 1919-1930 [see Frennberg and
Table
Summary a~~tisticsfor
monthly
Mean
Standard deviation
Nominal
0.754
0.0446
Real
0.467
0.0446
Excess
0.295
0.0445
I
nominal return\ on the value-weighted 1YlY:l lYYO:l2.
Min. -0.316 (1932:3) -0.319 (1932:3) 0.321 (193X)
Max. 0. I77 (19X3:2) 0.173 (1921:7) 0. I78 (iY31:7)
Skewness -0.788
(0.00) ~ 0.823 ( 0.00) 0.875 (0.00)
Swedish
Excess kurtosis 5.634
(0.00) 5.754 (0.00) 5.780 (0.00)
stock
index
T* 3.09 I (
Monthly returns. R,, are defined as In P,-In P, I. where P, is the value-weighted stock index (adjusted for inflation or the risk-free rate of return) at the end of period r. including dividends payed in period r. The T*-statistic ib the test of H,,: constant variance, against HI: at least one shift m variance. proposed by Hsu (1979). Prob. values are giben in parentheses and refer to the null hypotheses of normal distribution (for the skewness and excess kurtosis statistics) and constant variance (for the T*-test).
Hansson (1992)]. The consumer price figures are all monthly averages. To measure price changes as monthly averages rather than end of period measures, as for nominal stock returns, and the possibility of straightfoward measurement errors make them a bit unsuitable for calculating the real returns of stocks for a particular month. But for longer periods, spanning several years, these kind of errors should not play any significant role. Finally, we consider the so-called excess returns. They are calculated by subtracting the risk-free rate of return from the nominal stock return. The risk-free rate is the monthly return on short term money market instruments. Unfortunately, the short term money market in Sweden was born as late as in the early 1980s. Before 1980 there existed no market determined risk-free rate in Sweden. As a substitute the official discount rate of the Swedish central bank is used. Some basic properties of the data are worth noting before proceeding with the test procedure (see table 1). Our stock return data do neither match the normal distribution assumption ~ the skewness and kurtosis statistics ~ nor the identical distribution assumption - the T*-test ~ required by the strict random walk hypothesis. Thus the null hypothesis allowing non-normality and time varying volatility seems to be the adequate. It may be noted that the strong departure from normality indicated by the skewness and kurtosis statistics is due to the presence of a few ‘abnormal’ returns. Excluding 12 out of 864 months is sufficient for the sample to pass the normality test.4
‘However, these ‘abnormal’ returns are not due to data errors. They are related to well known events: March 1932 (-27.1”,,) when lvar Kreuger shot himself in Pals. February 1983 (+ 19.4”,,) when business conditions recovered in Sweden after a huge devaluation of the krona in October 1982, the October 1987 crash (-20.5”,,), or September 1990 ( - 21.6”,,)to mention a few.
P. Frrnnhvrg
and B. Hansson,
181
Swedish stock prices
Table 2 Actual
versus
theoretically
implied holding-period variances test. real returns, 1919~1990.
Holding-period
Investment horizon, q (months)
Actual
2 3 4 6 12 24 36 4x 60 72 84 96 108 120 132 144 180 216 240 300 360 480
0.0020 0.0047 0.0073 0.0101 0.0154 0.0368 0.0765 0.1068 0.1425 0.1671 0.1860 0.2008 0.2093 0.2221 0.2327 0.2380 0.2477 0.23 I8 0.2107 0.2174 0.1785 0.2242 0.4849
variance
and
Variance
ratio test
Implied
Variance
ratio
0.0040 0.0060 0.0080 0.0120 0.0241 0.0482 0.0723 0.0963 0.1204 0.1445 0.1686 0.1927 0.2 168 0.2408 0.2649 0.2890 0.3613 0.4335 0.48 17 0.602 1 0.7225 0.9634
1.171 1.215 I.253 I.278
I .x28 1.588 1.478 1.479 1.387 1.287 1.190 I.086 I.024 0.966 0.898 0.X57 0.64 1 0.486 0.45 1 0.296 0.310 0.485
variance
ratio
Prob. value 0.00 I 0.003 0.004 0.014 0.001 0.010 0.082 0.121 0.254 0.435 0.624 0.834 0.952 0.936 0.826 0.772 0.503 0.374 0.361 0.280 0.322 0.522
The first column shows the actual holding-period variance. The second column gives the implied holding-period variance. if the return process had followed a random walk with drift. i.e. with the drift and the variance equal to the actual mean and variance of the monthly returns (i.e. Var(y),,,=qVar( 1)). The third column is the variance ratio VR(q). The prob. values in the fourth column refer to a two-sided variance ratio test. Under H, the test statistic z(q) = I(gS’nt/J (VR(q) ~
t)l/IJ?tq)l
is asymptotically distributed N[O, I] where O(q) is the estimated [see Lo and MacKinlay (1989) for further details].
3.h. Act&
holding-period
variance
for VR(q)
variances jbr real returns
In table 2 the first column shows the actual holding-period variances of the real returns for different investment horizons. If monthly (one-period) stock returns are independent then the variance for the q-month horizon is q times the variance of monthly returns, i.e. q. 0.0020. As expected, the total actual variance first increases with q, though not perfectly proportional to q, but after q = 144 (12 years) the variance falls and reaches a minimum at q= 300 (25 years). Column 2 shows the implied holding period variance under the assumption of independent returns. If stock returns where random, i.e. if H, is true, then implied and actual variance should be roughly the same. If the q-period returns where positively autocorrelated then actual 2q-
182
P. Frennherg
and B. Hansson,
Swedish stock prices
period variance should be above the implied variance, and vice versa for negatively autocorrelated returns. As can be seen in table 2 the actual variance is generally above the implied variance for holding periods up to nine years (q =96), but for longer return horizons the relation is reversed. This result indicates positive autocorrelation for short horizons and negative autocorrelation for longer horizons. 3.~7. Variance ratio tests for real returns The results of the variance ratio test for real returns are shown in the last two columns in table 2. The variance ratio increases up to q=24 months and then it decreases monotonically, with one small exception, to reach a minimum at q= 300 months (25 years). The variance ratios are significantly above one at the 95’7” level for q =2 up to q= 24. Thus, H, is rejected. Since the variance ratios are above one we have indications of positive autocorrelation. The variance ratios below one, for qz 120, are never statistically significant, However, the declining variance ratio from q=48 to q=300 is a strong indication of negative autocorrelation for large qs. Our results are similar to those reported for the U.S. data by Poterba and Summers (1988). They found positive autocorrelation for holding periods shorter than one year and negative autocorrelation for investment horizons of two years and above. They reported a minimum variance ratio for an investment horizon of seven years while ours is reached at a horizon of 25 years. However, their study only covered holding periods up to eight years. 3.d.
Autoregressions
of multiperiod
real returns
Table 3a and 3b summarize the result of the autoregressions of the multiperiod real returns. We find that /l(q) is positive for investment horizons up to twelve months, with some significant values, but the explained variance or RZ is small. For longer investment horizons there are several significant values of /l(q) and R2 is above 300/,. The z-values are calculated following Richardson and Smith’s (1989) derivation of the distribution of J(q) under He. The point estimates of Fama and French (1988), from one to ten years, show no particular similarity with our evidence. Our beta estimates show no U-shaped pattern since there is significant positive autocorrelation for short horizons. However, starting with q= 12 there is a U-shaped pattern as in the Fama and French study, but the minimum is at eleven years instead of three. It has been noted by several authors that OLS estimates of p(q) tend to be biased downwards when using overlapping observations. In table 4 we compare the estimated p(q) and the implied Pimp(q) using eq. (6). The difference is small for qs up to 96 months and then negative. This matches
P. Frennherg and B. Hanson,
183
Swedish stock prices
Table 3 Investment horizon, q (months)
Real prob. value
8(q)
Rz-adj. (%) ~____
a. Multiperiod autoregressions of real and nominal 1919~1990: r(r,r+q)=x(q)+/I(q)r(-q,t)+e(t,f+q). 1 2 3 4 5 6 I 8 9 10 11 12 b. Multiperiod 24 36 48 60 72 84 96 108 120 132 144 180 216 240
0.168 0.068 0.073 0.119 0.159 0.213 0.228 0.211 0.188 0.151 0.096 0.033 autoregressions - 0.094 -0.175 -0.324 -0.338 -0.344 - 0.465 -0.562 - 0.805 - 0.850 -0.883 -0.871 ~ 0.798 - 0.622 - 0.386
0.000 0.047 0.129 0.014 0.007 0.002 0.001 0.006 0.026 0.074 0.292 0.737
8(q) stock returns
2.1 0.3 0.4 1.2 2.2 4.2 4.9 4.2 3.3 2.1 0.8 0.0
0.8 2.9 9.7 9.4 8.3 11.3 13.2 25.3 30.7 36.6 43.6 35.0 19.6 9.8
for q
R2-adj. (%)
for q up to 12 months,
0.182 0.076 0.09 1 0.150 0.197 0.256 0.280 0.271 0.257 0.232 0.192 0.144
of real and excess returns 0.504 0.315 0.112 0.145 0.183 0.101 0.068 0.016 0.017 0.021 0.033 0.101 0.183 0.549
Nominal prob. value
0.000 0.026 0.060 0.002 0.001 0.000 0.000 0.000 0.002 0.006 0.035 0.140
up to 240 months,
- 0.043 -0.176 -0.317 -0.235 - 0.048 0.024 0.037 - 0.097 -0.134 - 0.249 -0.326 -0.344 -0.398 -0.410
0.758 0.311 0.120 0.311 0.854 0.933 0.904 0.772 0.708 0.515 0.424 0.480 0.490 0.525
3.2 0.5 0.7 2.0 3.5 6.2 7.5 7.1 6.4 5.2 3.5 1.9 1919-1990. 0.1 3.7 11.9 6.0 0.1 0.0 0.0 0.6 1.6 8.1 18.6 22.1 30.9 40.7
OLS slopes of q-period returns regressed on lirst q-period lag, using a sample of overlapping qperiod returns starting in January 1919. The prob.values are calculated using Richardson and Smith’s (1989) finding under H, is distributed
that [&n - 2q + 1)1/I(q) (n = total N[0,(2q2+ 1)/3q].
number
of one-period-returns
= 864)
the Monte-Carlo simulations by Fama and French. However, b(q) is still negative. Finally we study the implied R2 statistics in table 4, which is the square of the implied beta. For investment horizons below one year, where we have found significant positive autocorrelation, R2 is never above 5%, i.e. of little practical significance for the investor. For longer horizons R2 increases and reaches a maximum above SOo/, at a horizon of twelve years. This is a very large number, which implies possibilities of timing for long term investors [see for example Samuelson (1989)]. Campbell (1991) has calculated the
184
Implied Investment horizon (4) (months)
/j(q)
compared
to actual
Table
4.
OLS
slopes
Estimated
Implied
P(4)
PC(/) VR(Zq).VR(q)
OLS
I
0.I68
2 3 4 6 I2 24 36 48 60 72 84 96 IOX I 20 I32 I44 I80 216 240
0.068 0.073 0.1 IX 0.2 I3 0.033 ~ 0.094 PO.175 ~ 0.324 PO.338 ~ 0.344 ~ 0.466 -0.562 - 0.805 ~O.XSO ~~ o.xx2 -0.x71 -0.79x ~ 0.622 Po.3x6
Sourw.~: Actual OLS I +/l(q)= VR(2q),VR(q).
and
implied
Implied
0.16X 0.067 0.04x 0.075 0.187 0.025 m~o.095 ~ 0. I67 ~0.309 PO.356 -0.395 ~ 0.479 -0.516 PO.594 PO.608 ~ 0.649 ~0.710 0.643 0.46X .- 0.314
slopes see Table 3. The and implied R’ is the square
R’. Real
implied of /j,,,Jy).
returns.
bias
lmphed R’
(1)i2)
(“J
~ 0.0003 0.0004 0.0248 0.0434 0.0259 0.0080 0.00 I8 ~ 0.0077 ~0.0154 0.0177 0.05 IO 0.0134 ~ 0.0458 PO.21 I5 ~ 0.2422 -0.2334 PO.1606 PO.1555 PO.1542 ~ 0.0722
2.8 0.4 0.2 0.6 3.5 0. I 0.9 2.8 9.5 12.7 15.6 22.9 26.6 35.3 37.0 42. I 50.4 41.3 21.9 9.9
/j(q) comes Estimation
from period
the relation is 1919P1990.
implied R2 using the bias adjusted beta-estimates from the Fama and French study (1988). The shape of the curve is similar to the Swedish one, but the American data peak already at a horizon of four to five years and the maximum R2 is only about 7?,,.
The whole test procedure for real returns has been applied to both nominal and excess returns. The results are shown in figs. 1 and 2 and in tables 3a and 3b. Nominal returns are a good alternative to real returns when analysing shorter investment horizons. As the investment horizon gets shorter the inflation measure becomes less accurate for reasons mentioned above in the data section. It is also true that the variance of short horizon inflation is very small compared to the variance in short horizon stock returns. As can be seen the estimated fi(q)s for nominal returns are all greater than the p(q)s for the real returns. Thus, the results of positive autocorrelation for short horizon stock returns can not be due to measurement errors in the inflation measure. Positive first-order autocorrelation for short horizon nominal stock returns has also been found in the U.S. data by
Nominal
Fig. I. 0t.s estimates of /j(q) for overlapping monthly nominal, real and excess return data for the Swedish stock market, 1919P1990. Solid lines represent a two-sided 95”,, confidence interval for /j(q) under H,, according to Richardson and Smith (1989). See also Kim et al. (198X) for a compartson wtth U.S. data.
Lo and MacKinlay (1988). Using weekly data for the period 1962:9-1985:12 and the variance ratio methodology, they found evidence of positive autocorrelation for several shorter investment horizons, though the magnitude of the correlation was lower than in our study of the Swedish stock market. For investors involved in long term asset allocation strategies, excess returns may be more interesting that real returns. We have already mentioned that our figures for excess returns are marred by the fact that the risk free rate of return has only been market determined for the last ten years. The results for excess returns (see figs. 1 and 2 and table 3b) are almost identical to the results for nominal and real returns for short investment horizons. Again the reason is that the variance in the risk-free rate is negligible compared to the variance in the stock returns. For longer investment horizons the similarities with real returns are weaker. There are still signs of negative autocorrelation but the statistical significance is very low. To summarize: we conclude that the evidence of positive autocorrelation for shorter return horizons is statistically highly significant and holds for both nominal and excess returns. The dependence emerges from the stock market itself. The signs of mean-reversion in the longer horizons found in the
P. Frennherg
186
and B. Hansson, Swedish stock prices
2 I.0 -1.6 -1.4 -1.2 -3$
I-0.6 -0.6 -0.4 -0.2 --
Fig. 2. Variance ratio 1919-1990. Estimated
of nominal, real and excess return from overlapping monthly data MacKinlay (1989).
data for the Swedish stock market, using the definitions of Lo and
real return data are considerably less pronounced in the excess return data. Does this indicate that the mean reversion is in the inflation data? The answer is no: the evidence shows that the inflation data are itself highly positively autocorrelated for all horizons q.’ 3.f.
Checking
,for robustness
To check for robustness we have results are the effects of the following and/or extreme subperiods. 3:f.I.
Nonsynchronous
in particular phenomena:
looked at whether our nonsynchronous trading
trading
The most common explanation of the phenomenon of positive first order autocorrelation in stock index return data is to consider it to be a statistical artifact, caused by infrequent or nonsynchronous trading [e.g. Scholes and Williams (1977), Atchinson et al. (1987) or Lo and MacKinlay (1988)]. The idea is that small capitalization stocks trade less frequently than larger stocks, which means that new information affects large firms’ stocks before small firms’ stocks. This lag leads to a positive serial correlation in stock 5Results available
from authors.
P. Frennherg
and B. Hansson,
Swedish stock prices
187
index returns. Since the trading volume on the Swedish stock market has been rather thin for long periods, especially during World War II and the following twenty years, could this perhaps explain the high positive autocorrelation observed for the one-month horizon returns in this study? We will argue that is not the case. First, autocorrelation caused by infrequent trading was originally put forward as an explanation of positive autocorrelation in daily data. We are using monthly data. Furthermore, we can also rely on the evidence from the following model presented by Lo and MacKinlay (1988): the autocorrelation due to nonsychronous trading is determined by the average probability of non-trading for a stock during a trading day. In their model all stocks are assumed to have equal probability of non-trading and the index under consideration is assumed to be equally weighted. The first-order nonsynchronous trading induced autocorrelation coefficient, p(l), is then simply the probability of non-trading [see Lo and MacKinlay (1988)]. Assuming that a month has on average 20 trading days, the first-order monthly autocorrelation coefficient induced by the nonsynchronous trading, p”(l), is given by:
(20+38p(l)+36~(2)+34~(33+~~~+2p(19)}, where p(j) by: /J(j)
is the jth-order
= P!
autocorrelation
coefficient
(7) for daily returns
given
(8)
where p is the (equal) probability for every individual stock of non-trading during a trading day. Our estimates of p”(1) is around 0.17 (see table 3a). By using (7) we have to presume an implied probability of non-trading of nearly 90% to explain our pm(l)-estimate with nonsynchronous trading. The median non-trading probability per trading day on the Stockholm stock exchange was 0.21 for the period 197771989. This leads to an induced first-order monthly autocorrelation of only 0.01. However, as mentioned in the data section, we use a value-weighted index, which is dominated by some 2C-30 large companies. This weakens the case for thin trading effects even further. A second and more serious objection to the nonsynchronous trading hypothesis is the fact that our index is based on bid prices, not the latest transaction data. As long as bid prices are adjusted to reflect current information there will be no nonsynchronous trading effect at all in our stock index. A third argument against the nonsynchronous trading hypothesis is given by studying a moving ten-year period estimation of B(l) (see fig. 3). The
JBF
G
188
-0.2 -Lower bound
End of sample period
Fig. 3. OLS estimates of /j(I) for monthly nominal returns. estimated over a muking ten year sample period starting with 1919:2~1929:1 and ending with 1981:1-199O:lZ. Solid lines represent a two-sided 95”,, confidence interval for /j(q) under H,, according to Richardson and Smith
(19X9).
highest /j( 1) coefficient is estimated for the period 1981-1990: a period when trading volume on the Swedish stock market rose far above its historical levels. In contrast, the lowest [I( 1) coefficient was estimated for the period 194G-1949: a period with very low trading volume. These results are in conflict with the nonsynchronous trading hypothesis.
A further check for robustness is to divide the data into two subperiods. This follows the approach by Kim et al. (1988), who argue that the evidence of mean reversion for the U.S. stock market is concentrated in the pre-war period. The data have been divided over two periods of equal size, 1919-1954 and 19.55-1990. The results for the multiperiod autoregressions arc presented in table 5. The signs of positive autocorrelation for short investment horizons are of the same size in both periods. The sign of mean reversion for longer investment horizons are also similar except that there is not such a distinct picture of mean reversion for the later period. The same procedure has also been performed with the variance ratio test with very similar results.” As a ‘Results
are available
on requests from the authors
P. Frennherg
and B. Hansson,
Table Multiperiod Investment horizon, q (months) 1
2 3 4 6 12 24 36 48 60 72 84 96 108 120 OLS
195412
Subperiod P(Y) 0.137 0.027 0.000 0.030 0.252 0.191 ~ 0.087 - 0.447 - 0.767 -0.722 ~ 0.497 -0.259 -0.237 -0.321 -0.505
autoregressions
1919-1954 Subperiod
R’-adj. (“A)
Prob. value 0.004 0.570 0.996 0.658 0.010 0.173 0.67 1 0.083 0.013 0.044 0.223 0.573 0.646 0.578 0.434
5
of real returns
1 (1919-1954)
1.7 0.0 0.0 0.0 6.0 3.3 0.5 20.7 61.5 59.4 36.2 12.4 15.4 19.0 27.7
189
Swedish stock prices
and
1955-1990.
2 (1955-1990)
B(Y) 0.193 0.099 0.131 0.192 0.184 -0.096 -0.088 0.131 0.120 0.047 -0.258 -0.780 - I.194 - 1.693 - 1.836
Prob.
R’-adj.
VdW
("0)
0.000 0.040 0.056 0.005 0.059 0.494 0.666 0.613 0.697 0.895 0.527 0.090 0.02 I 0.003 0.004
3.5 0.7 1.3 3.0 2.9 0.6 0.5 1.3 1.0 0.0 2.7 15.1 26.1 49.5 61.4
slopes of y-period real returns regressed on lirst q-period lag for two subsamples: 1919:1-and 1955:1~1990:12. The prob.values are calculated using Richardson and Smith’s (1989)
finding that [,/(rt - 2q + I )]/Qq) distributed N[0.(2$+ 1)/3q].
(n = total
number
of one-period-returns
=432)
under
H,
is
further check of the robustness of the results for different time periods we can again use the ten year moving estimates of p(1). As can be noted in fig. 3 [I(l) is definitely varying over time. It is positive for most of the sample periods and significantly different from zero for roughly half of the overlapping ten-year periods. These significant values are estimated for such different time periods as the 1930s the 1950s the 1960s and the 1980s. Thus, the phenomenon of positive autocorrelation for short horizon returns is not concentrated to any particular time period with very specific conditions, like the 1930s. To sum up: it seems that the basic findings - rejection of a random walk hypothesis in favour of a hypothesis suggesting positive autocorrelation for short return horizons and negative for long horizons - are reasonably robust.
4. Summary and concluding notes The results in this study suggest that Swedish stock returns are positively autocorrelated over short return horizons, i.e. one to twelve months, and negatively autocorrelated over longer return horizons, i.e. five years or more. Thus, the random walk hypothesis, interpreted as serial independence, is rejected. The basis for our conclusion are two common tests performed on a newly constructed set of monthly value-weighted Swedish stock return data
190
P. Frennherg
and B. Hansson,
Swedish stock prices
for the period 1919-1990. Our results are in line with recent research on the U.S. stock market. The results of positive short horizon autocorrelation holds for real as well as for nominal and excess return data. We have found that our results can not be explained by the nonsynchronous trading hypothesis, and the choice of sample period was only of marginal importance. Predictable short term returns could be a violation of the efficient market hypothesis [e.g. Fama (1970)]. However the low R2 statistics, less than five percent, indicates that the possibilities for economic profits are very limited. A more important implication of the short term autocorrelation is that estimation of stock volatility, a necessary input in option pricing, using the assumption of serial independence may lead to an underestimation of the true volatility. The conclusion that stock returns over longer periods show negative autocorrelation, also known as mean reversion, are less robust. For the period 1919-1990 both variance ratio tests and OLS estimates suggest negative autocorrelation, but the significance levels are low, especially for the low-powered variance ratio test. Divided into subperiods the negative autocorrelation for horizons from three to six years only show up in the early sample. However, for very long return horizons, ten years or more, and for real returns, the evidence seems to be in favour of a mean-reversion hypothesis. The presence of mean reversion in long period stock returns implies that the portfolio choice is not invariant to the investment horizon. Stocks are relatively less risky for long investment horizons and therefore more attractive to long term investors than for short term investors.’ Thus, the rule of thumb that investors with short investment horizon should have a smaller proportion of stocks than investors with long investment horizons, given the same level of risk aversion, might have an empirical explanation. ‘See for example
Samuelson
( 1989). Lofthouse
(1990) and Butler and Domian
(1991)
References Atchison, M.D., K.C. Butler and R.R. Simonds, 1987, Nonsynchronous security trading and market index autocorrelation. Journal of Finance 42, Ill&l 18. Berglund, T., K. Hedvall and E. Liljeblom, 1991. Predicting volatility of stock indexes for option pricing on a small security market, Working paper no. 34 (Stockholm School of Economics). Butler, K.C. and D.L. Domian, 1991, Risk, diversification and the investment horizon, Journal of Portfolio Management. Spring, 41-47. Campbell. J.Y.. 1991. A variance decomposition for stock returns, The Economic Journal 101, 1577179. Cochrane, J.Y., 1988, How bid is the random walk in GNP?. Journal of Political Economy 96, 8933920. Fama, E.F., 1970, Eflicient capital markets: A review of theory and empirical work. Journal of Finance 25, 3833417. Fama, E.F. and K.R. French, 1988, Permanent and temporary components of stock prices, Journal of Political Economy 96, 246273.
P. Frennberg nnd B. Hnnsson, Swedish stock prices
191
Frennberg, P. and B. Hansson, 1992, Computation of a monthly index for Swedish stock returns: 1919-1989, Scandinavian Economic History Review, forthcoming. Hsu, D.A., 1979. Detecting shifts of parameter in gamma sequences with applications to stock price and air traflic flow analysis, Journal of American Statistical Association 74, 31-40. Hsu, D.A., 1984, The behaviour of stock returns: Is it stationary or evolutionary?, Journal of Financial and Quantitative Analysis 19, 1 l-28. Jennergren, P. and P. Korsvold, 1974, Price formation in the Norwegian and Swedish stock markets Some random walk tests, Swedish Journal of Economics 76, 171~185. Kim, M.J.K. and C.R. Nelson and R. Startz. 1988, Mean reversion in stock prices‘? A reappraisal of the empirical evidence, National Bureau of Economic Research, Working Paper no. 2795. Lo. A.W. and A.C. MacKinlay, 1988, Stock market prices do not follow random walks: Evidence from a simple specification test, The Review of Financial Studies 1, 41-66. Lo, A.W. and A.C. MacKinlay, 1989, The size and power of the variance ratio test in finite samples, Journal of Econometrics 40, 2033238. Lofthouse, S., 1990, Sovereign portfolio management, Journal of International Security Markets, Summer, 133-137. Poterba, J.M. and L.H. Summers, 1988, Mean reversion in stock prices. Evidence and implications. Journal of Financial Economics 25, 3233348. Richardson, M. and T. Smith, 1989, Tests of financial models in the presence of overlapping observations, Review of Financial Studies 2. Richardson, M. and J.H. Stock, 1989, Drawing inference from statistics based on multiyear asset returns, Journal of Financial Economics 25, 3233348. Samuelson, P., 1989, The judgement of economic science on rational portfolio management: Indexing, timing, and long-horizon effects, The Journal of Portfolio Management 16, Fall, 412. Scholes, M. and J. Williams, 1977, Estimating betas from nonsynchronous data, Journal of Financial Economics 5. 309-328. Schwert, G.W., 1989, Why does stock market volatility change over time?, Journal of Finance 44, 1115~1153. Schwert, G.W., 1990, Indexes of U.S. stock prices from 1802 to 1987, Journal of Business 63, 399-426.