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Are stock price reversals really asymmetric? A note
Journalof z zViEl
Journal of Banking & Finance 20 (1996) 189-201
BANKING & FINANCE
Are stock price reversals really asymmetric? A note Gishan Dissanaike University of Cambridge, The Judge Institute of Management Studies, Mill Lane, Cambridge CB2 1RX, UK
Received 15 September 1994; accepted 15 September 1994
Abstract One of the enigmatic findings about the stock market overreaction effect is that it is 'asymmetric.' We argue that this apparent anomaly may be entirely illusory, resulting from the peculiar properties of returns. The test-period return on a contrarian portfolio is not always a satisfactory measure of the strength of price reversals. This also renders interportfolio comparisons about the symmetry of reversals more difficult. We develop an alternative measure, the Reversal Coefficient, which takes account of this deficiency. Some empirical corroboration is also provided. JEL classification: G1 Keywords: Overreaction;Anomalies; Contrarian strategies; Efficient markets
I. Introduction Several recent studies explore the theory that equity markets tend to be overly sensitive to financial news, such that, after a big rise or a big drop in market prices, there are predictable reversals. De Bondt and Thaler (1985, 1987) found that prior losers outperformed prior winners, a phenomenon which they christened the overreaction effect. Others have argued that the overreaction effect is merely another manifestation of the size effect (Zarowin, 1990), or that it can be explained by b i d - a s k errors (Conrad and Kaul, 1993) or incorrectly measured risk (Chan, 1988; Ball and Kothari, 1989). However, Chopra et al. (1992) detected an 0378-4266/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0378-4266(95)00028-3
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G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
overreaction effect even after making appropriate adjustments for size and risk. Therefore, the evidence on the overreaction hypothesis still remains controversial. This paper does not attempt to explore the important issues of size, risk, and overreaction. Rather, it focuses on one of the curious findings about stock price reversals, namely that they are asymmetric: 'The five-year price reversals for losers are more pronounced than for winners (about + 30 percent vs. about - 10 percent..).' i Similar observations have also been documented in studies of short-term price reversals: Atkins and Dyl (1990) found weak evidence that the stock market overreacts to good news and stronger evidence that the stock market overreacts to bad news. 2 De Bondt and Thaler (1989) review the short-term studies and conclude that '..the pattern of returns following..large one-day jumps is remarkably similar to that observed for long-term winners and losers. That is, there is a significant correction for losers but not for winners..' (p. 198). 3 This so-called asymmetry has posed somewhat of an enigma in the literature. In fact, Merton (1987) argued that it undermined the empirical evidence in favour of the overreaction hypothesis. He went on to state that it was 'difficult to see a clear theoretical explanation for overreaction being asymmetric' (p. 107). Some explanations for this phenomenon have, however, been discussed in the literature: De Bondt and Thaler (1985) argued that the asymmetry might be explained by differences in systematic risk, but in their subsequent (1987) paper they speculated that the asymmetry might occur because the reversal in earnings was larger for losers than for winners. Conrad and Kaul (1993), on the other hand, explain the asymmetry in terms of bid-ask biases. 4 In Section 2 we argue that this 'asymmetry' might potentially be explained by the peculiar properties of returns. That is, the apparent anomaly may be entirely illusory. We show that the test-period return on a contrarian portfolio is not always
I See De Bondt and Thaler (1989), p.194. 2 Lehmann (1990) also found that the mean return on the winner portfolio was about one half the magnitude of the mean return on the loser portfolio but he did not comment further on this finding. 3 Incidentally, Brown and Harlow (1988) also concluded that the 'tendency for the stock market to overreact is best regarded as an asymmetric, short term phenomenon' (emphasis added). Further, the phrase 'asymmetric overreaction' was used by Conrad and Kaul (1993, p.43), when referring to the work of De Bondt and Thaler (1987) and Chopra et al. (1992). 4 Another possible explanation is the Uncertain Information Hypothesis (UIH) which posits that information surprises (both positive and negative) increase uncertainty (see Brown et al., 1988). The UIH predicts that the average reaction following (short term) price declines should be stronger than that for (short term) price increases if investors display decreasing absolute risk-aversion. However, since the UIH also predicts that winners should not display negative test-period returns, it does not provide a complete explanation for the findings in the stock price reversal literature. In their follow-up paper, Brown et al. (1993) recognised that not all information surprises would increase uncertainty, which implies that winners could potentially display negative returns. However, the modified UIH no longer carries a presumption that the price reaction for losers should be greater than that for winners.
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
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a reliable measure of the strength of price reversals. This also renders inter-portfolio comparisons about the symmetry of reversals more difficult. We propose an alternative measure which takes account of these deficiencies. This measure is of use irrespective of whether the observed price reversals are due to overreaction or consistent with rational behaviour. In Section 3 we provide some empirical results, and we end with a summary in Section 4.
2. Asymmetric price reversals and asymmetric returns 2.1. Discussion
Price reversals have typically been identified by forming winner and loser portfolios on the basis of past returns and then evaluating their profitability in a subsequent test-period. The observed reversals were then deemed asymmetric simply because the test-period return on the loser portfolio exceeded the (absolute) return on the winner portfolio. That is, R L > I R w l . The symmetry of price reversals must of course be judged with reference to the price movement in the prior rank-period. This is especially so, given that one implication of the overreaction hypothesis is: 'The more extreme the initial price movement, the greater will be the subsequent adjustment' (De Bondt and Thaler, 1985). For example, if a loser portfolio experienced a larger price movement in the rank-period, relative to the winner portfolio, one might expect the subsequent price adjustment to be proportionately stronger too. However, more importantly, it has not been recognised that returns are themselves asymmetric; a price increase from 100 to 150 constitutes a return of + 50%, but a decrease from 150 to 100 implies a return of - 3 3 . 3 % which, in absolute terms, is much smaller. In other words, the two implied price-relatives are reciprocals of each other. 5 This may seem quite rudimentary. Nevertheless, as we will show, it has important implications for tests of the overreaction hypothesis and their interpretation. Assume, for example, that a loser and winner portfolio are both valued at £100 at the beginning of the rank period. The loser portfolio then experiences a £50 decline, and the winner portfolio an (identical) £50 increase, in the rank-period (see Fig. 1). Assume now that these loser and winner portfolios subsequently recover halfway, towards the values they had at the beginning of the rank-period. For them to have recovered thus, they should have displayed test-period returns of
s As an interesting aside, see Fisher (1922) on the Time Reversal Test for index numbers: '....the index number reckoned forward should be the reciprocal of that reckoned backward. Thus, if taking 1913 as a base and going forward to 1918, we find that, on the average, prices have doubled, then, by proceeding in the reverse direction, we ought to find the 1913 price level to be half that of 1918, from which we started as a base. ' (p. 64)
192
G. Dissanaike/Journal of Banking & Finance20 (1996) 189-201 Winner
150
100 ~ ' ~
~ ' 125
.50.~i
~
~
~
~
~uP 75
Loser
Fig. 1. The distinction between asymmetric price reverslas and asymmetric returns. + 50% and - 1 6 . 6 7 % , respectively. But if the portfolios displayed such a reversal pattern, according to the standard interpretation, it would be incorrectly called asymmetric! In other words, under ceteris paribus conditions in the rank-period, an asymmetric return pattern is precisely what one would expect if price reversals were symmetric. The problem originates in the fact that the test-period return is, in the first instance, an unsatisfactory measure of the strength of price reversals. 6 We therefore suggest a more reliable method for determining the strength of price reversals. Our method looks at the magnitude of the reversal in relation to rank-period performance; takes account of the return of the market portfolio; and also recognises the asymmetric nature of returns. We will first consider the problem of measuring reversals for individual securities and then proceed to consider the case of portfolios. 2.2. The single-security reversal coefficient Let Pt be the price of a stock at time t. We will assume, for the moment, that there are no dividends. There are two time periods, the rank-period and the test period. Assume that the stock is priced at 100p at the beginning of the rank-period, time t - 1. If, during the two time periods, the market appreciates by 10% and 25%, respectively, then for the stock to retain its position relative to the market, it should display prices of l l 0 p at the end of the rank-period and 137.5p at the end of the test-period. This price path is portrayed by the topmost line in Fig. 2. It represents the movement of the market portfolio during the two periods. But now assume that the stock does not follow the above price path. Assume instead that it declines, by 50%, to 5 0 p at the end of the first period, and that it
6 Using logarithmic returns will also not solve the problem. The rank-period log returns for the winner (loser) portfolio, in Fig. 1, would be +40.55% (-69.3%). The corresponding test-period returns would be -18.23% (+40.55%). We would not be able to conclude, on the basis of these returns, that the portfolios reversed halfway towards the values they had at the start, or, that the reversals were symmetric.
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
193
p r i c e Cpence) 140
130 120
R,,t=1.1
110
,,," ,,," ,..'"
100 90
80
70
/
60
.
0
50 40
.
t-1
t
t+l
Fig. 2. A graphical illustration of the reversal coefficient.
subsequently increases to 90p at the end of the second period. This price reversal is depicted by the solid line OB. The problem then is, how do we measure the strength of the reversal? For a start, we know that if the price had increased to 137.5p at time t + 1, instead of 90p, it would have reversed fully to what it was at the beginning of the first period (after adjusting for market movements). This reversal path is shown by the dotted line OA. On the other hand, if the price had not reversed at all-that is, if it had only retained its market rating in the second period - it would have to have attained a price of 62.5p at time t + 1, (i.e. 50p × 1.25). This reversal path is represented by the dotted line OC. Thus, if the price reversed fully, the magnitude of the reversal would be measured by the distance A - C , (~). The actual reversal, on the other hand, would be measured by the distance B - C , (h). Thus, the ratio h / ~ represents an appropriate measure of the strength of the price reversal, and we shall call this the single-security Reversal Coefficient, olit:
air
h
B- C
90.0 - 62.5
~
A - C
137.5 - 62.5
= 0.367.
(1)
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194
For practical purposes, it is more convenient to express the Reversal Coefficient in terms of returns rather than prices. This would also enable us to take account of dividends more conveniently. Thus, B - C =
-
OLit
-
P,_lRitRit+l -Pit_aRitRmt+l =
A -- C
(2)
Pit_lRmtRmt+ l - Pit_lRitRmt+ l
Rit+ 1 - Rmt+ 1
=
,
(3)
( RmtRmt+ 1 / R i t ) -- Rmt+ l
where
Rmt
one plus the return on the market portfolio in the rank period, and one plus the return on security i in the rank period, where the return is defined inclusive of dividends. The numerator in (3) is simply the actual test-period excess return, and the denominator is the test-period return which would have been necessary for the security to regain its initial value, relative to the market. There is, of course, nothing in the 'overreaction' hypothesis to suggest that prices should revert to what they were at the beginning of the ranking-period; the denominator, here, merely performs the role of a standardising factor. The Reversal Coefficient (air) can then be interpreted as follows:
Rit
=
=
oLit = 1,
the price of the security has reverted to its initial value, relative to the market. the price of the security has reverted (relative to the market) but it has not yet regained its initial value. the price of the security has reverted, relative to the market, but the magnitude of the reversal is greater than that which was necessary to regain its value at the beginning of the formation period. the price of the security has not reverted relative to the market.
0 < Olit < ], air > 1,
olit ~, O,
2.3. Portfolio Reversal Coefficients We are ultimately more interested in measuring the reversal of a portfolio rather than that of an individual security. We will assume that the portfolios are equal-weighted at the outset. We also assume a buy-and-hold method (BH) of computing test-period returns; the return on such a strategy is given by 1N[T T )
=
ElI-Ir,,- I-Ir ,, i \t=l
(4)
t=l
where N = the number of securities in the portfolio; T = the number of time periods (months); rmt = one plus the return on the market portfolio in month t; and rit = one plus the return on security i in month t. Portfolios are formed on the basis of Rank-Period Returns, computed using a multiplicative method 7 as in Eq. 5.
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201 0
0
I--I r , -
RPR, =
195
t= - T
1-I rmt.
(5)
t= --T
There are two alternative ways of devising a portfolio reversal coefficient. Firstly, we could treat the portfolio as if it were a security and define the portfolio reversal coefficient, C~pt, as follows:
%' =
r
(6)
H rmt t= ] N - T0 _ ~ E H ri t it=-T
T HI.rat t=l
Analogous to Eq. 3 above, the numerator in (6) is the actual test-period excess return, obtained by using Eq. 4, and the denominator is the test-period excess return which would have been necessary for the portfolio to regain its initial value, relative to the market. For computational convenience, the denominator in (6) can be written as
t= l =rlN[
[ z* i \t= - T 0
0
/7,2/FI [ i v i \ t = - T r~,- =N_Trmt
Trmt )]
0
+
, =H- r
(7)
rmt
T
I-I rmt ( ARP R ) t=l -
o
ARPR +
(s)
1-I r~, t~-T
where ARPR denotes the rank-period excess return on the portfolio. Note that A R P R is simply the average of the individual security rank-period returns obtained from Eq. 5, and is thus a figure routinely calculated in the overreaction studies. Hence, the only extra information required to compute o~p, is the return on the market in the rank and test periods. 7 Several researchers who investigated overreaction over 3 - 5 year windows used an arithmetic method to compute multi-period returns. However, Conrad and Kaul (1993) criticise the arithmetic method on the grounds that it is more susceptible to bid-ask biases. Further, Dissanaike (1994) argues that the arithmetic method is an unrealistic one, and shows that estimates of portfolio performance can be sensitive both to the method used to compute test-period returns and the method used to calculate rank-period returns.
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196
The second method of computing a portfolio reversal coefficient would be to compute a reversal coefficient for individual securities in a portfolio, and then average them. We call the average of the individual security reversal coefficients, ~/pt' where
1 N "gpt =
.
r i t - I--[rmt
r
(9)
- I-Irmt[RPRi] t=0 0
RPRi + l - I Fret t= - T
The Portfolio Reversal Coefficients can be computed for the two extreme portfolios, at any point in time after portfolio formation. The coefficients have several attractive features: They capture the magnitude of the subsequent price movement as a proportion of the initial price movement. This helps us to consider questions such as the following: If the initial price movement was due to, say, overreaction, how much of this movement was subsequently corrected? 8 We should stress, however, that we are not suggesting a change in the standard procedure used to detect price reversals. The test-period returns are useful, not least because they tell us whether the reversals would have offered investors excess profit opportunities. We are only suggesting that the Reversal Coefficients offer extra information about the strength of the price reversals, which can in turn give us a greater insight into the nature of stock price reversals. 9 The Reversal Coefficients also enable us to compare the coefficients of two or more portfolios. If the coefficients are positive and very similar, then the price reversals could be considered symmetric.
3. Some empirical results In order to demonstrate the practical usefulness of the Reversal Coefficient, we also present some empirical corroboration, using data for one test period. Monthly
8 The reversal coefficient is more suitable for use with extreme portfolios rather than median portfolios. This is because it becomes indeterminate as rank-period excess returns tend to zero. But this is not a problem because the overreaction studies typically focus on the extreme portfolios. 9 Although the Reversal Coefficient was defined in terms of market-adjusted returns, it can quite easily be modified to incorporate no market adjustment or alternative risk-adjustment assumptions (e.g. betas). The Reversal Coefficient also has potential uses in other areas of Finance and Accounting, where the event-study methodology is used. For example, it could probably be employed in studies which focus on the share price behaviour of firms before and after takeover. Another application may be in studies of earnings behaviour, where the Reversal Coefficient could be appropriately modified to include accounting earnings instead of stock prices.
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
197
returns were extracted from the London Share Price Database for all companies which were constituents of the FT 500 Index on 1st January 1984 and which had at least 48 months of return data prior to that date. A total of 455 companies met this criteria. A FT500 equal-weighted monthly return index was constructed from 1 / 1 / 8 0 to 1 / 1 / 8 9 to serve as the market index. 10 For each of the 455 companies, Rank-Period returns (RPR i) for a 48-month formation period were computed using Eq. 5. The companies were ranked on the basis of RPR. Those with the highest returns (top decile) were assigned to a ' w i n n e r ' portfolio and those in the bottom decile were assigned to a ' l o s e r ' portfolio. The performance of these portfolios was then tracked over the subsequent four years. it Table 1 shows that the loser portfolio outperformed the winner portfolio throughout the test-period. Forty eight months after portfolio formation, the return differential was over 250%, a differential which is both economically large and statistically significant from zero (t = 5.52). 12 More importantly, in agreement with previous findings, the return on the loser portfolio was greater - i n absolute terms - t h a n the return on the winner portfolio. Four years into the test-period, the loser portfolio registered a market-adjusted return of 156% while the winner portfolio posted a market-adjusted return of - 99%. Therefore, if the conventional interpretation were applied, it would indeed seem that the price reversals for losers were greater than those for winners. However, the Portfolio Reversal Coefficients (apt), contained in Table 2A, show very clearly that this is not the case: Four years after portfolio formation, the loser portfolio had recovered only 19.2% of the value it lost in the formation period, after adjusting for market movements. Per contra, the winner portfolio had lost nearly 54% of the value it gained. The fact that the price reversals for winners were stronger than for losers (and not vice versa), is further evidenced in Table 2B which contains Portfolio Reversal Coefficients computed using the second of our methods, (3'p,). The results are very similar to those contained in Table 2A. On average, after adjusting for market movements, losers regained 18% of the value they lost, and winners lost nearly 56% of the value they gained. The difference between the two coefficients was significantly different from zero at the 5% level in all the months for which M a n n - W h i t n e y U statistics were computed. The above results demonstrate very
10 This was done using FF500 membership data and LSPD returns. 11 When stocks are de-listed subsequent to portfolio formation, the proceeds (if any) are re-invested in the remaining securities (in proportion to their weights). 12 The F statistic was used to test the hypothesis that the variances of the two portfolios were equal. Because the hypothesis of equal variances was rejected at the 5% level, we used a separate variance t test, to test the null hypothesis that the mean returns on the two portfolio were equal against an alternative hypothesis of investor overreaction (a one-tailed test). We also used a Mann-Whitney nonparametric procedure to test the null hypothesis of equal population location parameters (see Siegel, 1956). The null hypothesis was rejected, at the 1% level, against a one-sided alternative hypothesis.
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
198
oJ
O0 O0
0
~
II
~
II
~
~
II
~
II
~
0 ¢-
0
1:
0 tT~ •"
0 O0
o
t"q
ir
~. I
~L
r-
~.
I
i
J
I "~
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
199
Table 2 Portfolio Reversal Coefficients (Type I and II) for W i n n e r and L o s e r Portfolios Portfolio
M o n t h s after Portfolio formation 12
24
30
36
42
48
A. Type I (apt) a Loser Winner
0.031 0.215
0.082 0.326
0.146 0.389
0.138 0.437
0.199 0.507
0.192 0.540
0.028 0.222
0.078 0.328
0.137 0.395
0.119 0.449
0.173 0.525
0.180 0.561
0.194 492.0 * * - 3.198
0.250 491.0 * - 2.4620
0.258 480.0 * - 2.2281
0.331 336.0 * * - 2.8902
0.352 240.0 * * - 3.458
0.381 232.0 * * - 3.5687
B. Type H ('rpt) a Loser Winner W-L M-WU b Z-value
The T y p e I (Type II) Coefficients are c o m p u t e d using Eq. 8 (9), respectively. Both coefficients are b a s e d on an F T 5 0 0 (equal-weighted) index. h This is the M a n n - W h i t n e y U statistic. Because the total sample size for this test exceeds 30, U is transformed into a normally distributed Z statistic. W e use a two-tailed test. The superscripts * and • * denote significance at the 5 % and 1% levels, respectively. a
clearly that stock price reversals for losers can be weaker than those for winners, in spite of the fact that the test-period return for losers exceeds the (absolute) return for winners. It is, therefore, entirely possible for aLT < Otwx and TLT < 7WT to hold, even though R L > I Rw I. Now, in principle, both market-adjusted returns and the Reversal Coefficients can be sensitive to the choice of market index. Hence, we also considered the effect of using the Financial Times 500 (value-weighted) index. It was found that the return asymmetry was even more pronounced than before -the Loser portfolio posted a return of 209.4% ( t = 4.86) and the Winner portfolio one of - 4 6 . 0 % (t = -2.73). But, the reversal coefficients (Type II) for the losers and winners were .286 and .332, suggesting that the price reversals were actually near symmetric. 13
4. S u m m a r y and conclusions Some authors have interpreted the asymmetric test-period returns of winners and losers to mean that price reversals are asymmetric. We have shown that this is not necessarily the case. That is, simply because R L > [ R w 1, it cannot be inferred that price reversals are stronger for losers than for winners. We developed an
13 This w a s also b o r n e out b y the alternative type of reversal coefficient (i.e. Type I).
200
G. Dissanaike/Journal of Banking & Finance 20 (1996) 189-201
alternative measure, the Reversal Coefficient, which could be used to determine the strength of price reversals. We should stress that we are not advocating a change in the standard procedure used to detect price reversals. The test-period returns are very useful, not least because they tell us whether the reversals would have offered investors excess profit opportunities. We are only suggesting that the reversal coefficients offer additional useful information not captured by the test-period returns -they provide an indication about the strength of the price reversals and can therefore provide a greater insight into the nature of stock price reversals, irrespective of whether the reversals are rational or not. We also provided some empirical corroboration in order to demonstrate the practical usefulness of the reversal coefficient: Using the Reversal Coefficient technique in conjunction with an equal-weighted index, we found that the price reversals for winners were stronger than those for losers, and not vice versa, even though the returns to losers were much larger than the returns to winners. Further tests ought to be conducted in order to find out whether this finding is sample specific. But if this result were found to obtain within other sample periods, it would have interesting ramifications because, following the initial findings by De Bondt and Thaler (1985), slightly more attention has been devoted to losers. ~4 We also used the Reversal Coefficient technique in combination with a value-weighted index, but, in this case, the price reversals were actually found to be near symmetric (although the return asymmetry was even more pronounced). Our conceptual analysis also shows that the relationship between market inefficiency and the ability to devise profitable investment strategies, is rather more subtle than has generally been acknowledged: Ceteris paribus, it is more profitable to buy shares which are undervalued rather than to short-sell shares which are overvalued. Put differently, buying losers may prove more profitable than short-selling winners, but this does not necessarily mean that the extent of mispricing for losers -insofar as it is mispricing -exceeds that for winners.
Acknowledgements I am indebted to Prof. Geoffrey Whittington for many insights and suggestions. Thanks are also due to Elroy Dimson and Geoff Meeks for useful comments, and to Joyce Wheeler for extracting the data. They share no responsibility for any remaining errors. A major part of this research was completed whilst I held a Research Studentship for Economics, awarded by Trinity College, a President's Scholarship, and an Overseas Research Studentship awarded by the Committee of 14 See, for example, Bremer and Sweeney(1991).
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
201
Vice-chancellors and Principals of the Universities of the UK. I also acknowledge a grant from the Eddington Fund of Trinity College, Cambridge.
References Atkins, A. and E. Dyl, 1990, Price reversals, bid-ask spreads and market efficiency, Journal of Financial and Quantitative Analysis 25, No. 4. Ball, R. and S.P. Kothari, 1989, Nonstationary expected returns: Implications for tests of market efficiency and serial correlation in returns, Journal of Financial Economics 25, 51-74. Bremer, Marc and Richard J. Sweeney, 1991, The reversal of large stock price decreases, Journal of Finance 46, 747-754. Brown, Keith C. and W.V. Harlow, 1988, Market overreaction: Magnitude and intensity, Journal of Portfolio Management, Winter, 6-13. Brown, Keith C. and W.V. Harlow, and Seha M. Tinic (1988), Risk aversion, uncertain information, and market efficiency, Journal of Financial Economics 22, December. Brown, Keith C. and W.V. Harlow, and Seha M. Tinic (1993), The risk and required return of common stock following major price innovations, Journal of Financial and Quantitative Analysis 28, No.I, March. Chan, K.C., 1988, On the contrarian investment strategy, Journal of Business 61, No. 2, April, 147-164. Chopra, Navin, Josef Lakonishok and Jay R. Ritter, 1992, Measuring abnormal performance: Do stocks overreact?, Journal of Financial Economics 31,235-268. Conrad, Jennifer and Gautam Kaul, 1993, The returns to long term winners and losers: Bid-ask biases or biases in computed returns, Journal of Finance, March. De Bondt, Werner F.M. and Richard H. Thaler, 1985, Does the stock market overreact?, Journal of Finance 40, No. 3, July, 793-805. De Bondt, Werner F.M. and Richard H. Thaler, 1987, Further evidence on investor overreaction and stock market seasonality, Journal of Finance 42, No. 3, July, 557-581. De Bondt, Werner F.M. and Richard H. Thaler, 1989, A mean-reverting walk down Wall Street, Journal of Economic Perspectives 3, No. 1, Winter, 189-202. Dissanaike, Gishan, 1994, On the computation of returns in tests of the stock market overreaction hypothesis, Journal of Banking and Finance 18, No. 6, December. Fisher, Irving, 1922, The making of index numbers: A study of their varieties, tests and reliability, (Houghton Mifflin, Boston, MA). Lehmann, Bruce, 1990, Fads, martingales, and market efficiency, Quarterly Journal of Economics CV, no. 1, 1-28. Merton, Robert C., 1987, On the current state of the stock market rationality hypothesis, in: Rudiger Dornbusch, Stanley Fischer and John Bossons, eds., Macroeconomics and finance; Essays in honor of Franco Modigliani, (MIT Press, Cambridge, MA). Siegel, Sidney, 1956, Nonparametric statistics for the behavioral sciences, (McGraw-Hill Book Co., New York). Zarowin, Paul, 1990, Size, seasonality, and stock market overreaction, Journal of Financial and Quantitative Analysis 25, No. 1, March, 113-125.
Journal of Banking & Finance 20 (1996) 189-201
BANKING & FINANCE
Are stock price reversals really asymmetric? A note Gishan Dissanaike University of Cambridge, The Judge Institute of Management Studies, Mill Lane, Cambridge CB2 1RX, UK
Received 15 September 1994; accepted 15 September 1994
Abstract One of the enigmatic findings about the stock market overreaction effect is that it is 'asymmetric.' We argue that this apparent anomaly may be entirely illusory, resulting from the peculiar properties of returns. The test-period return on a contrarian portfolio is not always a satisfactory measure of the strength of price reversals. This also renders interportfolio comparisons about the symmetry of reversals more difficult. We develop an alternative measure, the Reversal Coefficient, which takes account of this deficiency. Some empirical corroboration is also provided. JEL classification: G1 Keywords: Overreaction;Anomalies; Contrarian strategies; Efficient markets
I. Introduction Several recent studies explore the theory that equity markets tend to be overly sensitive to financial news, such that, after a big rise or a big drop in market prices, there are predictable reversals. De Bondt and Thaler (1985, 1987) found that prior losers outperformed prior winners, a phenomenon which they christened the overreaction effect. Others have argued that the overreaction effect is merely another manifestation of the size effect (Zarowin, 1990), or that it can be explained by b i d - a s k errors (Conrad and Kaul, 1993) or incorrectly measured risk (Chan, 1988; Ball and Kothari, 1989). However, Chopra et al. (1992) detected an 0378-4266/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0378-4266(95)00028-3
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G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
overreaction effect even after making appropriate adjustments for size and risk. Therefore, the evidence on the overreaction hypothesis still remains controversial. This paper does not attempt to explore the important issues of size, risk, and overreaction. Rather, it focuses on one of the curious findings about stock price reversals, namely that they are asymmetric: 'The five-year price reversals for losers are more pronounced than for winners (about + 30 percent vs. about - 10 percent..).' i Similar observations have also been documented in studies of short-term price reversals: Atkins and Dyl (1990) found weak evidence that the stock market overreacts to good news and stronger evidence that the stock market overreacts to bad news. 2 De Bondt and Thaler (1989) review the short-term studies and conclude that '..the pattern of returns following..large one-day jumps is remarkably similar to that observed for long-term winners and losers. That is, there is a significant correction for losers but not for winners..' (p. 198). 3 This so-called asymmetry has posed somewhat of an enigma in the literature. In fact, Merton (1987) argued that it undermined the empirical evidence in favour of the overreaction hypothesis. He went on to state that it was 'difficult to see a clear theoretical explanation for overreaction being asymmetric' (p. 107). Some explanations for this phenomenon have, however, been discussed in the literature: De Bondt and Thaler (1985) argued that the asymmetry might be explained by differences in systematic risk, but in their subsequent (1987) paper they speculated that the asymmetry might occur because the reversal in earnings was larger for losers than for winners. Conrad and Kaul (1993), on the other hand, explain the asymmetry in terms of bid-ask biases. 4 In Section 2 we argue that this 'asymmetry' might potentially be explained by the peculiar properties of returns. That is, the apparent anomaly may be entirely illusory. We show that the test-period return on a contrarian portfolio is not always
I See De Bondt and Thaler (1989), p.194. 2 Lehmann (1990) also found that the mean return on the winner portfolio was about one half the magnitude of the mean return on the loser portfolio but he did not comment further on this finding. 3 Incidentally, Brown and Harlow (1988) also concluded that the 'tendency for the stock market to overreact is best regarded as an asymmetric, short term phenomenon' (emphasis added). Further, the phrase 'asymmetric overreaction' was used by Conrad and Kaul (1993, p.43), when referring to the work of De Bondt and Thaler (1987) and Chopra et al. (1992). 4 Another possible explanation is the Uncertain Information Hypothesis (UIH) which posits that information surprises (both positive and negative) increase uncertainty (see Brown et al., 1988). The UIH predicts that the average reaction following (short term) price declines should be stronger than that for (short term) price increases if investors display decreasing absolute risk-aversion. However, since the UIH also predicts that winners should not display negative test-period returns, it does not provide a complete explanation for the findings in the stock price reversal literature. In their follow-up paper, Brown et al. (1993) recognised that not all information surprises would increase uncertainty, which implies that winners could potentially display negative returns. However, the modified UIH no longer carries a presumption that the price reaction for losers should be greater than that for winners.
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a reliable measure of the strength of price reversals. This also renders inter-portfolio comparisons about the symmetry of reversals more difficult. We propose an alternative measure which takes account of these deficiencies. This measure is of use irrespective of whether the observed price reversals are due to overreaction or consistent with rational behaviour. In Section 3 we provide some empirical results, and we end with a summary in Section 4.
2. Asymmetric price reversals and asymmetric returns 2.1. Discussion
Price reversals have typically been identified by forming winner and loser portfolios on the basis of past returns and then evaluating their profitability in a subsequent test-period. The observed reversals were then deemed asymmetric simply because the test-period return on the loser portfolio exceeded the (absolute) return on the winner portfolio. That is, R L > I R w l . The symmetry of price reversals must of course be judged with reference to the price movement in the prior rank-period. This is especially so, given that one implication of the overreaction hypothesis is: 'The more extreme the initial price movement, the greater will be the subsequent adjustment' (De Bondt and Thaler, 1985). For example, if a loser portfolio experienced a larger price movement in the rank-period, relative to the winner portfolio, one might expect the subsequent price adjustment to be proportionately stronger too. However, more importantly, it has not been recognised that returns are themselves asymmetric; a price increase from 100 to 150 constitutes a return of + 50%, but a decrease from 150 to 100 implies a return of - 3 3 . 3 % which, in absolute terms, is much smaller. In other words, the two implied price-relatives are reciprocals of each other. 5 This may seem quite rudimentary. Nevertheless, as we will show, it has important implications for tests of the overreaction hypothesis and their interpretation. Assume, for example, that a loser and winner portfolio are both valued at £100 at the beginning of the rank period. The loser portfolio then experiences a £50 decline, and the winner portfolio an (identical) £50 increase, in the rank-period (see Fig. 1). Assume now that these loser and winner portfolios subsequently recover halfway, towards the values they had at the beginning of the rank-period. For them to have recovered thus, they should have displayed test-period returns of
s As an interesting aside, see Fisher (1922) on the Time Reversal Test for index numbers: '....the index number reckoned forward should be the reciprocal of that reckoned backward. Thus, if taking 1913 as a base and going forward to 1918, we find that, on the average, prices have doubled, then, by proceeding in the reverse direction, we ought to find the 1913 price level to be half that of 1918, from which we started as a base. ' (p. 64)
192
G. Dissanaike/Journal of Banking & Finance20 (1996) 189-201 Winner
150
100 ~ ' ~
~ ' 125
.50.~i
~
~
~
~
~uP 75
Loser
Fig. 1. The distinction between asymmetric price reverslas and asymmetric returns. + 50% and - 1 6 . 6 7 % , respectively. But if the portfolios displayed such a reversal pattern, according to the standard interpretation, it would be incorrectly called asymmetric! In other words, under ceteris paribus conditions in the rank-period, an asymmetric return pattern is precisely what one would expect if price reversals were symmetric. The problem originates in the fact that the test-period return is, in the first instance, an unsatisfactory measure of the strength of price reversals. 6 We therefore suggest a more reliable method for determining the strength of price reversals. Our method looks at the magnitude of the reversal in relation to rank-period performance; takes account of the return of the market portfolio; and also recognises the asymmetric nature of returns. We will first consider the problem of measuring reversals for individual securities and then proceed to consider the case of portfolios. 2.2. The single-security reversal coefficient Let Pt be the price of a stock at time t. We will assume, for the moment, that there are no dividends. There are two time periods, the rank-period and the test period. Assume that the stock is priced at 100p at the beginning of the rank-period, time t - 1. If, during the two time periods, the market appreciates by 10% and 25%, respectively, then for the stock to retain its position relative to the market, it should display prices of l l 0 p at the end of the rank-period and 137.5p at the end of the test-period. This price path is portrayed by the topmost line in Fig. 2. It represents the movement of the market portfolio during the two periods. But now assume that the stock does not follow the above price path. Assume instead that it declines, by 50%, to 5 0 p at the end of the first period, and that it
6 Using logarithmic returns will also not solve the problem. The rank-period log returns for the winner (loser) portfolio, in Fig. 1, would be +40.55% (-69.3%). The corresponding test-period returns would be -18.23% (+40.55%). We would not be able to conclude, on the basis of these returns, that the portfolios reversed halfway towards the values they had at the start, or, that the reversals were symmetric.
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
193
p r i c e Cpence) 140
130 120
R,,t=1.1
110
,,," ,,," ,..'"
100 90
80
70
/
60
.
0
50 40
.
t-1
t
t+l
Fig. 2. A graphical illustration of the reversal coefficient.
subsequently increases to 90p at the end of the second period. This price reversal is depicted by the solid line OB. The problem then is, how do we measure the strength of the reversal? For a start, we know that if the price had increased to 137.5p at time t + 1, instead of 90p, it would have reversed fully to what it was at the beginning of the first period (after adjusting for market movements). This reversal path is shown by the dotted line OA. On the other hand, if the price had not reversed at all-that is, if it had only retained its market rating in the second period - it would have to have attained a price of 62.5p at time t + 1, (i.e. 50p × 1.25). This reversal path is represented by the dotted line OC. Thus, if the price reversed fully, the magnitude of the reversal would be measured by the distance A - C , (~). The actual reversal, on the other hand, would be measured by the distance B - C , (h). Thus, the ratio h / ~ represents an appropriate measure of the strength of the price reversal, and we shall call this the single-security Reversal Coefficient, olit:
air
h
B- C
90.0 - 62.5
~
A - C
137.5 - 62.5
= 0.367.
(1)
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194
For practical purposes, it is more convenient to express the Reversal Coefficient in terms of returns rather than prices. This would also enable us to take account of dividends more conveniently. Thus, B - C =
-
OLit
-
P,_lRitRit+l -Pit_aRitRmt+l =
A -- C
(2)
Pit_lRmtRmt+ l - Pit_lRitRmt+ l
Rit+ 1 - Rmt+ 1
=
,
(3)
( RmtRmt+ 1 / R i t ) -- Rmt+ l
where
Rmt
one plus the return on the market portfolio in the rank period, and one plus the return on security i in the rank period, where the return is defined inclusive of dividends. The numerator in (3) is simply the actual test-period excess return, and the denominator is the test-period return which would have been necessary for the security to regain its initial value, relative to the market. There is, of course, nothing in the 'overreaction' hypothesis to suggest that prices should revert to what they were at the beginning of the ranking-period; the denominator, here, merely performs the role of a standardising factor. The Reversal Coefficient (air) can then be interpreted as follows:
Rit
=
=
oLit = 1,
the price of the security has reverted to its initial value, relative to the market. the price of the security has reverted (relative to the market) but it has not yet regained its initial value. the price of the security has reverted, relative to the market, but the magnitude of the reversal is greater than that which was necessary to regain its value at the beginning of the formation period. the price of the security has not reverted relative to the market.
0 < Olit < ], air > 1,
olit ~, O,
2.3. Portfolio Reversal Coefficients We are ultimately more interested in measuring the reversal of a portfolio rather than that of an individual security. We will assume that the portfolios are equal-weighted at the outset. We also assume a buy-and-hold method (BH) of computing test-period returns; the return on such a strategy is given by 1N[T T )
=
ElI-Ir,,- I-Ir ,, i \t=l
(4)
t=l
where N = the number of securities in the portfolio; T = the number of time periods (months); rmt = one plus the return on the market portfolio in month t; and rit = one plus the return on security i in month t. Portfolios are formed on the basis of Rank-Period Returns, computed using a multiplicative method 7 as in Eq. 5.
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201 0
0
I--I r , -
RPR, =
195
t= - T
1-I rmt.
(5)
t= --T
There are two alternative ways of devising a portfolio reversal coefficient. Firstly, we could treat the portfolio as if it were a security and define the portfolio reversal coefficient, C~pt, as follows:
%' =
r
(6)
H rmt t= ] N - T0 _ ~ E H ri t it=-T
T HI.rat t=l
Analogous to Eq. 3 above, the numerator in (6) is the actual test-period excess return, obtained by using Eq. 4, and the denominator is the test-period excess return which would have been necessary for the portfolio to regain its initial value, relative to the market. For computational convenience, the denominator in (6) can be written as
t= l =rlN[
[ z* i \t= - T 0
0
/7,2/FI [ i v i \ t = - T r~,- =N_Trmt
Trmt )]
0
+
, =H- r
(7)
rmt
T
I-I rmt ( ARP R ) t=l -
o
ARPR +
(s)
1-I r~, t~-T
where ARPR denotes the rank-period excess return on the portfolio. Note that A R P R is simply the average of the individual security rank-period returns obtained from Eq. 5, and is thus a figure routinely calculated in the overreaction studies. Hence, the only extra information required to compute o~p, is the return on the market in the rank and test periods. 7 Several researchers who investigated overreaction over 3 - 5 year windows used an arithmetic method to compute multi-period returns. However, Conrad and Kaul (1993) criticise the arithmetic method on the grounds that it is more susceptible to bid-ask biases. Further, Dissanaike (1994) argues that the arithmetic method is an unrealistic one, and shows that estimates of portfolio performance can be sensitive both to the method used to compute test-period returns and the method used to calculate rank-period returns.
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196
The second method of computing a portfolio reversal coefficient would be to compute a reversal coefficient for individual securities in a portfolio, and then average them. We call the average of the individual security reversal coefficients, ~/pt' where
1 N "gpt =
.
r i t - I--[rmt
r
(9)
- I-Irmt[RPRi] t=0 0
RPRi + l - I Fret t= - T
The Portfolio Reversal Coefficients can be computed for the two extreme portfolios, at any point in time after portfolio formation. The coefficients have several attractive features: They capture the magnitude of the subsequent price movement as a proportion of the initial price movement. This helps us to consider questions such as the following: If the initial price movement was due to, say, overreaction, how much of this movement was subsequently corrected? 8 We should stress, however, that we are not suggesting a change in the standard procedure used to detect price reversals. The test-period returns are useful, not least because they tell us whether the reversals would have offered investors excess profit opportunities. We are only suggesting that the Reversal Coefficients offer extra information about the strength of the price reversals, which can in turn give us a greater insight into the nature of stock price reversals. 9 The Reversal Coefficients also enable us to compare the coefficients of two or more portfolios. If the coefficients are positive and very similar, then the price reversals could be considered symmetric.
3. Some empirical results In order to demonstrate the practical usefulness of the Reversal Coefficient, we also present some empirical corroboration, using data for one test period. Monthly
8 The reversal coefficient is more suitable for use with extreme portfolios rather than median portfolios. This is because it becomes indeterminate as rank-period excess returns tend to zero. But this is not a problem because the overreaction studies typically focus on the extreme portfolios. 9 Although the Reversal Coefficient was defined in terms of market-adjusted returns, it can quite easily be modified to incorporate no market adjustment or alternative risk-adjustment assumptions (e.g. betas). The Reversal Coefficient also has potential uses in other areas of Finance and Accounting, where the event-study methodology is used. For example, it could probably be employed in studies which focus on the share price behaviour of firms before and after takeover. Another application may be in studies of earnings behaviour, where the Reversal Coefficient could be appropriately modified to include accounting earnings instead of stock prices.
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
197
returns were extracted from the London Share Price Database for all companies which were constituents of the FT 500 Index on 1st January 1984 and which had at least 48 months of return data prior to that date. A total of 455 companies met this criteria. A FT500 equal-weighted monthly return index was constructed from 1 / 1 / 8 0 to 1 / 1 / 8 9 to serve as the market index. 10 For each of the 455 companies, Rank-Period returns (RPR i) for a 48-month formation period were computed using Eq. 5. The companies were ranked on the basis of RPR. Those with the highest returns (top decile) were assigned to a ' w i n n e r ' portfolio and those in the bottom decile were assigned to a ' l o s e r ' portfolio. The performance of these portfolios was then tracked over the subsequent four years. it Table 1 shows that the loser portfolio outperformed the winner portfolio throughout the test-period. Forty eight months after portfolio formation, the return differential was over 250%, a differential which is both economically large and statistically significant from zero (t = 5.52). 12 More importantly, in agreement with previous findings, the return on the loser portfolio was greater - i n absolute terms - t h a n the return on the winner portfolio. Four years into the test-period, the loser portfolio registered a market-adjusted return of 156% while the winner portfolio posted a market-adjusted return of - 99%. Therefore, if the conventional interpretation were applied, it would indeed seem that the price reversals for losers were greater than those for winners. However, the Portfolio Reversal Coefficients (apt), contained in Table 2A, show very clearly that this is not the case: Four years after portfolio formation, the loser portfolio had recovered only 19.2% of the value it lost in the formation period, after adjusting for market movements. Per contra, the winner portfolio had lost nearly 54% of the value it gained. The fact that the price reversals for winners were stronger than for losers (and not vice versa), is further evidenced in Table 2B which contains Portfolio Reversal Coefficients computed using the second of our methods, (3'p,). The results are very similar to those contained in Table 2A. On average, after adjusting for market movements, losers regained 18% of the value they lost, and winners lost nearly 56% of the value they gained. The difference between the two coefficients was significantly different from zero at the 5% level in all the months for which M a n n - W h i t n e y U statistics were computed. The above results demonstrate very
10 This was done using FF500 membership data and LSPD returns. 11 When stocks are de-listed subsequent to portfolio formation, the proceeds (if any) are re-invested in the remaining securities (in proportion to their weights). 12 The F statistic was used to test the hypothesis that the variances of the two portfolios were equal. Because the hypothesis of equal variances was rejected at the 5% level, we used a separate variance t test, to test the null hypothesis that the mean returns on the two portfolio were equal against an alternative hypothesis of investor overreaction (a one-tailed test). We also used a Mann-Whitney nonparametric procedure to test the null hypothesis of equal population location parameters (see Siegel, 1956). The null hypothesis was rejected, at the 1% level, against a one-sided alternative hypothesis.
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
198
oJ
O0 O0
0
~
II
~
II
~
~
II
~
II
~
0 ¢-
0
1:
0 tT~ •"
0 O0
o
t"q
ir
~. I
~L
r-
~.
I
i
J
I "~
G. Dissanaike /Journal of Banking & Finance 20 (1996) 189-201
199
Table 2 Portfolio Reversal Coefficients (Type I and II) for W i n n e r and L o s e r Portfolios Portfolio
M o n t h s after Portfolio formation 12
24
30
36
42
48
A. Type I (apt) a Loser Winner
0.031 0.215
0.082 0.326
0.146 0.389
0.138 0.437
0.199 0.507
0.192 0.540
0.028 0.222
0.078 0.328
0.137 0.395
0.119 0.449
0.173 0.525
0.180 0.561
0.194 492.0 * * - 3.198
0.250 491.0 * - 2.4620
0.258 480.0 * - 2.2281
0.331 336.0 * * - 2.8902
0.352 240.0 * * - 3.458
0.381 232.0 * * - 3.5687
B. Type H ('rpt) a Loser Winner W-L M-WU b Z-value
The T y p e I (Type II) Coefficients are c o m p u t e d using Eq. 8 (9), respectively. Both coefficients are b a s e d on an F T 5 0 0 (equal-weighted) index. h This is the M a n n - W h i t n e y U statistic. Because the total sample size for this test exceeds 30, U is transformed into a normally distributed Z statistic. W e use a two-tailed test. The superscripts * and • * denote significance at the 5 % and 1% levels, respectively. a
clearly that stock price reversals for losers can be weaker than those for winners, in spite of the fact that the test-period return for losers exceeds the (absolute) return for winners. It is, therefore, entirely possible for aLT < Otwx and TLT < 7WT to hold, even though R L > I Rw I. Now, in principle, both market-adjusted returns and the Reversal Coefficients can be sensitive to the choice of market index. Hence, we also considered the effect of using the Financial Times 500 (value-weighted) index. It was found that the return asymmetry was even more pronounced than before -the Loser portfolio posted a return of 209.4% ( t = 4.86) and the Winner portfolio one of - 4 6 . 0 % (t = -2.73). But, the reversal coefficients (Type II) for the losers and winners were .286 and .332, suggesting that the price reversals were actually near symmetric. 13
4. S u m m a r y and conclusions Some authors have interpreted the asymmetric test-period returns of winners and losers to mean that price reversals are asymmetric. We have shown that this is not necessarily the case. That is, simply because R L > [ R w 1, it cannot be inferred that price reversals are stronger for losers than for winners. We developed an
13 This w a s also b o r n e out b y the alternative type of reversal coefficient (i.e. Type I).
200
G. Dissanaike/Journal of Banking & Finance 20 (1996) 189-201
alternative measure, the Reversal Coefficient, which could be used to determine the strength of price reversals. We should stress that we are not advocating a change in the standard procedure used to detect price reversals. The test-period returns are very useful, not least because they tell us whether the reversals would have offered investors excess profit opportunities. We are only suggesting that the reversal coefficients offer additional useful information not captured by the test-period returns -they provide an indication about the strength of the price reversals and can therefore provide a greater insight into the nature of stock price reversals, irrespective of whether the reversals are rational or not. We also provided some empirical corroboration in order to demonstrate the practical usefulness of the reversal coefficient: Using the Reversal Coefficient technique in conjunction with an equal-weighted index, we found that the price reversals for winners were stronger than those for losers, and not vice versa, even though the returns to losers were much larger than the returns to winners. Further tests ought to be conducted in order to find out whether this finding is sample specific. But if this result were found to obtain within other sample periods, it would have interesting ramifications because, following the initial findings by De Bondt and Thaler (1985), slightly more attention has been devoted to losers. ~4 We also used the Reversal Coefficient technique in combination with a value-weighted index, but, in this case, the price reversals were actually found to be near symmetric (although the return asymmetry was even more pronounced). Our conceptual analysis also shows that the relationship between market inefficiency and the ability to devise profitable investment strategies, is rather more subtle than has generally been acknowledged: Ceteris paribus, it is more profitable to buy shares which are undervalued rather than to short-sell shares which are overvalued. Put differently, buying losers may prove more profitable than short-selling winners, but this does not necessarily mean that the extent of mispricing for losers -insofar as it is mispricing -exceeds that for winners.
Acknowledgements I am indebted to Prof. Geoffrey Whittington for many insights and suggestions. Thanks are also due to Elroy Dimson and Geoff Meeks for useful comments, and to Joyce Wheeler for extracting the data. They share no responsibility for any remaining errors. A major part of this research was completed whilst I held a Research Studentship for Economics, awarded by Trinity College, a President's Scholarship, and an Overseas Research Studentship awarded by the Committee of 14 See, for example, Bremer and Sweeney(1991).
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Vice-chancellors and Principals of the Universities of the UK. I also acknowledge a grant from the Eddington Fund of Trinity College, Cambridge.
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