Time Series Properties of Price Changes

Investors have used price charts and price patterns as tools for predicting future price movements for as long as there have been financial markets. It is not surprising, therefore, that the first studies of market efficiency focused on the relationship between price changes over time, to see if in fact such predictions were feasible. Some of this testing was spurred by the random walk theory of price movements, which contended that price changes over time followed a random walk. As the studies of the time series properties of prices have proliferated, the evidence can be classified into two classes - studies that focus on short-term (intraday, daily and weekly price movements) price behavior and research that examines long-term (annual and five-year returns) price movements.

a. Short term Price Movements

The notion that today's price change conveys information about tomorrow's price change is deep rooted in most investors' psyches. There are several ways in which this hypotheses can be tested in financial markets -

a. Serial correlation

The serial correlation measures the correlation between price changes in consecutive time periods, whether hourly, daily or weekly, and is a measure of how much the price change in any period depends upon the price change over the previous time period. A serial correlation of zero would therefore imply that price changes in consecutive time periods are uncorrelated with each other, and can thus be viewed as a rejection of the hypothesis that investors can learn about future price changes from past ones. A serial correlation which is positive, and statistically significant, could be viewed as evidence of price momentum in markets, and would suggest that returns in a period are more likely to be positive (negative) if the prior period's returns were positive (negative). A serial correlation which is negative, and statistically significant, could be evidence of price reversals, and would be consistent with a market where positive returns are more likely to follow negative returns and vice versa.

From the viewpoint of investment strategy, serial correlations can be exploited to earn excess returns. A positive serial correlation would be exploited by a strategy of buying after periods with positive returns and selling after periods with negative returns. A negative serial correlation would suggest a strategy of buying after periods with negative returns and selling after periods with positive returns. Since these strategies generate transactions costs, the correlations have to be large enough to allow investors to generate profits to cover these costs. It is therefore entirely possible that there be serial correlation in returns, without any opportunity to earn excess returns for most investors.

The earliest studies of serial correlation (Alexander (1964), Cootner (1962)and Fama (1965) all looked at large U.S. stocks and concluded that the serial correlation in stock prices was small. Fama, for instance, found that 8 of the 30 stocks listed in the Dow had negative serial correlations and that most of the serial correlations were less than 0.05. Other studies confirm these findings not only for smaller stocks in the United States, but also for other markets. For instance, Jennergren and Korsvold (1974) report low serial correlations for the Swedish equity market and Cootner (1961) conludes that serial correlations are low in commodity markets as well. While there may be statistical significance associated with some of these correlations, it is unlikely that there is enough correlation to generate excess returns.

The serial correlation in short period returns is affected by market liquidity and the presence of a bid-ask spread. Not all stocks in an index are liquid, and, in some cases, stocks may not trade during a period. When the stock trades in a subsequent period, the resulting price changes can create positive serial correlation. To see why, assume that the market is up strongly on day 1, but that three stocks in the index do not trade on that day. On day 2, if these stocks are traded, they are likely to go up in price to reflect the increase in the market the previous day. The net result is that you should expect to see positive serial correlation in daily or hourly returns in illiquid market indices.

The bid-ask spread creates a bias in the opposite direction, if transactions prices are used to compute returns, since prices have a equal chance of ending up at the bid or the ask price. The bounce that this induces in prices will result in negative serial correlations in returns. Roll (1984) provides a simple measure of this relationship,

Bid-Ask Spread = -V2 (Serial Covariance in returns) where the serial covariance in returns measures the covariance between return changes in consecutive time periods. For very short return intervals, this bias induced in serial correlations might dominate and create the mistaken view that price changes in consecutive time periods are negatively correlated.

b. Filter Rules

In a filter rule, an investor buys an investment if the price rises X% from a previous low and holds the investment until the price drops X% from a previous high. The magnitude of the change (X%) that triggers the trades can vary from filter rule to filter rule. with smaller changes resulting in more transactions per period and higher transactions costs. Figure 6.1 graphs out a typical filter rule.

Figure 6.1: Filter Rule

Price

Figure 6.1: Filter Rule

This strategy is based upon the assumption that price changes are serially correlated and that there is price momentum, i.e., stocks which have gone up strongly in the past are more likely to keep going up than go down. Table 6.4 summarizes results from a study on returns, before and after transactions costs, on a trading strategy based upon filter rules ranging from 0.5% to 20%. ( A 0.5% rule implies that a stock is bought when it rises 0.5% from a previous low and sold when it falls 0.5% from a prior high.)

Table 6.4: Returns on Filter Rule Strategies

Value of X

Return with

Return with Buy

# Transactions

Return after

strategy

and Hold

with strategy

transactions costs

0.5%

11.5%

10.4%

12,514

-103.6%

1.0%

5.5%

10.3%

8,660

-74.9%

2.0%

0..2%

10.3%

4,764

-45.2%

3.0%

-1.7%

10.1%

2,994

-30.5%

4.0%

0.1%

10.1%

2,013

-19.5%

5.0%

-1.9%

10.0%

1,484

-16.6%

6.0%

1.3%

9.7%

1,071

-9.4%

7.0%

0.8%

9.6%

828

-7.4%

8.0%

1.7%

9.6%

653

-5.0%

9.0%

1.9%

9.6%

539

-3.6%

10.0%

3.0%

9.6%

435

-1.4%

12.0%

5.3%

9.4%

289

2.3%

14.0%

3.9%

10.3%

224

1.4%

16.0%

4.2%

10.3%

172

2.3%

18.0%

3.6%

10.0%

139

2.0%

20.0%

4.3%

9.8%

110

3.0%

The only filter rule that beats the returns from the buy and hold strategy is the 0.5% rule,

The only filter rule that beats the returns from the buy and hold strategy is the 0.5% rule, but it does so before transactions costs. This strategy creates 12,514 trades during the period which generate enough transactions costs to wipe out the principal invested by the investor. While this test is an dated, it also illustrates a basic strategies that require frequent short term trading. Even though these strategies may earn excess returns prior to transactions costs, adjusting for these costs can wipe out the excess returns.

One popular indicator among investors that is a variant on the filter rule is the relative strength measure, which relates recent prices on stocks or other investments to either average prices over a specified period, say over six months, or to the price at the beginning of the period. Stocks that score high on the relative strength measure are considered good investments. This investment strategy is also based upon the assumption of price momentum.

c. Runs Tests

A runs test is a non-parametric variation on the serial correlation, and it is based upon a count of the number of runs, i.e., sequences of price increases or decreases, in the price changes. Thus, the following time series of price changes, where U is an increase and D is a decrease would result in the following runs -

UUU DD U DDD UU DD U D UU DD U DD UUU DD UU D UU D There were 18 runs in this price series of 33 periods. The actual number of runs in the price series is compared against the number that can be expected10 in a series of this length, assuming that price changes are random. If the actual number of runs is greater than the expected number, there is evidence of negative correlation in price changes. If it is lower, there is evidence of positive correlation. A study of price changes in the Dow 30 stocks, assuming daily, four-day, nine-day and sixteen day return intervals provided the following results -

DIFFERENCING INTERVAL

Daily

Four-day

Nine-day

Sixteen-day

Actual runs

735.1

175.7

74.6

41.6

Expected runs

759.8

175.8

75.3

41.7

Based upon these results, there is evidence of positive correlation in daily returns but no evidence of deviations from normality for longer return intervals.

Again, while the evidence is dated, it serves to illustrate the point that long strings of positive and negative changes are, by themselves, insufficient evidence that markets are not random, since such behavior is consistent with price changes following a random walk. It is

10 There are statistical tables that summarize the expected number of runs, assuming randomness, in a series of any length.

the recurrence of these strings that can be viewed as evidence against randomness in price behavior.

Long-term Price Movements

While most of the earlier studies of price behavior focused on shorter return intervals, more attention has been paid to price movements over longer periods (one-year to five-year) in recent years. Here, there is an interesting dichotomy in the results. When long term is defined as months rather than years, there seems to be a tendency towards positive serial correlation. Jegadeesh and Titman present evidence of what they call "price momentum" in stock prices over time periods of up to eight months when investors winner and loser stocks. However, when long term is defined in terms of years, there is substantial negative correlation returns, suggesting that markets reverse themselves over very long periods.

Fama and French examined five-year returns on stocks from 1931 to 1986 and present further evidence of this phenomenon. Studies that break down stocks on the basis of market value have found that the serial correlation is more negative in five-year returns than in one-year returns, and is much more negative for smaller stocks rather than larger stocks. Figure 6.2 summarizes one-year and five-years serial correlation by size class for stocks on the New York Stock Exchange.

Figure 6.2: Serial Correlation in Stock Returns

Figure 6.2: Serial Correlation in Stock Returns

123456789 10 Size class ¡smallest to biggest)

This phenomenon has also been examined in other markets, and the findings have been similar. There is evidence that returns reverse themselves over long time period.

Winner and Loser portfolios

Since there is evidence that prices reverse themselves in the long term for entire markets, it might be worth examining whether such price reversals occur on classes of stock within a market. For instance, are stocks that have gone up the most over the last period more likely to go down over the next period and vice versa? To isolate the effect of such price reversals on the extreme portfolios, DeBondt and Thaler constructed a winner portfolio of 35 stocks, which had gone up the most over the prior year, and a loser portfolio of 35 stocks, which had gone down the most over the prior year, each year from 1933 to 1978, and examined returns on these portfolios for the sixty months following the creation of the portfolio. Figure 6.3 summarizes the excess returns for winner and loser portfolios . Figure 6.3: Excess Returns for Winner and Loser Portfolios

Winner portfolio

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

Months after portfolio formation

Average of 46 yearly replications, starting every January between 1933 and 1978

FIGURE 23.23 Cumulative average residuals for winner and loser portfolios of 35 stocks, one to 60 months after portfolio formation; length of formation period: 5 years.

Winner portfolio

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

Months after portfolio formation

Average of 46 yearly replications, starting every January between 1933 and 1978

FIGURE 23.23 Cumulative average residuals for winner and loser portfolios of 35 stocks, one to 60 months after portfolio formation; length of formation period: 5 years.

This analysis suggests that loser portfolio clearly outperform winner portfolios in the sixty months following creation. This evidence is consistent with market overreaction and correction in long return intervals. Jegadeesh and Titman find the same phenomenon occurring, but present interesting evidence that the winner (loser) portfolios continue to go up (down) for up to eight months after they are created and it is in the subsequent periods that the reversals occur.

There are many, academics as well as practitioners, who suggest that these findings may be interesting but that they overstate potential returns on 'loser' portfolios. For instance, there is evidence that loser portfolios are more likely to contain low priced stocks (selling for less than $5), which generate higher transactions costs and are also more likely to offer heavily skewed returns, i.e., the excess returns come from a few stocks making phenomenal returns rather than from consistent performance. One study of the winner and loser portfolios attributes the bulk of the excess returns of loser portfolios to low-priced stocks and also finds that the results are sensitive to when the portfolios are created. Loser portfolios created every December earn significantly higher returns than portfolios created every June.

Speculative Bubbles, Crashes and Panics

Historians who have examined the behavior of financial markets over time have challenged the assumption of rationality that underlies much of efficient market theory. They point out to the frequency with speculative bubbles have formed in financial markers, as investors buy into fads or get-rich-quick schemes, and the crashes with these bubbles have ended, and suggest that there is nothing to prevent the recurrence of this phenomenon in today's financial markets. There is some evidence in the literature of irrationality on the part of market players.

a. Experimental Studies of Rationality

Some of the most interesting evidence on market efficiency and rationality in recent years has come from experimental studies. While most experimental studies suggest that traders are rational, there are some examples of irrational behavior in some of these studies.

One such study was done at the University of Arizona. In an experimental study, traders were told that a payout would be declared after each trading day, determined randomly from four possibilities - zero, eight, 28 or 60 cents. The average payout was 24 cents. Thus the share's expected value on the first trading day of a fifteen day experiment was $3.60 (24*15), the second day was $3.36 The traders were allowed to trade each day. The results of 60 such experiments is summarized in figure 6.4.

Figure 6.4: Trading Price by Trading Day

Figure 6.4: Trading Price by Trading Day

Trading Days

Trading Days

There is clear evidence here of a 'speculative bubble' forming during periods 3 to 5, where prices exceed expected values by a significant amount. The bubble ultimately bursts, and prices approach expected value by the end of the period. If this is feasible in a simple market, where every investor obtains the same information, it is clearly feasible in real financial markets, where there is much more differential information and much greater uncertainty about expected value.

It should be pointed out that some of the experiments were run with students, and some with Tucson businessmen, with 'real world' experience. The results were similar for both groups. Furthermore, when price curbs of 15 cents were introduced, the booms lasted even longer because traders knew that prices would not fall by more than 15 cents in a period. Thus, the notion that price limits can control speculative bubbles seems misguided.

b. Behavioral Finance

The irrationality sometimes exhibited by investors has given rise to a whole new area of finance called behavioral finance. Using evidence gathered from experimental psychology, researchers have tried to both model how investors react to information and predict how prices will change as a consequence. They have been far more successful at the first endeavor than the second. For instance, the evidence seems to suggest the following:

a. Investors do not like to admit their mistakes. Consequently, they tend to hold on to losing stocks far too long, or in some cases, double up their bets (investments) as stocks drop in value.

b. More information does not always lead to better investment decisions. Investors seem to suffer both from information overload and a tendency to react to the latest piece of information. Both result in investment decisions that lower returns in the long term.

If the evidence on how investors behave is so clear cut, you might ask, why are the predictions that emerge from these models so noisy? The answer, perhaps, is that any model that tries to forecast human foibles and irrationalities is, by its very nature, unlikely to be a stable one. Behavioral finance may emerge ultimately as a trump card in explaining why and how stock prices deviate from true value, but their role in devising investment strategy still remains questionable._

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