Random walk and efficient market hypothesis

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Method

By taking the view of an investor we need to face transactions cost and our objective is to make a better payoff after these transaction costs are paid, then we would if we just bought the stock at day one and hold on to it. We will assume a transaction cost of 0.075 % per transaction, which on a day where we both buy and short sell will imply a total transaction cost of 0.15 %. The transaction cost is based upon what a small or medium sized investor could face in Sweden at one of the largest internet brokers. A larger investor may face even lower transaction cost than we are using, but to make the result interesting for as many as possible we choose 0.075 %.
The data (closing prices of OMX Stockholm 30 Index, OMXS30) is obtained from Nasdaq OMX´s website, from 1986-09-03 until 2012-01-27. The data of the risk-free rate are ob-tained from the Riksbank‟s website. We are conducting tests of different moving averages and divide our sample into five subsets with a number of years in each set to see if we might find some patterns that changed during the years. Hence, we are only dealing with secondary data, but which we find reliable. Figure 3 shows the development of the Swedish stock market during the time period we are interested in.
We are dividing the full period into five smaller sub periods to be able to find differences between them. In table 2 we describe our different sub periods. And the diagrams below show how the market has moved in the different periods.
The diagrams below illustrate the trend for each period. By dividing our data into different sub periods we can identify how the market trend influence which of the trading rules that produce the highest return.
A moving average model with 1 day in the short moving average, 50 in the long moving average and 1 % band would then look like (1,50,1). We are using the same variable length moving average as Fiefield, Power and Sinclair (2005) did in eleven other European mar-kets. We are also testing (1,30,0) and (1,30,1) to see if a smaller number of days in our long moving average will be a better prediction of future returns.
We have found that variable length moving average is often used in similar studies done in the past and in other markets. We do not think that a more advanced model is always bet-ter to predict future market movements. The more simple the model the easier it is to un-derstand and use for an investor, that is another reason why variable length moving average is popular. With fewer parameters the human error factors heavily diminish and the risk of wrongly interpret the data and the result. The different moving averages we are using are shown in table 3. The first column displays the amount of days in the short moving aver-age, the second column shows the amount of days in the long moving average and the third column shows the band. Each row is a separate model we are using.
The return on a given buy day are calculated by the given formula:
Where Rit is the return on the index at day t, ln is the natural logarithm, Pit is the closing price on index at day t and Pit-1 is the closing price on the index the day before day t.
The return on a given sell day (in case of short selling) is then the opposite:
We are then testing the statistical significance on each sub period and the whole period by making a linear regression on the excess return on the index versus the excess return on our trading, considering transaction costs. The excess return is the return on the invest-ment reduced by the return on the risk-free asset. The regression is made on the excess re-turn on the index and the excess return on the trading strategy. So it is the two different excess returns that are compared. The excess return is calculated as follows:
Where Rft is equal to the daily risk-free rate at time t. We are using the Swedish 30-days T-bills interest as our risk-free rate.
We are formulating a null hypothesis which we are testing.
H0 = The Swedish stock market is in the weak form of efficient market hypothesis.
HA = The Swedish stock market is not in the weak form of efficient market hypothesis.
Our regression estimates if the result is reliable or not and we are using t-tests with α = 0.1,
α = 0.05 and α= 0.01 to find the statistical significance.
We will also be testing for a random walk in our data set. We are using a unit root-test to find if our time series are following a random walk which would imply that technical trad-ing will not be able to predict future price movements.

Empirical findings and analysis

Table 4 and 5 displays the results, logarithmic return, for our twelve tested trading rules. The data is separated in five sub periods, and the last column display the result for the full time period 1987-2012. Table 4 displays the excess return and column 2 indicate if the row is displaying the total excess return for the period or the daily average excess return for the period. In the first row, total excess return:
While the second row is showing the average daily excess return:
As shown in table 4, each moving average indicates excess return in the last column but only a couple of them have statistical significance with alpha 1 % and 5 %, mostly the shorter ones. Another interesting trend is that the smaller moving averages was better than the larger ones during our first sub periods but during the last sub periods the larger mod-els have been the better ones, although this conclusion lack statistical significance.
In table 4 the excess return compared to the buy-and-hold strategy is presented. For each moving average the sum is presented in the first row and the average daily excess return is presented in the second row for each sub period and the entire period. The excess return is calculated, as mentioned earlier, as the return with trading reduced by the return for the buy-and-hold strategy. Hence, if the excess return is minus it does not necessarily implies that a portfolio using that moving average has decreased, but it has performed worse than the buy-and-hold strategy.
During both of the sub periods with clear upward trends (1993-1997 and 2003-2007), we have a similar pattern where no model beats the buy-and-hold strategy and yields excess re-turn. But then there are opposite results when the stock market lacks a clear trend and in our third sub period (1998-2003) all the moving averages shows an excess return, in the first period (1986-1993) all but the largest moving average (2,200,0/1) shows excess return and in the last period (2008-2012) it is only the smallest moving average (1,30,0/1) that does not yield an excess return. It is not surprising that trading stocks actively is less bene-ficial during bullish markets since it is increasing the “risk” of missing upturns in the stock market. A bad signal does not only generate a negative payoff, is also decreases the invest-ment due to transaction costs. When the market is not in a bullish trend, perhaps if it is in a bearish trend, it is beneficial to move out of the market for some time and even to be short selling assets. A good short sell is in a way twice as good as a regular investment since the investor does not only avoiding the drop but also making money out of it. When compar-ing a trading strategy with the buy-and-hold strategy on the same asset it is only the days out of the market, or in our case the days we are short selling, that may yield a positive ex-cess return. On a given buy day our portfolio increases/decreases exactly as much as the benchmark which does not yield an excess return. That is the main reason that it is difficult to yield excess returns during bullish markets, even though a positive payoff is still ob-tained.
Hence, the Swedish stock market seems to have changed. The best moving average for predicting future returns have changed from a small moving average to a larger moving av-erage, but once again this lack statistical evidence. For an investor trying to create an excess return with these methods this is not favorable. Since which moving average that works best is constantly changing the investor cannot know for certain which will be the best for the next few years. Although the pattern does not change that fast, the model that was best in one period is also in the top in the next period hence the investor can choose the one that was best in the previous period.
The strategy that has yield the largest excess return during the entire period is (1,50,0) and is statistical significant with a confidence interval of 99 %. Both (1,30,0) and (1,30,1) did very well in the first period (1987-1992) but comparing the first period with the total period there is a clear pattern of decline even though they were both performing well under the third period (1998-2002). The exact opposite goes for the larger moving averages which did not perform well during the early 90‟s but has been outperforming shorter ones during the two last periods (2003-2012). Using 200 days in the long moving average combined with a one percent band has proven to yield the highest excess return since 2003.
Table 5 shows the average daily return on a given buy day on the first row and a given sell day on the second row for each model and sub period. The return on sell days is the return for an investor whom has chosen to short sell, which means that the return on the index is the opposite of the table. As seen in the last column for each and every model the average return on a buy day have been better than on a sell day during the whole period. That is not surprising though since the market has increased a lot since 1987. The negative returns (loss for investors) are more likely to appear when a sell signal has occurred than when a buy signal has occurred which make short selling less attractive. In other words if you only choose between going short or going long it is preferable to go long. Although going short have in many cases proved to yield positive payoff as well. For most models it is during 1993-1997 and 2003-2007 that short selling has yield a negative return, the periods with bullish markets. For most models, all but (2,200,0) and (2,200,1), short selling has yield an average positive return for the investor during the whole period.
As in table 4 we can confirm that the two shortest models (1,30,0) and (1,30,1) have not been beneficial in the last two periods (2003-2007 and 2008-2012), where neither the buy nor the sell signals have generated a positive payoff in the last period (2008-2012). The model that yield the best excess return (1,50,0) has only two periods with negative pay off for short selling, period 1993-1997 and 2003-2007. Furthermore all the buy signals generat-ed by (1,50,0) have a positive payoff. In the three periods 1987-1992, 1998-2002 and 2008-2012 the sell signals have generated a positive return and the average return have been larg-er then on the buy days. This implies that short selling is an important part of the model.
If our moving average models would not be able to predict future returns there would not be a difference on average return on a given buy day or a given sell day. Since in table 5 we find that in almost every model yields a positive payoff on both buy and sell days (all but two models) there certainly is a difference which supports our results that technical analysis in fact may predict future returns.
Consider that two investments were made when the first signal was generated by the (1,50,0) strategy, one which would be using a buy-and-hold strategy and the other would be using the signals generated by the moving average. During the next approximately 24 years which our study is covering the return from the buy-and-hold has increased by 8.7 times the original investment. The trading strategy on the other hand would have increased by 27.3 times the original investment. Hence it has been a huge difference for investors who have traded their investment with a good strategy comparing those who have let their in-vestment be still for the entire period.
If a market is inefficient that implies that an asset is not always priced at its true value, it may be both underpriced and overpriced. Since assets may be wrongly priced it is not only possible for an investor to make an excess return, investors might lose more/gain less than warranted by their risk profile. That is one of the reasons that it is preferable to invest in a more efficient market, due to a higher level of risk control in a more efficient market.
But according to these result, the most of the excess return seems to appears in the first pe-riod. Has technical trading not been profitable since then? We choose to test our strategies after the two first period and tests from 1998-2012 to find out if our trading rules have provided a statistical significant excess return during that period.
According to our test we find that some of our models have provided an excess return with statistical significance. The result is presented in table 6, it shows in which level of alpha each model is statistical significant.
Hence, technical trading rule has been efficient even after 1992.
We perform a test to find if our time series has a unit root. If a unit root exists it would not be possible to use technical trading to make an excess return, since the data would follow a random walk. We are using the formula:
Where we formulate the hypothesis:
H0: (unit root exists, hence, random walk).
HA: (no unit root exists, hence, no random walk).
Our beta differs slightly from 1 but without statistical significance at the 90 % confidence interval. So we cannot reject that there is a unit root in our full data series which is contra-dicting to our previous results. With a unit root the random walk hypothesis holds which would then imply that the weak form of efficient market hypothesis also holds.
On the other hand when testing the last three years instead of the entire time period we find that our β < 1 which is statistical significant at the 95 % confidence interval. Hence, the market does not seem to follow a random walk at all times which make it possible to create an excess return in between. We test for a unit root in all of our sub periods and find that we can actually reject that there is a unit root in the first and last sub period at α = 0.1. The other three sub periods has a beta that differs from 1 but not with enough statistical significance. Although the forth sub period and the full period is almost rejected as well, with α = 0.11 they would be rejected hence there is no clear evidence for a unit root.
Our result after testing the profitability of technical trading is therefore supported by our unit root test. The result that it is possible to create an excess return on the Swedish market by technical trading is also consistent with the result of the Metghalchi et al. (2008) study on data until 2004.
For a market to become efficient it is, ironically, important for people to believe in an inef-ficient market. When people try to exploit mispricing in the market they drive the market to efficiency. On the other hand if people tend to believe that the all information is fully re-flected and do not try to exploit mispricing assets might stay mispriced leading to ineffi-ciency. The wheel then goes round and round and the market will never be fully inefficient or efficient.
The technology with computers and high frequency traders are making the market more ef-ficient which is most likely one of the reasons that our largest excess return is in the first sub period. Today it is easier for investors to gain information at lower cost (both in terms of time and money) and making analysis becomes easier in a way for the investors. The al-gorithmic traders are offering a more liquid market which makes it less volatile and pushes prices faster to their true values exploiting mispricing.
If a market is inefficient this implies that an asset is not always priced at its true value, it may be both underpriced and overpriced. Since assets may be wrongly priced it is not only possible for an investor to make an excess return, investors might lose more/gain less than warranted by their risk profile. That is one of the reasons that it is preferable to invest in a more efficient market, due to a higher level of risk control in a more efficient market.
In an efficient market where assets are priced at its true value bubbles should not occur nor would financial crashes on the stock market. Of course a market could rise or fall quite heavily at times, but the tops would not be as high and the bottoms not as low as the case in an inefficient market. In bubbles investors tend to pay overprice compared to a true val-ue, in crashes investors tend to sell at underprice. That a market is not strong efficient is understandable since not every investor has the same information and different investors tend to evaluate the same information differently. Although that alone would not lead to these bubbles and crashes. It is argued that during bubbles investors follow the crowd, eve-ryone believes in future rise which during some time might be right, until the price hits the ceiling. When the top is reached the mood of the crowd suddenly changes and the inves-tors starts to sell, perhaps even in panic, which then pushes the price below its true value. Neither the rise nor the fall should happen in an efficient market. The only time a stock would move, significantly, is when new information is revealed. In a perfect market stock charts would look like they were drawn with a ruler, a flat line where the stock makes sud-den jumps up or down. But even if the market is not perfect, in the weak form of efficient market hypothesis the stock chart would be less blurry then they tend to be today.

Table of Contents
1 Introduction
2 Theoretical framework
2.1 Random walk and efficient market hypothesis
2.2 What is technical analysis?
2.3 Previous research
3 Method
4 Empirical findings and analysis
5 Conclusion
6 References
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