In his book, A Random Walk Down Wall Street, the author Burton Malkiel carried out an experiment in which he asked his students to toss a coin repetitively and mark down the results on a piece of graph paper. If the coin toss came up heads then they noted an increase, if tails then a decrease. After numerous coin tosses the chart constructed closely resembled the movements of a share price. The Princeton professor then showed the piece of graph paper to a Technical Analyst and asked him to give a forecast. The unsuspecting analyst assumed it revealed the price fluctuations of a security, and proceeded to analyse the chart and make recommendations. He concluded the share was about to go up and advised to buy right now. When Burton Malkiel told him the chart didn’t show the movements of a share price, but was actually the results of coin tosses the analyst was understandably very sore. However, to Malkiel it proved that technical analysis was a spurious practice and that markets moved randomly*.
Anyone who looks at the daily chart of the AUD/USD will see long series of green up- days, with sequences of 4, 5 or 6 green candles in a row. Compared to other securities the AUD/USD seems particularly prone to trending strongly. In fact since 2001 it has risen a staggering 122% from an all time low of 0.47750 to recent all time highs of 1.0582. Surely, given Malkiel’s experiment, this sort of bias to the upside would be unlikely to occur from the results of a toss of coin? If indeed financial market’s move randomly wouldn’t the price be expected to ‘revert to mean’ over time? Isn’t that also the theoretical underpinning of the sage Wall Street advice to ‘buy low and sell high’? If that is so how does this explain the Aussie’s relentless rise?
In this article we ask whether there is a significant bias towards a prevalence of up-days in AUD/USD or whether the random walkers are correct and the Aussie’s rise can be explained as a purely chance occurrence. This is part of a long standing debate between the school of thought who believe the art of technical analysis – or trend following – is of considerable merit in predicting market change and reverent critics like Malkiel, who refute the effectiveness and accuracy of this forecasting method.
‘A G’Day for the Dollar?’
In a purely random series there is no overall probability of a certain type of day occurring more frequently than another type of day. But is this the case with AUSD/USD?
Our experiment used all available data sampled over a 20 year timespan from between 04/04/91 to 04/04/11. The total number of 5138 days were sorted into categories of ‘up’ (close higher than previous close) ‘down’ (close less than previous close) and ‘unchanged’ and the breakdown was as follows:
Up days = 2658
Down days = 2407
Unchanged days = 73
Firstly, lets ignore the unchanged days for a moment and focus on the up days and down day results. We see there is a definite bias towards market rise – 2658 up compared to 2407 down days – a difference of 251. This may not seem like a large amount, but it is a difference of around 9%. Given there is a 50/50 chance of either spinning heads or tails we would expect a result of 2533/2532 – or thereabouts at least. But what would be the chance of getting 2658 heads and 2407 as in the results of our experiment?
In fact, the probability of getting 251 more heads than tails in the sample is actually remarkably low. There is only a 0.024%** chance of obtaining 2658 or more ‘heads’ in a purely randomly generated sample. This low figure demonstrates that it is highly unlikely that the data for AUD/USD was randomly generated. According to scientifically acceptable levels of statistical significance this strongly refutes Burton Malkiel’s random walk hypothesis.
Now, if we look at the actual sequences of days we can ask a further question pertinent to the debate. What is the probability of one type of day following another? If markets are truly random there shouldn’t be any difference in the chance of one day following another other. Ignoring unchanged days there are only 4 possible outcomes for a certain day following another day, these are as follows:
1) Down (defined as a lower close from the previous day) followed by up (defined as a higher close than the day before)
2) Up followed by up,
3) Down followed by down and
4) Up followed by down.
Below are the numbers of days for each of the possible outcomes using AUD/USD:
1) = 1254
2) = 1358
3) = 1118
4) = 1258
If the currency pair moved in a completely random way then the number of times each outcome occurred should be equal. Given there are only 4 possible outcomes and they are all equally likely to appear there would be a ¼ – or 25% chance of any single one occurring randomly.
The percentages for the outcomes in the data for the AUS/USD however are as follows:
1) = 25.14%
2) = 27.23%
3) = 22.41%
4) = 25.22%
Outcomes 1) and 4) show a very close correspondence to random expectations, however 2) and 3) are not as close to the 25% expected. Is the deviation statistically significant, however?
The probabilities of the experiment results happening in a purely randomly generated data set are as follows:
|Exactly the outcome||Less than the outcome||More than the outcome|
The first column is the chance of the outcome occurring exactly (e.g. getting precisely 1254 – no more or less), the second number is the probability of less than the result occurring (e.g. there is a 59.59% chance of getting less than 1254 from a random data set) and the third number is the chance of a randomly generated set producing a figure higher than the result (e.g. there is a 41.68% chance of getting more than 1254).
As mentioned above columns for outcomes 1) and 4) are quite close to that expected from a purely randomly generated data series, however 2) and 3) lie at the extremes. These results show that there is < 0.00% chance that a random generated set of results would produce the result in row 3), of 1118 down followed by down days.
The very low probability in the sequences of up followed by up, and down followed by down days passes the generally accepted minimum requirement for proving statistical significance of alpha 5% and 1% .
Two Up? Too right!
The experiment proves that it is highly unlikely the long uninterrupted trend in the Aussie Dollar can be explained using a random walk model of the markets. There is a statistically significant bias in the data for the general occurrence of up days and up days followed by up days as well as a statistically significant dearth of the inverse. Clearly the Aussie’s like to go walkabout in a not so random line!
* Malkiel has since revised his view and has accepted that some markets do reveal non-random characteristics.
**Probabilities were generated using the binomial probability calculator at the following link:http://faculty.vassar.edu/lowry/binomialX.html