In his classic, A Random Walk Down Wall Street, Burton Malkiel argues that markets follow a random walk. To illustrate what he means by random walk, he provides example charts generated by tossing a coin. Depending on which way the coin fell, the market would move up or down. An anecdote in the text describes a chart analyst's excited reaction to these faux price charts. Malkiel's point here regarding chartist techniques is not without merit, but we have some other concerns with Malkiel's stance.
Malkiel presents the example of a random walk as background for the following argument against technical analysis:
1. If the market moves in a random walk, then it is impossible to use available price data to predict future prices.
2. The market moves in a random walk.
3. It is impossible to use available price data to predict future prices. (from 1 and 2)
4. If it is impossible to use past prices to predict future prices, then it is impossible to trade profitably making decisions based on available price data.
5. It is impossible to trade profitably making decisions based on available price data. (from 3 and 4)
Statement 1 is undeniable and we'll go along with statement 2 for the time being. Statement 3 is certainly valid from 1 and 2. Statement 5 clearly follows from 3 and 4. Statement 4, however, is false and causes the argument to reach a false conclusion in 5. Let's take a look at why statement 4 is false.
Constructing a random walk
To construct a model of a random walk, we begin with a hypothetical instrument trading at price level 100. At the end of one period, the price will move either up or down based on a random process such as the toss of a coin. If the coin lands heads, the new price becomes the prior price multiplied by 1.01, an increase of one per cent. If the coin lands tails, the new price becomes the prior price divided by 1.01, which exactly offsets the effect of a move up. To mimimize the effects of rounding, all price values are taken to the 10,000ths place.
Possible price paths
The following table shows all of the possible price paths through a trading week.
Possible Weekly Price Paths
**EDIT BY NAKED SHORTS: PLEASE SEE ATTACHED PICTURE**
Note that certain outcomes can happen in only one way. For example a price of 104.0604 on Friday can only occur if the market moves up on each day during the week. The odds of such a week happening are 1/16. Other outcomes are far more likely. A price level on Friday of 100.0000, no net change for the week, is the most likely outcome. The odds of a week resulting in no net change are 6/16.
Profiting from randomness
We now have a random walk model for price change which makes it impossible to use available data to predict future prices. If Malkiel is correct, no set of price based rules will be effective in profitably trading this imaginary market. Is this the case? To test this theory, we have a simple set of four rules for trading based solely on price data.
1. Enter a long position (buy) on Monday.
2. Exit the position (sell) as soon as there is a profit of 1%.
3. If today is the third down day of the week, exit the position.
4. If the position is still open on Friday, exit the position.
The next table lists every possible week that could materialize under this random walk model, as well as the outcomes of following the four rules.
Entry Price Market Action Rule to Exit Exit Price Profit
100.0000 UP,UP,UP,UP 2 101.0000 1.0000
100.0000 UP,UP,UP,DOWN 2 101.0000 1.0000
100.0000 UP,UP,DOWN,UP 2 101.0000 1.0000
100.0000 UP,UP,DOWN,DOWN 2 101.0000 1.0000
100.0000 UP,DOWN,UP,UP 2 101.0000 1.0000
100.0000 UP,DOWN,UP,DOWN 2 101.0000 1.0000
100.0000 UP,DOWN,DOWN,UP 2 101.0000 1.0000
100.0000 UP,DOWN,DOWN,DOWN 2 101.0000 1.0000
100.0000 DOWN,UP,UP,UP 2 101.0000 1.0000
100.0000 DOWN,UP,UP,DOWN 2 101.0000 1.0000
100.0000 DOWN,UP,DOWN,UP 4 100.0000 0.0000
100.0000 DOWN,UP,DOWN,DOWN 3 98.0296 -1.9704
100.0000 DOWN,DOWN,UP,UP 4 100.0000 0.0000
100.0000 DOWN,DOWN,UP,DOWN 3 98.0296 -1.9704
100.0000 DOWN,DOWN,DOWN,UP 3 97.0590 -2.9410
100.0000 DOWN,DOWN,DOWN,DOWN 3 97.0590 -2.9410
By taking a total of the profit column and dividing by 16, we can determine that following this trading system on such a random market would yield 0.0111 points on average, per trade. So here we have a case where prices in the future are entirely independent of prices in the past and present, making future price prediction entirely impossible. Even so, a set of price based trading rules can be (slightly) profitable.
Some might point out that the profits in this model seem small. This is largely because we opted to keep our example as simple as possible. More profitable methods of trading a random walk exist, but such methods would require an inconveniently large outcome table and other modifications.
Also, there were other models we could have used to construct a random walk. We chose the one we used because there is no bias up or down, and because it satisfies the requirement that prior changes have no predictive power for future changes.
Malkiel doesn't actually present his argument in a formal structure. The structure we present is based on our understanding of the random walk argument presented by Malkiel and others against technical analysis and is not intended to distort his case.
Implications and Conclusions
As our example clearly shows, making trading decisions based on available price data can be profitable even if that price data does not facilitate prediction making. Hence, even if the random walk theory were correct, this still would not invalidate the practice of trading according to rules based on price information. We certainly do not recommend trading from a set of rules based on a questionable model of the markets, such as random walk theory. However, the example we provide here serves as evidence that given a proper model for market movement, a system based on price is a solid way of generating profit.