Developing Stock Trading Strategies. Part-1

Table of Contents
1.   Introduction
2.   Comparison of trading strategies
3.   Return per trade
4.   Average return per trade
5.   More about average return
6.   Growth coefficient
7.   Distribution of returns
8.   Risk of trading
9.   More about risk of trading

This publication is for short-term traders, i.e. for traders who hold stocks for one to eight days. Short-term trading assumes buying and selling stocks often. After two to four months a trader will have good statistics and he or she can start an analysis of trading results. What are the main questions which should be answered from this analysis?

- How can I increase the profitability of my strategy and decrease the risk of trading?

No doubt it is better to ask these questions before using any trading strategy. We will consider methods of estimating profitability and risk of trading strategies, optimally dividing trading capital, using stop and limit orders and many other problems related to stock trading.

Consider two hypothetical trading strategies. Suppose you use half of your trading capital to buy stocks selected by your secret system and sell them on the next day. The other half of your capital you use to sell short some specific stocks and close positions on the next day.

In the course of one month you make 20 trades using the first method (let us call it strategy #1) and 20 trades using the second method (strategy #2). You decide to analyze your trading results and make a table which shows the returns (in %) for every trade you made.

#	Return per trade in % Strategy 1	Return per trade in % Strategy 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20	+3 +2 +3 -5 +6 +8 -9 +5 +6 +9 +1 -5 -2 +0 -3 +4 +7 +2 -4 +3	+4 -5 +6 +9 -16 +15 +4 -19 +14 +2 +9 -10 +8 +15 -16 +8 -9 +8 +16 -5

The next figure graphically presents the results of trading for these strategies.

Which strategy is better and how can the trading capital be divided between these strategies in order to obtain the maximal profit with minimal risk? These are typical trader's questions and we will outline methods of solving them and similar problems.

The first thing you would probably do is calculate of the average return per trade. Adding up the numbers from the columns and dividing the results by 20 (the number of trades) you obtain the average returns per trade for these strategies

Does this mean that the second strategy is better? No, it does not! The answer is clear if you calculate the total return for this time period. A definition of the total return for any given time period is very simple. If your starting capital is equal to C0 and after some period of time it becomes C1 then the total return for this period is equal to

Surprisingly, you can discover that the total returns for the described results are equal to

What happened? The average return per trade for the first strategy is smaller but the total return is larger! Many questions immediately arise after this "analysis":

- How should we use these strategies to obtain the maximum profit with minimal risk?

To answer these questions let us introduce some basic definitions of trading statistics and then outline the solution to these problems.

Return per Trade
Suppose you bought N shares of a stock at the price P0 and sold them at the price P1. Brokerage commissions are equal to COM. When you buy, you paid a cost price

Suppose you made n trades with returns R1, R2, R3, ..., Rn. One can define an average return per trade Rav

This calculations can be easily performed using any spreadsheet such as MS Excel, Origin, ... .

You can easily check that the described definition of the average return is not perfect. Let us consider a simple case.

Suppose you made two trades. In the first trade you have gained 50% and in the second trade you have lost 50%. Using described definition you can find that the average return is equal to zero. In practice you have lost 25%! Let us consider this contradiction in details.

Suppose your starting capital is equal to $100. After the first trade you made 50% and your capital became

So you have lost $25, which is equal to -25%. It seems that the average return is equal to -25%, not 0%.

This contradiction reflects the fact that you used all your money for every trade. If after the first trade you had withdrawn $50 (your profit) and used $100 (not $150) for the second trade you would have lost $50 (not $75) and the average return would have been zero.

In the case when you start trading with a loss ($50) and you add $50 to your trading account and you gain 50% in the second trade the average return will be equal to zero. To use this trading method you should have some cash reserve so as to an spend equal amount of money in every trade to buy stocks. It is a good idea to use a part of your margin for this reserve.

However, very few traders use this system for trading. What can we do when a trader uses all his trading capital to buy stocks every day? How can we estimate the average return per trade?

where Sale1 and Cost1 represent the sale and cost of trade #1. This ratio we call the growth coefficient. If the growth coefficient is larger than one you are a winner. If the growth coefficient is less than one you are a loser in the given trade.

If K1, K2, ... are the growth coefficients for trade #1, trade #2, ... then the total growth coefficient can be written as a product

In our previous example the growth coefficient for the first trade K1 = 1.5 and for the second trade K2 = 0.5. The total growth coefficient which reflects the change of your trading capital is equal to

which correctly corresponds to the real change of the trading capital. For n trades you can calculate the average growth coefficient Kav per trade as

These calculations can be easily performed by using any scientific calculator. The total growth coefficient for n trades can be calculated as

In our example Kav = (1.5 * 0.5) ^ 1/2 = 0.866, which is less than 1. It is easily to check that

However, the average returns per trade Rav can be used to characterize the trading strategies. Why? Because for small profits and losses the results of using the growth coefficients and the average returns are close to each other. As an example let us consider a set of trades with returns

Kav =
(0.95*1.07*0.99*1.02*0.97*1.05*1*1.02*0.9*1.11*0.98*1.05*1.03*0.99*1.02)^(1/15)
= 1.009

which corresponds to 0.9%. This is very close to the calculated value of the average return = 1%. So, one can use the average return per trade if the return per trades are small.

Let us return to the analysis of two trading strategies described previously. Using the definition of the average growth coefficient one can obtain that for these strategies

So, the average growth coefficient is less for the second strategy and this is the reason why the total return using this strategy is less.

If the number of trades is large it is a good idea to analyze the trading performance by using a histogram. Histogram (or bar diagram) shows the number of trades falling in a given interval of returns. A histogram for returns per trade for one of our trading strategies is shown in the next figure

As an example, we have considered distribution of returns for our Low Risk Trading Strategy (see more details in http://www.stta-consulting.com) from January 1996 to April 2000. The bars represent the number of trades for given interval of returns. The largest bar represents the number of trades with returns between 0 and 5%. Other numbers are shown in the Table.

Return Range, %

Number of Stocks

Return Range, %

Number of Stocks

0 < R< 5

5 < R< 10

10 < R< 15

15 < R< 20

20 < R< 25

25 < R< 30

30 < R< 35

35 < R< 40

249

174

127

-5 < R< 0

-10 < R< -5

-15 < R< -10

-20 < R< -15

-25 < R< -20

-30 < R< -25

-35 < R< -30

-40 < R< -35

171

For this distribution the average return per trade is 4.76%. The width of histogram is related to a very important statistical characteristic: the standard deviation or risk.

The larger the standard deviation, the wider the distribution of returns. A wider distribution increases the probability of negative returns, as shown in the next figure.

Distributions of returns per trade for Rav = 3% and for different standard deviations

Therefore, one can conclude that a wider distribution is related to a higher risk of trading. This is why the standard distribution of returns is called the risk of trading. One can also say that risk is a characteristic of volatility of returns.

The smaller the risk-to-return ratio, the better the trading strategy. If this ratio is less than 3 one can say that a trading strategy is very good. We would avoid any trading strategy for which the risk-to-return ration is larger than 5. For distribution in Fig. 1.2 the risk-to-return ratio is equal to 2.6, which indicates low level of risk for the considered strategy.

Returning back to our hypothetical trading strategies one can estimate the risk to return ratios for these strategies. For the first strategy this ratio is equal to 3.2. For the second strategy it is equal to 5.9. It is clear that the second strategy is extremely risky, and the partion of trading capital for using this strategy should be very small.

How small? This question will be answered when we will consider the theory of trading portfolio.

The definition of risk introduced in the previous section is the simplest possible. It was based on using the average return per trade. This method is strightforward and for many cases it is suficient for comparing different trading strategies.

However, we have mentioned that this method can give false results if returns per trade have a high volatility (risk). One can easily see that the larger the risk, the larger the difference between estimated total returns using average returns per trade or the average growth coefficients. Therefore, for highly volatile trading strategies one should use the growth coefficients K.

Using the growth coefficients is simple when traders buy and sell stocks every day. Some strategies assume specific stock selections and there are many days when traders wait for opportunities by just watching the market. The number of stocks that should be bought is not constant.

In this case comparison of the average returns per trade contains very little information because the number of trades for the strategies is different and the annual returns will be also different even for equal average returns per trade.

One of the solutions to this problem is considering returns for a longer period of time. One month, for example. The only disadvantage of this method is the longer period of time required to collect good statistics.

Another problem is defining the risk when using the growth coefficients. Mathematical calculation become very complicated and it is beyond the topic of this publication. Here, we will use a tried and true definition of risk via standard deviations of returns per trade in %. In most cases this approach is sufficient for comparing trading strategies. If we feel that some calculations require the growth coefficients we will use them and we will insert some comments about estimation of risk.

The main goal of this section to remind you that using average return per trade can slightly overestimate the total returns and this overestimation is larger for more volatile trading strategies.