How to Win the Stock Market Game

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9 These data can be presented graphically. 9 Dependence of weekly returns on the SP 500 change for hypothetical strategy Using any graphical program you can plot the dependence of weekly returns on the SP 500 change and using a linear fitting program draw the fitting line as in shown in Figure. The correlation coefficient c is the parameter for quantitative description of deviations of data points from the fitting line. The range of change of c is from -1 to +1. The larger the scattering of the points about the fitting curve the smaller the correlation coefficient. The correlation coefficient is positive when positive change of some parameter (SP 500 change in our example) corresponds to positive change of the other parameter (weekly returns in our case). The equation for calculating the correlation coefficient can be written as where X and Y are some random variable (returns as an example); S are the standard deviations of the corresponding set of returns; N is the number of points in the data set. For our example the correlation coefficient is equal to This correlation is very high. Usually the correlation coefficients are falling in the range (-0.1, 0.2). We have to note that to correctly calculate the correlation coefficients of trading returns one needs to compare X and Y for the same period of time. If a trader buys and sells stocks every day he can compare daily returns (calculated for the same days) for different strategies. If a trader buys stocks and sells them in 2-3 days he can consider weekly or monthly returns. Correlation coefficients are very important for the market analysis. Many stocks have very high correlations. As an example let us present the correlation between one days price changes of MSFT and INTC.

10 10 Correlation between one days price change of INTC and MSFT The presented data are gathered from the 1988 to 1999 year period. The correlation coefficient c = 0.361, which is very high for one day price change correlation. It reflects simultaneous buying and selling these stocks by mutual fund traders. Note that correlation depends on time frame. The next Figure shows the correlation between ten days (two weeks) price changes of MSFT and INTC. Correlation between ten day price change of INTC and MSFT The ten day price change correlation is slightly weaker than the one day price change correlation. The calculation correlation coefficient is equal to

11 Efficient Trading Portfolio The theory of efficient portfolio was developed by Harry Markowitz in (H.M.Markowitz, "Portfolio Selection," Journal of Finance, 7, 77-91, 1952.) Markowitz considered portfolio diversification and showed how an investor can reduce the risk of investment by intelligently dividing investment capital. Let us outline the main ideas of Markowitz's theory and tray to apply this theory to trading portfolio. Consider a simple example. Suppose, you use two trading strategies. The average daily returns of these strategies are equal to R1 and R2. The standard deviations of these returns (risks) are s1 and s2. Let q1 and q2 be parts of your capital using these strategies. q1 + q2 = 1 Problem: Solution: Find q1 and q2 to minimize risk of trading. Using the theory of probabilities one can show that the average daily return for this portfolio is equal to R = q1*r1 + q2*r2 The squared standard deviation (variance) of the average return can be calculated from the equation s 2 = (q1*s1) 2 + (q2*s2) 2 + 2*c*q1*s1*q2*s2 where c is the correlation coefficient for the returns R1 and R2. To solve this problem it is good idea to draw the graph R, s for different values of q1. As an example consider the two strategies described in Section 2. The daily returns (calculated from the growth coefficients) and risks for these strategies are equal to R1 = 1.4% s1 = 5.0% R2 = 1.3% s2 = 11.2 % c = 0.09 The correlation coefficient for these returns is equal to The next figure shows the return-risk plot for different values of q1. 11

13 We calculated return R and standard deviation s (risk) for various values of q1 - part of the capital employed for purchase using the first strategy. The next figure shows the return - risk plot for various values of q1. 13 Return - risk plot for various values of q1 for strategy described in the text You can see that minimal risk is observed when q1 = 0.4, i.e. 40% of trading capital should be spend for strategy #1. Let us plot the risk to return ratio as a function of q1. The risk to return ratio as a function of q1 for strategy described in the text You can see that the minimum of the risk to return ratio one can observe when q1 = 0.47, not 0.4. At this value of q1 the risk to return ratio is almost 40% less than the ratio in the case where the whole capital is employed using only one strategy. In our opinion, this is the optimal distribution of the trading capital between these two strategies. In the table we show the returns, risks and risk to return ratios for strategy #1, #2 and for efficient trading portfolio with minimal risk to return ratio. Average return, % Risk, % Risk/Return Strategy # Strategy # Efficient Portfolio q1 = 47%

14 One can see that using the optimal distribution of the trading capital slightly reduces the average returns and substantially reduces the risk to return ratio. Sometimes a trader encounters the problem of estimating the correlation coefficient for two strategies. It happens when a trader buys stocks randomly. It is not possible to construct a table of returns with exact correspondence of returns of the first and the second strategy. One day he buys stocks following the first strategy and does not buy stocks following the second strategy. In this case the correlation coefficient cannot be calculated using the equation shown above. This definition is only true for simultaneous stock purchasing. What can we do in this case? One solution is to consider a longer period of time, as we mentioned before. However, a simple estimation can be performed even for a short period of time. This problem will be considered in the next section. 14

16 In this example there are only two returns (Jan 3, Jan 10), which can be compared and be used for calculating the correlation coefficient. Here we will consider the influence of correlation coefficients on the calculation of the efficient portfolio. As an example, consider two trading strategies (#1 and #2) with returns and risks: R1 = 3.55 % s1 = 11.6 % R2 = 2.94 % s2 = 9.9 % Suppose that the correlation coefficient is unknown. Our practice shows that the correlation coefficients are usually small and their absolute values are less than Let us consider three cases with c = -0.15, c = 0 and c = We calculated returns R and standard deviations S (risk) for various values of q1 - part of the capital used for purchase of the first strategy. The next figure shows the risk/return plot as a function of q1 for various values of the correlation coefficient. 16 Return - risk plot for various values of q1 and the correlation coefficients for the strategies described in the text. One can see that the minimum of the graphs are very close to each other. The next table shows the results. c q1 R S S/R As one might expect, the values of "efficient" returns R are also close to each other, but the risks S depend on the correlation coefficient substantially. One can observe the lowest risk for negative values of the correlation coefficient.

18 18 The probabilities (in %) of 50% drops in the trading capital for different values of average returns and risk-to-return ratios One can see that for risk to return ratios less than 4 the probability of losing 50% of the trading capital is very small. For risk/return > 5 this probability is high. The probability is higher for the larger values of the average returns. Conclusion: A trader should avoid strategies with large values of average returns if the risk to return ratios for these strategies are larger than 5. Influence of Commissions We have mentioned that when a trader is losing his capital the situation becomes worse and worse because the influence of the brokerage commissions becomes larger. As an example consider a trading method, which yields 2% return per day and commissions are equal to 1% of initial trading capital. If the capital drops as much as 50% then commissions become 2% and the trading system stops working because the average return per day becomes 0%. We have calculated the probabilities of a 50% capital drop for this case for different values of risk to return ratios. To compare the data obtained we have also calculated the probabilities of 50% capital drop for an average daily return = 1% (no commissions have been considered). For initial trading capital the returns of these strategies are equal but the first strategy becomes worse when the capital becomes smaller than its initial value and becomes better when the capital becomes larger than the initial capital. Mathematically the return can be written as R = Ro - commissions/capital * 100% where R is a real return and Ro is a return without commissions. The next figure shows the results of calculations.

20 Distributions of Annual Returns 20 Is everything truly bad if the risk to return ratio is large? No, it is not. For large values of risk to return ratios a trader has a chance to be a lucky winner. The larger the risk to return ratio, the broader the distribution of annual returns or annual capital growth. Annual capital growth = (Capital after 1 year) / (Initial Capital) We calculated the distribution of the annual capital growths for the strategy with the average daily return = 0.7% and the brokerage commissions = \$20. The initial trading capital was supposed = \$5,000. The results of calculations are shown in the next figure for two values of the risk to return ratios. Distributions of annual capital growths for the strategy described in the text The average annual capital growths are equal in both cases (3.0 or 200%) but the distributions are very different. One can see that for a risk to return ratio = 6 the chance of a large loss of capital is much larger than for the risk to return ratio = 3. However, the chance of annual gain larger than 10 (> 900%) is much greater. This strategy is good for traders who like risk and can afford losing the whole capital to have a chance to be a big winner. It is like a lottery with a much larger probability of being a winner. When to give up In the previous section we calculated the annual capital growth and supposed that the trader did not stop trading even when his capital had become less than 50%. This makes sense only in the case when the influence of brokerage commissions is small even for reduced capital and the trading strategy is still working well. Let us analyze the strategy of the previous section in detail. The brokerage commissions were supposed = \$20, which is 0.4% for the capital = \$5,000 and 0.8% for the capital = \$2,500.

21 So, after a 50% drop the strategy for a small capital becomes unprofitable because the average return is equal to 0.7%. For a risk to return ratio = 6 the probability of touching the 50% level is equal to 16.5%. After touching the 50% level a trader should give up, switch to more profitable strategy, or add money for trading. The chance of winning with the amount of capital = \$2,500 is very small. The next figure shows the distribution of the annual capital growths after touching the 50% level. 21 Distribution of the annual capital growths after touching the 50% level. Initial capital = \$5,000; commissions = \$20; risk/return ratio = 6; average daily return = 0.7% One can see that the chance of losing the entire capital is quite high. The average annual capital growth after touching the 50% level (\$2,500) is equal to 0.39 or \$1950. Therefore, after touching the 50% level the trader will lose more money by the end of the year. The situation is completely different when the trader started with \$10,000. The next figure shows the distribution of the annual capital growths after touching the 50% level in this more favorable case. Distribution of the annual capital growths after touching the 50% level. Initial capital = \$10,000; commissions = \$20; risk/return ratio = 6; average daily return = 0.7% One can see that there is a good chance of finishing the year with a zero or even positive result. At least the chance of retaining more than 50% of the original trading capital is much larger than the chance of losing the rest of money by the end of the year. The average annual capital growth after touching the 50% level is equal to Therefore, after touching the 50% level the trader will compensate for some losses by the end of the year.

29 Problem: 29 Suppose, a trader has \$5,000 to buy stocks and he does not use margin. The brokerage round trip commissions are equal to \$20. The average return per trade (after taking into account the bid-ask spread) is equal to R (%). The returns have distribution with the standard deviation (risk) s. What is the optimal number of stocks N a trader should buy to minimize the risk to return ratio? Solution: To buy one stock a trader can spend (5000 / N - 10) dollars. The average return R1 per one stock will be equal to R1 = 100% * [(5000 / N - 10) R/100-10] / (5000 / N) = 100% * [( N) R/100-10N] / 5000 This is also equal to the average return Rav of the entire capital because we consider return in %. Rav = R1 One can see that increasing N reduces the average return and this reduction is larger for a larger value of brokerage commissions. The total risk S is decreased by 1/SQRT (N) as we discussed previously. S = s/sqrt (N) The risk to return ratio can be written as S/R = 5000*s/{SQRT (N)*100% * [( N) R/100-10N]} The problem is reduced to the problem of finding the minimum of the function S/R. The value of s does not shift the position of the maximum and for simplicity we can take s = 1. The function S/R is not simple and we will plot the dependence of the ratio S/R for different values of N and R. The risk to return ratio as a function of number of stocks

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