252 to calculate the annual return) your trading capital will be
TC' = A' + B' + C'For every part of capital (A, B, or C) there were N/3 trades.Ntr = N/3If you hold stocks for four days the number of trades would N/4, etc. Let us define the growthcoefficient k per one tradek = <X1/X0>where X0 is the stock price at the moment of purchase and X1 is the stock price at the momentof selling. The angular brackets denote averaging. Every part of trading capital after Ntr tradeswill be increased asA' = AkNtrB' = BkNtrC' = CkNtrUsing these equation one can easily obtain that after N days your trading capital will beTC' = TC*kNtr= TC*kN/3The last equation can be written asTC' = TC*[k1/3]NSo, the coefficient K = k1/3 can be considered as the growth coefficient per one day forthe whole trading capital. If you hold stocks 4 days K = k1/4 , etc. In general, if you hold stocksn days the coefficient K is equal toK = k1/nThe best trading strategy is the strategy with the maximal daily growth coefficient.As an example, consider trading oversold stocks (see Chapter 6 for details). If you buysome oversold stocks at the price CLO1 and sell them at the prices OPE2, OPE3, OPE4, or OPE5
54your average returns are shown in Table 11.1. In this table the returns are shown for variouslevels of stock price drops during day #1.Table 11.1 The average returns (in %) for oversold stocks (selling at the market opening onthe days #2, #3, #4, and #5) for different levels of stock price drops during the day #1.CLO1 < CLO0
CLO1 < 0.95*CLO0
CLO1 < 0.9*CLO0
<OPE2 - CLO1>
0.50
0.85
1.06
<OPE3 - CLO1>
1.17
2.77
4.81
<OPE4 - CLO1>
1.61
3.67
5.77
<OPE5 - CLO1>
1.91
3.90
5.93
These data can be rewritten in terms of growth coefficients kk are presented in Table 11.2.
per one trade. Calculated
Table 11.2 The growth coefficients per one trade k for oversold stocks (selling at the marketopening on days #2, #3, #4, and #5) for various levels of stock price drops during day #1.CLO1 < CLO0
CLO1 < 0.95*CLO0
CLO1 < 0.9*CLO0
<OPE2 - CLO1>
1.0050
1.0085
1.0106
<OPE3 - CLO1>
1.0117
1.0277
1.0481
<OPE4 - CLO1>
1.0161
1.0367
1.0577
<OPE5 - CLO1>
1.0191
1.0390
1.0593
Now we are ready to consider four strategies: selling at OPE2, at OPE3, at OPE4 andOPE5. For the first strategy you hold stocks 1 day (n = 1) , for the second strategy you holdstocks 2 days (n = 2), etc. Using the equation for the growth coefficient per one day for thewhole trading capital K = k1/n one can obtain these coefficients for the four mentioned strategiesfor every level of stock price drops. (Actually, twelve trading strategies will be considered.)Table 11.3 The growth coefficient per one day for the whole trading capital K for oversoldstocks (selling at the market opening on days #2, #3, #4, and #5) for variuos levels of stockprice drops during the day #1.CLO1 < CLO0
CLO1 < 0.95*CLO0
CLO1 < 0.9*CLO0
<OPE2 - CLO1>
1.0050
1.0085
1.0106
<OPE3 - CLO1>
1.0058
1.0138
1.0238
<OPE4 - CLO1>
1.0053
1.0121
1.0189
<OPE5 - CLO1>
1.0047
1.0096
1.0145
One can see that for all levels of stock price drops during day #1 the best strategy isselling at the market opening on day #3.However, in the basic trading strategy we recommend selling on the morning of day #4.Why? Because we should consider transaction costs. They substantially reduce the averagereturns and growth coefficients will be different. The absolute return for the strategy OPE4 CLO1 is higher, and you have a better chance of reducing the influence of transaction costs.If you are an experienced trader and your trading capital is large (> $30 000) yourtransaction costs can be small. In this case, you can use the more profitable strategy OPE3 CLO1 due to small transaction costs.This section is an introduction to the general approach to trading strategy analysis. Youcan try to analyze other strategies using the ideas in this section. We did not consider risk
55analysis. Mathematically, such analysis is much more complicated. Contact us if you want toknow more about this type of analysis.11.2
How many stocks to buy?
In the basic trading strategy we recommended dividing your trading capital into 6 partsand using the whole part to buy shares of stock. You could find other advice - try to buy equalnumbers of shares. In this case you will use a small part of your capital to buy low priced stocksand large part of your capital to buy high priced stocks.There is a contradiction between these advises. Which is better?Theoretically, it is better to follow the first advice: every day use equal amounts ofmoney for two stock purchases. In this case you will buy a larger number of shares of lowerprice stock. The average return will suffer if you spend more money buying higher price stocks.Let us show the results of computer analysis.We calculated the average return per two stock purchase for the basic trading strategyfor different levels of stock price. Suppose, you do not like stocks which cost less than $12. Youwill select stocks from the list of potentially bullish stocks which cost more than $12. Your returnwill be substantially smaller: 2.92% instead of 3.25% with the regular limit of $10. Table 11.4shows the average returns per purchase depending on the stock price limits.Table 11.4 The average returns per purchase depending on the stock price limitsPrice limit
Average return per purchase, %
10
3.25
11
3.08
12
2.92
13
2.69
15
2.61
If you are an experienced trader and your capital is large you can use the basic tradingstrategy. For beginners, it is better to be more selective in stocks and not place large capital insmall stocks. They are very volatile and a couple of bad trades can kill you. Try to buy smallernumbers of shares of small stocks untill you get some good trading experience. Do not be upsetif you see large price moves for low priced stocks. They can also go down. Slowly but surely.This old advice is very good for beginners.11.3 Upgrades and downgradesWe will describe a very simple trading strategy. This strategy is good for traders who areable to watch the market for the whole trading day.Every trading day you can find the list of stocks which are upgraded or downgraded byvarious brokerage firms. You can find this list, for example, at http://biz.yahoo.com/c/u.htmlweb page provided by Briefing.com. At this site, the information is updated three times daily ondays on which U.S. stocks markets are open (9:30am, 12pm, 4pm Eastern).The ratings are listed in descending order from positive to negative. Many firms includelong-term (L-T), intermediate term (I-T), and near term (N-T) classifications within their ratingssystems.Onecanfindmoreinformationaboutequityratingsystemsathttp://biz.yahoo.com/f/bc.html#ratings.Let's consider upgraded stocks. The prices of these stocks go up very quickly after theupgrade announcement. If you are not a professional it is very hard to catch these moves.However, there is a good chance to win in this game. Wait for the price pullback after the firstrally. Never buy at the current maximum day price. Buy this stock when pullback is over. Sellthis stock on the morning of the next day.
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Figure 11.1 Price and volume change for upgraded stocks.Figure 11.1 illustrates the price change of upgraded stocks during the day before andafter the upgrade announcement. (1996-1997 year statistics) You can see that profit is not largebut this game is rather safe.11.4 Trading tipsThese trading tips are the result of our own trading experience. Our mistakes wereanalyzed, some conclusion have been formulated and written down as tips.Follow the strategy! If you bought a stock and it goes down the next day while analysisshows this stock is still bullish - buy more shares of this stock and consider the purchase anindependent trade.Do not trade stocks with a large bid-ask spread. If some stock looks very attractive - try tobuy it at the price between bid and ask. Use the limit order in this case. Remember: atransaction cost of more than 3% will kill any good strategy.If you see an unusually large positive price change during the first or second day, sell thisstock early.It is helpful to read the headline news about the stock. It is very good when bad news hadcome out already. Do not be afraid of a bad earnings report, analysts downgrading, or badnews about an industry. Stock can go up anyway. Short sellers will close their positions andbargain hunters are always ready to take a risk.Try to buy equal amounts of shares of stocks. The smaller the price, the larger the risk. Donot put a huge amount of money in small stocks.Use on-line brokers to reduce commissions. If you are buying 100 - 200 shares of stock,commissions should not be larger than $10.
57
Do not hold stocks for a long time if theyare on the downtrend. If you missed thelocal maximum the stock may shortlydrop, and your loss will be large.
It is better to have a real time quote service. Some brokers give you this service for free.Write your results in a spreadsheet. Calculate your average return R and standard deviationS. Try to keep the ratio of S/R as small as possible. If it is increasing over time, think aboutyour mistakes.You are not a computer and your psychology can make results better or worse thanaverage. Many times you will sell stocks too early and you will not buy some stocks justbecause you do not like them. It is OK when your decisions are based on your experienceor on additional analysis. It is bad if they are influenced by pure fear. Analyze yourmistakes. S/R ratio is a good tool for this analysis.If you are very sure about a stock, you can keep it longer. But we don't have any idea howyou can be very sure. The only sure thing in the market is uncertainty.Our advice:If you want to hold stocks longer - place a stop order closely to current price.
58Chapter 12
Trading portfolio12.1 Theory of efficient portfolio12.2 Using the theory of efficient portfolio12.1 Theory of efficient portfolioThe theory of efficient portfolio was developed by Harry Markowitz in 1952.(H.M.Markowitz, "Portfolio Selection," Journal of Finance, 7, 77 - 91, 1952.) Markowitzconsidered portfolio diversification and showed how an investor can reduce the risk ofinvestment by choosing stocks that do not move together.Let us outline the main ideas of Markowitz's theory. Consider a simple example. Suppose,you bought two stocks. The average returns of these stocks are equal to R1 and R2. Thestandard deviations of these returns are σ1 and σ2. Let q1 and q2 be parts of your capitalinvested in these stocks.q1 + q2 = 1Using the theory of probabilities one can show that the average return for this portfolio is equaltoR = q1*R1 + q2*R2The squared standard deviation (variance) of the average return can be calculated from theequationσ2 = (q1∗σ1)2 + (q2∗σ2)2 + 2*c*q1∗σ1*q2∗σ2where c is the correlation coefficient for the returns R1 and R2. (See Chapter 2 to learn moreabout correlation coefficients.)The problem: find q1 and q2 which give return R with minimal risk σ.To solve this problem it is good idea to draw the graph R , σ for different values of q1and q2. Let us show an example. Suppose every day you buy two stocks from some list ofbullish stocks. The first stock from the list is very bullish and its return R1 is two times largerthen the return R2 of the second stock. However, the risk of trading the first stock is higher thanthe risk of trading the second stock.R1 = 1.0 +/- 3.0 %R2 = 0.5 +/- 2.0 %Suppose the correlation coefficient for these returns is equal to zero.c= 0Figure 12.1 shows R , σ / R plot for this trading method.
59
Figure 12.1 Return-Risk plot for the trading portfolio described in the text.This plot shows the answer to the problem. The risk is minimal if the part of tradingcapital used to buy the first stock from the list is equal to 0.33. The risk is two times less thanfor the strategy when you spend the whole capital to buy the first stock only.So, the trading portfolio which provides the minimal risk consists of two stocks. 1/3 ofthe capital is in the first stock and the 2/3 of the capital is in the second stock. The expectedreturn for this portfolio is smaller than maximal expected value and the trader can adjust hisholdings depending on how much risk he can afford. People who like getting rich quickly canincrease position in the first stock. If you want a more peaceful life you can use q1= 1/3 and q2= 2/3.This is the main idea of building portfolio depending on risk. If you trade more securitiesthe Return-Risk plot becomes more complicated. It is not a single line but some complicatedfigure. Special computer methods of analysis of such plots have been developed. In our notes,we will consider some simple cases only to demonstrate the general ideas, and we will use thesemethods for building a trading portfolio with minimal risk.12.2 Using the theory of efficient portfolioLet us consider some practical use of the theory of effective portfolio. For the basictrading strategy we recommended buying two stocks every day and using equal amounts ofmoney for each stock. The first stock is the stock from the list of potentially bullish stocks withmaximal price drop during trading day #1. The second stock is the stock with maximal pricedrop from the rest of the list. In Chapter 5 we showed that the average return per purchase forthe first and the second stocks are equal toR1 = 3.55 +/- 11.6 %R2 = 2.94 +/- 9.9 %The correlation coefficient for these returns can be calculated using the equation fromsection 2.4. Figure 12.2 shows R1 versus R2. One can see that the correlation is small.Calculations show that the correlation coefficient c = 0.165.
60
Figure 12.2 Return per purchase for the first and for the second stocks selected from the list ofpotentially bullish stocks.Using equations from Section 12.1 we calculated return R and standard deviation σ (risk)for variuos values of q1 - part of the capital used for purchase of the first stock. Figure 12.3shows the return - risk plot for various values of q1.
Figure 12.3 Return - risk plot for various values of q1 for the basic trading strategy.You can see that minimal risk is observed when q1 = 0.4, i.e. 40% of trading capitalshould be spend for the stock from the list of potentially bullish stocks with maximal price dropduring trading day #1. The total return is smaller but this is what the trader should decide forhimself: lower risk or better return.Our opinion: the absolute value of risk is not a good characteristic of trading strategy. Itis more important to study the risk to return ratios. Minimal value of this ratio is the maincriterion of the best strategy. Let us plot the risk to return ratio as a function of q1.
61
Figure 12.4 The risk to return ratio as a function of q1 for the basic trading strategy.You can see that the minimum of the risk to return ratio one can observe is when q1 =0.47, i.e. it is close to the recommended value of q1 = 0.5. So, to obtain the minimal risk toreturn ratio for the basic trading strategy one should spend equal dollar amounts to buy twostocks from the list of potentially bullish stocks with minimal price drop during the next day afteranalysis.ConclusionSo, the text is over. We hope you learned something new and useful. Many questionswere not described or were described briefly because we tried to keep things as simple aspossible. However, many other stock trading problems can be described without artificialcomplexity. We would appreciate if our readers show us the problems which are worth beingdiscussed in the newer version of the text.We wrote an analytical text. This not a guide to "how to get rich quickly." Nevertheless,this text can be useful to improve your trading performance. Our goal was to show how toanalyze the trading strategies. We hope this text will help you to avoid many expensive tradingmistakes.We think this text can be used for further development of various trading strategies. Wedescribed general approaches to stock trading analysis. Similar methods can be applied to theanalysis of trading options, futures, bonds, etc.This is an interactive book. Write us and we will include your questions and our answersin the text. If your question is difficult to answer, give us some time to do additional researchand analysis. We expect our common work will be useful for all of us.With Best Wishes,Vladimir Daragan
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