From the data presented in Table 4.3.2 you can see that for a
28larger drop of stock price the predicted return is higher. The closing price change betweensucceeding days for oversold stocks is larger than "normal" (about 0.1%) even for the 5-th day.You can say that a good strategy is very simple: buy such stocks and hold them for atime period longer than 5 days, since average daily growth is larger thanreturn of the market.Yes, you are right. This is not a bad strategy if you are thinking about getting themaximum from one trade. But your annual return can suffer in this case, because daily pricegrowth decreases with time. You might be better off switching to other stocks. Then you cancatch the larger price moves, which occur during the first days after analysis. Later we willconsider this problem in detail.It may be interesting to consider the probabilities of daily growth. In other words, we willcalculate the probability of CLOm > CLOn. These data you can see in Table 4.3.Table 4.3 Probability of stock price growth.Potentially BullishStocks
C1 < C0
C1 < 0.95C0
C1 < 0.9C0
C2 > C1
0.563
0.561
0.608
0.608
C3 > C2
0.539
0.561
0.594
0.610
C4 > C3
0.529
0.534
0.528
0.511
C5 > C4
0.534
0.544
0.527
0.532
The higher the price drop on day #1, the higher the probability of price growth during thefollowing days. Note the decrease in probability between the second and the third lines of thistable. It means the probability of growth on the forth day becomes significantly lower than thatfor the first two trading days.It seems that we are ready to state a profitable trading strategy. No, we are not ready!We have to perform a more detailed analysis of stock performance during the days immediatelyfollowing purchase. The most important questions are the overnight price moves and themaximal daily prices. This problem will be considered in the next section.4.4 Overnight MovesLet us recall some notations. OPE1 is the stock price at the market opening on day #1,the next day after the day of the analysis. OPE2 is the stock price atthe market opening on day #2, etc.<O2-C1> = <(OPE2 - CLO1)/CLO1> 100% is the return if you buy stock at the market closingon day #1 and sell it at the market opening on day #2.Now we are ready to understand the results which are presented in Table 4.4.Table 4.4
Average overnight price change in %
Pot. Bullish Stocks
C1 < C0
C1 < 0.95C0
C1 < 0.9C0
<O2-C1>
0.54
0.50
0.85
1.06
<O3-C2>
0.49
0.59
1.20
2.10
<O4-C3>
0.41
0.53
0.90
1.11
<O5-C4>
0.39
0.48
0.67
1.11
The presented results show that overnight stock price moves can be very large. The morea stock is oversold, the larger the overnight move. For oversold stocks when C1 < 0.95C0 theovernight price moves are the largest during two first nights. So, do not sell stocks beforemarket closing. The best time to sell is the morning.For traders the relationship between closing price and the maximum price of the next dayis important. This value can be used for planning limit orders for the next day of trading. Youcan also catch the maximum price by monitoring the stock for the whole trading day. This
29monitoring is quite boring and nonprofessionals cannot watch the computer screen for the wholeday. Nevertheless, we will show some numbers to have an idea of what you could get by doingthis job. Let us introduce the notations:Mn-Cm = 100%(MAXn - CLOm)/CLOmThis value shows maximum stock gains on the n-th day with respect to closing on the mth day. After averaging, we obtain:Table 4.5 Maximum price gain during the trading day in %Potentially BullishStocks
C1 < C0
C1 < 0.95C0
C1 < 0.9C0
<M2-C1>
3.54
3.82
6.27
9.42
<M3-C2>
3.29
3.72
5.54
6.93
<M4-C3>
3.11
3.49
4.70
5.14
<M5-C4>
3.07
3.43
4.53
6.46
Some numbers look fantastic! You can obtain the biggest gain during day #2. But howcan we guess the maximum during the trading day? What strategy can utilize these possiblegains? These questions will be considered in Chapter 8.In the next chapter the basic trading strategy will be considered. This is a strategy whenyou buy two potentially bullish stocks with the maximum price drop during day #1. Theproblems related to the risk and returns, transaction costs, and correlation with the marketperformance will be considered.
30Chapter 5
Basic Trading Strategy5.1 Statistical Analysis of the Basic Trading Strategy5.2 Transaction cost and the Basic Trading Strategy5.3 How dangerous is this game?5.4 Correlation with the market5.1 Statistical Analysis of the Basic Trading StrategyBasic Trading StrategyThis strategy is very simple. Maybe it is not the best, but a simplicity and highprofitability allow us to call this trading strategy the Basic Trading Strategy. Let's describe thegeneral idea of this strategy.1. We should prepare the list of potentially bullish stocks after market closing on day #0.These stocks are partially oversold in the 16 - 32 day time frames. The stocks from the list maynot be oversold on the day of analysis for all possible time frames. We did not take into accountthe behavior of these stocks during the last 4 days. This is why we call these stocks potentiallybullish stocks. You can find more details about stock selection method in Section 4.2.2. Before market closing on day #1 one should check the prices of all stocks from the list andbuy two stocks with maximal % price drop during day #1. Mathematically, this can be writtenasCLO1/CLO0 -----> minimal
(*)
Note that in this strategy the stocks can rise during trading day #1. The only important thing inthis case: their rises should be minimal among the stocks from the list. The equation (*)describes this statement.3. During the next two days (#2 and #3) one should hold the stocks. At the market openingon day # 4 these stocks should be sold.Trading CapitalWe have performed computer analysis of this strategy and found a high profitability usingthis method. An important part of this strategy is dividing the trading capital between stocks.Days #2 and #3 are the days of holding stocks which were bought on day #1. If we donot buy other stocks during these days we will lose possible profits. So as to be able to buystocks every day the trading capital should be divided into three equal parts. Every part of thecapital should be used to buy two stocks. Let's illustrate this idea graphically. A, B, and C arethe part of the trading capital.
31
Figure 5.1 The time diagram for the Basic Trading StrategySo, the sumA + B + C = Trading Capital = TCwill be the current trading capital. At the beginning of trading, the current trading capital isequal to Initial Trading Capital and the parts A, B, and C are equal to,A = B = C = (Initial Trading Capital)/3Part A should be used on day #1 to buy two stocks. This part becomes A' on day #4 and shouldbe used to buy new stocks. Correspondingly, part B should be used to buy stocks on day #2,#5, etc.day #1
A
day #2
B
day #3
day #4C
day #5A'
day#6B'
C'
The computer analysis has been performed for the case when each part (A, B, or C) wasdivided equally between two stocks.So, if you buy two stocks which cost $20 and $30 and your available part of the capital(A, for example) is equal to $10 000, you should buy5000 / 20 = 250 shares of the first stock and5000 / 30 = 166 shares of the second stock.So, you should use 1/6 part of your capital to buy one stock. To keep transaction costssmall, the capital should be about $30 000 or more to use the advantages of this strategy.The values of A, B, and C change with time. This difference can characterize the risk ofthe strategy.ReturnsHere we will show how to determine the average returns for the basic trading strategy.One can define many types of returns. Let's mention some of them.Average Return per purchase:This is an average return per daily two stock purchase. It can be calculated asReturn per purchase = 100 * <(X' - X)/X >
32where X' represents the sell prices of two stocks and X represents the cost prices of two stockspurchased. It should be noted that this is not a daily return, since the stock holding period isequal to 3 days.Average Daily Return:This return is related to the current trading capital TC. It can be estimated asADR = Average Daily Return = 100 *<(X' - X)/TC>For estimation, one can suppose that the trading capital is approximately equal to 3*X.In this case, one can write a simple equation for the average daily returnADR = Average Daily Return = (Return per purchase)/3Note: This equation is approximate. It is valid only in the case of small daily returns (< 2%).The average daily returns are small for any strategy and one can use this equation as a goodestimation of the returns.StatisticsLet us show some statistical data for the basic trading strategy. The computer analysiswas performed during the period from November, 1995 to May, 1999.Number of trading days analyzed = 861Average return per purchase (%) = 3.25 +/- 8.2Risk/Return Ratio for return per purchase = 2.53Average daily return (%) = 1.08 +/- 2.74Probability of positive pure daily returns, % 67.8Return (Risk/Return) for the first stock selected: 3.55 (3.28)Return (Risk/Return) for the second stock selected: 2.94 (3.35)One can see that return for the first stock selected is much higher than the return perpurchase. However, the risk to return ratio when you buy two stocks every day is much lower.This is why the Basic Trading Strategy assumes buying two stocks every day.5.2
Transaction cost and the Basic Trading Strategy
In the previous section all returns have been calculated assuming zero transaction cost.This is not realistic, even in the case of large trading capital when the brokerage commissionsare relatively very small. We mentioned before that bid-ask spreads can substantially reducetrading returns. Traders try to catch small stock growth and even 0.5% of transaction cost cankill a bad trading strategy.The transaction cost will be considered in % of the part of capital for daily stockpurchasing. We will calculate an expected annual return (EAR) as a function of the transactioncost. One can write a simple equation which is related EAR to the average daily return (DR)EAR = [(ADR - TC/3)/100 + 1) ^ 252Here, TC is the transaction cost in %. It was supposed that there are 252 trading daysper year. We divided transaction cost by 3 because we calculate TC as a percentage of the partof the trading capital using for daily stock purchasing.
33The Figure 5.2 shows the dependence of the expected annual return on the transactioncost. The average daily return was taken as 1.08 %, as we found in the previous section.
Figure 5.2 Expected Annual Return as a function of the transaction cost. The average dailyreturn is equal to 1.08%.From Figure 5.2 one can see that a transaction cost of more than 3% renders the annualreturn of the Basic Trading Strategy close to zero. Higher transaction costs make annual returnseven negative. This is why it is necessary to chose stocks with small bid-ask spreads and to usebrokerage firms with minimal commissions.5.3
How dangerous is this game?
We described risk as the standard deviation of the set of returns. For mathematicians thisdefinition is clear. However, it would be better to present another interpretation of risk. Fortraders it is very important to know what is the probability of large price drops for selectedstocks.Consider the Basic Trading Strategy. What is the probability for selected stocks to fall 30%? Or -20%? The next table shows some results of computer analysis.Probability of price decline more than -30%
0.46%
Probability of price decline more than -20%
1.39%
Probability of price decline more than -10%
4.18%
Probability of price growth more than 30%
1.10%
Probability of price growth more than 20%
3.72%
Probability of price growth more than 10%
14.6%
So, the probabilities of large price drops are about 3 times less than probabilities of largeprice growth. It is worth showing this result graphically.
34
Figure 5.3 The probabilities of large price drops and growth for selected stocks for the BasicTrading Strategy. Statistical data for 1995 - 1999.Attention! The probability of large price drops for selected stocks is small, but it is notequal to zero. To avoid a large decline of your trading capital - diversify!5.4 Correlation with the marketHow does the Basic Trading Strategy work when market is falling? Not so good, not sobad. Let's illustrate this by some examples.We performed calculations of returns per purchase and changes in the S&P 500 index(OPE4 - CLO1) for 1995 - 1999 period. Figure 5.4 shows the correlation of return per purchaseand the change in the S&P 500 index during the same period of time.
Figure 5.4 The return per purchase for the Basic Trading Strategy depending on the change inthe S&P 500 index.The straight line is the result of linear fitting. One can see that the average return perpurchase becomes negative when the change in the S&P 500 index is very negative (less than 2%). The correlation coefficient is equal to 0.245.The correlation is relatively small, but you can not expect a large profit when the marketis falling. So, if you think the market is overbought - stay away or do not buy large number ofshares. The risk of losing money is very high.
35Chapter 6
Lower Risk Strategies6.1 Stock are oversold - how much?6.2 Low risk strategy6.3 Very low risk strategy6.4 Return per trade and monthly return6.1 Stock are oversold - how much?In the previous chapter you read about the Basic Trading Strategy. We like this strategy,but there are many people who cannot affort the risk related to this trading method. During theFall of 1998 the trading portfolio for this strategy dropped about 50%! There was a nicerecovery during next two quarters, but a 50% drop can repel many traders. It is hard to trust astrategy which allows such swings. Many people would give up trading stocks after such aterrible quarter.There are many trading strategies with much lower risk. The main idea of thesestrategies is trading our selected potentially bullish stocks. The differences between strategieslie in the methods of stock selection at the market closing on day #1, i.e. on the next day afterthe analysis.To reduce risk one should be very selective choosing stocks to buy. The simplest idea isto buy stocks with some level of price drop during day #1. In the Basic Trading Strategy onebuys stocks with maximum price decline during the day #1. Sometimes the maximum pricedrop is not large. Usually this is an indication of strong market growth during day #1. You canavoid buying stocks if they do not decline significantly. For example, you can consider onlystocks with a price decline of more than 5 or 10%. This method can be illustrated by thefollowing scheme.Table 6.1
Low risk trading strategies
1. Before market closing on the next day after the analysis check the stocks in thepotentially bullish stock list.2. Find two stocks with maximum price (in %) decline during day #1, i.e., the ratioCLO1/CLO0 should be minimal.3. If the price decline of both stocks is more than 10%, buy these stocks. Percentage ofdecline we will call the level of selection.4. Hold these stocks for two days.5. Sell the stocks at the market opening on day #4.The average profit from one purchase can be very substantial. The Figure 6.1 illustratesthe average price change for various levels of selections.
36
Figure 6.1 Average price changes relative to CLO1 price of potentially bullish stocks with variousprice levels CLO1 (selection levels). Vertical bars show the average minimal and maximal dailyprices. Black circles are the average opening and closing stock prices.From this figure one can conclude that the OPE4 price is the optimum price for sellingstocks. If you hold stocks longer the probability of growth is close to the growth probability ofaverage stock. One can also see: the larger the price drop during day #1, the larger thepotential return. If stocks fall more than 10% during day #1 (CLO1 < 0.9 CLO0) the averageminimal price during day #4 is larger than CLO1. This does not mean that you will always be awinner in this case. There are broad distributions for all the mentioned prices, but your chanceof winning is rather great.When you select a trading strategy you should remember about average annual numberof trades. If the potentially bullish stocks become oversold, i.e. their prices dropped during day#1, the number of such stocks can be very small. The total number of trades per year will alsobe small, and the annual return will be low. We have mentioned this before, but this idea is soimportant and it is worth repeating it once again. The next table shows the probability of findingoversold stocks at various selection levels.Table 6.2
The probabilities of findings an oversold stock and average returns per trade (in %)at various selection levels.
Selection Level
C1 < C0
C1 < 0.95C0
C1 < 0.9C0
Probability of findingselected stocks
0.98
0.76
0.37
Average return pertrade (OPE4-CLO1)
1.6
3.7
5.8
To estimate the profitability of various strategies one should multiply the return per tradeR by the probability P to find selected stocks. The results are presented in Table 6.3.
37Table 6.3 Profitability of trading strategies at various selection levels.Selection Level
C1 < C0
C1 < 0.95C0
C1 < 0.9C0
R*P
1.57
2.81
2.15
So, the strategy with selection level -5% (C1 < 0.95C0) is the most profitable amongthese strategies. However, the risk of this strategy is higher then the risk of the strategy withthe selection level -10%. The risk to return ratios for these strategies are equal to 3.1 and 2.3respectively. This is why we prefer the trading strategy with the selection level = -10% and it iscalled the low risk strategy.We think it will be interesting to see your chance of winning when using differentstrategies. In Table 6.4 you can find the probabilities of stock price growth from CLO1 to OPE4.Table 6.4
Probabilities of OPE4 > CLO1 or OPE5 > CLO1 for oversold stocks.
Selection Level
C1 < C0
C1 < 0.95C0
C1 < 0.9C0
P(OPE4 > CLO1)
0.60
0.67
0.71
P(OPE5 > CLO1)
0.61
0.67
0.68
This table illustrates once again why the OPE4 price is an optimal point to sell stocks.Selling stocks at the price OPE5 can be less profitable for some strategies because of the drop inthe growth probability.In the next section we will consider some statistical data for the low risk trading strategywith the selection level 10%.6.2 Low risk strategyThe main idea of the low risk strategy was described in Table 6.1. Let us recall that forthis strategy one should find two stocks from the list of potentially bullish stocks with a pricedrop of more than 10% during day #1. Calculations for the period 1995 - 1999 showed that thisprobability is equal to 13%. Instead of 861 trading days during this period there were only 112days when two stocks from the list dropped more than 10%. This why annual return is muchsmaller than for the Basic Trading Strategy.Why do we recommend buying two stocks? The reason is simple: diversification. We arestrongly urge diversification. There is always a chance of a large price drop even for thisselection method. Mathematically this chance can be described using risk to return ratios. Forreturns per purchase of two stocks this ratio is equal to 1.5. For one stock from the selection listthis ratio is equal to 1.9 - 2.1. This is 33% larger. If you want to reduce risk, buy two stocks.Your profit will be smaller, but risk will be also smaller. The next table presents some statisticsfor the low risk trading strategy (1995 - 1999).Number of trading days = 112Average return per purchase (%) = 6.95 +/- 10.5Risk/Return Ratio for return per purchase = 1.51Probability of positive pure daily returns = 77.7 %Return(Risk/Return) for the first stock selected:
7.0 (2.1)
Return(Risk/Return) for the second stock selected: 6.9
(1.93)
During November 1995 - May 1999 the trading portfolio was increased 6.08 times with a1.5% transaction cost. The S&P 500 index during same period of time increased 2.28 times. It isnot a bad result for a strategy with low risk.
38However, there is risk of losing money even with this strategy. Let us show the results ofcalculating of the probabilities of large price moves, just as we did for the Basic Trading Strategyin the previous chapter.Probability of price decline more than -30%
0.45%
Probability of price decline more than -20%
1.3%
Probability of price decline more than -10%
4.01%
Probability of price growth more than 30%
3.1%
Probability of price growth more than 20%
8.9%
Probability of price growth more than 10%
26.8%
The probability of large price drops are close to those for the Basic Trading Strategy.However, the probabilities of large price upside moves are much higher.How can we reduce the probability of large price drops? This will be described in the nextsection.6.3 Very low risk strategyWho likes risk? We don't. At least, when we trade our own accounts. We do not like stockprice drops on the next day after purchase. We become very upset when the stocks go down onthe next day. We blame ourselves and dream about canceling the purchase and startingover today to buy stocks after two days of large price drops.What is happening when you buy stocks after a two day price drop? It does not happenoften, but return may be much greater. Consider the results of computer analysis. The buyingprice is CLO2 for this strategy. In the tables the average returns X - CLO2 are presented fordifferent values of drop coefficients K1 and K2 so thatCLO1 < K1 *CLO0CLO2 < K2 *CLO1Tables 6.5
Returns and risk/return ratios in the case of two day stock price drops.OPE3 - CLO2, %K2 = 1.00
K2 = 0.95
K2 = 0.90
K1 = 1.00
0.45 (7.3)
0.70 (5.9)
1.26 (4.3)
K1 = 0.95
1.09 (3.8)
1.29 (3.2)
2.46 (1.9)
K1 = 0.90
2.17 (2.1)
2.71 (1.7)
3.74 (1.4)
CLO3 - CLO2, %K2 = 1.00
K2 = 0.95
K2 = 0.90
K1 = 1.00
0.43 (14)
1.26 (6.3)
2.16 (4.7)
K1 = 0.95
1.59 (5.0)
2.31 (3.7)
4.06 (2.6)
K1 = 0.90
3.25 (2.3)
4.67 (1.7)
4.97 (1.7)
OPE4 - CLO2, %K2 = 1.00
K2 = 0.95
K2 = 0.90
K1 = 1.00
1.06 (6.8)
2.72 (3.3)
4.71 (2.5)
K1 = 0.95
2.90 (3.2)
4.40 (2.3)
7.16 (1.7)
K1 = 0.90
4.86 (2.1)
7.29 (1.4)
8.70 (1.3)
39CLO4 - CLO2, %K2 = 1.00
K2 = 0.95
K2 = 0.90
K1 = 1.00
1.11 (7.7)
3.02 (3.6)
5.26 (2.6)
K1 = 0.95
3.04 (3.7)
4.90 (2.5)
7.61 (1.9)
K1 = 0.90
4.93 (2.5)
7.59 (1.8)
8.77 (1.7)
OPE5 - CLO2, %K2 = 1.00
K2 = 0.95
K2 = 0.90
K1 = 1.00
1.63 (5.6)
3.84 (3.0)
6.19 (2.4)
K1 = 0.95
3.87 (3.1)
5.85 (2.3)
8.60 (1.9)
K1 = 0.90
6.59 (2.1)
9.71 (1.5)
11.5 (1.4)
CLO5 - CLO2, %K2 = 1.00
K2 = 0.95
K2 = 0.90
K1 = 1.00
1.58 (6.3)
3.32 (3.8)
4.74 (3.8)
K1 = 0.95
3.51 (3.8)
4.98 (3.0)
5.89 (3.6)
K1 = 0.90
6.17 (2.5)
8.41 (2.0)
8.92 (2.5)
OK! The tables look very attractive. The only problem is how often you will find stockswith two-day drops in a row during day #1 and #2? The next table shows these probabilities.Table 6.6
Probabilities (in %) of finding some stocks in the list of potentially bullish stocks withtwo day price drops.K2 = 1.00
K2 = 0.95
K2 = 0.90
K1 = 1.00
90.2
51.8
18.3
K1 = 0.95
50.5
25.1
9.6
K1 = 0.90
19.7
9.6
3.9
The best trades when you find stocks with two day price drops of 10% you can makeabout once a month. Drops of -5%, -10% or -10%, -5% you can find on the average once intwo weeks. The stocks which were bought on the evening of the day #2 should be sold on themorning of the day #5. This will give a large return and a low risk to return ratio.This trading method is boring. Day after day you will do analysis just to recognize thatthere are no stocks to buy. The market will go up, but you will be waiting and watching as otherpeople make money. If you don't like such stock fishing you can use this analysis to draw a veryimportant conclusion - if you bought a stock on day #1 and at the end of day #2 the stock pricewent down, then you should buy more shares of this stock. The odds are on your side.
40Chapter 7
You bought and ...7.1 Further price drop7.2 Moving up7.3 When to sell7.1 Further price dropConsider strategy CLO1 - OPE4. Suppose, you bought shares of stock at CLO1 price inthe case of CLO1 < K1*CLO0. The stock price went down so thatCLO2 < K2*CLO1You double your position in this stock: you are buying additional shares. How about your profitand risk to return ratios?The answer is in Table 7.1. The probabilities of CLO2 < K2*CLO1 in the case of CLO1 <K1*CLO0 are shown, as well as returns for described strategy. For comparison, the simpletrading strategy (CLO1 - OPE4, if CLO1 < K1*CLO0) is shown as Return0.Table 7.1Returns (in %) of strategies of trading oversold stocks. In parentheses is the risk/return ratio.Position is open at CLO1 if CLO1 < K1*CLO0. The number of shares is doubled at CLO2 if CLO2< K2*CLO1. The whole position is closed at OPE4. For comparison, the returns of simple tradingstrategy (CLO1 - OPE4, if CLO1 < K1*CLO0) are shown as Return0K1
K2
Probability
Return
Return0
1
1
43.9
2.41 (3.4)
1.61 (5.5)
1
0.95
10.5
2.06 (4.1)
1.61 (5.5)
1
0.9
2.5
1.79 (4.8)
1.61 (5.5)
0.95
1
39.2
4.69 (2.3)
3.67 (3.1)
0.95
0.95
15.1
4.40 (2.5)
3.67 (3.1)
0.95
0.9
4.7
4.04 (2.8)
3.67 (3.1)
0.9
1
39.2
6.96 (1.8)
5.77 (2.3)
0.9
0.95
17.6
6.72 (1.9)
5.77 (2.3)
0.9
0.9
7
6.34 (2.0)
5.77 (2.3)
You can see that in all cases increasing positions after falling