How to Win the Stock Market Game

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1

H ow t o W in t h e St ock M a r k e t Ga m e D e ve lopin g Shor t - Te r m St ock Tr a ding St r a t e gie s by Vladim ir Daragan

PART 1 Ta ble of Con t e n t s 1.

I nt roduct ion

2.

Com parison of t rading st rat egies

3.

Ret urn per t rade

4.

Average ret urn per t rade

5.

More about average ret urn

6.

Growt h coefficient

7.

Dist ribut ion of ret urns

8.

Risk of t rading

9.

More about risk of t rading

10. Correlat ion coefficient 11. Efficient t rading port folio I nt r oduct ion This publicat ion is for short - t erm t raders, i.e. for t raders who hold st ocks for one t o eight days. Short - t erm t rading assum es buying and selling st ocks oft en. Aft er t wo t o four m ont hs a t rader will have good st at ist ics and he or she can st art an analysis of t rading result s. What are t he m ain quest ions, which should be answered from t his analysis? - I s m y t rading st rat egy profit able? - I s m y t rading st rat egy safe? - How can I increase t he profit abilit y of m y st rat egy and decrease t he risk of t rading? No doubt it is bet t er t o ask t hese quest ions before using any t rading st rat egy. We will consider m et hods of est im at ing profit abilit y and risk of t rading st rat egies, opt im ally dividing t rading capit al, using st op and lim it orders and m any ot her problem s relat ed t o st ock t rading. Com pa r ison of Tr a din g St r a t e gie s Consider t wo hypot het ical t rading st rat egies. Suppose you use half of your t rading capit al t o buy st ocks select ed by your secret syst em and sell t hem on t he next day. The ot her half of your capit al you use t o sell short som e specific st ocks and close posit ions on t he next day.

2 I n t he course of one m ont h you m ake 20 t rades using t he first m et hod ( let us call it st rat egy # 1) and 20 t rades using t he second m et hod ( st rat egy # 2) . You decide t o analyze your t rading result s and m ake a t able, which shows t he ret urns ( in % ) for every t rade you m ade. #

Ret urn per t rade in % St rat egy 1

Ret urn per t rade in % St rat egy 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 present s t he result s of t rading for t hese st rat egies.

Re t ur ns pe r t r a de s for t w o hypot he t ica l t r a ding st r a t e gie s Which st rat egy is bet t er and how can t he t rading capit al be divided bet ween t hese st rat egies in order t o obt ain t he m axim al profit wit h m inim al risk? These are t ypical t rader's quest ions and we will out line m et hods of solving t hem and sim ilar problem s.

3 The first t hing you would probably do is calculat e of t he average ret urn per t rade. Adding up t he num bers from t he colum ns and dividing t he result s by 20 ( t he num ber of t rades) you obt ain t he average ret urns per t rade for t hese st rat egies Ra v1 = 1 .5 5 % Ra v2 = 1 .9 % Does t his m ean t hat t he second st rat egy is bet t er? No, it does not ! The answer is clear if you calculat e t he t ot al ret urn for t his t im e period. A definit ion of t he t ot al ret urn for any given t im e period is very sim ple. I f your st art ing capit al is equal t o C0 and aft er som e period of t im e it becom es C1 t hen t he t ot al ret urn for t his period is equal t o Tot a l Re t ur n = ( C1 - C0 ) / C0 * 1 0 0 % Surprisingly, you can discover t hat t he t ot al ret urns for t he described result s are equal to Tot a l Re t ur n1 = 3 3 % Tot a l Re t ur n2 = 2 9 .3 % What happened? The average ret urn per t rade for t he first st rat egy is sm aller but t he t ot al ret urn is larger! Many quest ions im m ediat ely arise aft er t his " analysis": -

Can we use t he average ret urn per t rade t o charact erize a t rading st rat egy? Should we swit ch t o t he first st rat egy? How should we divide t he t rading capit al bet ween t hese st rat egies? How should we use t hese st rat egies t o obt ain t he m axim um profit w it h m inim al risk?

To answer t hese quest ions let us int roduce som e basic definit ions of t rading st at ist ics and t hen out line t he solut ion t o t hese problem s. Re t ur n pe r Tr a de Suppose you bought N shares of a st ock at t he price P0 and sold t hem at t he price P1 . Brokerage com m issions are equal t o COM . When you buy, you paid a cost price Cost = P0 * N + COM When you sell you receive a sale price Sa le = P1 * N - COM Your ret urn R for t he t rade ( in % ) is equal t o R = ( Sa le - Cost ) / Cost * 1 0 0 %

Ave r a ge Re t ur n pe r Tr a de Suppose you m ade n t rades wit h ret urns R1 , R2 , R3 , ..., Rn. One can define an average ret urn per t rade Ra v Ra v = ( R1 + R2 + R3 + ... + Rn) / n

4 This calculat ions can be easily perform ed using any spreadsheet such as MS Excel, Origin, ... . M or e a bout a ve r a ge r e t ur n You can easily check t hat t he described definit ion of t he average ret urn is not perfect . Let us consider a sim ple case. Suppose you m ade t wo t rades. I n t he first t rade you have gained 50% and in t he second t rade you have lost 50% . Using described definit ion you can find t hat t he average ret urn is equal t o zero. I n pract ice you have lost 25% ! Let us consider t his cont radict ion in det ails. Suppose your st art ing capit al is equal t o $100. Aft er t he first t rade you m ade 50% and your capit al becam e $ 1 0 0 * 1 .5 = $ 1 5 0 Aft er t he second t rade when you lost 50% your capit al becam e $ 1 5 0 * 0 .5 = $ 7 5 So you have lost $25, which is equal t o - 25% . I t seem s t hat t he average ret urn is equal t o - 25% , not 0% . This cont radict ion reflect s t he fact t hat you used all your m oney for every t rade. I f aft er t he first t rade you had wit hdrawn $50 ( your profit ) and used $100 ( not $150) for t he second t rade you would have lost $50 ( not $75) and t he average ret urn would have been zero. I n t he case when you st art t rading w it h a loss ( $50) and you add $50 t o your t rading account and you gain 50% in t he second t rade t he average ret urn will be equal t o zero. To use t his t rading m et hod you should have som e cash reserve so as t o an spend equal am ount of m oney in every t rade t o buy st ocks. I t is a good idea t o use a part of your m argin for t his reserve. How ever, very few t raders use t his syst em for t rading. What can we do when a t rader uses all his t rading capit al t o buy st ocks every day? How can w e est im at e t he average ret urn per t rade? I n t his case one needs t o consider t he concept of growt h coefficient s. Gr ow t h Coe fficie nt Suppose a t rader m ade n t rades. For t rade # 1 K1 = Sa le 1 / Cost 1 where Sa le 1 and Cost 1 represent t he sale and cost of t rade # 1. This rat io we call t he growt h coefficient . I f t he grow t h coefficient is larger t han one you are a winner. I f t he growt h coefficient is less t han one you are a loser in t he given t rade. I f K1 , K2 , ... are t he growt h coefficient s for t rade # 1 , t rade # 2 , ... t hen t he t ot al growt h coefficient can be writ t en as a product K = K1 * K2 * K3 * ... I n our previous exam ple t he growt h coefficient for t he first t rade K1 = 1 .5 and for t he second t rade K2 = 0 .5 . The t ot al growt h coefficient , which reflect s t he change of your t rading capit al is equal t o K = 1 .5 * 0 .5 = 0 .7 5

5 which correct ly corresponds t o t he real change of t he t rading capit al. For n t rades you can calculat e t he average growt h coefficient Ka v per t rade as Ka v = ( K1 * K2 * K3 * ...) ^ ( 1 / n ) These calculat ions can be easily perform ed by using any scient ific calculat or. The t ot al growt h coefficient for n t rades can be calculat ed as K = Ka v ^ n I n our exam ple Ka v = ( 1 .5 * 0 .5 ) ^ 1 / 2 = 0 .8 6 6 , which is less t han 1. I t is easily t o check t hat 0 .8 6 6 ^ 2 = 0 .8 6 6 * 0 .8 6 6 = 0 .7 5 How ever, t he average ret urns per t rade Ra v can be used t o charact erize t he t rading st rat egies. Why? Because for sm all profit s and losses t he result s of using t he growt h coefficient s and t he average ret urns are close t o each ot her. As an exam ple let us consider a set of t rades wit h ret urns R1 = - 5% R2 = + 7% R3 = - 1% R4 = + 2% R5 = - 3% R6 = + 5% R7 = + 0% R8 = + 2% R9 = - 10% R10 = + 11% R11 = - 2% R12 = 5% R13 = + 3% R14 = - 1% R15 = 2% The average ret urn is equal t o Ra v = ( - 5 + 7 - 1 + 2 - 3 + 5 + 0 + 2 - 1 0 + 1 1 - 2 + 5 + 3 - 1 + 2 ) / 1 5 = + 1 % The average growt h coefficient is equal t o Ka v = ( 0 .9 5 * 1 .0 7 * 0 .9 9 * 1 .0 2 * 0 .9 7 * 1 .0 5 * 1 * 1 .0 2 * 0 .9 * 1 .1 1 * 0 .9 8 * 1 .0 5 * 1 .0 3 * 0 .9 9 * 1 .0 2 ) ^ ( 1 / 1 5 ) = 1 .0 0 9

which corresponds t o 0 .9 % . This is very close t o t he calculat ed value of t he average ret urn = 1 % . So, one can use t he average ret urn per t rade if t he ret urn per t rades are sm all. Let us ret urn t o t he analysis of t w o t rading st rat egies described previously. Using t he definit ion of t he average grow t h coefficient one can obt ain t hat for t hese st rat egies Ka v1 = 1 .0 1 4 Ka v2 = 1 .0 1 3 So, t he average growt h coefficient is less for t he second st rat egy and t his is t he reason why t he t ot al ret urn using t his st rat egy is less.

6 D ist r ibut ion of r e t ur n s I f t he num ber of t rades is large it is a good idea t o analyze t he t rading perform ance by using a hist ogram . Hist ogram ( or bar diagram ) shows t he num ber of t rades falling in a given int erval of ret urns. A hist ogram for ret urns per t rade for one of our t rading st rat egies is shown in t he next figure

H ist ogr a m of r e t ur ns pe r t r a de s for t he Low Risk Tr a ding St r a t e gy As an exam ple, w e have considered dist ribut ion of ret urns for our Low Risk Trading St rat egy ( see m ore det ails in ht t p: / / www.st t a- consult ing.com ) from January 1996 t o April 2000. The bars represent t he num ber of t rades for given int erval of ret urns. The largest bar represent s t he num ber of t rades wit h ret urns bet ween 0 and 5% . Ot her num bers are shown in t he Table. Ret urn Range, %

Num ber of St ocks

Ret urn Range, %

Num ber of St ocks

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 72 47 25 17 4

- 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 85 46 17 5 6 1 3

For t his dist ribut ion t he average ret urn per t rade is 4.76% . The widt h of hist ogram is relat ed t o a very im port ant st at ist ical charact erist ic: t he st andard deviat ion or risk. Risk of t r a din g To calculat e t he st andard deviat ion one can use t he equat ion

7 The larger t he st andard deviat ion, t he wider t he dist ribut ion of ret urns. A wider dist ribut ion increases t he probabilit y of negat ive ret urns, as show n in t he next figure.

D ist r ibut ions of r e t ur ns pe r t r a de for Ra v = 3 % a n d for diffe r e nt st a nda r d de via t ions Therefore, one can conclude t hat a wider dist ribut ion is relat ed t o a higher risk of t rading. This is why t he st andard dist ribut ion of ret urns is called t he risk of t rading. One can also say t hat risk is a charact erist ic of volat ilit y of ret urns. An im port ant charact erist ic of any t rading st rat egy is Risk - t o- Re t ur n Ra t io = s/ Ra v The sm aller t he risk- t o- ret urn rat io, t he bet t er t he t rading st rat egy. I f t his rat io is less t han 3 one can say t hat a t rading st rat egy is very good. We would avoid any t rading st rat egy for which t he risk- t o- ret urn rat ion is larger t han 5. For dist ribut ion in Fig. 1.2 t he risk- t o- ret urn rat io is equal t o 2.6, which indicat es low level of risk for t he considered st rat egy. Ret urning back t o our hypot het ical t rading st rat egies one can est im at e t he risk t o ret urn rat ios for t hese st rat egies. For t he first st rat egy t his rat io is equal t o 3.2. For t he second st rat egy it is equal t o 5.9. I t is clear t hat t he second st rat egy is ext rem ely risky, and t he port ion of t rading capit al for using t his st rat egy should be very sm all. How sm all? This quest ion will be answered when we will consider t he t heory of t rading port folio. M or e a bout r isk of t r a ding The definit ion of risk int roduced in t he previous sect ion is t he sim plest possible. I t w as based on using t he average ret urn per t rade. This m et hod is st raight forward and for m any cases it is sufficient for com paring different t rading st rat egies. However, we have m ent ioned t hat t his m et hod can give false result s if ret urns per t rade have a high volat ilit y ( risk) . One can easily see t hat t he larger t he risk, t he larger t he difference bet ween est im at ed t ot al ret urns using average ret urns per t rade or t he average growt h coefficient s. Therefore, for highly volat ile t rading st rat egies one should use t he growt h coefficient s K. Using t he grow t h coefficient s is sim ple when t raders buy and sell st ocks every day. Som e st rat egies assum e specific st ock select ions and t here are m any days when t raders wait for opport unit ies by j ust wat ching t he m arket . The num ber of st ocks t hat should be bought is not const ant .

8 I n t his case com parison of t he average ret urns per t rade cont ains very lit t le inform at ion because t he num ber of t rades for t he st rat egies is different and t he annual ret urns w ill be also different even for equal average ret urns per t rade. One of t he solut ions t o t his problem is considering ret urns for a longer period of t im e. One m ont h, for exam ple. The only disadvant age of t his m et hod is t he longer period of t im e required t o collect good st at ist ics. Anot her problem is defining t he risk when using t he growt h coefficient s. Mat hem at ical calculat ion becom e very com plicat ed and it is beyond t he t opic of t his publicat ion. I f you feel st rong in m at h you can w rit e us ( service@st t a- consult ing.com ) and we will recom m end you som e reading about t his t opic. Here, we will use a t ried and t rue definit ion of risk via st andard deviat ions of ret urns per t rade in % . I n m ost cases t his approach is sufficient for com paring t rading st rat egies. I f we feel t hat som e calculat ions require t he growt h coefficient s w e will use t hem and we w ill insert som e com m ent s about est im at ion of risk. The m ain goal of t his sect ion t o rem ind you t hat using average ret urn per t rade can slight ly overest im at e t he t ot al ret urns and t his overest im at ion is larger for m ore volat ile t rading st rat egies. Cor r e la t ion Coe fficie nt Before st art ing a descript ion of how t o build an efficient t rading port folio we need t o int roduce a new param et er: correlat ion coefficient . Let us st art wit h a sim ple exam ple. Suppose you t rade st ocks using t he follow ing st rat egy. You buy st ocks every week on Monday using your secret select ion syst em and sell t hem on Friday. During a week t he st ock m arket ( SP 500 I ndex) can go up or dow n. Aft er 3 m ont h of t rading you find t hat your result are st rongly correlat ed wit h t he m arket perform ance. You have excellent ret urns for week when t he m arket is up and you are a loser when m arket goes down. You decide t o describe t his correlat ion m at hem at ically. How t o do t his? You need t o place your weekly ret urns in a spreadsheet t oget her wit h t he change of SP 500 during t his week. You can get som et hing like t his: Weekly Ret urn, %

Change of SP 500, %

13

1

-5

-3

16

1

4

3.2

20

5

21

5.6

-9

-3

-8

- 1.2

2

-1

8

6

7

-2

26

3

9 These dat a can be present ed graphically.

D e pe nde n ce of w e e k ly r e t ur ns on t he SP 5 0 0 ch a nge for hypot he t ica l st r a t e gy Using any graphical program you can plot t he dependence of weekly ret urns on t he SP 500 change and using a linear fit t ing program draw t he fit t ing line as in shown in Figure. The correlat ion coefficient c is t he param et er for quant it at ive descript ion of deviat ions of dat a point s from t he fit t ing line. The range of change of c is from - 1 t o + 1. The larger t he scat t ering of t he point s about t he fit t ing curve t he sm aller t he correlat ion coefficient . The correlat ion coefficient is posit ive when posit ive change of som e param et er ( SP 500 change in our exam ple) corresponds t o posit ive change of t he ot her param et er ( weekly ret urns in our case) . The equat ion for calculat ing t he correlat ion coefficient can be writ t en as

where X and Y are som e random variable ( ret urns as an exam ple) ; S are t he st andard deviat ions of t he corresponding set of ret urns; N is t he num ber of point s in t he dat a set . For our exam ple t he correlat ion coefficient is equal t o 0.71. This correlat ion is very high. Usually t he correlat ion coefficient s are falling in t he range ( - 0.1, 0.2) . We have t o not e t hat t o correct ly calculat e t he correlat ion coefficient s of t rading ret urns one needs t o com pare X and Y for t he sam e period of t im e. I f a t rader buys and sells st ocks every day he can com pare daily ret urns ( calculat ed for t he sam e days) for different st rat egies. I f a t rader buys st ocks and sells t hem in 2- 3 days he can consider weekly or m ont hly ret urns. Correlat ion coefficient s are very im port ant for t he m arket analysis. Many st ocks have very high correlat ions. As an exam ple let us present t he correlat ion bet ween one days price changes of MSFT and I NTC.

10

Cor r e la t ion be t w e e n one da ys pr ice cha n ge of I N TC a n d M SFT The present ed dat a are gat hered from t he 1988 t o 1999 year period. The correlat ion coefficient c = 0 .3 6 1 , which is very high for one day price change correlat ion. I t reflect s sim ult aneous buying and selling t hese st ocks by m ut ual fund t raders. Not e t hat correlat ion depends on t im e fram e. The next Figure shows t he correlat ion bet ween t en days ( t wo weeks) price changes of MSFT and I NTC.

Cor r e la t ion be t w e e n t e n da y pr ice cha nge of I N TC a nd M SFT The t en day price change correlat ion is slight ly weaker t han t he one day price change correlat ion. The calculat ion correlat ion coefficient is equal t o 0.327.

11 Efficie nt Tr a din g Por t folio The t heory of efficient port folio was developed by Harry Markowit z in 1952. ( H.M.Markow it z, " Port folio Select ion," Journal of Finance, 7 , 77 - 91, 1952.) Markowit z considered port folio diversificat ion and showed how an invest or can reduce t he risk of invest m ent by int elligent ly dividing invest m ent capit al. Let us out line t he m ain ideas of Markow it z's t heory and t ray t o apply t his t heory t o t rading port folio. Consider a sim ple exam ple. Suppose, you use t wo t rading st rat egies. The average daily ret urns of t hese st rat egies are equal t o R1 and R2 . The st andard deviat ions of t hese ret urns ( risks) are s1 and s2 . Let q1 and q2 be part s of your capit al using t hese st rat egies. q1 + q2 = 1 Pr oble m : Find q1 and q2 t o m inim ize risk of t rading. Solut ion: Using t he t heory of probabilit ies one can show t hat t he average daily ret urn for t his port folio is equal t o R = q1 * R1 + q2 * R2 The squared st andard deviat ion ( variance) of t he average ret urn can be calculat ed from t he equat ion s 2 = ( q1 * s1 ) 2 + ( q2 * s2 ) 2 + 2 * c* q1 * s1 * q2 * s2 where c is t he correlat ion coefficient for t he ret urns R1 and R2 . To solve t his problem it is good idea t o draw t he graph R, s for different values of q1 . As an exam ple consider t he t w o st rat egies described in Sect ion 2. The daily ret urns ( calculat ed from t he growt h coefficient s) and risks for t hese st rat egies are equal t o R1 = 1 .4 % R2 = 1 .3 %

s1 = 5 .0 % s2 = 1 1 .2 %

The correlat ion coefficient for t hese ret urns is equal t o c = 0 .0 9 The next figure shows t he ret urn- risk plot for different values of q1 .

12

Re t ur n- Risk plot for t he t r a ding por t folio de scr ibe d in t he t e x t This plot shows t he answ er t o t he problem . The risk is m inim al if t he part of t rading capit al used t o buy t he first st ock from t he list is equal t o 0.86. The risk is equal t o 4.7, which is less t han for t he st rat egy when t he w hole capit al is em ployed using t he first t rading st rat egy only. So, t he t rading port folio, which provides t he m inim al risk, should be divided bet ween t he t w o st rat egies. 86% of t he capit al should be used for t he first st rat egy and t he 14% of t he capit al m ust be used for t he second st rat egy. The expect ed ret urn for t his port folio is sm aller t han m axim al expect ed value, and t he t rader can adj ust his holdings depending on how m uch risk he can afford. People, who like get t ing rich quickly, can use t he first st rat egy only. I f you want a m ore peaceful life you can use q1 = 0.86 and q2 = 0.14, i.e. about 1/ 6 of your t rading capit al should be used for t he second st rat egy. This is t he m ain idea of building port folio depending on risk. I f you t rade m ore securit ies t he Ret urn- Risk plot becom es m ore com plicat ed. I t is not a single line but a com plicat ed figure. Special com put er m et hods of analysis of such plot s have been developed. I n our publicat ion, we consider som e sim ple cases only t o dem onst rat e t he general ideas. We have t o not e t hat t he absolut e value of risk is not a good charact erist ic of t rading st rat egy. I t is m ore im port ant t o st udy t he risk t o ret urn rat ios. Minim al value of t his rat io is t he m ain crit erion of t he best st rat egy. I n t his exam ple t he m inim um of t he risk t o ret urn rat io is also t he value q1 = 0.86. But t his is not always t rue. The next exam ple is an illust rat ion of t his st at em ent . Let us consider a case when a t rader uses t wo st rat egies ( # 1 and # 2) wit h ret urns and risks, which are equal t o R1 = 3 .5 5 % R2 = 2 .9 4 %

s1 = 1 1 .6 % s2 = 9 .9 %

The correlat ion coefficient for t he ret urns is equal t o c = 0 .1 6 5 This is a pract ical exam ple relat ed t o using our Basic Trading St rat egy ( look for det ails at ht t p: / / www.st t a- consult ing.com ) .

13 We calculat ed ret urn R and st andard deviat ion s ( risk) for various values of q1 - part of t he capit al em ployed for purchase using t he first st rat egy. The next figure shows t he ret urn risk plot for various values of q1 .

Re t ur n - r isk plot for va r ious va lue s of q1 for st r a t e gy de scr ibe d in t he t e x t You can see t hat m inim al risk is observed when q1 = 0.4, i.e. 40% of t rading capit al should be spend for st rat egy # 1. Let us plot t he risk t o ret urn rat io as a funct ion of q1 .

The r isk t o r e t ur n r a t io a s a fu nct ion of q1 for st r a t e gy de scr ibe d in t he t e x t You can see t hat t he m inim um of t he risk t o ret urn rat io one can observe when q1 = 0.47, not 0.4. At t his value of q1 t he risk t o ret urn rat io is alm ost 40% less t han t he rat io in t he case where t he whole capit al is em ployed using only one st rat egy. I n our opinion, t his is t he opt im al dist ribut ion of t he t rading capit al bet ween t hese t wo st rat egies. I n t he t able we show t he ret urns, risks and risk t o ret urn rat ios for st rat egy # 1, # 2 and for efficient t rading port folio wit h m inim al risk t o ret urn rat io. Average ret urn, %

Risk, %

Risk/ Ret urn

St rat egy # 1

3.55

11.6

3.27

St rat egy # 2

2.94

9.9

3.37

Efficient Port folio q1 = 47%

3.2

8.2

2.5

14

One can see t hat using t he opt im al dist ribut ion of t he t rading capit al slight ly reduces t he average ret urns and subst ant ially reduces t he risk t o ret urn rat io. Som et im es a t rader encount ers t he problem of est im at ing t he correlat ion coefficient for t w o st rat egies. I t happens when a t rader buys st ock s random ly. I t is not possible t o const ruct a t able of ret urns wit h exact correspondence of ret urns of t he first and t he second st rat egy. One day he buys st ocks following t he first st rat egy and does not buy st ocks following t he second st rat egy. I n t his case t he correlat ion coefficient cannot be calculat ed using t he equat ion shown above. This definit ion is only t rue for sim ult aneous st ock purchasing. What can we do in t his case? One solut ion is t o consider a longer period of t im e, as we m ent ioned before. However, a sim ple est im at ion can be perform ed even for a short period of t im e. This problem w ill be considered in t he next sect ion.

15 PART 2 Ta ble of Con t e n t s 1.

Efficient port folio and correlat ion coefficient

2.

Probabilit y of 50% capit al drop

3.

I nfluence of com m issions

4.

Dist ribut ion of annual ret urns

5.

When t o give up

6.

Cash reserve

7.

I s you st rat egy profit able?

8.

Using t rading st rat egy and psychology of t rading

9.

Trading period and annual ret urn

10. Theory of diversificat ion

Efficie nt por t folio a nd t he cor r e la t ion coe fficie nt . I t is relat ively easily t o calculat e t he average ret urns and t he risk for any st rat egy when a t rader has m ade 40 and m ore t rades. I f a t rader uses t wo st rat egies he m ight be int erest ed in calculat ing opt im al dist ribut ion of t he capit al bet w een t hese st rat egies. We have m ent ioned t hat t o correct ly use t he t heory of efficient port folio one needs t o know t he average ret urns, risks ( st andard deviat ions) and t he correlat ion coefficient . We also m ent ioned t hat calculat ing t he correlat ion coefficient can be difficult and som et im es im possible w hen a t rader uses a st rat egy t hat allows buying and selling of st ocks random ly, i.e. t he purchases and sales can be m ade on different days. The next t able shows an exam ple of such st rat egies. I t is supposed t hat t he t rader buys and sells t he st ocks in t he course of one day. Dat e Jan 3 Jan 4 Jan 5 Jan 6 Jan 7 Jan 10 Jan 11 Jan 12 Jan 13 Jan 14

Ret urn per purchase for St rat egy # 1 + 5.5% + 2.5%

Ret urn per purchase for St rat egy # 2 - 3.5% - 5%

- 3.2% + 1.1%

+ 8% 9.5%

+ 15.0% - 7.6% - 5.4%

16 I n t his exam ple t here are only t wo ret urns ( Jan 3, Jan 10) , which can be com pared and be used for calculat ing t he correlat ion coefficient . Here we will consider t he influence of correlat ion coefficient s on t he calculat ion of t he efficient port folio. As an exam ple, consider t w o t rading st rat egies ( # 1 and # 2) wit h ret urns and risks: R1 = 3 .5 5 % s1 = 1 1 .6 % R2 = 2 .9 4 % s2 = 9 .9 % Suppose t hat t he correlat ion coefficient is unknown. Our pract ice shows t hat t he correlat ion coefficient s are usually sm all and t heir absolut e values are less t han 0.15. Let us consider t hree cases wit h c = - 0.15, c = 0 and c = 0.15. We calculat ed ret urns R and st andard deviat ions S ( risk) for various values of q1 - part of t he capit al used for purchase of t he first st rat egy. The next figure shows t he risk/ ret urn plot as a funct ion of q1 for various values of t he correlat ion coefficient .

Re t ur n - r isk plot for va r ious va lue s of q1 a n d t he cor r e la t ion coe fficie nt s for t he st r a t e gie s de scr ibe d in t he t e x t . One can see t hat t he m inim um of t he graphs are very close t o each ot her. The next t able shows t he result s. c - 0.15 0 0.15

q1 0.55 0.56 0.58

R 3.28 3.28 3.29

S 7.69 8.34 8.96

S/ R 2.35 2.57 2.72

As one m ight expect , t he values of "efficient " ret urns R are also close t o each ot her, but t he risks S depend on t he correlat ion coefficient subst ant ially. One can observe t he lowest risk for negat ive values of t he correlat ion coefficient .

17 Conclusion : The com posit ion of t he efficient port folio does not subst ant ially depend on t he correlat ion coefficient s if t hey are sm all. Negat ive correlat ion coefficient s yield less risk t han posit ive ones. One can obt ain negat ive correlat ion coefficient s using, for exam ple, t wo " opposit e st rat egies" : buying long and selling short . I f a t rader has a good st ock select ion syst em for t hese st rat egies he can obt ain a good average ret urn wit h sm aller risk. Pr oba bilit y of 5 0 % ca pit a l dr op How safe is st ock t rading? Can you lose m ore t han 50% of your t rading capit al t rading st ocks? I s it possible t o find a st rat egy w it h low probabilit y of such disast er? Unfort unat ely, a t rader can lose 50 and m ore percent using any aut hent ic t rading st rat egy. The general rule is quit e sim ple: t he larger your average profit per t rade, t he large t he probabilit y of losing a large part of your t rading capit al. We will t ry t o develop som e m et hods, which allow you t o reduce t he probabilit y of large losses, but t here is no way t o m ake t his probabilit y equal t o zero. I f a t rader loses 50% of his capit al it can be a real disast er. I f he or she st art s spending a sm all am ount of m oney for buying st ocks, t he brokerage com m issions can play a very significant role. As t he percent age allot t ed t o com m issions increases, t he t ot al ret urn suffers. I t can be quit e difficult for t he t rader t o ret urn t o his init ial level of t rading capit al. Let us st art by analyzing t he sim plest possible st rat egy. Pr oble m : Suppose a t rader buys one st ock every day and his daily average ret urn is equal t o R. The st andard deviat ion of t hese ret urns ( risk) is equal t o s. What is t he probabilit y of losing 50 or m ore percent of t he init ial t rading capit al in t he course of one year? Solut ion:

Suppose that during one given year a trader makes about 250 trades. Suppose also that the distribution of return can be described by gaussian curve. (Generally this is not true. For a good strategy the distribution is not symmetric and the right wing of the distribution curve is higher than the left wing. However this approximation is good enough for purposes of comparing different trading strategies and estimating the probabilities of the large losses.) We w ill not present t he equat ion t hat allows t hese calculat ions t o be perform ed. I t is a st andard problem from gam e t heory. As always you can writ e us t o find out m ore about t his problem . Here we will present t he result of t he calculat ions. One t hing we do have t o not e: we use t he growt h coefficient s t o calculat e t he annual ret urn and t he probabilit y of large drops in t he t rading capit al. The next figure shows t he result s of calculat ing t hese probabilit ies ( in % ) for different values of t he average ret urns and risk- t o- ret urn rat ios.

18

The pr oba bilit ie s ( in % ) of 5 0 % dr ops in t he t r a ding ca pit a l for diffe r e nt va lue s of a ve r a ge r e t ur ns a n d r isk - t o- r e t ur n r a t ios One can see t hat for risk t o ret urn rat ios less t han 4 t he probabilit y of losing 50% of t he t rading capit al is very sm all. For risk/ ret urn > 5 t his probabilit y is high. The probabilit y is higher for t he larger values of t he average ret urns. Conclusion : A t rader should avoid st rat egies w it h large values of average ret urns if t he risk t o ret urn rat ios for t hese st rat egies are larger t han 5. I nflue nce of Com m issions We have m ent ioned t hat w hen a t rader is losing his capit al t he sit uat ion becom es worse and worse because t he influence of t he brokerage com m issions becom es larger. As an exam ple consider a t rading m et hod, which yields 2% ret urn per day and com m issions are equal t o 1% of init ial t rading capit al. I f t he capit al drops as m uch as 50% t hen com m issions becom e 2% and t he t rading syst em st ops w orking because t he average ret urn per day becom es 0% . We have calculat ed t he probabilit ies of a 50% capit al drop for t his case for different values of risk t o ret urn rat ios. To com pare t he dat a obt ained we have also calculat ed t he probabilit ies of 50% capit al drop for an average daily ret urn = 1% ( no com m issions have been considered) . For init ial t rading capit al t he ret urns of t hese st rat egies are equal but t he first st rat egy becom es worse when t he capit al becom es sm aller t han it s init ial value and becom es bet t er when t he capit al becom es larger t han t he init ial capit al. Mat hem at ically t he ret urn can be writ t en as R = Ro - com m ission s/ ca pit a l * 1 0 0 % where R is a real ret urn and Ro is a ret urn wit hout com m issions. The next figure shows t he result s of calculat ions.

19

The probabilit ies ( in % ) of 50% drops in t he t rading capit al for different values of t he average ret urns and risk- t o- ret urn rat ios. Filled sym bols show t he case when com m issions/ ( init ia l ca pit a l) = 1% and Ro = 2% . Open sym bols show t he case when Ro = 1% and com m issions = 0. One can see from t his figure t hat t aking int o account t he brokerage com m issions subst ant ially increases t he probabilit y of a 50% capit al drop. For considered case t he st rat egy even wit h risk t o r e t u r n r a t io = 4 is very dangerous. The probabilit y of losing 50% of t he t rading capit al is larger t han 20% when t he risk of ret urn rat ios are m ore t han 4. Let us consider a m ore realist ic case. Suppose one t rader has $10,000 for t rading and a second t rader has $5,000. The round t rip com m issions are equal t o $20. This is 0.2% of t he init ial capit al for t he first t rader and 0.4% for t he second t rader. Bot h t raders use a st rat egy wit h t he average daily ret urn = 0.7% . What are t he probabilit ies of losing 50% of t he t rading capit al for t hese t raders depending on t he risk t o ret urn rat ios? The answer is illust rat ed in t he next figure.

The probabilit ies ( in % ) of 50% drops in t he t rading capit al for different values of t he average ret urns and risk- t o- ret urn rat ios. Open sym bols represent t he first t rader ( $10,000 t rading capit al) . Filled sym bols represent t he second t rader ( $5,000 t rading capit al) . See det ails in t he t ext . From t he figure one can see t he increase in t he probabilit ies of losing 50% of t he t rading capit al for sm aller capit al. For risk t o ret urn rat ios great er t han 5 t hese probabilit ies becom e very large for sm all t rading capit als. Once again: avoid t rading st rat egies wit h risk t o ret urn rat ios > 5.

20 D ist r ibut ions of An n ua l Re t ur ns I s everyt hing t ruly bad if t he risk t o ret urn rat io is large? No, it is not . For large values of risk t o ret urn rat ios a t rader has a chance t o be a lucky winner. The larger t he risk t o ret urn rat io, t he broader t he dist ribut ion of annual ret urns or annual capit al growt h. Ann ua l ca pit a l gr ow t h = ( Ca pit a l a ft e r 1 ye a r ) / ( I nit ia l Ca pit a l) We calculat ed t he dist ribut ion of t he annual capit al growt hs for t he st rat egy wit h t he average daily ret urn = 0.7% and t he brokerage com m issions = $20. The init ial t rading capit al was supposed = $5,000. The result s of calculat ions are shown in t he next figure for t wo values of t he risk t o ret urn rat ios.

D ist r ibut ions of a n nu a l ca pit a l gr ow t hs for t he st r a t e gy de scr ibe d in t he t e x t The average annual capit al growt hs are equal in bot h cases ( 3.0 or 200% ) but t he dist ribut ions are very different . One can see t hat for a risk t o ret urn rat io = 6 t he chance of a large loss of capit al is m uch larger t han for t he risk t o ret urn rat io = 3. However, t he chance of annual gain larger t han 10 ( > 900% ) is m uch great er. This st rat egy is good for t raders w ho like risk and can afford losing t he whole capit al t o have a chance t o be a big winner. I t is like a lot t ery wit h a m uch larger probabilit y of being a winner. W he n t o give up I n t he previous sect ion we calculat ed t he annual capit al grow t h and supposed t hat t he t rader did not st op t rading even when his capit al had becom e less t han 50% . This m akes sense only in t he case w hen t he influence of brokerage com m issions is sm all even for reduced capit al and t he t rading st rat egy is st ill working well. Let us analyze t he st rat egy of t he previous sect ion in det ail. The brokerage com m issions were supposed = $20, which is 0.4% for t he capit al = $5,000 and 0.8% for t he capit al = $2,500.

21 So, aft er a 50% drop t he st rat egy for a sm all capit al becom es unprofit able because t he average ret urn is equal t o 0.7% . For a risk t o ret urn rat io = 6 t he probabilit y of t ouching t he 50% level is equal t o 16.5% . Aft er t ouching t he 50% level a t rader should give up, swit ch t o m ore profit able st rat egy, or add m oney for t rading. The chance of winning w it h t he am ount of capit al = $2,500 is very sm all. The next figure shows t he dist ribut ion of t he annual capit al growt hs aft er t ouching t he 50% level.

D ist r ibut ion of t he a nnua l ca pit a l gr ow t hs a ft e r t ouching t he 5 0 % le ve l. I nit ia l ca pit a l = $ 5 ,0 0 0 ; com m issions = $ 2 0 ; r isk / r e t ur n r a t io = 6 ; a ve r a ge da ily r e t ur n = 0 .7 % One can see t hat t he chance of losing t he ent ire capit al is quit e high. The average annual capit al grow t h aft er t ouching t he 50% level ( $2,500) is equal t o 0.39 or $1950. Therefore, aft er t ouching t he 50% level t he t rader w ill lose m ore m oney by t he end of t he year. The sit uat ion is com plet ely different w hen t he t rader st art ed wit h $10,000. The next figure shows t he dist ribut ion of t he annual capit al growt hs aft er t ouching t he 50% level in t his m ore favorable case.

D ist r ibut ion of t he a nnua l ca pit a l gr ow t hs a ft e r t ouching t he 5 0 % le ve l. I nit ia l ca pit a l = $ 1 0 ,0 0 0 ; com m issions = $ 2 0 ; r isk / r e t ur n r a t io = 6 ; a ve r a ge da ily r e t ur n = 0 .7 % One can see t hat t here is a good chance of finishing t he year w it h a zero or even posit ive result . At least t he chance of ret aining m ore t han 50% of t he original t rading capit al is m uch larger t han t he chance of losing t he rest of m oney by t he end of t he year. The average annual capit al growt h aft er t ouching t he 50% level is equal t o 0.83. Therefore, aft er t ouching t he 50% level t he t rader will com pensat e for som e losses by t he end of t he year.

22 Conclusion : Do not give up aft er losing a large port ion of your t rading capit al if your st rat egy is st ill profit able. Ca sh Re se r ve We have m ent ioned t hat aft er a large capit al drop a t rader can st art t hinking about using his of her reserve capit al. This m akes sense when t he st rat egy is profit able and adding reserve capit al can help t o fight t he larger cont ribut ion of t he brokerage com m issions. As an exam ple we consider t he sit uat ion described in t he previous sect ion. Let us writ e again som e param et ers: I nit ial t rading capit al = $5,000 Average daily ret urn = 0.7% ( wit hout com m issions) Brokerage com m issions = $20 ( roundt rip) Risk/ Ret urn = 3 Reserve capit al = $2,500 will be added if t he m ain t rading capit al drops m ore t han 50% . The next figure shows t he dist ribut ion of t he annual capit al growt hs for t his t rading m et hod.

D ist r ibut ion of t he a nnua l ca pit a l gr ow t hs a ft e r t ouching t he 5 0 % le ve l. I nit ia l ca pit a l = $ 5 ,0 0 0 ; com m issions = $ 2 0 ; r isk / r e t ur n r a t io = 6 ; a ve r a ge da ily r e t ur n = 0 .7 % . Re se r ve ca pit a l of $ 2 ,5 0 0 ha s be e n use d a ft e r t he 5 0 % dr op of t he init ia l ca pit a l The average annual capit al growt h aft er t ouching t he 50% level for t his t rading m et hod is equal t o 1.63 or $8,150, which is larger t han $7,500 ( $5,000 + $2,500) . Therefore, using reserve t rading capit al can help t o com pensat e som e losses aft er a 50% capit al drop. Let 's consider a im port ant pract ical problem . We were t alking about using reserve capit al ( $2,500) only in t he case when t he m ain capit al ( $5,000) drops m ore t han 50% . What will happen if we use t he reserve from t he very beginning, i.e. we will use $7,500 for t rading wit hout any cash reserve? Will t he average annual ret urn be larger in t his case? Yes, it will. Let us show t he result s of calculat ions. I f a t rader uses $5,000 as his m ain capit al and adds $2,500 if t he capit al drops m ore t han 50% t hen in one year he will have on average $15,100. I f a t rader used $7,500 from t he beginning t his figure will be t ransform ed t o $29,340, which is alm ost t wo t im es larger t han for t he first m et hod of t rading. I f com m issions do not play any role t he difference bet ween t hese t wo m et hods is sm aller.

23 As an exam ple consider t he described m et hods in t he case of zero com m issions. Suppose t hat t he average daily ret urn is equal t o 0.7% . I n t his case using $5,000 and $2,500 as a reserve yields an average of $28,000. Using t he ent ire $7,500 yields $43,000. This is about a 30% difference. Therefore, if a t rader has a w inning st rat egy it is bet t er t o use all capit al for t rading t han t o keep som e cash for reserve. This becom es even m ore im port ant when brokerage com m issions play a subst ant ial role. You can say t hat t his conclusion is in cont radict ion t o our previous st at em ent , where we said how good it is t o have a cash reserve t o add t o t he t rading capit al when t he lat t er drops t o som e crit ical level. The answer is sim ple. I f a t rader is sure t hat a st rat egy is profit able t hen it is bet t er t o use t he ent ire t rading capit al t o buy st ocks ut ilizing t his st rat egy. However, t here are m any sit uat ions when a t rader is not sure about t he profit abilit y of a given st rat egy. He m ight st art t rading using a new st rat egy and aft er som e t im e he decides t o put m ore m oney int o playing t his gam e. This is a t ypical case w hen cash reserve can be very useful for increasing t rading capit al, part icularly when t he t rading capit al drops t o a crit ical level as t he brokerage com m issions st art playing a subst ant ial role. The reader m ight ask us again: if t he t rading capit al drops why should we put m ore m oney int o playing losing gam e? You can find t he answer t o t his quest ion in t he next sect ion. I s your st r a t e gy pr ofit a ble ? Suppose a t rader m akes 20 t rades using som e st rat egy and loses 5% of his capit al. Does it m ean t hat t he st rat egy is bad? No, not necessarily. This problem is relat ed t o t he det erm inat ion of t he average ret urn per t rade. Let us consider an im port ant exam ple. The next figure represent s t he ret urns on 20 hypot het ical t rades.

Ba r gr a ph of t he 2 0 r e t ur ns pe r t r a de de scr ibe d in t he t e x t Using growt h coefficient s we calculat ed t he t ot al ret urn, which is det erm ined by t ot a l r e t ur n = ( cur r e nt ca pit a l - init ia l ca pit a l) / ( init ia l ca pit a l) * 1 0 0 % For t he considered case t he t ot al ret urn is negat ive and is equal t o - 5% . We have calculat ed t his num ber using t he growt h coefficient s. The calculat ed average ret urn per t rade is

24 also negat ive, and it is equal t o - 0.1% wit h t he st andard deviat ion ( risk) = 5.4% . The average growt h coefficient is less t han 1, which also indicat es t he average loss per t rade. Should t he t rader abandon t his st rat egy? The answer is no. The st rat egy seem s t o be profit able and a t rader should cont inue using it . Using t he equat ions present ed in part 1 of t his publicat ion gives t he wrong answer and can lead t o t he wrong conclusion. To underst and t his st at em ent let us consider t he dist ribut ion of t he ret urns per t rade. Usually t his dist ribut ion is asym m et ric. The right wing of t he dist ribut ion is higher t han t he left one. This is relat ed t o nat ural lim it of losses: you cannot lose m ore t han 100% . However, let us for sim plicit y consider t he sym m et ry dist ribut ion, which can be described by t he gaussian curve. This dist ribut ion is also called a norm al dist ribut ion and it is present ed in t he next figure.

N or m a l dist r ibut ion. s is t he st a nda r d de via t ion The st andard deviat ion s of t his dist ribut ion ( risk) charact erizes t he widt h of t he curve. I f one cut s t he cent ral part of t he norm al dist ribut ion wit h t he widt h 2s t hen t he probabilit y of finding an event ( ret urn per t rade in our case) wit hin t hese lim it s is equal t o 67% . The probabilit y of finding a ret urn per t rade wit hin t he 4s lim it s is equal t o 95% . Therefore, t he probabilit y t o find t he t rades wit h posit ive or negat ive ret urns, which are out of 4s lim it s is equal t o 5% . Low e r lim it = a ve r a ge r e t ur n - 2 s Uppe r lim it = a ve r a ge r e t ur n - 2 s The ret urn on t he last t rade of our exam ple is equal t o - 20% . I t is out of 2s and even 4s lim it s. The probabilit y of such losses is very low and considering - 20% loss in t he sam e way as ot her ret urns would be a m ist ake. What can be done? Com plet ely neglect ing t his negat ive ret urn would also be a m ist ake. This t rade should be considered separat ely. There are m any w ays t o recalculat e t he average ret urn for given st rat egy. Consider a sim plest case, one where t he large negat ive ret urn has occurred on a day when t he m arket drop is m ore t han 5% . Such event s are very rare. One can find such drops one or t w o t im es per year. We can assum e t hat t he probabilit y of such drops is about 1/ 100, not 1/ 20 as for ot her ret urns. I n t his case t he average ret urn can be calculat ed as Ra v = 0 .9 9 R1 + 0 .0 1 R2

25 where R1 is t he average ret urn calculat ed for t he first 19 t rades and R2 = - 20% is t he ret urn for t he last t rade relat ed t o t he large m arket drop. I n our exam ple R1 = 1% and Ra v = 0.79% . The st andard deviat ion can be left equal t o 5.4% . This m et hod of calculat ing t he average ret urns is not m at hem at ically perfect but it reflect s real sit uat ions in t he m arket and can be used for crude est im at ions of average ret urns. Therefore, one can consider t his st rat egy as profit able and despit e loss of som e m oney it is wort h cont inuing t rading ut ilizing t his st rat egy. Aft er t he t rader has m ade m ore t rades it would be a good idea t o recalculat e t he average ret urn and m ake t he final conclusion based on m ore st at ist ical dat a. We should also not e t hat t his com plicat ion is relat ed exclusively t o sm all st at ist ics. I f a t rader m akes 50 and m ore t rades he m ust t ake int o account all t rades wit hout any special considerat ions. Using Tr a ding St r a t e gie s a n d Tr a ding Psychology This short sect ion is very im port ant . We wrot e t his sect ion aft er analysis of our own m ist akes and we hope a reader will learn from our experience how t o avoid som e t ypical m ist akes. Suppose a t rader perform s a com put er analysis and develops a good st rat egy, which requires holding st ocks for 5 days aft er purchase. The st rat egy has an excellent hist orical ret urn and behaves well during bull and bear m arket s. However, when t he t rader st art s using t he st rat egy he discovers t hat t he average ret urn for real t rading is m uch w orse. Should t he t rader swit ch t o anot her st rat egy? Before m aking such a decision t he t rader should analyze why he or she is losing m oney. Let us consider a t ypical sit uat ion. Consider hypot het ical dist ribut ions of hist orical ret urns and real ret urns. They are shown in t he next figure.

H ypot he t ica l dist r ibu t ions of t he hist or ica l a nd r e a l t r a ding r e t ur ns This figure shows a t ypical t rader's m ist ake. One can see t hat large posit ive ret urns ( > 10% ) are m uch m ore probable t han large negat ive ret urns. However, in real t rading t he probabilit y of large ret urns is quit e low. Does t his m ean t hat t he st rat egy st ops working as soon as a t rader st art s using it ? Usually, t his is not t rue. I n of m ost cases t raders do not follow st rat egy. I f t hey see a profit

26 10% t hey t ry t o sell st ocks or use st op orders t o lock in a profit . Usually t he st op orders are execut ed and t he t rader never has ret urns m ore t han 10 - 15% . On t he ot her hand if t raders see a loss of about 10% t hey t ry t o hold a st ock longer in hope of a recovery and t he loss can becom e even larger. This is w hy t he dist ribut ion of ret urns shift s t o t he left side and t he average ret urn is m uch sm aller t han hist orical ret urn. Conclusion : I f you find a profit able st rat egy - follow it and const ant ly analyze your m ist akes. Tr a din g Pe r iod a nd Annua l Re t ur n To calculat e t he average annual ret urn one needs t o use t he average daily growt h coefficient calculat ed for t he whole t rading capit al. Let us rem ind t he reader t hat t his coefficient should be calculat ed as an average rat io Ki = < C ( i) / C ( i- 1 ) > where C( i) is t he value of t he capit al at t he end of i- t h t rading day. The average growt h coefficient m ust be calculat ed as t he geom et ric average, i.e. for n t rading days Ka v - t he a ve r a ge da ily gr ow t h coe fficie nt Ka v = ( K1 * K2 * ... * Kn) ^ ( 1 / n) How should one calculat e t he average daily growt h coefficient if a t rader holds t he st ocks for 2 days, spending t he ent ire capit al t o buy st ocks? This m et hod of t rading can be present ed graphically as BUY

H OLD

SELL

BUY

H OLD

SELL BUY

Suppose t hat t he average growt h coefficient per t rade ( not per day! ) is equal t o k . This can be int erpret ed as t he average growt h coefficient per t wo days. I n t his case t he average growt h coefficient per day Kav can be calculat ed as t he square root of k Ka v = k ^ ( 1 / 2 ) The num ber of days st ocks are held we will call t he t rading period. I f a t rader holds st ocks for N days t hen t he average ret urn per day can be writ t en as Ka v - t he average daily growt h coefficient k - t he average growt h coefficient per t rade N - holding ( t rading) period Ka v = k ^ ( 1/ N) The average annual capit al growt h Ka nn ua l ( t he rat io of capit al at t he end of t he year t o t he init ial capit al) can be calculat ed as Ka nn ua l = k ^ ( 2 5 0 / N ) We supposed t hat t he num ber of t rading days per year is equal t o 250. One can see t hat annual ret urn is larger for a larger value of k and it is sm aller for a larger num ber of N . I n ot her

27 words, for a given value of ret urn per t rade t he annual ret urn will suffer if t he st ock holding period is large. Which is bet t er: holding st ocks for a short er period of t im e t o have m ore t rades per year or holding st ocks for a longer t im e t o have a larger ret urn per t rade k ? The next graph illust rat es t he dependence of t he annual growt h coefficient on k and N .

The a n nua l ca pit a l gr ow t h K( a nnua l) a s a fu nct ion of t he a ve r a ge gr ow t h coe fficie nt pe r t r a de k for va r ious st ock h olding pe r iods N Using t his graph one can conclude t hat t o have an annual capit al growt h equal t o about 10 ( 900% annual ret urn) one should use any of follow ing st rat egies: N = 1 a nd k = 1 .0 1 N = 2 a nd k = 1 .0 2 N = 3 a nd k = 1 .0 3 I n t he first case one should t rade st ocks every day, which subst ant ially increases t he t ot al brokerage com m issions. Let us consider t his im port ant problem in det ail. Suppose t hat t he round t rip brokerage com m issions and bid- ask spread equal ( on average) 1.5% of t he capit al used t o buy a st ock. I n t his case t he first st rat egy becom es unprofit able and profit s from ot her t wo st rat egies are sharply reduced. One needs t o have m uch larger profit per t rade t o have a large annual ret urn. For annual capit al growt h about 10 t he st rat egies wit h N = 1, 2, 3 should have growt h coefficient s per t rade k as large as N = 1 a nd k = 1 .0 2 5 N = 2 a nd k = 1 .0 3 5 N = 3 a nd k = 1 .0 4 5 I f a t rader uses a st rat egy wit h a t rading period of t wo and m ore days he can divide his t rading capit al t o buy st ocks every day. I n t his case t he risk of t rading will be m uch lower. This im port ant problem w ill be considered in t he next sect ion.

28 The or y of D ive r sifica t ion Suppose t hat a t rader uses a st rat egy wit h t he holding period N = 2. He buys st ocks and sells t hem on t he day aft er t om orrow. For t his st rat egy t here is an opport unit y t o divide t he t rading capit al in half and buy st ocks every day, as show n in t he next t able Fir st ha lf of ca pit a l Se con d ha lf of ca pit a l

BUY

H OLD BUY

SELL

BUY

H OLD

H OLD SELL

SELL BUY

BUY H OLD

H OLD SELL BUY

SELL BUY H OLD

Every half of t he capit al will have t he average annual growt h coefficient Ka nn ua l ( 1 / 2 ) = k ^ ( 2 5 0 / 2 ) and it is easily t o calculat e t he annual growt h coefficient ( annual capit al growt h) for t he ent ire capit al Ka nnu a l. Ka nn ua l = ( Ca pit a l a ft e r 1 ye a r ) / ( I nit ia l Ca pit a l) Let CAP ( 0 ) denot es t he init ial capit al and CAP ( 2 5 0 ) t he capit al aft er 1 year t rading. One can writ e Ka nn ua l = CAP ( 2 5 0 ) / CAP ( 0 ) = Ka nnu a l ( 1 / 2 ) = k ^ ( 2 5 0 / 2 ) Correspondingly for N = 3 one can writ e Ka nn ua l = CAP ( 2 5 0 ) / CAP ( 0 ) = k ^ ( 2 5 0 / 3 ) and so on. One can see t hat t he form ula for annual capit al growt h does not depend on capit al division. The only difference is t he larger influence of brokerage com m issions. However, if we consider t he risk of t rading when t he capit al is divided we can conclude t hat t his m et hod of t rading has a great advant age! To calculat e t he risk for t he st rat egy wit h N = 2 ( as an exam ple) one can use an equat ion S = SQRT ( 1 / 4 * s^ 2 + 1 / 4 * s^ 2 ) = s * SQRT ( 1 / 2 ) where SQRT is a not at ion for t he square root funct ion and s is t he risk of a t rade. I t w as supposed t hat t he t rader buys one st ock per day. For any N one can obt ain S = s / SQRT ( N ) The larger N is, t he sm aller t he risk of t rading. This is relat ed t o dividing capit al diversificat ion. However, t he m ore you divide your capit al, t he m ore you need t o pay com m issions. Mat hem at ically, t his problem is ident ical t o t he problem of buying m ore st ocks every day. The risk will be sm aller, but t he t rader has t o pay m ore com m issions and t he t ot al ret urn can be sm aller. What is t he opt im al capit al division for obt aining t he m inim al risk t o ret urn rat io? Let us consider an exam ple, which can help t o underst and how t o invest igat e t his problem .

29 Pr oble m : Suppose, a t rader has $5,000 t o buy st ocks and he does not use m argin. The brokerage round t rip com m issions are equal t o $20. The average ret urn per t rade ( aft er t aking int o account t he bid- ask spread) is equal t o R ( % ) . The ret urns have dist ribut ion wit h t he st andard deviat ion ( risk) s. What is t he opt im al num ber of st ocks N a t rader should buy t o m inim ize t he risk t o ret urn rat io? Solut ion: To buy one st ock a t rader can spend ( 5000 / N - 10) dollars. The average ret urn R1 per one st ock will be equal t o

R1 = 1 0 0 % * [ ( 5 0 0 0 / N - 1 0 ) R/ 1 0 0 - 1 0 ] / ( 5 0 0 0 / N ) = 1 0 0 % * [ ( 5 0 0 0 - 1 0 N ) R/ 1 0 0 - 1 0 N ] / 5 0 0 0

This is also equal t o t he average ret urn Ra v of t he ent ire capit al because we consider ret urn in % . Ra v = R1 One can see t hat increasing N reduces t he average ret urn and t his reduct ion is larger for a larger value of brokerage com m issions. The t ot al risk S is decreased by 1 / SQRT ( N ) as we discussed previously. S = s/ SQRT ( N ) The risk t o ret urn rat io can be writ t en as S/ R = 5 0 0 0 * s/ { SQRT ( N ) * 1 0 0 % * [ ( 5 0 0 0 - 1 0 N ) R/ 1 0 0 - 1 0 N ] } The problem is reduced t o t he problem of finding t he m inim um of t he funct ion S/ R. The value of s does not shift t he posit ion of t he m axim um and for sim plicit y we can t ake s = 1. The funct ion S/ R is not sim ple and we will plot t he dependence of t he rat io S/ R for different values of N and R.

The r isk t o r e t ur n r a t io a s a funct ion of num be r of st ock s

30 One can see t hat for t he average ret urn per t rade R = 1% and for t he $5000 capit al t he opt im al num ber of st ocks when t he risk t o ret urn rat io is m inim al is equal t o 1.5. Therefore, a t rader should buy 1 st ock one day, 2 st ocks anot her day, ... For R = 1.5% t he opt im al num ber of st ocks is equal t o 2.5. I f R = 2% t hen N = 3.5. The problem is solved. Let us consider t he " value of paym ent " for lower risk t o ret urn rat io. Ca se 1 . R = 1 % I f a t rader buys one st ock he or she w ould pay $20 in com m issions and t he average profit is equal t o ( 5000 - 10) * 0.01 - 10 = $39.9 I n t his case t he opt im al num ber of st ocks t o buy is equal t o N = 1.5. The average round t rip com m issions are equal t o $20 * 1.5 = $30 The average profit can be calculat ed as ( 5000 - 15) * 0.01 - 15 = $34.85 Therefore, a t rader is losing about $5 per day when he buys 1.5 st ocks. This is equal t o 13% of t he profit . This is t he paym ent for lower risk. Ca se 2 . R = 1 .5 % I f a t rader buys one st ock he or she w ould pay $20 in com m issions and t he average profit is equal t o ( 5000 - 10) * 0.015 - 10 = $64.85 I n t his case t he opt im al num ber of st ocks t o buy is equal t o N = 2.5. The average round t rip com m issions are equal t o $20 * 2.5 = $50 The average profit can be calculat ed as ( 5000 - 25) * 0.015 - 25 = $49.63 Therefore, t he t rader pays about $15 for lower risk. This is equal t o 23% of t he profit . Ca se 3 . R = 2 % I f a t rader buys one st ock t he average profit is equal t o ( 5000 - 10) * 0.02 - 10 = $89.8 I n t his case N = 3.5. The average round t rip com m issions are equal t o $20 * 3.5 = $70 The average profit is equal t o ( 5000 - 35) * 0.02 - 35 = $64.3 Therefore, t he t rader pays about $25 for lower risk. This is equal t o 28% of t he profit . You can ask why we should lose so m uch in profit t o have lower risk? The answer is sim ple. This is t he only way t o survive in t he m arket wit h a sm all init ial t rading capit al. Aft er a couple of m ont hs of successful t rading your capit al will be larger and t he influence of com m issions will be m uch sm aller.

31 PART 3 Ta ble of Con t e n t s 1. Random walk and st op- lim it st rat egy 2. Non- random walk and st op- lim it st rat egy 3. How t o m ake a profit in t he non- random m arket 4. What can be wrong 5. St ops and real t rading Ra ndom w a lk a nd st op- lim it st r a t e gy One of t he m ost popular st ock m arket t heories is t he random walk m odel. I t is assum ed t hat for short periods of t im e st ock prices change random ly. So, probabilit ies of growt h and decline are equal. One of t he m ost im port ant t heorem s of t he random walk m odel can be form ulat ed for t he st ock m arket as: For any init ial st ock price and for any lim it price t he probabilit y of hit t ing t he lim it price is equal t o 1. From t he first point of view m aking profit is very easy: buy st ock, place t he lim it order t o sell above t he st ock price and wait . Sooner or lat er t he lim it will be t ouched and you can m ake a profit . This is wrong! There is no chance of m aking any profit in t he random m arket . You should rem em ber t hat t he st ock price can also t ouch t he level = 0 ( or very low price) and t he gam e is over. To show how t o analyze t he t rading st rat egies in t he random m arket consider st op- lim it st rat egy. The st op level can be equal t o zero, which corresponds t o a gam e wit hout any st op. S L _______________| _________| ___________| _________________ st op cur r e nt lim it price On t he diagram , S and L are t he differences bet ween t he current st ock price and t he st op and lim it order levels. The sim plest st rat egy is t o buy som e st ock and wait unt il t he st ock price t ouches t he st op or lim it levels ( prices) . I f t he st op level is t ouched first - you are a loser. I f t he st ock price t ouches t he lim it level - you sell t he st ock wit h a profit and you are a winner. I t can be shown t hat for t his m odel t he probabilit y of t ouching t he st op level can be writ t en as P ( S) = L / ( S + L) One can find t he derivat ion of t his equat ion in William Feller's book I nt roduct ion t o Probabilit y Theory and I t s Applicat ion. The probabilit y of t ouching t he lim it level is equal t o P ( L) = S / ( S + L) One can check t hat P ( L) = 1 – P ( S) . The average ret urn per t rade can be writ t en as R = L* P ( L) - S* P ( S) = 0

32 This st rat egy gives zero average ret urn in t he case of zero com m issions. I t can be shown t hat t he variance of t he ret urns ( t he squared st andard deviat ion or " squared risk") is equal t o s2 = S * L So, t he larger t he deviat ions of t he st op or lim it order levels from t he current st ock price, t he larger t he risk of t his t rading st rat egy. You can consider m any ot her t rading st rat egies but it can be shown t hat for random walk price changes t he average ret urns are always equal t o zero for zero t ransact ion cost s. I n real life you w ill be a loser because of brokerage com m issions and bid- ask spreads. N on- r a n dom w a lk a n d st op- lim it st r a t e gy The sit uat ion changes when t he probabilit y of growt h is larger t han t he probabilit y of decline. I n t his case t he probabilit y of t ouching t he lim it level can be higher t han t he probabilit y of t ouching t he st op level and your average ret urn is posit ive. Consider a sim ple case. Suppose t hat t he st ock price is equal t o $100 and t he price is changing by one dollar st eps. As in t he previous sect ion let L and S be t he deviat ions of t he lim it and st op orders from t he st ock price. Denot e by p t he probabilit y of st ock price gain and by q t he probabilit y of st ock price decline. The sum of t hese probabilit ies is equal t o 1. p + q = 1 The probabilit y of t ouching t he st op level can be writ t en as P ( S) = [ 1 - ( p/ q) ^ L] /

[ 1 - ( p/ q) ^ ( S+ L) ]

The probabilit y of t ouching t he lim it level can be calculat ed from P ( L) = 1 – P ( S) The average ret urn can be calculat ed from t he equat ion R = L* P ( L) - S* P ( S) What are t he opt im al lim it and st op orders in t he case when p is not equal t o q? The next figure shows t he result s of calculat ion of t he average ret urns per t rade for various values of t he st op and lim it orders.

33

Ave r a ge r e t ur n pe r t r a de for st op- lim it st r a t e gy for t w o va lue s of t he gr ow t h pr oba bilit y p. S is t he de via t ion of t he st op le ve l fr om t he cur r e nt st ock pr ice in % From t his figure one can draw t wo very im port ant conclusions: - if a t rader select s st ocks wit h a grow t h probabilit y larger t han 50% t hen t he average ret urn is posit ive and t he st op levels m ust be far away from t he current st ock price t o obt ain t he m axim al ret urn. - if a t rader select st ocks wit h a grow t h probabilit y sm aller t han 50% t hen t he average ret urn is negat ive and t he st op levels m ust be as close as possible t o t he current st ock price t o m inim ize losses. There is an im port ant t heorem for t he non- random walk m odel: if t he lim it level is equal t o infinit y ( no lim it ) t hen t he probabilit y of t ouching t he st op level is equal t o: P = 1 P = ( q/ p) ^ S

if p < = q if p > q

Therefore, if p > q ( bullish st ock) t here is a chance t hat t he st op level will never be t ouched. This probabilit y is larger for larger S and sm aller rat io q/ p. H ow t o m a k e a pr ofit in t he non- r a ndom m a r k e t I t is hard t o im agine t hat all st ocks have a probabilit y of growt h exact ly equal t o 50% . Many st ocks have a growt h probabilit y of 55% , 60% and m ore. How ever, t he st ock m arket does not grow w it h a high probabilit y and t his m eans t here m any st ock s wit h low growt h probabilit ies 45% , 40% and less. The probabilit ies depend on t echnically overbought or oversold condit ions, good or bad fundam ent als, int erest from t raders, et c. How can we apply st op- lim it st rat egy for t his m arket t o obt ain a profit ?

34 To answer t his quest ion, consider a sim ple case. Suppose t hat t he "m arket " consist s of t w o st ocks. One st ock has t he growt h probabilit y p1 = 0.55 and t he second st ock has t he growt h probabilit y p2 = 0 .4 5 . We buy t he t wo st ocks using equal am ount s of m oney for each st ock. We also place st op loss orders and place m ent al lim it orders t o sell if w e see som e profit . Denot e by S and L t he differences bet ween t he current st ock price and t he st op and lim it order levels. The next figure shows t he average ret urns per t rade for t his st rat egy depending on t he levels of st op and lim it orders.

The a ve r a ge r e t ur n a s a funct ion of t he le ve ls of lim it or de r for va r ious st op loss or de r le ve ls. D e t a ils of t his t r a ding st r a t e gy a r e de scr ibe d in t he t e x t I t is im port ant t o not e t hat using st op loss orders which are far away from t he current st ock price t oget her w it h close lim it orders provide negat ive ret urns. The winning st rat egy is using " t ight " st ops and large lim it t arget s. W ha t ca n be w r ong From t he previous sect ion one can conclude t hat m aking m oney in t he st ock m arket is quit e easy. All you need is t o have enough m oney t o buy a dozen of st ocks, place st ops and lim it s and wait for a nice profit . However, t he m aj orit y of t raders are losing m oney. What is wrong? I n t he previous sect ion we supposed t hat st ocks have grow t h probabilit ies, which are st able. So, if p = 0.55 at t he m om ent of purchase it w ill be equal t o 0.55 even aft er sharp price growt h. This is wrong. One can find a st ock wit h p = 0.55 and even larger but aft er a couple of days t his probabilit y usually becom es close t o 0.5 or even becom es less t hen 0.5, which is t he reason for price fluct uat ions. The wait ing t im e when your lim it order w ill be execut ed can be very long. Many t im es you will sell st ock s early wit h sm aller profit . I f you select st ocks random ly, you will m ost ly select st ocks wit h growt h probabilit ies close t o 50% . I f we consider t he m odel described in t he previous sect ion wit h p1 = 0.49 and p2 = 0.51 t hen t his schem e will not work. Brokerage com m issions and bid- ask spreads will eat up your profit . What can be done? You need t o select st ocks wit h high probabilit ies of growt h and opt im ize t he st op order levels. I n t he next sect ion we present t he result s of com put er analysis of t he real m arket t o show m et hods for developing a winning t rading st rat egy.

35 St ops a n d r e a l t r a ding Suppose you random ly select a st ock, buy it and want t o place a st op loss order t o prevent your t rading capit al from a large loss. What is t he probabilit y of t ouching your st op order level? How does it depend on t he st ock volat ilit y? How will t he probabilit y of t ouching st op change in t w o days? Five days? To answ er t hese quest ions w e have perform ed a com put er analysis of t he 11 years hist ory of 300 random ly select ed act ively t raded st ocks. The st op order will be execut ed during a designat ed t im e period if t he m inim al st ock price for t his period is lower t han your st op order level. Suppose you bought a st ock at t he m arket closing at t he price CLO. Denot e by MI N t he m inim al st ock price during a cert ain period of t im e. We will st udy t he dist ribut ion of t he difference x = M I N - CLO which charact erizes t he m axim um of your possible loss on t he t rade for t he considered period of t im e. The next figure illust rat es t his st at em ent .

The diffe r e nce M I N - CLO cha r a ct e r ize s t he m a x im a l loss fr om t he t r a de dur ing 1 1 t r a din g da ys a ft e r t h e st ock pu r cha se The average value of M I N - CLO depends on t he st ock's volat ilit y. There are m any ways of defining t he volat ilit y. We w ill consider t he average am plit udes of daily price change A = < M AX - M I N > / 2 where t he averaging has been perform ed for a one- m ont h period. I t is obvious t hat t he average value of < MI N - CLO> m ust depend on t he am plit ude A. We have shown t hat < M I N - CLO> is approxim at ely equal t o negat ive value of A, i.e. < M I N - CLO> = - A where M I N is t he m inim al st ock price during t he next day aft er purchase. The correlat ion coefficient of t his linear dependence is equal t o 0.5. For a longer period of t im e t he value of < M I N - CLO> can be m uch less t han - A.

36 You cannot expect t hat placing a st op order at a level, which is slight ly less t han - A will be safe enough t o avoid selling t he st ock due t o daily st ock price fluct uat ions. The dist ribut ion of x = M I N - CLO is rat her broad. The next figure shows t hese dist ribut ions for one and t en days aft er st ock purchase.

D ist r ibut ion of ( M I N - CLO) / A for r a ndom ly se le ct e d st ock s One can see t hat t he dist ribut ions of M I N - CLO are rat her broad and non- sym m et rical. There is a lit t le chance t hat t he m inim al st ock price will always be higher t han t he purchase price CLO. The dist ribut ion becom e broader for longer st ock holding periods. The average ( m ean of t he dist ribut ion) becom es m ore and m ore negat ive. The next figure shows t he t im e dependence of t he average values < ( M I N - CLO) / A> and t he st andard deviat ions of t he dist ribut ion of t his value.

37 Tim e de pe nde nce of t he a ve r a ge va lue s < ( M I N - CLO) / A> a nd t he st a nda r d de via t ions of t he dist r ibut ion of t his r a t io One can see a st rong increase in t he widt h of dist ribut ion wit h t im e. The crucial quest ion is t he probabilit y of t ouching t he st op levels. We perform ed t he calculat ions of t hese probabilit ies for various st op levels. The next figure present s t he result s.

Pr oba bilit ie s of t ouching st op le ve ls dur ing 1 , 2 , 4 a nd 8 da ys for r a ndom ly se le ct e d st ock s. A is t he da ily pr ice a m plit ude de fine d in t he t e x t . One can see t hat if t he st op loss order is placed at - A level t hen t he probabilit y of execut ion of t he st op order during t he next day aft er t he purchase is equal t o 45% . The probabilit y becom es equal t o 60% if you hold t he st ock for t wo days. I f you hold t he st ock for 8 days t hen t he probabilit y of execut ing t he st op order becom es 80% ! I f you place t he st op at t he - 3 A level t han t he probabilit y of st op order execut ion rem ains less t han 50% even if you hold t he st ock for 8 days. Using t his plot you can calculat e t he average loss per t rade. Let us offer you an exam ple. Suppose you bought a st ock at $100. The daily price am plit ude is equal t o $3. Suppose you hold t he st ock for 4 days. To calculat e t he average loss due t o execut ion of t he st op order you need t o m ult iply t he difference CLO - STOP by t he probabilit y of t ouching t he st op level. AVERAGE LOSS = ( CLO - STOP) * PROBABI LI TY Here is a t able t hat can help you t o select t he opt im al st op loss level. STOP

CLO - STOP

PROBABI LI TY

AVERAGE LOSS

97

3

0.71

2.13

94

6

0.49

2.94

91

9

0.32

2.88

88

12

0.19

2.28

85

15

0.11

1.65

38 I t is int erest ing t hat t he worst decision you can m ake for t his t rade is placing t he st op at t he $94 level ( - 6% ) . Your average loss is m axim al at t his point ! For pract ical purposes w e publish t he figure, which shows dependence of average loss on t he st op order level in A unit s. 1, 2, 4, and 8 days of holding st ock are considered.

D e pe nde n ce of t he a ve r a ge loss on t he st op or de r le ve l in A un it s for 1 , 2 , 4 , a nd 8 da ys st ock holdin g From t his figure one can conclude t hat t o m inim ize t he average loss from t he st op orders you need t o place st ops eit her very close or very far away t o t he current price. Your decision should be based on t he num ber of days you are going t o hold t he st ock. One again: all calculat ions have been m ade for random ly select ed st ocks. For t hese st ocks t he average growt h probabilit ies are close t o 50% . What will happen if we consider specifically select ed st ocks when t he growt h probabilit y is not equal t o 50% ? Before st art ing t o look at t his im port ant quest ion we need t o consider som e t echnical param et ers, which are very helpful in st ock select ion. They allow t o select st ocks wit h t he highest grow t h probabilit ies and develop very profit able t rading st rat egies. These param et ers have been int roduced in our book Short - Term Trading Analysis. ( See link t o Text Level- 2 on t he page ht t p: / / www.st t a- consult ing.com ) . I n t he next sect ions w e repeat t hese definit ions.

39 PART 4 Ta ble of Con t e n t s 1. St ock price t rends 2. Deviat ion param et ers 3. Ret urns of overbought and oversold st ocks 4. Opt im al st ops for oversold st ocks 5. St op st rat egy for inexperienced t raders 6. St op st rat egy for an average t rader 7. St ock volat ilit y 8. Trading st rat egy using lim it orders 9. Lim it s, st ops and risk 10. I ncreasing average ret urn St ock pr ice t r e nds Trend is a sim ple int uit ive t erm . I f a st ock price is increasing one can say t he t rend of t his st ock is posit ive. I f a st ock is declining - t he t rend is negat ive. You can also say: a st ock has m om ent um . This is st andard t erm inology. I n m any books on t echnical analysis of t he st ock m arket you can read about m om ent um invest m ent s: buy st ocks wit h t he highest grow t h rat es. We need t o define t rends m at hem at ically. The sim plest m et hod is using t he linear fit of st ock price- t im e dependence.

D iffe r e nt t r e nds Consider N t rading days and draw t he fit t ing line L ( i) t hrough t he point s P i , where P i is t he closing st ock price on t he i- t h day.

40 1 2 3 ... N | ___| ___| ___| ___| ___| ___| ___| ___| ___| ___| ___| ___| ___| ___| ___| ___| P 1 P 2 P 3 ... PN The N - t h day is t he day of int erest , t he day of t he analysis. L ( i) = A + Bi

i= 1 , 2 , ..., N

The coefficient B is t he slope of t he fit t ing line. The slope can be considered as t he price t rend of t he st ock. I t charact erizes t he average daily price change ( in dollars) during one day. However, t his value is not perfect when com paring t rends of different st ocks. We will define a t rend as t he average price change in % during one t rading day. Mat hem at ically t his can be writ t en as Tr e nd = T = < ∆P/ P> * 1 0 0 %

This definit ion has one disadvant age. What price P should be used in t his equat ion? Using t he current st ock price is not a good idea. I n t his case, t rend will be very volat ile. I t is bet t er t o use a m ore st able price charact erist ic: t he value of t he linear fit t ing line on day # N t he day of t he analysis P = A + B* N The final equat ion for t rend looks like T = B/ ( A + B* N ) * 1 0 0 % This equat ion we used in our com put er analysis of t he st ock m arket . Such a definit ion of t rend subst ant ially reduces t he influence of price fluct uat ions during t he final days. The next figure shows an exam ple of t rend calculat ions for a 16 days t im e fram e.

Ex a m ple of st ock t r e nd ca lcula t ions for a 1 6 da ys t im e fr a m e Anot her way t o define t he t rend is by using logarit hm ic scaling. This is a good idea if you st udy a st ock's long hist ory of price change, for exam ple, from 1 t o 100 dollars. For short - t erm t rading t his change is unrealist ic and we will keep t hings as sim ple as possible. So, we will use sim ple linear fit s, and t rends are relat ed t o t he slope of t he fit t ing lines.

41 D e via t ion pa r a m e t e r s ( D - pa r a m e t e r s) The deviat ion or D- param et er, which will be defined here, is crucial for short - t erm st ock t rading. Briefly, t his is a charact erist ic of deviat ion of t he current st ock price from t he fit t ing line. This charact erist ic is im port ant for t he definit ion of oversold or overbought st ocks. A sim plest idea of t he definit ion of overbought and oversold st ock s is t he locat ion of t he st ock closing price relat ive t o t he t rading range. We assum e t hat when t he st ock price is in t he t rading range, t he st ock perform ance is norm al. I f t he price is out of t his range, t he st ock m ay be oversold or overbought .

I llust r a t ion for t he de finit ion of D - pa r a m e t e r We define t rading range as a channel bet ween support and resist ance lines. Usually, t hese lines are det erm ined int uit ively from t he st ock price chart s. This is not a good way if you use t he com put er t o analyze st ock price perform ance. The line drawing depends on t he t rader's skill and t his m et hod can be used only for represent at ion of st ock perform ance in t he past . Som e t raders use m inim al and m axim al prices t o draw t he support and resist ance lines, ot hers like closing prices. There is a problem of what t o do wit h point s, which are far away from t he t rading channel, et c. 2 We suggest using t he st andard deviat ion σ ( σ = ) for t he definit ion of t he support and resist ance lines. Mat hem at ically t his can be writ t en as Su ppor t lin e = A + Bi- σ Re sist a nce line = A + Bi + σ where A + Bi is t he equat ion of t he linear fit t ing st ock price t im e dependence; i is t he day num ber. One should not ice t hat t he definit ion of t he support and resist ance lines depends on t he num ber of t rading days being considered. You should always indicat e what t im e fram e has been used. To charact erize t he deviat ion of t he st ock closing price from t he fit t ing line we have int roduced a new st ock price charact erist ic: deviat ion, or D- param et er. D = ( P N - A - BN ) / σ

where N is t he num ber of days which w ere used for linear fit t ing, P N is t he closing st ock price on day # N ( t he day of t he analysis) , A and B are t he linear fit t ing param et ers, and σ is t he st andard deviat ion of closing prices from t he fit t ing line.

42 Let us explain t he m eaning of t his equat ion. The difference ( P N - A - BN ) is t he deviat ion of t he current st ock price from t he fit t ing line. D - param et er shows t he value of t his deviat ion in σ unit s. D < - 1 t he st ock m a y be ove r sold D > + 1 t he st ock m a y be ove r bought One can say t he words " oversold" and " overbought " should be used t oget her wit h t he num ber of t rading days considered. So it m ore precise t o say: t he st ock is oversold in t he 30 days t im e fram e. Here we have t o not e t hat com parison D wit h + 1 or - 1 is not opt im al and depends on t he t im e fram e. I n t he next sect ion we will consider an approach, which allows us t o calculat e t he values of t he D and T param et ers so as t o opt im ize t he select ion of oversold and overbought st ocks. Re t ur ns of ove r bought a nd ove r sold st ock s To check t he hypot hesis about our crit eria of overbought and oversold condit ions one needs t o calculat e t he average ret urns for st ocks for which t he D param et er is negat ive or posit ive. Here we have t o not e t hat oversold and overbought condit ions becom e m ore pronounced if one also considers t he t rends T. I t is obvious ( and t his has been checked by com put er analysis) t hat if t he t rend is negat ive and D is also negat ive t he oversold condit ion becom es st ronger. How can we det erm ine t he range of t he D and T param et ers for oversold st ocks? Let us st art w it h an analysis of dist ribut ions of t hese param et ers. 16 days st ock price hist ory will be considered in t his sect ion, i.e. for calculat ion of D and T one needs t o download st ocks prices for t he last 16 t rading days. On t he next figure t he dist ribut ions of T ( 16) and D ( 16) are shown. Lat er we will drop t he ( 16) not at ion, but one needs t o rem em ber t hat for ot her hist orical t im e fram es t he dist ribut ions have subst ant ially different shapes.

D ist r ibu t ion of t h e de via t ion s ( D ) a n d t r e n ds ( T) ca lcu la t e d for a 1 6 da y s st ock pr ice h ist or y.

From t he first point of view we need t o consider st ocks wit h very low values of T and D t o be sure t hat t hese st ocks are oversold. Yes, t heoret ically t his is t rue. Com put er analysis shows t hat t he sm aller T and D , t he larger t he probabilit y of posit ive ret urns. Ret urns of st ocks wit h T < - 1% and D < - 1.5 are very large.

43 I n pract ice, select ing st ocks wit h such ext rem e values of deviat ions and t rends yields sm aller annual ret urns. The num ber of t hese st ocks is sm all a t rader is able t o find such st ocks only one or t wo t im es a m ont h. As we analyzed in t he previous sect ions, t he annual ret urn will be very sm all. I t is m ore effect ive t o select st ocks wit h soft er condit ions but t he probabilit y of finding such st ocks is higher. We suggest defining oversold st ocks as st ocks for which T < Ta v - ( St a nda r d D e via t ion of T dist r ibut ion) D < D a v - ( St a nda r d D e via t ion of D dist r ibut ion ) Correspondingly t he overbought st ocks are t he st ocks for which T > Ta v + ( St a nda r d D e via t ion of T dist r ibut ion) D > D a v + ( St a nda r d D e via t ion of D dist r ibut ion ) We will st udy t he average ret urns Re t ur n = [ CLO ( N ) - CLO] / CLO * 1 0 0 % where CLO is t he closing st ock price on t he day of t he analysis and CLO( N) is t he closing st ock price on day # N aft er t he day of t he analysis. The next schem e illust rat es our definit ions. | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ a ....| ....| ....| ....| ....| ....| ....| ....| . ...| ....| 1 6 da y hist or y for D a nd T ca lcula t ions N= 1 2 3 4 5 6 7 8 9 Here, a denot es t he day of t he analysis ( N = 0) . The dist ribut ions of t he D and T param et ers are shown on t he next figure. The st andard deviat ion of dist ribut ion of t he D - param et ers is equal t o 1.185. For t he T param et er t he st andard deviat ion is equal t o 0.558. Therefore, we will define oversold and overbought st ocks wit hin 16 days fram e as Ove r sold st ock s: T < - 0 .5 5 8 1 a n d D < - 1 .1 8 5 Ove r bought st ock s: T > 0 .5 5 8 1 a nd D > 1 .1 8 5 The next figure show s t he average ret urns of t he oversold and overbought st ocks as a funct ion of num ber N of days following t he day of t he analysis. For com parison w e present t he average ret urns of random ly select ed st ocks

44

The a ve r a ge r e t ur ns ( CLO ( N ) - CLO) / CLO * 1 0 0 % of t he ove r sold a nd ove r bought st ock s a s a funct ion of num be r N of da ys follow ing t he da y of t he a na lysis. The a ve r a ge r e t ur ns of r a ndom ly se le ct e d st ock s ( bla ck squ a r e s) a r e pr e se nt e d for com pa r ison. Let us form ulat e som e conclusions from t his plot . - The average ret urn of random ly select ed st ocks is a linear funct ion of t im e w it h a posit ive slope. This is relat ed t o t he bull m arket of t he 90's, when t he st ock price hist ory was st udied. - The average ret urns of t he oversold st ocks is m uch larger t han t he average ret urns of t he overbought st ocks. This effect is m ore pronounced for short periods of st ock holding. - I f a t rader buys oversold st ocks or sells short overbought st ocks he( she) should not hold t hese st ocks for a long t im e. I t is bet t er t o close posit ion in t hree t o five days and sw it ch t o ot her st ocks w it h higher pot ent ial short - t erm ret urns. Opt im a l st ops for ove r sold st ock s Now we are ready t o consider an analysis of opt im al st op levels for specifically select ed st ocks. As an exam ple we will consider oversold and overbought st ocks w it hin 16 day t im e fram e. Let us st udy t he correlat ion of deviat ion ( D ) and t rend ( T) param et ers w it h t he m inim al st ock price during N following days aft er t he day of analysis. The next figure present s t he average values of ( M I N - CLO) / A in t he sam e way as we did previously. Here, A is t he average daily st ock price change as we defined before. The open squares show result s for random ly select ed st ocks t o com pare t hese dat a w it h our previous analysis.

45

The a ve r a ge va lue s of ( M I N - CLO) / A for ove r sold a nd ove r bought st ock s w it hin 1 6 da ys t im e fr a m e . Ope n squ a r e s show r e sult s for r a ndom ly se le ct e d st ock s I t is int erest ing t o not e t hat for oversold st ocks t he m inim al prices for t he first four days aft er t he day of t he analysis are very close t o t he m inim al prices of random ly select ed st ocks. This is despit e t he large posit ive m ove of t he closing price of t hese st ocks. This phenom enon is relat ed t o t he higher volat ilit y of oversold st ocks during t he first days of t rading aft er large drops in t he st ock price. Many people get upset over t hese st ocks and cont inue selling. On t he ot her hand, bot t om fishers buy t hese st ocks, and in t he end t he bulls win t his gam e. A t rader should be prepared t o overcom e t he difficult y of observing drops in st ock price during t he com pet it ion bet ween bulls and bears. One needs t o have pat ience and wait for a posit ive price m ove t o sell t he st ock wit h a profit . St at ist ically such an approach is a winning gam e, but one should rem em ber t hat st at ist ics always assum e a dist ribut ion of ret urn and possible losses are likely. Let us consider very im port ant quest ion: how can w e place an opt im al st op loss order t o m inim ize losses and t o obt ain a good average ret urn? The next figure present s t he result s of calculat ion of t he average ret urns ( t hey were defined previously) as a funct ion of num ber of st ock holding days at various levels of st ops. The param et er S is defined as S = STOP - CLO

46

Ave r a ge r e t ur ns a s a fu nct ion of nu m be r of st ock holding da ys a t va r ious le ve ls of st ops. Ove r sold st ock s a r e conside r e d. From t his figure one can conclude t hat for oversold st ocks using any st ops decreases t he average ret urn. The worst t hing w hat a t rader can do is place st ops close t o t he - A level. I f a t rader holds st ocks for a period less t han 4 days it is wort h considering st ops, w hich are placed very close t o t he CLO price. The change of t he average ret urn is not so significant . The next figure present s t he average ret urns as a funct ion of levels of st ops for a four days st ock holding period.

Ave r a ge r e t ur ns a s a funct ion of

le ve ls of st ops for a four da ys st ock holding pe r iod

One can see t hat t he best ret urns are obt ained when a t rader uses st ops, which are locat ed very far away from t he purchase price or t rades st ocks wit hout any st ops. This cont radict s t he popular opinion t hat profit able t rading wit hout st ops is im possible. How one can handle t rading wit hout st ops? I t is possible only in one case: t h e t r a din g ca pit a l

47 m ust be la r ge e nough t o be a ble t o bu y m a ny st ock s. I n t his case, even a 50% price drop of one or t wo st ock s will not kill a t rader. Ot herwise, t rading w it hout st ops is financial suicide. For a sm all capit al - one which allows holding t wo t o four st ocks in t he port folio it is bet t er t o use st ops and place t hem eit her very close t o t he purchase price or lower t han - 5A level. The average ret urn will be less, but t he t rader will survive. Let us repeat once again: t rading st ocks w it hout st ops is possible only for experienced t raders, who are very sure about t heir st ock select ion. They m ust use well- t est ed st rat egy and buy m any st ocks so as t o m inim ize t he risk of a sharp drop in t heir t rading capit al. What happens if a t rader m ake m ist akes and buys st ock wit h very low growt h pot ent ial? This can happen due t o a bear m arket , choosing a wrong indust ry, or j ust bad luck. This problem will be considered in t he next sect ion. St op st r a t e gy for ine x pe r ie nce d t r a de r s Let us suppose t hat a t rader is a novice in t he st ock m arket . He is t rying t o beat t he m arket by using a t rading st rat egy, which seem s t o be profit able because it allows him t o t rade like m any ot her t raders. This is a t ypical m ist ake. The m aj orit y of t raders are losing m oney and any st rat egy, which is sim ilar t o ot her people's st rat egies, will not be very profit able. The problem is how t o survive in t he st ock m arket gam e w hile t est ing a new st rat egy? Suppose t hat our novice uses popular m om ent um st rat egy: he buys st ocks wit h t he highest price rise during t he last days of t he rise. We can sim ulat e such a st rat egy by considerat ion of overbought st ock s. As in t he previous sect ions w e will consider a 16 days hist ory period for calculat ion of t he D and T param et ers. The sim plest way t o survive in t he st ock m ark et is by using st op loss orders. The next figure shows average ret urns as a funct ion of num ber of st ock holding days at various levels of st ops. The no- st op- loss- orders st rat egy is shown for com parison.

Ave r a ge r e t ur ns a s a fu nct ion of nu m be r of st ock holding da ys a t va r ious le ve ls of st ops. Ove r bought st ock s a r e conside r e d Let us list t he conclusions t hat can be drawn from analysis of t his figure. - For very short periods of st ock holding ( one t o four days) using st ops does not earn ret urns worse t han t he ret urns from t he no- st op- loss- orders st rat egy. - The best result s are obt ained using very t ight st ops.

48 - For longer periods of st ock holding closer st ops ( about 1 - 2 A) earn ret urns w orse t han t hose of t he no- st op- loss- orders st rat egy. - I f you do not like t ight st ops and you are going t o hold st ocks for a long period of t im e t hen it is bet t er t o use st ops which are lower t han - 5A. Let us suppose t hat our novice has got som e experience, has underst ood t hat his st rat egy was bad, and now his underst anding allow s him t o choose st ocks wit h about 50% probabilit y of growt h. This case will be considered in t he next sect ion. St op st r a t e gy for a n a ve r a ge t r a de r The case, which we are going t o consider, is close t o t he st op- lim it st rat egy for a m ixt ure of st ocks wit h bot h high and low probabilit ies of growt h. This problem was t heoret ically considered in one of t he previous sect ions. How can we apply t his st rat egy in pract ice? I t is hard t o place sim ult aneously t he lim it and st op orders for one st ock. So, we will consider a st rat egy, which assum es selling st ocks aft er N days of holding. The problem is t he det erm inat ion of t he opt im al st ops t o cut st ocks wit h low probabilit ies of growt h. Consider t he average ret urns of t he st rat egy as a funct ion of st op levels. We will assum e t hat 50% of select ed st ocks are oversold and 50% of t he st ock s are overbought . This is a good m odel for t he average st ock select ion.

Ave r a ge r e t ur ns a s a fu nct ion of nu m be r of st ock holding da ys a t va r ious le ve ls of st ops. A m ix t ur e of ove r sold a n d ove r bought st ock s is conside r e d The figure is very sim ilar t o t he figure in t he previous sect ion. The largest ret urn can be obt ained if t he t rader uses very t ight st ops. The result s in t his case will be m uch bet t er t han t he ret urn wit hout using any st ops. St op orders t hat are far away from t he purchase price produce bet t er ret urns t han st ops around t he - A level. St ock vola t ilit y All previous result s have been present ed via t he daily st ock price am plit udes A, which can be a charact erist ic of st ock volat ilit y. You probably know m any ot her definit ions of t his param et er. I n t his sect ion we will consider m et hods of calculat ions of st ock volat ilit y and will show t hat st ock volat ilit y is a funct ion of t he T and D param et ers.

49 There are hundreds m et hods of defining t he st ock volat ilit y. The am plit ude A is m easured in dollars, and it is bet t er t o int roduce a new param et er t o com pare one st ock t o anot her. Let us define t he st ock volat ilit y in t wo different ways V 1 ( t ) = ( M AX ( t ) – M I N ( t ) ) / ( M AX ( t ) + M I N ( t ) ) * 1 0 0 % V 2 ( t ) = < ( M AX ( t ) – M I N ( t - 1 ) ) / ( M AX ( t ) + M I N ( t ) ) * 1 0 0 % > where V1 ( t ) and V 2 ( t ) are st ock volat ilit ies referring t o day t . V 1 ( t ) describes relat ive volat ilit y during t he day t and V2 ( t ) describes relat ive volat ilit y during days t and t - 1 . Aft er averaging one can writ e V1 = < V1 ( t) > V2 = < V2 ( t) > where t he angular bracket s < ...> denot e t he averaging over som e period of t im e. I n t his work we perform one m ont h averaging. The first param et er V 1 ( t ) can be also writ t en via t he daily am plit ude A ( t ) . V1 ( t) = A ( t) / P ( t) * 1 0 0 % where P ( t ) is t he average price during t rading day t P ( t ) = ( M AX ( t ) + M I N ( t ) ) / 2 The next figure illust rat es t hese definit ions.

I llust r a t ion for t he de finit ions of st ock vola t ilit ie s The values of V 1 and V 2 are very close t o each ot her. The next figure shows V1 and V2 for 250 st ocks.

50

I llust r a t ion of sim ila r it y of V1 a nd V2 One can conclude t hat t he daily am plit ude A, or t he relat ive volat ilit y V 1 which is relat ed t o t his value, is a good charact erist ic of t he short - t erm st ock volat ilit y. There m any ot her definit ions of st ock volat ilit ies relat ed t o t he closing prices only. However, for our purposes, for which w e need t o st udy st op orders, it is necessary t o consider m inim al and m axim al daily prices. To give you som e idea about t he values of t he daily am plit udes we w ill show t he values of V 1 for som e act ive st ocks. The next t able shows st ocks wit h large V1 values. Tick e r

V1 , %

VTSS

3.27

ASND

2.67

QCOM

2.48

DI GI

2.47

KLAC

2.46

LSI

2.46

XLNX

2.43

AOL

2.43

PSFT

2.37

MU

2.33

DELL

2.32

QNTM

2.30

EMC

2.27

PMTC

2.26

NXTL

2.23

AMAT

2.20

ORCL

2.18

COMS

2.18

CHRS

2.09

51 BGEN

2.08

BBBY

2.08

AMD

2.07

SUNW

2.05

I NGR

2.04

LLTC

2.03

NOVL

2.02

The next t able shows st ocks wit h sm all V1 values. Tick e r

V1 , %

DOW

0.88

SBC

0.88

AI G

0.88

CHV

0.85

DOV

0.84

AI T

0.83

GI S

0.82

AN

0.82

XOM

0.82

GSX

0.80

MHP

0.80

CLX

0.79

MMM

0.79

TX

0.79

SBH

0.77

GRN

0.77

MMC

0.76

VO

0.76

SPC

0.75

ED

0.74

BTI

0.72

RD

0.58

BBV

0.57

BP

0.56

SC

0.54

AEG

0.44

To use t his t able t o est im at e t he daily am plit ude A one needs t o m ult iply t he V 1 value by t he current price and divide t he result by 100. However, using average values of V 1 can be dangerous if a t rader is going t o buy oversold or overbought st ock s. The volat ilit ies of t hese st ocks are higher. The next figure shows t he average values of t he relat ive volat ilit ies V1 calculat ed for random ly select ed st ocks and for overbought and oversold st ocks.

52

The a ve r a ge va lue s of t he r e la t ive vola t ilit ie s V 1 ca lcula t e d for r a ndom ly se le ct e d st ock s a nd for ove r bought a nd ove r sold st ock s. Ve r t ica l ba r s show t he st a nda r d de via t ions of t he dist r ibut ions of V 1 One can see t han t he m axim al relat ive volat ilit ies V1 are observed for oversold st ocks. Tr a din g st r a t e gy using lim it or de r s Using lim it orders t o sell is very popular am ong novices. They buy st ocks, place t he sell lim it order and wait for t he st ocks t o t ouch t his lim it . Unfort unat ely, t his is a not a good st rat egy. There is a non- zero probabilit y of com plet e disast er. The wait for t he lim it t o be t ouched can be very long and during t his t im e t he st ock price can go t o very low levels. Using t he non- random walk m odel it can be shown t hat t he lim it order will never be execut ed wit h probabilit y P P = ( p/ q) ^ L

if

q > p

Let us rem ind t he reader t hat p is t he growt h probabilit y and q = 1 of decline. L is t he difference bet ween t he lim it order level and t he current of t his m odel have been considered previously. Using lim it s can reduce t he average ret urns per t rade even in t he case st ocks. The next figure shows t he result s of calculat ing t he dependencies of t he num ber of st ock holding days for different levels of lim it s L. L = LI M – CLO

p is t he probabilit y st ock price. Det ails of buying oversold average ret urns on

53

The de pe n de ncie s of a ve r a ge r e t ur ns on t he num be r of st ock holding da ys for diffe r e nt le ve ls of lim it s One can see t hat t he worst result s are obt ained when a t rader uses lim it s, which are very close t o t he price of purchase. Lim it s, st ops a nd r isk We have finished our short discussion of t he influence of st op and lim it orders on t he average ret urn per t rade. However, we can be asked: well, lim it s and st ops reduce t he average ret urns. How about t he risk? Maybe it is bet t er t o have a sm aller ret urn but sm aller risk. We agree wit h t his argum ent . The only point t hat needs t o be clarified is t he relat ionship bet ween t he average ret urn and t he risk. We m ent ioned earlier t hat t he best t rading st rat egy is t he st rat egy wit h t he m inim al risk/ ret urn rat io. I n t his sect ion we consider t he influence of st ops and lim it s on t his rat io. The t rading of oversold st ocks will be considered as an exam ple. We calculat ed t he average ret urns, risk ( st andard deviat ion of t he ret urn dist ribut ion) and t he risk/ ret urn rat ios for various st op and lim it values for a four days st ock holding period. The next figure shows t he result of t hese calculat ions for t he sell- lim it st rat egy described previously.

54

Ave r a ge r e t ur ns, r isk ( st a nda r d de via t ion of t he r e t u r n dist r ibu t ion ) a nd t he r isk / r e t urn r a t ios for va r ious lim it va lue s for a four da ys st ock holdin g pe r iod. Tr a ding ove r sold st ock ha s be e n conside r e d From t his figure one can m ake a very im port ant conclusion: t he risk t o ret urn rat io varies very lit t le if t he level of t he sell lim it order is larger t han 2.5A. Let us rem ind t he reader t hat A is t he average daily st ock price am plit ude. Therefore, using lim it orders for oversold st ocks wit h high probabilit ies of growt h is not a bad idea. The risk t o ret urn rat ios for t he st rat egy wit h st op loss orders have a m ore com plicat ed dependence on t he st op levels. The next figure shows t he result s of calculat ing t he average ret urns, risk, and risk t o ret urn rat ios for different levels of st op orders. As in t he previous case, t he st rat egy of a four days st ock holding of oversold st ocks has been considered.

55

Ave r a ge r e t ur ns, r isk ( st a nda r d de via t ion of t he r e t u r n dist r ibu t ion ) a nd t he r isk / r e t ur n r a t ios for va r ious st op va lue s for a four da ys st ock holding pe r iod. Tr a ding ove r sold st ock ha s be e n conside r e d From t he first point of view one can conclude t hat t he best st rat egy is placing st op loss orders very close t o t he purchase price. The risk t o ret urn rat io of t his st rat egy is t he lowest . Theoret ically t his is correct . However, experienced t raders know very well t hat t hese st ops cannot prevent losses from negat ive overnight gaps. The opening st ock price can be m uch lower t han t he closing price of t he previous day, and t he st op order will be execut ed at very low level. The m orning flow of sell orders in event of bad news can cause execut ion of t he st op loss order at an alm ost m inim al price during t he early selling off. The average ret urn can be m uch lower t han expect ed. The t rader m ust also rem em ber about bid- ask spreads and brokerage com m issions, which also reduce t he average ret urn. I t seem s t hat t he opt im al placing of t he st op loss order is lower t han - 5A. The risk t o ret urn rat io is close t o t he rat io wit hout st op orders, t he average ret urn does not suffer m uch, and t his st op can prevent a big loss of t he t rading capit al. How m uch lower? I t depends on t rader's habit s and experience. The average ret urns and t he risk t o ret urn rat ios do not change m uch aft er - 5A. I t is m uch m ore im port ant t o not place a st op loss order in t he vicinit y of - 2A, where t he risk t o ret urn rat io is m axim al. We have considered som e st rat egies, which allow us t o obt ain t he m axim al average ret urn while m inim izing t he dow nward risk. How ever, if you look closely at t he absolut e values of ret urns you can conclude t hat t hese ret urns are sm all and com parable t o t he t ransact ion cost ( bid- ask spreads and brokerage com m issions) . How can we increase t he absolut e values of ret urns? This quest ion will be considered in t he next sect ion.

56 I ncr e a sin g a ve r a ge r e t ur n There are m any ways t o im prove t rading st rat egies. We have considered opt im izat ion of st ock holding periods, opt im al division of t rading capit al, and using st ops and lim it s t o sell st ocks. However, t he m ain source of obt aining bet t er ret urn is a good st ock select ion. Buying oversold st ocks is good st rat egy, but it can be im proved if a t rader m akes st ronger select ion. Let us illust rat e t his idea by t he next exam ple. Select ion of oversold st ock s w it hin a 16 days t im e int erval allows us t o obt ain an average ret urn about 2.2% , as described in t he sect ion " Ret urns of overbought and oversold st ocks" . I t is possible t o obt ain a m uch bet t er ret urn if one buys st ocks on t he next day if st ock prices decline st ill furt her. Denot e ( t - 1) t he day of analysis, i.e. t he day when t he list of oversold st ocks is obt ained. Day ( t ) is t he day of st ock purchase. We consider buying st ocks at t he m arket closing, i.e. t he purchase price is equal t o CLO ( t ) . The st ocks will be sold on day N at t he price CLO ( t + N) . The next schem e illust rat es t hese not at ions. _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ | _ _ a | ....p| ....| ....| ....| ....| ....| ....| ....| ....| ....| ...

1 6 da y hist or y for D a n d T ca lcula t ions

N= 0

1

2

3

4

5

6

Here, a = t - 1 denot es t he day of t he analysis, p = t denot es t he day of t he purchase ( N = 0) . The next figure shows t he average ret urns R = [ CLO ( t + N ) – CLO ( t ) ] / CLO ( t ) * 1 0 0 % as a funct ion of N. We consider t wo cases. One is select ing st ocks, w hich dropped during day t ( t he next day aft er t he day of t he analysis) m ore t han - 2A, where A is t he daily price am plit ude, which was defined previously. The second case is buying st ocks, which rose during day t m ore t han 2A.

The a ve r a ge r e t ur ns of spe cia lly se le ct e d ove r sold st ock s a s a fu nct ion of t he st ock holdin g da ys One can see t hat select ing st ocks wit h large price drops subst ant ially increases t he average ret urn and t he st rat egy becom es m uch m ore profit able.

57 I f one buys st ock s, which st art ed rising, in t heir price during day t t hen t he average ret urn is close t o zero. This is an exam ple in which m om ent um st rat egy ( buying rising st ocks) is not working. I n t his sect ion we have considered im proving t he average ret urn per t rade. However, we have m ent ioned before t hat a large ret urn per t rade does not always m eans a large annual ret urn. I f t he num ber of t rading days when one is able t o find t hese st ocks is sm all t hen t he annual ret urn will be sm aller. One needs t o opt im ize crit eria of st ock select ion t o st rike a balance bet ween t he num bers of st ocks per year, which can be found for t rading, t he st ock holding period, and t he average ret urn per t rade. Exam ples of balanced t rading st rat egies wit h low risk t o ret urn rat ios can be found in our ot her publicat ion " Short - t erm t rading analysis" or t he Text Level- 2 on t he websit e ht t p: / / www.st t a- consult ing.com . We have now com plet ed our descript ion of t he analyt ical m et hods, which can be used for im proving t rading st rat egies. We hope t hat our publicat ion will help you t o perform t he analysis of your ow n t rading st rat egy. We have t ried t o describe very com plicat ed quest ions as sim ply as possible. For t his reasons m any im port ant point s were but briefly described. We are always ready t o help you t o clarify our ideas. Feel free t o ask us any quest ions. We would also very m uch appreciat e it if you writ e us about ot her problem s relat ed t o st ock t rading which you feel wort h analyzing.