Aomaly Correctio by Optimal Tradig Frequecy Yiqiao Yi Columbia Uiversity September 9, 206 Abstract Uder the assumptio that security prices follow radom walk, we look at price versus differet movig averages. Differet periods of movig averages give ivestor differet sigals ad we assume that a ratioal ivestor would wat to buy more whe the price goes dow. This paper provides a theoretical model for a ivestor to systematically buy heavy whe the security prices go dow.
Cotets Itroductio 3 2 Mathematical Model 5 2. Movig Averages........................ 5 2.2 Price to Movig Averages................... 5 2.3 Tradig Frequecy....................... 6 2.4 Optimal Tradig Frequecy.................. 7 3 Coclusio 8 3. Discussio of Applicatios................... 8 4 Appedix 4. Proof of Theorem 2.4...................... 2
Itroductio Security prices are geerally uderstood as radom walk. Although some scholars still doubt the cocept of efficiet market hypothesis, there are scholars like Malkiel show us empirical evidece that we do observe data i favor of efficiet market hypothesis. He has show us that professioal ivestmet maagers do ot outperform their idex bechmarks ad has provided us evidece that prices do digest all available iformatio [4]. I this paper, we would use Malkiel s argumet as a major assumptio. That is, we assume market is efficiet over the log ru ad security prices do reflect all available iformatio out there. This is also to say that the first order derivative of price (price chages) follows biomial distributio (big probabilities that absolute values of retur are small ad small probabilities that absolute values of retur are big). Ituitively speakig, we do ot argue whether idividual should sped his time or eergy to pick stocks o his ow because it is too costly to do pick small probabilistic evets. Istead, we are providig a model to correct the aomaly i the market. Jegadeesh ad Titma (2007) has foud evidece that strategies of buyig stocks performed well i the past ad sellig stocks that performed poorly i the past geerate sigificat positive returs i 3- ad 2- moth holdig period [3]. Similar ideas are preseted by De Bodt ad Thaler (985, 987) [][2]. This up-mius-dow mometum factor i the security market provides aother importat groud work for us. If the strategies itroduced by these scholars are worth ruig (which proved that they are), there would be ratioal ivestors out there who wat to try these strategies. This provide liquidities for our model. If o oe is willig to buy high sell low, it does ot make sese for us to argue to buy low sell high. Hece, we would assume that there is some liquidities out there provided, that is, the market exists. Aother thig that we eed to pay attetio to is the cocept of aomaly. A value strategy ca be a aomaly if sigificat alpha ca be foud. A mometum strategy ca be a aomaly if its alpha is sigificat. We defie aomalies amog a pool of expected returs to be the oes caot be tolerated by a ivestor subjectively. This is a defiitio that has ot bee discussed before. As a matter of fact, this defiitio is difficult to start ay quatitative approach because of the adjective subjective. We set up this defiitio o purpose because we will itroduce a tradig frequecy δ to be associated with the ivestor s tolerace of the security prices ad hece we ca start to have some discussios about aomalies. The questios we wat to aswer i this paper origiate from the assumptios above. If market is efficiet ad icorporates all publicly available iformatio, it is ratioal to ifer that a market grows by a various amout of factors out there, say GDP, NGDP, iterest rates, fudametals, techicals, algorithms, ad etc. sice all of the iformatio are public. The we would ideally see a relatively straighteed lie of prices throughout the history if we compare to other idicators. However, we observed two very differet variatios betwee security market ad ecoomic market. Scholars ca fid factors ad use pricig models to fid 3
out why the security market is so volatile, hece, they ca explai the reaso why we have such observatios, i.e., there are a lot more spikes i the price chart of stock market tha there are i GDP chart (see Figure ad 2). Is it all possible to simply buy the market whe market is low ad just let the market grow with the ecoomy? If so, how do we do it i a cosistet ad ratioal sese that the lower the market dips the more we buy? Figure : The retur after iterest rate with $ ivestmet i the market from July of 926 to Jue of 206. Source: From Ke Frech Data Library. Figure 2: The GDP i US Dollars from 960 to 205. Source: From World Bak Library Curret GDP (i Dollars) for Uited States. 4
2 Mathematical Model 2. Movig Averages We look at price as a time-series fuctio. That is, for each time ode t, we have a executed price p. This ca be see as a discrete fuctio with a map from time to price. We ca deote price as p t. Aythig ca happe ay time i the market ad a idividual price level is usually very volatile. We icorporate the cocept of movig average so that we ca look at the average price of some period i the past, which allows us to overlook the tradig oise i the market. Defiitio 2.. Simple movig average (SMA) takes the average of prices traded time period(s) i the past,. SMA = We ca also implemet the cocept of expoetial movig average, which is to add a certai amout of weight to the curret prices whe calculatig the averages. 2.2 Price to Movig Averages Based o the movig averages of the prices, we ca establish the cocept of price-to-movig averages. Defiitio 2.2. Price-to-Movig Average is the ratio of price over a selected period of movig averages. For periods, we have. SMA = Pt We ca iterpret this defiitio by our assumptios. We assumed that price follows radom walk. The averages of a radom walk should also be radom walk. That is to say, we have two series of umbers followig biomial distributios, which implies that the ratio should also be a biomial distributio. The data had a average expected retur of 0.65%. If we imagie this to be the iitial x-axis, we would observe a biomial distributio for price-to-movig averages for all three data series. 5
Figure 3: The graph shows returs after iterest rate with $ ivestmet i the market alog with three other movig averages with time periods to be 50-day, 00-day, ad 50-day from July of 926 to Jue of 206. Source: From Ke Frech Data Library. Figure 4: The graph shows price-to-movig averages with time periods to be 50-day, 00-day, ad 50-day from July of 926 to Jue of 206. Source: From Ke Frech Data Library. 2.3 Tradig Frequecy We defied i the itroductio sectio of this paper that we call aomalies to be the oes amog a pool of observatios that caot be tolerated by ivestors subjectively. This cocept is associated with a tradig frequecy δ. A ivestor could be greedy or moody whe he made a ivestmet decisio. We do ot care. We summarize all of those emotios ito δ. 6
Emotioally impatiet ivestors ted to be less educated ad would buy very ofte, hece a high tradig frequecy (δ is big). Emotioally patiet ivestors ted to be more educated ad would buy oce i a while, hece a low tradig frequecy (δ is small). Based o each price-to-movig average, we ca calculate stadard deviatio, σ. For every p t SMA, we have a calculated σ t,. From biomial distributio, we kow that the probability that p t SMA is high is small ad the probability that p t SMA is low is big. Hece, for the observatio space X = (t, ) ad a particular evet k = (i, j), we have i, j R, P r(p t SMA = p i SMA j) = P r(x = k) ( ) ( ) = p k ( p) k! k Z, where = k k k!( k)! The iterpretatio is the followig. There exists some i ad j such that the probability that the price-to-movig average equals to this particular i ad j is defied by the biomial equatio. The ext questio we eed to thik about is how do we uderstad or eve quatify how may particular pairs of (i, j) we eed to choose. The aswer to this questio is depedet o idividual ivestor, which is associated with tradig frequecy δ. Moreover, the tradig frequecy δ is depedet o stadard deviatio σ t,. I other words, we eed the followig defiitio. Defiitio 2.3. The particular evet k is the product of δ ad total observatios with (t, ) (t,) (t h R, δ = P r(σ t, hσ t,) =, ) (hσt,) (t,) σt, while σ t, = ((p t SMA i) i= (p t SMA i)) 2 This equatio looks complicated, yet it is easy to uderstad. The amout of evet k happes i a way that is depedet o tradig frequecy δ. Tradig frequecy δ is defied as a probability. It is a probability calculated to be the amout of times stadard deviatio of priceto-movig averages to be bigger tha a ideal amout over the amout of times of all observatios. I other words, we are lookig at a certai area o the two tails of the biomial distributio. We are more iterested i takig care of the left tail, i.e., whe to buy istead of whe to sell. i= 2.4 Optimal Tradig Frequecy How ofte do we have to trade? Each of us has a uique risk tolerace as well as expected retur for our ivestmet. The each of us has a particular k we are usig whe makig ivestmet decisios. This is to say, we eed to fid the right amout of k aroud the price whe the price-to-movig average is the lowest. The bigger k a ivestor has the earlier he eeds to start buyig. 7
The first goal is to fid the optimal price-to-movig averages level. We eed to take the first order derivative of price-to-movig averages ad set the result to zero. We have the followig theorem. Theorem 2.4., let be a small uit of time period, whe Pt = +, we have a critical price to act o it. The iterpretatio is fairly ituitive. We look at a series of price ad the price goes up or dow. We calculate the average of prices i a certai period. The we have a series of average prices. We look at the ratio of price over movig averages ad we treat the ratio to be a ew umber series with some arbitrary mea. We calculate the sum of prices i the past certai of time odes. The optimal price level occurs at a time whe the sum of the averages at two time odes does ot chage at all (or chage tiy little). Ituitively speakig, this is a time whe the price is goig dow too much ad ow it is startig to bouce back, causig the sum of the past prices to be relatively stable. For example, we calculate the sum of hudreds of prices i the past. If the price keeps goig dow, we are supposed to get a sum smaller ad smaller, yet oe day the sum stays the same all of the sudde. The this price ca be critical to buy ad vice versa. The complete proof is show i the Appedix 4.. I practice, we will probably fid some price areas as critical price level. That is to say, we would ot fid a perfect equivalet sums of prices i the data. We would probably see a rage of prices that satisfy the coditio. The we ca apply tradig frequecy δ. The bigger the δ is the bigger the time ode is. Therefore, a cosistet ivestor who is true to his ow preferece would start to buy earlier ad vice versa (sell earlier). The smaller the δ is the smaller the time odes is. A cosistet ivestor would buy later ad vice versa. 3 Coclusio I this paper we preset a method to pick a critical price by usig movig averages ad ivestor tradig frequecy. We assume price follows a radom walk ad we assume sufficiet liquidities for to make a market. We provide a model such that a ivestor ca fid a critical price to be local miimum or local maximum to act o. The essece of this paper rests i the goal of gettig rid of the ups ad dows of the equity market. We preset this model hopig to help equity market to be less volatile. 3. Discussio of Applicatios I practice, this model defiitely ca pick up some good price rages. I Figure 5, we have show the daily price plot for S&P 500 from Jauary 2000 to Jauary 206 from Yahoo Data Library. We calculated the movig averages o 0-, 20-, 30-, 50-, 00-, 200-, ad 300-day periods. For less tha 0% tradig frequecy, we calculate the buy sigal for each price-tomovig averages, which are preseted as spikes o x-axis. The more priceto-movig averages based o differet time periods satisfy the coditio 8
the higher the spikes are. The higher the spikes are the heavier a ivestor should buy. We observe that the spikes pretty much covered both fiacial crisis (200 ad 2008) ad the model successfully tells us, by usig oly past data at each time ode, that whe to start buyig ad whe to buy heavy. 9
0 Figure 5: The graph shows S&P 500 from Jauary 2000 to Jauary 206. The model takes 0-, 20-, 30-, 50-, 00-, 200-, 300-day to be movig average periods ad calculate the price-to-movig average ratio by ivestor tradig frequecy to be less tha 0%. The spikes from x-axis are buy sigal calculated based o critical price level. The higher the spikes are the heavier a ivestor should buy. Source: From Yahoo Fiace Library.
4 Appedix 4. Proof of Theorem 2.4 Proof: We have price-to-movig average to be the followig. For periods, we have SMA = Pt We wat to kow what time is the price the lowest. Thus, we wat to take derivative i respect with time t. The set the first order derivative to be zero ad we have ( SMA ) t ( (+ ) ( The we eed = lim 0 ( lim 0 )( lim 0 ( t lim 0 ( t + + ) ( + + + t P t+ )( Pt+ P t+ ) t P t ) = 0 = 0 Pt ) = 0 Pt ) = 0 ( lim 0 t ) ) () ( + ) ( + ) = 0 ( (+ ) ( ) () ( ) + ) = 0 Sice + = whe 0 (the price this secod is very close to the price ext secod), we ca factor out oe of them ( (+ ) ( ) ( ( ) ( + ) = 0 ) + ) = 0 Hece, whe ( Pt) = ( + ), we have the optimal price i the market that has the first order derivative to be zero (or approximately zero), which shows that this price is a local miimum/maximum (a critical price or a actioable price). Q.E.D.
Refereces [] De Bodt, Werer F.M., ad Thalor, Richard (985), Does the Stock Market Overreact? Joural of Fiace, 40, 793-805. [2] De Bodt, Werer F.M., ad Thalor, Richard (987), Further Evidece of Ivestor Overreactio ad Stock Market Seasoality, Joural of Fiace, 42, 557-58. [3] Jegadeesh, Narasimha, ad Titma, Sherida (993), Returs to Buyig Wiers ad Sellig Losers: Implicatios for Stock Market Efficiecy, The Joural of Fiace, 48(), 65-9. [4] Malkiel, Burto G. (2007), Reflectios o the Efficiet Market Hypothesis: 30 Years Later, The Fiacial Review, 40, -9. 2