Itroducto The paper presets Costat Rebalaced Portfolo frst troduced by Thomas Cover. There are several weakesses of ths approach. Oe s that t s extremely hard to fd the optmal weghts ad the secod weakess s that dowtred market CRP teds to tred dow. To deal wth the frst weakess Cover troduces the Uversal Portfolo oto ad ths research paper addresses the secod weakess. I order to cope wth tred we propose to have both short ad log stocks the portfolo ad at the ed of each day rebalace the portfolo. The caveat s, the weght of the wealth vested short ad log stocks should ot be equal. The results show that Modfed CRP beats the portfolo that cossts of stocks that have egatve ad moderate retur.
Ivestmet Theory Overvew It was the goal of dvdual, atos ad socetes sce the daw of huma race to accumulate wealth, sometmes to satsfy basc eeds for exstece, ad sometmes just for the sake of wealth accumulato. I recet cetury, facal markets became creasgly attractve for dvduals ad sttutos alke, ad are crucal part of etre ecoomy, due to ther role facltatg the rase of captal, hedgg, accumulato of wealth, ad teratoal trade. Facal Market s a market for a facal strumet, whch buyers ad sellers fd each other ad create or exchage facal assets. ometmes these are orgazed a partcular place ad/or sttuto, but ofte they exst more broadly through commucato amog dspersed buyers ad sellers, cludg baks, over log dstaces. [] I recet years, wth the veto of the teret, dvdual vestors who would ot have meas to stock markets, research ad deas otherwse became very actve partcpats the stock markets. Numerous theores were proposed by ecoomsts, mathematcas ad computer scetsts wth ultmate goal md: How to maxmze the retur ad/or mmze the rsk? Oe of those famous theores, whch s wdely used today, s a Portfolo Theory, proposed by Harry Markowtz Portfolo electo artcle, that was publshed The Joural of Face 95, ad later was publshed as a book 959 [, 3]. Corerstoe
of the Portfolo Theory s the dversfcato ad t argues that by carefully creatg portfolo, oe ca maxmze the expected retur ad mmze the rsk. everal schools emerged that took varous approaches to terpret the everyday stock market teractos. Whle oe beleve that t s possble to use statstcs ad past data to uderstad the stock market, others, such as propoets of Effcet Market Hypothess Theory argue that stock prce already corporates all the formato that s avalable, ad that whe vestor buys or sells the stock, hs success wll deped o chace ad essetally models bult by the vestor have o predctve power. Yet aother questo, whch puzzled vestors ad actve traders for decades, s how to predct future stock prce wth some degree of accuracy? There are two ma aalytcal approaches to attempt to predct the stock prce. - Fudametal Aalyss: A fudametal aalyss reles o the statstcs of macroecoomcs data such as terest rates, flato rates as well as compay s facal status[4]. - Techcal Aalyss: A techcal aalyss reles o the aalyss of hstorcal prce ad volume data. Ths research paper s goal s to fd a successful strategy to crease the odds the stock market. For that I wll use log-ormal portfolo theory, whch was evolved from the Iformato Theory cocepts. Whle the theory optmzes the retur, t stll follows the market s overall tred ad susceptble to systemc rsk.
Iformato Theory Overvew I would lke to remark to the reader that ths secto s adapted from the book Elemets of Iformato Theory by Thomas Cover. Iformato Theory, as a brach appled mathematcs ad electrcal egeerg was fouded 948 by Claude hao. Itally formato theory was set to aswer questos related to data compresso ad trasmsso. However, sce ts cepto formato theory applcatos ca be foud may dfferet felds: cludg but ot lmted to Physcs, Computer cece, Mathematcs, ad Ecoomcs. To quatfy the ucertaty of a radom varable formato theory, we use etropy. The Etropy H() of a dscrete radom varable s defed by: H ( ) = p( x)log p( x) x ce the etropy s expressed bts, the log base s to be. Expected value of radom varable of g() s wrtte: E g( ) = g( x) p( x) p x The defto above, whch defes the etropy of a sgle varable, ca also be exteded to a par of radom varables.
The jot etropy H(, Y) of a par of dscrete radom varables (, Y) wth a jot dstrbuto p(x, y) s defed as H (, Y ) = p( x, y)log p( x, y) x y Y Codtoal etropy of a radom varable s aother mportat cocept that eeds to be defed. If (, Y) ~ p(x,y), the codtoal etropy H(Y ) s defed as H(Y ) = p ( x) H ( Y = x) x = ( x) x = x y Y p p( y x)log p( y x) y Y p ( y x)log p( y x) Iformato Theory ad Portfolo Maagemet I A ew terpretato of Iformato Rate publshed 956, Joh Larry Kelly Jr. showed that If the put symbols to a commucato chael represet the outcomes of a chace evet o whch bets are avalable at odds cosstet wth ther probabltes, a gambler ca use the kowledge gve hm by the receved symbols to cause hs moey to grow expoetally [5]. Later 96 Edward Thorp showed the practcal use of the theory ad that t could be used Ivestmet decso makg.
The tock Vector A stock market ca be represeted as a vector of stocks = (,,... m ) 0, =,,,m where s the prce relatve. (The rato of the prce of stock at the ed of the day to the prce at the begg of the day). I example f prce of stock s up by 4% at the ed of the day, the =. 04. The Portfolo A portfolo b=( b, b,... b ), where b 0, b = s the wealth allocato across the ( m stocks. The Portfolo relatve s defed as the rato of the wealth at the ed of the day to the wealth at the begg of the day ad ca be represeted as. Thus f the stock vector s ad a portfolo we are usg s b the: = b t The goal of dvdual as well as sttutoal vestor s to maxmze. Curretly the theory that s used by experts s Mea-varace approach, whch s based o maxmzg expected value of ad was troduced by harpe ad Markowtz. Other wdely used cocepts are CAPM (captal asset prcg model), effcet froter, Moder Portfolo Theory. These portfolo selecto models however, are used for the log term decso makg. If a trader trades or rebalaces hs portfolo a more frequet bass, for stace every day, the the behavor of the product of wealth relatves s determed by the expected logarthm of the wealth relatve ad ot by the expected value.
The Growth Rate The growth rate of a stock market portfolo b wth respect to a stock dstrbuto F(x) s defed as t t W ( b, F) = logb xdf( x) = E(logb ) Whe the logarthm s to base, the growth rate s also called the doublg rate. ce the am s to maxmze the wealth, the the optmal growth rate ca be defed as follows: * W ( F) = maxw ( b, F) where b = b The Log-optmal Portfolo If portfolo Theorem. * b acheves the maxmum of W(b, F) t s called a log-optmal portfolo. Let,..., be..d ~F(x). Let * = = b * t s the wealth relatve after days. The * * log W ( F) or = * W *
Proof for the theorem s gve as follows: * * t * log = log b W ( F ) = Theorem The log-optmal portfolo for a stock market ~ F satsfes the followg codtos: j E ( ) = * t b f b * > 0 j f b * = 0 Ths theorem s called Kuh-Tucker characterzato of the log-optmal portfolo. j Asymptotc Optmalty of Log-Optmal Portfolo It ca be show wth probablty that the fal wealth expected log ca be maxmzed by log-optmal portfolo. Let * = = b * t s the wealth relatve after days for vestor usg log optmal portfolo. Let = = b t s the wealth relatve after days of a vestor who s usg ay other strategy. The: E log = W E log * * Thomas Cover hs book provdes the proof for the theorem, whch s as follows:
max E log = max E = logb t = t max logb (,,..., = E ) = = E logb * t = W * Thus the best results are acheved by a costat portfolo strategy. The ext theorem shows that log-optmal portfolo wll perform as good as or better tha ay other portfolo strategy. Theorem: Let * = = b * t s the wealth relatve after days for vestor usg log optmal portfolo. Let = = b t s the wealth relatve after days of a vestor who s usg ay other strategy. The: lm sup log 0 * wth probablty.
Iformatoal Edge If the theory s the frst half of the puzzle, the formatoal edge s the secod part that s ecessary to succeed the stock market challege. Why do we eed formatoal edge, ad most mportatly how do we acheve t? If you had the formato about the certa compay that you have today, 0 years ago, would t help to make your md? If early etes we kew how bg the teret would be the future, would we pck Amazo, Ebay stocks as soo as they were out IPO? If we had the same formato about facal health of the compaes, such as Ct Group, AIG or Bak of Amerca, several years ago, would we avod the stocks? I recet tervew to BusessWeek Chrs emeuk - portfolo maager - at TIAA- CREF ackowledges the mportace of formatoal edge ad how vtal t s for the compay stock to be cluded to hs portfolo. To ga formatoal edge, portfolo maager travels to the compaes headquarters ad as he otes You'd be surprsed how much publc formato gets overlooked ad s ot carefully aalyzed. [6] o how ca oe acheve formatoal edge? The most commo approaches are two: Fudametal Aalyss ad Techcal Aalyss. As t was metoed earler, fudametal aalyss looks at the overall macroecoomc codtos as well as facal health of the compay. Techcal aalyss, o the other had, attempts to predct the stock movemet based o hstorcal data. I ths research paper, I use techcal aalyss, amely autocorrelato ad apply the Costat Rebalace Portfolo theory to practce.
Data ad Results For ths research, I collected 0 years daly data for 0 blue chp stocks. everal strateges were tested ad compared. The followg table demostrates the buy ad hold strategy for 0 blue chp stocks over the 0 year perod. Buy ad Hold gle tock trategy tock tock Adjusted prce stock prce (/03/000) (/3/009) Retur % KO $ 45.03 $ 57.00 7% AA $ 3.59 $ 6. -5% WMT $ 59.46 $ 53.45-0% CAT $ 8.6 $ 56.60 04% BA $ 33.9 $ 54.3 63% T $ 30.06 $ 7.6-8% PFE $ 3.57 $ 8.9-3% MCD $ 3.57 $ 6.44 9% MFT $ 47.64 $ 30.48-36% GE $ 37.36 $ 5.3-60% The data for the research was obtaed from face.yahoo.com, ad the stock prce reflects the splts ad dvdeds adjustmet. If a vestor purchased share of each compay, the retur 0 years would be 9%. If a vestor o the other had, vested 0% of hs wealth to each of these stocks, the retur o vestmet would be equal to 30%. Next several examples demostrate Costat Rebalaced Portfolo (CRP) approach results. Accordg to CRP, whch s based o Iformato Theory, at the ed of each tradg day redstrbute the wealth to all stocks gve the vector: b=( b, b,... b ) ( m
Theory propoets ofte cte the followg example to show that t holds water. Cosder tock whose prce stays costat ad tock whose prce doubles oe day ad halves aother. If b = [, ] The the wealth wll grow expoetally [6]: = = ( + ) = 3 4.. 3 = ( t+ = 3 + t+ = 3 ) = 3 4 t 3 To observe how CRP wll affect the wealth over tme, we take 0 stocks our portfolo ad redstrbute the wealth at the ed of each day. Our b = [... ]. 0 0 If we dsregard the commssos ad fees, the at the ed of the 009, our portfolo retur s 39%. It s clear that the strategy outperforms the buy ad hold strategy where our wealth were equally dstrbuted amog 0 stocks. However, f we would allocate all of our wealth 3 stocks: BA, CAT ad MCD /3/000, we ca observe that our CRP strategy would ot be superor to buy ad hold, because the retur o vestmet o three stocks s hgher tha the retur o CRP.
Whe we assg equal weghts to each stock our portfolo, t s called uform rebalaced portfolo. ce the weghts ca be easly adjusted, we are ot costraed to keep equal amout of wealth each stock. mply adjustg our portfolo b: b = [0., 0, 0., 0., 0.3, 0, 0, 0., 0, 0] b = [KO, AA, WMT, CAT, BA, T, PFE, MCD, MFT, GE] Rebalacg our portfolo at the ed of each tradg day, we wll acheve 8% retur o our tal vestmet. Ths clearly beats the best buy ad hold strategy the portfolo, whch would be Caterpllar (NYE: CAT). Aother agle that CRP wll be a better opto tha buy ad hold or ay other strateges s whe we have pcked losers our portfolo. Cosder a scearo where we pcked two stocks our portfolo, whch were Alcoa (NYE: AA), ad Geeral Electrc (NYE: GE). If we had decded to equally dvde our wealth to two stocks ad apply buy ad hold strategy, the at the ed of the vestmet perod we would lose 55% of our wealth. Now f we use CRP ad apply varous weghts, the our retur o vestmet wll stll be egatve but CRP outperforms buy ad hold strategy. The followg table summarzes portfolo retur whe dfferet weghts are appled. AA GE ROI 0% 90% -55.3% 0% 80% -5.56% 30% 70% -48.38% 40% 60% -45.96% 50% 50% -44.38% 60% 40% -43.74% 70% 30% -44.08% 80% 0% -45.38% 90% 0% -47.58%
As we ca see from the table, the best result s acheved whe two stocks are costatly rebalaced at 60% to Alcoa stock ad 40% to GE rato. The followg graphs represet the wealth crease or decrease for buy ad hold vs. CRP. CRP vs. Buy&Hold 60%_40% AA GE
Uform CRP vs. Buy & Hold Uform CRP B&H
We ca see from the graph that CRP outperformg the Buy & Hold strategy both scearos, that s maxmzes the reveue ad mmzes the rsk. There are also two mportat thgs ca be derved by aalyzg the graph: - There exsts a portfolo b, whch s the optmal weght ad f oe uses t, he or she ca maxmze the retur o vestmet. - ecod observato dcates that CRP follows the tred ad yelds better retur whe the portfolo s tredg up, ad mmzes the loss whe the portfolo treds dow. The secod observato s of partcular terest to me ad ths paper I preset a algorthm to break the tred. There s always a fe le betwee greed ad fear. Whe we see stocks our portfolo gog up, we do t sell them because we thk stock wll tred up further ad afrad we lose a opportuty. Whe the stocks go dow, we are scared that the stocks ca go dow further ad we sell the stock ad after a whle realze that stock evetually made ts way up. Oe of the weakesses of CRP offered by Cover s that t teds to follow the tred. That s f stock goes up, CRP wll beat the best performg stock, ad f the portfolo goes dow t wll ot perform worse tha other strateges. o ts rsk ad reward trade off CRP s more geared towards the reward sde. The stock market crash of late 008 ad early 009 was oe of the worst stock market crashes the hstory ad wped out 40-50% of vestors wealth. The followg chage s proposed to CRP, whch my opo wll balace the rsk ad reward.
Cosder a portfolo vector wth both log ad short postos. The results show that havg a short posto the portfolo reduces the rsk substatally, wthout havg much effect o portfolo growth. The followg graph compares three strateges Buy ad Hold, CRP ad Balaced CRP. Uform CRP B&H Log_hort - Buy ad Hold strategy assumes that vestor vests 0% of hs wealth ad holds the stocks for 0 years - Ivestor, who uses CRP method, costatly rebalaces the portfolo ad keeps 0% of hs wealth each stock. - Ivestor, who uses Balaced CRP method, keeps 6% of hs wealth each stock as a log posto ad 4% as a short posto.
Cosder aother example: durg the perod of 0/4/007 03/06/009, &P 500, lost 55%. The followg table demostrates how each strategy, B&H, CRP ad Balaced CRP performed durg ths perod. Balaced CRP B&H CRP Ths clearly shows that troducg short sale to the portfolo reduces the systemc rsk. Last but ot the least, the followg results compare CRP, B&H ad Balaced CRP for par of stocks. Frst par s two stocks the portfolo that outperformed other stocks the portfolo. That s CAT ad MCD, 04% ad 9% retur respectvely. ecod par s two stocks that uderperformed other stocks the portfolo. That s AA whch lost 5% ad GE whch lost 60%. Thrd par s two stocks T ad KO whose performace was -8% ad 7% respectvely.
AA&GE ot allowed s allowed KO&T ot allowed s allowed
CAT&MCD ot allowed s allowed As t ca be observed from these three graphs, a portfolo that allows the short sellg, balaces the Rsk & Reward ad s especally favorable whe the stock market tredg dow.
Cocluso A powerful vestmet strategy based o Iformato Theory was proposed by Thomas Cover. The purpose of ths strategy s to costatly rebalace portfolo ad thus beat the market. The weakesses of ths approach are:. To fd the optmal weght,. CRP follows the tred. If short sale s allowed the portfolo, the t wll balace the rsk reward ad maybe extremely favorable durg the dowward tred the stock market.
. Markowtz H.M., 95. Portfolo electo. The Joural of Face 7 (). 77-9.. Markowtz H.M., Portfolo electo: Effcet Dversfcato of Ivestmets. Wley, New York. 3. Pe-Chag Cha, Che-Hao Lu, 006. A TK type fuzzy rule based system for stock prce predcto. Elsever 4. Kelly Jr. 956. A ew terpretato of Iformato Rate. Bell Laboratores 5. http://www.busessweek.com/vestor/cotet/jul007/p00707_739878. htm) 6. chapre Rob Foudatos of Mache Learg. 003 Lecture Notes. 7. (http://www-persoal.umch.edu/~aladear/glossary/f.html