Comparison of Supervised & Unsupervised Learning in βs Estimation between Stocks and the S&P500

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1 Comparson of Supervsed & Unsupervsed Learnng n βs Esmaon beween Socks and he S&P500 J. We, Y. Hassd, J. Edery, A. Becker, Sanford Unversy T I. INTRODUCTION HE goal of our proec s o analyze he relaonshps beween socks whn he S&P500, and use varous machne learnng echnques ha we ve suded n class o do so. Ideally, he analyss wll lead us o he dscovery of effecve ways o calculae he βs beween socks and he S&P500, whch would ulmaely allow us o oban a proper hedge rao for radng he sock vs. he S&P500. To premse, for mos of our analyses, we wll use he log reurns of he daa, as n fnancal modelng of daa ypcally people do no look a he absolue values of he daa bu nsead a he percen reurned for he asse (.e. S / S - ), and akng he logarhm allows us o nroduce some lneary o he problem. A he end of our proec, we modeled he relaonshp beween he socks wh he S&P500 usng a sochasc dfferenal equaon, and by focusng on he dosyncrac componen of hs model we were able o deec a sgnal o rade off of. We end our proec wh an analyss of our varous radng sraeges and her respecve Prof and Losses (PnL). Our daa comes couresy of EvA Hedge Fund n San Francsco, CA, and consss of nra-day daa (specfcally, 5 mnue me nervals) of he S&P500 and s componens. A. Movaon: Abou he β The β s he coeffcen beween he marke componen and he consuen componen, namely can be expressed as he lnear relaonshp beween: LogRe (S Marke ) ~ β * LogRe (S Componen ) The β n urn has several naural nerpreaons as well as uses hough all of he nerpreaons and uses are nmaely relaed. The naural nerpreaon s ha he β can be used as a hedge rao beween he consuen componen (.e. he sock or he secor of socks) and he marke componen (.e. he S&P500 ndex). Ths rao ells us he amoun a rader would have o buy/sell he marke/sock o reman rsk neural. Anoher use for he β s for forecasng and predcon. By modelng he lnear relaonshp beween he log reurns, one could hen use he esmaed β o forecas fuure values of he me seres. By dong so, one could deermne when he sock s radng oo cheap or rch o he marke, and hen ake advanage of hs neffcency by nang a rade. Ths dfference beween he sock and he marke can be used as a sgnal n hgh-frequency radng, and s he man movaon behnd he sochasc dfferenal equaon proposed by Avellaneda e al, and whch s explaned n more deal n he followng secon. B. Back-esng mehodology: Modelng We used a quanave approach o sock prcng based on relave performance whn ndusry secors or PCA facors, whch has been presened n []. The sock prces are noed where s me, and where he ndces are noed. In he case of supervsed learnng, represens he prce of he facor used o span he marke. The sock reurns are modeled accordng o he followng sochasc dfferenal equaon: I s composed of: a sysemac componen, drven by he reurns of he ndces An dosyncrac componen. I s assumed o be he ncremen of a saonary sochasc process whch models prce flucuaons correspondng o over-reacons or oher dosyncrac flucuaons n he sock prce whch are no refleced he ndusry secor. The componen s modeled as an Ornsen- Uhlembeck process,.e. follows hs model: where s an ndcaor of he mean-reverson speed. Ths process s saonary and auo-regressve. In parcular, he ncremen has uncondonal mean zero. C. Back-esng mehodology: radng sraegy We focus only on he process dmensonless varable: and defne he

2 where and. These values are compued wh a sldng wndow before he me of esmaon. The s-score measures he dsance o equlbrum n uns sandard devaons of, he conegraed resdual whch s compued from. In oher words, descrbes how far away a gven sock s from he heorecal equlbrum value assocaed wh our model. Our basc radng sgnal based on mean-reverson s where he hedge values are deermned emprcally. In pracce, we used and. Such a radng sraegy s llusraed by he fgure below: II. SUPERVISED LEARNING A. The dfferen norms The man dffcules wh our daase concern s dsrbuon. Over he wo-year perod we have, we cover a perod of bg growh and he fnancal crss. The frs problem s ha our daase may no be sable,.e. he beas n hose dfferen perods may dffer, due o fundamenal marke condons. Wh he analogy of house prces, we can magne ha he house prces have fundamenally changed beween hose perods, meanng ha may no be possble o learn a model from one perod and apply o he oher. The oher dffculy s ha he dsrbuon s no Gaussan, here are a lo of umps and bg varaons, and a T- dsrbuon seems o be a beer f. Modelng he daase dsrbuon Acual Frequency Fed Gaussan Fed Suden In order o deal wh hose problems, we have seleced dfferen obecve funcon o mnmze. The usual one nvolves he L2-norm, bu does no seem robus o he non-gaussan case, and does no adap o he fundamenal changes n he daase. The L-norm has he propery of beng much more robus o oulers. For example, n a smple D seng, usng he L2- norm, we know he Maxmum Lkelhood esmaor of a seres of number s he mean. However usng he L-norm, hs esmae s he medan, whch s more robus o oulers. where he axs represens he me, he black lne s and he red lne s he PnL. Noce ha he PnL flucuaes only on specfc mes, correspondng o pons where crosses one of he hedges defne above. Noce also ha he PnL may drop because we used wo addonal cu-offs whch represen sop-loss hedges, a whch we close our poson, despe he fac ha he conegraed resdual does no reurns o he mean. Ths accouns for umps n he sock value whch aren aken no accoun n our mean-reverng model, bu whch breach our rsk lms. By closng our poson, resuls n a money loss and explans he drops of he PnL, bu s necessary n pracce because of rsk lms. In order o deal wh he nsably and he fundamenal changes, we can dvde our daase no wo dfferen clusers: one for he hghly volale days, and one for he oher days. We can use a K-means or a mxure of Gaussans o separae he dfferen days. We have observed ha durng he crss, we had much more volale days han before, and n pracce one could eher urn off her radng sraeges durng hgh-volaly days, or one could desgn a separae radng sraegy ha handles hghly-volale days well and swch beween he wo. In our seng, we esed one sraegy whch uses he same β for boh perods of me (hgh-volaly and low-volaly) and a separae sraegy whch only predcs durng low-volaly days. To calculae volaly, we were careful o only use half-a-days worh of

3 Tes Error daa, as n pracce he obecve s o guess wheher or no a day wll have low-volaly and hgh-volaly as early no he day as possble. Hence, hs naurally nroduces a possble msclassfcaon error rae whch we calculaed o be 5.78%. Once we had compued all half-day volales, runnng K-means gave us: Gaussans for he weghs and he L-norm for example. Below s a summary of he dfferen supervsed mehods we used as well as her respecve obecve funcons: Norm L2-Norm L-Norm Mxure mn mn,, Obecve Y F 2 mn, Y F 2 2 Y F B. Choosng he norms and parameers We need o choose dfferen parameers such as he rao of Tes Sze o Tranng Sze, he numbers of facors we use or he overall me nerval we can use. Le s plo he es error/ranng error or he nfluence of each facor o choose. From whch we can see a very clear volaly cluserng effec when llumnang all hgh-volaly days: Tes Error Tranng Sze/Tes sze Fg.. Tes error wh respec o he rao Tranng Sze / Tes se sze usng he usual L2-norm. The mprovemen n he Tes error s small afer a rao of 0, whch we wll use for he backesng. Furhermore, he Cross-valdaon error s farly sable over me wh he oal se sze, meanng ha we can expec he same performance for dfferen me nerval. Fnally, we have assessed he mporance of each facor by he ncrease n error when s excluded. The fnancal secor s he mos mporan, as he fnancals played an mporan role durng he crss, and he Telecom secor s he leas mporan. Usng he mean squared error (MSE) as a measuremen for srengh of predcon, we compare he wo sraeges and fnd ha he MSE for he frs sraegy (same β for all days) was.73 mes hgher han he MSE for he second sraegy (only forecasng on low-volaly days). However, despe hese posve resuls, for he purposes of PnL and back-esng we chose o mplemen he naïve sraegy of usng he same β for all days. Fnally, we noe ha we could also combne he mehods, usng Mxure of C. Back-esng resuls In order o backes our models, we have used he radng sraegy descrbed n par B. The resuls are very dfferen dependng on he perod we choose, and our sraegy s no able o yeld good reurns n a conssen manner. Durng he crss, and precsely a he me of Lehman s falure, we have a huge loss. On average we ge a yearly reurn of 3% a year, whch s no ha bad for such a perod, and a Sharpe rao of 2.8 over en days, whch s he sandard. The mxure of Gaussans mehod gves he bes reurns and Sharpe Rao.

4 B. Resuls Our graphs show he egenvalues of he PCA componens for he L and L2 norms. As expeced, he par of he varance explaned by he frs componens s hgher wh he L2 norm han wh he L norm. Fg. 2. PnL comparson of L, L2, and Mxure of Gaussans supervsed learnng mehods. Mxure of Gaussans (red) performed bes. III. UNSUPERVISED LEARNING A. Movaon Though supervsed learnng provdes a suffcen framework o fnd he βs and rade, we would lke our sraegy o rely on ndces chosen o be he mos relevan ones o explan he consdered log reurn sock, raher han pre-defned. To address hs problem, we used he Prncpal Componen Analyss, whch enables us o denfy he drvng forces of he marke and predc he evoluon of he sock n erms of very few ndces whch explan mos of he marke's varance. The advanage of he Prncpal Componen Analyss as an unsupervsed learnng echnque s ha we make no assumpon on he predcor varables; he algorhm fnds self he lnear facors ha bes explaned he response. We defned he ranng and es error of our model: Error L2 N M Y 2 F The graphs below represen he dfference of a parcular sock and s esmaed value agans he number of facors used n he esmaon, averaged over all he S&P500 socks. As expeced, he error decreases when he number of facors ncreases., Error L Y F N M, The commonly used L2-PCA maxmzes: arg max W W T X 2 When he maxmzaon problem s expressed n erms of he L2 norm, he resul s unforunaely very sensve n presence of oulers and could resul n a skewed esmaon of he beas. Thus, we modfed he maxmzaon problem n erms of he L norm, whch provdes a Robus Prncpal Componen Analyss. arg max W W T Though he radonal L2-PCA was performed hrough R s sandard package, we used he pcapp package n R o perform he compuaon of he L-PCA. I provdes he Robus Prncpal Componens usng he Grd search algorhm, presened n [2]. X C. Back Tesng of sraegy In order o es f we should use he L norm raher han he L2 norm, we performed a back-esng sraegy on he socks usng he wo dfferen βs esmaons. We ran he PCA on 2/3 of he daa o esmae he βs and he facors, we hen execue our sascal arbrage sraegy on he es se wh esmaed βs and facors.

5 IV. CONCLUSION For our proec, we nvesgaed he relave performance of several supervsed learnng mehods (L-regresson, L2- regresson, and Mxure of Gaussans), as well as several unsupervsed learnng mehods (L-PCA and L2-PCA). Because of he fa-aled naure of many fnancal me seres, here s a hgher endency for oulers and nuon ells us ha by usng a more robus sasc namely he L- norm we could have more success n our modelng. Usng PnL and Sharpe Rao as a merc for deermnng he effecveness of a gven β, we deermned ha for boh supervsed and unsupervsed learnng mehods, he L-norm ndeed performed beer and n fac had smlar reurns for boh cases. More specfcally, n he case of supervsed learnng mehods, we saw ha he Mxure of Gaussans for he weghs wh he L-norm obecve funcon had a yearly reurn of abou 3%, whle he L-PCA for he unsupervsed learnng mehod had a yearly reurn of abou 2.7%. For fuure mprovemens, we would lke o re-mplemen our back-esng algorhm so ha res o es for md-day volaly and ake hs no accoun before decdng o rade. We beleve ha by elmnang hghly volale days (such as he perod surroundng he Lehman ncden) we can mprove our overall reurns and also mprove upon he conssency of our reurns. REFERENCES [] Avellaneda, M. and Lee, J.-H. (2008). Sascal Arbrage n he U.S. Eques Marke ( July) [2] Croux C, Flzmoser P, Olvera M (2007). Algorhms for Proeconpursu Robus Prncpal Componen Analyss. Chemomercs and Inellgen Laboraory Sysems The graph on he op represens he Prof and Losses (PnL) of he radng sraegy presened n he nroducon, sarng a.00. The graph on he boom presens he dsrbuon of Sharpe raos obaned over he dfferen S&P500 socks (5 facors were used n he PCA). We obaned ha he L norm has a beer PnL and Sharpe rao dsrbuon n many cases. In our example, he PnL s.27 for L norm compared o.06 for L2 norm afer 50 eraons and he mean of he Sharpe raos are of 2.8 for he L norm and 2.77 for L2 norm. Ths resul s very encouragng and proves ha our nuon ha he oulers affec negavely he esmaon of he βs was rue.