Learning the population dynamics of technical trading strategies

Nicholas Murphy1, ∗ and Tim Gebbie1, †

1Department of Statistical Sciences, University of Cape Town, South Africa

We use an adversarial expert based online learning algorithm to learn the optimal parametersrequired to maximise wealth trading zero-cost portfolio strategies. The learning algorithm is usedto determine the relative population dynamics of technical trading strategies that can survivehistorical back-testing as well as form an overall aggregated portfolio trading strategy from theset of underlying trading strategies implemented on daily and intraday Johannesburg StockExchange data. The resulting population time-series are investigated using unsupervised learningfor dimensionality reduction and visualisation. A key contribution is that the overall aggregatedtrading strategies are tested for statistical arbitrage using a novel hypothesis test proposed byJarrow et al. [23] on both daily sampled and intraday time-scales. The (low frequency) dailysampled strategies fail the arbitrage tests after costs, while the (high frequency) intraday sampledstrategies are not falsified as statistical arbitrages after costs. The estimates of trading strategysuccess, cost of trading and slippage are considered along with an offline benchmark portfolioalgorithm for performance comparison. The work aims to explore and better understand theinterplay between different technical trading strategies from a data-informed perspective.

I INTRODUCTION

The problem of selecting plausible trading strategiesand allocating wealth among these strategies to maximizewealth over multiple decision periods can be a difficulttask. An approach to combining strategy selection withwealth maximisation is to use online or sequential ma-chine learning algorithms [1]. Online portfolio selectionalgorithms attempt to automate a sequence of trading de-cisions among a set of stocks with the goal of maximizingreturn in the long run. Such algorithms typically use his-torical market data to determine, at the beginning of atrading period, a way to distribute their current wealthamong a set of stocks. These types of algorithms can usemany more features than merely price, so called “side-information”, but the principle remains that same.

The attraction of this approach is that the investordoes not need to have any knowledge about the un-derlying distributions that could be generating thestock prices (or even if they exist). The investor is leftto “learn” the optimal portfolio to achieve maximumwealth using past data directly [1].

Cover [2] introduced a “follow-the-winner” online in-vestment algorithm1 called the Universal Portfolio (UP)algorithm2. The basic idea of the UP algorithm is to allo-cate capital to a set of experts characterised by differentportfolios or trading strategies; and to then let them runwhile at each iterative step to shift capital from losers to

∗Electronic address: mrpnic004@myuct.ac.za†Electronic address: tim.gebbie@uct.ac.za1 follow-the-winner algorithms give greater weightings to better

performing experts or stocks2 The algorithm was later refined by Cover and Ordentlich [4] (see

Section II B)

winners to find a final aggregate wealth.

Here our “experts” will be similarly characterised by aportfolio (or trading strategy) where a particular agentmakes decisions independently of all other experts. TheUP algorithm holds parametrized constant rebalancedportfolio (CRP) strategies as it’s underlying experts.We will have a more generalised approach to generatingexperts. The algorithm provides a method to effectivelydistribute wealth among all the CRP experts such thatthe average log-performance of the strategy approachesthe best constant rebalanced portfolio (BCRP) which isthe hindsight strategy chosen which gives the maximumreturn of all such strategies in the long run. The keyinnovation that Cover [2] provided was a mathematicalproof for this claim based on an arbitrary sequences ofergodic and stationary stock return vectors.

If some log-optimal portfolio exists such that no otherinvestment strategy has a greater asymptotic averagegrowth then to achieve this one must have full knowledgeof the underlying distribution and of the generatingprocess to achieve such optimality [2–5]. Such knowledgeis unlikely in the context of financial markets. However,strategies which achieve an average growth rate whichasymptotically approximates that of the log-optimalstrategy are possible when the underlying asset returnprocess is sufficiently close to being stationary andergodic. Such a strategy is called universally consistent.Gyorfi et al. [5] proposed a universally consistentportfolio strategy and provided empirical evidence ofa strategy based on nearest-neighbour based expertswhich reflects such asymptotic log-optimality.

The idea is to match current price dynamics withsimilar historical dynamics (pattern matching) using anearest-neighbour search algorithm to select parametersfor experts. The pattern-matching algorithm was ex-tended by Loonat and Gebbie [6] in order to implement

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a zero-cost (long/short) portfolio selection algorithm viaa quadratic approximation in the context statistical arbi-trage trading. The algorithm was also re-cast to replicatenear-real-time applications using look-up libraries learntoff-line. However, there is a computational cost associ-ated with coupling creation of off-line pattern libraries;more specifically, the algorithms are not truly online.

A key objective in the implementation of online learn-ing in this work is that the underlying experts are on-line too; they can be sequentially computed on a movingfinite data-window using parameters from the previoustime-step. An online approach rather than having theneed to search data-histories or make comparisons witha library of patterns learnt off-line.

Here we ignore the pattern matching step in theaforementioned algorithm and rather propose our ownexpert generating algorithm using tools from technicalanalysis. Concretely, we replace the pattern-matchingexpert generating algorithm with a selection of technicaltrading strategies.

Technical analysis indicators are popular tools fromtechnical analysis used to generate trading strategies[7, 8]. They claim to be able to exploit statistically mea-surable short-term market opportunities in stock pricesand volume by studying recurring patterns in historicalmarket data [9–11]. What differentiates technical anal-ysis from traditional time-series analysis is that it tendto place an emphasis on recurring patterns rather thanrecurring statistical properties of times-series.

This work does not address the question: Which, ifany, technical analysis based methods have useful infor-mation for trading? Rather we aim to bag a collectionof technical experts and allows them to compete in anadversarial manner using the online learning algorithm.This then allows us to consider whether the resulting ag-gregate strategy can: first, pass reasonable tests for sta-tistical arbitrage, and second, has a relatively low proba-bility of being the result of back-test overfitting. Can thestrategy be considered a statistical arbitrage, and can itgeneralise well out-of-sample.

Traditionally, technical analysis has been a visualactivity whereby traders study the patterns and trendsin charts based on price or volume data, and usethese diagnostic tools in conjunction with a varietyof qualitative market features and news flow to maketrading decisions. Perhaps not dissimilar to the rela-tionship between alchemy and chemistry, or astrologyand astronomy, but in the setting of financial markets.Although many studies have criticised the lack of asolid mathematical foundation for many of the proposedtechnical analysis indicators [12, 13, 15]. There hasbeen an abundance of academic literature utilisingtechnical analysis for the purpose of trading and severalstudies have attempted to develop and test indicatorsin a more mathematically, statistically and numericallysound manner [13, 14, 16]. However, much of this workneeds to be viewed with some suspicion - it is extremely

unlikely that this or that particular strategy or approachwas not the result of some sort of back-test overfitting[17–19]. However, that is not to say that there maynot be some merit in non-linear time-series analysis orpattern-matching in financial markets.

Here we will be concerned with the idea of understand-ing whether the collective population of technical expertscan through time, lead to dynamics that can be consid-ered a statistical arbitrage [20] with a reasonably lowprobability of back-test overfitting [19].

Can we generate wealth (before costs) using the onlineaggregation of technical strategies? Then, what broadgroups of strategies will emerge as being successful (herein the sense of positive trading profits with declining vari-ance in losses)? Costs are always a plausible explana-tion for any apparently profitable trading strategy (see[6]), and after costs, the high-likelihood that there wasa healthy amount of data-overfitting; particularly giventhat we only have single price paths from history andhave little or no knowledge about the true probability ofthe particular path that has been measured.

Rather than considering various debates relating thetechnicalities of market efficiency where one is concernedwith the expeditiousness of market prices to incorporatenew information at any time where information is explic-itly exogenous; we restrict ourselves to market efficiencyin the sense used by Fischer Black [13, 21, 22]. This isthe situation where some of the short-term informationis in fact noise, and that this type of noise is a fundamen-tal property of real markets. Although market efficiencymay plausible hold over the longer term, on the short-term there may be departures that are amenable to testsfor statistical arbitrage [23], departures that create in-centives to trade, and more importantly departures thatmay not be easily traded out of the market dues to vari-ous asymmetries in costs and market access.

In order to analyse whether the overall back-testedstrategy depicts a candidate statistical arbitrage, we im-plement a test first proposed by Hogan et al. [20] andthen further refined by Jarrow et al. [23]. Hogan etal. provided a plausible technical definition of statisti-cal arbitrage based on a vanishing probability of loss andsemi-variance in the trading profits, and then use this topropose a test for statistical arbitrage using a Bonferronitest [20]. This test methodology was extended and gen-eralized by Jarrow et al. [23] by computing a so-calledMin-t statistic to then use a Monte Carlo procedure tomake inferences regarding a carefully defined no statisti-cal arbitrage null hypothesis.

This is analogous to evaluating market efficiency inthe sense of the Noisy efficient market hypothesis [21]whereby a failure to reject the no statistical arbitragenull hypothesis will result in concluding that the marketis in fact sufficiently efficient and no persistent anomaliescan be consistently exploited by trading strategies overthe long-term.

Traders will always be inclined to employ strategies

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which depict a statistical arbitrage and especiallystrategies which have a probability of loss that declinesto zero quickly as such traders will often have limitedcapital and short horizons over which they must providesatisfactory returns (profits) [23].

We make the effort here to be very clear that we do notattempt to identify profitable (technical) trading strate-gies, but rather we will generate a large population ofstrategies (experts) constructed from various technicaltrading rules and combinations of the associated parame-ters of these rules which will in turn represent the expertsin the online learning algorithm. The experts who accu-mulate the greatest wealth during each trading periodwill receive more wealth in the following trading periodand thus contribute more to the final aggregated portfo-lio.

This requires us to introduce a methodology to trans-form the signals generated by the technical trading ex-perts into a set of portfolio weights (controls) for eachexpert in a given time period t to compute the expertwealth’s. We then provide the equity curves for the indi-vidual experts portfolios (the accumulated trading profitthrough time). This can be best thought of as some sortof “fund-of-funds” over the underlying collection of trad-ing strategies. This is a meta-expert that aggregates ex-perts that represent all the individual technical tradingrules. The overall meta-expert strategy performance isachieved by the online learning algorithm.

We perform a back-test of the algorithm on twodifferent data sets over two separate time periods;one using daily data over a six year period, and theother using a mixture of intraday and daily data overa two month period. A selection of the fifteen mostliquid stocks which constitute the Johannesburg StockExchange (JSE) Top 40 shares is utilised for the twoseparate implementations3.

The overall strategy performance is compared to theBCRP strategy to form a benchmark comparison to eval-uate the success of our strategy. The overall strategy isthen tested for statistical arbitrage to find that in both adaily, and intraday-daily data implementation, the strat-egy depicts a statistical arbitrage.

It must be cautioned that this is prior to accountingfor costs which can be expected to shift the bias. The keypoint here (as in [6]) is that is does seem that plausiblestatistical arbitrages are detected on the meso-scaleand short-term, however, for reasonable costing, it maywell be that these statistical arbitrages exists becausethey cannot be profitable traded out of the system. Inaddition to this, the overall strategy’s probability of lossis computed to get an idea of the convergence of suchlosses to zero; most traders will be concerned about

3 More details on the data sets can be found in Section IV

short run losses.

The paper will proceed as follows: Section II explainsthe construction of the algorithm including details ofhow the experts are generated, how their correspondingtrading signals are transformed into portfolio weightsand a step-by-step break-down of the learning algorithm.In Section III, we introduce the concept of a statisticalarbitrage and the methodology for implementing astatistical arbitrage test for a trading strategy, and forcomputing probability of loss of a strategy. Section IVdescribes the data utilised in the study. All resultsof implementations of our algorithm on the data andin-depth analysis is presented in Section V. Section VIstates all final conclusions from the experiments andpossible future work.

In summary, we are able to show that on a daily sam-pled time-scale there is most likely little value in the ag-gregate trading of the technical strategies of the typeconsidered here. However, on intraday times-scale thingslook slightly different. Even with reasonable costs ac-counted for there still seems to be the possibility thatprice based technical trading cannot be ruled out as of-fering an avenue for statistical arbitrage. Considerablecare is still required to ensure that one is accounting forthe full complexity of market realities that may makeit practically difficult to arbitrage these sorts of appar-ent trading opportunities out of the market as they maybe the results of top-down structure and order-flow itselfrather than some notion of information inefficiency; thesignature of noise trading.

II LEARNING TECHNICALTRADING

Rather than using a back-test in the attempt to findthe single most profitable strategy, we produce a largepopulation of trading strategies (experts) and use anadaptive algorithm to aggregate the performances of theexperts to arrive at a final portfolio to be traded. Theidea of the online learning algorithm is to consider a pop-ulation of experts created using a large set of technicaltrading strategies generated from a variety of parame-ters and to form an aggregated portfolio of stocks to betraded by considering the wealth performance of the ex-pert population. During each trading period, expertstrade and execute buy (1), sell (-1) and hold (0) signalsindependent of one another based on each of their indi-vidual strategies. The signals are then transformed intoa set of portfolio weights (controls) such that the overallsum of weights in the portfolio sums to zero (zero-cost)and the strategy becomes self-funding. We also requirethat the portfolio is unit leveraged. Based on each in-dividual expert’s accumulated wealth up until the sometime time t, a final aggregate portfolio for the next pe-

4

riod t+ 1 is formed by creating a performance weightedcombination of the experts. Experts who perform betterin period t will have a larger relative contribution towardthe aggregated portfolio to be implemented in period t+1than those who perform poorly. Below, we describe themethodology for generating the expert population.

A Expert Generating Algorithm

A.1. Technical trading

Technical trading refers to the practice of usingtrading rules derived from technical analysis indicatorsto generate trading signals. Here, indicator’s refer tomathematical formulas based on prices (OHLC), volumetraded or a combination of both (OHLCV). An abun-dance of technical indicators and associated trading ruleshave been developed over the years with mixed success.Indicators perform differently under different marketconditions which is why traders will often use multipleindicators to confirm the signal that one indicator giveson a stock with another indicators signal. Thus, inpractice and various studies in the literature, manytrading rules generated from indicators are typicallyback tested on a significant amount (typically thousandsof data points) of historical data to find the rules thatperform the best4. It is for this reason that we considera diverse set of technical trading rules. In addition tothe set of technical trading strategies, we implementthree other popular portfolio selection algorithms each ofwhich has been adapted to generate zero-cost portfoliocontrols. An explanation of these three algorithms isprovided in Appendix D while each of the technicalstrategies are described in Appendix C.

In order to produce the broad population of experts,we consider combinations among a set of four model pa-rameters. The first of these parameters is the underlyingstrategy of a given expert, ω, which corresponds to theset of technical trading strategies and trend-followingstrategies where the total number of different tradingrules is denoted by W . Each of the rules require atmost two parameters each time a buy, sell or hold signalis computed at some time period t. The parametersrepresent the number of short and long-term look-backperiods of indicators used in the rules. We will denotethe vector of short-term parameters by n1 and thelong-term parameters by n2 which make up two of thefour model parameters. Let L = |n1| and K = |n2|5be the number of short-term and long-term look-backparameters respectively. Also, we denote the number of

4 This is also known as data mining5 | · | denotes the dimension of a vector

trading rules which utilise one parameter by W1 and thenumber of trading rules utilising two parameters by W2.

The final model parameter, denoted by c, refers toobject clusters where c(i) is the ith object cluster and Cis the number of object clusters. We will consider fourobject clusters (C = 4); the trivial cluster which containsall the stocks and the three major sectors of stocks onthe JSE, namely, Resources, Industrials and Financials6.The algorithm will loop over all combinations of thesefour model parameters to create a buy, sell or hold signalat each time period t. Each combination of ω(i), c(i),n1(i) and n2(i) will represent an expert. Thus, someexperts may trade all the stocks (trivial clusters) andothers subsets of the stocks (Resources, Industrials andFinancials). It is important to note that for rules with 2parameters, the loop over the long-term parameters willonly activate at indices for which n1(k) < n2(k). Here, krepresents the loop index over the long-term parameters.The total number of experts, Ω, is then given by

Ω = no. of experts with 1 parameter

+ no. of experts with 2 parameter

= C · L ·W1 + C ·W2 ·∑[∑

(n2 > max(n1)) :∑(n2 > min(n1))]

We will denote each expert’s strategy7 by hnt which isan (m+ 1)× 1 vector representing the portfolio weightsof the nth expert for all m stocks and the risk-free assetat time t. Here, m refers to the chosen number of stocksto be passed into the expert generating algorithm. Fromthe set of m stocks, it is important to note that eachexpert will not necessarily trade all m stocks (unless theexpert trades the trivial cluster), since of those m stocks,only a hand full of stocks will fall into a given sector con-stituency. This implies that even though we specify eachexperts strategy (hnt ) to be an (m + 1) × 1, we will justset the controls to zero for the stocks which the expertdoes not trade. Denote the expert control matrix by Ht

made up of all n experts’ strategies at time t for all mstocks i.e. Ht = [h1

t , . . . ,hnt ]. In order to choose the m

stocks to be traded, we take the m most liquid stocksover a specified of days, say δliq days. We use averagedaily volume (ADV) as a proxy for liquidity8. ADV issimply the average volume traded for a given stock overa period of time. The ADV for stock m over the past δliq

6 See AppendixE E.1 for a breakdown of the three sectors into theirconstituents for daily data and AppendixE E.2 for the breakdownof intraday data

7 When we refer to strategy, we are talking about the weights ofthe stocks in the expert’s portfolio. We will also refer to theseweights as controls.

8 Other indicators of liquidity do exist such as the width of thebid-ask spread and market depth however ADV provides a simpleapproximation of liquidity.

5

periods is

ADVm =1

δliq

δliq∑t=1

Vmt (1)

where Vmt is the volume of the mth stock at period t.

The m stocks with the largest ADV will then be fed intothe algorithm for trading.

A.2. Transforming signals into weights

In this section we describe how each individualexpert’s set of trading signals at each time period t aretransformed into a corresponding set of portfolio weights(controls) which constitute the expert’s strategy (hnt ).As mentioned above, an expert will not necessarily tradeall m stocks, however, for the purpose of generalitywe refer to the stocks traded by a given expert as meven though the weights of many of these m stocks willbe zero for multiple periods as the expert will only beconsidering a subset of these stocks depending on whichobject cluster the expert trades.

Suppose it is currently time period t and the nth

expert is trading m stocks. Given that there are mstocks in the portfolio, m trading signals will need to beproduced at each trading period. The risk-free assetspurpose will be solely to balance the portfolio giventhe set of trading signals. Given the signals for thecurrent time period t and previous period t− 1, all holdsignals at time t are replaced with the correspondingnon-zero signals from time t − 1 as the expert retainshis position in these stocks9. All non-hold signals attime t are not replaced as the expert has taken a newposition in the stock completely from the previousperiod. This implies that when the position in a givenstock was sort at period t−1 for example and the currentsignal is long then he takes a long position in the stockrather than neutralising his position. Before computingthe portfolio controls, we compute a combined signalvector made up of signals from time period t − 1 andtime t using the idea discussed above. We will referto these combined set of signals as output signals forthe purpose of brevity. We then consider four possi-ble cases of the output signals at time t for a given expert:

1. All output signals are hold (0)2. All output signals are non-negative (0 or 1)3. All output signals are non-positive (0 or -1)4. There are combinations of buy, sell and hold

signals (0, 1 and -1) in the set of output signals

9 Only the position is retained from the previous period(long/short) not the magnitude of the weight held in the stock

Due to the fact that cases 2 and 3 exist, we need toinclude a risk free asset in the portfolio so that we canenforce the self-financing constraint; the controls mustsum to zero

∑i wi = 0. We refer to such portfolios

as zero-cost portfolios. Additionally, we implement aleverage constraint by ensuring that the absolute valueof the controls sum to unity:

∑i |wi| = 1.

To compute the controls for case 1, we set all stockweights and the risk free asset weight to zero so that theexpert does not allocate any capital in this case sincethe output signals are all zero.

For case 2, we compute the standard deviations of theset of stocks which resulted in buy (positive) signals fromthe output signals using their closing prices over the last120 trading periods (working days) in order to give alarger weight to the more volatile stocks. Let the numberof buy signals from the set of output signals be denotedby nb and denote the vector of standard deviations ofstocks to be bought by σ+. Then the weight allocatedto stocks with positive signals is given by

w = 0.5 · 1∑i σ+(i)

· σ+ (2)

where the lowest value of σ+(i) corresponds to theleast volatile stock and vice versa for large σ+(i). Thisequation ensures that

∑i wi = 0.5. We then short the

risk free asset with a weight of a half (wrf = −0.5). Thisallows us to borrow using the risk free asset and purchasethe corresponding stocks in which we are taking a longposition.

Case 3 is similar to Case 2 above, however instead ofhaving positive output signals, all output signals are neg-ative. Again we compute standard deviations of the setof stocks which resulted in sell (negative) signals fromthe output signals using their closing prices over the last120 trading periods. Let the number of sell signals fromthe set of output signals be denoted by ns and denotethe vector of standard deviations of stocks to be soldby σ−. Then the weight allocated to stocks which haveshort positions is given by

w = −0.5 · 1∑i σ−(i)

· σ− (3)

We then take a long position in the risk free asset witha weight of one half (wrf = 0.5).

For case 4, we use the similar methodology to thatdiscussed above in Case 2 and 3. To compute the weightsfor the short assets we use the formula

w = −0.5 · 1∑i σ−(i)

· σ− (4)

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Similarly, for the long assets we have

w = 0.5 · 1∑i σ+(i)

· σ+ (5)

We then set the risk free rate to be equal to∑i wi in

order to enforce the self-financing and fully investedconstraints. Finally, for assets which had hold signals,the weights are set to zero.

In Section V, we implement 2 methods for transform-ing the signals into controls. The method describedabove is what we will refer to as the volatility loadingmethod for transforming signals into controls and thesecond method we consider is called the inverse volatilityloading method. The inverse volatility loading methodis defined similarly to the method described above,however, instead of multiplying through by the volatilityvector in each of the above cases, we multiply throughby the inverse of the volatility vector (element-wiseinverses). We will not implement the inverse volatil-ity loading method in this study as the results are similar.

Appendix B Algorithm 1 shows the algorithm outlinefor the Expert Generating Algorithm. The Expert Gen-erating Algorithm calls the controls function which trans-forms trading signals into portfolio controls. The controlsfunction is made up of two parts, first to compute theoutput signals as discussed in Section II A A.2. which isoutlined in Appendix B Algorithm 2 and the second partis to transform the output signals into portfolio controlswhich is outlined in Appendix B Algorithm 3.

B Online Learning Algorithm

Given that we now have a population of experts eachwith their own controls, hnt , we implement the onlinelearning algorithm to aggregate the experts strategies attime t based on their performance and form a final singleportfolio to be used in the following period t + 1 whichwe denote bt. The aggregation scheme used is inspiredby the Universal Portfolio (UP) strategy taken from thework done by [2, 4] and a modified version proposedby [5]. Although, due to the fact that we have severaldifferent base experts as defined by the different tradingstrategies rather than Cover’s [2] constant rebalancedUP strategy, our algorithm is better defined as a meta-learning algorithm [24]. We use the subscript t since theportfolio is created using information only available attime t even though the portfolio is implemented in thefollowing time period. The algorithm will run from theinitial time tmin

10 which is taken to be 2 until terminaltime T . tmin is required to ensure there is sufficient data

10 this is the time at which trading commences

to compute a return for the first active trading day. Wemust point out here that experts will only actively beginmaking trading decisions once there is sufficient data tosatisfy their look-back parameter(s) and subsequently,since the shortest look-back parameter is 4 periods, thefirst trading decisions will only be made during day5. The idea is to take in m stock’s OHLCV valuesat each time period which we will denote by Xt. Wethen compute the price relatives at each time period

t given by xt = (x1,t, . . . , xm,t) where xm,t =P c

m,t

P cm,t−1

and where P cm,t is the closing price of stock m at timeperiod t. Expert controls are generated from the pricerelatives for the current period t to form the expertcontrol matrix Ht. From the corresponding expertcontrol matrix, the algorithm will then compute theexpert performance Sht which is the associated wealthof all n experts at time t. Denote the nth expert’swealth at time t by Shnt . We then form the finalaggregated portfolio, denoted by bt, by aggregatingthe expert’s wealth using the agent mixture update rules.

The relatively simplistic learning algorithm is incre-mentally implemented online but offline it can be paral-lelised across experts. Given the expert controls from theExpert Generating Algorithm (Ht), the online learningalgorithm is implemented by carrying out the followingsteps [6]:

1. Update portfolio wealth: Given the portfoliocontrol bm,t−1 for the mth asset at time t − 1, weupdate the portfolio wealth for the tth period

∆St =

M∑m=1

bm,t−1(xm,t+1 − 1) + 1 (6)

St = St−1∆St (7)

St represents the compounded cumulative wealth ofthe overall aggregate portfolio and S = S1, . . . , Stwill denote the corresponding vector of aggregateportfolio wealth’s over time. Here the realised pricerelatives for the tth period and the mth asset, xm,t,are combined with the portfolio controls for the pre-vious period to obtain the realised portfolio returnsfor the current period t. ∆St−1 is in fact the profitsand losses for the current trading period t. Thus,we will use it to update the algorithms overall cu-mulative profits and losses which is given by

PLt = PLt−1 + ∆St − 1 (8)

2. Update expert wealth: The expert controls Ht

were determined at the end of time-period t− 1 fortime period t by the expert generating algorithm forΩ experts and M objects about which the expertsmake expert capital allocation decisions. At thenend of the tth time period the performance of eachexpert n, Shnt , can be computed from the change

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in the price relatives xm,t for the each of the Mobjects in the investment universe considered usingthe closing prices at the start, P cm,t−1, and the end

of the tth time increment, P cm,t, using the expertcontrols.

∆Shnt =

[M∑m=1

hnt (xm,t − 1)

]+ 1 (9)

Shnt = Shnt−1 ·∆Shnt (10)

3. Update expert mixtures: We consider a UP ex-pert mixture update rule and will generically referto the online update as a function g.

3.1. Universal Portfolio [2, 5, 6] inspired mixturecontrols: the mixture of the nth expert for thenext time increment, t+1, is equivalent to theaccumulated expert wealth up until time t andwill be used as the update feature for the nextunrealised increment subsequent appropriatenormalisation

qn,t+1 = Shnt (11)

4. Re-normalise expert mixtures: As mentionedpreviously, we will consider experts such that theleverage is set to unity for zero-cost portfolios: 1.)∑n qn = 0 and 2.) ν =

∑n |qn| = 1. We will not

consider the long-only experts (absolute experts asin Loonat and Gebbie [6]), but only consider ex-perts whom satisfy the prior two conditions whichwe will refer to as active experts. This in fact allowsfor shorting of one expert against another; then dueto the nature of the mixture controls the resultingportfolio becomes self-funding.

qn,t+1 =qn,t+1 − 1

N

∑Ωn=1 qn,t+1∑Ω

n=1 |qn,t+1 − 1N

∑Ωn=1 qn,t+1|

(12)

5. Update portfolio controls: The portfolio con-trols bm,t are updated at the end of time period tfor time period t+ 1 using the expert mixture con-trols qn,t+1 from the updated learning algorithmand the vector of expert controls hnt for each ex-pert n from the expert generating algorithms usinginformation from time period t and average over alln experts.

bm,t+1 =∑n

qn,t+1hnt (13)

The strategy is to implement the portfolio controls,wait until the end of the increment, measure the features(OHLCV values), update the experts and then re-applythe learning algorithm to compute the expert mixturesand portfolio controls for the next time increment. Moredetails about the various components of the algorithm

are provided in Appendix B.

C Algorithm implementation forintraday-daily trading

Intraday trading poses a whole set of new issuesthat need to be considered and isn’t as easy as simplyplugging the data into the algorithm and treating it asif it is similar to daily data just sampled at more regularintervals11. Trading on an intraday time scale containsvery different dynamics to trading on a daily time scale.

We implement the learning algorithm on a combi-nations of daily and intraday data whereby decisionsmade on the daily time scale are made completelyindependent of those made on the intraday time scalebut the dynamics of the associated wealth’s generatedby the processes are fused together. The best way tothink about it is to consider the experts as tradingthroughout the day, making decisions based solely onintraday data while compounding their wealth, andonce a trading decision is made at the final time bar,the expert makes one last trading decision on that daybased on daily historic OHLCV data (the look-backperiods will be based on passed trading days and noton time bars for that day). The daily trading decisioncan be thought of as just being the last time bar ofthe day where we are just using different data to makethe decision. The methodology for each of the intradayand daily trading mechanisms are almost exactly asexplained in Section II B above however there are acouple of alterations to the algorithm. As in the dailydata implementation, a given expert will begin makingtrading decisions as soon as there is a sufficient amountof data available to them. The idea is to begin thealgorithm from day two so that there will be sufficientdata to compute a return on the daily time scale. Wethen loop over the intraday time bars from 9:15am to4:30pm on each given day.

We will refer to trading using a combination ofdaily and intraday data as intraday-daily trading. Tointroduce some notation for intraday-daily trading, letShFt,tI

12 be the expert wealth vector for all n experts

for the tthI time bar on the tth day and denote by HFt,tI

the associated expert control matrix. The superscript Frefers to the ’fused’ daily and intraday matrices. Morespecifically, HF

t,tI will contain the 88 intraday expert

11 The main issue with this approach is the deviation in the pricesat the end of day t and the start of day t−1. Often the deviationis significant which causes returns to blow up.

12 Expert wealth is computed as before with SHFn,t,tI

=

SHFn,t,tI−1 · dShFn,t,tI

where dShFn,t,tIis the nth experts return

at time bar tI on day t.

8

controls followed by the end of day expert controls basedon daily closing OHLCV data for each given day overthe trading horizon. Denote TI as the final time barin a day (4:30pm). An experts wealth accumulated upuntil the final time bar TI

13 on day t, ShFn,t,TI+1, is

calculated from the nth column of the expert controlmatrix, denoted hFn,t,TI

, from the previous period TIand is computed solely from intraday data. Overallportfolio controls for the final intraday trade on dayt (bt,TI+1) are computed as before along with theoverall portfolio wealth St,TI+1. This position is helduntil the close of the days trading when the closingprices of the m stocks PC

t is revealed. Once the closingprices are realised, the final intraday position is closed.That is, an offsetting trade of −bt,TI+1 is made at the

prices PCt . This profit/loss is then compounded onto

St,TI+1. Thus, no intraday positions are held overnight.The experts will then make one final trading decisionbased on the daily OHLCV data given that the closingprice is revealed and will look-back on daily historicOHLCV data to make these decisions. The nth expert’swealth ShFn,t,TI+2 is updated using controls hFn,t,TI+1.The corresponding portfolio controls for all m stocksare computed for the daily trading decision on day tto be implemented at time t + 1 (bt+1), the returns

(price relatives) for day t is computed (rt =PC

t

PCt−1

) and

the wealth is compounded onto the cumulative wealthachieved is STI+2 = St,TI+1 · (bt · (xt − 1) + 1) where

St,TI+1 = St,TI· (bTI

(xTI− 1)) with rTI

=PC

TI

PCTI−1

. The

daily position bt+1 is then held until the end of thefollowing day or possibly further into the future (untilnew daily data portfolio allocations are made). Thiscompletes trading for day t.

At the beginning of day t + 1, the expert wealthShFn,t+1,1

14 is set back to 1. Setting experts wealthback to 1 at the beginning of the day rather thancompounding on the wealth from the previous day isdue to the fact that learning on intraday data betweendays is not possible due to the fact that conditions inthe market have completely changed. Trading will beginby computing expert controls hFn,t+1,2 for the secondtime bar, however all experts will not have enoughdata to begin trading since the shortest look-backparameter is 3 and hence controls will all be set tozero. As the trading day proceeds, experts will beginproducing non-zero controls as soon as they have enoughdata to satisfy the amount of data needed for a givenlook-back parameter. Something to note here is that

13 TI will always be equal to 88 as there are 88 5 minute time barsbetween 9:15am and 4:30pm

14 We do not start the trading day at the first time bar tI = 1 sincewe need to compute a return which requires 2 data points.

due to the fact that the STeFI index15 (risk-free asset)is only posted daily, we utilise the same STeFI valuefor trading throughout the day. Finally, in order todifferentiate between daily OHLCV data and intradayOHLCV data, we will denote them as Xd and XI

respectively. The algorithm outline for the intraday-daily algorithm is illustrated in Appendix B Algorithm 5.

D Online Portfolio BenchmarkAlgorithm

To get an idea of how well our online algorithm per-forms, we compare it’s performance to that of the of-fline BCRP. As mentioned previously, the hindsight CRPstrategy chosen which gives the maximum return of allsuch strategies in the long run. To find the portfolio con-trols of such a strategy, we perform a brute force MonteCarlo approach to generate 5000 random CRP strate-gies on the entire history of price relatives and choosethe BCRP strategy to be the one that returns the maxi-mal terminal portfolio wealth. As a note here, the CRPstrategies we consider are long-only.

E Transaction Costs and MarketFrictions

Apart from the (direct) transaction fees (commissions)charged by exchanges for the trading of stocks, there arevarious other costs (indirect) that need to be consideredwhen trading such assets. Each time a stock is boughtor sold there are unavoidable costs and it is imperativethat a trader takes into account these costs. The otherthree most important component of these transactioncosts besides commissions charged by exchanges are thespread16, price impact and opportunity cost [26].

To estimate indirect transaction costs (TC) for eachperiod t, we will consider is the square-root formula [27]

TC = Spread + σ.

√n

ADV(14)

where:

1. Volatility of the returns of a stock (σ): SeeSection II E E.1. below.

2. Average daily volume of the stock (ADV):ADV is computed using the previous 90 days trad-ing volumes for daily trading and the previous 5

15 See Section IV for more details on the STeFI index16 spread = best ask price minus best bid price

9

days intraday trading volume for intraday-dailytrading.

3. Number of shares traded (n): The number ofshares traded (n) is taken to be five tenths of a ba-sis point of ADV for each stock per day for dailytrading. The number of stocks traded for intraday-daily trading is 1% of ADV for the entire portfolioper day which is then split evenly among all ac-tive trading periods during the day and betweenall 15 stocks to arrive at a final value of 0.00012(1%/85/15) of ADV per stock per trading period.

4. Spread: Spread is assumed to be 1bps per day(1%% /pd) for daily trading. For intraday-dailytrading, we assume 12bps per day17 which wethen split evenly over the day to incur a cost of0.0012/85bps per time bar.

The use of the square-root rule in practice dates backmany years and is used as a pre-trade transaction costestimate [27]. The first term in Eq. (14) can be regardedas the term representing the slippage18 or temporaryprice impact and results due to our demand for liquidity[28]. This cost will only impact the price at which weexecute our trade at and not the market price (hencesubsequent transactions). The second term in Eq. (14)is the (transient) price impact which will not only affectthe price of the first transaction but also the price ofsubsequent transactions by other traders in the market,however the impact decays over time as a power-law[29]. In the following subsection, we will discuss how thevolatility (σ) is estimated for the square-root formula.Technically, σ, n and ADV in Eq. (14) should each bedefined by a vector representing the volatilities, numberof stocks traded and ADV of each stock in the portfoliorespectively however for the sake of generality we willwrite it as a constant thus representing the volatility fora single portfolio stock.

In addition to the indirect costs associated withslippage and price impact as accounted for by thesquare-root formula, we include direct costs such asthe borrowing of trading capital, the cost of regulatorycapital and the various fees associated with tradingon the JSE [6]. Such costs will also account for smallfees incurred in incidences where short-selling has takenplace. For the daily data implementation, we assume atotal direct cost of 4bps per day. For the intraday-dailyimplementation a total direct cost of 70bps per day isassumed (following Loonat and Gebbie [6]) which we

17 We follow the conservative approach taken by Loonat and Gebbie[6] as opposed to the moderate approach where 9bps per day isassumed

18 Slippage is often calculated as the difference between the priceat which an order for a stock is placed and the price at whichthe trade is executed

then split evenly over each days active trading periods(85 time bars since first expert only starts trading af-ter the 5th time bar) to get a cost of 70bps/85 per period.

For daily trading, we recover an average daily transac-tion cost of roughly 9.38bps which is approximately thesame as the assumed 10bps by Loonat and Gebbie [6].Loonat and Gebbie argue that for intraday trading, itis difficult to avoid a direct and indirect cost of about50-80bps per day in each case leaving a conservative esti-mate of total costs to be approximately 160bps per day.We realise an overall average cost per period of 1.5bpswhile the average cost per day assuming we trade for 85periods throughout each day is roughly 130bps (85*1.5)for intraday-daily trading.

E.1. Volatility Estimation for TransactionCosts

In this section we will discuss different methods forcalculating the estimates for volatility (σ) for daily andintraday data in the square-root formula (Eq. (14)).

1.a. Daily Data Estimation

The volatility of daily prices at each day t is taken tobe the standard deviation of closing prices over the last90 days. If 90 days have not passed then the standarddeviation will be taken over the number of days availableso far.

1.b. Intraday Data Estimation

The volatility for each intraday time bar tI on day t isdependent on the time of day. For the first 15 time bars,the volatility is taken to be a forecast of a GARCH(1,1)model which has been fitted on the last 60 returns ofthe previous day t − 1. The reason for this choice isthat the market is very volatile during the openinghour as well as the fact that there will be relatively fewdata points to utilise when computing the volatility.The rest of the days volatility estimates are computedusing the Realized Volatility (RV) method [30]. RVis one of the more popular methods for estimatingvolatility of high-frequency returns19 computed fromtick data. The measure estimates volatility by summingup intraday squared returns at short intervals (eg. 5minutes). Andersen et al [30] propose this estimate forvolatility at higher frequencies and derive it by showingthat RV is an approximate of quadratic variation underthe assumption that log returns are a continuous timestochastic process with zero mean and no jumps. Theidea is to show that the RV converges to the continuous

19 Most commonly refers to returns over intervals shorter than oneday. This could be minutes, seconds or even milliseconds.

10

time volatility (quadratic variation) [31], which we willnow demonstrate.

Assume that the instantaneous returns of observed logstock prices (pt) with unobservant latent volatility (σt)scaled continuously through time by a standard Wienerprocess (dWt) can be generated by the continuous timemartingale [31]

dpt = σtdWt (15)

The it follows that the conditional variance of the sin-gle period returns, rt+1 = pt+1 − pt is given by

σ2t =

∫ t+1

t

σ2sds (16)

Eq. (16) is also known as the integrated volatility forthe period t to t+ 1.

Suppose the sampling frequency of the tick data intoregularly spaced time intervals is denoted by f such thatbetween period t−1 and t there are f continuously com-pounded returns. Then

rt+1/f = pt+1/f − pt (17)

Hence, we get the Realised Volatility (RV) based on fintraday returns between periods t+ 1 and t as

RVt+1 =

f∑i=1

r2t+i/f (18)

The argument here is that provided we sample at fre-quent enough time steps (f), the volatility can be ob-served theoretically from the sample path of the returnprocess and hence [31, 32]

limf→∞

(∫ t+1

t

σ2sds−

f∑i=1

r2t+i/f

)= 0 (19)

which says that the RV of a sequence of returns asymp-totically approaches the integrated volatility and hencethe RV is a reasonable estimate of current volatility lev-els.

III TESTING FORSTATISTICAL ARBITRAGE

To test the overall trading strategy for statisticalarbitrage, we implement a novel statistical test originallyproposed by [20] and later modified by [23] by applyingit to the overall strategy’s profit and losses PL. The ideais to axiomatically define the conditions under whicha statistical arbitrage exists and assume a parametric

model for incremental trading profits in order to forma null hypothesis derived from the union of severalsub-hypotheses which are formulated to facilitate em-pirical tests of statistical arbitrage. The modified test,proposed by [23], called the Min-t test, is derived from aset of restrictions imposed on the parameters defined bythe statistical arbitrage null hypothesis and is applied toa given trading strategy to test for statistical arbitrage.The Min-t statistic is argued to provide a much moreefficient and powerful statistical test compared to theBonferroni inequality used in [20]. The lack of statisticalpower is reduced when the number of sub-hypothesesincreases and as a result, the Bonferroni approach isunable to reject an incorrect null hypothesis leading toa large Type II error.

To set the scene and introduce the concept of a sta-tistical arbitrage, suppose that in some economy, a stock(portfolio)20 st and a money market account Bt

21 aretraded. Let the stochastic process (x(t), y(t) : t ≥ 0) rep-resent a zero initial cost trading strategy that trades x(t)units of some portfolio (st) and y(t) units of a moneymarket account at a given time t. Denote the cumulativetrading profits at time t by Vt. Let the time series of dis-counted cumulative trading profits generated by the trad-ing strategy be denoted by ν(t1), ν(t2), . . . , ν(tT ) where

ν(ti) =Vti

Btifor each i = 1, . . . , T . Denote the increments

of the discounted cumulative profits at each time i by∆νi = ν(ti) − ν(ti−1). Then, a statistical arbitrage isdefined as:

Definition 1 (Statistical Arbitrage [20, 23]). A statisti-cal arbitrage is a zero-cost, self-financing trading strategy(x(t) : t ≥ 0) with cumulative discounted trading profitsν(t) such that

1. ν(0) = 0

2. limt→∞

EP[ν(t)] > 0

3. limt→∞

P[ν(t) < 0] = 0

4. limt→∞

V ar[∆ν(t)|∆ν(t) < 0] = 0

In other words, a statistical arbitrage is a tradingstrategy that 1) has zero initial cost, 2) in the limit haspositive expected discounted cumulative profits, 3) inthe limit has a probability of loss that converges to zeroand 4) variance of negative incremental trading profitsconverge to zero in the limit. It is clear that deterministicarbitrage stemming from traditional financial mathe-matics is in fact a special case of statistical arbitrage [33].

20 In our study, we will be considering a portfolio21 The money market account is initialised at one unit of a currency

i.e. B0 = 1.

11

In order to test for statistical arbitrage, assume thatthe incremental discounted trading profits evolve overtime according to the process

∆νi = µiθ + σiλzi (20)

where i = 1, . . . , T . There are two cases to consider forthe innovations: 1) zi i.i.d N(0,1) normal uncorrelatedrandom variables satisfying z0 = 0 or 2) zi follows anMA(1) process given by:

zi = εi + φεi−1 (21)

in which case the innovations are non-normal and cor-related. Here, εi are i.i.d. N(0,1) normal uncorrelatedrandom variables. It is also assumed that and ∆ν0 = 0and in case of our algorithm νtmin

= 0. We will refer thefirst model as the unconstrained mean (UM) model andthe second model as the and unconstrained mean withcorrelation (UMC) model. Furthermore, we refer to themodel with θ = 0 as the constrained mean (CM) whichassumes constant incremental profits over time and hencehas an incremental profit process given by:

∆νi = µ+ σiλzi (22)

The discounted cumulative trading profits for the UMmodel at terminal time T discounted back to the initialtime which are generated by a trading strategy are givenby

ν(T ) =

T∑i=1

∆νi ∼ N(µ

T∑i=1

iθ, σ2T∑i=1

i2λ)

(23)

From Eq. (23), it is straightforward to show that thelog-likelihood function for the discounted incrementaltrading profits is given by

`(µ, σ2, λ, θ|∆ν) = logL(µ, σ2, λ, θ|∆ν)

= −1

2

T∑i=1

log(σ2i2λ)

− 1

2σ2

T∑i=1

1

i2λ(∆νi − µiθ)2 (24)

The probability of a trading strategy generating a lossafter n periods is as follows [23]

PrLoss after n periods = Φ

(−µ∑ni=1 i

θ

σ(1 + φ)√∑n

i=1 i2λ

)(25)

where Φ(·) denotes the cumulative standard normal dis-tribution function. For the CM model, Eq. (25) is easilyadjusted by setting φ and θ equal to zero. This prob-ability converges to zero at a rate that is faster than

exponential.As mentioned previously, to facilitate empirical tests

of statistical arbitrage under Definition 1, a set of sub-hypotheses are formulated to impose a set of restrictionson the parameters of the underlying process driving dis-counted cumulative incremental trading profits and areas follows:

Proposition 1 (UM Model Hypothesis [23]). Under thefour axioms defined in Definition 1, a trading strategygenerates a statistical arbitrage under the UM model ifthe discounted incremental trading profits satisfy the in-tersection of the following four sub-hypotheses jointly:

1. H1 : µ > 0

2. H2 : −λ > 0 or θ − λ > 0

3. H3 : θ − λ+ 12 > 0

4. H4 : θ + 1 > 0

An intersection of the above sub-hypotheses defines astatistical arbitrage and as by De Morgan’s Laws22, thenull hypothesis of no statistical arbitrage is defined bya union of the sub-hypotheses. Hence, the no statisti-cal arbitrage null hypothesis is the set of sub-hypotheseswhich are taken to be the complement of each of thesub-hypotheses in Proposition 1:

Proposition 2 (UM Model Alternative Hypothesis [20,23]). Under the four axioms defined in Definition 1, atrading strategy does not generate a statistical arbitrageif the discounted incremental trading profits satisfy anyone of the following four sub-hypotheses:

1. H1 : µ ≤ 0

2. H2 : −λ ≤ 0 or θ − λ ≤ 0

3. H3 : θ − λ+ 12 ≤ 0

4. H4 : θ + 1 ≤ 0

The null hypothesis is not rejected provided that a sin-gle sub-hypothesis holds. The Min-t test is then used totest the above null hypothesis of no statistical arbitrageby considering each sub-hypothesis separately using the

t-statistics t(µ), t(−λ), t(θ− λ), t(θ− λ+0.5), and t(θ+1)where the hats denote the Maximum Likelihood Esti-mates (MLE) of the parameters. The Min-t statistic isdefined as [23]

Min-t = Mint(µ), t(θ − λ), t(θ − λ+ 0.5),

Max[t(−λ), t(θ + 1)] (26)

22 This states that the complement of the intersection of sets is thesame as the union of their complements.

12

The intuition is that the Min-t statistic returns thesmallest test statistic which is the sub-hypothesis whichis closest to being accepted. The no statistical arbitragenull is then rejected if Min-t > tc where tc depends onthe significance level of the test which we will refer to asα. Since the probability of rejecting cannot exceed thesignificance level α, we have the following condition forthe probability of rejecting the null at the α significancelevel

PrMin-t > tc|µ, λ, θ, σ ≤ α (27)

What remains is for us to compute the critical valuetc. We will implement a Monte Carlo simulation proce-dure to compute tc which we describe in more detail inSection III A step 5 below.

A Outline of the Statistical ArbitrageTest Procedure

The steps involved in testing for statistical arbitrageare outlined below:

1. Trading increments ∆νi: From the vector of cu-mulative trading profits and losses, compute theincrements (∆ν1, . . . ,∆νT ) where ∆νi = ν(ti) −ν(ti−1).

2. Perform MLE: Compute the likelihood functionas given in Eq. (24) and maximize it to find the

estimates of the four parameters, namely, µ, σ, θ

and λ. The log-likelihood function will obviouslybe adjusted depending on whether the CM (θ = 0)or UM test is implemented. We will only considerthe CM test in this study. Since MATLAB’s built-in constrained optimization algorithm23 only per-forms minimization, we minimize the negative ofthe log-likelihood function.

3. Standard errors: From the estimated parame-ters in the MLE step above, compute the negativeHessian estimated at the MLE estimates which isindeed the Fisher Information (FI) matrix denotedby I(Θ). In order to compute the Hessian, the ana-lytical partial derivatives are derived from Eq. (24).Standard errors are then taken to be the squareroots of the diagonal elements of the inverse of I(Θ)since the inverse of the Fisher information matrix isan asymptotic estimator of the covariance matrix.

4. Min-t statistic: Compute the t-statistics foreach of the sub-hypotheses which are given by

t(µ), t(−λ), t(θ − λ), t(θ − λ + 0.5), and t(θ + 1)and hence the resulting Min-t statistic given by

23 Here we are referring to MATLAB’s fmincon function

Eq. (26). Obviously t(θ − λ), t(θ − λ + 0.5) and

t(θ + 1) will not need to be considered for the CMtest.

5. Critical values: Compute the critical value at theα significance level using the Monte Carlo proce-dure (uncorrelated normal errors) and Bootstrap-ping (correlated non-normal errors)

(a) CM model

First, simulate 5000 different profit process us-ing Eq. (22) with (µ, λ, σ2) = (0, 0, 0.01)24.For each of the 5000 profit processes, performMLE to get estimated parameters, the asso-ciated t-statistics and finally the Min-t statis-tics. tc is the taken to be the 1-α quantile ofthe resulting distribution of Min-t values.

6. P-values: Compute the empirical probability ofrejecting the null hypothesis at the α significancelevel using Eq. (27) by utilising the critical valuefrom the previous step and the simulated Min-tstatistics.

7. n-Period Probability of Loss: Compute proba-bility of loss after n periods for each n = 1, . . . , Tand observe the number of trading periods it takesfor the probability of loss to converge to zero (orbelow 5% as in the literature). This is doneby computing the MLE estimates for the vector(∆ν1,∆ν2, . . .∆νn) for each given n and substitut-ing these estimates into Eq. (25).

IV THE DATA

The two data sets utilised for the purpose of back-testing the online algorithm are outlined below.

A Daily Data

The daily data is sourced from Thomson Reuters andcontains data corresponding to all stocks listed on theJSE Top 4025. The data set consists of data for 42stocks over the period 01-01-2005 to 29-04-2016 howeverwe will only utilise the stocks which traded more than60% of the time over this period. Removing such stocksleaves us with a total of 31 stocks. The data comprisesof the opening price (P o), closing price (P c), lowest price(P l), the highest price (Ph) and daily traded volume (V )

24 tc is maximized when µ and λ are zero. σ2 is set equal to 0.01to approximate the empirical MLE estimate [23].

25 Please refer to Appendix E E.1 for the full names of theBloomberg ticker symbols.

13

(OHLCV). In additions to these 31 stocks, we also requirea risk-free asset for balancing the portfolio. We make thechoice of trading the Short Term Fixed Interest (STeFI)index. The STeFI benchmark is a proprietary index thatmeasures the performance of Short Term Fixed Inter-est or money market investment instruments in SouthAfrica. It is constructed by Alexander Forbes (and for-merly by the South African Futures Exchange (SAFEX))and has become the industry benchmark for short-termcash equivalent investments (up to 12 months) [34].

B Intraday-Daily Data

Bloomberg is the source of all tick (intraday) dataused in this paper. The data set consists of 30 of theTop 40 stocks on the JSE from 02-01-2018 to 29-06-2018.The data is then sampled at 5 minute intervals to cre-ate an OHLCV entry for all 5 minute intervals over the6 month period. We remove the first 10 minutes andlast 20 minutes of the continuous trading session (9:00-16:50) as the market is relatively illiquid and volatile dur-ing these times which may lead to spurious trade deci-sions. We are thus left with 88 OHLCV entries for eachstock on any given day. In addition to the intraday data,daily OHLCV data for the specified period is requiredfor the last transaction on any given day. Again, wemake use of the STeFI index as the risk-free asset. Thedata was sourced from a Bloomberg terminal using the RBloomberg API, Rblpapi, and all data processing is donein MATLAB to get the data into the required form forthe learning algorithm.

V RESULTS AND ANALYSIS

A Daily Data

In the section below, we implement the various algo-rithms described above in order to plot a series of graphsfor daily JSE Top 40 data as discussed in Section IVabove. We will plot five different graphs: first is theoverall portfolio wealth over time which corresponds toSt as described above, second, the cumulative profitand losses over time PLt, third, the overall portfoliocontrols corresponds to bt over time, fourth, the relativepopulation wealth of experts corresponds to the wealthaccumulated over time by each of the experts competingfor wealth in the algorithm Sht and finally, the relativepopulation wealth of the strategies takes the mean overall experts for each given trading strategy to get anaccumulated wealth path for each technical trading rule.

For the purpose of testing the learning algorithm,we will identify the 15 most liquid stocks over oneyear prior to the start of active trading. The stocks,

ranked by liquidity, are as follows: FSRJ.J, OMLJ.J,CFRJ.J, MTNJ.J, SLMJ.J, NTCJ.J, BILJ.J, SBKJ.J,WHLJ.J, AGLJ.J, SOLJ.J, GRTJ.J, INPJ.J, MNDJ.Jand RMHJ.J.

A.1. No Transaction Costs

Figure 1: Overall cumulative portfolio wealth (S) for dailydata with no transaction costs (blue) and the benchmarkBCRP strategy (orange). The figure inset illustrate the asso-ciated profits and losses (PL) of the strategy.

Barring transaction costs, it’s clear that the portfoliomakes a favourable cumulative returns on equity over the6 year period as is evident in Figure 1. The performanceof the algorithm is not too far off that of the benchmarkBCRP strategy which is promising as the originalliterature proves that the algorithm should track sucha benchmark. The figure inset in Figure 1 illustratesthat the provides consistent positive trading profits overthe entire trading horizon. Figure 2(a) shows the expertwealth for all Ω experts and Figure 2(b) shows the meanexpert wealth for each strategy. These figures illustratethat on average, the underlying experts perform fairlypoorly compared to the overall strategy however thereis evidence that some experts make satisfactory returnsover the period.

Table II and Table III provide the group summarystatistics of the terminal wealth’s of experts and of theexpert’s profits and losses over the entire trading horizonrespectively where experts are grouped based on their

14

(a) (b)

Figure 2: Figure 2(a) illustrates the expert wealth (Sh) for all Ω experts for daily data with no transaction costs. Figure 2(b)illustrates the mean expert wealth of all experts for each trading strategy (ω(i)) for daily data with no transaction costs.

underlying strategy (ω(i)). The online Z-Anticor26 al-gorithm produces the best expert (maximum terminalwealth) followed closely by the slow stochastic rule whileZ-Anticor also produces experts with the greatest meanterminal wealth over all experts (column 2). Addition-ally, Z-Anticor produces expert’s with wealth’s that varythe most (highest standard deviation). Williams %R pro-duces the worst expert by quite a long way (minimumterminal wealth). The trading rule with the worst meanterminal wealth and worst mean ranking are SAR andslow stochastic respectively. With regards to profits andlosses (Table III), the momentum rule (MOM) producesthe expert with the greatest profit in a single period.SAR followed by Anti-Z-BCRP produce the worst andsecond worst mean profit/loss per trading period respec-tively whereas Z-Anticor and Z-BCRP achieve the bestmean profit/loss per trading period.

Figure 3(a) illustrates the 2-D plot of the latent spaceof a Variational Autoencoder (VAE) for the time series’of wealth’s of all the experts and is coloured by objectcluster. It is not surprising that the expert’s wealth timeseries’ show quite well defined clusters in terms of thestock which experts trade in their portfolio as the stocksthat each expert trades will be directly related to thedecisions they make given the incoming data and hencethe corresponding returns (wealth) they achieve.

26 Please refer to Appendix C and Appendix D for a detailed de-scription of the various trading rules mentioned in the tablesTable II and Table III.

To provide some sort of comparison, in Figure 3(b) weplot the same results as above but this time we colourthe experts in terms of their underlying strategy ω(i).The VAE seems to be able to pick up much clearersimilarities (dissimilarities) between the experts based onthe stocks they trade compared to which strategy theyutilise providing evidence that the achieved wealth hasa much stronger dependence on the stock choice ratherthan the chosen strategy. This may be an importantpoint to consider and gives an indication that it may beworth considering more sophisticated ways to choose thestocks to trade rather than developing more (profitable)strategies. A discussion on the features that shouldbe considered by a quantitative investment manager inassessing an assets usefulness is provided in Samo andHendricks [35].

Next, we implement the CM test for statistical arbi-trage on the daily cumulative profits and losses (PL) forthe strategy without transaction costs. In order to havea result that is synonymous with [23], we choose a periodof 400 days to test our strategy. We test the realisedprofits and losses for the 400 day period stretching fromthe 30th trading day until the 430th trading day. This isto allow for the algorithm to initiate and leave enoughtime for majority of the experts to have sufficient datato begin making trading decisions. Having simulated the5000 different Min-t statistics as in Section III A step 5(a) using simulations of the profit process in Eq. (22),Figure 4 illustrates the histogram of Min-t values. Thecritical value tc is then computed as the 0.95-quantile of

15

Strategy Mean (mean rank) St. Dev. Min Max

EMA X-over 0.8739 (673.6343) 0.1767 0.5216 1.4493Ichimoku Kijun Sen 0.9508 (623.3194) 0.2313 0.5424 1.5427MACD 0.9504 (657.7639) 0.1750 0.5601 1.6065Moving Ave X-over 0.8895 (632.6944) 0.1930 0.5206 1.4505ACC 1.0994 (736.5833) 0.3131 0.5283 1.9921BOLL 1.0499 (569.1944) 0.3536 0.6076 1.7746Fast Stochastic 0.9995 (778.6111) 0.3699 0.6006 1.8555MARSI 1.0723 (639.3611) 0.2081 0.6947 1.6917MOM 1.0403 (681.4444) 0.1353 0.7349 1.3595Online Anti-Z-BCRP 0.7579 (731.9444) 0.1935 0.4649 1.0924Online Z-Anticor 1.3155 (694.5278) 0.4388 0.6363 2.3886Online Z-BCRP 1.2818 (652.8611) 0.2637 0.8561 1.8341PROC 0.8963 (718.0833) 0.1631 0.6305 1.2161RSI 1.1339 (757.3889) 0.2544 0.6440 1.7059SAR 0.7314 (654.1111) 0.0619 0.6683 0.8683Slow Stochastic 1.1135 (793.2222) 0.3302 0.6955 2.1023Williams %R 0.9416 (728.6944) 0.3150 0.4662 1.5131

Table II: Group summary statistics of the overall rankings of experts grouped by their underlying strategy (ω(i) wherei = 1, . . . , 17) for the daily trading. In brackets next to mean are the mean overall ranking of experts within the group of theirunderlying strategy.

Strategy Mean St. Dev. Min Max

EMA X-over -0.00010 0.00633 -0.09745 0.08074Ichimoku Kijun Sen -0.00004 0.00723 -0.10467 0.06157MACD -0.00003 0.00725 -0.15993 0.08074Moving Ave X-over -0.00009 0.00644 -0.15993 0.11482ACC 0.00007 0.00760 -0.15993 0.08028BOLL 0.00002 0.00711 -0.06457 0.06480Fast Stochastic -0.00001 0.00847 -0.06469 0.06279MARSI 0.00006 0.00612 -0.06788 0.06527MOM 0.00004 0.00603 -0.06051 0.15820Online Anti-Z-BCRP -0.00022 0.00773 -0.09847 0.09336Online Z-Anticor 0.00021 0.00759 -0.06475 0.09773Online Z-BCRP 0.00021 0.00771 -0.09336 0.09847PROC -0.00007 0.00733 -0.10467 0.09745RSI 0.00010 0.00666 -0.06460 0.09745SAR -0.00023 0.00724 -0.10467 0.08724Slow Stochastic 0.00009 0.00809 -0.06480 0.06820Williams %R -0.00006 0.00815 -0.06820 0.06317

Table III: Group summary statistics of the expert’s profits and losses per period grouped by their underlying strategy (ω(i)where i = 1, . . . , 17).

the simulated distribution which refers to a significancelevel of α = 5% and is illustrated by the red vertical line.The resulting critical value is tc = 0.7263. The Min-tresulting from the realised incremental profits and lossesof the overall strategy is 3.0183 (vertical green line). ByEq. (27), we recover a p-value of zero. Thus, we canconclude that there is significant evidence to reject thenull of no statistical arbitrage at the 5% significance level.

In addition to testing for statistical arbitrage, we also

report the number of days it takes for the probability ofloss of the strategy to decline below 5% using Eq. (27)adjusted for the case of the CM model. As discussed in7, for each n = 1, . . . , T , we perform MLE for ∆ν1:n toget the parameter estimates. We then substitute theseestimates into Eq. (27) to get an estimate of the proba-bility of loss for the nth period. This is all done in termsof the CM model. The figure inset of Figure 4 illustratesthe probability of loss for each of the first 25 trading dayswhere we compute the probability of loss of the profit and

16

(a) (b)

Figure 3: Figure 3(a) and show the latent space of Variational Autoencoder on the time series’ of expert wealth’s implementedusing Keras in Python. In Figure 3(a) experts are coloured by which of the 4 object clusters they trade whereas in Figure 3(b),experts are coloured by their underlying trading strategy ω(i).

loss process from the first trading period up to the nth

period for each n = 1, . . . , 25. As is evident from the fig-ure inset, it takes roughly 10 periods for the probabilityof loss to converge below 5%.

A.2. Transaction Costs

In this section we reproduce the results from above butthis time including transaction costs for daily tradingas discussed in Section II E. Once direct and indirect(Eq. (14)) costs have been computed, the idea is tosubtract off the transaction cost from St and PLt oneach day t and compound the resulting profit/loss ontothe cumulative wealth and profits and losses respectivelyup until day t.

It is clear from the inset of Figure 5, which illustratesthe profits and losses (PL) of the overall strategy lesstransaction costs for each period, that consistent lossesare incurred when transaction costs are incorporated.Furthermore, there is no evidence to reject the no sta-tistical arbitrage null hypothesis as the Min-t statisticresulting from the overall strategy is well below the crit-ical value at the 95th percentile of the histogram as il-lustrated in Figure 6. In addition to this, although theprobability of loss of the strategy with transaction costsincluded initially converges to zero, soon after it jumpsup to a probability of one and remains there for the restof the periods. This is illustrated in the inset of Figure 6.

Considering the above evidence contained in Figure 5,

Figure 6 and it’s associated figure inset, the overall strat-egy does not survive historical back tests in terms ofprofitability when transaction costs are considered andmay not be well suited for an investor utilising daily datawhom has a limited time to make adequate profits.

B Intraday-Daily Data

Below we report the results of the algorithm imple-mentation for a combination of intraday and daily JSEdata. We run the algorithm on the the OHLCV data of15 most liquid stocks from a set of 30 of the JSE Top4027. Liquidity is calculated in terms of average dailytrade volume for the first 4 days of the period 02-01-2018to 09-03-2018. The set of 15 stocks is as follows: FSR:SJ,GRT:SJ, SLM:SJ, BGA:SJ, SBK:SJ, WHL:SJ, CFR:SJ,MTN:SJ, DSY:SJ, IMP:SJ, APN:SJ, RMH:SJ, AGL:SJ,VOD:SJ and BIL:SJ. The remaining 40 days’ data forthe aforementioned period is utilised to run the learningalgorithm on. As in the daily data implementation, weagain analyse the two cases of trading with and withouttransaction costs which we report in the following twosubsections below.

27 See Appendix E E.2 for the list of the 30 stocks along with theirBloomberg ticker symbols

17

Figure 4: Histogram of the 5000 simulated Min-t statisticsresulting from the CM test implemented on the simulatedincremental process given in Eq. (22) along with the Min-tstatistic (green) for the overall strategy’s profit and loss se-quence over the 400 day period stretching from the 30th trad-ing day until the 430th trading day without any account fortransactions costs. The figure inset displays the probability ofloss for each of the first 30 trading days where we compute theprobability of loss of the profit and loss process from the firsttrading period up to the nth period for each n = 1, . . . , 25.

B.1. No Transaction Costs

Without transaction costs, the cumulative wealthachieved by the overall strategy, illustrated in Figure 7evolves similar to an exponential function over time. Theassociated profits and losses are displayed in the figureinset of Figure 7. Incremental profits and losses areobviously a lot smaller compared to the daily data casesresulting in a much smoother function in comparison tothe daily data case (Figure 1).

Table IV is the intraday-daily analogue of Table II.In this case, the exponential moving crossover strat-egy (EMA X-over) produces the expert with the great-est wealth and acceleration (ACC) the expert with theleast terminal wealth. Exponential moving crossover alsoproduces experts with the highest variation in terminalwealth’s. Price rate of change (PROC) is by far the strat-egy with the best mean ranking experts among all expertshowever Z-BCRP produces experts with highest meanterminal wealth.

Again, as for the daily data case, we implement atest for statistical arbitrage for intraday-daily trad-

Figure 5: Overall cumulative portfolio wealth (S) for dailydata with transaction costs. The figure inset illustrates theprofits and losses (PL) for overall strategy for daily data withtransaction costs.

ing without transaction costs for 400 trading periodsstarting from the 6th time bar of the 2nd tradingday28 using the intraday-daily profit and loss sequence(PL). Figure 8 illustrates the histogram of simulatedMin-t values with the 0.95-percentile of the simulateddistribution representing the critical value tc (red)and the Min-t (green) resulting from the incrementalprofits and losses of the overall strategy resulting fromthe learning algorithm. The resulting critical value is0.7234 and the Min-t value is 4.2052. Thus, there isstrong evidence to reject the null hypothesis of no statis-tical arbitrage as the resulting p-value is identical to zero.

The figure inset of Figure 8 illustrates the probabilityof loss for each of the first 25 periods of the 400 periodsas discussed in the above paragraph. It takes 13 periodsfor the probability of loss to converge to zero which ishighly desirable.

B.2. Transaction Costs

We now report the results of the algorithm run on thesame intraday-daily data as in the subsection above but

28 This corresponds to the trading period within which the veryfirst trading decisions are made.

18

Strategy Mean (mean rank) St. Dev. Min Max

EMA X-over 1.0024 (662.7639) 0.0094 0.9801 1.0375Ichimoku Kijun Sen 0.9989 (710.3750) 0.0085 0.9663 1.0303MACD 0.9995 (684.8704) 0.0067 0.9720 1.0202Moving Ave X-over 1.0012 (708.7824) 0.0058 0.9766 1.0204ACC 0.9953 (831.3333) 0.0079 0.9646 1.0048BOLL 0.9974 (712.9722) 0.0069 0.9787 1.0089Fast Stochastic 0.9991 (711.4167) 0.0040 0.9871 1.0085MARSI 0.9973 (736.2500) 0.0062 0.9824 1.0094MOM 0.9982 (723.1389) 0.0087 0.9700 1.0082Online Anti-Z-BCRP 0.9980 (597.3056) 0.0062 0.9828 1.0103Online Z-Anticor 1.0015 (655.7778) 0.0058 0.9896 1.0180Online Z-BCRP 1.0031 (566.5833) 0.0069 0.9898 1.0149PROC 0.9980 (445.1389) 0.0064 0.9814 1.0140RSI 0.9997 (535.5833) 0.0065 0.9861 1.0171SAR 0.9945 (499.7222) 0.0053 0.9790 1.0005Slow Stochastic 1.0007 (508.5278) 0.0048 0.9927 1.0173Williams %R 1.0020 (536) 0.0034 0.9957 1.0133

Table IV: Group summary statistics of the overall rankings of experts grouped by their underlying strategy (ω(i) wherei = 1, . . . , 17) for intraday-daily trading. In brackets are the mean overall ranking of experts utilising each strategy.

Figure 6: Histogram of the 5000 simulated Min-t statisticsresulting from the CM model and the incremental processgiven in Eq. (22) along with the Min-t statistic (green) for theoverall strategy’s profit and loss sequence over the 400 dayperiod stretching from the 30th trading day until the 430th

trading day with transactions costs incorporated. Also illus-trated is the critical value at the 5% significance level (red).The figure inset shows the probability of the overall tradingstrategy generating a loss for each of the first 400 tradingdays.

Figure 7: The overall cumulative portfolio wealth (S) forintraday-daily data with no transaction costs. The figure insetillustrates the associated profits and losses.

this time with transaction costs incorporated as outlinedin Section II E. Figure 9 and the figure inset illustrate theoverall cumulative portfolio wealth (S) and profits andlosses (PL) respectively for intraday-daily trading withtransaction costs. For comparative reasons, the axes areset to be equivalent to those in the case of no transaction

19

Figure 8: Histogram of the 5000 simulated Min-t statisticsresulting from the CM model and the incremental processgiven in Eq. (22) for the first 400 trading periods for intraday-daily profits and losses without taking into account transac-tion costs along with the Min-t statistic for the overall strat-egy (green) and the critical value at the 5% significance level(red). The figure inset shows the probability of the overalltrading strategy generating a loss after n periods for eachn = 5, . . . , 25 of the intraday-daily profit and loss process(PL) taken from the 5th time bar of the second day whenactive trading commences.

costs (Figure 9 and the figure inset). Surprisingly, evenwith a total daily trading cost (direct and indirect) ofroughly 130bps, which is a fairly conservative approach,the algorithm is able to make satisfactory returns whichis in contrast to the daily trading case (Figure 5).Furthermore, Figure 10 provides significant evidencethat the no statistical arbitrage null hypothesis can berejected and has an almost identical Min-t statistic (4.32in the transaction costs case compared to 4.21) to that ofthe case of no transaction costs (Figure 8). What is evenmore comforting is the fact that even when transactioncosts are considered, the probability of loss per tradingperiod converges to zero albeit slightly slower (26 tradingperiods) than the case of no transaction costs (13 tradingperiods as illustrated in the inset of figure Figure 8).

The above results for intraday-daily trading are incomplete contrast to the case of daily trading with trans-action costs whereby the no statistical arbitrage nullcould not be rejected, the probability of loss did notconverge to zero and remain there, and trading profitssteadily declined over the trading horizon. This suggeststhat the proposed algorithm may be much better suited

Figure 9: The overall cumulative portfolio wealth (S) forintraday-daily data with transaction costs. The figure insetillustrates the associated profits and losses.

to trading at higher frequencies. This is not surprisingand is in complete agreement with Schulmeister [36] whoargues that the profitability of technical trading strate-gies had declined over from 1960 before becoming un-profitable from the 1990’s. A substantial set of technicaltrading strategies are then implemented on 30-minutedata and the evidence suggests that such strategies re-turned adequate profits between 1983 and 2007 howeverthe profits declined slightly between 2000 and 2007 com-pared to the 1980’s and 1990’s. This suggests that mar-kets may have become more efficient and even the possi-bility that stock price and volume trends have shifted toeven higher frequencies than 30 minutes [36]. This sup-ports the choice to trade the algorithm proposed in thispaper on at least 5-minute OHLCV data and reinforcesour conclusion that ultimately, the most desirable im-plementation of the algorithm would be in volume-timewhich is best suited for high frequency trading.

VI CONCLUSION

We have developed a learning algorithm built froma base of technical trading strategies for the purposeof trading equities on the JSE that is able to providefavourable returns when ignoring transaction costs, un-der both daily and intraday trading conditions. The re-turns are reduced when transaction costs are consideredin the daily setting, however there is sufficient evidence to

20

Figure 10: Histogram of the 5000 simulated Min-t statisticsresulting from the CM model and the incremental processgiven in Eq. (22) for the first 400 trading periods for intraday-daily profit and losses less transaction costs along with theMin-t statistic for the overall strategy (green) and the criticalvalue at the 5% significance level (red).

suggest that the proposed algorithm is really well suitedto intraday trading.

This is reinforced by the fact that there exists meaning-ful evidence to reject a carefully defined null hypothesisof no statistical arbitrage in the overall trading strategyeven when a reasonably conservative view is taken on in-traday trading costs. We are also able to show that it inboth the daily and intraday-daily data implementationsthat the probability of loss declines below 5% relativelyquickly which strongly suggests that the algorithm is wellsuited for a trader whose preference or requirement is tomake adequate returns in the short-run.

The superior performance of the algorithm for intra-day trading is in agreement with Schulmeister [36] whoconcluded that daily profitability of a large set of techni-cal trading strategies has steadily declined since 1960 andhas been unprofitable since the onset of the 1990’s. Thisis in contrast to trading 30-minute data between 1983 and2007 which has produced decent average gross returnsbut this has slowly declined since the early 2000’s. Inconclusion, the proposed algorithm is much better suitedto trading at higher frequencies.

We are however cognisant of the fact that intradaytrading will also typically required a large componentof accumulated trading profits to finance frictions, con-cretely to fund direct, indirect and business model costs[6]. For this reason we are careful to remain sceptical

with this class of algorithms long-run performance whentrading with real money in a live trading environment.The current design of the algorithm is not yet readyto be traded on live market data, however with someeffort it is easily transferable to such use cases giventhe sequential nature of the algorithm and its inherentability to receive and adapt to new incoming data whilemaking appropriate trading decisions based on the newdata. Concretely, the algorithm should be deployedin the context of volume-time trading rather than thecalendar time context considered in this work.

Possible future work includes implementing the algo-rithm in volume-time which will be best suited for dealingwith a high frequency implementation of the proposedalgorithm given the intermittent nature of order-flow.We also propose replacing the learning algorithm withan online (adaptive) neural network that has the abil-ity to predict optimal holding times of stocks. Anotherinteresting line of work that has been considered is tomodel the population of trading experts as competing ina predator-prey environment [37, 38]. This was a an ini-tial key motivation for the research project, to find whichcollections of technical trading strategies can be groupedcollectively and how these would interact with each other.This includes utilising cluster analysis to group togetheror separate trading experts based on their similarities anddissimilarities and hence make appropriate inferences re-garding their interactions and behaviours at the level ofcollective and emergent dynamics.

VII ACKNOWLEDGEMENTS

NM and TG would like to thank the Statistical FinanceResearch Group in Department of Statistical Sciences forvarious useful discussion relating to the work. In par-ticular we would like to thank Etienne Pienaar, LionelYelibi and Duncan Saffy. We thank Michael Gant forhis help with developing some of the strategies and fornumerous valuable discussions with regards to the statis-tical arbitrage test. We would also like thank GuenterSchmidt for various insightful comments and discussionrelating to the testing and validation of online learningalgorithms and optimal conversion problems. TG wouldlike to thank UCT FRC for funding (UCT fund 459282).

VIII APPENDIX

A Variable Definitions

Variable Definition

21

ω parameter of underlying strategy of a givenexpert which corresponds to one of the tech-nical trading strategies or trend-followingstrategies. ω(i) is then the ith strategy fromthe set of all strategies

W1 number of trading rules which utilise oneparameter

W2 number of trading rules utilising twoparameters

n1 vector of short-term look-back parametersn2 vector of long-term look-back parametersL number of short-term look-back parametersK number of short-term look-back parametersc object clusters parameter where c(i) is the ith

object clusterC number of object clustersΩ total number of expertsADV average daily volumeδliq number of days to look-back to choose the m

most liquid stocksm number of stocks to be passed into the expert

generating algorithm ranked by their liquidityhnt nth expert’s strategy (m+1 portfolio controls)

for period tHt expert control matrix by made up of all n

experts’ strategies at time t for all m stocksi.e. Ht = [h1

t , . . . ,hnt ]

Sht wealth of all n experts at time tShn

t nth expert’s wealth at time tbt final overall aggregate portfolio to be used in

the following period t+ 1 which we denoteS vector of the overall aggregate portfolio com-

pounded wealth over timeSt overall aggregate portfolio compounded

wealth at time tPL vector of the overall aggregate portfolio cu-

mulative profits and lossesPLt the overall aggregate portfolio cumulative

profits and losses at time tnb (ns) number of buy (sell) signals from the set of

output signalsσ+ (σ−) vector of standard deviations of stocks to be

bought (sold)w m+ 1 vector of weights allocated to m stocks

and the risk-free assetwrf weight allocated to the risk-free assettmin time period at which trading commences (ei-

ther daily/intraday)T terminal time of trading (generally daily but

can be either daily/intraday)Xt m×5 matrix of stock’s OHLCV values at each

time period t (either daily/intraday)Xd daily OHLCV dataXI intraday OHLCV dataxt vector of m+ 1 price relatives at time period

tPc

t vector of closing prices for all m stocks at timeperiod t

P cm,t closing price of stock m at time period ttI tthI time bar of a given dayTI final (terminal) time bar of a given day

(4:30pm)

HFt,TI

fused intraday-daily expert control matrixhFn,t,TI

nth expert’s controls for all m assets at timebar tI on the tth day

ShFt,tI fused intraday-daily expert wealth vector for

all n experts for the tthI time bar on the tth

dayShF

n,t,TI+1 nth expert’s wealth at time bar tI on the tth

dayσ volatility of the returns of a stockν(t) cumulative discounted trading profits/losses

at time t∆νt cumulative discounted trading profit/loss in-

crement at time t i.e. ∆νt = ν(t) − ν(t) − 1Min-t statistic used for accepting/rejecting the no

statistical arbitrage null hypothesis

Table V: Variable definitions

B Online Learning Algorithm

Algorithm 1 Expert Generating Algorithm

Require:1: 1. OHLCV prices up to current time Xt

2. short-term parameters n1

3. long-term parameters n2

4. set of strategies to be considered ω

5. set of cluster indices to be considered c

6. current portfolio controls bt

7. past agent-controls Ht−1

8. current agent-controls Ht

2: Expert index = 03:

4: for t = tmin to T do5: for c = 1 to C do6: for w = 1 to W do7: Define wth strategy as string and convert

to function8:

9: for ` = 1 to L do10: `1 = n1(`)11: for k = 1 to K do12: if wth strategy only has 1 param

then13: break14: end if15: k1 = n1(k)16: if k1 > `1 then17: Expert index =

Expert index+ 118: Call controls function to

compute weights for wth

strategy → hc,`,kn,t = w19: else20: continue21: end if

22

22: end for23: if Strategy has 1 parameter then24: Call controls function to compute

weights for wth strategy → hc,`n,t =w

25: end if26: end for27: end for28: end for29: end for

Algorithm 2 Compute output signals

Require:1: 1. past expert controls hnt−1

2. current period signals s2: initialise combined signals: output s = zeros(size(s))3: if all previous controls (hnt−1) were NaN’s or zeros

then4: output s = s5: else6: for i = 1 to length(s) do7: if s(i) == 0 then8: output s(i) = sign(hnt−1(i))9: else

10: output s(i) = s(i)11: end if12: end for13: end if

Algorithm 3 Transform signals to controls: volatilityloading

Require:1: 1. OHLCV prices up to current time Xt

2. output s2:

3: extract closing prices from Xt: Pc

4:

5: if all output s equal zero then6: w =zeros(length(output s)+1)7:

8: else if output s ≥ 0 then9: compute standard deviations of closing prices

over last 120 days for stocks where elements ofoutput s > 0:

10:

11: vol+ = std(Pc(output s > 0,end-119)12: w = [0.5 · output s;−0.5]13: w(output s > 0) = (1/sum(vol+)) ∗

w(output s > 0) · vol+

14: else if output s ≤ 0 then15: compute standard deviations of closing prices

over last 120 days for stocks where elements ofoutput s < 0:

16:

17: vol− = std(Pc(output s < 0,end-119)18: w = [0.5 · output s; 0.5]19: w(output s < 0) = (1/sum(vol−)) ∗

w(output s < 0) · vol−

20: else21: compute standard deviations of closing prices

over last 120 days for stocks where elements ofoutput s > 0 and where output s < 0:

22:

23: vol+ = std(Pc(output s > 0,end-119)24: vol− = std(Pc(output s < 0,end-119)25: w = [output s; 0]

26: w(output s > 0) = 0.5 ·(

1

sum(abs(vol+))

)·vol+·

w(output s > 0)

27: w(output s < 0) = 0.5 ·(

1

sum(abs(vol−))

)·vol−·

w(output s < 0)28: w(end) = sum(w)29: end if30: return w

(xt,Ht) bm =∑

n qnhnt

Sn =∑

m hnt rm + 1

S =∑

n qnSn

S =∑

m bmrm + 1

rm=xm−1

qn

bm

rm=xm−1

qn

Figure 11: Relationship between the components of the On-line Learning Algorithm

Algorithm 4 Online Learning Algorithm

Require:1: 1. updated agent-controls Ht+1

2. current price relatives xt

3. current portfolio controls bt

4. current agent-controls Ht

5. past agent-wealth Sht−1

6. past portfolio wealth St−1

2: for t = tmin to T do3: Update portfolio wealth:4: St = St−1(bt(xT

t − 1) + 1)5:

6: Update expert wealth’s:7: Shnt = Shnt−1(hnt (xT

t − 1) + 1)8:

9: Update expert mixtures:10: qn,t+1 = Shnt11:

12: Renormalise the expert mixtures:

23

13:

qn,t+1 =

∑n qn,t+1 = 1, qn,t+1 ≥ 0∑n |qn,t+1| = 1,

∑n qn,t+1 = 0

14: Update the portfolio:15: bt+1 =

∑n qn,t+1h

nt+1

16:

17: Leverage corrections:18: if (ν =

∑m |bm,t|) 6= 1 then

19: Renormalise controls:20: bn,t+1 = 1

ν bn,t+1

21: Renormalise mixtures:22: qn,t+1 = 1

ν qn,t+1

23: end if24: end for25: return (bt+1,Shnt ,St,qn,t+1)

Algorithm 5 Intraday-Daily Algorithm

Require:1: 1. OHLCV daily prices Xd

2. OHLCV intraday time bars XI

3. vector indicating index for start of each dayuniqueday

2: initialise daily price relatives: retd = repeat(1,m+1)3:

4: for t = 2 to T do5:

6: initialise intraday price relatives for t-th day:7: retI = repeat(1,m+ 1)8:

9: initialise expert wealth’s:10: ShF (uniqueday(t)+t-1) = repeat(1)11:

12: repeat daily STEFI for all time bars:13: STEFII = repeat(STEFI(t),uniqueday(t+1) -

uniqueday(t))14:

15: for tI = uniqueday(t)+1 to uniqueday(t + 1)-1do

16:

17: get closing prices for time bars tI − 1 and tI :18: Pc

I

19:

20: compute price relatives for current time barand append to previous period price relatives:

21: retI = [retI ; PcI(t)/P

cI(t− 1)]

22:

23: run expert generating algorithm:24: expert gen(t0, t, retI)25:

26: run online learning algorithm:27: online learn(t0, t, retI)28:

29: end for

30:

31: get closing prices for day t − 1 and day t: Pct−1

and Pct

32:

33: compute price relatives for day t:34: retd = [retd; P

cd(t)/P

cd(t− 1)]

35:

36: run expert generating algorithm:expert gen(t, uniqueday(t+ 1))

37:

38: run online learning algorithm:39: online learn(t, uniqueday(t+ 1))40:

41: end for42:

43: for c = 1 to C do44: for w = 1 to W do45: Define wth strategy as string from file name

and convert to function46:

47: for ` = 1 to L do48: `1 = n1(`)49: for k = 1 to K do50: if wth only has 1 parameter then51: break52: end if53: k1 = n1(k)54: if k1 > `1 then55: Expert index = Expert index + 156: Call controls function to

compute weights for wth strategy

→ hn,c,`,kt = w57: else58: continue59: end if60: end for61: if Strategy has 1 parameter then62: Call controls function to compute

weights for wth strategy → hn,c,`t = w63: end if64: end for65: end for66: end for

C Technical Indicators and TradingRules

We follow [9, 14] in introducing and describing someof the more popular technical analysis indicators as wellas a few others that are widely available. We also pro-vide some trading rules which use technical indicators togenerate buy, sell and hold signals.

24

C.1 Simple Moving Average

The most common moving average is the simple mov-ing average (SMA). The SMA is the mean of a time series(typically of closing prices) over the last n trading daysand is usually updated every trading period to take intoaccount more recent data and drop older values. Thesmaller the value of n, the closer the moving average willfit to the price data

1.a SMA Indicator: SMAct(n)

SMAt(Pc, n) =

1

n

n−1∑i=0

P ct−i (28)

C.2 Exponential Moving Average

The exponential moving average (EMA) makes use oftoday’s close price, yesterdays moving average value anda smoothing factor (α). The smoothing factor will de-termine how quickly the exponential moving average re-sponds to current market prices [14]. A simple movingaverage is used for the initial EMA value.

2.a EMA Indicator: EMAct(n)

EMA(Pc, n) = αP ct + (1− α)EMA(P ct−1, n) (29)

where α = 2n+1

C.3 Highest High

Highest high is the greatest high price in the last nperiods.

HH(n) = max(P hn ) (30)

where the vector with high prices of last n periods isgiven by P h

n = (Pht−n, Pht−n+1, P

ht−n+2, . . . , P

ht )

C.4 Lowest Low

Lowest low is the smallest low price in the last n peri-ods.

LL(n) = min(P ln) (31)

where the vector with low prices of last n periods is givenby P l

n = (P lt−n, Plt−n+1, P

lt−n+2, . . . , P

lt )

C.5 Moving Average Crossover Trading Rule

The moving average crossover rule uses 2 SMA’s, ashort SMA and a longer SMA. A buy signal occurs whenthe faster (shorter) moving average crosses above theslower (longer) moving average, and a sell signal occurswhen the shorter moving average crosses below the longermoving average.

Figure 12: Moving average crossover trading rule imple-mented on 8 months of Anglo American PLC daily closingprices from 01-01-2007 to 30-08-2007. Green and red verticallines represent buy and sell signals respectively.

C.6 Exponential Moving Average CrossoverRule

The calculation is identical to the Moving AverageCrossover rule above however instead of using a SMA,an EMA is used.

C.7 Ichimoku Kinko Hyo

The Ichimoku Kinko Hyo (at a glance equilibriumchart) system consists of five lines and the Kumo (cloud)[39–41]. The five lines all work in concert to produce theend result. The size of the Kumo is an indication of thecurrent market volatility, where a wider Kumo is a morevolatile market.

7.a Ichimoku Kinko Hyo Indicators

1. Tenkan-sen (Conversion Line): (HH(n1) +LL(n1))/2

2. Kijun-sen (Base Line): (HH(n2) + LL(n2))/2

3. Chikou Span (Lagging Span): Close plotted n2

days in the past

4. Senkou Span A (Leading Span A): (Conver-sion Line + Base Line)/2

5. Senkou Span B (Leading Span B): (HH(n3) +LL(n3))/2

25

6. Komu (Cloud): Area between the Leading SpanA and the Leading Span B form the Cloud

Ichimoku uses three key time periods for it’s input pa-rameters: Typically, n1 = 7, n2 = 22, and n3 = 44. Wewill keep the n1 parameter fixed at 7 but vary the other2 parameters.

7.b Ichimoku Kinko Hyo Strategies

The Kijun Sen Cross strategy is one of the most pow-erful and reliable trading strategies within the Ichimokusystem due to the fact that it can be used on nearly alltime frames with exceptional results [41].

Decision Condition

Buy Kijun Sen crosses the closing price curve fromthe bottom up

Sell Kijun Sen crosses the closing price curve fromthe top down

Hold otherwise

Table VI: Ichimoku Kijun Sen Cross strategy

The buy and sell (bullish and bearish) signals are clas-sified into three major classifications: strong, neutral andweak. The strength is determined by where the crossoveroccurs in relation to the cloud.

Figure 13: Ichimoku Kijun Sen Cross strategy trading ruleimplemented on 9 months of Anglo American PLC closingprices from 01-01-2007 to 30-09-2007. The log of the closingprice is used for illustrative purposes. Green and red candlesticks refer to bullish and bearish trading days respectively.When the Leading span 1 is greater than the Leading span2, the outlook is bullish (green cloud) and vice versa for abearish outlook (red cloud). The number next to each of thevertical green and red lines indicates the strength of the buyand sell signals respectively.

C.8 Momentum

8.a Momentum Indicator: MOMt(n)

Momentum gives the change in the closing price over thepast n periods

MOMt(n) = P ct − P ct−n (32)

C.9 Momentum Trading Rule

Decision Condition

Buy MOMt−1(n) ≤ EMAt(MOMt(n), λ) &MOMt(n) > EMAt(MOMt(n), λ)

Sell MOMt−1(n) ≥ EMAt(MOMt(n), λ) &MOMt(n) < EMAt(MOMt(n), λ)

Hold otherwise

Table VII: Momentum trading rule

C.10 Acceleration

10.a Acceleration Indicator: ACCt(n)

Acceleration measures the change in momentum betweentwo consecutive periods t and t− 1

ACCt(n) = MOMt(n)−MOMt−1(n) (33)

10.b Acceleration Trading Rule

Decision Condition

Buy ACCELt−1(n) + 1 ≤ 0 & ACCELt(n) + 1 > 0

Sell ACCELt−1(n) + 1 ≥ 0 & ACCELt(n) + 1 < 0

Hold otherwise

Table VIII: Acceleration trading rule

C.11 Moving Average Convergence/DivergenceOscillator

The Moving Average Convergence/Divergence(MACD) oscillator is a momentum indicator developedby Gerald Appel and attempts to determine whethertraders are accumulating stocks or distributing stocks.It is calculated by computing the difference between ashort-term and a long-term moving average. The idea isthen to compute the signal line, which is accomplishedby taking an exponential moving average of the MACDdetermines instances at which to buy (oversold) and sell

26

(oversold) when used in conjunction with the MACD[11].

11.a MACD Indicators: MACDt(n1, n2)

The MACD indicator is computed using the followingsteps:

1. LongEMAt = EMAt(Pc, n2)

2. ShortEMAt = EMAt(Pc, n1)

3. MACDt(n1, n2) = ShortEMAt − LongEMAt

4. SignalLinet(n1, n2, n3) =EMAt(MACDt(n2, n1), n3)

5. MACDSt(n1, n2, n3) = MACDt(n1, n2) −SignalLinet(n1, n2, n3)

11.b MACD Trading Rule

Decision Condition

Buy MACDt−1(n2, n1) ≤ MACDSt(n2, n1, n3) &MACDt(n2, n1) > MACDSt(n2, n1n3)

Sell MACDt−1(n2, n1) ≥ MACDSt(n2, n1, n3) &MACDt(n2, n1) < MACDSt(n2, n1, n3)

Hold otherwise

Table IX: MACD trading rule

Figure 14: MACD trading rule implemented on 9 monthsof Anglo American PLC closing prices from 01-01-2007 to 30-08-2007. Green and red vertical lines represent buy and sellsignals respectively.

C.12 Fast Stochastics

12.a Fast Stochastic Indicators

Fast%Kt(n) :

Fast%Kt(n) =P ct − LL(n)

HH(n)− LL(n)(34)

Fast%Dt(n) :

Fast%Dt(n) = SMAt(Fast%Kt(n), 3) (35)

12.b Fast Stochastic Trading Rule

Decision Condition

Buy Fast%Kt−1(n) ≤ Fast%Dt(n) &Fast%Kt(n) > Fast%Dt(n)

Sell Fast%Kt−1(n) ≥ Fast%Dt(n) &Fast%Kt(n) < Fast%Dt(n)

Hold otherwise

Table X: Fast stochastic trading rule

C.13 Slow Stochastics

13.a Slow Stochastic Indicators

Slow%Kt(n):

Slow%Kt(n) = SMAt(Fast%Kt(n), 3) (36)

Slow%Dt(n):

Slow%Dt(n) = SMAt(Slow%Kt(n), 3) (37)

13.b Slow Stochastic Trading Rule

Decision Condition

Buy Slow%Kt−1(n) ≤ Slow%Dt(3) &Slow%Kt(n) > Slow%Dt(3)

Sell Slow%Kt−1(n) ≥ Slow%Dt(3) &Slow%Kt(n) < Slow%Dt(3)

Hold otherwise

Table XI: Slow stochastic trading rule

C.14 Relative Strength Index

Relative Strength Index (RSI) compares the days thatstock prices finish up (closing price higher than the pre-vious day) against those periods that stock prices finishdown (closing price lower than the previous day) [9].

14.a RSI Indicator: RSIt(n)

RSIt(n) = 100− 100

1 + SMAt(Pupn ,n1)

SMAt(P dwnn ,n1)

(38)

27

where [9]

P upt =

P ct if P ct−1 < P ctNaN otherwise

(39)

P dwnt =

P ct if P ct−1 > P ctNaN otherwise

(40)

and

P upn = (P up

t−n, Pupt−n+1, . . . , P

upt ) (41)

P dwnn = (P dwn

t−n , Pdwnt−n+1, . . . , P

dwnt ) (42)

14.b RSI Trading Rule

Decision Condition

Buy RSIt−1(n) ≤ 30 & RSIt(n) > 30

Sell RSIt−1(n) ≥ 70 & RSIt(n) < 70

Hold otherwise

Table XII: Relative strength index (RSI) trading rule

Figure 15: RSI trading rule implemented on 9 months ofAnglo American PLC closing prices from 01-01-2007 to 30-08-2007. Green and red vertical lines represent buy and sellsignals respectively.

C.15 Moving Average Relative Strength Index

Moving Average Relative Strength Index (MARSI) isan indicator that smooths out the action of RSI indicator[42].

15.a MARSI Indicator: MARSIt(n1, n2)

MARSI is calculated by simply taking an n2-period SMAof the RSI indicator.

MARSIt(n1, n2) = SMA(RSIt(n1), n2) (43)

15.b MARSI Trading Rule

Rather buying or selling when RSI crosses upper andlower thresholds (30 and 70) as in the RSI trading ruleabove, buy and sell signals are generated when the SMAof the MARSI crosses above or below the thresholds [42].

Decision Condition

Buy MARSIt−1(n) ≤ 30 & MARSIt(n) > 30

Sell MARSIt−1(n) ≥ 70 & MARSIt(n) < 70

Hold otherwise

Table XIII: MARSI trading rule

C.16 Bollinger Band

Bollinger bands uses a SMA (Bollmt (n)) as it’s referencepoint (known as the median band) with regards to the

upper and lower Bollinger bands (Bollut (n) and Bolldt (n)respectively) which are calculated as functions of stan-dard deviations (s). When the closing price crosses above(below) the upper (lower) Bollinger band, it is a sign thatthe market is overbought (oversold).

16.a Bollinger Band Indicator: Bollmt (n)

Bollmt (n) = SMAct(n)

Upper Bollinger band: Bollmt (n) + sσ2t (n)

Lower Bollinger band: Bollmt (n)− sσ2t (n)

(44)

where s is chosen to be 2.

16.b Bollinger Trading Rule

Decision Condition

Buy P ct−1 ≥ Bolldt (n) & P c

t ≥ Bollut (n)

Sell P ct−1 ≤ Bolldt (n) & P c

t > Bollut (n)

Hold otherwise

Table XIV: Bollinger trading rule

28

C.17 Price Rate-Of-Change

17.a PROC Indicator: PROCt(n)

The rate of change of the time series of closing prices P ctover the last n periods expressed as a percentage

PROCt(n) = 100 ·P ct − P ct−nP ct−n

(45)

17.b PROC Trading Rule

Decision Condition

Buy PROCt−1(n) ≤ 0 & PROCt(n) > 0

Sell PROCt−1(n) ≥ 0 & PROCt(n) < 0

Hold otherwise

Table XV: Price rate of change (PROC) trading rule

C.18 Williams %R

18.a Williams %R: Willt(n)

Williams Percent Range (Williams %R) is calculated sim-ilarly to the fast stochastic oscillator and shows the levelof the close relative to the highest high in the last n pe-riods

Willt(n) =HH(n)− P ct

HH(n)− LL(n)· (−100) (46)

18.b Williams %R Trading Rule

Decision Condition

Buy Willt−1(n) ≥ −20 & Willt(n) < −80

Sell Willt−1(n) ≤ −20 & Willt(n) > −80

Hold otherwise

Table XVI: Williams %R trading rule

C.19 Parabolic SAR

Parabolic Stop and Reverse (SAR), developed by J.Wells Wilder, is a trend indicator formed by a parabolicline made up of dots at each time step [43]. The dots areformed using the most recent Extreme Price and an ac-celeration factor (AF), 0.02, which increases each time anew Extreme Price (EP) is reached. The AF has a maxi-mum value of 0.2 to prevent it from getting too large. Ex-treme Price represents the highest (lowest) value reachedby the price in the current up-trend (down-trend). The

acceleration factor determines where in relation to theprice the parabolic line will appear by increasing by thevalue of the AF each time a new EP is observed and thusaffects the rate of change of the Parabolic SAR.

19.a SAR Indicator: SAR(n)

The steps involved in calculating the SAR indicator areas follows:

1. initialise variables: trend is initially set to 1 (up-trend), EP to zero, AF AF0 to 0.02, SAR0 to theclosing price at time zero, lastHigh to high price attime zero (Ph0 ) and lastLow to the low price at timezero (P l0))

2. update parameters: EP, lastHigh, lastLow and AFbased on where the current high is in relation tothe lastHigh (up-trend) or where the current low isin relation to the lastLow (down-trend)

3. compute the next period SAR value: update timet+ 1 SAR value, SARt+1, using Eq. (47)

4. modify the SAR value and the parameters for achange in trend: modify the SARt+1 value, AF,EP lastLow, lastHigh and the trend based on thetrend and it’s value in relation to the current lowP lt and current high Ph0

5. go to next time period and return to step 2

Below is the formula for the Parabolic SAR for time t+1calculated using the previous value at time t:

SARt+1 = SARt + α(EP− SARt) (47)

19.b SAR Trading Rule

Decision Condition

Buy SARt−1 ≥ P ct−1 & SARt < P c

t

Sell SARt−1 ≤ P ct−1 & SARt > P c

t

Hold otherwise

Table XVII: SAR trading rule

D Trend Following and Contrarian MeanReversion Strategies

The zero-cost BCRP (trend following), zero-cost anti-BCRP and zero-cost anti-correlation (both contrarianmean reverting) algorithms are explained in more detailin the following subsections.

29

D.1 Zero-Cost BCRP

Zero-cost BCRP is the zero-cost long/short version ofthe BCRP strategy and is a trend following algorithmin that long positions are taken in stocks during upwardtrends while short positions are taken during downwardtrends. The idea is to first find the portfolio controlsthat maximize the expected utility of wealth using all in-sample price relatives according to a given constant levelof risk aversion. The resulting portfolio equation is whatis known to be the Mutual Fund Separation Theorem[44]. The second set of portfolio controls in the mutualfund separation theorem (active controls) is what we willuse as the set of controls for the zero-cost BCRP strategyand are given by:

b =1

γΣ−1

[E[R]− 1

1ᵀΣ−1 E[R]

1ᵀΣ−11

](48)

where Σ−1 is the inverse of the covariance matrix of re-turns for all m stocks, E[R] is vector of expected returnsof the stocks, 1 is a vector of ones of length m and γ isthe risk aversion parameter. Eq. (48) is the risky Op-timal Tactical Portfolio (OTP) that takes optimal riskybets given the risk aversion γ. The risk aversion is se-lected during each period t such that the controls areunit leverage and hence

∑mi=1 |bi| = 1.

The covariance matrix and expected returns for eachperiod t are computed using the set of price relatives fromtoday back ` days (short-term look-back parameter).

D.2 Zero-Cost Anti-BCRP

Zero-cost anti-BCRP is exactly the same as zero-costBCRP except that we reverse the sign of the expectedreturns vector such that E[R] = −E[R].

D.3 Zero-Cost Anti-Correlation

Zero-cost anti-correlation (Z-Anticor) is an adaptedversion of the Anticor algorithm developed in [45]. Thefirst step is to extract the price relatives for the two mostrecent sequential windows each of length `. Let µ`2 andµ`1 denote the average log-returns of the ` price relatives

in the most recent window (xt−`+1t ) and the price rela-

tives in the window prior to that (xt−2`+1t−` ) respectively.

Also, let the lagged covariance matrix and lagged corre-lation matrix be defined as follows:

Σ` =1

`− 1[(xt−2`+1

t−` − 1)− 1ᵀµ`1]ᵀ[(xt−`+1t − 1)− 1ᵀµ`2]

(49)

P`ij =Σ`ij√Σ`ijΣ

`ij

(50)

Z-Anticor then computes the claim that each pair ofstocks have on one another, denoted claim`

i→j , which isthe claim of stock j on stock i. This is the extent towhich we want to shift our allocation from stock i tostock j [45]. claim`

i→j exists and is thus non-zero if andonly if µ2 > µ1 and Pij > 0. The claim is then calculatedas

claim`i→j = P`ij + max(−P`ii, 0) + max(−P`jj , 0) (51)

The adaptation we propose for long/short portfoliosfor the amount of transfer that take places from stock ito stock j is given by:

transfer`i→j =1

3claim`

i→j (52)

Finally, we calculate the expert control for the ith stockin period t+ 1 as follows:

hnt+1(i) = hnt (i) +∑i

[transfer`j→i − transfer`i→j ] (53)

Each control is then normalised in order to ensure unitleverage on the set of controls.

E Data Related Appendices

E.1 JSE TOP 40 Sector Constituents: DailyData

The daily data is sourced from Thomson Reuters.The companies and their associated Reuters InstrumentCode (RIC) of the three major sectors in the JSE Top40 are [52]:

Resources: JSE-RESI (J210)Anglo American Platinum Ltd (AMSJ.J), Anglo Ameri-can PLC (AGLJ.J), AngloGold Ashanti Ltd (ANGJ.J),BHP Billiton PLC (BILJ.J), Mondi Ltd (MNDJ.J),Mondi PLC (MNPJ.J), Sasol Ltd (SOLJ.J).

Industrials: JSE-INDI (J211)Aspen Pharmacare Holdings Ltd (APNJ.J), BidvestGroup Ltd (BVTJ.J), British American TobaccoPLC (BTIJ.J), Compagnie Financiere RichemontSA (CFRJ.J), Capital & Counties Properties PLC(CCOJ.J), Growthpoint Properties Ltd (GRTJ.J), IntuProperties PLC (ITUJ.J), Mediclinic International Ltd(MEIJ.J), MTN Group Ltd (MTNJ.J), Naspers Ltd(NPNJ.J), Remgro Ltd (REMJ.J), Redifine PropertiesLtd (RDFJ.J), SABMiller PLC (SABJ.J), ShopriteHoldings Ltd (SHPJ.J), Steinhoff International Holdings(SNHJ.J), Tiger Brands Ltd (TBSJ.J), Vodacom Group

30

Ltd (VODJ.J), Woolworths Holdings Ltd (WHLJ.J), MrPrice Group Ltd (MRPJ.J), Netcare Ltd (NTCJ.J).

Financials: JSE-FINI (J212)Discovery Holdings Ltd (DSYJ.J), Firstrand Ltd(FSRJ.J), Investec Ltd (INLJ.J), Investec PLC(INPJ.J), Nedbank Group Ltd (NEDJ.J), Old Mu-tual PLC (OMLJ.J), RMB Holdings Ltd (RMHJ.J),Rand Merchant Investment Holdings Ltd (RMIJ.J),Sanlam Ltd (SLMJ.J), Standard Bank Group Ltd(SBKJ.J), Brait SE (BATJ.J), Barclays Africa GroupLtd (BGAJ.J), Capitec Bank Holdings Ltd (CPIJ.J),Fortress REIT Ltd (B) (FFBJ.J), Fortress REIT Ltd(A) (FFAJ.J), Reinet Investments SCA (REIJ.J).

E.2 JSE TOP 40 Sector Constituents: IntradayData

Below are 30 of the JSE Top 40 stocks as of the 30June 2018. All the tick and daily data from 01-01-2018to 30-06-2018 is sourced from Bloomberg.Resources: JSE-RESI (J210)Anglo American PLC (AGL:SJ), AngloGold AshantiLtd (ANG:SJ), African Rainbow Minerals Ltd (ARI:SJ),BHP Billiton PLC (BIL:SJ), Exxaro Resources Ltd(EXX:SJ), Impala Platinum Holdings Ltd (IMP:SJ),

Mondi Ltd (MND:SJ), Sasol Ltd (SOL:SJ).

Industrials: JSE-INDI (J211)Aspen Pharmacare Holdings Ltd (APN:SJ), BidvestGroup Ltd (BVT:SJ), British American TobaccoPLC (BTI:SJ), Compagnie Financiere RichemontSA (CFR:SJ), Capital & Counties Properties PLC(CCO:SJ), Growthpoint Properties Ltd (GRT:SJ), IntuProperties PLC (ITU:SJ), MTN Group Ltd (MTN:SJ),Naspers Ltd (NPN:SJ), Tiger Brands Ltd (TBSJ.J),Shoprite Holdings Ltd (SHP:SJ), Vodacom Group Ltd(VOD:SJ), Woolworths Holdings Ltd (WHL:SJ).

Financials: JSE-FINI (J212)Discovery Holdings Ltd (DSY:SJ), Firstrand Ltd(FSR:SJ), Investec Ltd (INL:SJ), Nedbank Group Ltd(NED:SJ), RMB Holdings Ltd (RMH:SJ), Sanlam Ltd(SLM:SJ), Standard Bank Group Ltd (SBK:SJ), Bar-clays Africa Group Ltd (BGA:SJ).

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