DDeeply Equal-Weighted Subset Portfolios
Sang Il Lee ∗ DeepAllocation TechnologiesJune 26, 2020
Abstract
The high sensitivity of optimized portfolios to estimation errors has prevented theirpractical application. To mitigate this sensitivity, we propose a new portfolio modelcalled a Deeply Equal-Weighted Subset Portfolio (DEWSP). DEWSP is a subset oftop- N ranked assets in an asset universe, the members of which are selected based onthe predicted returns from deep learning algorithms and are equally weighted. Herein,we evaluate the performance of DEWSPs of different sizes N in comparison with theperformance of other types of portfolios such as optimized portfolios and historicallyequal-weighed subset portfolios (HEWSPs), which are subsets of top- N ranked assetsbased on the historical mean returns. We found the following advantages of DEWSPs:First, DEWSPs provides an improvement rate of 0.24% to 5.15% in terms of monthlySharpe ratio compared to the benchmark, HEWSPs. In addition, DEWSPs are builtusing a purely data-driven approach rather than relying on the efforts of experts.DEWSPs can also target the relative risk and return to the baseline of the EWP of anasset universe by adjusting the size N . Finally, the DEWSP allocation mechanism istransparent and intuitive. These advantages make DEWSP competitive in practice. Despite the significant success of deep learning, its application to stock trading remains ex-tremely challenging owing to the volatile movements of stock prices, making it difficult todefine the input values and understand how to apply the output values. Machine learningmodels are built on a training set and are tested on a disjointed test set to prove theirgeneralization capability, and are commonly applied in various applications such as imageprocessing, image recognition, speech recognition, and Internet searches. However, this ap-proach is limited when applied to the financial field owing to the time-evolving propertiesof the financial markets, for example, structural breaks at occasional time points [1, 2],volatility clustering [3], and time-varying mean returns [4]. Furthermore, the time orderingof financial data prevents the use of cross-validation as a reliable estimate of the ensemble ∗ Electronic address: [email protected] a r X i v : . [ q -f i n . P M ] J un eneralization error. As a result, the performance of financial time-series models tends tobe extremely sensitive to pre-specified periods, showing the high power of in-sample (IS)prediction and the poor power of out-of-sample (OOS) prediction [5]. This hampers thepractical use of portfolio optimization techniques because an optimization is prone to the‘garbage in, garbage out’ phenomenon, in which biases occur in a portfolio selection unlesspredictions are adjusted suitably for an estimation error. To mitigate this problem, we builta new model called a Deeply Equal-Weight Subset Portfolios (DEWSPs) that combines deeplearning techniques with an equal-weight strategy. Portfolio theory
The mean-variance portfolio (MVP) theory, pioneered by Markowitz(1952) [6], has long been recognized as the cornerstone of modern portfolio theory (MPT).It provides a mathematical framework for determining a set of portfolios with a maximizedexpected return per unit of risk; in addition, the return and risk of a set are drawn as a line,called an efficient frontier, on a risk-return plane. However, despite the theoretical advancesin portfolio models including the MVP and its extensions, their practical use remains prob-lematic owing to the difficulty in estimating reliable expected returns, which critically affectthe performance of the portfolio [7]. For example, MVPs are not necessarily well-diversified[8], portfolio optimizers are often “error maximizers” [9], and a mean–variance optimizationcan produce extreme or non-intuitive weights for some of the assets in the portfolio [10, 11].Many studies have attempted to apply improved estimation procedures and mitigate theestimation error problem. These include Bayesian methods [12, 13], shrinkage methods [14][15, 16], a factor structure imposed on the returns [17], and the combination of a tangencyportfolio, a risk-free rate, and a global minimum variance portfolio [18].
Equal-weight portfolio (EWP)
There is a growing body of evidence showing that the useof simple rules of thumb is more successful than optimization. The most well-known exampleis the EWP, also called 1 /N naive diversification, which is free of parameter uncertainty andhas the following properties: It never shorts any assets, it avoids a concentration, and upon arebalancing of the dates, it sells high and buys low, thus exploiting a possible mean-reversioneffect [19]. The strength of an EWP is well known experimentally [7, 12, 20, 21]. DeMiguelet al.’s study [20] is particularly convincing because the authors evaluated 14 models on 7empirical datasets. They found that none of the 14 models consistently outperform a 1 /N EWP. Tu and Zhou (2011) [22] showed the combination of an EWP and more sophisticatedmodes [6, 12, 17, 18] is a way to improve performance. The importance of the EW ap-proach lies in its simplicity and widespread use. In addition, Bernartzi and Thaler (2001)[23] demonstrated that EW diversification is ingrained in human behavior by finding that aconsiderable fraction of participants equally distribute their contributions across the avail-able investment opportunities. This implies that investment decisions tend to use intuitionto choose a security and do not necessarily rely on sophisticated formal techniques. Investorscan execute an EW strategy with a large universe but extremely low transaction costs usingequally weighted exchange traded funds (ETFs), for example, Direxion NASDAQ-100 EqualWeighted Index Shares, First Trust Dow 30 Equal Weight ETF, and Goldman Sachs EqualWeight U.S. Large Cap Equity ETF.
DEWSP
DEWSPs are constructed by incorporating deep learning techniques into an EW2trategy. The building procedure consists of three steps: First, the 1-month ahead return ofassets is forecasted using deep learning algorithms. Second, assets are ranked in descendingorder based on the forecasts. Finally, subset portfolios are constructed with top- N rankedassets that are equally weighted. Our contribution is as follows: • DEWSP is fully data-driven based on hyperparameter optimization. The entire pro-cess is automatic without the views of human experts in building the models, whichcontributes to reduced costs in terms of portfolio management. We also use publicdata on the prices and volume, which can be publicly obtained from various Web sites.Thus, DEWSPs are easily reproducible. • DEWSPs show an increase in their risk and return from the baseline of the EWPwith a decrease in the number of assets. This means that it is possible to controlthe aggressiveness of DEWSPs in terms of their risk-return tradeoff. This mitigatesdifficulties in understanding the black-box portfolio optimization and in tailoring therisk and return of ranked portfolios based on financial factors (e.g., based on size, value,and leverage).
Related papers
This study covers stock prediction using deep learning methods and ranked-portfolios. Deep learning models are on the rise, showing impressive results in modelingthe complex behavior of financial data. Examples include stock prediction based on longshort-term memory (LSTM) networks [24], deep portfolios based on deep autoencoders [25],threshold-based portfolios using recurrent neural networks [26], deep factor models usingdeep feed-forward networks [27], a time-varying multi-factor model using LSTM networks[28], and an enhancing standard factor model using deep learning [29].Ranked portfolios are widely used with varying degrees of complexity, and their ba-sic premise is the same: ranking stocks-based on factors such as their value, momentum,quality, size, low risk, and a combination of these factors, and then selecting a particularproportion of the top-ranked stocks to add to the portfolio. These include portfolios rankedin terms of size and book-to-market [30], portfolios ranked on value and momentum factors[31], portfolios ranked on time-series momentum [32], and portfolios ranked on binary clas-sification using returns predicted through deep learning [24].The remainder of this paper is organized as follows: In Section 2, we describe the dataand preprocessing methods applied. In Section 3, we describe the experimental setting andimplementation. In Section 4, we provide the experimental results and compare differentportfolio models. Finally, some concluding remarks are offered in section 5.
Small portfolios are considered for an easier analysis, and are important for several practicalreasons [33]: First, it is difficult for small investors to acquire and continuously monitor alarge portfolio. Second, large investors need to identify a threshold where the cost exceeds the3enefit of risk reduction from diversification. Third, large portfolios amplify the estimationerrors during the optimization process. To select a small but well-diversified universe, we referto the most commonly applied classification system, i.e., the Global Industry ClassificationStandard (GICS). The asset universe consists of 22 diversified stocks in Standard and Poor’s500 index (S&P 500) that belong to 11 different GICS sectors: • Energy : ExxonMobil (XOM) and Chevron (CVX),
Utilities : Duke Energy (DUK)and Consolidated Edison (ED),
Materials : Sherwin-Williams (SHW) and DuPont(DD),
Industrials : Boeing (BA) and Union Pacific (UNP),
Consumer Discre-tionary : Amazon (AMZN) and McDonald’s (MCD)
Consumer Staples : Coca-Cola(KO) and Procter & Gamble (PG)
Healthcare : United Health Group (UNH) andJohnson & Johnson (JNJ)
Financials : Berkshire Hathaway (BRK-B) and JPMor-gan Chase (JPM)
Information Technology Sector : Apple (AAPL) and Microsoft(MSFT),
Communication Services : Facebook (FB) and Alphabet (GOOG),
RealEstate : American Tower (AMT) and Simon Property Group (SPG).We use data from Yahoo Finance from January 1997 to October 2019, which is the com-mon period of data availability. The monthly stock dataset contains five attributes: openprice, high price, low price, adjusted close price, and volume (OHLCV). The last of the dailyOHLCV datasets per month is used as the raw dataset. For each experiment, we split thedata into an in-sample (70%) period and an out-of-sample (30%) period. The in-sample dataare divided again into a training dataset (50%) for developing the prediction models and avalidation set (50%) for evaluating its predictive ability.
Technical indicators
A technical analysis is a method for forecasting price movementsusing past prices and volume and includes a variety of forecasting techniques such as a chartanalysis, cycle analysis, and computerized technical trading systems.A technical analysis has a long history of widespread use by participants in speculativemarkets [34, 35, 36, 37, 38, 39], and there is a large body of academic evidence demonstratingthe usefulness of such analysis, including theoretical support [40] and empirical evidence[41, 42], as well as the role of such analysis in out-of-sample equity premium predictability[43, 44, 45]. We used a full set of 14 technical indicators based on 3 types of popular technicalstrategies, i.e., the moving average crossover, momentum, and volume rules: • The time-series momentum indicator, MOM( m ), is the generation of a buy signalwhen the price is higher than the historical price. Its validation is supported by theobservation that the “trend” effect persists for approximately 1 year and then partiallyreverses over a longer timeframe. Here, MOM t ( m ) at time t is defined as follows:MOM t ( m ) = (cid:40) , if P t ≥ P t − m − , otherwise . (1)where P t is the index value at time t , and m is the look-back period. We use m = 1, 3,6, 9 and 12, which are respectively labeled as MOM t (1M), MOM t (3M), MOM t (6M),MOM t (9M), and MOM t (12M). 4 The moving average indicator, MA( s, l ), provides a signal for an upward or downwardtrend. A buy signal is generated when the short-term moving average crosses above thelong-term moving average because this represents the beginning of an upward trend. Asell signal is generated when the short-term moving average falls below the long-termmoving average because this represents the beginning of a downward trend.Let us define a simple moving average of the index as follows:MA
Pj,t = (1 /j ) j − (cid:88) i =0 P t − m for j = s or l, (2)where s and l are the look-back periods for short and long moving averages. Themoving average indicator MA t ( s, l ) is then designed as follows:MA t ( s, l ) = (cid:40) , if MA Ps,t ≥ MA Pl,t − , otherwise . (3)The six moving average indicators are constructed for s =1, 2, and 3, and for l = 9 and12, which are symbolized as MA(1M-9M), MA(1M-12M), MA(2M-9M), MA(2M-12M),MA(3M-9M), and MA(3M-12M). • The volume indicator, VOL( s, l ), indicates a strong market trend if the recent stockmarket volume and stock price increase. Let us define the on-balance volume (OBV)as follows: OBV t = t (cid:88) k =1 V OL k D k , (4)where V OL k is a measure of the trading volume (i.e., number of shares traded) duringperiod k , and D k is a binary variable: D k = (cid:40) , if P k ≥ P k − − , otherwise . (5)The value of OBV t conceptionally measures both positive and negative volume basedon the belief that changes in volume can predict a stock movement. The volume-basedindicator is then defined as the difference between the moving averages with an s -periodand an l -period:VOL( s, l ) = (cid:40) , if MA OBV s,t ≥ MA OBV l,t − , otherwise . (6)Here, MA OBV j,t = (1 /j ) (cid:80) j − i =0 OBV t − i is the moving average of OBV t for j = s or l . Thesix moving average indicators are constructed for s =1, 2, and 3 and for l = 9 and12, which are symbolized as follows: VOL(1M-9M), VOL(1M-12M), VOL(2M-9M),VOL(1M-12M), VOL(3M-9M) and VOL(3M-12M).5 Frameworks
For a comparative analysis, we also built three different types of portfolios, which are distinctin terms of their optimization or estimation process. All portfolios are built on the followingassumptions: (1) all stocks are infinitely divisible; (2) there are no restrictions on the buyingor selling of any selected portfolio; (3) there is no friction (e.g., transaction costs, taxation,commissions, or liquidity); and (4) it is possible to buy and sell stocks at the closing pricesat any time t . We adapt a periodic rebalancing strategy in which the investor adjusts theweights in the investor’s portfolio at the close price on the last business day of every month. List of portfolios considered : • DEWSP: This is a subset of portfolios that consist of the top N -th ranked assets amongall N assets based on their expected returns. • EW whole portfolio (EWWP): This is a traditional EWP of all assets N , and can beviewed as a special case of DEWSP when N = N . Because there are no parameterestimations, it serves as the baseline for an evaluation of the risk and return of theDEWPs of different sizes. • Historically EW subset portfolios (HEWSPs): Like DEWSPs, HEWSPs are top-rankedsubset portfolios, although their expected returns are estimated as a historical averageover the training and validation (HEWSP-TV) and historical average over the valida-tion (HEWSP-V). This reveals the effect of the return prediction of the DEWSPs. • Randomly EW subset portfolios (REWSPs): These are subsets of portfolios consistingof N assets selected randomly, without the use of a ranking method. A comparisonbetween REWSPs and DEWSPs and HEWSPs reveals the effect of the estimatedreturn prediction. • Maximum Sharpe ratio portfolios (MSRPs): These are complete portfolios that aremaximized to achieve the highest Sharpe ratio, and are mathematically defined asfollows: max w t w Tt µ t / (cid:112) w Tt Σ t w t s.t. w Tt = 1 , and w i,t ≥ , ∀ i, (7)where µ t is a vector of N predicted returns, w t = ( w ,t , . . . , w N ,t ) T is a vector ofportfolio weights, Σ t is a covariance matrix of the asset returns, N = (1 , . . . , T is an N -dimensional vector, and w Tt µ t and w Tt (cid:80) t w t are the portfolio return and variance,respectively. Because µ t and Σ t are unknown in practice, we replace them with (cid:98) µ t from deep learning algorithms and (cid:98) Σ t from an in-sample dataset. A comparison withDEWSPs reveals the effect of the estimation error on a portfolio optimization. • Minimum variance portfolios (MVPs): These are complete portfolios optimized for thelowest volatility, and solve the following constrained minimization problem:min w w Tt Σ t w t s.t. w Tt = 1 , and w i,t ≥ , ∀ i. (8)6 comparison with DEWSPs reveals the effectiveness of optimization under the con-dition of no estimation errors. A multilayer feedforward neural network (FFNN) was used in this study. We used Tree-structured Parzen Estimator (TPE) approach [46] for automated hyperparameter tuningand Table 1 presents the list of hyperparameters and their values. Each optimization runwas initialized with randomly selected points, after which it proceeded sequentially for atotal of 50 function evaluations. During one evaluation run, the FFNN was trained over anin-sample training data. The mean squared error (MSE) is calculated on a validation setper function evaluation, early stopping was applied when there is no improvement on thevalidation accuracy after 10 continuous epochs.
We used two popular regularization methods, i.e., a dropout and batch normalization (BN).A dropout [47] is a simple way to prevent co-adaptation among hidden nodes of a deepfeed-forward neural network by dropping out randomly selected hidden nodes. In recentyears, BN [48] has replaced a dropout in modern neural network architectures, and uses thedistribution of the summed input to a specific neuron over a mini-batch of training casesto compute the mean and variance, which are then used to normalize the summed input tothat neuron for each training case. A dropout and BN layers were employed for all hiddenlayers.
Average percent change (APC)
The APC measures the rate of change in a DEWSPreturn and the volatility as size N increases from N = 1 to N = N to see the rate of changefrom the baseline of N = N to N = 1, and is defined as follows:APC x = 1 N − N − (cid:88) N =1 x N − x N +1 x N +1 , (9)where x is r t or σ t . Average Sharpe ratio improvement rate (ASRIR)
ASRIR measures the relative im-provement of the DEWSPs as compared to the HEWSP benchmark in terms of the Sharperatio (SR), and is defined as follows:ASRIR = 1 N N (cid:88) N =1 x N DEWSP − x N HEWSP-TV/T x N HEWSP-TV/T , (10)where x is the SR of DEWSPs and HEWSPs of the same size N .7able 1: List of hyperparameters and range of each hyperparameter.Hyperparamter Considered values/functionsNumber of Hidden Layers {
2, 3 } Number of Hidden Units {
2, 4, 8, 16 } Standard deviation { } Dropout { } Batch Size {
28, 64, 128 } Optimizer { RMSProp, ADAM, SGD (no momentum) } Activation Function Hidden layer: { tanh, ReLU, sigmoid } , Output layer: LinearLearning Rate { } Number of Epochs { } Number of layers : number of layers of a neural network.
Number of hidden units : numberof units in the hidden layers of a neural network.
Standard deviation : standard deviationof a random normal initializer.
Dropout : dropout rates.
Bath size : number of samples perbatch.
Activation : sigmoid function σ ( z ) = 1 / (1 + e − z ), hyperbolic tangent function tanh( z ) =( e z − e − z ) / ( e z − e − z ), and rectified linear unit (ReLU) function ReLU( z ) = max(0 , z ). LearningRate : learning rate of the back-propagation algorithm.
The Number of Epochs : number ofiterations over all training data.
Optimizer : stochastic gradient descent (SGD) [49], RMSProp[50], and ADAM [49]
We examined the portfolio performance over both IS and OOS periods for three differentuniverses: a total of 22 stocks (Exp. I), with 11 stocks consisting of the first stocks of eachsector on the list (Exp. II), and the other 11 stocks (Exp. III). The following observationwas made based on the empirical simulation results. • The left side of Figure 1 graphically shows the realized risk and return points of theportfolios on the risk-return plane. Each color represents a different type of portfolio,and different points with the same color represent different sizes. A comparison ofDEWSPs and HEWSPs with the REWSPs of a (seemingly) random pattern indicatesthat the prediction-based ranking assets can be used to construct portfolios with in-creasing return and volatility as N decreases. In Table 2, APC r s and APC σ s indicatequantitative measurements of the increase over Exp. I, II, and III, and APC r / APC σ shows the degree of trade-off between the return and risk. • We also found ASRIRs of 21 .
15, 27 .
04, and 13 .
09% for Exp. I, II, and III, respectively,indicating the superiority of DEWSP during the IS period. • MVP, as expected, achieves the least volatility of 0 .
99, and MSRP achieves the high-est Sharpe ratio of 0.65 ( µ = 0.026 and σ = 0.040), which outperform those of the8 .03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 Risk (Realized volatility) R e a li z e d r e t u r n DEWPHEWP-THEWP-TVRandPy=0.51xMSRPMVP 0.03 0.04 0.05 0.06 0.07
Risk (Realized volatility) R e a li z e d r e t u r n DEWPHEWP-THEWP-TVRandPy=0.455xMSRPMVP
Figure 1: Realized risk vs. return of six different types of portfolios for the in-sample (left)and out-of-sample (right) experiments. The dotted lines specify the maximum SR estimate.Table 2: Performance evaluation results of DEWSPs over in-sample period.Metrics (%) Exp. I Exp. II Exp. IIIAPC r σ r / APC σ • The DEWSPs are built using 1-month ahead predicted returns from the trained model,and the HEWSPs are built using the average historical return over the in-sample period. • The computation results are summarized on the right side of Figure 1 and in Table3. As with the IS experiment, the return and volatility of the DEWSPs and HEWSPsstill show an increasing pattern with the positive APC values. This allows us to tailorthe portfolio’s return and risk for investment purposes.Table 3: Performance evaluation results of DEWSPs over the out-of-sample period.Metrics (%) Exp. I Exp. II Exp. IIIAPC r σ r / APC σ The ASRIRs ranged from 0.24 to 5.15% indicate that the DEWSPs outperform thehistorical models in terms of the monthly SR. The values are small compared to those ofthe in-sample ASRIRs, but indicate promising results. First, we can beat the HEWSPbenchmark, and second, we can tailor the return and volatility of the portfolios relativeto the baseline of the EWWP. • Although the MVP without a parameter estimation still gives the least volatility at0 . .
44% ( µ = 0 .
014 and σ = 0 . Despite the significant success of machine learning in numerous fields, stock prediction isstill severely limited owing to its seasonal, non-stationary, and unpredictable nature. Con-sequently, portfolio models are inevitably exposed to the risk of estimation errors, whichhinders their performance.To cope with such risk, we have proposed a new DEWSP model by incorporating deep-learning-based predictions into the framework of the EW strategy. We empirically demon-strated that DEWSPs can be used to target the levels of portfolio return and risk relativeto the baseline of the EWWPs by adjusting the number of assets, and that its mechanismis clear in terms of the risk-return trade-off. We also showed that DEWSPs are superior toHEWSPs in terms of the SR and that the mean-variance optimization amplifies the estima-tion error dramatically, which results in a substantially worse Sharpe ratio. To summarize,DEWSPs are attractive from an implementation perspective, i.e., the use of public stockdata, a transparent mechanism based on a risk-return trade-off, automatic hyperparameteroptimization, the existence of a baseline of the EWP and a benchmark of the HEWSP, thecapability of building portfolios using small numbers of assets (with expandability to largeassets), and a simple incorporation of deep learning algorithms into the portfolio scheme.
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