# portfolio_selection_via_subset_resampling__a310af33.pdf

Portfolio Selection via Subset Resampling

Weiwei Shen, , Jun Wang

School of Computer Science and Software Engineering East China Normal University, Shanghai, China GE Global Research Center, Niskayuna, NY, USA, realsww@gmail.com, wongjun@gmail.com

As the cornerstone of the modern portfolio theory, Markowitz s mean-variance optimization is a major model adopted in portfolio management. However, the estimation errors in its input parameters substantially deteriorate its performance in practice. Specifically, loss could be huge when the number of assets for investment is not much smaller than the sample size of historical data. To hasten the applicability of Markowitz s portfolio optimization to large portfolios, in this paper, we propose a new portfolio strategy via subset resampling. Through resampling subsets of the original large universe of assets, we construct the associated subset portfolios with more accurately estimated parameters without requiring additional data. By aggregating a number of constructed subset portfolios, we attain a well-diversified portfolio of all assets. To investigate its performance, we first analyze its corresponding efficient frontiers by simulation, provide analysis on the hyperparameter selection, and then empirically compare its out-of-sample performance with those of various competing strategies on diversified datasets. Experimental results corroborate that the proposed portfolio strategy has marked superiority in extensive evaluation criteria.

1 Introduction Portfolio selection has taken on increasing significance in finance for managing a wide range of assets, such as mutual funds, pension funds and university endowments (Brandt 2010). After more than half a century since the seminal work by (Markowitz 1952), the mean-variance framework remains prevalent and represents the most broadly chosen approach in both industry and academia for portfolio selection (Kolm, T ut unc u, and Fabozzi 2014). Its popularity has called for research on complete comprehension and practical implementation. Briefly, the mean-variance framework formulates the risk-return tradeoff of assets via the means and the covariance matrix of asset returns. It intuitively implies that among available portfolios of assets that can achieve a return target, investors should choose the portfolio with the lowest volatility. However, the classical mean-variance portfolio often performs poorly out of sample, as its input parameters, the first two moments of asset returns, are difficult to estimate accurately (Best and Grauer 1991). Moreover, such a situation

Copyright c 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.

worsens when historical return data are limited. In particular, realized loss could be unbounded when the number of assets is larger than the sample size of data (Tu and Zhou 2011). On the other hand, stationarity of the model parameters is another main concern (Broadie 1993). Return data from thirty years ago might have little bearing on returns this year, so parameters are unlikely to be stationary over a long period of time. While the estimation errors decrease when more data are used, more data generally require a longer time horizon. The dilemma between stationarity of parameters and estimation errors exaggerates the challenges in dealing with large portfolios. One well-known attempt to overcome the estimation problem without demanding more data is the resampled efficient portfolio proposed in (Michaud 1989). Specifically, the basic concept of Michaud s resampled efficient portfolio comprises of generating resamples of asset returns by a parametric bootstrap procedure, computing the mean-variance portfolio for each resample, and finally averaging over the obtained portfolios to account for the parameter uncertainty. It aims to lessen the impact of estimation risk on portfolio weights and to obtain a more balanced asset allocation, thereby improving the portfolio performance (Michaud and Michaud 2008). However, due to its mixed and ambiguous testing results, its effectiveness has been continually called into question (Markowitz and Usmen 2003; Harvey et al. 2010; Wolf 2013). Mostly, its generalization performance shows limited or no improvement over Markowitz s mean-variance portfolio (Scherer 2002). While the development of new resampling based portfolio strategies has relatively stagnated since Michaud s work (Becker, G urtler, and Hibbeln 2015), ensemble methods have achieved remarkable success in advancing performance of existing algorithms and meanwhile enriching the methodology on its own (Dietterich 2000; Zhou 2012). As the resampled efficient portfolio could be viewed as an application of the bootstrap aggregating algorithm (Bagging) in order to enhance stability and mitigate variance (Breiman 1996), exploring new ensemble methods and leveraging them into portfolio selection problems deserves a fresh attempt. The illuminating papers by (Ho 1998) and (Kleiner et al. 2014) have provided a pathway to applying new ensemble methods into portfolio problems. The former presents the random subspace method (RSM) in a classification context.

Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence (AAAI-17)

Weak classifiers are constructed in random subspaces of the data feature space. Those classifiers are subsequently combined by simple majority voting in the final decision rule. The latter proposes a method called Bag of Little Bootstraps (BLB) for massive data to assess the quality of estimators. BLB first randomly selects subsets of the data, and then performs a bootstrap on each subset by constructing weighted resamples of the subset. Such two methods respectively show how subset resampling attacks the challenges embedded in the two sides of a typical data matrix, i.e., the small sample problem when data are high-dimensional and the big data problem when data are massive. Inspired by the aforementioned two subset resampling based methods, to facilitate the applicability of Markowitz s portfolio optimization to large portfolios, in this paper, we propose a new portfolio selection strategy via subset resampling. In particular, we focus on the problem when the number of assets is large and the number of return data is critical. We resample subsets of the whole portfolio so that the sizes of data for estimating the corresponding covariance matrices have been relatively increased. We then can construct the associated subset optimal portfolios with more accurate covariances. Finally, we attain a well-diversified portfolio of all assets through aggregating a number of prolonged subset optimal portfolios. We offer analysis on the hyperparameter selection and conduct extensive empirical comparisons over ten peer strategies on four representative real-world datasets. The experimental results lucidly demonstrate the superiority of the proposed portfolio strategy.

2 Background and Related Work In this section, we review portfolio selection problems with emphasis on the mean-variance portfolio, and then rehash motivating ensemble methods in machine learning. The mean-variance portfolio assumes that investment decisions on getting a diversified portfolio mainly depend on the means and the covariances of asset returns. In practice, investors need to estimate both input parameters and plug them into an optimization routine to get estimated optimal portfolio weights. However, as the estimation errors in parameters are amplified by optimization and then propagate into the solution of the optimization, extreme portfolio weights and a lack of diversification are commonly observed. This phenomenon that has eventually ruined the out-of-sample performance of Markowitz s portfolio is coined as error maximization by (Michaud 1989). Tremendous efforts have been expended to handle the estimation risk on the parameter uncertainty. Among them, as the return means are extremely difficult to estimate accurately (Merton 1980), (Scherer 2011) emphasizes that without incorporating the expected return the yield minimum-variance portfolio often performs better out of sample. (Jorin 1986) suggests using the Bayes-Stein shrinkage estimator for the return mean estimation. (Ledoit and Wolf 2004) propose a robust and effective shrinkage estimator for the covariance matrix estimation. (Fan, Zhang, and Yu 2012) and (Shen, Wang, and Ma 2014) show superior portfolio performances when various types of norm regularities are incorporated into the mean-variance framework. (Kan and Zhou 2007) and (Tu

and Zhou 2011) propose three-fund and four-fund blending portfolios to further improve the models based on the Bayes Stein shrinkage estimator, respectively. More comprehensive reviews may be referred to (Brandt 2010). Meanwhile, the impressive records of applying machine learning algorithms into numerous regimes spark their applications and adoptions in finance. Specifically, over years machine learning researchers have made significant contributions in designing portfolio selection strategies from many novel aspects. Among them, (Cover 1991) and (Blum and Kalai 1999) propose and analyze the constant-rebalanced portfolio with or without transaction costs. (Borodin, El Yaniv, and Gogan 2004) consider learning the best asset by exploiting the market volatility and the statistical relationship between assets. (Agarwal et al. 2006) use a Newton step based method to compute the portfolio for the next iteration in the universal portfolio context. (Li et al. 2012) focus on a market timing portfolio of determining a passive or an aggressive trading strategy. (Shen et al. 2015) and (Shen and Wang 2016) propose to employ the bandit learning framework to attack portfolio problems. Illustration over a wide range of portfolio strategies from machine learning may be found in the survey by (Li and Hoi 2014). On the other hand, ensemble methods in machine learning virtually share the same theme as portfolio selection: namely, diversification (Derbeko, El-Yaniv, and Meir 2002; Zhou 2012). The level of diversity of weak learners determines the generalization quality of the aggregated learner. Similar to making investment decisions in assets, if we had access to a learner with perfect generalization performance, then there would be no necessity to appeal to ensemble techniques (Dietterich 2000; Rokach 2010). Among ensemble methods, bagging and the random subspace method are particularly efficacious in improving weak learners when training sample sizes are small (Polikar 2006). Also, it is known in machine learning that bagging is useless for learners having a decreasing learning curve, i.e., that the generalization error of the base learner decreases with an increase in the training sample size (Skurichina and Duin 2002). In addition, (Karoui and Purdom 2016) recently show that the bootstrap method cannot improve the estimation accuracy of the covariance matrix when the sample size is critical. Thus, it is unsurprising to see the comments made by researchers in finance on the ineffectiveness of Michaud s bagging based resampled efficient portfolio (Scherer 2002; Harvey et al. 2010). In contrast, the random subspace method is known to be instrumental in weak learners obtained on small and critical training sample sizes when the learning curve is decreasing. It bolsters the performance of learners which suffer from the curse of dimensionality (Ho 1998; Skurichina and Duin 2002). Hence, to overcome the error maximization in portfolio selection due to a small training sample size, we resort to investigating algorithms based on subset resampling rather than bootstrapping.

3 Methodology In this section, we first introduce the notations and finance terms used in this paper. Then we describe the proposed portfolio selection strategy via subset resampling. Finally we

discuss the properties and behaviors of the new method in constructing efficient frontiers by a simulation study.

Notation In a frictionless, self-financing, discrete-time and finite horizon investment environment, we denote a series of trading periods as tk = kΔt, k = 0, . . . , m, where Δt represents one day, one week or one month, depending on the rebalance interval. For simplicity, we use k for short as the index to indicate the trading period at time tk hereafter. From time tk 1 to tk the gross return vector of n risky assets accessible to investors is Rk = (Rk,1, . . . , Rk,n) . The gross return Rk,i for the i-th asset is computed as Rk,i = Sk,i/Sk 1,i, where Sk,i and Sk 1,i represent the prices of the i-th asset at time tk and tk 1, respectively. Let us respectively denote by μk and Σk the vector of the means and the covariance matrix of the n asset returns at time tk. Denote ωk = (ωk,1, . . . , ωk,n) as the vector of the portfolio weights reflecting the investment decision at time tk. The i-th element of ωk specifies the invested percentage of wealth in the i-th asset. The sum of the portfolio weights equals one, i.e., ω k 1 = 1, where 1 stands for the n 1 vector of ones. ωk,i > 0 means that investors take a long position of the i-th asset. In contrast, ωk,i < 0 indicates a short sale of the i-th asset, where investors liquidate the borrowed i-th asset for investment in other assets. If the price of the borrowed asset bounces back, investors, who need to buy back and return the borrowed asset, will suffer from a loss. The maximum loss for a long position is the total amount of invested wealth and the maximum loss of a short sale position could theoretically be infinite. Given a gross return Rk and a portfolio weight ωk 1, the realized portfolio net return rk from time tk 1 to tk is computed as rk = R k ωk 1 1.

Subset Resampling Portfolio One common formulation of the classical long-short meanvariance optimization can be written as

min ω k 1=1 ω k Σkωk s.t. μ k ωk Rk, (1)

where investors attempt to minimize the risk represented by the total variance of the portfolio while achieving a return target Rk. In practice, both the covariance matrix Σk and the means of returns μk are unknown, thereby requiring estimates. In our study, we apply a subset resampling method to reduce the impact of the estimation errors for large portfolios with a short history of return data. The subset resampling portfolio (SSR) presented in Algorithm 1 is straightforward to implement. At time tτ, let us assume the available historical asset returns are {Rk}τ k=1 with τ n and the return target of investors is Rτ. Instead of taking bootstrap samples of the returns of all assets and computing the estimated optimal portfolio weights for all assets simultaneously as (Michaud and Michaud 2008), SSR averages over multiple estimated optimal weights for small portfolios. More formally, given a subset size b < n, SSR uniformly at random samples s subsets of size b from the original n assets. Denote by Ij {1, . . . , n} the corresponding index set with |Ij| = b for j = 1, . . . , s. Denote by

Algorithm 1 Subset Resampling Portfolio

1: Inputs: τ: number of periods for estimation; {Rk}τ k=1: historical return data; Rτ+1: one out of sample return; n: number of assets; b: subset size; s: number of sampled subsets; Rτ: return target; 2: for j = 1 s do 3: Randomly sample a set Ij of b indices from {1, . . . , n} without replacement; 4: Select the associated return data as {Rj,b k }τ k=1;

5: Compute the sample covariance matrix as ˆΣ j,b τ ; 6: Compute the sample means of returns as ˆμj,b τ ; 7: Compute the optimal subset portfolio weights ˆωj,b τ by solving the mean-variance optimization (1) based on

the estimated parameters ˆΣ j,b τ and ˆμj,b τ ; 8: Construct the weights for the whole portfolio ˆωj τ: 9: for i = 1 n do 10: ˆωj τ,i = ˆωj,b τ,i I{i Ij}; 11: Aggregate the constructed portfolio weights based on s resamples as ˆωτ = s 1 s j=1 ˆωj τ; 12: Compute the realized out-of-sample portfolio net return ˆrτ+1 = R τ+1 ˆωτ 1; 13: Outputs: The vector of portfolio weights ˆωτ and the realized out-of-sample portfolio return ˆrτ+1.

{Rj,b k }τ k=1 the associated return data for subset j. First, for each subset, SSR calculates the associated b b sample co-

variance matrix as ˆΣ j,b τ and the b 1 sample means of returns as ˆμj,b τ . Second, SSR computes the optimal subset portfolio weights ˆωj,b τ by solving the mean-variance optimization (1) based on the estimated mean ˆμj,b τ and covariance matrix ˆΣ j,b τ . Third, SSR averages over portfolio weights from all the subsets to generate the weights for the whole portfolio as ˆωτ = s 1 s j=1 ˆωj τ, where ˆωj τ = (ˆωj τ,1, . . . , ˆωj τ,n) is the prolonged n 1 vector with ˆωj τ,i = ˆωj,b τ,i I{i Ij} for i = 1, . . . , n, and the symbol I( ) stands for an indicator function. Accordingly, the realized out-of-sample portfolio net return ˆrτ+1 from time tτ to tτ+1 is ˆrτ+1 = R τ+1 ˆωτ 1.

Discussions SSR partially sacrifices diversification benefits to alleviate the estimation risk on the parameter uncertainty. The pivotal hyperparameters that determine its performance characteristics are the subset size b and the number of subsets s. To enjoy the advantages in mitigating the estimation errors, we need to choose b < min(n, τ). However, a tradeoff exists when we determine b. The smaller the subset size is, the more accurate the estimation of the b b sample covariance matrix would be, yet the more diversification benefits would be lost, and vice versa. On the other hand, another tradeoff between computational costs and diversification benefits exists when we determine the number of subsets s. In principle, a large number of subsets would surely be preferred. However, that could increase the computational burden to be infeasible.

While general results for the optimal hyperparameters are unavailable as they would heavily rely on the underlying return dynamics of assets, our following study offers some insights and suggestions. In Figure 1, we investigate the empirical performance characteristics of SSR with respect to the two hyperparameters in constructing efficient frontiers by simulation. Briefly, an efficient frontier plots the best risk-return tradeoff curve that a set of assets could possibly achieve by following one particular strategy. An efficient frontier dominates another when the former lies on the upper left of the latter. Following (Broadie 1993), three types of efficient frontiers are illustrated and compared: estimated, true and actual efficient frontiers. An estimated efficient frontier represents in-sample performance of a portfolio strategy, an actual efficient frontier shows outof-sample performance, and a true efficient frontier stands for the best performance that could be achieved. See the long version of this work for details. First, Figures 1(a), 1(e) and 1(i) show that the estimated efficient frontiers always become more overoptimistic as the subset size b increases, and the efficient frontiers directly computed by sample moments are more volatile. Second, Figures 1(b), 1(f) and 1(j) demonstrate that the actual efficient frontiers have a typical U-shape performance due to the tradeoff between diversification benefits and estimation errors, and the results from the subset size b = n0.8, which is conceptually about 5 data points per asset, are fairly stable. Third, Figures 1(c), 1(d), 1(g), 1(h), 1(k) and 1(l) illustrate that both the estimated and the actual efficient frontiers are less sensitive to the number of subsets s, and the effects from the subset size b is more important than the number of subsets s. In sum, the study suggests that users of SSR should focus on testing its performance by tuning the subset size b and choose a large number of subsets s to increase the diversification benefits. Besides cross-validation types of hyperparameter tuning, we suggest two possible guidelines of electing b: Users could start with b that gives no fewer than 5 data points per asset and combine this rule with the classical result that 30 assets often form a well-diversified portfolio.

4 Experiments

In this section, we fist introduce the tested datasets, the competing portfolios and the evaluation metrics. Then we conduct comparison studies and report the experimental results.

In our experiments, we intentionally choose diversified and large datasets to fairly evaluate our new strategy. Fama and French datasets: In the finance community, the Fama and French datasets have been widely recognized as high-quality and standard evaluation protocols (Fama and French 1992). Based on various types of financial segments of the U.S. stock market, the datasets contain carefully constructed portfolios from historical data. In general, they have an extensive coverage of assets classes and span a long period. In our experiments, the FF100 dataset includes monthly returns of 100 assets over forty years.

Table 1: Summary of the testing datasets

# Dataset Frequency Time Period m n

1. FF100 Monthly 07/01/1963 - 12/31/2004 498 100 2. ETF139 Weekly 01/01/2008 - 10/30/2012 252 139 3. EQ181 Weekly 01/01/2008 - 10/30/2012 252 181 4. SP434 Daily 09/05/2001 - 08/09/2013 2999 434

Real-world market datasets: Three real-world datasets are used in our experiments: ETF139, EQ181 and SP434. Specifically, ETF139 contains 139 exchange-traded funds (ETF). Due to the advantages of having clear tax and fee structures, high liquidity and diversity, ETFs have become popularized among investors. EQ181 is constructed from the individual equities from the large-cap segment of the pool of the Russell Top 200 Index. We exclude those stocks with missing historical data from the start of our testing periods and finally receive a total of 181 assets. Likewise, for the SP434 dataset, we filter the daily return data of the 500 firms listed in the S&P 500 Index and retain 434 assets. Table 1 summarizes these two types of benchmarks. They implicitly underline different perspectives in performance assessment. On the one hand, FF100 and SP434 highlight the long-term performance due to their long period of time spans and limited selection bias. On the other hand, ETF139 and EQ181 reflect the vicissitude market environment after the recent financial crisis starting from 2007. The four datasets have diverse trading frequencies: monthly, weekly and daily. Thus, through empirical evaluations on those datasets, we can thoroughly understand the performance of each strategy.

Competing Portfolio Strategies To comprehensively assess the proposed strategy, we consider ten state-of-the-art competing portfolios: (a) Equallyweighted portfolio (EW): EW is a naive yet robust strategy. It has been shown outperforming 14 sophisticated models across seven real-world datasets as well as one simulated dataset at monthly frequency of 2000 years (De Miguel, Garlappi, and Uppal 2009). Thus, EW is suggested to serve as the first touchstone in portfolio research. (b) Value-weighted portfolio (VW): VW forms a market mimicking passive portfolio. Most active mutual fund managers have the difficulty in outperforming passive benchmarks such as market indcies even before netting out fees (Fama and French 2010). (c) Minimum-variance portfolio (MV): MV based on sample moments has been shown outperforming the classical mean-variance portfolio in different markets and time spans (Scherer 2011). (d) Resampled efficient portfolio by (Michaud 1989) (RES): Since its inception, RES remains the most quoted portfolio strategy based on resampling. (e) Two-fund portfolio by (Tu and Zhou 2011) (TZT): TZT infuses the classical mean-variance and the EW portfolios to achieve both estimation error mitigation and wealth growth. (f) Three-fund portfolio by (Kan and Zhou 2007) (KZT): KZT encompasses the risk-free asset, the mean-variance and MV portfolios as to cancel out estimation errors in the meanvariance portfolio by incorporating the MV portfolio. (g) Four-fund portfolio by (Tu and Zhou 2011) (TZF): TZF

(a) Estimated, s = 1000

(b) Actual, s = 1000

(c) Estimated, b = n0.4

(d) Actual, b = n0.4

(e) Estimated, s = 2000

(f) Actual, s = 2000

(g) Estimated, b = n0.6

(h) Actual, b = n0.6

(i) Estimated, s = 4000

(j) Actual, s = 4000

(k) Estimated, b = n0.8

(l) Actual, b = n0.8

Figure 1: SSR efficient frontiers based on different hyperparameters in a simulation study. The whole dataset SP434 with n = 434 assets is used to estimate the means and the covariance matrix as inputs for simulation. See Table 1 for more information about SP434. By assuming the asset return follows a corresponding multivariate normal distribution, τ = 500 return data for training are then synthesized by simulation.

is formed by mixing the KZT and the EW portfolios. Their study shows that it performs comparably with EW in some special cases and better in general. (h) Two-fund portfolio by (Kan, Wang, and Zhou 2016) (KWZ): KWZ is an updated version of TZT particularly for portfolios only with risky assets. It is targeting at outperforming EW. (i) Covariance shrinkage estimator based portfolio by (Ledoit and Wolf 2004) (SKC): SKC hinges on a popular and robust shrinkage estimator for the covariance matrix estimation that works well even when the training sample size is critical. (j) On-line passive aggressive mean reversion portfolio by (Li et al. 2012) (PAMR): PAMR from machine learning researchers has been shown robustly outperforming 12 portfolio strategies on six datasets.

Performance Metrics

Following (De Miguel, Garlappi, and Uppal 2009), we employ the rolling-horizon setting for the sequential out-of-sample performance evaluation. From time tτ to tm 1, at each rebalance time th for h = τ, . . . , m 1, we first use the return data {Rk}h k=h τ+1 to determine the portfolio weights ˆωh. Second, we compute the realized out-of-sample net return ˆrh+1 for the subsequent trading period. Then, we evaluate the out-of-sample characteristics of portfolios by four standard metrics in finance (Brandt 2010): (i) Sharpe Ratio (SR): SR is a common risk-adjusted return measure for a portfolio strat-

egy considering both return and risk. Simply, SR is calculated as the portfolio return normalized by its standard deviation: SR = r/ σ with r as the mean of portfolio net returns and σ as the standard deviation:

k=τ+1 ˆrk, σ =

k=τ+1 ( r ˆrk)2. (2)

To compare strategies based on different rebalancing frequencies, we report the annualized Sharpe ratio as

HSR, where the scaling factor H represents the number of rebalancing times per year. In calculation, we use H = 12, 52 and 252 for monthly, weekly and daily rebalances, respectively. (ii) Volatility (VO): VO is a broadly computed quantitative risk measure in finance. Similar to SR, we report the annualized volatility using the scaling factor H and the standard deviations of returns σ in equation (2). (iii) Turnover Rate (TO): TO is a crucial metric to quantify the volume of rebalancing. Due to the existence of market frictions such as transaction costs and taxes, a high TO leads to extra trading costs and could drastically degrade after-cost return performance. TO is computed by

TO = 1 m τ 1

k=τ ||ˆωk+ ˆωk+1||1, (3)

where ˆωk+ is the portfolio weight vector before rebalancing at tk+1 and 1 denotes l1-norm. Briefly, the above equation

Table 2: Portfolio performance of strategies

Dataset Metrics SSR EW VW MV RES TZT KZT TZF KWZ SKC PAMR

FF100 SR 1.51 0.93 1.03 0.62 0.62 0.83 0.11 1.04 0.68 1.36 0.43 p-value 0.00 1.00 0.01 0.18 0.16 0.36 0.00 0.30 0.29 0.02 0.00 VO (%) 12.64 18.29 17.98 29.90 30.03 21.17 344.16 18.81 40.11 13.51 27.71 TO (%) 38.95 2.37 0.00 781.11 802.13 544.12 25245.53 232.76 126.60 55.75 133.81 MDD (%) 33.71 37.28 37.03 81.09 81.15 58.62 100 44.06 88.25 34.98 54.16 ETF139 SR 1.61 0.51 0.50 0.73 0.65 0.42 0.97 0.32 -0.27 1.13 0.86 p-value 0.10 1.00 0.55 0.83 0.83 0.82 0.14 0.51 0.19 0.49 0.21 VO (%) 2.09 21.20 21.06 3.82 3.89 21.78 56.46 18.60 22.57 2.67 36.85 TO (%) 25.54 1.08 0.00 801.99 825.08 403.91 10520.42 338.78 5762.82 20.21 153.38 MDD (%) 1.27 22.35 22.51 4.42 4.59 24.83 19.35 20.44 28.11 1.40 34.33 EQ181 SR 1.76 0.97 0.97 0.80 0.80 1.34 -0.24 1.00 1.96 1.48 0.30 p-value 0.57 1.00 0.84 0.90 0.86 0.41 0.41 0.87 0.43 0.76 0.21 VO (%) 7.79 15.43 15.29 23.72 23.87 17.45 45.50 15.07 45.57 8.87 22.86 TO (%) 15.70 1.85 0.00 552.75 568.96 140.72 1332.92 55.33 114.74 27.34 140.48 MDD (%) 3.07 9.19 9.17 9.97 10.43 10.21 22.18 8.45 16.81 6.62 18.7 SP434 SR 1.92 0.78 0.72 1.28 1.27 -0.10 0.40 0.82 0.46 1.27 1.14 p-value 0.00 1.00 0.52 0.23 0.23 0.08 0.41 0.70 0.46 0.23 0.26 VO (%) 6.98 20.16 18.26 13.18 13.18 61.12 564.66 17.14 138.27 7.78 61.12 TO (%) 6.56 1.50 0.00 131.73 144.76 3816.62 5911.32 57.59 17.21 24.9 130.39 MDD (%) 18.14 46.34 44.71 15.29 16.01 100 100 39.49 99.99 26.63 65.30

Note: We use short sliding widows τ = 120, 150, 200 and 500 for the four datasets, respectively. SSR, MV, RES and SKC implement the minimum-variance portfolio without return targets. For SSR, b = n0.7 and s = 15k. The p-values under the SR results quantify the statistical significance of the difference in SR between two comparing portfolios. EW is the benchmark portfolio. As returns are hardly i.i.d., the studentized circular block bootstrapping methodology in (Ledoit and Wolf 2008) is used to compute the p-values with 5000 bootstrap resamples and a block with the size of 5.

generates an average absolute value of the rebalancing trades across all the assets and over all the trading periods. (iv) Maximum Drawdown (MDD): MDD is one of the topmost risk measures for money management professionals, as large drawdowns often lead to fund redemptions (Magdon-Ismail and Atiya 2004). MDD is computed as the maximum drop of the cumulative wealth over a tested time period:

MDD = max k [τ,m](Mk Wk), (4)

with the running maximum of the cumulative wealth Mk and the cumulative wealth Wj at time tj obtained by

Mk = max j [τ,k] Wj with Wj =

l=τ+1 (1 + ˆrl), (5)

where we assume investors start with one dollar.

Results Table 2 presents the overall performance of the compared 11 portfolios across the four tested benchmarks. SSR apparently outperforms the classical Markowitz s type portfolio MV and Michaud s bagging based portfolio RES. Consistent with the observations in (Becker, G urtler, and Hibbeln 2015), MV and RES have almost identical performance, and the bootstrap resampling steps in Michaud s portfolio seem futile. In most of the cases, SSR has higher SRs, lower VOs and lower MDDs than the challenging baselines EW and VW, the upto-date portfolio blending strategies TZT, KZT and TZF, the market timing portfolio PAMR, as well as the shrinkage estimator based portfolio SKC. In sum, SSR behaves as a robust strategy with high return and low risk. Notably, VW and EW are intrinsically designed as passive strategies for low turnover rates. For example, VW mimics the market trend in a passive way and requires no rebalancing, resulting in zero TOs constantly. After excluding those two

passive strategies, as a moderately active trading strategy, SSR achieves the lowest TOs among all the remaining nine portfolio strategies. It has been recognized that high turnover rates stem from the sensitivity of portfolio optimization solutions to the estimation errors in input parameters (Best and Grauer 1991). Hence, portfolios with high estimation risks are sensitive to changes in the market and could have inflated before-cost returns. For instance, although KWZ has a little higher SR than SSR on EQ181, its TO is ten times larger than that of SSR, which inevitably leads to huge trading costs. This observation in turn indicates that in a multi-period trading setup the sacrificed diversification benefits in the subset resampling step are compensated not only by diminished estimation errors but also by low turnover rates. More experimental results with analysis are included in the long version of this work.

5 Conclusions and Discussions

In this paper, we develop a new portfolio selection strategy via subset resampling to promote the applicability of Markowitz s portfolio optimization to a large pool of assets. By sacrificing some diversification benefits, the new strategy receives the compensation of largely ameliorated estimation errors. We provide analysis on the hyperparameter selection and offer extensive comparison studies with ten representative portfolio strategies on four diversified datasets. As the work extends the depth of the portfolio research via ensemble methods, we believe that it represents an effort in leveraging successfully applied machine learning algorithms into long-lasting finance problems. Our future work includes explicitly incorporating the consideration of market frictions such as transaction costs and taxes into more sophisticated ensemble learning algorithms (Rokach 2010; Shen and Wang 2015).

References Agarwal, A.; Hazan, E.; Kale, S.; and Schapire, R. E. 2006. Algorithms for portfolio management based on the Newton method. In Proceedings of the 23rd International Conference on Machine Learning, 9 16. Becker, F.; G urtler, M.; and Hibbeln, M. 2015. Markowitz versus Michaud: Portfolio optimization strategies reconsidered. The European Journal of Finance 21(4):269 291. Best, M. J., and Grauer, R. R. 1991. On the sensitivity of meanvariance-efficient portfolios to changes in asset means: Some analytical and computational results. Review of Financial Studies 4(2):315 342. Blum, A., and Kalai, A. 1999. Universal portfolios with and without transaction costs. Machine Learning 35(3):193 205. Borodin, A.; El-Yaniv, R.; and Gogan, V. 2004. Can we learn to beat the best stock? Journal of Artificial Intelligence 21:579 594. Brandt, M. W. 2010. Portfolio choice problems. In Ait-Sahalia, Y., and Hansen, L. P., eds., Handbooks of Financial Econometrics. Elsevier. 269 336. Breiman, L. 1996. Bagging predictors. Machine learning 24(2):123 140. Broadie, M. 1993. Computing efficient frontiers using estimated parameters. Annals of Operations Research 45(1):21 58. Cover, T. M. 1991. Universal portfolios. Mathematical Finance 1(1):1 29. De Miguel, V.; Garlappi, L.; and Uppal, R. 2009. Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? The Review of Financial Study 22:1915 1953. Derbeko, P.; El-Yaniv, R.; and Meir, R. 2002. Variance optimized bagging. In European Conference on Machine Learning, 60 72. Springer. Dietterich, T. G. 2000. Ensemble methods in machine learning. In International Workshop on Multiple Classifier Systems, 1 15. Fama, E. F., and French, K. R. 1992. The cross-section of expected stock returns. Journal of Finance 47(2):427 465. Fama, E. F., and French, K. R. 2010. Luck versus skill in the crosssection of mutual fund returns. The Journal of Finance 65(5):1915 1947. Fan, J.; Zhang, J.; and Yu, K. 2012. Asset allocation and risk assessment with gross exposure constraints for vast portfolios. Journal of American Statistical Association 412 428. Harvey, C. R.; Liechty, J. C.; Liechty, M. W.; and M uller, P. 2010. Portfolio selection with higher moments. Quantitative Finance 10(5):469 485. Ho, T. K. 1998. The random subspace method for constructing decision forests. IEEE Transactions on Pattern Analysis and Machine Intelligence 20(8):832 844. Jorin, P. 1986. Bayes-Stein estimation for portfolio analysis. Journal of Financial and Quantitative analysis 21:279 292. Kan, R., and Zhou, G. 2007. Optimal portfolio choice with parameter uncertainty. Journal of Financial and Quantitative Analysis 42(03):621 656. Kan, R.; Wang, X.; and Zhou, G. 2016. Optimal portfolio selection with and without risk-free asset. Available at SSRN 2729429. Karoui, N. E., and Purdom, E. 2016. The bootstrap, covariance matrices and PCA in moderate and high-dimensions. ar Xiv preprint ar Xiv:1608.00948. Kleiner, A.; Talwalkar, A.; Sarkar, P.; and Jordan, M. I. 2014. A scalable bootstrap for massive data. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76(4):795 816.

Kolm, P. N.; T ut unc u, R.; and Fabozzi, F. J. 2014. 60 years of portfolio optimization: Practical challenges and current trends. European Journal of Operational Research 234(2):356 371. Ledoit, O., and Wolf, M. 2004. Honey, I shrunk the sample covariance matrix. The Journal of Portfolio Management 30(4):110 119. Ledoit, O., and Wolf, M. 2008. Robust performance hypothesis testing with the Sharpe ratio. Journal of Empirical Finance 15:850 859. Li, B., and Hoi, S. C. 2014. Online portfolio selection: A survey. ACM Computing Survey 46(3):35. Li, B.; Hoi, P.; Hoi, S. C.; and Gopalkrishnan, V. 2012. PAMR: Passive aggressive mean reversion strategy for portfolio selection. Machine Learning 87:221 258. Magdon-Ismail, M., and Atiya, A. F. 2004. Maximum drawdown. Risk Magazine 17(10):99 102. Markowitz, H., and Usmen, N. 2003. Resampled frontiers versus diffuse Bayes: An experiment. Journal Of Investment Management 1(4):9 25. Markowitz, H. 1952. Portfolio selection. Journal of Finance 7:77 91. Merton, R. C. 1980. On estimating the expected return on the market: An exploratory investigation. Journal of Financial Economics 8:323 361. Michaud, R. O., and Michaud, R. O. 2008. Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization and Asset Allocation. Oxford University Press. Michaud, R. O. 1989. The Markowitz optimization enigma: is optimized optimal? Financial Analysts Journal 45(1):31 42. Polikar, R. 2006. Ensemble based systems in decision making. IEEE Circuits and Systems Magazine 6(3):21 45. Rokach, L. 2010. Ensemble-based classifiers. Artificial Intelligence Review 33(1-2):1 39. Scherer, B. 2002. Portfolio resampling: Review and critique. Financial Analysts Journal 58(6):98 109. Scherer, B. 2011. A note on the returns from minimum variance investing. Journal of Empirical Finance 18(4):652 660. Shen, W., and Wang, J. 2015. Transaction costs-aware portfolio optimization via fast L owner-John ellipsoid approximation. In Proceedings of the 29th AAAI Conference on Artificial Intelligence, 1854 1860. Shen, W., and Wang, J. 2016. Portfolio blending via Thompson sampling. In Proceedings of the 25th International Joint Conference on Artificial Intelligence, 1983 1989. Shen, W.; Wang, J.; Jiang, Y.-G.; and Zha, H. 2015. Portfolio choices with orthogonal bandit learning. In Proceedings of the 24th International Joint Conference on Artificial Intelligence, 974 980. Shen, W.; Wang, J.; and Ma, S. 2014. Doubly regularized portfolio with risk minimization. In Proceedings of the 28th AAAI Conference on Artificial Intelligence, 1286 1292. Skurichina, M., and Duin, R. P. W. 2002. Bagging, boosting and the random subspace method for linear classifiers. Pattern Analysis & Applications 5(2):121 135. Tu, J., and Zhou, G. 2011. Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies. Journal of Financial Economics 99:204 215. Wolf, M. 2013. Resampling vs. shrinkage for benchmarked managers. Wilmott Magazine 76 81. Zhou, Z.-H. 2012. Ensemble Methods: Foundations and Algorithms. CRC press.