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Diversified reward-risk parity in portfolio construction

Jaehyung Choia,1,∗, Hyangju Kimb, Young Shin Kimc

aGoldman Sachs & Co., New York, USAbDepartment of Applied Mathematics and Statistics, Stony Brook University, New York,

USAcCollege of Business, Stony Brook University, New York, USA

Abstract

We introduce diversified risk parity embedded with various reward-risk measuresand more generic allocation rules for portfolio construction. We empirically testadvanced reward-risk parity strategies and compare their performance with anequally-weighted risk portfolio in various asset universes. The reward-risk par-ity strategies we tested exhibit consistent outperformance evidenced by higheraverage returns, Sharpe ratios, and Calmar ratios. The alternative allocationsalso reflect less downside risks in Value-at-Risk, conditional Value-at-Risk, andmaximum drawdown. In addition to the enhanced performance and reward-risk profile, transaction costs can be reduced by lowering turnover rates. TheCarhart four-factor analysis also indicates that the diversified reward-risk parityallocations gain superior performance.

Keywords: portfolio construction, asset allocation, risk parity, reward-riskmeasures, classical tempered stable distribution, ARMA, GARCHJEL classification: G11, G12, C58

1. Introduction

Since Markowitz’s seminal paper (Markowitz (1952)), portfolio constructionhas been one of the most intriguing topics in finance. Not confined to academia,it has been settled as a crucial investment procedure for practitioners who seek

∗Corresponding authorEmail addresses: jj.jaehyung.choi@gmail.com (Jaehyung Choi),

hyangju.kim@stonybrook.edu (Hyangju Kim), aaron.kim@stonybrook.edu (Young ShinKim)

1Disclosure: The opinions and statements expressed in this paper are those of the authorand may be different to views or opinions otherwise held or expressed by or within GoldmanSachs. The content of this paper is for information purposes only and is not investmentadvice or advice of any other kind. None of the author, Goldman Sachs, or its affiliates,officers, employees, or representatives accepts any liability whatsoever in connection with anyof the content of this paper or for any action or inaction of any person taken in reliance uponsuch content or any part thereof.

Preprint submitted to Elsevier September 16, 2021

higher profitability and control financial risks under various investment con-straints in prevailing market conditions. In particular, the Markowitz frameworkmakes the portfolio construction process evolve into a more mathematical andquantitative decision-making process. Markowitz’s portfolio construction pro-cess is basically solving an optimization problem to ‘optimally’ allocate capitalby maximizing or minimizing objective functions that reflect investors’ targetreturns and risk appetites. Finding extrema in the optimization problems pro-vides the optimal solution to the investors. For example, the global minimumvariance portfolio (Markowitz (1952)) tries to minimize a variance or to maxi-mize an expected return at a given level of risk-return profiles. Similarly, Tobin(1958) and Sharpe (1964) also demonstrated that the highest Sharpe ratio is anoptimal point on the efficient frontier in the presence of the risk-free asset.

However, there are several drawbacks to the Markowitz portfolio. First,the Markowitz portfolios are sensitive to changes in model input parameters(Michaud (1989)). For example, even when the Markowitz model inputs – amultivariate expected return vector or a covariance matrix of an investmentuniverse – are slightly changed, the resulting portfolio allocation to each con-stituent can vary abruptly. As the model parameters are updated with newly-arrived market data to portfolio optimization tools, the Markowitz frameworkpotentially demands massive portfolio reallocation such that causes expensivetransaction costs which erode portfolio performance. Second, the existence ofoptimal solutions under investment constraints is not always guaranteed. It isoften hard to find the optimal solution satisfying a given constraint set in fea-sible domains. Although these disadvantages can be alleviated by the help ofshrinkage estimation (Ledoit and Wolf (2003)), regularization (DeMiguel et al.(2009)), and constraint impact (Jagannathan and Ma (2003)), the limitationscould be still persistent.

A more fundamental and philosophical question to the Markowitz frame-work is that the portfolio variance is not a representative of genuine risks toinvestment portfolios because larger positive returns contribute to increase prof-itability and volatility simultaneously. As potential solutions, other types ofobjective functions have been suggested for resolving the shortcomings and is-sues in the Markowitz-type optimization. Genuine portfolio risk measures havebeen used as objective functions in order to incorporate actual downside port-folio risks into asset allocation. For example, finding portfolio weights such thatminimize Value-at-Risk (Gaivoronski and Pflug (2005)), conditional Value-at-Risk (Rockafellar and Uryasev (2000); Krokhmal et al. (2002)), and maximumdrawdown instead of standard deviation is more compatible and consistent withinvestment insights on downside risks.

Since the financial crisis in 2008, risk parity strategies have received increas-ing attention from both academia and industry. While the traditional ‘well-diversified’ portfolios failed substantially in this market turmoil, the risk paritystrategies relatively performed well. This is because the risk parity portfolio fo-cuses on the allocation of risk in order to construct a truly diversified portfolio.For example, Bruder and Roncalli (2012) introduced the portfolio constructionby matching the risk contribution of each asset to the same level.

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However, it could be difficult for some practitioners to implement risk bud-geting methodologies in real world because setting the equal risk budgetingbased on a given risk measure would be computationally and practically de-manding to achieve. For example, some risk parity methods require partialderivative of risk measures with respect to weight in order to calculate the riskcontribution (Stoyanov et al. (2006); Kim et al. (2012)). Other risk parity port-folio construction methods such as maximum diversification ratio (Choueifaty et al.(2008)) and maximum decorrelation (Christoffersen et al. (2010)) also need ro-bust matrix inversion and regularization techniques rooted from matrix anal-ysis and statistics if concerning covariance matrix and correlation matrix arehigh-dimensional. For larger asset universes, the situation becomes much morechallenging because of the curse of dimensionality. Furthermore, more complexmodels are more prone to sensitivity to input parameter changes.

To mitigate these technical difficulties in risk parity methodologies, simpleand heuristic approaches to risk parity also have been documented. Startingwith the equally-weighted portfolio which is apparently the simplest portfolio,the market capitalization-weighted portfolio (Sharpe (1964)) is another exampleof simple diversification methods. Fernholz and Shay (1982) also suggested an-other market capitalization-based allocation in which the weight of each assetis a power of its market capitalization. It is plausible to exploit other ref-erential data such as fundamental weights based on accounting information(Arnott et al. (2005)). Additionally, the equal risk contribution (ERC) strate-gies (Maillard et al. (2010)) bet against market exposures via beta in the capitalasset pricing model (CAPM). The diversified risk parity (DRP) portfolios sug-gested by Maillard et al. (2010) adopt the standard deviation of each asset asthe portfolio weights. Similar to the DRP portfolios, the diversified minimumvariance (DMV) portfolio allocates weights inversely-proportional to variance.Still, these approaches are not based on reward-risk measures nor traditionalrisk measures of each individual asset.

In this paper, we suggest heuristic allocation methodologies based on variousreward-risk measures and general forms of allocation functions. The portfolioconstruction rules are empirically tested in multiple investment universes. Thesealternative portfolios as the generalization of the ERC allocation and the DRPallocation provide various advantages in performance and risk profile over theequally-weighted portfolio. However, this paper is not for a horse race-typecomparison between various diversified reward-risk parity portfolios and thebenchmark portfolio. Instead, this paper extends the scope of diversified riskparity to reward-risk measures and more generic allocation rules in order forinvestors to consider diverse options to construct their portfolios that are morein line with their investment goals and risk tolerances.

In the next section, alternative allocation rules using diversified reward-riskparity are introduced. In section 3, datasets and methodologies for this studyare given. In section 4, we present performance and reward-risk profiles ofalternative allocations. In section 5, the Carhart four-factor analysis on theportfolio performance is conducted. In section 6, we conclude the paper.

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2. Theoretical backgrounds

In this section, we visit definitions of reward-risk measures that are buildingblocks for portfolio construction by diversified reward-risk parity. Additionally,we present risk models for calculating the reward-risk measures. And then, thediversified reward-risk parity is introduced as the generalization of diversifiedrisk parity to reward-risk measures and various allocation rules.

2.1. Reward-risk measures

Throughout this paper, various reward-risk measures are employed for port-folio construction, performance assessment, and risk management. We shortlyvisit definitions of these measures for further usages.

2.1.1. Volatility and variance

Volatility assesses return fluctuations of a given financial time series, andwe compute the volatility as either of two different definitions: the standarddeviation from historical returns and the conditional volatility from ARMA-GARCH models. Additionally, we also consider variances from the two volatilitymeasures.

2.1.2. Sharpe ratio

Sharpe ratio (Sharpe (1964)) is one of the most well-known reward-risk ratiosin finance and also broadly used by practitioners. Sharpe ratio is defined as theratio of expected return to standard deviation of a return time series r:

SR =E[r − rf ]

σ[r − rf ](1)

where E[·] is the expectation value, σ[·] is the standard deviation, and rf isthe risk-free rate. In Markowitz framework, the tangency portfolio exhibitsthe highest Sharpe ratio among all feasible portfolios under the existence of therisk-free asset (Markowitz (1952); Sharpe (1964)). From investors’ point of view,assets with higher Sharpe ratios are more preferred in portfolio construction.

Similar to the usage of two definitions in volatility, two different Sharperatio definitions are used for portfolio construction and performance assess-ment. One is the traditional Sharpe ratio definition that is the ratio of em-pirical average return to historical standard deviation. This is mentioned asmarginal/unconditional Sharpe ratio in the paper. Another is conditional Sharperatio calculated as the ratio of conditional mean to conditional volatility com-puted by ARMA-GARCH models.

2.1.3. Maximum drawdown

Maximum drawdown (MDD) is the worst consecutive loss in a specified timeperiod. The maximum drawdown of a portfolio is given by

MDD = − minτ∈(0,T )

(

mint∈(0,τ)

r(t, τ))

(2)

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where r(t, t′) is the return during the time period between t and t′. It is the worstrealized performance since the inception of a portfolio during a given investmenthorizon. Apparently, assets with lower maximum drawdowns are favored byinvestors. Moreover, maximum drawdown also encodes the information on timeevolution of a return series opposite to other quantile-based risk measures. Forexample, even when we use a given return series for risk calculation, differentmaximum drawdowns are obtained by different sequential orders of the timeseries.

2.1.4. Calmar ratio

Back to the definition of Sharpe ratio, Eq. (1), the denominator of the ratiois the standard deviation which penalizes assets with larger volatility. How-ever, losses are not the only source of volatility. Although positive returns arethe origins of portfolio profitability, such returns also contribute to increasingvolatility. Hence, the separation of volatility from downside risks is crucial inconsidering performance measures.

Calmar ratio (Young (1991)) is the ratio of return to maximum drawdown:

Calmar =R

MDD(3)

where R is the realized cumulative return, and MDD is the realized maximumdrawdown of Eq. (2) within a given investment time horizon. Investors preferassets with higher Calmar ratios to assets with lower Calmar ratios.

In the original definition of Calmar ratio (Young (1991)), the time windowfor return and maximum drawdown was given to 36 months. In this paper, theCalmar ratio is computed from the realized return and the maximum drawdownin the six-month window for portfolio construction. In order to assess portfolioperformance and risk, the Calmar ratio presents the ratio of annualized return tomaximum drawdown from the whole period since the inception of the portfolio.

2.1.5. Value-at-Risk and conditional Value-at-Risk

Value-at-Risk (VaR) is the loss at a given quantile of a performance distri-bution. For a given return time series r, VaR of (1 − η)100% (0 < η < 1) isdefined as

VaR(

(1− η)100%)

= −inf{l|P (r − rf > l) ≤ 1− η} (4)

where rf is the risk-free rate and P is the cumulative distribution function of agiven underlying probability distribution. VaR describes quantile risks of returndistributions as a single number computed from historical data or simulateddata.

Despite its simplicity, several shortcomings of VaR still exist. First ofall, non-subadditivity for VaR implies that portfolio diversification could bringworse VaR values than the weighted sum of VaRs from its components. Itis obviously counter-intuitive to the investment insight that diversification canreduce investment risks.

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Rockafellar and Uryasev (2000, 2002) suggested conditional Value-at-Risk(CVaR), also known as average Value-at-Risk. CVaR is the average loss ofextreme losses worse than a given quantile loss. By using the definition of VaRin Eq. (4), the CVaR at (1− η)100% is defined as

CVaR(

(1 − η)100%)

=1

η

∫ η

0

VaR(

(1− ζ)100%)

dζ (5)

where 0 < η < 1. When a portfolio suffers from severe losses worse than athreshold, a CVaR value indicates the average of such extreme losses.

One advantage of CVaR over VaR is its coherency (Artzner et al. (1999)).Opposite to the characteristics of VaR, CVaR fulfills not only the subadditivitybut also other properties of coherent risk measures. In addition to the coherencyof CVaR, much information on the downside tail is incorporated into CVaR. Forexample, CVaR values of two portfolios would be different even if VaRs of thoseportfolios are the same. It is obvious that portfolios with thicker downside tailsexhibit larger CVaR values than portfolios with thinner downside tails.

2.1.6. Stable tail adjusted return ratio

As the similar intuition of Calmar ratio in Eq. (3), Martin et al. (2003)suggested the stable tail adjusted return (STAR) ratio in order to enhance theconcept of Sharpe ratio. Instead of adopting the standard deviation in Sharperatio, Eq. (1), the ratio is penalized by CVaR which represents genuine downsiderisks. For a time series r, the definition of STAR ratio at (1−η)100% (0 < η < 1)is given by

STAR(

(1 − η)100%)

=E[r − rf ]

CVaR(

(1 − η)100%) (6)

where rf is the risk-free rate. It is straightforward that the standard deviationin the definition of Sharpe ratio is replaced with CVaR defined in Eq. (5). Byusing the STAR ratio, assets with less downside tail risks exhibit larger STARratios. Obviously, portfolios with higher ratios are more preferred.

2.1.7. Rachev ratio

Although the STAR ratio in Eq. (6) removes the ambiguity between volatil-ity and downside risk, the expected return in the nominator of the STAR ra-tio definition still includes redundancy regarding the downside risks. Extremedownward events penalize the ratio not only by increasing the CVaR in thedenominator but also by decreasing the expected return in the nominator. It isnecessary to divide downside risks from the reward measure.

Biglova et al. (2004) introduced Rachev ratio to segregate contributions oflosses from the reward part. Instead of employing the expected return, Rachevratio is defined as the ratio of expected upward tail gain to downward tail loss.It is represented with the ratio of two CVaRs from an original return time

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series r used in the denominator and its sign-inverted return time series in thenominator:

Rachev((1− η)100%, (1− ζ)100%) =CVaR

(

(1 − η)100%)

for (rf − r)

CVaR(

(1− ζ)100%)

for (r − rf )(7)

where 0 < η < 1, 0 < ζ < 1, and rf is the risk-free rate. It is obvious that anasset with a lower downside risk and a higher upside reward is more preferred.

2.2. Risk models

In order to test the robustness of diversified reward-risk parity to risk modelchoice, we apply three different risk models for reward-risk calculation. The firstrisk model for reward-risk measures is the ARMA(1,1)-GARCH(1,1) model withclassical tempered stable (CTS) innovations suggested by Kim et al. (2010a),Kim et al. (2010b), and Kim et al. (2011). Since this model captures autocor-relation, volatility clustering, skewness, and heavy-tailedness of a financial timeseries, various applications of this model are found in portfolio management(Tsuchida et al. (2012); Beck et al. (2013); Georgiev et al. (2015); Anand et al.(2016)) and momentum strategy (Choi et al. (2015)).

In CTS(α,C+, C−, λ+, λ−,m) distributions, m is the location parameter, αis the tail index, C+ and C− are the scale parameters, λ+ and λ− are the decayrate of the tails. The characteristic function of a CTS distribution is in thefollowing form:

φ(u) = exp(

ium− iuΓ(1− α)(C+λ1−α+ − C−λ

1−α−

)

+ Γ(−α)(

C+

(

(λ+ − iu)α − λα+

)

+ C−

(

(λ− + iu)α − λα−

))

)

(8)

where C+, C−, λ+, λ− are positive, α ∈ (0, 2), m ∈ R and Γ is the gammafunction.

As described in Kim et al. (2011), the parameters of the ARMA(1,1)-GARCH(1,1)-CTS model are estimated from the following steps. With an assumption thatresiduals of the ARMA(1,1)-GARCH(1,1) model are Student’s t -distributed,the ARMA-GARCH parameters are estimated from maximum likelihood esti-mation (MLE). The innovations obtained from the previous step are appliedto CTS model parameter estimation by fast Fourier transformation and MLE(Rachev et al. (2011)). Using all the estimated model parameters, we can calcu-late various reward-risk measures. For more details, check Kim et al. (2010a,b,2011) and references therein.

The second risk model for reward-risk measure calculation is a simple variantof the first model by replacing the CTS distribution as the innovation model withthe normal tempered stable (NTS) distribution suggested by Barndorff-Nielsen and Levendorskii(2001) and Barndorff-Nielsen and Shephard (2001). In this case, the risk modelis the ARMA(1,1)-GARCH(1,1)-NTS model which is used in the literature(Anand et al. (2016, 2017); Kurosaki and Kim (2019)). The third risk modelis the multivariate extension of the innovation distribution in the second model.

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With the multivariate normal tempered stable (MNTS) innovations (Kim et al.(2012); Kim (2015, 2020)), we utilize DCC GARCH model (Engle and Sheppard(2001)) instead of the GARCH model to capture time-varying covariance ma-trix. In this case, the risk model for reward-risk measure calculation is theARMA(1,1)-DCC GARCH(1,1)-MNTS model.

2.3. Risk parity and diversified reward-risk allocation

One simplest risk parity portfolio is the equal risk contribution (ERC) port-folios (Maillard et al. (2010)). In an investment universe with N assets, theallocation weight of each asset in the ERC portfolio is given by

wi =β−1i

∑Ni=1 β

−1i

(9)

where βi is the CAPM beta of the i-th asset. Considering the financial interpre-tation of the CAPM beta, the ERC portfolio penalizes exposures to the marketportfolio. By betting against the beta, it tries to reduce asset-level correlationsto the benchmark.

Maillard et al. (2010) also suggested the diversified risk parity (DRP) al-location based on standard deviation. The weights in the DRP portfolio areallocated with the inverse of volatility:

wi =σ−1i

∑N

i=1 σ−1i

(10)

where σi is the standard deviation of the returns for the i-th asset. In thisportfolio construction, less volatile investment vehicles are more preferred tomore volatile ones.

Similarly, it is also possible to leverage variance instead of standard devia-tion. The diversified minimum variance (DMV) portfolio allocates more weightson assets with smaller variance:

wi =σ−2i

∑N

i=1 σ−2i

(11)

where σi is the standard deviation of the returns for the i-th asset. The DRPportfolio and the DMV portfolio penalize the volatility of each asset in the formof standard deviation and variance, respectively.

In principle, the portfolio construction rules based on the heuristic risk par-ity such as Eq. (9), Eq. (10), and Eq. (11) simply assign more weights onassets with smaller risks than assets with larger risks based on the assumptionthat CAPM beta and volatility are the proxies of risks. By favoring less riskyassets over more risky assets, the allocation rules decide the extent of how muchpenalization is given to each asset based on risk measures. In these allocationschemes, the investors are risk-averse.

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It is natural to consider reward-risk measures in similar diversified risk parityportfolio construction. The various allocation weights mentioned above can begeneralized into the following form:

wi =ρ−1i

∑N

i=1 ρ−1i

(12)

where ρ is a reward-risk measure of interest. For example, the ERC portfolioof Eq. (9) corresponds to ρ = β. In case of the DRP allocation and the DMVallocation, the weights are decided by ρ = σ and ρ = σ2 in Eq. (10) and Eq.(11), respectively.

As a further generalization of Eq. (12), we extend diversified reward-riskparity allocations by a using more generic allocation rule Φ such as

wi =Φ(ρi)

∑Ni=1 Φ(ρi)

. (13)

The previous examples are special cases of this generalization. For example, theERC allocation and the DRP allocation rules reflect Φ(ρ) = ρ−1. The DMVportfolio construction is considered in two different ways: the inverse squarefunction of Φ(ρ) = ρ−2 with ρ = σ or the inverse function Φ(ρ) = ρ−1 withρ = σ2. It is also noteworthy that the equally-weighted portfolio is a specialcase of a constant Φ.

The allocation rule Φ in Eq. (13) defines how assets are assigned based onreward-risk measures. In principle, Φ can be any functions to model investors’investment preference, risk aversion, and investment horizon. For example, someinvestors prefer assets with better reward-risk measures in the past becausethey are more trend-followers. However, another market participant penalizesthe same assets with belief and prediction that such trends will be diminishedand reverted. Combining these two approaches, the others would invest his/hercapitals to the assets with large reward-risk measures until the criteria hit thethreshold. After then, weights on the assets beyond (below) the limit startto be decreased (increased). Furthermore, different lengths of investment win-dows would demand different allocation rules Φ even for the same reward-riskmeasures (Choi (2014)).

In this paper, we confine Φ to monotonically decreasing or increasing func-tion of ρ. Additionally, let us assume that we are risk-aversion investors. Firstof all, these assumptions pursue the simplicity and consistency of implemen-tation across various reward-risk measures. Instead of adopting very bespokeΦ, diversified reward-risk parity relies on more straightforward allocation rules.Moreover, the simple allocation rule makes clearer and more robust explanationson the validity of the new diversification method than any bespoke allocationrules.

With Φ as monotonically decreasing or increasing function of ρ, it is possibleto exploit various functions as long as the forms of the functions are consistentwith characteristics of reward-risk measures. Decision between monotonically

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decreasing or increasing functions used for candidates to Φ needs further con-siderations on which measures are adopted as ρ. The nature and characteristicsof ρ in Eq. (13) are the main factors to eligible forms of Φ that are consistentwith the financial meaning and definition of each reward-risk measure. For ex-ample, when investors seek to minimize risks in the portfolio, risky assets arepenalized in allocation. In order to reduce the exposures on such risky assets innew reward-risk parity methods, an allocation rule Φ needs to be monotonicallydecreasing functions of risk measures ρ. Contrary to risk measures, when purereward measures are employed in portfolio construction via Eq. (13), investorsprefer assets with larger reward-risk metrics and avoid assets with lower reward-risk measures. In this sense, Φ for reward measures should be a monotonicallyincreasing function.

Reward-risk measures are categorized into two classes in the following ways.The first class of reward-risk measures includes reward-risk ratio measures suchas Sharpe ratio, Calmar ratio, STAR ratio, and Rachev ratio. Since higher ratiomeasures are more preferable and lower ratio measures are more avoidable, Φ(ρi)should be a monotonically increasing function of ρi.

The simplest candidate for diversified reward-risk parity is proportional tothe measures given by

wi =ρi

∑N

i=1 ρi(14)

where ρi is a reward-risk measure of the i-th asset. Other variations such as anexponent of a reward-risk measure ρ, similar to Fernholz and Shay (1982), arealso possible.

Another linear allocation rule is the linear version of Eq. (14):

wi =a+ bρi

∑N

i=1(a+ bρi)(15)

where a and b are non-negative constants. It is straightforward to check that theequal weight allocation is a special case of this generalized reward-risk paritywhen b = 0 in Eq. (15).

Additionally, Eq. (15) is decomposed into the equally-weighted allocationpart and the deviation from the mean reward-risk measure:

wi =1

N+

b(ρi − ρ)∑N

i=1(a+ bρi)(16)

where a and b are non-negative constants, and ρ is the average reward-risk mea-sure. In this expression, the linear allocation of a reward-risk measure fluctuatesaround the equally-weighted allocation.

There is the possibility of reward-risk ratio measures to be negative valuesthat cause short-selling in the portfolio. Some investors do not short-sell assetsor are not allowed to be in short positions. Opposite to the Markowitz opti-mization, constraints on no short-selling in diversified reward-risk parity can

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be directly implemented in more straightforward ways by explicitly imposingρi|+ = max(ρi, 0) instead of using just raw ρi. For example, the introductionof the quasi-linear allocation rule to Eq. (14) and Eq. (15) represent the assetallocations with no short-selling constraints in the following forms:

wi =(a+ bρi)|+

∑N

i=1(a+ bρi)|+(17)

or

wi =a+ bρi|+

∑N

i=1(a+ bρi|+)(18)

where a and b are non-negative constants and ρi|+ = max(ρi, 0). Each weightin Eq. (17) is floored at zero. Meanwhile, the minimal non-zero allocation isguaranteed in Eq. (18).

Another class of reward-risk measures includes pure risk measures such asvolatility, variance, maximum drawdown, VaR, and CVaR. Since these met-rics represent volatility or losses, assets with larger measures are less favorable.Weights on assets with larger risk measures need to be smaller than those onassets with smaller risk measures.

There are the four simplest ways of achieving the penalization of risks inallocation. The first implementation is taking the inverse of risk measures as aweighting scheme. Given a risk measure, the diversified reward-risk parity of apure risk measure ρ is defined as

wi =ρ−1i

∑Ni=1 ρ

−1i

(19)

where ρi is the risk measure of i-th asset.Another way of penalizing risk measures in portfolio construction is as fol-

lows:

wi =a− bρi

∑N

i=1(a− bρi)(20)

where a and b are non-negative constants. If a = 1, b = 1, and the measureis one of VaR, CVaR, and maximum drawdown, 1 − ρ can be considered theexpected portfolio value after severe losses if there is no correlation betweenassets.

The third way of the risk penalization is to linearize Eq. (19) in the followingform:

wi =a+ bρ−1

i∑N

i=1(a+ bρ−1i )

(21)

where a and b are non-negative constants. Similar to Eq. (14) and Eq. (15),Eq. (20) and Eq. (21) also cover the equally-weighted allocation as a specialcase of b = 0.

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Lastly, it is possible to consider the no short-selling constraint by usingρi|+ = max(ρi, 0), which is the risk version of Eq. (17) and Eq. (18), althoughsuch cases are relatively rare for risk measures.

3. Dataset and methodology

In this section, we introduce investment universes for empirically testingdiversified reward-risk parity portfolio construction. After then, the portfolioconstruction procedure for empirical tests is explained.

3.1. Dataset

In order to test diversified reward-risk parity allocation across various assetclasses, we cover global equity benchmarks, largest mutual funds and ETFs,SPDR U.S. sector ETFs, and Dow Jones Industrial Average components. Ad-justed daily price datasets of these asset universes were downloaded from YahooFinance.

3.1.1. Global equity benchmarks

Daily price data for global equity benchmarks in the period between January1st, 1999, and December 31st, 2020 were downloaded. The tickers in the indexuniverse are found in Table 1.

3.1.2. Largest mutual funds and ETFs

The list of largest mutual funds and ETFs starts from the top 25 largestmutual funds and ETFs list at Market Watch2. Excluding money market fundsand government securities funds, there are thirteen mutual funds and four ETFsbetween January 1st, 1999 and December 31st, 2020. The list of tickers in theuniverse is given in Table 2.

3.1.3. U.S. equity markets: SPDR sector ETFs

Nine SPDR U.S. sector ETF tickers in the universe are listed in Table 3.These nine tickers have existed since the inception of the SPDR sector ETFseries. The time window for each time series is chosen as the period betweenJanuary 1st, 1999, and December 31th, 2020.

3.1.4. U.S. equity markets: Dow Jones Industrial Average

The constituents of the Dow Jones Industrial Average universe cover theperiod from January 1st, 1999 and December 31st, 2020. The tickers withcomponent changes are listed in Table 4.

2https://www.marketwatch.com/tools/mutual-fund/top25largest

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Table 1: Ticker information for global stock benchmarks

Ticker Name Minimum Date Maximum Date

000001.SS Sse Composite Index 1999-01-01 2020-12-31399001.SZ Shenzhen Component 1999-01-01 2020-12-31IMOEX.ME Moex Russia Index 2013-03-05 2020-12-31ˆAORD All Ordinaries 1999-01-01 2020-12-31ˆAXJO S&P/Asx 200 1999-01-01 2020-12-31ˆBFX Bel 20 1999-01-01 2020-12-31ˆBSESN S&P Bse Sensex 1999-01-01 2020-12-31ˆBUK100P Cboe Uk 100 2010-09-17 2020-12-31ˆBVSP Ibovespa 1999-01-01 2020-12-31ˆDJI Dow Jones Industrial Average 1999-01-01 2020-12-31ˆFCHI Cac 40 1999-01-01 2020-12-31ˆFTSE Ftse 100 1999-01-01 2020-12-31ˆGDAXI Dax Performance-Index 1999-01-01 2020-12-31ˆGSPC S&P 500 1999-01-01 2020-12-31ˆGSPTSE S&P/Tsx Composite Index 1999-01-01 2020-12-31ˆHSI Hang Seng Index 1999-01-01 2020-12-31ˆIPSA S&P/Clx Ipsa 2002-01-02 2019-06-14ˆIXIC Nasdaq Composite 1999-01-01 2020-12-31ˆJKSE Composite Index 1999-01-01 2020-12-31ˆJN0U.JO Top 40 Usd Net Tri Index 2017-09-26 2020-12-31ˆKLSE Ftse Bursa Malaysia Klci 1999-01-01 2020-12-31ˆKS11 Kospi Composite Index 1999-01-01 2020-12-31ˆMERV Merval 1999-01-01 2020-12-31ˆMXX Ipc Mexico 1999-01-01 2020-12-31ˆN100 Euronext 100 1999-12-31 2020-12-31ˆN225 Nikkei 225 1999-01-01 2020-12-31ˆNYA Nyse Composite (Dj) 1999-01-01 2020-12-31ˆRUT Russell 2000 1999-01-01 2020-12-31ˆSTI Sti Index 1999-01-01 2020-12-31ˆSTOXX50E Estx 50 Pr.Eur 2007-03-30 2020-12-31ˆTA125.TA Ta-125 1999-01-01 2020-12-31ˆTWII Tsec Weighted Index 1999-01-01 2020-12-31ˆXAX Nyse Amex Composite Index 1999-01-01 2020-12-31

Ticker information for global stock benchmarks

3.1.5. Risk-free rates

In order to calculate reward-risk measures introduced in the previous section,the yield of the 91-day U.S. Treasury bill is employed as the risk-free rate forglobal equity benchmarks, largest mutual funds, SPDR U.S. sector ETFs, andthe Dow Jones Industrial Average components. In Yahoo Finance, the tickerof the 91-day U.S. Treasury bill is ˆIRX. The period of the daily data is fromJanuary 1st, 1999 to December 31st, 2020.

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Table 2: Ticker information for largest mutual funds and ETFs

Ticker Name Minimum Date Maximum Date

AGTHX American Funds The Growth Fund of America Class A 1999-01-01 2020-12-31FCNTX Fidelity Contrafund Fund 1999-01-01 2020-12-31FXAIX Fidelity 500 Index Fund 2011-05-04 2020-12-31IVV iShares Core S\&P 500 ETF 2000-05-19 2020-12-31QQQ Invesco QQQ Trust 1999-03-10 2020-12-31VBTLX Vanguard Total Bond Market Index Fund Admiral Shares 2001-11-12 2020-12-31VFIAX Vanguard 500 Index Fund Admiral Shares 2000-11-13 2020-12-31VGTSX Vanguard Total International Stock Index Fund Investor Shares 1999-01-01 2020-12-31VIIIX Vanguard Institutional Index Fund Institutional Plus Shares 1999-01-01 2020-12-31VINIX Vanguard Institutional Index Fund Institutional Shares 1999-01-01 2020-12-31VITSX Vanguard Total Stock Market Index Fund Institutional Shares 1999-01-01 2020-12-31VOO Vanguard S\&P 500 ETF 2010-09-09 2020-12-31VSMPX Vanguard Total Stock Market Index Fund Institutional Plus Shares 2015-04-28 2020-12-31VTBIX Vanguard Total Bond Market II Index Fund Investor Shares 2011-01-21 2020-12-31VTI Vanguard Total Stock Market Index Fund ETF Shares 2001-06-15 2020-12-31VTSAX Vanguard Total Stock Market Index Fund Admiral Shares 2000-11-13 2020-12-31VTSMX Vanguard Total Stock Market Index Fund Investor Shares 1999-01-01 2020-12-31

Ticker information for largest mutual funds and ETFs

Table 3: Ticker information for SPDR U.S. sector ETFs

Ticker Name Minimum Date Maximum Date

XLB Materials Select Sector Spdr Fund 1999-01-01 2020-12-31XLE Energy Select Sector Spdr Fund 1999-01-01 2020-12-31XLF Financial Select Sector Spdr Fund 1999-01-01 2020-12-31XLI Industrial Select Sector Spdr Fund 1999-01-01 2020-12-31XLK Technology Select Sector Spdr Fund 1999-01-01 2020-12-31XLP Consumer Staples Select Sector Spdr Fund 1999-01-01 2020-12-31XLU Utilities Select Sector Spdr Fund 1999-01-01 2020-12-31XLV Health Care Select Sector Spdr Fund 1999-01-01 2020-12-31XLY Consumer Discretionary Select Sector Spdr Fund 1999-01-01 2020-12-31

Ticker information for SPDR U.S. sector ETFs

3.2. Portfolio construction process

On a reference date, daily price data during the last six months are usedfor computing reward-risk measures. As mentioned in the previous section,these measures, except for maximum drawdown and Calmar ratio, are calcu-lated by using three different risk models: the ARMA(1,1)-GARCH(1,1) modelswith CTS and NTS innovations, and the ARMA(1,1)-DCC GARCH(1,1)-MNTSmodel. Maximum drawdown and Calmar ratio are directly computed from his-torical return data.

With reward-risk metrics from the past six months of the portfolio con-struction date, portfolio weights are decided by each allocation rule based ondiversified reward-risk parity. In order to fairly compare performance with theequally-weighted portfolio, the short-selling caused by negative metrics is notallowed.

The allocation rules used for portfolio construction are given in Table 5. Firstof all, the equally-weighted portfolio is the benchmark portfolio. For reward-risk

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ratio measures such as Calmar ratio, marginal/conditional Sharpe ratio, STARratio, and Rachev ratio, two allocation rules are used: ρi|+ and 1 + ρi|+. Thefirst rule is flooring at zero that considers no short-selling and also correspondsto the case of a = 0, b = 1 in Eq. (18). The second rule is the linear version ofthe first rule that is the case of a = 1, b = 1 in Eq. (18). The constant termallows investors to impose positive weights across all the assets in the universe.,i.e., at least non-zero weights to all the assets in the universe opposite to Eq.(17).

For pure risk measures such as maximum drawdown, unconditional/conditionalvolatility/variance, VaR, and CVaR, the allocation rules of 1/ρi and 1− ρi areexploited for penalizing downside risks during the portfolio construction. Theformer allocation is the case of a = 0, b = 1 in Eq. (21) and the latter is a = 1,b = 1 in Eq. (20). The flooring to avoid short-selling is not considered for riskmeasures because negative risk measures are usually not common.

The alternative portfolios constructed by the diversification rules of reward-risk measures mentioned above are being held for the next six months. After sixmonths, the portfolios are rebalanced based on the metrics at that moment. Theportfolio construction process occurs every month, and one-sixth of the overallportfolio is replaced by the processes described in the previous paragraphs.

4. Results

In this section, we provide performance results and risk profiles of variousdiversified reward-risk parity portfolios in the investment universes introducedin the previous section. First of all, we visit each investment universe to as-sess reward-risk profiles of the alternative portfolio construction methods. Andthen we summarize common features and characteristics found in the diversi-fied reward-risk parity portfolios across the various markets. For brevity, wemainly represent the results by the reward-risk measures from the ARMA(1,1)-GARCH(1,1) model with the CTS distribution through the section. However,we also summarize the outcomes of diversified reward-risk parity from the othertwo risk models.

4.1. Global equity benchmarks

According to Table 6, diversified reward-risk parity portfolios outperformthe equally-weighted portfolio in the universe of global equity benchmarks. Themost lucrative portfolio in the universe is constructed from the raw allocationrule of Calmar ratio that achieves annually 9.92% under the standard deviationof 14.94% while the equally-weighted portfolio earns annually 6.70% with thevolatility of 12.91%. Similar to the raw allocation, the linear version of the di-versified Calmar ratio allocation also gains outperformance with respect to theequally-weighted portfolio. Both raw and linear Calmar ratio strategies obtainincreases in average return and standard deviation. Although the volatility isincreased by 16.42% and 2.32% with respect to the benchmark, there is larger

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improvement in profitability by 48.06% and 10.75%, respectively. Addition-ally, the linear diversification of Calmar ratio is less exposed to extreme eventsrepresented with thinner tails than the benchmark.

Sharpe ratio is also useful to construct outperforming portfolios from di-versified reward-risk parity. By using the raw allocation rule, unconditionaland conditional Sharpe ratio-based strategies achieve the annualized profitabil-ity (volatility) of 9.38% (14.69%) and 8.29% (13.15%), respectively. While theannualized return fluctuation is increased by 13.79% and 1.86% with respect tothe equal weight allocation, these Sharpe ratio portfolios obtain 40.00%- and23.73%-larger profits than the benchmark. Similar to the raw allocation, thelinear diversification rules of these Sharpe ratios also earn higher returns of6.77–6.81% with standard deviations of 12.92–12.93%. Except for the raw al-location rule of marginal Sharpe ratio, the return distributions of other threeSharpe ratio portfolios are in more desirable shapes with more left-skewed buthigher averages than that of the benchmark.

Other diversified reward-risk ratio parity strategies also exhibit more en-hanced performances than the equally-weighted portfolio does. In particular,the raw diversification rules tend to perform stronger than the linear alloca-tion rules. Annualized returns of the STAR ratio portfolios are in the rangeof 6.75–8.44% which are above the benchmark profitability, and volatility lev-els of 12.92–13.25% are comparable with that of the equally-weighted portfolio.For example, 26%-higher annual profits are earned by the STAR ratio (90%)portfolio, which is the best performing portfolio among STAR ratio, althoughthe volatility of the portfolio is increased by only 2.6% with respect to theequally-weighted portfolio. The same pattern of much improved profitabilitywith slightly increased volatility is also observed in the performance of diver-sified Rachev ratio portfolios. It is straightforward that higher moments ofthe portfolio performance for these ratio-based portfolios are more favorable,i.e., the return distributions are more left-skewed and the kurtosis values arecomparable in size with the enhanced average returns.

Similar to the diversified reward-risk ratio portfolios, it is noteworthy toconfirm from Table 7 that diversified risk-based allocations, in particular, linearversions of diversified risk measures are also useful for constructing portfolioswith similar-sized profits and reduced volatility levels with respect to the equalweight strategy. For example, linearly-diversified VaR and CVaR parity portfo-lios achieve comparable average returns of 6.68–6.70%, and the performance ofthese strategies is less volatile than that of the benchmark. The same pattern inperformance is found in the volatility/variance-based portfolios. Being in linewith the previous example, the linearly-diversified maximum drawdown parityprovides slightly weaker but less volatile performance. Meanwhile, inverse allo-cation rules of pure risk measures construct portfolios with lower profitabilityand slightly reduced volatility levels.

As reported in Table 6 and Table 7, reward-risk profiles of the diversifiedreward-risk parity portfolios are improved with respect to those of the equally-weighted allocation. Larger Sharpe ratios and Calmar ratios are earned bythe diversified reward-risk ratio strategies. Among these alternative portfolios,

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the linear diversification allocations are less exposed to downside risks in maxi-mum drawdown, VaR, and CVaR. Additionally, the quantile ratios of VaR andCVaR also indicate that these alternative reward-risk portfolios are equippedwith thinner downside tails. The pure risk-based constructions are less risky invarious risk measures than the equal weight allocation. However, Sharpe ratiosof the diversified risk parity portfolios are slightly reduced with respect to thebenchmark.

It is noteworthy that portfolio turnover rates can be improved by the di-versified reward-risk parity allocations. In particular, the linear diversificationrules, regardless of measure types, generally reflect lower transaction costs thanother allocation rules of corresponding measures. While the benchmark portfoliorebalances 3.81% of its positions on average, the turnover rate of the linearly-diversified maximum drawdown strategy is 3.61%. Additionally, turnover ratesof the linear VaR and CVaR allocations are slightly decreased to 3.76–3.79%.Similarly, the standard deviation/variance portfolios obtain less turnover ratesof 3.79–3.81%. Other linear ratio-based portfolios from STAR ratio, Rachevratio, and Sharpe ratio also replace positions at moderate levels of 5.07–9.62%.Meanwhile, the linear Calmar ratio strategy yields the high transaction cost of20.78% among the ratio-based portfolios.

Opposite to the linear allocations, turnover rates of the raw and inverseallocation rules are relatively higher than those of the benchmark or the lineardiversification allocations. For example, the portfolios constructed from inverserisk metrics replace slightly higher portions of portfolios. The inverse maximumdrawdown allocation substitutes 9.73% of its notional on average. Turnoverrates of the diversified volatility and variance (VaR and CVaR) portfolios arein the range of 8.40–13.97% (14.20–16.64%). Meanwhile, alternative reward-risk parity strategies constructed by the raw allocation rules tend to mark evenhigher transaction costs than the benchmark. Relatively moderate transactioncosts of 11.80–18.19% are charged by the diversified Rachev ratio portfolios. Thetwo Sharpe ratio-based portfolios replace 58.98% and 63.57% of the notionalson average. The worst transaction costs of 65.25–66.14% are yielded by theCalmar ratio-based and STAR ratio-based allocations.

4.2. Largest mutual funds and ETFs

Similar to the global equity benchmark universe, summary statistics in Table8 indicate that diversified reward-risk ratio parity allocations in the largestmutual funds and ETFs universe produce superior portfolios in average returnto the equal weight allocation. In particular, the most profitable portfoliosare constructed from the asset allocation rules based on Rachev ratio. Whilethe benchmark allocation yields the annualized return (standard deviation) of7.86% (17.78%), raw and linear diversification rules by Rachev ratio (99%, 99%)earn the profitability (volatility) of 7.91–7.93% (17.86–17.90%) per year. Thesame pattern with higher profits of 7.89–7.91% and slightly larger volatility of17.89–17.96% is also found in Rachev ratio (95%, 95%) and Rachev ratio (90%,90%) strategies. Comparable-sized annual returns of 7.77–7.87% and standarddeviations of 17.83–17.95% are obtained by the raw and linear diversification

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rules of the Rachev ratios using the 50%-quantile in the nominator. Returndistributions of all the diversified Rachev ratio parity allocations are more left-skewed but slightly heavier-tailed with larger kurtosis.

Portfolio construction by linearly-diversified reward-risk ratio parity consis-tently outperforms the equally-weighted portfolio. First of all, average returnsof the linear STAR ratio strategies are improved to 7.87–7.88%, and volatilitylevels of the portfolio performances are comparable with that of the equally-weighted portfolio. Additionally, linearly-diversified allocations by marginaland conditional Sharpe ratios achieve the profitability (volatility) of 7.84–7.86%(17.75–17.93%). In particular, the unconditional Sharpe ratio portfolio is lessvolatile than the benchmark allocation. Meanwhile, the raw allocations of thereward-risk ratios in the mutual fund universe provide much weaker profits ingeneral. Moreover, the return distributions of these raw ratio-based diversifica-tion portfolios are less desirable because the returns are more right-skewed andheavier-tailed than that of the benchmark.

In Table 9, diversified reward-risk parity using pure risk measures is also agood allocation scheme in the largest mutual funds and ETFs universe. Similarto the cases of the reward-risk ratio measures explained above, linear alloca-tions of pure risk measures such as volatility/variance, VaR, and CVaR are notonly slightly improved in performance but also less volatile in annualized returnthan the equally-weighted portfolio. For example, linearly-diversified VaR andCVaR strategies earn annually 7.84–7.86% with the volatility of 17.71–17.75%.Average annualized returns of diversified volatility portfolios are around 7.86%,and the portfolio performance fluctuates between 17.76% and 17.78% per year.In addition, the maximum drawdown allocation obtains a slightly decreasedaverage return and standard deviation. Meanwhile, inverse allocations of riskmetrics in general gain significantly poorer profitability although the returns ofthese strategies are much less volatile than the benchmark. For example, profitsand volatility levels of inversely-diversified VaR and CVaR portfolios are about25% decreased simultaneously.

Reward-risk measures in Table 8 and Table 9 indicate that the outperfor-mance of the diversified reward-risk parity allocations is achieved with takingless downside risks. For example, all the linear risk diversification portfolios ex-hibit not only larger marginal/conditional Sharpe ratios and Calmar ratios butalso more improved maximum drawdowns, VaRs, CVaRs, and quantile ratiosof VaR and CVaR than the benchmark. Additionally, the inverse risk measureallocations usually generate relatively enhanced downside risks in maximumdrawdown, VaR, CVaR, VaR ratio, and CVaR ratio. The similar pattern is alsoobserved for the diversified ratio portfolios. The Rachev ratio (99%, 99%) rulegains the performance with better Sharpe ratios, Calmar ratio, and risk profiles.Lower VaR and CVaR values are recorded by the diversified Rachev ratio (95%,95%) and Rachev ratio (90%, 90%) portfolios. Moreover, the STAR ratio andSharpe ratio allocations also obtain comparable or higher Sharpe ratios as wellas better risk profiles with improved maximum drawdown, VaR and CVaR, VaRratio, and CVaR ratio from the daily portfolio returns indicating less exposuresto downside risks than the equally-weighted portfolio.

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Turnover rates can be reduced by exploiting diversified reward-risk parity. Ingeneral, linearly-diversified metrics tend to construct portfolios with lower trans-action costs. Comparing with the benchmark turnover of 2.98%, the linear VaRand CVaR allocations are the best portfolios for reducing average turnover ratesto 2.96–2.99%. The four portfolios from the linear diversification of volatilityand variance with turnover rates of 2.97–2.98% are also superior in transactioncost to the benchmark. Comparable turnover rates of 3.00% and 3.06% areobtained by the linear Sharpe ratio and maximum drawdown strategies, respec-tively. In addition, transaction costs of the diversified Rachev ratio portfoliosare in the range of 4.44–6.24%. Meanwhile, the average turnover rate of thelinear Calmar ratio strategy is 14.58%, relatively much higher than other linearstrategies.

However, portfolio turnover rates of the diversification rules by the rawreward-risk ratio measures and the inverse risk metrics are worse than thoseof the linear versions and the benchmark. The inverse allocation of maximumdrawdown costs 6.45% of its notionals on average, which is the best portfolio intransaction cost among the raw and inverse allocations. The diversified volatil-ity/variance (VaR/CVaR) portfolios replace the positions at moderate levelsof 7.87–11.41% (13.07–15.09%). Among the raw diversification strategies, theRachev ratio portfolios are the best turnover rate portfolios of 9.03–13.61%.However, the Rachev ratio strategies are exception cases for high transactioncosts of the raw ratio allocations. About a half of portfolio notionals on averageis replaced by the raw allocations of reward-risk measures: the Calmar ratioportfolio of 50.01%, the marginal and conditional Sharpe ratios of 44.81% and56.61%, respectively. The raw allocations by STAR ratio are heavily rebalancedwith turnover rates of 57.61–58.31%, the worst substitution percentages amongall the portfolios.

4.3. U.S. equity markets: SPDR U.S. sector ETFs

Table 10 reports that all diversified reward-risk ratio allocations achieveimproved performance over the benchmark allocation in the SPDR U.S. sectorETF universe. While the equally-weighted portfolio obtains 8.91% per year,the most lucrative portfolio is constructed from the raw diversification rule ofCalmar ratio, allowing the annual profit of 9.73% and the standard deviationof 19.45%. The profitability is increased by 10% but the volatility is increasedby about 4% with respect to the benchmark. Among all linear allocations ofreward-risk ratios, the strongest profitability of 9.16% is also achieved by thelinear diversification of Calmar ratio. Moreover, with the improved performance,the volatility of the linear Calmar ratio strategy is actually reduced with respectto the benchmark.

All diversified Sharpe ratio portfolios also dominate the benchmark perfor-mance. Raw and linear allocation rules by unconditional Sharpe ratio annuallyearn 9.43% and 8.93% on average, respectively. Additionally, two conditionalSharpe ratio portfolios of 9.27% and 8.96% outperform the benchmark portfolio.Although the standard deviation of the raw allocation by marginal Sharpe ratiois slightly higher than that of the equal weight portfolio, all the other Sharpe

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ratio portfolios are less volatile than the benchmark. Moreover, the diversifiedSharpe ratio strategies are less exposed to extreme events with smaller kurtosis.

Similar to the Sharpe ratio portfolios, other diversified reward-risk ratioparity portfolios using STAR ratio and Rachev ratio mark outperformance overthe benchmarks. The diversification rules of STAR ratio earn superior profits of8.92–9.23% to the equally-weighted allocation. Except for Rachev ratio (99%,99%), all diversified Rachev ratio allocations exhibit improved profitability withrespect to the benchmarks. All these reward-risk ratio portfolios are less volatilein performance and thinner-tailed in return distribution. In particular, the rawallocation rules of Rachev ratio exhibit not only stronger performance but alsolower volatility levels than the linear versions.

As found in Table 11, diversified reward-risk parity portfolios constructedby VaR, CVaR, volatility, variance, and maximum drawdown are as good asor better in profitability than the benchmark. For example, diversified VaRand CVaR allocations mark comparable annualized profits of 8.75–8.93% andmuch lower volatility levels of 17.61–18.69%. Similarly, diversified risk paritystrategies of standard deviation and variance earn annualized returns (standarddeviation) of 8.57–8.91% (17.68–18.71%). Similar-sized average returns of 8.88–8.89% and lower standard deviations of 17.91–18.40% are obtained by the rawand inverse allocation rules of maximum drawdown. Across all the diversifiedrisk parity portfolios, the linear versions of the allocations consistently gainstronger performance, and the inverse allocations of the metrics are helpfulfor substantially reducing the volatility of the performance. Moreover, thesealternative strategies are more negatively skewed in return distribution.

Reward-risk measures in Table 10 and Table 11 indicate that the portfoliosconstructed by the diversified reward-risk parity are less exposed to downsiderisks than the equally-weighted portfolio. Considering the outperformance ofthe alternative strategies mentioned in the previous paragraphs, the low-risk as-pects of the diversified reward-risk allocations are intriguing. First of all, everystrategy exhibits a lower maximum drawdown than the benchmark allocation.Additionally, Sharpe ratios and Calmar ratios for most of the reward-risk parityportfolios are enhanced although very small numbers of the alternative portfo-lios fail to achieve higher reward-risk ratios than the equally-sized allocationdoes. Besides these improved ratio statics, the diversified reward-risk parityallocations, except for the raw Sharpe ratio and Calmar ratio allocations, arealso less riskier in maximum drawdown, VaR, CVaR, and quantile ratios forVaR and CVaR than the equally-weighted allocation.

Saving transaction costs is another advantage of implementing diversifiedreward-risk parity portfolios in the SPDR U.S. sector ETF universe. In general,turnover rates of the linearized measure-based allocations are comparable withor more improved than that of the equally-weighted portfolio, 2.89%. For ex-ample, the linear allocation rules by maximum drawdown and marginal Sharperatio decrease transaction costs of the portfolios to 2.45% and 2.55% of port-folio notionals, respectively. The linear versions of the marginal/conditionalvolatility and variance parity also replace 2.86–2.89% of the portfolios. Thelinearly-diversified VaR and CVaR strategies rebalance 2.80–2.83% on average.

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Reminding the outperformance of the alternative allocations constructed bySTAR ratio and Rachev ratio, slightly higher turnover rates of 3.67–6.42% arealso endurable. Meanwhile, the transaction cost of the linear Calmar ratiostrategy is 18.80% on average, relatively much more expensive than other lineardiversification strategies.

However, the raw ratio allocations and the inverse risk allocations tend toyield more adverse turnover rates. The inverse allocations of risk measures markrelatively reasonable transaction costs. For example, the maximum drawdownand volatility/variance measures rebalance 5.22–10.61% of portfolios, and the in-verse VaR and CVaR portfolios have turnover rates of 11.04–14.10%. Among theratio measures, the raw diversification of Rachev ratios is moderate in turnoverrate with 10.13–16.35%. The situation of other portfolios by reward-risk ratiomeasures is worse. The unconditional and conditional Sharpe ratio portfolioswith the raw allocation rule mark turnover rates of 50.06% and 61.51% on av-erage, substantially more inferior than those of the other portfolios. The rawCalmar ratio strategy replaces 55.96% on average. Comparing with all the di-versified reward-risk parity portfolios including the benchmark case, the rawSTAR ratio allocations are the worst portfolio rebalancing of 62.70–63.67% onaverage.

4.4. U.S. equity markets: Dow Jones Industrial Average

According to Table 12, it is noteworthy that diversified reward-risk par-ity based on ratio measures achieve enhanced performances than the equally-weighted portfolio of 9.84% per year in U.S. Dow Jones Industrial Averageuniverse. In particular, the Calmar ratio strategies are the most lucrative ones.The raw and linear allocation rules of Calmar ratio earn annually 10.32% and10.06% on average, respectively. Additionally, the performance of both alloca-tions is less volatile with standard deviations of 18.68% and 19.49% than thebenchmark of 19.82%. Moreover, the return distributions are more desirablewith more negative skewness and smaller kurtosis than that of the benchmark.

Similar to the Calmar ratio portfolios, diversified Sharpe ratio allocations arealso useful to construct portfolios that outperform the equally-weighted portfo-lio in average return and volatility. All four Sharpe ratio strategies consistentlyincrease average returns and decrease volatility levels. For example, the diver-sified conditional Sharpe ratio parity achieves annualized 10.01% on average,the second strongest portfolio in profitability among all other portfolios includ-ing the benchmark. Additionally, the outperformance is based on smaller returnfluctuations in the range of 18.67–19.74%. Furthermore, the return distributionsof the diversified Sharpe ratio strategies are thinner-tailed with smaller kurto-sis indicating that the portfolios are less exposed to the probability of extremelosses.

Diversified reward-risk parity in STAR ratio and Rachev ratio is also moreprofitable with reduced volatility levels. Similar to the previous cases, the rawdiversification rules of the reward-risk ratio measures tend to outperform the lin-ear allocations of the same measures. For example, the raw STAR ratio (90%)-based parity is the best STAR ratio portfolio with generating annually 10.05%

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on average. Moreover, comparing with the benchmark volatility of 19.82%, thestandard deviation of the STAR ratio (90%) allocation is decreased to 19.00%.The same pattern is also observed for all other raw and linear allocations ofSTAR ratio and Rachev ratio. The profitability of all the other diversified STAR(Rachev) ratio strategies is stronger in the range of 9.83–9.94% (9.81–9.87%)with lower volatility levels of 19.03–19.76% (19.53–19.77%). All the diversifiedSTAR ratio and Rachev ratio portfolios are less exposed to extreme events withlower kurtosis values than the benchmark is.

In Table 13, diversified reward-risk parity allocations by pure risk measuressuch as VaR, CVaR, standard deviation, variance, and maximum drawdownearn comparable or weaker profitability than the equally-weighted allocation.The performance of linearly-diversified risk parity strategies is generally moreprofitable than inversely-diversified risk metric strategies. Meanwhile, the in-verse risk metric portfolios are less volatile than the benchmark and the linearrisk parity portfolios. For example, the performance of the linearly-diversifiedVaR and CVaR strategies is in the comparable range of 9.79–9.83% while thevolatility levels are decreased to 19.65–19.75%. Although profits of the inverseVaR and CVaR portfolios decrease, volatility levels of these portfolios are alsoreduced. The similar pattern is found in the performance of the diversified stan-dard deviation and variance allocations. However, there is notable reduction inannualized performance for the maximum drawdown allocations.

Reward-risk profiles reported in Table 12 and Table 13 support that the di-versified reward-risk parity portfolios are less risky in various reward-risk mea-sures than the benchmark portfolio. Most diversified reward-risk parity rulesachieve higher unconditional Sharpe ratios and Calmar ratios than the equally-weighted allocation. Many alternative portfolios also gain higher conditionalSharpe ratios. These diversified parity portfolios, except for the raw alloca-tions of Sharpe ratio and Calmar ratio, are less exposed to downside tail risks.Besides, the portfolios constructed from the linear allocations of Calmar ratio,Sharpe ratio, STAR ratio, and risk measures exhibit not only lower maximumdrawdowns but also smaller VaRs and CVaRs than the benchmark. Among thevarious portfolios, every risk measure of the diversified maximum drawdownand variance portfolios is at the lowest level. In particular, Sharpe ratios andCalmar ratios of the maximum drawdown portfolios are greater than those ofthe benchmark, and the maximum drawdown strategies are less risky in VaR,CVaR, and maximum drawdown than the benchmark. Moreover, the inverseVaR and CVaR allocations also yield smaller downside risk measures.

It is noticeable that turnover rates of the linearly-diversified reward-riskparity allocations in the Dow Jones Industrial Average universe are improved.Comparing with the benchmark turnover rate of 6.75%, the linear diversifica-tion rules by Sharpe ratio, maximum drawdown, STAR ratio, standard devia-tion, variance, VaR, and CVaR are helpful to construct portfolios with reducedturnover rates of 5.49–6.69%. In particular, the linear allocations of maximumdrawdown and Sharpe ratio exhibit the lowest average turnover rates of 5.49%and 5.79%, respectively. It is a reminder from the previous paragraphs that thelinear Sharpe ratio strategy is one of the best performers, and all the risk mea-

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sures are smaller than those of the benchmark. Meanwhile, the turnover rateof the linearized Calmar ratio strategy is 26.33%, the worst among the linearallocation strategies.

However, the raw allocations of reward-risk ratio measures or the inverse al-locations of pure risk measures are the worst portfolios in turnover. Among thesetwo allocation types, the raw allocation rules of reward-risk ratios tend to yieldhigher turnovers. The inverse diversification rules of volatility/variance andmaximum drawdown yield 9.00–18.73% and 12.29%, respectively. In addition,turnover rates of 14.20–17.18% are charged by the inverse allocations of VaRand CVaR. Among the raw allocations, the Rachev ratio portfolios rebalancethe assets in the portfolios with relatively low turnover rates of 13.75–18.90%.Meanwhile, the strategies from Calmar ratio and marginal/conditional Sharperatios replace 66.07%, 58.46%, and 66.84% of notionals, respectively. The worstperformers in turnover rate are all the STAR ratio portfolios of 68.58–69.20%.

4.5. Overall results

Regardless of asset class and market, the diversified reward-risk parity strate-gies outperform the traditional equally-weighted allocation. In many marketuniverses, the most profitable portfolio is constructed from the raw allocationsof Calmar ratio and Sharpe ratio. However, these strategies are not the bestdiversification rules from the viewpoint of risk management. For example, theportfolios are heavily exposed to downside tail risks. This risky aspect of theportfolio performance is exemplified by the finding that the pure risk measuressuch as VaR and CVaR of the portfolios are substantially poorer than the corre-sponding measures of the benchmark strategy. Additionally, the turnover ratesare also noticeably worse than that of the equally-weighted portfolio in eachasset class. Apparently, the higher turnover rate imposes massive portfolio re-balancing that implies expensive transaction costs enough to erode considerableportions of the outstanding portfolio performance.

In this perspective, using STAR ratio or Rachev ratio in portfolio construc-tion with the raw allocation rule is a good alternative in order to pursue thebalance between performance and risk. The raw diversification rules by STARratio and Rachev ratio are two of the most profitable portfolio constructionmethods that consistently outperform the equally-weighted allocation in everymarket covered in the paper. Moreover, higher moments such as standard devia-tion and kurtosis of the performance are also improved. Besides the moments ofreturn distributions, improved reward-risk profiles of the portfolios indicate thatthe portfolios are more desirable to construct not only for profitability but alsofor risk management. Sharpe ratios and Calmar ratios of these strategies areincreased, and maximum drawdowns tend to be smaller than the benchmarkdrawdown. In addition, VaR and CVaR of the portfolios are slightly largerthan or comparable with the risk measures of the equally-weighted allocations.Moreover, turnover rates of the diversified Rachev ratio portfolios are moder-ately greater in all the asset classes. Meanwhile, the worst turnover rates amongall the alternative portfolios are obtained by the STAR ratio parity strategies.

23

The higher transaction costs caused by massive rebalancing are regarded as thetrade-off for achieving the enhanced profitability and risk management.

Similar to the raw allocation of Rachev ratio, the diversified reward-risk ratioparity with the linear allocation rule is the most well-balanced allocation acrossall the facets of portfolio management such as performance, downside risks, andtransaction costs. For example, the linearly-diversified STAR ratio portfoliosnot only outperform the equally-weighted portfolio but also are less volatile inreturn than the benchmark in all the market universes. Lower kurtosis valuesof the portfolios indicate that the portfolios are less exposed to heavy-tailedevents than the equally-weighted portfolio. In addition to the higher momentsof returns, greater Sharpe ratios of the portfolio performances also supportthe superiority in profitability of the linearly-diversified STAR ratio allocationrules. Moreover, the portfolios are less riskier in various risk measures than thebenchmark. In various asset classes, VaR, CVaR, and maximum drawdown ofthe STAR ratio strategies are decreased. Lastly, turnover rates of the allocationsare also reduced with respect to the equally-weighted allocation. Based on thesmaller turnover rates, transaction costs can be saved even though the portfoliosearn the more profitable performances under the alleviated downside risks. Thesimilar patterns are also observed in the performance of other linearly-diversifiedreward-risk parity portfolios. The improved reward-risk profiles accompaniedwith the outperformance at the lower transaction costs are also consistent withthe low volatility anomaly (Blitz et al. (2007); Baker et al. (2011)).

Diversified reward-risk parity using pure risk measures is also good at man-aging severe downside tail risks and avoiding high turnover rates. In particular,the linearly-diversified pure risk measures construct portfolios with comparableaverage returns and slightly lower volatility levels. For example, the portfolioperformance characteristics of the VaR and CVaR allocations are almost sim-ilar to those of the linearly-diversified STAR ratio portfolios. Moreover, thesestrategies with enhanced Shape ratios and Calmar ratios also exhibit reduceddownside risks in various risk measures. An additional advantage of these lin-earized risk parity allocations is taking low transaction costs. Similarly, thelow-risk nature is also found in the inversely-diversified risk parity portfolios.Although the portfolios tends to be less lucrative than any other alternativeportfolios, the portfolio performance yields much lower volatility levels thanother portfolios including the benchmark. Additionally, Sharpe ratios and Cal-mar ratios of the inversely-diversified risk parity strategies are also improved,and these risk-based allocations are significantly less riskier in VaR, CVaR, andmaximum drawdown than the other allocations. Meanwhile, the transactioncosts are moderately worse than that of the benchmark portfolio.

It is also noteworthy that the diversified reward-risk parity is robust tomodel selection. Through this section, we present the portfolio performanceand risk profiles of the diversified reward-risk parity portfolios constructed basedon reward-risk measures calculated only from the ARMA-GARCH-CTS model.However, the results is not dependent on the choice of the risk model. When theARMA-GARCH-NTS model and the ARMA-DCC GARCH-MNTS model areused for reward-risk calculation, the similar patterns in performance and risk

24

profiles of the diversified reward-risk parity portfolios are also found. Reward-risk measures from these models also construct the diversified reward-risk parityportfolios with enhanced performance, improved downside risks, and reducedtransaction costs although the reward-risk profiles of those portfolios by theNTS model and the MNTS model are not presented in the paper for brevity.Additionally, the same portfolio characteristics of a given reward-risk measureunder an allocation rules are observed regardless of risk model.

5. Factor analysis

In this section, we analyze the performance and risk profiles of the diversifiedreward-risk parity portfolios with the Carhart four-factor model. The factoranalysis on the returns of the alternative portfolios in SPDR U.S. sector ETFuniverse is conducted in order to seek better understandings on the diversifiedreward-risk parity portfolio construction.

In the previous section, it is found that the diversification portfolios of thereward-risk measures achieve the higher average returns, and are also exposed tothe lower downside tail risks. The improved portfolio profitability and reward-risk profiles support the existence of the low volatility anomaly (Blitz et al.(2007); Baker et al. (2011)).

In order to scrutinize the low volatility anomaly observed at the diversifiedreward-risk allocations, the factor analysis on the portfolio returns is indispens-able. Among various factor models, one of the most commonly-used factormodels is the Carhart four-factor model (Carhart (1997)) that exploits four dif-ferent factors: market factor (Mkt), size factor (SMB), value factor (HML), andmomentum factor (MOM). The first three factors are also known as the Fama-French three factors (Fama and French (1993, 1996)). The factor analysis onportfolio returns rp is conducted as the linear regression with respect to the fourCarhart factors:

rp = αp + βp,MktfMkt + βp,SMBfSMB + βp,HMLfHML + βp,MOMfMOM + ǫp (22)

where αp is the intercept, ǫp is the residual, and βp,i is the factor loading forthe Carhart factor fi. The historical data of the four factors were downloadedfrom K. R. French’s data library at Dartmouth.

Regression results of the Carhart four-factor analysis on the portfolio per-formance in the SPDR U.S. sector ETF universe are given in Table 14. Itis noteworthy that the diversified reward-risk parity allocations still outper-form the equally-weighted allocation even after controlling the Carhart factors.Regression intercepts of most alternative portfolios are greater than the bench-mark intercept. For example, all the diversified risk-based parity strategiesgain statistically significant and larger factor-adjusted returns than that of thebenchmark. In particular, the inverse risk allocations achieve almost 14.29–40.48%-higher alphas with statistical significance at 1%. Intercepts of all theraw diversification strategies of Rachev ratio, except for Rachev ratio (99%,

25

99%), are maximally 9.37%-greater than that of the benchmark. The linearly-diversified reward-risk portfolios mark slightly higher factor-adjusted returnsthan the benchmark. These larger intercepts are also consistent with the port-folio performance in section 4. Meanwhile, the raw ratio allocations by Sharperatio, Calmar ratio, and STAR ratio exhibit relatively smaller alphas in thefactor analysis. For example, the Carhart four-factor intercept of the Sharperatio-based allocation is 32% decreased than that of the equally-weighted port-folio. Considering the outperformance of the diversified raw reward-risk ratiostrategies reported in the previous section, it is obvious that large portions ofthe profitability earned by these raw reward-risk ratio portfolios are originatedfrom the factor exposures.

The patterns of the intercepts described in the above paragraph are relatedto the factor exposures. Although the alternative allocations and the bench-mark allocation exhibit positive (negative) exposures on the market (size) andvalue (momentum) factors, each group of portfolios are exposed to the factorsin different ways. The inverse risk strategies, which are the most profitable per-formers in factor-adjusted return, are 5–10% less exposed to the market factorand the value factor, but about 20% more exposed to the size factor than theother portfolios including the benchmark. While the market exposures of theraw Rachev ratio allocations are comparable, the exposures to the size factor andthe value factor are 10-15% decreased. The factor exposures of the linearizedreward-risk allocations are similar to the factor structure of the equally-weightedallocation. Moreover, these factor loadings are statistically significant at 1%,and magnitude differences in factor loading between the benchmark and thealternative allocations are relatively small. Furthermore, high R2 values in theregression results of these portfolios indicate that the portfolio performances arewell-explained by the Carhart factor model.

Meanwhile, the performances of the raw diversification portfolios from thereward-risk ratios such as Sharpe ratio, Calmar ratio, and STAR ratio exhibitdifferent patterns in factor exposure with the factor analysis results of any otherallocations. Opposite to other alternative portfolios, the momentum exposuresof the diversified reward-risk ratio strategies are not only positive but also sta-tistically significant at 1% while the factor loadings on the other three factorsare also statistically significant at 1%. This is related to the dependence on theaverage return in the definition of these reward-risk ratio measures. Althoughthe exposure to the market factor is in similar size with the benchmark, boththe size factor and the value factor have much lower factor loadings. Addi-tionally, the exposures to the momentum factor are not only more increased inmagnitude but also statistically significant. Although the R2 value of the linearregression is still high, it is relatively lower than the other regression results.

It is also noteworthy that the factor structure for the performance of thelinearly-diversified Calmar ratio portfolio is unique. The intercept, the marketexposure, and the value exposure are in similar size with the benchmark andother linearized reward-risk portfolios. However, similar to the raw reward-riskratio portfolios, the factor loading on the size factor is about 15% reduced, andthe momentum exposure is doubled and positive. The R2 value is still high

26

enough to explain the portfolio performance with the Carhart factor model.

6. Conclusion

In this paper, we introduce alternative reward-risk parity strategies basedon the diversification by reward-risk measures. The diversified reward-risk par-ity is the two-folded generalization of applying reward-risk measures and moregeneric allocation rules to traditional diversified risk parity portfolios such asthe DRP portfolio and the ERC portfolio. With a certain allocation rule, theequally-weighted diversification is also a special case of the diversified reward-risk parity portfolio. In order to capture autocorrelation, volatility clustering,asymmetry, and heavy tails of financial time series, the reward-risk measures arecalculated from three different risk models: the ARMA(1,1)-GARCH(1,1) modelwith CTS/NTS innovations and the ARMA(1,1)-DCC GARCH(1,1) model withMNTS innovations.

Regardless of financial markets, the new heuristic reward-risk parity port-folios outperform the equally-weighted portfolio. The diversified reward-riskparity allocations are more profitable in annualized return than the benchmark.In particular, the diversification rules by Calmar ratio and Sharpe ratio are thebest ways of allocating capital to increase portfolio profits. Additionally, thereward-risk portfolios constructed from Rachev ratio and STAR ratio are alsoimproved in profitability with respect to the benchmark. Other diversified riskallocations by volatility/variance, VaR, CVaR, and maximum drawdown exhibitcomparable average returns and volatility measures.

The outperformance of the diversified reward-risk parity strategies is achievedby taking less downside risks. Many alternative portfolios in diverse asset classesare consistently less exposed to various risks represented by standard deviation,VaR, CVaR, and maximum drawdown than any other portfolios including theequally-weighted portfolio. Moreover, the skewness and kurtosis of the port-folio performances are improved. The Sharpe ratios and Calmar ratios of thealternative portfolios are also greater than those of the other portfolios.

In addition, the diversified reward-risk parity is robust to risk model choice.When the ARMA(1,1)-GARCH(1,1) model with NTS innovations and the ARMA(1,1)-DCC GARCH(1,1) model with MNTS innovations are used for calculatingreward-risk measures in order to construct the diversified reward-risk parityportfolios, we also observe the same patterns in portfolio performance and riskprofiles with the portfolios from the CTS reward-risk measures.

The Carhart four-factor analysis also supports the outperformance of the di-versified reward-risk parity portfolios. After controlling the Carhart factors, theportfolio performances still remain more profitable than the equally-weightedportfolio. With high R2 values, the larger regression intercepts are obtained bythe diversified reward-risk allocations. The inverse risk parity portfolios achievethe largest factor-adjusted returns among the alternative portfolios, and the rawRachev ratio portfolios also mark stronger alphas than the benchmark. Addi-tionally, the intercepts of the linearly-diversified reward-risk portfolios consis-

27

tently exceed that of the benchmark. Contrary to the profitability, the lowestintercepts are acquired by the raw allocations of Sharpe ratio and Calmar ratio.

In conclusion, the portfolio construction based on diversified reward-riskparity not only outperforms the equally-weighted portfolio but also providesenhanced risk management. The alternative portfolios are evidence for the lowvolatility anomaly.

Acknowledgement

The formulation of the key idea and its implementation were completed in2014–2015 while Jaehyung Choi was attending the State University of New Yorkat Stony Brook. The initial draft based on Bloomberg datasets by 2012 was alsowritten at that time. Major results were recently reproduced with up-to-datedatasets, and the contents of the initial draft were modified with minor revisionsin accordance with the new datasets.

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Table 4: Ticker information for Dow Jones Industrial Average

Ticker Name Minimum Date Maximum Date

AA Alcoa Corporation 1999-01-01 2013-09-23AAPL Apple Inc. 2015-03-19 2020-12-31AIG American International Group, Inc. 2004-04-08 2008-09-22AMGN Amgen Inc. 2020-08-31 2020-12-31AXP American Express Company 1999-01-01 2020-12-31BA The Boeing Company 1999-01-01 2020-12-31BAC Bank Of America Corporation 2008-02-19 2013-09-23C Citigroup Inc. 1999-01-01 2009-06-08CAT Caterpillar Inc. 1999-01-01 2020-12-31CRM Salesforce.Com, Inc. 2020-08-31 2020-12-31CSCO Cisco Systems, Inc. 2009-06-08 2020-12-31DD Dupont De Nemours, Inc. 1999-01-01 2019-04-02DIS The Walt Disney Company 1999-01-01 2020-12-31DOW Dow Inc. 2019-04-02 2020-12-31GE General Electric Company 1999-01-01 2018-06-26GM General Motors Company 1999-01-01 2009-06-08GS The Goldman Sachs Group, Inc. 2013-09-23 2020-12-31GT The Goodyear Tire & Rubber Company 1999-01-01 1999-11-01HD The Home Depot, Inc. 1999-11-01 2020-12-31HON Honeywell International Inc. 1999-01-01 2008-02-19HON Honeywell International Inc. 2020-08-31 2020-12-31HPQ Hp Inc. 1999-01-01 2013-09-23IBM International Business Machines Corporation 1999-01-01 2020-12-31INTC Intel Corporation 1999-11-01 2020-12-31IP International Paper Company 1999-01-01 2004-04-08JNJ Johnson & Johnson 1999-01-01 2020-12-31JPM Jpmorgan Chase & Co. 1999-01-01 2020-12-31KO The Coca-Cola Company 1999-01-01 2020-12-31KRFT Kraft Foods Inc. 2008-09-22 2012-09-24MCD Mcdonald’S Corporation 1999-01-01 2020-12-31MMM 3M Company 1999-01-01 2020-12-31MO Altria Group, Inc. 1999-01-01 2008-02-19MRK Merck & Co., Inc. 1999-01-01 2020-12-31MSFT Microsoft Corporation 1999-11-01 2020-12-31NKE Nike, Inc. 2013-09-23 2020-12-31PFE Pfizer Inc. 2004-04-08 2020-08-31PG The Procter & Gamble Company 1999-01-01 2020-12-31RTX Raytheon Technologies Corporation 2020-04-06 2020-08-31T At&T Inc. 1999-11-01 2004-04-08T At&T Inc. 2005-11-21 2015-03-19TRV The Travelers Companies, Inc. 2009-06-08 2020-12-31UNH Unitedhealth Group Incorporated 2012-09-24 2020-12-31UTX United Technologies Corporation 1999-01-01 2020-04-06V Visa Inc. 2013-09-23 2020-12-31VZ Verizon Communications Inc. 2004-04-08 2020-12-31WBA Walgreens Boots Alliance, Inc. 2018-06-26 2020-12-31WMT Walmart Inc. 1999-01-01 2020-12-31XOM Exxon Mobil Corporation 1999-01-01 2020-08-31

Ticker information for Dow Jones Industrial Average

32

Table 5: Allocation rules by reward-risk and risk measures

Measure Rule

Equal Weigtht 1Reward-risk ratio ρi|+

1 + ρi|+Risk 1/ρi

1− ρi

33

Table 6: Summary statistics for ratio portfolios in global equity benchmarks

Rule Measure Mean Std Skew Kurt. Cumul. Sharpe CSharpe Calmar Max DD VaR(95%) CVaR(95%) VaR95/99 CVaR95/99 Turnover

1 Equal Weight 6.7047 12.9093 -0.7144 10.2438 268.8017 0.5194 1.5745 0.1254 53.4596 0.6987 0.9374 0.6537 0.6972 3.8070

ρ|+ Sharpe 9.3762 14.6899 -0.6829 13.2473 532.2622 0.6383 5.2029 0.1571 59.6910 0.8969 1.3217 0.5708 0.6506 58.9799

ρ|+ Cond. Sharpe 8.2860 13.1492 -0.7976 10.2933 420.4911 0.6302 2.4552 0.1562 53.0430 0.7405 1.0377 0.6106 0.6783 63.5673

ρ|+ Calmar 9.9172 14.9423 -0.6256 11.4051 607.2031 0.6637 4.8674 0.1698 58.4161 1.0285 1.5178 0.5692 0.6500 65.7492

ρ|+ STAR(90%) 8.4420 13.2496 -0.7992 10.2773 437.2571 0.6372 2.4924 0.1569 53.7941 0.7413 1.0403 0.6098 0.6767 66.1352

ρ|+ STAR(95%) 8.4231 13.2231 -0.7931 10.2187 435.4351 0.6370 2.4742 0.1571 53.6053 0.7420 1.0413 0.6103 0.6754 65.6250

ρ|+ STAR(99%) 8.3669 13.1847 -0.7904 9.9958 429.4078 0.6346 2.3971 0.1569 53.3119 0.7454 1.0445 0.6113 0.6769 65.2535

ρ|+ R(50%,90%) 7.0855 13.0067 -0.7403 10.4852 300.2271 0.5448 1.7668 0.1328 53.3484 0.7042 0.9628 0.6365 0.6842 18.1892

ρ|+ R(50%,95%) 7.0363 13.0070 -0.7318 10.3807 295.8891 0.5410 1.7348 0.1320 53.2952 0.7070 0.9676 0.6351 0.6845 17.2495

ρ|+ R(50%,99%) 6.9386 13.0190 -0.7218 10.1733 287.2433 0.5330 1.6212 0.1303 53.2525 0.7043 0.9533 0.6464 0.6895 17.1484

ρ|+ R(90%,90%) 6.9670 12.9647 -0.7373 10.4171 290.3125 0.5374 1.8025 0.1302 53.5268 0.6934 0.9384 0.6468 0.6896 12.8468

ρ|+ R(95%,95%) 6.9214 12.9559 -0.7332 10.3433 286.4699 0.5342 1.8292 0.1293 53.5405 0.6907 0.9310 0.6499 0.6934 11.6388

ρ|+ R(99%,99%) 6.8776 12.9477 -0.7395 10.2949 282.8105 0.5312 1.8596 0.1282 53.6525 0.6878 0.9280 0.6497 0.6922 11.8010

1 + ρ|+ Sharpe 6.7715 12.9218 -0.7194 10.2696 274.1721 0.5240 1.6348 0.1264 53.5613 0.6936 0.9262 0.6573 0.7018 3.4403

1 + ρ|+ Cond. Sharpe 6.8121 12.9284 -0.7272 10.3495 277.4948 0.5269 1.5941 0.1273 53.5131 0.6981 0.9416 0.6491 0.6932 6.9454

1 + ρ|+ Calmar 7.4191 13.2023 -0.7536 10.0414 328.5960 0.5620 1.9173 0.1353 54.8343 0.6961 0.9601 0.6279 0.6806 20.7751

1 + ρ|+ STAR(90%) 6.7987 12.9315 -0.7248 10.3599 276.3446 0.5257 1.5875 0.1269 53.5863 0.6965 0.9345 0.6537 0.6969 6.2032

1 + ρ|+ STAR(95%) 6.7770 12.9249 -0.7220 10.3246 274.5973 0.5243 1.5891 0.1266 53.5434 0.7002 0.9412 0.6515 0.6966 5.3731

1 + ρ|+ STAR(99%) 6.7488 12.9177 -0.7209 10.2908 272.3424 0.5224 1.5870 0.1261 53.5011 0.6979 0.9350 0.6549 0.6985 4.5878

1 + ρ|+ R(50%,90%) 6.8221 12.9409 -0.7250 10.3471 278.2090 0.5272 1.6345 0.1276 53.4828 0.6978 0.9365 0.6536 0.6968 7.2078

1 + ρ|+ R(50%,95%) 6.7919 12.9366 -0.7207 10.2962 275.7068 0.5250 1.6202 0.1271 53.4545 0.6997 0.9395 0.6518 0.6983 6.2601

1 + ρ|+ R(50%,99%) 6.7506 12.9322 -0.7174 10.2383 272.3167 0.5220 1.5835 0.1263 53.4352 0.7104 0.9702 0.6345 0.6904 5.3847

1 + ρ|+ R(90%,90%) 6.8375 12.9372 -0.7272 10.3479 279.5298 0.5285 1.6960 0.1277 53.5231 0.6927 0.9305 0.6540 0.6948 7.4977

1 + ρ|+ R(95%,95%) 6.8138 12.9322 -0.7247 10.3036 277.5883 0.5269 1.6964 0.1273 53.5201 0.6987 0.9471 0.6441 0.6903 6.8751

1 + ρ|+ R(99%,99%) 6.7929 12.9265 -0.7283 10.2792 275.9075 0.5255 1.7113 0.1268 53.5727 0.6930 0.9314 0.6524 0.6959 6.9849

Summary statistics for 6/6 monthly diversified reward-risk portfolios in global equity benchmarks are given in the table. Mean and standard deviationare annualized numbers. Cumulative return is in percentage scale. Reward-risk and risk measures for 6/6 monthly diversified reward-risk portfolios inglobal equity benchmarks are also given. Sharpe ratio and conditional Sharpe ratio are annualized and Maximum drawdown (MDD) is in percentagescale. VaR and CVaR are represented in daily percentage scale. VaR 95/99 and CVaR 95/99 are from the ratio of 95% to 99%. The turnover rateis in percentage scale.

34

Table 7: Summary statistics for risk portfolios in global equity benchmarks

Rule Measure Mean Std Skew Kurt. Cumul. Sharpe CSharpe Calmar Max DD VaR(95%) CVaR(95%) VaR95/99 CVaR95/99 Turnover

1 Equal Weight 6.7047 12.9093 -0.7144 10.2438 268.8017 0.5194 1.5745 0.1254 53.4596 0.6987 0.9374 0.6537 0.6972 3.8070

1/ρ Max Drawdown 6.4232 12.7036 -0.7204 11.3681 248.4699 0.5056 2.2444 0.1197 53.6662 0.6556 0.8879 0.6446 0.6923 9.7322

1/ρ Std. 6.0447 12.3782 -0.7395 11.8440 223.2563 0.4883 1.6376 0.1152 52.4912 0.6660 0.8953 0.6519 0.6960 8.3991

1/ρ Std. Sq. 6.0271 12.3394 -0.7538 12.2472 222.3447 0.4884 1.9211 0.1169 51.5543 0.6679 0.9021 0.6474 0.6933 11.3151

1/ρ Cond. Std. 6.1468 12.3089 -0.7787 12.0417 231.3004 0.4994 1.6120 0.1181 52.0399 0.6512 0.8740 0.6531 0.6974 12.4685

1/ρ Cond. Std. Sq. 6.2569 12.2231 -0.7837 11.2425 240.3136 0.5119 1.6693 0.1228 50.9318 0.6491 0.8713 0.6529 0.6974 13.9714

1/ρ VaR(90%) 6.4285 12.3430 -0.7963 12.4326 252.3791 0.5208 1.9874 0.1234 52.0999 0.6398 0.8618 0.6503 0.6946 16.6410

1/ρ VaR(95%) 6.3270 12.3197 -0.7954 12.2503 244.7339 0.5136 1.9158 0.1217 52.0005 0.6479 0.8837 0.6390 0.6857 14.4937

1/ρ VaR(99%) 6.2121 12.3376 -0.7976 12.1206 235.8790 0.5035 1.7687 0.1194 52.0297 0.6503 0.8763 0.6502 0.6935 14.2253

1/ρ CVaR(90%) 6.2990 12.3361 -0.7830 12.0964 242.4521 0.5106 1.8286 0.1211 52.0003 0.6483 0.8769 0.6464 0.6915 14.5664

1/ρ CVaR(95%) 6.2514 12.3306 -0.7786 12.0201 238.8993 0.5070 1.7633 0.1202 52.0172 0.6527 0.8845 0.6450 0.6896 14.2034

1/ρ CVaR(99%) 6.1669 12.3559 -0.7755 11.8963 232.3641 0.4991 1.6040 0.1184 52.0819 0.6561 0.8888 0.6456 0.6891 15.1420

1− ρ Max Drawdown 6.5477 12.8365 -0.7182 10.6199 256.8953 0.5101 1.7423 0.1220 53.6746 0.6784 0.9101 0.6538 0.6977 3.6139

1− ρ Std. 6.6921 12.9012 -0.7143 10.2634 267.8382 0.5187 1.5801 0.1252 53.4424 0.6976 0.9362 0.6534 0.6973 3.7850

1− ρ Std. Sq. 6.7045 12.9090 -0.7144 10.2443 268.7680 0.5194 1.5734 0.1254 53.4589 0.7017 0.9469 0.6475 0.6945 3.8060

1− ρ Cond. Std. 6.6944 12.8997 -0.7145 10.2582 268.0522 0.5190 1.5888 0.1253 53.4322 0.6991 0.9414 0.6500 0.6951 3.7850

1− ρ Cond. Std. Sq. 6.7045 12.9089 -0.7144 10.2440 268.7733 0.5194 1.5817 0.1254 53.4584 0.6985 0.9368 0.6538 0.6979 3.8056

1− ρ VaR(90%) 6.6966 12.8975 -0.7158 10.2684 268.2553 0.5192 1.5829 0.1254 53.4213 0.6964 0.9315 0.6564 0.6998 3.7666

1− ρ VaR(95%) 6.6924 12.8941 -0.7160 10.2747 267.9523 0.5190 1.5834 0.1253 53.4112 0.6965 0.9322 0.6558 0.6994 3.7622

1− ρ VaR(99%) 6.6828 12.8877 -0.7160 10.2851 267.2225 0.5185 1.5724 0.1252 53.3897 0.6950 0.9291 0.6571 0.6999 3.7632

1− ρ CVaR(90%) 6.6906 12.8931 -0.7159 10.2760 267.8116 0.5189 1.5802 0.1253 53.4070 0.6966 0.9324 0.6557 0.6995 3.7620

1− ρ CVaR(95%) 6.6863 12.8902 -0.7159 10.2807 267.4920 0.5187 1.5692 0.1252 53.3973 0.6968 0.9362 0.6525 0.6960 3.7625

1− ρ CVaR(99%) 6.6757 12.8849 -0.7152 10.2875 266.6830 0.5181 1.5727 0.1251 53.3738 0.6957 0.9322 0.6549 0.6983 3.7896

Summary statistics for 6/6 monthly diversified reward-risk portfolios in global equity benchmarks are given in the table. Mean and standard deviationare annualized numbers. Cumulative return is in percentage scale. Reward-risk and risk measures for 6/6 monthly diversified reward-risk portfolios inglobal equity benchmarks are also given. Sharpe ratio and conditional Sharpe ratio are annualized and Maximum drawdown (MDD) is in percentagescale. VaR and CVaR are represented in daily percentage scale. VaR 95/99 and CVaR 95/99 are from the ratio of 95% to 99%. The turnover rateis in percentage scale.

35

Table 8: Summary statistics for ratio portfolios in largest mutual funds and ETFs

Rule Measure Mean Std Skew Kurt. Cumul. Sharpe CSharpe Calmar Max DD VaR(95%) CVaR(95%) VaR95/99 CVaR95/99 Turnover

1 Equal Weight 7.8627 17.7827 -0.2005 8.7977 285.0864 0.4422 3.8654 0.1513 51.9683 0.7282 1.0346 0.5953 0.6833 2.9803

ρ|+ Sharpe 6.0911 14.7655 -0.2910 7.2564 192.5621 0.4125 4.9865 0.1221 49.8850 0.4515 0.6440 0.5919 0.6829 44.8140

ρ|+ Cond. Sharpe 7.4320 16.0285 -0.3176 8.9579 274.1358 0.4637 6.8349 0.1459 50.9485 0.4215 0.6257 0.5610 0.6617 56.6123

ρ|+ Calmar 5.9008 14.5912 -0.2793 7.8198 182.4017 0.4044 4.9131 0.1193 49.4522 0.4251 0.6052 0.5935 0.6842 50.0108

ρ|+ STAR(90%) 7.4448 16.0476 -0.3148 9.0673 274.9204 0.4639 6.8536 0.1465 50.8145 0.4220 0.6268 0.5606 0.6615 58.1340

ρ|+ STAR(95%) 7.4406 16.0362 -0.3147 9.1054 274.7314 0.4640 6.8218 0.1463 50.8428 0.4252 0.6312 0.5609 0.6617 57.7916

ρ|+ STAR(99%) 7.4277 16.0169 -0.3129 9.1442 273.9407 0.4637 6.7993 0.1457 50.9870 0.4281 0.6351 0.5614 0.6620 57.6121

ρ|+ R(50%,90%) 7.8339 17.9481 -0.2159 9.1471 280.2742 0.4365 4.5258 0.1498 52.2879 0.6460 0.9261 0.5880 0.6786 13.6054

ρ|+ R(50%,95%) 7.8191 17.9222 -0.2137 9.1536 279.4292 0.4363 4.5219 0.1494 52.3400 0.6454 0.9249 0.5883 0.6788 12.9050

ρ|+ R(50%,99%) 7.7722 17.8668 -0.2091 9.1556 276.4455 0.4350 4.5511 0.1479 52.5411 0.6386 0.9159 0.5877 0.6784 12.9546

ρ|+ R(90%,90%) 7.8982 17.9570 -0.2132 9.1284 285.4146 0.4398 4.2835 0.1519 51.9878 0.6759 0.9658 0.5906 0.6802 9.8702

ρ|+ R(95%,95%) 7.9090 17.9323 -0.2119 9.1480 286.6940 0.4410 4.2301 0.1524 51.8815 0.6817 0.9735 0.5911 0.6806 9.0274

ρ|+ R(99%,99%) 7.9299 17.8997 -0.2112 9.2325 288.9129 0.4430 4.1776 0.1535 51.6456 0.6879 0.9817 0.5916 0.6809 9.4552

1 + ρ|+ Sharpe 7.8420 17.7629 -0.1995 8.7042 283.6659 0.4415 3.9207 0.1507 52.0498 0.7209 1.0250 0.5947 0.6829 3.0009

1 + ρ|+ Cond. Sharpe 7.8705 17.8553 -0.2087 8.8184 284.6417 0.4408 4.0689 0.1513 52.0334 0.7024 1.0006 0.5931 0.6819 5.7527

1 + ρ|+ Calmar 7.4471 17.4207 -0.1910 8.3615 257.0775 0.4275 4.1458 0.1417 52.5574 0.6736 0.9593 0.5934 0.6822 14.5837

1 + ρ|+ STAR(90%) 7.8760 17.8673 -0.2074 8.8728 284.9230 0.4408 4.0270 0.1514 52.0363 0.7071 1.0069 0.5934 0.6820 5.4630

1 + ρ|+ STAR(95%) 7.8730 17.8467 -0.2060 8.8559 284.9865 0.4411 4.0046 0.1513 52.0202 0.7101 1.0107 0.5938 0.6823 4.6673

1 + ρ|+ STAR(99%) 7.8690 17.8215 -0.2041 8.8313 285.0293 0.4415 3.9600 0.1513 52.0059 0.7155 1.0177 0.5944 0.6827 3.8978

1 + ρ|+ R(50%,90%) 7.8695 17.8981 -0.2073 8.9662 283.9291 0.4397 4.0782 0.1511 52.0944 0.7006 0.9983 0.5929 0.6818 6.0596

1 + ρ|+ R(50%,95%) 7.8629 17.8693 -0.2056 8.9431 283.8221 0.4400 4.0547 0.1510 52.0880 0.7039 1.0026 0.5932 0.6819 5.2203

1 + ρ|+ R(50%,99%) 7.8518 17.8301 -0.2032 8.9039 283.4933 0.4404 4.0059 0.1507 52.0975 0.7089 1.0091 0.5937 0.6822 4.4438

1 + ρ|+ R(90%,90%) 7.8906 17.9076 -0.2084 9.0074 285.5409 0.4406 4.0918 0.1518 51.9869 0.6999 0.9976 0.5927 0.6817 6.2412

1 + ρ|+ R(95%,95%) 7.8949 17.8856 -0.2075 9.0068 286.2182 0.4414 4.0470 0.1520 51.9243 0.7043 1.0033 0.5931 0.6818 5.7455

1 + ρ|+ R(99%,99%) 7.9073 17.8628 -0.2069 9.0425 287.5941 0.4427 4.0092 0.1527 51.7803 0.7085 1.0089 0.5934 0.6820 5.9990

Summary statistics for 6/6 monthly diversified reward-risk portfolios in largest mutual funds and ETFs are given in the table. Mean and standarddeviation are annualized numbers. Cumulative return is in percentage scale. Reward-risk and risk measures for 6/6 monthly diversified reward-riskportfolios in largest mutual funds and ETFs are also given. Sharpe ratio and conditional Sharpe ratio are annualized and Maximum drawdown(MDD) is in percentage scale. VaR and CVaR are represented in daily percentage scale. VaR 95/99 and CVaR 95/99 are from the ratio of 95% to99%. The turnover rate is in percentage scale.

36

Table 9: Summary statistics for risk portfolios in largest mutual funds and ETFs

Rule Measure Mean Std Skew Kurt. Cumul. Sharpe CSharpe Calmar Max DD VaR(95%) CVaR(95%) VaR95/99 CVaR95/99 Turnover

1 Equal Weight 7.8627 17.7827 -0.2005 8.7977 285.0864 0.4422 3.8654 0.1513 51.9683 0.7282 1.0346 0.5953 0.6833 2.9803

1/ρ Max Drawdown 7.5988 16.8275 -0.2586 8.9457 277.0198 0.4516 4.6024 0.1565 48.5555 0.6192 0.8882 0.5875 0.6784 6.4521

1/ρ Std. 6.0125 13.6515 -0.2864 8.8283 197.6332 0.4404 4.8586 0.1334 45.0772 0.4772 0.6849 0.5871 0.6784 7.8727

1/ρ Std. Sq. 7.0013 16.0905 -0.3049 9.7919 240.3645 0.4351 5.5537 0.1492 46.9102 0.5191 0.7550 0.5766 0.6715 8.0083

1/ρ Cond. Std. 6.0631 13.5317 -0.2989 8.3831 201.9347 0.4481 5.0357 0.1358 44.6482 0.4680 0.6738 0.5846 0.6767 11.4079

1/ρ Cond. Std. Sq. 7.2722 15.9850 -0.3000 8.2495 262.0763 0.4549 4.2345 0.1522 47.7960 0.6605 0.9426 0.5917 0.6809 9.6953

1/ρ VaR(90%) 6.1657 14.0408 -0.3401 9.1737 204.0171 0.4391 5.8361 0.1376 44.8045 0.4292 0.6262 0.5743 0.6701 15.0874

1/ρ VaR(95%) 6.1599 13.8915 -0.3285 8.8879 205.0094 0.4434 5.7526 0.1376 44.7716 0.4283 0.6241 0.5752 0.6707 13.4573

1/ρ VaR(99%) 6.0755 13.6260 -0.3151 8.8222 201.9092 0.4459 5.7276 0.1354 44.8838 0.4184 0.6091 0.5760 0.6713 13.0743

1/ρ CVaR(90%) 6.1208 13.8015 -0.3240 8.8753 203.2853 0.4435 5.7699 0.1365 44.8449 0.4234 0.6171 0.5750 0.6707 13.4026

1/ρ CVaR(95%) 6.0881 13.6882 -0.3181 8.8384 202.1639 0.4448 5.7410 0.1356 44.8826 0.4203 0.6120 0.5758 0.6711 13.1482

1/ρ CVaR(99%) 6.0029 13.4864 -0.3070 8.7953 198.4488 0.4451 5.7371 0.1333 45.0373 0.4107 0.5972 0.5768 0.6719 14.0207

1− ρ Max Drawdown 7.8029 17.4546 -0.2168 8.8403 284.9234 0.4470 3.9758 0.1539 50.6986 0.7052 1.0036 0.5939 0.6823 3.0578

1− ρ Std. 7.8571 17.7578 -0.2015 8.8045 284.9949 0.4425 3.8714 0.1515 51.8691 0.7269 1.0330 0.5951 0.6832 2.9662

1− ρ Std. Sq. 7.8627 17.7820 -0.2005 8.7982 285.0923 0.4422 3.8589 0.1513 51.9647 0.7286 1.0351 0.5953 0.6833 2.9797

1− ρ Cond. Std. 7.8576 17.7580 -0.2015 8.7964 285.0263 0.4425 3.8677 0.1514 51.8863 0.7272 1.0334 0.5951 0.6831 2.9693

1− ρ Cond. Std. Sq. 7.8627 17.7820 -0.2005 8.7977 285.0934 0.4422 3.8647 0.1513 51.9658 0.7282 1.0347 0.5952 0.6832 2.9799

1− ρ VaR(90%) 7.8589 17.7532 -0.2020 8.7994 285.2077 0.4427 3.8647 0.1516 51.8547 0.7272 1.0332 0.5952 0.6832 2.9610

1− ρ VaR(95%) 7.8571 17.7433 -0.2024 8.7988 285.2092 0.4428 3.8787 0.1516 51.8214 0.7260 1.0317 0.5951 0.6832 2.9603

1− ρ VaR(99%) 7.8523 17.7230 -0.2032 8.7969 285.1117 0.4431 3.8793 0.1517 51.7639 0.7252 1.0306 0.5951 0.6831 2.9675

1− ρ CVaR(90%) 7.8561 17.7398 -0.2025 8.7984 285.1756 0.4429 3.8760 0.1516 51.8135 0.7259 1.0315 0.5951 0.6832 2.9615

1− ρ CVaR(95%) 7.8541 17.7308 -0.2029 8.7976 285.1297 0.4430 3.8589 0.1517 51.7877 0.7266 1.0320 0.5955 0.6835 2.9647

1− ρ CVaR(99%) 7.8478 17.7115 -0.2034 8.7949 284.8945 0.4431 3.8623 0.1517 51.7468 0.7256 1.0307 0.5954 0.6834 2.9946

Summary statistics for 6/6 monthly diversified reward-risk portfolios in largest mutual funds and ETFs are given in the table. Mean and standarddeviation are annualized numbers. Cumulative return is in percentage scale. Reward-risk and risk measures for 6/6 monthly diversified reward-riskportfolios in largest mutual funds and ETFs are also given. Sharpe ratio and conditional Sharpe ratio are annualized and Maximum drawdown(MDD) is in percentage scale. VaR and CVaR are represented in daily percentage scale. VaR 95/99 and CVaR 95/99 are from the ratio of 95% to99%. The turnover rate is in percentage scale.

37

Table 10: Summary statistics for ratio portfolios in SPDR U.S. sector ETFs

Rule Measure Mean Std Skew Kurt. Cumul. Sharpe CSharpe Calmar Max DD VaR(95%) CVaR(95%) VaR95/99 CVaR95/99 Turnover

1 Equal Weight 8.9077 18.7304 -0.2507 11.3279 364.0831 0.4756 3.3735 0.1681 52.9871 0.9047 1.2740 0.6025 0.6882 2.8911

ρ|+ Sharpe 9.4266 19.2272 -0.2044 13.1232 408.3901 0.4903 2.2055 0.1809 52.1133 1.0063 1.3958 0.6152 0.6965 50.0645

ρ|+ Cond. Sharpe 9.2708 18.5885 -0.1595 9.7554 404.7413 0.4987 4.2096 0.1921 48.2588 0.8394 1.1922 0.5955 0.6840 61.5056

ρ|+ Calmar 9.7269 19.4450 -0.1831 13.7636 437.3571 0.5002 1.5579 0.1884 51.6383 1.2011 1.6501 0.6233 0.7022 55.9618

ρ|+ STAR(90%) 9.2330 18.5955 -0.1569 9.6928 400.5232 0.4965 4.2331 0.1915 48.2033 0.8413 1.1957 0.5949 0.6836 63.6676

ρ|+ STAR(95%) 9.2250 18.5915 -0.1587 9.7097 399.7458 0.4962 4.2443 0.1916 48.1481 0.8415 1.1961 0.5949 0.6836 63.1621

ρ|+ STAR(99%) 9.2111 18.5926 -0.1610 9.7232 398.2223 0.4954 4.2322 0.1921 47.9612 0.8447 1.2001 0.5952 0.6838 62.6961

ρ|+ R(50%,90%) 9.0896 18.6044 -0.2255 10.9099 385.0684 0.4886 3.5838 0.1772 51.2895 0.8766 1.2387 0.5997 0.6863 16.3461

ρ|+ R(50%,95%) 9.0675 18.6150 -0.2274 10.9386 382.5584 0.4871 3.5613 0.1766 51.3447 0.8787 1.2411 0.6000 0.6866 15.4927

ρ|+ R(50%,99%) 9.0365 18.6327 -0.2293 10.9822 379.0211 0.4850 3.5282 0.1761 51.3201 0.8812 1.2433 0.6009 0.6872 15.0973

ρ|+ R(90%,90%) 8.9810 18.6828 -0.2408 11.0258 372.3876 0.4807 3.5298 0.1719 52.2585 0.8843 1.2489 0.6001 0.6866 11.3268

ρ|+ R(95%,95%) 8.9279 18.7157 -0.2458 11.0644 366.3830 0.4770 3.5151 0.1699 52.5333 0.8866 1.2520 0.6002 0.6866 10.2508

ρ|+ R(99%,99%) 8.8437 18.7737 -0.2529 11.1001 356.9491 0.4711 3.5051 0.1671 52.9174 0.8871 1.2525 0.6004 0.6867 10.1297

1 + ρ|+ Sharpe 8.9284 18.7156 -0.2491 11.3280 366.4382 0.4771 3.3939 0.1688 52.9042 0.8997 1.2675 0.6022 0.6880 2.5509

1 + ρ|+ Cond. Sharpe 8.9606 18.6849 -0.2436 11.1679 370.2603 0.4796 3.4448 0.1703 52.6301 0.8938 1.2601 0.6016 0.6876 6.0918

1 + ρ|+ Calmar 9.1584 18.6460 -0.2309 11.2975 391.4454 0.4912 3.4831 0.1749 52.3558 0.8749 1.2358 0.6000 0.6864 18.7973

1 + ρ|+ STAR(90%) 8.9213 18.7013 -0.2444 11.1947 366.0000 0.4770 3.4236 0.1692 52.7273 0.8968 1.2642 0.6017 0.6876 5.3936

1 + ρ|+ STAR(95%) 8.9241 18.7053 -0.2458 11.2264 366.1950 0.4771 3.4140 0.1691 52.7794 0.8981 1.2657 0.6019 0.6878 4.5222

1 + ρ|+ STAR(99%) 8.9228 18.7117 -0.2475 11.2627 365.9506 0.4769 3.4003 0.1689 52.8332 0.9001 1.2681 0.6021 0.6879 3.6761

1 + ρ|+ R(50%,90%) 8.9526 18.6866 -0.2414 11.1707 369.4135 0.4791 3.4416 0.1706 52.4901 0.8944 1.2610 0.6015 0.6875 6.2314

1 + ρ|+ R(50%,95%) 8.9400 18.6947 -0.2433 11.2025 367.9918 0.4782 3.4161 0.1701 52.5675 0.8967 1.2639 0.6017 0.6876 5.3119

1 + ρ|+ R(50%,99%) 8.9291 18.7049 -0.2455 11.2449 366.7212 0.4774 3.4079 0.1696 52.6455 0.8986 1.2659 0.6022 0.6880 4.3651

1 + ρ|+ R(90%,90%) 8.9389 18.7052 -0.2449 11.1607 367.6863 0.4779 3.4494 0.1698 52.6421 0.8939 1.2608 0.6012 0.6872 6.4167

1 + ρ|+ R(95%,95%) 8.9092 18.7227 -0.2478 11.1842 364.4004 0.4758 3.4418 0.1688 52.7785 0.8944 1.2612 0.6014 0.6875 5.8332

1 + ρ|+ R(99%,99%) 8.8689 18.7514 -0.2519 11.2053 359.8583 0.4730 3.4424 0.1675 52.9625 0.8953 1.2625 0.6014 0.6874 5.7539

Summary statistics for 6/6 monthly diversified reward-risk portfolios in SPDR U.S. sector ETFs are given in the table. Mean and standard deviationare annualized numbers. Cumulative return is in percentage scale. Reward-risk and risk measures for 6/6 monthly diversified reward-risk portfoliosin SPDR U.S. sector ETFs are also given. Sharpe ratio and conditional Sharpe ratio are annualized and Maximum drawdown (MDD) is in percentagescale. VaR and CVaR are represented in daily percentage scale. VaR 95/99 and CVaR 95/99 are from the ratio of 95% to 99%. The turnover rateis in percentage scale.

38

Table 11: Summary statistics for risk portfolios in SPDR U.S. sector ETFs

Rule Measure Mean Std Skew Kurt. Cumul. Sharpe CSharpe Calmar Max DD VaR(95%) CVaR(95%) VaR95/99 CVaR95/99 Turnover

1 Equal Weight 8.9077 18.7304 -0.2507 11.3279 364.0831 0.4756 3.3735 0.1681 52.9871 0.9047 1.2740 0.6025 0.6882 2.8911

1/ρ Max Drawdown 8.8770 17.9138 -0.2559 11.5192 376.1365 0.4955 3.6274 0.1784 49.7528 0.8268 1.1713 0.5976 0.6847 7.5628

1/ρ Std. 8.8598 17.6782 -0.2314 11.5614 378.7282 0.5012 3.5152 0.1793 49.4175 0.8410 1.1887 0.5994 0.6860 5.2194

1/ρ Std. Sq. 8.8044 17.2657 -0.2511 11.7990 380.4512 0.5099 3.6842 0.1878 46.8733 0.8057 1.1423 0.5969 0.6843 8.2580

1/ρ Cond. Std. 8.7716 17.6250 -0.2387 11.6255 370.6933 0.4977 3.5848 0.1789 49.0220 0.8253 1.1671 0.5991 0.6858 8.8240

1/ρ Cond. Std. Sq. 8.5738 17.3148 -0.2519 11.9705 356.4042 0.4952 3.7443 0.1840 46.5943 0.7951 1.1256 0.5981 0.6853 10.6069

1/ρ VaR(90%) 8.9266 17.6232 -0.2273 11.3085 386.6496 0.5065 3.7073 0.1840 48.5081 0.8166 1.1574 0.5972 0.6845 14.1175

1/ρ VaR(95%) 8.8470 17.6138 -0.2322 11.4358 378.5825 0.5023 3.6718 0.1812 48.8290 0.8188 1.1599 0.5977 0.6849 11.6321

1/ρ VaR(99%) 8.7706 17.6314 -0.2373 11.5071 370.4816 0.4974 3.6308 0.1795 48.8571 0.8224 1.1641 0.5982 0.6852 11.0381

1/ρ CVaR(90%) 8.7731 17.6271 -0.2332 11.4324 370.8105 0.4977 3.6369 0.1797 48.8078 0.8212 1.1631 0.5978 0.6849 11.7053

1/ρ CVaR(95%) 8.7756 17.6281 -0.2357 11.4835 371.0592 0.4978 3.6444 0.1797 48.8304 0.8213 1.1627 0.5982 0.6852 11.1837

1/ρ CVaR(99%) 8.7515 17.6341 -0.2385 11.5478 368.5077 0.4963 3.6053 0.1795 48.7648 0.8248 1.1666 0.5989 0.6858 11.8009

1− ρ Max Drawdown 8.8933 18.3985 -0.2569 11.4859 368.8347 0.4834 3.5629 0.1717 51.7998 0.8460 1.1945 0.6003 0.6868 2.4452

1− ρ Std. 8.9067 18.7085 -0.2509 11.3348 364.3940 0.4761 3.3849 0.1683 52.9068 0.9011 1.2690 0.6025 0.6881 2.8667

1− ρ Std. Sq. 8.9074 18.7293 -0.2507 11.3284 364.0899 0.4756 3.3745 0.1681 52.9827 0.9043 1.2733 0.6026 0.6883 2.8896

1− ρ Cond. Std. 8.9067 18.7062 -0.2510 11.3318 364.4397 0.4761 3.3837 0.1684 52.8902 0.9019 1.2700 0.6026 0.6882 2.8575

1− ρ Cond. Std. Sq. 8.9077 18.7291 -0.2507 11.3280 364.1126 0.4756 3.3790 0.1681 52.9809 0.9039 1.2729 0.6025 0.6882 2.8891

1− ρ VaR(90%) 8.9072 18.6968 -0.2509 11.3290 364.6693 0.4764 3.3838 0.1686 52.8307 0.9017 1.2698 0.6025 0.6882 2.8276

1− ρ VaR(95%) 8.9062 18.6878 -0.2511 11.3299 364.7261 0.4766 3.3847 0.1687 52.7934 0.9006 1.2684 0.6024 0.6882 2.8189

1− ρ VaR(99%) 8.9029 18.6695 -0.2514 11.3318 364.7470 0.4769 3.3818 0.1689 52.7166 0.8993 1.2665 0.6025 0.6882 2.8042

1− ρ CVaR(90%) 8.9052 18.6847 -0.2511 11.3303 364.7019 0.4766 3.3917 0.1687 52.7807 0.8999 1.2674 0.6024 0.6881 2.8161

1− ρ CVaR(95%) 8.9039 18.6766 -0.2512 11.3311 364.7255 0.4767 3.3866 0.1688 52.7461 0.8997 1.2670 0.6025 0.6882 2.8092

1− ρ CVaR(99%) 8.8999 18.6593 -0.2515 11.3338 364.6414 0.4770 3.3867 0.1690 52.6713 0.8980 1.2648 0.6024 0.6881 2.8024

Summary statistics for 6/6 monthly diversified reward-risk portfolios in SPDR U.S. sector ETFs are given in the table. Mean and standard deviationare annualized numbers. Cumulative return is in percentage scale. Reward-risk and risk measures for 6/6 monthly diversified reward-risk portfoliosin SPDR U.S. sector ETFs are also given. Sharpe ratio and conditional Sharpe ratio are annualized and Maximum drawdown (MDD) is in percentagescale. VaR and CVaR are represented in daily percentage scale. VaR 95/99 and CVaR 95/99 are from the ratio of 95% to 99%. The turnover rateis in percentage scale.

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Table 12: Summary statistics for ratio portfolios in Dow Jones Industrial Average

Rule Measure Mean Std Skew Kurt. Cumul. Sharpe CSharpe Calmar Max DD VaR(95%) CVaR(95%) VaR95/99 CVaR95/99 Turnover

1 Equal Weight 9.8419 19.8158 -0.0612 10.0569 442.6942 0.4967 3.0121 0.1702 57.8115 0.9079 1.2647 0.6120 0.6942 6.7482

ρ|+ Sharpe 9.5866 18.6657 -0.1058 9.5262 438.7364 0.5136 2.5313 0.2178 44.0164 1.1694 1.6180 0.6175 0.6984 58.4629

ρ|+ Cond. Sharpe 10.0084 18.9900 0.0777 9.4165 482.4326 0.5270 2.8412 0.2146 46.6281 1.0165 1.4015 0.6206 0.7011 66.8401

ρ|+ Calmar 10.3212 18.6839 -0.0926 8.9297 530.3241 0.5524 2.4968 0.2616 39.4614 1.4439 1.9913 0.6202 0.7007 66.0731

ρ|+ STAR(90%) 10.0513 18.9987 0.0734 9.2650 487.5955 0.5291 3.0306 0.2171 46.3046 0.9850 1.3596 0.6197 0.7006 69.2023

ρ|+ STAR(95%) 9.9439 19.0295 0.0709 9.2443 473.4976 0.5226 3.0290 0.2147 46.3236 0.9835 1.3592 0.6186 0.6998 68.9955

ρ|+ STAR(99%) 9.8527 19.0448 0.0688 9.2903 461.9973 0.5173 3.0236 0.2126 46.3333 0.9803 1.3551 0.6184 0.6995 68.5758

ρ|+ R(50%,90%) 9.8592 19.5309 -0.0313 9.8770 451.3793 0.5048 2.9391 0.1780 55.3774 0.9333 1.2985 0.6129 0.6948 18.7071

ρ|+ R(50%,95%) 9.8020 19.5537 -0.0335 9.8904 444.1117 0.5013 2.9308 0.1771 55.3396 0.9340 1.2994 0.6129 0.6949 18.2920

ρ|+ R(50%,99%) 9.7292 19.5769 -0.0358 9.9513 435.1502 0.4970 2.8570 0.1764 55.1599 0.9427 1.3107 0.6135 0.6953 18.8973

ρ|+ R(90%,90%) 9.8749 19.6521 -0.0504 9.8852 450.3813 0.5025 2.9520 0.1749 56.4495 0.9247 1.2871 0.6126 0.6946 13.7517

ρ|+ R(95%,95%) 9.8542 19.6910 -0.0559 9.8861 447.0405 0.5004 2.9514 0.1739 56.6507 0.9225 1.2838 0.6127 0.6947 13.0349

ρ|+ R(99%,99%) 9.8703 19.7406 -0.0621 9.8815 447.7668 0.5000 2.9025 0.1734 56.9068 0.9227 1.2836 0.6130 0.6949 14.0656

1 + ρ|+ Sharpe 9.8625 19.7864 -0.0631 10.0528 445.7884 0.4984 3.0481 0.1711 57.6373 0.9037 1.2599 0.6112 0.6937 5.7905

1 + ρ|+ Cond. Sharpe 9.8409 19.7445 -0.0554 10.0114 444.2366 0.4984 3.0487 0.1717 57.3114 0.9021 1.2568 0.6118 0.6941 8.5247

1 + ρ|+ Calmar 10.0568 19.4868 -0.0712 9.6921 476.2547 0.5161 3.2301 0.1829 54.9985 0.8789 1.2275 0.6096 0.6932 26.3331

1 + ρ|+ STAR(90%) 9.8494 19.7620 -0.0578 10.0215 444.8199 0.4984 3.0450 0.1714 57.4759 0.9026 1.2574 0.6119 0.6942 8.0382

1 + ρ|+ STAR(95%) 9.8330 19.7745 -0.0587 10.0247 442.6113 0.4973 3.0526 0.1709 57.5398 0.9019 1.2564 0.6119 0.6943 7.5883

1 + ρ|+ STAR(99%) 9.8340 19.7865 -0.0596 10.0370 442.4429 0.4970 3.0299 0.1707 57.6201 0.9056 1.2615 0.6119 0.6942 6.9639

1 + ρ|+ R(50%,90%) 9.8461 19.7214 -0.0530 9.9986 445.3800 0.4993 2.9885 0.1724 57.0988 0.9150 1.2741 0.6123 0.6944 8.7535

1 + ρ|+ R(50%,95%) 9.8242 19.7396 -0.0548 10.0095 442.4157 0.4977 3.0018 0.1718 57.1821 0.9130 1.2713 0.6123 0.6945 8.1285

1 + ρ|+ R(50%,99%) 9.8101 19.7602 -0.0569 10.0308 440.2824 0.4965 2.9766 0.1712 57.2862 0.9151 1.2743 0.6122 0.6944 7.4632

1 + ρ|+ R(90%,90%) 9.8565 19.7297 -0.0562 9.9690 446.3864 0.4996 2.9872 0.1725 57.1419 0.9156 1.2751 0.6122 0.6943 9.1059

1 + ρ|+ R(95%,95%) 9.8456 19.7498 -0.0589 9.9695 444.6470 0.4985 2.9762 0.1720 57.2379 0.9156 1.2750 0.6122 0.6944 8.7983

1 + ρ|+ R(99%,99%) 9.8555 19.7742 -0.0620 9.9653 445.2435 0.4984 2.9548 0.1718 57.3568 0.9155 1.2744 0.6125 0.6946 9.3972

Summary statistics for 6/6 monthly diversified reward-risk portfolios in Dow Jones Industrial Average are given in the table. Mean and standarddeviation are annualized numbers. Cumulative return is in percentage scale. Reward-risk and risk measures for 6/6 monthly diversified reward-riskportfolios in Dow Jones Industrial Average are also given. Sharpe ratio and conditional Sharpe ratio are annualized and Maximum drawdown (MDD)is in percentage scale. VaR and CVaR are represented in daily percentage scale. VaR 95/99 and CVaR 95/99 are from the ratio of 95% to 99%. Theturnover rate is in percentage scale.

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Table 13: Summary statistics for risk portfolios in Dow Jones Industrial Average

Rule Measure Mean Std Skew Kurt. Cumul. Sharpe CSharpe Calmar Max DD VaR(95%) CVaR(95%) VaR95/99 CVaR95/99 Turnover

1 Equal Weight 9.8419 19.8158 -0.0612 10.0569 442.6942 0.4967 3.0121 0.1702 57.8115 0.9079 1.2647 0.6120 0.6942 6.7482

1/ρ Max Drawdown 9.4613 18.4023 -0.0469 10.6753 430.0339 0.5141 3.5396 0.1878 50.3783 0.7981 1.1160 0.6088 0.6925 12.2917

1/ρ Std. 9.5437 18.3332 -0.0245 10.7354 441.0092 0.5206 3.0292 0.1870 51.0456 0.8418 1.1737 0.6114 0.6927 8.9994

1/ρ Std. Sq. 9.4198 17.4372 -0.0064 11.3465 445.2943 0.5402 3.2444 0.2030 46.4064 0.7900 1.1036 0.6100 0.6910 12.8717

1/ρ Cond. Std. 9.4399 18.2613 -0.0358 10.7230 430.5641 0.5169 3.2093 0.1869 50.5150 0.8147 1.1365 0.6108 0.6933 13.4904

1/ρ Cond. Std. Sq. 9.3563 17.4882 -0.0352 11.4752 436.8391 0.5350 3.5107 0.2031 46.0674 0.7654 1.0719 0.6077 0.6904 18.7332

1/ρ VaR(90%) 9.5110 18.1583 -0.0247 10.5880 440.9307 0.5238 3.2120 0.1908 49.8469 0.8213 1.1463 0.6104 0.6927 15.6314

1/ρ VaR(95%) 9.5415 18.1811 -0.0288 10.6298 443.9930 0.5248 3.1987 0.1910 49.9530 0.8196 1.1435 0.6107 0.6929 14.2009

1/ρ VaR(99%) 9.4654 18.2269 -0.0310 10.6545 434.2170 0.5193 3.1218 0.1898 49.8591 0.8238 1.1490 0.6110 0.6929 14.8284

1/ρ CVaR(90%) 9.4971 18.1974 -0.0287 10.6387 438.4918 0.5219 3.1654 0.1904 49.8862 0.8220 1.1468 0.6108 0.6929 14.3708

1/ρ CVaR(95%) 9.4641 18.2186 -0.0301 10.6451 434.2231 0.5195 3.1600 0.1898 49.8731 0.8209 1.1449 0.6111 0.6930 14.8309

1/ρ CVaR(99%) 9.4278 18.2491 -0.0301 10.6794 429.4567 0.5166 3.0752 0.1894 49.7705 0.8271 1.1527 0.6117 0.6934 17.1826

1− ρ Max Drawdown 9.4455 19.1010 -0.0686 10.3003 413.5284 0.4945 3.1826 0.1754 53.8515 0.8530 1.1892 0.6113 0.6939 5.4928

1− ρ Std. 9.8257 19.7671 -0.0616 10.0639 441.9357 0.4971 3.0129 0.1707 57.5739 0.9062 1.2621 0.6120 0.6943 6.6901

1− ρ Std. Sq. 9.8401 19.8121 -0.0613 10.0568 442.5553 0.4967 3.0065 0.1703 57.7927 0.9085 1.2654 0.6120 0.6942 6.7434

1− ρ Cond. Std. 9.8262 19.7654 -0.0613 10.0610 442.0431 0.4971 3.0068 0.1708 57.5152 0.9067 1.2628 0.6121 0.6943 6.6768

1− ρ Cond. Std. Sq. 9.8406 19.8118 -0.0612 10.0562 442.6330 0.4967 3.0018 0.1703 57.7828 0.9093 1.2663 0.6122 0.6944 6.7424

1− ρ VaR(90%) 9.8290 19.7489 -0.0607 10.0559 442.7365 0.4977 3.0053 0.1712 57.4134 0.9075 1.2638 0.6122 0.6944 6.6446

1− ρ VaR(95%) 9.8237 19.7292 -0.0608 10.0575 442.5703 0.4979 3.0098 0.1715 57.2924 0.9063 1.2621 0.6122 0.6944 6.6197

1− ρ VaR(99%) 9.8066 19.6849 -0.0610 10.0656 441.5851 0.4982 3.0025 0.1720 57.0046 0.9055 1.2609 0.6123 0.6945 6.5649

1− ρ CVaR(90%) 9.8194 19.7209 -0.0609 10.0597 442.2705 0.4979 3.0156 0.1716 57.2365 0.9056 1.2613 0.6121 0.6944 6.6068

1− ρ CVaR(95%) 9.8119 19.7012 -0.0610 10.0633 441.8486 0.4980 2.9928 0.1718 57.1099 0.9070 1.2631 0.6122 0.6944 6.5839

1− ρ CVaR(99%) 9.7869 19.6531 -0.0613 10.0783 440.0460 0.4980 2.9948 0.1724 56.7837 0.9055 1.2609 0.6122 0.6944 6.5810

Summary statistics for 6/6 monthly diversified reward-risk portfolios in Dow Jones Industrial Average are given in the table. Mean and standarddeviation are annualized numbers. Cumulative return is in percentage scale. Reward-risk and risk measures for 6/6 monthly diversified reward-riskportfolios in Dow Jones Industrial Average are also given. Sharpe ratio and conditional Sharpe ratio are annualized and Maximum drawdown (MDD)is in percentage scale. VaR and CVaR are represented in daily percentage scale. VaR 95/99 and CVaR 95/99 are from the ratio of 95% to 99%. Theturnover rate is in percentage scale.

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Table 14: Carhart 4-factor regression for all portfolios in SPDR U.S. sector ETFs

Rule Measure α(%) βMkt βSMB βHML βMOM R2

1 Equal Weight 0.1260∗ 0.9219∗∗ -0.1730∗∗ 0.1979∗∗ -0.0186 0.9573ρ|+ Sharpe 0.0864 0.9268∗∗ -0.1006∗∗ 0.1063∗∗ 0.1394∗∗ 0.8524ρ|+ Cond. Sharpe 0.1244 0.9132∗∗ -0.1263∗∗ 0.0762∗∗ 0.0569∗∗ 0.9184ρ|+ Calmar 0.1007 0.9229∗∗ -0.0839∗ 0.1139∗∗ 0.1540∗∗ 0.8400ρ|+ STAR(90%) 0.1207 0.9127∗∗ -0.1261∗∗ 0.0727∗∗ 0.0586∗∗ 0.9180ρ|+ STAR(95%) 0.1208 0.9123∗∗ -0.1263∗∗ 0.0731∗∗ 0.0575∗∗ 0.9181ρ|+ STAR(99%) 0.1202 0.9120∗∗ -0.1263∗∗ 0.0722∗∗ 0.0561∗∗ 0.9178ρ|+ R(50%,90%) 0.1378∗ 0.9179∗∗ -0.1667∗∗ 0.1684∗∗ -0.0017 0.9544ρ|+ R(50%,95%) 0.1367∗ 0.9180∗∗ -0.1673∗∗ 0.1691∗∗ -0.0041 0.9547ρ|+ R(50%,99%) 0.1351∗ 0.9184∗∗ -0.1681∗∗ 0.1686∗∗ -0.0077 0.9548ρ|+ R(90%,90%) 0.1313∗ 0.9200∗∗ -0.1725∗∗ 0.1807∗∗ -0.0114 0.9570ρ|+ R(95%,95%) 0.1278∗ 0.9207∗∗ -0.1743∗∗ 0.1835∗∗ -0.0156 0.9577ρ|+ R(99%,99%) 0.1222∗ 0.9223∗∗ -0.1778∗∗ 0.1837∗∗ -0.0225 0.95861 + ρ|+ Sharpe 0.1260∗ 0.9220∗∗ -0.1716∗∗ 0.1966∗∗ -0.0138 0.95691 + ρ|+ Cond. Sharpe 0.1286∗ 0.9212∗∗ -0.1709∗∗ 0.1895∗∗ -0.0115 0.95681 + ρ|+ Calmar 0.1246∗ 0.9217∗∗ -0.1495∗∗ 0.1858∗∗ 0.0346∗∗ 0.94421 + ρ|+ STAR(90%) 0.1256∗ 0.9216∗∗ -0.1716∗∗ 0.1916∗∗ -0.0133 0.95721 + ρ|+ STAR(95%) 0.1262∗ 0.9216∗∗ -0.1718∗∗ 0.1928∗∗ -0.0143 0.95721 + ρ|+ STAR(99%) 0.1265∗ 0.9216∗∗ -0.1720∗∗ 0.1943∗∗ -0.0156 0.95721 + ρ|+ R(50%,90%) 0.1286∗ 0.9208∗∗ -0.1713∗∗ 0.1879∗∗ -0.0127 0.95701 + ρ|+ R(50%,95%) 0.1279∗ 0.9210∗∗ -0.1716∗∗ 0.1895∗∗ -0.0141 0.95711 + ρ|+ R(50%,99%) 0.1274∗ 0.9212∗∗ -0.1720∗∗ 0.1912∗∗ -0.0159 0.95711 + ρ|+ R(90%,90%) 0.1281∗ 0.9211∗∗ -0.1728∗∗ 0.1888∗∗ -0.0147 0.95751 + ρ|+ R(95%,95%) 0.1261∗ 0.9215∗∗ -0.1737∗∗ 0.1905∗∗ -0.0169 0.95781 + ρ|+ R(99%,99%) 0.1235∗ 0.9223∗∗ -0.1755∗∗ 0.1905∗∗ -0.0205 0.95831/ρ Max Drawdown 0.1440∗ 0.8939∗∗ -0.1874∗∗ 0.1947∗∗ 0.0198 0.94641/ρ Std. 0.1694∗∗ 0.8734∗∗ -0.2033∗∗ 0.1826∗∗ -0.0029 0.94581/ρ Std. Sq. 0.1770∗∗ 0.8550∗∗ -0.2089∗∗ 0.1978∗∗ 0.0166 0.93521/ρ Cond. Std. 0.1606∗∗ 0.8746∗∗ -0.2040∗∗ 0.1790∗∗ 0.0024 0.94501/ρ Cond. Std. Sq. 0.1504∗ 0.8610∗∗ -0.2065∗∗ 0.1950∗∗ 0.0246 0.93311/ρ VaR(90%) 0.1702∗∗ 0.8747∗∗ -0.2037∗∗ 0.1665∗∗ 0.0118 0.94511/ρ VaR(95%) 0.1653∗∗ 0.8743∗∗ -0.2035∗∗ 0.1704∗∗ 0.0077 0.94551/ρ VaR(99%) 0.1600∗∗ 0.8745∗∗ -0.2037∗∗ 0.1725∗∗ 0.0034 0.94571/ρ CVaR(90%) 0.1588∗∗ 0.8749∗∗ -0.2028∗∗ 0.1717∗∗ 0.0064 0.94551/ρ CVaR(95%) 0.1600∗∗ 0.8746∗∗ -0.2035∗∗ 0.1720∗∗ 0.0045 0.94571/ρ CVaR(99%) 0.1590∗∗ 0.8745∗∗ -0.2041∗∗ 0.1719∗∗ 0.0016 0.94551− ρ Max Drawdown 0.1322∗ 0.9118∗∗ -0.1801∗∗ 0.1964∗∗ -0.0009 0.95301− ρ Std. 0.1267∗ 0.9211∗∗ -0.1734∗∗ 0.1978∗∗ -0.0179 0.95711− ρ Std. Sq. 0.1260∗ 0.9219∗∗ -0.1730∗∗ 0.1979∗∗ -0.0185 0.95721− ρ Cond. Std. 0.1267∗ 0.9211∗∗ -0.1734∗∗ 0.1979∗∗ -0.0178 0.95701− ρ Cond. Std. Sq. 0.1261∗ 0.9219∗∗ -0.1730∗∗ 0.1980∗∗ -0.0185 0.95721− ρ VaR(90%) 0.1269∗ 0.9208∗∗ -0.1735∗∗ 0.1974∗∗ -0.0175 0.95701− ρ VaR(95%) 0.1271∗ 0.9205∗∗ -0.1736∗∗ 0.1973∗∗ -0.0173 0.95691− ρ VaR(99%) 0.1275∗ 0.9199∗∗ -0.1740∗∗ 0.1970∗∗ -0.0169 0.95681− ρ CVaR(90%) 0.1272∗ 0.9204∗∗ -0.1737∗∗ 0.1973∗∗ -0.0172 0.95691− ρ CVaR(95%) 0.1273∗ 0.9201∗∗ -0.1739∗∗ 0.1971∗∗ -0.0170 0.95681− ρ CVaR(99%) 0.1276∗ 0.9195∗∗ -0.1742∗∗ 0.1968∗∗ -0.0167 0.9567

The Carhart four-factor analysis on the alternative portfolios in SPDR U.S. sector ETFs isgiven in the table. Intercept is in monthly scale.

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