Spanning analysis of stock market anomalies under Prospect Stochastic Dominance
aa r X i v : . [ q -f i n . P M ] A p r Spanning analysis of stock market anomalies underProspect Stochastic Dominance
Stelios Arvanitis, Olivier Scaillet, Nikolas TopaloglouThis version: April 2020
Abstract
We develop and implement methods for determining whether introducing new se-curities or relaxing investment constraints improves the investment opportunity set forprospect investors. We formulate a new testing procedure for prospect spanning for twonested portfolio sets based on subsampling and Linear Programming. In an application,we use the prospect spanning framework to evaluate whether well-known anomalies arespanned by standard factors. We find that of the strategies considered, many expandthe opportunity set of the prospect type investors, thus have real economic value forthem. In-sample and out-of-sample results prove remarkably consistent in identifyinggenuine anomalies for prospect investors.
Keywords and phrases : Nonparametric test, prospect stochastic dominance effi-ciency, prospect spanning, market anomaly, Linear Programming.
JEL Classification:
C12, C14, C44, C58, D81, G11, G40.
Traditional models in economics and finance assume that investors evaluate portfolios ac-cording to the expected utility framework. The theoretical motivation for this goes back to1on Neumann and Morgenstern (1944). Nevertheles, experimental and empirical work hasshown that people systematically violate Expected Utility theory when choosing among riskyassets. Prospect theory, first described by Kahneman and Tversky (1979) (see also Tverskyand Kahneman (1992)), is widely viewed as a better description of how people evaluate riskin experimental settings. While the theory contains many remarkable insights, it has provenchallenging to apply these insights in asset pricing, and it is only recently that there has beenreal progress in doing so (Barberis et al. (2019)). Barberis and Thaler (2003) and Barberis(2013) are excellent reviews on behavioral finance and prospect theory.Stock market anomalies are key drivers of innovation in asset pricing. These are tradableportfolio strategies, usually constructed as long-short portfolios based on the top and bottomdeciles of sorted stocks, according to specific characteristics (anomalies). Under the standardMean-Variance (MV) paradigm, establishing a cross-sectional return pattern as an anomalyinvolves testing for pricing based on a factor model. If factors are traded, spanning regressionsrelate to MV criterion. Arbitrage pricing stipulates that a portfolio of factors is MV-efficientand no other portfolio can achieve a higher Sharpe Ratio (SR). In that sense, an anomaly is astrategy that exhibits higher SR and should be traded away. However, we can question MVspanning for portfolio selection if returns do not follow elliptical distributions, or investorpreferences depend on more than the first two moments of the return distribution. Moreover,experimental evidence (Baucells and Heukamp (2006)) suggests that investors do not alwaysact as risk averters. Instead, under certain circumstances, they behave in a much morecomplex fashion, exhibiting characteristics of both risk-loving and risk-averting. They behavedifferently on gains and losses, and they are more sensitive to losses than to gains (lossaversion). The relevant utility function could be concave for gains and convex for losses(S-Shaped).The present study contributes to this literature by introducing, operationalizing andapplying prospect spanning tests for portfolio analysis. The general research question iswhether a given investment possibility set K , namely the benchmark set, contains portfolios2hich prospect dominates all alternatives in an expanded investment possibility set L .Stochastic spanning (Arvanitis et al. (2019)) is a model-free alternative to MV spanningof Huberman and Kandel (1987) (see also Jobson and Korkie (1989), De Roon, Neyman,and Werker (2001)). Spanning occurs if introducing new securities or relaxing investmentconstraints does not improve the investment possibility set for a given class of investors.MV spanning checks if the mean-variance frontier of a set of assets is identical to the mean-variance frontier of a larger set made of those assets plus additional assets (Kan and Zhou(2012), Penaranda and Sentana (2012)). Here we investigate such a problem for investorswith prospect type preferences which are interested in the whole return distributions gener-ated by two sets of assets, namely stochastic dominance. First, we introduce the concept ofprospect spanning, which is consistent with prospect type investors. We propose a theoreti-cal measure for prospect spanning based on stochastic dominance and derive the exact limitdistribution for the associated empirical test statistic for a general class of dynamic processes.To check prospect spanning on data, we develop consistent and feasible test procedures basedon subsampling and Linear Programming (LP).Similarly to Arvanitis et al. (2019), it is easy to see that if the prospect efficient set is non-empty, a prospect spanning set is essentially any superset of the former. As such, we can usea prospect spanning set to provide an outer approximation of the efficient set. This is usefulin at least two ways. First, if the spanning set is small enough, the problem of optimal choiceis reduced to a potentially simpler problem. Indeed, a spanning set is a reduction of theoriginal portfolio set without loss of investment opportunities for any investor with S-shapedpreferences. Secondly, if an algorithm for the choice of non-trivial canditate spanning sets isavailable, we can use this to construct decreasing sequences of prospect spanning sets thatappropriately converge to the efficient set. Given the complexity of the prospect efficient set(see for example Ingersoll (2016)) such an approach can be useful for the determination ofits properties.The second contribution of the paper is to examine if we can explain well-known stock3arket anomalies by standard factor models for prospect investors. To do so, we test iftrading strategies are genuine violations of standard factor models. More precisely, in the in-sample analysis, we use the prospect spanning test in order to check whether a portfolio setoriginating from a standard factor model, K , spans the same set augmented with a marketanomaly, L . This check could be of significant relevance to the empirical analysis of financialmarkets. If the hypothesis of prospect spanning holds, the particular market anomaly can beexplained by the factor model. Then the trading strategy that is identified in the literature asmarket anomaly may not be an attractive investment opportunity for prospect investors. Onthe contrary, if the hypothesis is not true, then the anomaly expands the opportunity set forprospect investors, and is useful to that extent. We also examine whether the cross-sectionalpatterns that found to expand the set of factors in-sample, maintain this abnormal returnout-of-sample. Therefore, we use out-of-sample backtesting experiments as an independentcriterion for robustness of in-sample test results (Harvey et al. (2016)). It turns out thatprospect spanning tests produce remarkably consistent results both in- and out-of-sample inidentifying trading strategies as genuine market anomalies for prospect investors. Thus, ourframework helps validating stock market anomalies for prospect preferences.Benartzi and Thaler (1995) utilize prospect theory to present an approach called myopicloss aversion which consists of two behavioural concepts, namely loss aversion and mentalaccounting. Barberis et al. (2001) study asset prices in an economy where investors derivedirect utility not only from consumption but also from fluctuations in the value of theirfinancial wealth. They are loss averse over these fluctuations and how loss averse they aredepends on their prior investment performance. The design of their model is influenced byprospect theory. Barberis and Huang (2008) study the pricing of financial securities wheninvestors make decisions according to cumulative prospect theory. Several other papersconfirm that positively skewed stocks have lower average returns (Boyer, Mitton, and Vorkink(2010), Bali, Cakici, and Whitelaw (2011), Kumar (2009), Conrad, Dittmar, and Ghysels(2013)). Barberis and Xiong (2009, 2012) and Ingersoll and Jin (2013) show that theoretical4nvestment models based on S-Shape utility maximisers help to understand the dispositioneffect found empirically in many studies (see e.g. Odean (1988), Grinblatt and Han (2005),Frazzini (2006), Calvet, Campbell, and Sodini (2009)). Kyle, Ou-Yang, and Xiong (2006)provide a formal framework to analyze the liquidation decisions of economic agents underprospect theory. He and Zhou (2011) study the impact of prospect theory on optimal riskyexposures in portfolio choice through an analytical treatment. Ebert and Strack (2015) setup a general version of prospect theory and prove that probability weighting implies skewnesspreference in the small. Barberis et al. (2016) test the hypothesis that, when thinking aboutallocating money to a stock, investors mentally represent the stock by the distribution ofits past returns and then evaluate this distribution in the way described by prospect theory.Moreover, Barberis et al. (2019) present a model of asset prices in which investors evaluaterisk according to prospect theory and examine its ability to explain prominent stock marketanomalies. In our paper, we test whether well-known factor models span the augmenteduniverse with a prominent stock market anomaly, and if not, whether the result is supportedout-of sample.The paper is organised as follows. In Section 2, we review the definition of prospectstochastic dominance relation and we define the relevant concept of prospect spanning. Weprovide with a representation based on a class of S-shaped utility functions without assumingdifferentiability. Using an empirical approximation of the latter, we construct a test for thenull hypothesis of spanning based on subsampling. The construction is based on the limitingnull distribution of the test statistic which has the form of a saddle type point of a relevantGaussian process. Under a weak condition on the structure of the parameter contact sets,we show that the test is asymptotically exact and consistent. This is weaker than theparameter extreme point comparisons used in Arvanitis, Scaillet and Topaloglou (2019) toobtain exactness in large samples.In Section 3, we provide with a numerical approximation of the statistic that is based onthe utility representation derived before. The utility functions are univariate, and normal-5zed. We use a finite set of increasing piecewise-linear functions, restricted to the boundedempirical supports, that are constructed as convex mixtures of appropriate "ramp functions”( in the spirit of Russel and Seo (1989)) in our representation. For every such utility func-tion, we solve two embedded linear maximization problems. This is an improvement over theimplementation in Arvanitis and Topaloglou (2017) and Arvanitis, Scaillet and Topaloglou(2019) where they formulate tests in terms of Mixed-Integer Programming (MIP) problems.MIP problems are NP-complete, and far more difficult to solve. Our numerical approxima-tions are simple and fast since they are based on standard LP. They suit better resamplingmethods, which otherwise become quickly computationally demanding in empirical work.In Section 4, we perform an empirical application where we use the prospect spanningtests to evaluate stock market anomalies using standard factor models. We consider threesuch models that build on the pioneer three-factor model of Fama and French (1993): thefour-factor model of Hou, Xue and Zhang (2015), the five-factor model of Fama and French(2015), and the four-factor model of Stambaugh and Yuan (2017). Given the extensiveset of results produced under alternative spanning criteria, the analysis is confined to 11well-known strategies used to construct Stambaugh-Yuan factors, along with 7 extra (18overall) that attracted significant attention, namely Betting against Beta, Quality minusJunk, Size, Growth Option, Value (Book to Market), Idiosyncratic Volatility and Profitabil-ity. The 11 anomalies used in Stambaugh and Yuan (2017) are realigned appropriately toyield positive average returns. In particular, anomaly variables that relate to investmentactivity (Asset Growth, Investment to Assets, Net Stock Issues, Composite Equity Issue,Accruals) are defined low-minus-high decile portfolio returns, rather than high-minus-low.All the other anomalies are constructed as high-minus-low decile portfolio returns. These18 trading strategies constitute our playing field for comparing spanning test results. Yet,we emphasize that this paper is not intended to compare factor models in terms of theirability to capture the cross-section of expected returns under prospect preferences. Instead,we use alternative factor models as a robustness check for testing the consistency of in- and6ut-of-sample results under the prospect spanning framework. Each factor model is our ini-tial system of investment coordinates which we take as a granted opportunity set, withoutquestioning its asset pricing validity. We view here the factors solely as investable assets(since they correspond to tradable strategies based on asset portfolios), and similarly for theanomalies. The anomalies might be labelled by other authors as factors if indeed priced inthe cross-section, but we do not address such a research question in this paper.Finally, Section 5 concludes the paper. In Appendix A, we provide a short description ofthe stock market anomalies used in the empirical application. In Appendix B, we also providea short description of the performance measure used in the out-of-sample analysis. We givein a separate Online Appendix: i) the limiting properties of the testing procedures undersequences of local alternatives, ii) a Monte Carlo study of the finite sample properties of thetest, iii) the proofs of the main results, as well as auxiliary lemmata and their proofs, iv)summary statistics of the factor and anomaly returns over our sample period from January1974 to December 2016, and v) additional empirical results on out-of-sample analysis ofmarket anomalies. The theory of stochastic dominance (SD) gives a systematic framework for analyzing in-vestor behavior under uncertainty (see Chapter 4 of Danthine and Donaldson (2014) foran introduction oriented towards finance). Stochastic dominance ranks portfolios based ongeneral regularity conditions for decision making under risk (see Hadar and Russell (1969),Hanoch and Levy (1969), and Rothschild and Stiglitz (1970)). SD uses a distribution-freeassumption framework which allows for nonparametric statistical estimation and inferencemethods. We can see SD as a flexible model-free alternative to mean-variance dominanceof Modern Portfolio Theory (Markowitz (1952)). The mean-variance criterion is consistent7ith Expected Utility for elliptical distributions such as the normal distribution (Chamber-lain (1983), Owen and Rabinovitch (1983), Berk (1997)) but has limited economic meaningwhen we cannot completely characterize the probability distribution by its location andscale. Simaan (1993), Athayde and Flores (2004), and Mencia and Sentana (2009) developa mean-variance-skewness framework based on generalizations of elliptical distributions thatare fully characterized by their first three moments. SD presents a further generalizationthat accounts for all moments of the return distributions without necessarily assuming aparticular family of distributions.Inspired by previous work, Levy and Levy (2002) formulate the notions of prospectstochastic dominance (PSD) (see also Levy and Wiener (1998), Levy and Levy (2004)) andMarkowitz stochastic dominance (MSD). Those notions extend the well-know first degreestochastic dominance (FSD) and second degree stochastic dominance (SSD). PSD and MSDinvestigates choices by investors who have S-shaped utility functions and reverse S-shapedutility functions. Arvanitis and Topaloglou (2017) develop consistent tests for PSD and MSDefficiency which is an extension to the case where full diversication is allowed. Arvanitis,Scaillet and Topaloglou (2019) investigate MSD spanning. This paper extends those worksto prospect spanning, which is consistent with prospect preferences.
Given a probability space (Ω , F , P ) , suppose that F denotes the cdf of some probabilitymeasure on R n . Let G ( z, λ, F ) be R R n { λ T u ≤ z } dF ( u ) , i.e., the cdf of the linear transformation x ∈ R n → λ T x where λ assumes its values in L , which denotes the portfolio space. Wesuppose that the portfolio space is a closed non-empty subset of S = { λ ∈ R n + : T λ = 1 , } ,possibly formulated by further economic, legal restrictions, etc. In many applications, wehave that L = S . We denote by K a distinguished subcollection of L and generic elements8f L by λ, κ , etc. In order to define the concepts of PSD and subsequently of stochasticspanning, we consider J ( z , z , λ ; F ) := R z z G ( u, λ, F ) du. Definition 1. κ weakly Prospect-dominates λ , written as κ < P λ , iff we have the inequal-ities P ( z, λ, κ, F ) := J ( z, , κ, F ) − J ( z, , λ, F ) ≤ , ∀ z ∈ R − and P ( z, λ, κ, F ) := J (0 , z, κ, F ) − J (0 , z, λ, F ) ≤ , ∀ z ∈ R ++ . Given the stochastic dominance relation above, stochastic spanning occurs when aug-mentation of the portfolio space does not enhance investment opportunities, or equivalently,investment opportunities are not lost when the portfolio space is reduced. The followingdefinition clarifies the concept w.r.t. the Prospect dominance relation.
Definition 2. K Prospect-spans L ( K < P L ) iff for any λ ∈ L , ∃ κ ∈ K : κ < P λ . If K = { κ } , the element κ of the singleton K is termed as Prospect super-efficient.The efficient set of the dominance relation is the subset of L that contains the maximalelements. The efficient set is a spanning subset of the portfolio space. Thereby, any supersetof the efficient set is also a spanning subset of L . We can consider a spanning set as an outerapproximation of the efficient set. Given a candidate spanning set exists, the question iswhether this actually spans the portfolio space. If a method for answering such a questionalso exists, we can accurately approximate the efficient set via the choice of finer spanningsubsets of the portfolio space. This is important in the context of decision theory andinvestment choice.Hence, the question we address here is: given a candidate K , is K < P L ? The followinglemma provides an analytical characterization by means of nested optimizations, which iskey for a numerical implementation on real data and statistical inference. Lemma 3.
Suppose that K is closed. Then K < P L iff we get the condition ρ ( F ) :=max i =1 , sup λ ∈ L sup z ∈ A i inf κ ∈ K P i ( z, λ, κ, F ) = 0 , where A = R − , A = R ++ . Moreover, we get that κ isProspect super-efficient iff sup λ ∈ L max i =1 , sup z ∈ A i P i ( z, λ, κ, F ) = 0 . .2 Representation By Utility Functions We provide an expected utility characterization of spanning. Aside the economic interpre-tation, this is key to the numerical LP implementation of the inferential procedures that weconstruct in the next section. In doing so, we generalize the utility characterization of PSDin Levy and Levy (2002), in the sense that we do not require differentiability of the utilities.Our approach is in the spirit of the Russel and Seo (1989) representations for the secondorder stochastic dominance. We rely on utilities represented as unions of graphs of convexmixtures of appropriate “ramp functions” on each half-line.To this end, we denote with W − , W + , the sets of Borel probability measures on thereal line with supports that are closed subsets of R − and R + , respectively, with existingfirst moments and uniformly integrable. The latter requirement is convenient yet harm-less since orderings are invariant to utility rescalings. Those sets are convex, and closedw.r.t. the topology of weak convergence and their union contains the set of degenerate mea-sures. Define V − := n v w : R − → R , v w ( u ) = R R − [ z u ≤ z + u z ≤ u ≤ ] dw ( z ) , w ∈ W − o , and V + := n v w : R + → R , v w ( u ) = R R + [ u ≤ u ≤ z + z z ≤ u< + ∞ ] dw ( z ) , w ∈ W + o . Every elementof V + is increasing and concave, and dually every element of V − is increasing and convex.Furthermore, any function defined by the union of the graph of an arbitrary element of V + with the graph of an arbitrary element of V − is the graph of an S-shaped utility function asdefined by Levy and Levy (2002). Such a utility function is concave for gains and convexfor losses. Denote the set of S-shaped utility functions obtained by such graph unions as V .Thereby, V := v : R → R , v ( u ) = v w ( u ) , u ≤ v w ( u ) , u ≥ , where v w ∈ V − , v w ∈ V + . Lemma 4.
We have ρ ( F ) = max i =1 , sup v w ∈ V i [sup λ ∈ L E λ [1 u ∈ A i v w ( u )] − sup κ ∈ K E κ [1 u ∈ A i v w ( u )]] , here E λ denotes expectation w.r.t. G ( z, λ, F ) . If the hypotheses of Lemma 3 hold and K isconvex, then K < P L iff, sup v ∈ V [sup λ ∈ L E λ [ v ] − sup κ ∈ K E κ [ v ]] = 0 . The fist part of the lemma connects the functional that represents spanning to the afore-mentioned classes of utilities. This is exploited below in order to obtain feasible numericalformulations based on LP. Those formulations are reminiscent of the LP programs devel-oped in the early papers of testing for SSD efficiency of a given portfolio by Post (2003) andKuosmanen (2004). The second part of Lemma 4 crystalizes the intuitive characterizationof spanning w.r.t. investment opportunities. It states that spanning holds if and only if thereduction of investment opportunities from L to K does not reduce optimal choices uniformlyw.r.t. this class of preferences. We cannot directly rely on Lemma 3 for empirical work if F is unknown and/or the optimiza-tions are infeasible. We construct a feasible statistical test for the null hypothesis of K < P L by utilizing an empirical approximation of F and by building feasible and fast optimisationswith LP. The null and alternative hypotheses take the following forms: H : ρ ( F ) = 0 , and H a : ρ ( F ) > . In the special case of super-efficiency, the hypotheses write as in Arvanitisand Topaloglou (2017).We consider a process ( Y t ) t ∈ Z taking values in R n . Y i,t denotes the i th element of Y t . The sample path of size T is the random element ( Y t ) t =1 ,...,T . In our empirical fi-nance framework, it represents returns of n financial assets upon which we can constructportfolios via convex combinations. F is the cdf of Y and F T is the empirical cdf as-sociated with the random element ( Y t ) t =1 ,...,T . Under our assumptions below, F T is aconsistent estimator of F , so we consider the following test statistic ρ T := √ T ρ ( F T ) = √ T max i =1 , sup λ ∈ L sup z ∈ A i inf κ ∈ K P i ( z, λ, κ, F T ) , which is the scaled empirical analog of ρ ( F ) . As already mentioned, when K is a singleton, the test statistic coincides with theone used in Arvanitis and Topaloglou (2017). The following assumption enables the deriva-11ion of the limit distribution of ρ T under H and is weaker than Assumption 2 in Arvanitis,Scaillet and Topaloglou (2019). Assumption 5. F is absolutely continuous w.r.t. the Lebesgue measure on R n with convexsupport that is bounded from below, and for some < δ , E h k Y k δ i < + ∞ . ( Y t ) t ∈ Z is a -mixing with mixing coefficients a T = O ( T − a ) for some a > η , < η < , as T → ∞ . The lower bound hypothesis is harmless in our empirical finance framework since we areusing financial returns. The mixing part is readily implied by concepts such as geometric er-godicity which holds for many stationary models used in the context of financial econometricsunder parameter restrictions and restrictions on the properties of the underlying innovationprocesses. Examples are the strictly stationary versions of (possibly multivariate) ARMA orseveral GARCH and stochastic volatility type of models (see Francq and Zakoian (2011) forseveral examples). Counter-examples are models that exhibit long memory, etc. The mo-ment condition is established in the aforementioned models via restrictions on the propertiesof building blocks and the parameters of the processes involved.For the derivation of the limit theory of ρ T under the null hypothesis, we consider the con-tact sets Γ i = (cid:8) λ ∈ L , κ ∈ K (cid:23) λ , z ∈ A i : P i ( z, λ, κ, F ) = 0 (cid:9) , where K (cid:23) λ := { κ ∈ K : κ < P λ } which under the null contains elements different from λ for any element of L − K . Forany i , the set Γ i is non empty since Γ ⋆i := { ( κ, κ, z ) , κ ∈ K , z ∈ A i } ⊆ Γ i . Furthermore, ( λ, κ, ∈ Γ , ∀ λ, κ . Since due to Assumption 5 z := inf λ,Y λ ′ Y exists, for all z ≤ z , ( λ, κ, z ) ∈ Γ i , ∀ λ ∈ L , κ ∈ K (cid:23) λ for the i that corresponds to the sign of z . In what follows,we denote convergence in distribution by . Proposition 6.
Suppose that K is closed, Assumption 5 holds and that H is true. Then as T → ∞ , ρ T ρ ∞ , where ρ ∞ := max i =1 , sup λ sup z inf κ P i ( z, λ, κ, G F ) , ( λ, z, κ ) ∈ Γ i , and G F is a centered Gaussian process with covariance kernel given byCov ( G F ( x ) , G F ( y )) = P t ∈ Z Cov (cid:0) { Y ≤ x } , { Y t ≤ y } (cid:1) and P almost surely uniformly continuoussample paths defined on R n . Z + ∞ G λF ( u ) du = Z + ∞ X t ∈ Z Cov (cid:0) { λ T Y ≤ u } , { λ Tr Y t ≤ u } (cid:1) du ≤ ∞ X t =0 √ a T Z + ∞ p − G ( u, λ, F ) du < + ∞ , andVar Z −∞ G λF ( u ) du = Z −∞ X t ∈ Z Cov (cid:0) { λ T Y ≤ u } , { λ Tr Y t ≤ u } (cid:1) du ≤ ∞ X t =0 √ a T Z −∞ p G ( u, λ, F ) du< + ∞ , where the first inequalities in each of the previous expressions follow from inequal-ity 1.12b in Rio (2000), and the second ones follow from Assumption 5 (see also p. 196 ofHorvath et al. (2006)).Since F and Γ i are unknown in practice, we use the results of the previous lemma toconstruct a decision procedure based on subsampling, in the spirit of Linton, Post and Whang(2014) (see also Linton, Maasoumi, and Whang (2005)). Algorithm 7.
This consists of the following steps:
1. Evaluate ρ T at the original sample value.2. For < b T ≤ T , generate subsample valuesfrom the original observations ( Y l ) l = t,...t + b T − for all t = 1 , , . . . , T − b T + 1 .3. Evaluate the test statistic on each subsample valuethereby obtaining ρ T,b T ,t for all t = 1 , , . . . , T − b T + 1 .4. Approximate the cdf of the asymptotic distribution under the null of ρ T by s T,b ( y ) = T − b T +1 P T − b T +1 t =1 ρ T,b T ,t ≤ y ) and calculate its − α quantile q T,b T (1 − α ) = inf y { s T,b ( y ) ≥ − α } , for the significance level < α < . .5. Reject the null hypothesis H if ρ T > q T,b T (1 − α ) . The partitioning used to get the results in Proposition 6 directly leads to the consideration of subsamplingas a resampling procedure. A testing procedure based on (block) bootstrap as in Scaillet and Topaloglou(2010), can, due to the form of the recentering, be consistent, but can be too conservative asymptotically,and thereby suffer from a lack of power compared to the subsampling under particular local alternatives(see also the relevant discussion in Arvanitis et al. (2019)). The potential of asymptotic exactness for thesubsampling test justifies the particular resampling choice for inference.
13n order to derive the limit theory for the testing procedure, namely its asymptotic exactnessand consistency stated in the next theorem, we first use the following standard assumptionthat restricts the asymptotic behaviour of b T governing the size b T + 1 of each subsample. Assumption 8.
Suppose that ( b T ) , possibly depending on ( Y t ) t =1 ,...,T , satisfies the condition P ( l T ≤ b T ≤ u T ) → , where ( l T ) and ( u T ) are real sequences such that ≤ l T ≤ u T for all T , l T → ∞ and u T T → as T → ∞ . Theorem 9.
Suppose Assumptions 5 and 8 hold. For the testing procedure described inAlgorithm 7, we have that1. If H is true, and for λ ∈ L − K , inf Y λ T r Y ≤ there exists ( κ, z ) ∈ K (cid:23) λ × R ++ with ( λ, κ, z ) ∈ Γ and that if ( λ, κ ⋆ , z ⋆ ) ∈ Γ for κ ⋆ = κ then z ⋆ = z , then for all α ∈ (0 , . T →∞ P ( ρ T > q T,b T (1 − α )) = α.
2. If H a is true then lim T →∞ P ( ρ T > q T,b T (1 − α )) = 1 . When for λ ∈ L − K , inf Y λ T r Y ≤ then due to Assumption 5 for any contact triple ( λ, κ, z ) ∈ Γ we have that P ( z, λ, κ, G F ) must be non-degenerate. Whenever z correspondssolely to the particular κ , we obtain that ρ ∞ is non-degenerate and if its cdf jumps at theinfimum of its support, then the jump magnitude is bounded above by . . Hence in thiscase the test is asymptotically exact for all the usual choices of the significance level sincethe probability of rejection under the null hypothesis, i.e., the size of the test, reaches α in large samples. We combine Proposition 6 above and Theorem 3.5.1 of Politis, Romanoand Wolf (1999) in the proof of the exactness statement, namely point 1 of Theorem 9. Toget exactness, the condition imposed on L − K is significantly weaker than the assumptionon the relation between the extreme points of L and K adopted by Arvanitis, Scaillet andTopaloglou (2019). It amounts to the existence of a spanned portfolio whose support is notstrictly positive and so that, in the event of positive returns, there exists an elementaryincreasing and concave utility for positive returns and a unique portfolio such that the14atter dominates the former and we are indifferent between the two portfolios with thisparticular utility. Besides, the test is also consistent since the probability of rejection underthe alternative hypothesis, i.e., the power of the test, reaches 1 in large samples. We show inthe proof of the consistency statement, namely point 2 of Theorem 9, that the test statisticdiverges to + ∞ under the alternative hypothesis when T goes to + ∞ .We opt for the “bias correction” regression analysis of Arvanitis et al. (2019) to reducethe sensitivity of the quantile estimates q T,b T (1 − α ) on the choice of b T in empirically real-istic dimensions for n and T (see also Arvanitis, Scaillet and Topaloglou (2019) for furtherevidence on its better finite sample properties). Specifically, given α , we compute the quan-tiles q T,b T (1 − α ) for a “reasonable” range of b T . Next, we estimate the intercept and slopeof the following regression line by OLS: q T,b T (1 − α ) = γ T, − α + γ T, − α ( b T ) − + ν T ;1 − α,b T . Finally, we estimate the bias-corrected (1 − α ) -quantile as the OLS predicted value for b T = T : q BCT (1 − α ) := ˆ γ T, − α + ˆ γ T, − α ( T ) − . Since q T,b T (1 − α ) converges in probabilityto q ( ρ ∞ , − α ) and ( b T ) − converges to zero as T → , ˆ γ T, − α converges in probability to q ( ρ ∞ , − α ) and the asymptotic properties are not affected.In the Online Appendix, we also show that under further assumptions, the test is asymp-totically locally unbiased under given sequences of local alternatives. Besides, the MonteCarlo analysis reported in the Online Appendix shows that the test performs well with anempirical size close to 5% and an empirical power above 90% for a significance level α = 5% . In this section, we exploit the results of Lemma 4 in order to provide with a finitary approx-imation of the test statistic. We rely on this to provide with a numerical implementationbased on LP below. We denote expectation w.r.t. the empirical measure by E F T . Let R − denote max i =1 ,...,n Range (cid:0) Y i,t Y i,t ≤ (cid:1) t =1 ,...,T = [ x, . Partition R − into n equally spacedvalues as x = z < · · · < z n = 0 , where z n := x − n − n − x , n = 1 , · · · , n ; n ≥ . Fur-15hermore, partition the interval [0 , , as < n − < · · · < n − n − < , n ≥ . Similarly, R + := max i =1 ,...,n Range (cid:0) Y i,t Y i,t ≥ (cid:1) t =1 ,...,T = [0 , x ] . Partition R + into p equally spaced val-ues as z < · · · < z p = x , where z p := p − p − x , n = 1 , · · · , p ; p ≥ , and again partitionthe interval [0 , , as < p − < · · · < p − p − < , p ≥ . Using the above, we consider thetest statistic: ρ ⋆T := √ T max i =1 , sup v ∈ V ⋆i (cid:20) sup λ ∈ L E F T (cid:2) v (cid:0) λ T Y (cid:1)(cid:3) − sup κ ∈ K E F T (cid:2) v (cid:0) κ T Y (cid:1)(cid:3)(cid:21) , (1)where the set of utility functions for negative returns is: V ⋆ − := ( v : v ( u ) = n X n =1 w n [ z n x ≤ u ≤ z n + u z n ≤ u ≤ ] , ( w , . . . , w n ) ∈ W − ) , W − := ( (w , . . . , w n ) ∈ (cid:26) , − , · · · , n − − , (cid:27) n : n X n=1 w n = 1 ) , and the set of utility functions for positive returns is: V ⋆ + := ( v : v ( u ) = p X p =1 w p (cid:2) u ≤ u ≤ z p + z p z p ≤ u ≤ x (cid:3) , ( w , . . . , w p ) ∈ W + ) , W + := ( (w , . . . , w p ) ∈ (cid:26) , − , · · · , p − − , (cid:27) p : p X p=1 w p = 1 ) . We obtain the following result on the approximation of ρ T by ρ ⋆T . Proposition 10.
When the support of F is also bounded from above, as n , n , p , p → ∞ ,we have ρ ⋆T → ρ T , P a.s. Our feasible computational strategy builds on LP formulations for the numerical evalu-ation using the previous finitary approximation of the test statistic.We have a set of convex utility functions of the form: v ( u ) = P n n =1 w n max( u, z n ) forthe negative part. For every v ∈ V ⋆ − , we have at most n line segments with knots at n possible outcome levels. Then, we can enumerate all n = n − Q n − i =1 ( n + i − elements16f V ⋆ − . Our application in Section 4 uses n = 10 , and n = 5 , which gives n = 715 distinct utility functions, and a total of 1430 small LP problems for the two embeddedmaximisation problems in (1). Solving (1) yields simultaneously the optimal factor portfolio κ , and the optimal augmented portfolio λ that maximize the expected utility. Below, wegive the mathematical formulation for the first optimization problem sup λ ∈ Λ E F N (cid:2) u (cid:0) λ T Y (cid:1)(cid:3) ,that yields the optimal augmented portfolio λ . The same formulation is used for the secondoptimization sup κ ∈ κ E F N (cid:2) u (cid:0) κ T Y (cid:1)(cid:3) .Let us define: c ,n := P n m = n ( c ,m − c ,m +1 ) z m , c ,n := P n m = n w m , and N := { n = 1 , · · · , n : w n > } S { n } . For any given u ∈ V − , sup λ ∈ Λ E F N (cid:2) u (cid:0) λ T Y (cid:1)(cid:3) isthe optimal value of the objective function of the following LP problem in canonical form: max T − T X t =1 y t (2)s.t., for t = 1 , · · · , T, n ∈ N , i = 1 , · · · , M,y t ≤ λ T Y t c ,n + Q − t + Q + t , y t ≤ c ,n + Q − t + Q + t ,Q − t ≥ c ,n − λ T Y t c ,n , Q + t ≥ λ T Y t c ,n − c ,n , Q − t ≥ , Q + t , ≥ , M X i =1 λ i = 1 , λ i ≥ , and y t being free . We have a set of concave utility functions of the form: v ( u ) = P p p =1 w p min( u, z p ) , for thepositive part. Again, for every v ∈ V ⋆ + , we have at most p line segments with knots at p possible outcome levels. As before, the number of elements of V ⋆ + is p = p − Q p − i =1 ( p + i −
1) = 1430 , for p = 10 and p = 5 .Let us define: c ,p := P p m = p ( c ,m − c ,m +1 ) z m , c ,p := P p m = p w m , and P := { p = 1 , · · · , p : w p > } S { p } . For any given u ∈ V + , sup λ ∈ Λ E F N (cid:2) u (cid:0) λ T Y (cid:1)(cid:3) is the17ptimal value of the objective function of the following LP problem in canonical form: max T − T X t =1 y t (3)s.t., for t = 1 , · · · , T, n ∈ P , i = 1 , · · · , M,y t ≤ λ T Y t c ,p , y t ≤ c ,p , M X i =1 λ i = 1 λ i ≥ , and y t being free . The total run time for each computation does not exceed one minute when we use adesktop PC with a 3.6 GHz, 6-core Intel i7 processor, with 16 GB of RAM, using MATLABand GAMS with the Gurobi optimization solver.
In the empirical application, we examine if we can explain well-known stock market anomaliesby standard factors within a new breed of asset pricing models, for prospect type investorpreferences. For this purpose, we use the prospect spanning tests, both in- and out-of-sample.
We start with a benchmark factor model from a set of models that have generated supportin the recent literature, and we ask whether a characteristic identified in the literature asstock market anomaly, is a market anomaly for prospect investors. To answer this question,we consider three models that build on the pioneer three-factor model of Fama and French(1993): the four-factor model of Hou, Xue and Zhang (2015), the five-factor model of Famaand French (2015), and the four-factor model of Stambaugh and Yuan (2017). Fama andFrench (1993) aim to capture the part of average stock returns left unexplained in CAPM ofSharpe (1964) and Lintner (1965) by including, in addition to the market factor, two extrarisk factors relating to size (measured by market equity) and the ratio of book-to-market18quity. In addition to the market excess return, the influential three-factor model of Famaand French (1993) includes a book-to-market or "value" factor, HML, and a size factor,SMB, based on market capitalization. Motivated by Miller and Modigliani (1961), Famaand French (2015) five-factor model (henceforth, FF-5) augments the original Fama-Frenchthree-factor model by two extra factors, one for profitability and another for investment.Hou, Xue and Zhang (2015) consider a four-factor model (dubbed the q-factor model) thatincludes the original market and size factors of Fama and French (1993) augmented by aprofitability and investment factor. Stambaugh and Yuan (2017) consider a four-factor model(henceforth, M-4) including the standard market and size factors along with two compositefactors for investment and profitability. To construct the composite factors, they combineinformation from 11 market anomalies relating to investment and profitability measures. Weuse alternative factor models as a robustness check, namely for testing the consistency ofin- and out-of-sample results under the prospect preferences, and not for a horse race incross-sectional asset pricing.The stock market anomalies we examine in this paper have a long history in the relevantliterature. A common theme in the original papers that first highlighted these patterns,is that they all challenge the rational asset pricing paradigm as they exhibit returns thatare not in line with the risks taken. However, notwithstanding whether they are causedby sentiment (a catch-all term that stand for all kinds of irrational decision-making) or bymarket frictions (e.g. margin requirements), it is also acknowledged that most of them persistbecause they cannot be “arbitraged” away. From the perspective of the Arbitrage PricingTheory this implies that arbitrageurs cannot trade against them without exposing themselvesto significant risks. In this paper, we test the 11 strategies used to construct Stambaugh-Yuan factors, along with Betting against Beta, Quality minus Junk, Size, Growth Option,Value (Book to Market), Idiosyncratic volatility and Profitability. The 11 anomalies used inStambaugh and Yuan (2017) are Accruals, Asset Growth, Composite Equity Issue, Distress,Growth Profitability Premium, Investment to Assets, Momentum, Net Operating Assets,19et Stock Issues, O-Score, and Return on Assets. They are realigned appropriately to yieldpositive average returns. In particular, anomaly variables that relate to investment activity(Asset Growth, Investment to Assets, Net Stock Issues, Composite Equity Isues, Aaccruals)are defined low-minus-high decile portfolio returns, rather than high-minus-low, as in Houet al. (2015). All the other anomalies are constructed as high-minus-low decile portfolioreturns. A short description of the 18 market anomalies that we study in the paper is givenin Appendix A (see Stambaugh and Yuan (2017) for further details). Returns of the Famaand French 5 factors were downloaded from Kenneth French’s site. The dataset consistsof all monthly observations from January 1974 until December 2016. M-4 factor returnsand anomaly spread return series were downloaded from the websites of Robert Stambaughand AQR. In the Online Appendix, we report summary statistics of the factor and anomalyreturns over our sample period.
In this section, we test in-sample the null hypothesis that the set of standard factors prospectspans the set enlarged with a particular market anomaly. We test separately for the Famaand French 5 factors, the Stambaugh-Yuan 4 factors as well as Hou-Xue-Zhang 4 factors,with respect to each one of the 18 additional anomalies. We get the subsampling distributionof the test statistic for subsample size b T ∈ { T . , T . , T . , T . } . Using OLS regression onthe empirical quantiles q T,b T (1 − α ) for a significance level α = 5% , we get the estimate q BCT for the bias-corrected critical value. We reject spanning if the test statistic ρ ⋆T is higher thanthe regression estimate q BCT .Tables 1-3 report the test statistics ρ ⋆T as well as the regression estimates q BCT when wetest for spanning of the alternative factor models w.r.t. each one of the 18 market anomalies.20able 1: Test statistics: Fama and French (FF-5) Factors
Variable Test statistic ρ ⋆T Regression estimates q BCT
ResultAccruals 0.0016 0.0025 SpanningAsset Growth 0.0 0.0 SpanningComposite Equity Issue 0.0015 0.0003 Reject SpanningDistress 0.0045 0.0005 Reject SpanningGrowth Profitability Premium 0.0015 0.0012 Reject SpanningInvestment to Assets 0.0014 0.0001 Reject SpanningMomentum 0.0696 0.0204 Reject SpanningNet Operating Assets 0.0268 0.0009 Reject SpanningNet Stock Issues 0.0011 0.0003 Reject SpanningO-Score 0.0129 0.0092 Reject SpanningReturn on Assets 0.0024 0.0047 SpanningBetting against Beta 0.0235 0.0176 Reject SpanningQuality minus Junk 0.0088 0.0061 Reject SpanningSize 0.0 0.0 SpanningGrowth Option 0.0 0.0 SpanningValue (Book to Market) 0.1921 0.1878 Reject SpanningIdiosyncratic Volatility 01959 0.0100 Reject SpanningProfitability 0.0 0.0 Spanning
Entries report the test statistics ρ ⋆T and the regression estimates q BCT for spanning of theFama and French (FF-5) model with respect to each one of the 18 market anomalies. Wereject spanning at significance level α = 5% if ρ ⋆T > q BCT . The dataset spans the period fromJanuary, 1974 to December, 2016. 21able 2: Test statistics: Stambaugh-Yuan (M-4) Factors
Variable Test statistic ρ ⋆T Regression estimates q BCT
ResultAccruals 0.0081 0.0083 SpanningAsset Growth 0.0057 0.0069 SpanningComposite Equity Issue 0.0143 0.078 Reject SpanningDistress 0.0533 0.0020 Reject SpanningGrowth Profitability Premium 0.0113 0.0049 Reject SpanningInvestment to Assets 0.0116 0.0164 Reject SpanningMomentum 0.1189 0.1143 Reject SpanningNet Operating Assets 0.0653 0.0071 Reject SpanningNet Stock Issues 0.0145 0.0073 Reject SpanningO-Score 0.0133 0.0122 Reject SpanningReturn on Assets 0.0012 0.0015 SpanningBetting against Beta 0.0755 0.0703 Reject SpanningQuality minus Junk 0.0374 0.0099 Reject SpanningSize 0.0 0.0 SpanningGrowth Option 0.0 0.0 SpanningValue (Book to Market) 0.2939 0.2817 Reject SpanningIdiosyncratic Volatility 0.2593 0.1039 Reject SpanningProfitability 0.0 0.0 Spanning
Entries report the test statistics ρ ⋆T and the regression estimates q BCT for spanning of theStambaugh-Yuan (M-4) model with respect to each one of the 18 market anomalies. Wereject spanning at significance level α = 5% if ρ ⋆T > q BCT . The dataset spans the period fromJanuary, 1974 to December, 2016. 22able 3: Test statistics: Hou-Xue-Zhang (q) Factors
Variable Test statistic ρ ⋆T Regression estimates q BCT
ResultAccruals 0.0106 0.0039 Reject SpanninAsset Growth 0.0176 0.0101 Reject SpanningComposite Equity Issue 0.0163 0.0159 Reject SpanningDistress 0.0386 0.0133 Reject SpanningGrowth Profitability Premium 0.0084 0.0038 Reject SpanningInvestment to Assets 0.0157 0.0123 Reject SpanningMomentum 0.0835 0.0305 Reject SpanningNet Operating Assets 0.0449 0.0059 Reject SpanningNet Stock Issues 0.0178 0.0170 Reject SpanningO-Score 0.0140 0.0109 Reject SpanningReturn on Assets 0.0235 0.0321 SpanningBetting against Beta 0.0404 0.0424 SpanningQuality minus Junk 0.0304 0.0177 Reject SpanningSize 0.0 0.0 SpanningGrowth Option 0.0029 0.0 Reject SpanningValue (Book to Market) 0.2045 0.1878 Reject SpanningIdiosyncratic Volatility 0.2386 0.0101 Reject SpanningProfitability 0.0 0.0 Spanning
Entries report the test statistics ρ ⋆T and the regression estimates q BCT for spanning of theHou-Xue-Zhang (q) model with respect to each one of the 18 market anomalies. We rejectspanning at significance level α = 5% if ρ ⋆T > q BCT . The dataset spans the period fromJanuary, 1974 to December, 2016.We observe that the FF-5 model spans 6 out of 18 market anomalies, that is, Accruals,Asset Growth, Return on Assets, Size, Growth Option, and Profitability. The M-4 modelspans the same 6 market anomalies, while the q model spans Return on Assets, Bettingagainst Beta, Size, and Profitability. Thus, in most cases, optimal portfolios based on theinvestment opportunity set that includes a market anomaly is not spanned by the corre-sponding optimal portfolio strategies based on the original factors. We also observe thatReturn on Assets, Size, and Profitability are spanned by all the factor models, indicatingthe robustness of these characteristics being not considered as genuine market anomalies byprospect investors. 23 .3 Out-of-Sample Analysis
In this section, we examine whether the inclusion of a market anomaly in the investmentopportunity set benefits to prospect investors out-of-sample. Although we reject the nullhypothesis of prospect spanning in most cases for the in-sample tests, it is not known apriori whether an optimal augmented portfolio also outperforms an optimal portfolio madeof factors only in an out-of-sample analysis. This is because by construction we form theseportfolios at time t , based on the information prevailing at time t , while we reap the portfolioreturns over [ t, t + 1] (next month). The out-of-sample test is a real-time exercise mimickingthe way that a real-time investor acts.Each time the hypothesized portfolio manager with prospect preferences forms optimalportfolios from two separate asset universes: the first universe consists only of factors froma factor model (FF-5, M-4, q), the set K . The second universe is the respective set offactors augmented by a single trading (spread) strategy, the set L . Portfolio managersare assumed to solve portfolio optimization problems, motivated by the prospect spanningframework, effectively looking for a portfolio picked from the augmented universe L thatprospect stochastically dominates all portfolios of the respective factor universe K ..The rejection of the prospect spanning hypothesis implies that there exists at leastone portfolio in L build from the factors (of each particular factor model) and one mar-ket anomaly, which is weakly prefered to every factor portfolio in K by at least one S-shapedutility function (see Definition 2). Such a portfolio is by construction efficient w.r.t. K (seeDefinition 2.1 in Linton et al. (2014) for the SSD case which we can easily generalize to ourPSD case). The empirical version of such a portfolio is the optimal portfolio λ that maxi-mizes ρ T for the particular sample value. In what follows, and given this characterization,we analyze the performance of such empirically optimal PSD portfolios through time, com-pared to the performance of the optimal factor portfolios solely derived from K by prospectinvestors.We resort to backtesting experiments on a rolling horizon basis. The rolling windows24over the 516 months period from 01/1974 to 12/2016. At each month, we use the datafrom the previous 25 years (300 monthly observations) to calibrate the procedure. We solvethe resulting optimization problem for the prospect stochastic spanning test and record theoptimal portfolios. The clock is advanced and we determine the realized returns of theoptimal portfolios from the actual returns of the various assets. Then we repeat the sameprocedure for the next time period and we compute the ex post realized returns over theperiod from 01/1999 to 12/2016 (216 months) for both portfolios.We compute a number of commonly used performance measures: the average return(Mean), the standard deviation (SD) of returns, the Sharpe ratio, the downside Sharpe ratio(D. Sharpe ratio) of Ziemba (2005), the upside potential and downside risk (UP) ratio ofSortino and van der Meer (1991), the opportunity cost of Simaan (2013), and a measure ofthe portfolio risk-adjusted returns net of transaction costs (Return Loss) of DeMiguel et al.(2009). The downside Sharpe and UP ratios are considered to be more appropriate measuresof performance than the typical Sharpe ratio given the asymmetric return distribution of theanomalies. For the calculation of the opportunity cost, we use the following utility functionwhich satisfies the curvature of prospect theory (S-shaped): U ( R ) = R α if R ≥ or − γ ( − R ) β if R < , where γ is the coefficient of loss aversion (usually γ = 2 . ) and α, β < . Weprovide a short description of those performance measures in Appendix B. In the next lines,we only detail the results of the out-of-sample tests for the Momentum market anomaly.The latter is well documented on diverse markets and asset classes (Asness, Moskowitz, andPedersen (2013)). In the Online Appendix, we report the performance measures for the 5Fama and French, the 4 Stambaugh and Yuan and the 4 Hou-Xue-Zhang optimal factorportfolios, and the optimal augmented portfolios for all the other market anomalies that wetest.Table 4 reports the performance measures for the Momentum anomaly under each factormodel (Panels A, B and C, respectively). These performance measures supplement theevidence obtained from the in-sample analysis. We observe that the Mean, the Sharpe ratio,25ownside Sharpe ratio and UP ratio of the optimal augmented portfolio are improved withrespect to the optimal factor portfolio. Although these measures are based on the firsttwo moments, they support the in-sample result that the set enlarged with the momentumanomaly is not spanned by any factor model. The same is true when we take into accounttransaction costs. The Return Loss is always positive. The opportunity cost measure takesinto account the entire distribution of returns under a given characterization of preferences.We observe that augmenting the factors by Momentum increases the performance of theoptimal portfolio with respect to each factor model. The optimal weight of Momentumvaries from 40% to 99%, indicating the superior performance of this characteristic.In the Online Appendix, we present analogous Tables for the other market anomalies.Interestingly, based on the opportunity cost, enlarging the factor set by a market anomalyincreases the performance of an optimal portfolio in 12 out of the 18 cases with respect to FF-5 factors (Composite Equity Issue, Distress, Growth Profitability Premium, Investment toAssets, Momentum, Net Operating Assets, O-Score, Net Stock Issues, Betting against Beta,Quality minus Junk, Value, and Idiosyncratic Volatility), in 10 cases with respect to M-4factors (Composite Equity Issue, Distress, Investment to Assets, Momentum, Net OperatingAssets, Net Stock Issues, Betting against Beta, Quality minus Junk, Value, and IdiosyncraticVolatility) and in 14 cases with respect to q factors (Accruals, Asset Growth, CompositeEquity Issue, Distress, Growth Profitability Premium, Investment to Assets, Momentum,Net Operating Assets, O-Score, Net Stock Issues, Betting against Beta, Quality minus Junk,Size, Value, and Idiosyncratic Volatility). For all these additional market anomalies, wefind a positive opportunity cost θ . One needs to give a positive return equal to θ to aninvestor who does not include the anomalies in her portfolio so that she becomes as happyas an investor who includes them. The computation of the opportunity cost requires thecomputation of the expected utility and hence the use of the probability density function ofportfolio returns. Thus, the calculated opportunity cost has taken into account the higherorder moments in contrast to the Sharpe ratios. Therefore, the opportunity cost estimates26rovide further convincing evidence for the diversification benefits of the inclusion of themarket anomalies given their deviation from normality.Additionally, although the rest of the performance measures depend mostly on the firsttwo moments of the return distribution, they give consistent results. The Return Lossmeasure that takes into account transaction costs, is positive in all the above cases. Thisreflects an increase in risk-adjusted performance (i.e., an increase in expected return per unitof risk) and hence expands the investment opportunities of prospect investors. The same istrue for the UP ratio. Finally, the Sharpe ratio and the downside Sharpe ratio agree thatthe performance of the optimal portfolios augmented with the above market anomalies isimproved, although the differences are small in some cases.The analysis indicates that the Composite Equity Issue, Distress, Investment to As-sets, Momentum, Net Operating Assets, Net Stock Issues, Quality minus Junk, Value, andIdiosyncratic Volatility emerge as unambiguously genuine market anomalies under all fac-tor sets, both in- and out-of-sample. Prospect investors would benefit from including thesecharacteristics in their portfolios, expanding the investment opportunity set offered by factorportfolios. We stress that the prospect spanning approach is particularly robust in-sampleand out-of-sample. The remarkable consistency of in-sample and out-of-sample results offersgood incentives for adopting such an approach when exploring instances of apparent marketinefficiency.To sum up, the in-sample spanning tests, as well as the out-of-sample analysis givenby the performance measures, indicate that in most cases (depending on the factor modelused) the investment universe augmented with a market anomaly dominates the 5 Fama andFrench, the 4 Stambaugh and Yuan, and the 4 Hou-Xue-Zhang factors, yielding diversifica-tion benefits and providing better investment opportunities for investors with prospect typepreferences towards risk. 27able 4: Performance measures. The case of the Momentum anomaly. Panel A Panel B Panel CFF-5 + anom. M-4 + anom. q + anom.Mean 0.0056 0.0062 0.0044 0.0048 0.0073 0.0072SD 0.0358 0.0370 0.0388 0.0409 0.0808 0.0385Sharpe ratio 0.1507 0.1604 0.1063 0.1117 0.0879 0.1814D. Sharpe ratio 0.1622 0.1706 0.1078 0.1108 0.0868 0.1995UP ratio 0.6401 0.6693 0.5646 0.5853 0.5348 0.6769Return Loss 0.0351% 0.0205% 0.3723%Opportunity Cost α = β = 0 . α = β = 0 . α = β = 0 . Descriptive statistics of the weight allocation of the optimal portfoliosMean Std. Dev. Skewness KurtosisFF-5 Factors Market 0.5955 0.1507 -3.3717 10.9074SMB 0.0 0.0 - -HML 0.0 0.0 - -RMW 0.0 0.0 - -CMA 0.0 0.0 - -Momentum 0.4045 0.1507 3.3717 10.9074M-4 Factors Market 0.5331 0.2255 -1.6812 1.5383SMB 0.0 0.0 - -MGMT1 0.0020 0.0113 7.4184 59.9621PERf1 0.0 0.0 - -Momentum 0.4648 0.2273 1.6464 1.4817q Factors Market 0.0028 0.0411 14.6969 216.000ME 0.0 0.0 - -IA 0.0 0.0 - -ROE 0.0 0.0 - -Momentum 0.9972 0.0411 -14.6969 216Entries report the performance measures (Mean, Standard Deviation, Sharpe ratio, DownsideSharpe ratio, UP ratio, Returns Loss and Opportunity Cost) for the factor optimal portfolios,as well as the augmented with the Momentum optimal portfolio. The dataset spans theperiod from January, 1999 to December, 2016. Panel A report measures for the case ofthe FF-5 factors. Panel B for the case of the M-4 factors, while panel C for the case ofthe q factors. In the second half, the Table exhibits the descriptive statistics of the weightallocation of the optimal augmented portfolios.
In this paper, we develop and implement methods for determining whether introducing newsecurities or relaxing investment constraints improves the investment opportunity set for28rospect investors. We develop a testing procedure for prospect spanning for two nestedportfolio sets based on subsampling and standard LP.In the empirics, we apply the prospect spanning framework to asset prices in whichinvestors evaluate risk according to prospect theory and examine its ability to explain 18 well-known stock market anomalies. The setting deploys prospect theory in a fully nonparametricway. We find that of the strategies considered, many expand the opportunity set of theprospect investors, thus have real economic value for them.Most importantly, we show that the prospect spanning approach is particularly robustbetween in-sample and out-of-sample applications. The paper contributes to a current strandof literature aiming to reevaluate published anomalies and discern those with real economiccontent for prospect investors. From a practitioner perspective, this robust framework forestablishing investment opportunities for prospect investors can be of real value, especiallyin the case of quantitative investment funds that combine talent, capital and computationalpower to the purpose of exploiting the existing anomalies and discovering new ones.
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PPENDIX A: Description of Stock Market Anomalies
Below we provide the origin and a short description of the 18 market anomalies used inthe empirical application.1. Accruals: Sloan (1996) argues that investors tend to overestimate in their earningsexpectations the persistence of the earnings’ component that is due to accruals. As a result,firms with low accruals earn on average abnormally higher returns than firms with highaccruals.2. Asset Growth: Cooper, Gulen, and Schill (2008) maintain that investors tend tooverreact positively right after asset expansions. According to the authors, this behaviorcauses firms with high growth in their total assets to exhibit relatively lower returns overthe subsequent fiscal years.3. Composite Equity Issues: Daniel and Titman (2006) base their analysis on a measureof equity issuance that they devised finding that equity issuers tend to underperform non-issuer firms.4. Distress: Campbell, Hilscher, and Szilagyi (2008) find that firms with high defaultprobability tend to exhibit lower subsequent returns. This pattern is counter-intuitive inthe context of rational asset pricing, given that according to the standard models high riskentails high expected return and vice versa.5. Gross Profitability Premium: Novy-Marx (2013) argues that gross profit is the mostobjective profitability metric. As a result, firms with the strongest gross profit have onaverage higher returns than the less profitable ones.6. Investment to Assets: Titman, Wei, and Xie (2004) argue that investors are put offby empire-building managers who over-invest. For this reason, firms showing a significantincrease in gross property, plant, equipment or inventories tend to underperform the market.7. Momentum: Momentum (Jegadeesh and Titman (1993)) is perhaps the most cited38nomaly in asset pricing. Since Carhart factor model (1997), it has been included in variousreduced-form models of the SDF as a factor. The momentum effect is attributed to sentimentand describes the pattern of “winner” stocks gaining higher subsequent returns and “loser”stocks relatively lower.8. Net Operating Assets: Hirshleifer et al. (2004) suggest that investors often neglectinformation about cash profitability and focus instead on accounting profitability. Becauseof this bias, firms with high net operating assets (measured as the cumulative differencebetween operating income and free cash flow) get to have negative long-run stock returns.9. Net Stock Issues: Ritter (1991) and Loughran and Ritter (1995) indicate that eq-uity issuers underperform non-issuers with similar characteristics. Fama and French (2008)demonstrate that net stock issues are negatively correlated with subsequent returns.10. O-Score: This anomaly coincides with the distress anomaly we mentioned earlier.In this case, the spread portfolios are constructed from stock ranking based on the O-score(Ohlson (1980)) to measure distress likelihood.11. Return on Assets: Chen, Novy-Marx, and Zhang (2010) associate high past returnon assets with abnormally high subsequent returns. Return on assets is measured as theratio of quarterly earnings to last quarter’s assets.12. Betting against Beta: Black, Jensen and Scholes (1972) showed that low (high) betastocks have consistently positive (negative) risk-adjusted returns. Frazzini and Pedersen(2014) propose an investment strategy (“betting-against-beta” (BAB)) that exploits thisanomaly by buying low-beta stocks and shorting high-beta stocks. Because of its robustness,this anomaly is currently one of the most widely examined APT violations.13. Quality minus Junk: Asness, Frazzini and Pedersen (2013) show that high-qualitystocks (safe, profitable, growing, and well managed) exhibit high risk-adjusted returns. Theauthors attribute this pattern to mispricing.14. Size: The market capitalization. is computed as the log of the product of price pershare and number of shares outstanding, computed at the end of the previous month.395. Growth Option: Growth Option measure represents the residual future-oriented firmgrowth potential. This future (yet-to-be exercised) growth option measure is calculatedas the % of a firm’s market value (V) arising from future-oriented growth opportunities(PVGO/V). It is inferred by subtracting from the current market value of the firm (V) theperpetual discounted stream of expected operating cash flows under a no-further growthpolicy (see, e.g., Kester (1984), Anderson and Garcia-Feijoo (2006), Berk, Green, and Naik(1999)).16. Value (Book to market): The log of book value of equity scaled by market valueof equity, computed following Fama and French (1992) and Fama and French (2008); firmswith negative book value are excluded from the analysis.17. Idiosyncratic Volatility: Standard deviation of the residuals from a firm-level regres-sion of daily stock returns on the daily Fama-French three factors using data from the pastmonth. See Ang et al. (2006).18. Profitability.: It is measured as revenue minus cost of goods sold at time t, divided byassets at time t-1. Stocks with high profitability ratios tend to outperform on a risk-adjustedbasis (Novy-Marx (2013), Novy-Marx and Velikov (2015)). Recent research suggests thatprofitability is one of the stock return anomalies that has the largest economic significance(see Novy-Marx (2013)).
APPENDIX B: Description of Performance Measures
For the downside Sharpe ratio, first we need to calculate the downside variance (ormore precisely the downside risk), σ P − = P Tt =1 ( x t − ¯ x ) − T − , where the benchmark ¯ x is zero, andthe x t taken are those returns of portfolio P at month t below ¯ x , i.e., those t of the T months with losses. To get the total variance, we use twice the downside variance namely σ P − so that the downside Sharpe ratio is, S P = ¯ R p − ¯ R f √ σ P − , where ¯ R p is the average periodreturn of portfolio P and ¯ R f is the average risk free rate. The UP ratio compares the40pside potential to the shortfall risk over a specific target (benchmark) and is computedas follows. Let R t be the realized monthly return of portfolio P for t = 1 , ..., T of thebacktesting period, where T = 216 is the number of experiments performed and let ρ t berespectively the return of the benchmark (risk free rate) for the same period. Then, we have,UP ratio = K P Kt =1 max[0 ,R t − ρ t ] √ K P Kt =1 (max[0 ,ρ t − R t ]) . It is obvious that the numerator of the above ratio is theaverage excess return over the benchmark and so reflects upside potential. In the same way,the denominator measures downside risk, i.e. shortfall risk over the benchmark.Next, we use the concept of opportunity cost presented in Simaan (2013) to analyse theeconomic significance of the performance difference of the two optimal portfolios. Let R Aug and R F be the realized returns of the optimal augmented and the optimal factors portfolios,respectively. Then, the opportunity cost θ is defined as the return that needs to be addedto (or subtracted from) the optimal factors portfolio return R F , so that the investor isindifferent (in utility terms) between the strategies imposed by the two different investmentopportunity sets, i.e., E [ U (1 + R F + θ )] = E [ U (1 + R Aug )] . A positive (negative) opportunity cost implies that the investor is better (worse) off if theinvestment opportunity set allows for the market anomaly factor prospect type investing.The opportunity cost takes into account the entire probability density function of assetreturns and hence it is suitable to evaluate strategies even when the asset return distributionis not normal. For the calculation of the opportunity cost, we use the following utilityfunction which satisfies the curvature of prospect theory (S-shaped): U ( R ) = R α if R ≥ or − γ ( − R ) β if R < , where γ is the coefficient of loss aversion (usually γ = 2 . ) and α, β < .Finally, we evaluate the performance of the two portfolios under the risk-adjusted (netof transaction costs) returns measure, proposed by DeMiguel et al. (2009) which indicatesthe way that the proportional transaction cost, generated by the portfolio turnover, affectsthe portfolio returns. Let trc be the proportional transaction cost, and R P,t +1 the realizedreturn of portfolio P at time t + 1 . The change in the net of transaction cost wealth N W P ofportfolio P through time is, N W
P,t +1 = N W
P,t (1 + R P,t +1 )[1 − trc × P Ni =1 ( | w P,i,t +1 − w P,i,t | ) . RT C
P,t +1 = NW P,t +1 NW P,t − . Let µ F and µ Aug be the out-of-sample mean of monthly
RT C factros and the Augmented optimalportfolio, respectively, and σ F and σ Aug be the corresponding standard deviations. Then,the return-loss measure is, R Loss = µ Aug σ Aug × σ F − µ F ,,