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Rocciolo, F., Gheno, A. , & Brooks, C. (2022). Explaining abnormal returns in stock markets: An alpha-neutral version of the CAPM. International Review of Financial Analysis, 82, [102143]. https://doi.org/10.1016/j.irfa.2022.102143 Publisher's PDF, also known as Version of record License (if available): CC BY Link to published version (if available): 10.1016/j.irfa.2022.102143 Link to publication record in Explore Bristol Research PDF-document This is the final published version of the article (version of record). It first appeared online via Elsevier at https://doi.org/10.1016/j.irfa.2022.102143. Please refer to any applicable terms of use of the publisher. University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ International Review of Financial Analysis 82 (2022) 102143 Contents lists available at ScienceDirect International Review of Financial Analysis journal homepage: www.elsevier.com/locate/irfa Explaining abnormal returns in stock markets: An alpha-neutral version of the CAPM b c a, Francesco Rocciolo , Andrea Gheno , Chris Brooks University of Bristol, 15-19 Tyndalls Park Road, Bristol, BS8 1PQ, UK Imperial College London, Exhibition Rd, South Kensington, London SW7 2BX, UK Department of Business Studies, University of Rome III, Via Silvio D’Amico, 77, 00145, Roma RM, Italy A R T I C L E I N F O A B S T R A C T JEL classification: This paper develops a behavioural asset pricing model in which traders are not fully rational as is commonly G12 assumed in the literature. The model derived is underpinned by the notion that agents’ preferences are affected G40 by their degree of optimism or pessimism regarding future market states. It is characterized by a representation G41 consistent with the Capital Asset Pricing Model, augmented by a behavioural bias that yields a simple and Keywords: intuitive economic explanation of the abnormal returns typically left unexplained by benchmark models. The Asset pricing model results we provide show how the factor introduced is able to absorb the ‘‘abnormal" returns that are not Behavioural asset pricing captured by the traditional CAPM, thereby reducing the pricing errors in the asset pricing model to statistical Optimism/pessimism insignificance. Abnormal returns 1. Introduction Jensen (1968), fail to result in parameter estimates that are jointly indistinguishable from zero. Despite all of the critiques cited above, the CAPM still remains During the last 50 years, a substantial part of the research effort in both theoretical and empirical asset pricing has been focused on the a model most entrusted by both practitioners and academics (Fama disclosure of patterns in average stock returns which are not described & French, 1996a, 1996b). At the same time, however, such strong by the Sharpe (1964), Lintner (1965), and Mossin (1966) capital asset evidence against the CAPM, underlying the paucity of the explanatory pricing model (CAPM) and are thus referred to as ‘‘anomalies’’ in power of a single-factor model, has driven scholars to engage in a the asset pricing literature. Within this body of work, we might note huge effort to develop new multifactor models. In particular, develop- the findings of patterns between stock returns and firms’ characteris- ments in the asset pricing literature have given rise to two different tics, long term reversals (De Bondt & Thaler, 1985) and momentum approaches to the problem. The first, purely empirical, includes multi- (Jegadeesh & Titman, 1993), the discovery of an excessively flat re- factor models which can be seen as different specifications of Ross’ asset lationship between average returns and market beta, the scarcity pricing theory (Ross, 1976), such as the most praised Fama and French of explanatory power of the latter, which sometimes even manifests (1993) – henceforth FF – three-factor model, Carhart’s 1997 four- itself in a negative relationship (Fama & French, 1992; Lakonishok factor model, the liquidity-adjusted CAPMs of Pástor and Stambaugh & Shapiro, 1986), and the instability of market beta over time (Guo, (2003) and Acharya and Pedersen (2005), and, more recently, the Fama Wu, & Yu, 2017; Jagannathan & Wang, 1996). Moreover, the CAPM and French (2015) five-factor model. As for the second approach, we is fully rejected from a statistical point of view, in that the model have a stream of literature that collects all of the natural extensions intercepts generated from time series regressions on actual data (also of the classic CAPM through a relaxation of some of its underlying known in the literature as Jensen’s alphas after the seminal paper of assumptions, such as Black’s 1972 zero-beta CAPM, Merton’s 1973 Corresponding author. E-mail addresses: f.rocciolo@imperial.ac.uk (F. Rocciolo), andrea.gheno@uniroma3.it (A. Gheno), chris.brooks@bristol.ac.uk (C. Brooks). Relevant studies include those that relate expected returns to size (Banz, 1981), book-to-market-equity (Rosenberg, Reid, & Lanstein, 1985), the earnings-price ratio (Basu, 1977, 1983), debt-equity ratio (Bhandari, 1988), profitability (Fama & French, 2006; Novy-Marx, 2013) and investment (Fama & French, 2006; Titman, Wei, & Xie, 2004). A review of these anomalies can be found in Fama and French (2008). Friend and Blume (1970), Black, Jensen, and Scholes (1972), Blume and Friend (1973), Reinganum (1981), Stambaugh (1982), Fama and French (1992). See for instance Jensen (1968), Friend and Blume (1970), Black et al. (1972), Blume and Friend (1973), Fama and MacBeth (1973), Reinganum (1981), Stambaugh (1982), Gibbons, Ross, and Shanken (1989), Fama and French (1992, 1996b). https://doi.org/10.1016/j.irfa.2022.102143 Received 11 October 2021; Received in revised form 15 March 2022; Accepted 1 April 2022 Available online 20 April 2022 1057-5219/© 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 intertemporal (I)CAPM, Kraus and Litzenberger’s 1976 three-moment made in this direction, and such an approach results in specifications CAPM, Jagannathan and Wang’s 1996 conditional (C)CAPM, consump- that are challenging to test on actual data. tion/investment based CAPMs (Breeden, 1979; Cochrane, 1991) and In order to avoid such issues, we make use of an order of prefer- Dittmar’s 2002 four-moment CAPM. The connection between the two ences adjusted for optimism proposed by Rocciolo, Gheno, and Brooks approaches lies in the interpretation of empirical multifactor models as (2019), which is both simple and characterized by high descriptive different specifications of equilibrium models. For instance, Maio and power. We justify this choice as a compromise between the repre- Santa-Clara (2012, 2017) analyse the conditions that must be satisfied sentativeness of agents’ behaviour and analytical tractability in that by a multifactor model in order for it to be justifiable by the ICAPM. the employment of such preferences permits the maintenance of the Whether we want to interpret multifactor models as equilibrium linearity of the asset pricing model and its expression in terms of models or not, all these specifications have in common that they the beta terminology of the original CAPM. This is a feature that is are mercilessly rejected from a statistical point of view in terms of typically not achievable when other models such as prospect theory Jensen’s alpha, as shown in many empirical applications. Fama and are employed. Moreover, the S-shaped value function typically assumed French (2015), for instance, argue that their five-factor model performs in prospect theory seems unqualified in describing agents’ behaviour better than their three-factor model (FF, 1993) but still shows alphas when they face ‘‘mixed’’ prospects — i.e., prospects characterized by that are jointly significantly different from zero. Similarly, Harvey and both gains and losses (Levy & Levy, 2002). Conversely, optimism- Siddique (2000), Dittmar (2002), Messis, Alexandridis, and Zapranis adjusted preferences, accounting explicitly for the possible skewness (2021), Lewellen and Nagel (2006), and Maio and Santa-Clara (2012), of the prospects, describe these kinds of situations well. In this sense, the model that we are going to derive is similar in spirit to the present studies respectively on the conditional three-moment CAPM, four-moment CAPM, CAPM with asymmetric and constant systemic three-moment CAPM, in which investors’ attitude towards skewness is implicitly taken into account (as well as its extension to the fourth risk, conditional consumption CAPM, CCAPM, and ICAPM, finding similar results in terms of the significance of the intercepts. moment) in an optimism-adjustment to the utility function. The funda- mental difference, however, with respect to the models cited above, is Apart from the standard view stating that other risk factors are to be included in the evaluation, the ‘‘behaviouralist" interpretation argues given by the fact that the latter inevitably end up as multifactor models while our specification, as we will show, preserves a single factor that the return component left unexplained by the model should be attributed to some departure from the hypothesis of agents’ full ratio- representation in terms of beta and consistency with the traditional CAPM. nality (Barberis & Thaler, 2003). Common explanations that have been advanced include investors’ over-reactions to bad economic news and Moreover, the CAPM derived provides a clear economic interpre- tation of Jensen’s alpha that is also consistent with the empirical market seasonality (De Bondt & Thaler, 1987), under- and over-reaction to public (Barberis, Shleifer, & Vishny, 1998) and private (Daniel, evidence reported in Diether et al. (2002). It also provides, through the introduction of market sentiment into the specification, new evidence Hirshleifer, & Subrahmanyam, 2001) information, optimism/pessimism (Diether, Malloy, & Scherbina, 2002), narrow framing and loss aver- concerning the empirical validity of the CAPM. The results shown are strongly consistent with the underlying theory, which, as we will sion (Barberis, Huang, & Santos, 2001) and, more recently, ambiguity aversion (Guidolin & Liu, 2016). demonstrate, outperforms the currently most celebrated asset pricing models such as the Fama–French three- and five-factor models. More Interestingly, another common feature of both multifactor models specifically, the test that we conduct on a large sample of portfolios and equilibrium models, which in particular originates from consid- sorted by size, book-to-market, investment, and operating profitability, ering the former as specifications of the latter, is that most are un- shows, independently from the asset considered, pricing errors that derpinned by the hypothesis of fully rational agents, represented by are jointly indistinguishable from zero. We thus provide new evidence the usage of a von Neumann–Morgenstern (VNM) order of preferences. that, contrary to the common view, when the CAPM is corrected for As showed by Cochrane (2009), in fact, the CAPM, and thus the market component in explaining the cross section of stocks’ expected the departure from full rationality of agents’ behaviour, it is still alive and well. The series of diagnostic tests we run for confirmation gives returns, can be derived directly by using different types of VNM utility functions. A serious issue, and one that in our view is still not suitably robustness to our findings. The remainder of the paper is organized as follows. In the next considered in the asset pricing literature, is that such preferences do not properly describe the actual behaviour of individuals. As shown in section we outline the optimism-adjusted preferences framework used in the derivation of our behavioural capital asset pricing model. Sec- a large number of studies in decision-making under risk, in fact, VNM preferences are not able to capture a wide range of features that have tion 3 explores the datasets and the econometric techniques employed in order to obtain the results summarized in Section 4. Finally, Section 5 been shown to characterize the behaviour of agents, including, just to name few, the under- and over-weighting of probabilities, loss aversion concludes. and narrow framing. In the light of these considerations, in this paper we introduce a 2. The model different version of the CAPM in which agents are boundedly-rational in the sense that they behave not as they theoretically should but as In this section we proceed to the derivation of our asset pricing model under conditions departing from full rationality. We start by the empirical evidence shows that they do. In particular, we focus our attention on the inclusion of probability weights and the extent to introducing the system of preferences that characterize the agents in our economy. This is necessary since the representation of how agents which agents are optimistic or pessimistic in the asset pricing model. make choices in the market will act as the basic framework in the These, in our view, represent the most compelling, and somehow derivation of the model. encompassing, departures from rationality. It is now a commonly held view that the use of the prospect theory of Kahneman and Tversky 2.1. Optimism-adjusted preferences (1979) and Tversky and Kahneman (1992) is warranted. However, the employment of such preferences in asset pricing leads to a considerable Let us consider an agent characterized by a VNM utility function 𝑢(𝑥) loss of analytical tractability, as one can appreciate from the attempts and let 𝑋 be a prospect represented by a finite number of outcomes See, for instance, Allais (1953), Kahneman and Tversky (1979) and Tversky and Kahneman (1992) for probability weighting and loss aversion, See for instance Barberis and Huang (2008), He and Zhou (2011) and and Thaler (1999) for mental accounting. De Giorgi, Hens, and Levy (2011). 2 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 𝑥 , each of which has an assigned probability 𝑝 . Then let 𝛾 be a Tversky, 1979) – with the expected utility paradigm and with the 𝑗 𝑗 positive real number in [0, 1] representative of the agent’s degree of advantage of a very simple mathematical representation. In fact, since optimism and 𝜙(𝜎 , 𝛾) ∈ [0, 1] be a real positive function of the degree the weighting function 𝜙(𝛾, 𝜎 ) is deterministic and independent from 𝑋 𝑋 the final outcomes of the prospect, we can rewrite Eq. (1) as of optimism 𝛾 and of the standard deviation of the outcomes 𝜎 , such [ ] that 𝜙(𝜎 , 𝛾) ∈ ( , 1] and increasing in the variance of the outcomes ∑ ∑ 1 1 2 𝑈(𝑥, 𝜎 , 𝛾) = 2 𝑢(𝑥 )𝑝 [1 − 𝜙(𝛾, 𝜎 )] + 𝑢(𝑥 )𝑝 𝜙(𝛾, 𝜎 ) (2) 𝑥 𝑗 𝑗 𝑥 𝑗 𝑗 𝑥 𝜎 if 𝛾 ∈ ( , 1], 𝜙(𝜎 , 𝛾) ∈ [0, ) and decreasing in the variance of the 2 2 𝑥 ≤𝑥̄ 𝑥 >𝑥̄ 𝑗 𝑗 1 1 1 2 2 outcomes 𝜎 if 𝛾 ∈ [0, ), and 𝜙(𝜎 , 𝛾) = ∀ 𝜎 if 𝛾 = . 𝑋 𝑋 2 2 2 where 𝜙(𝛾, 𝜎 ) can be interpreted in this representation as a function Following Rocciolo et al. (2019), an optimist, represented by a value 𝑋 which assigns different weights to the objective probabilities according of the parameter 𝛾 ∈ ( , 1], can be described as an agent who assigns to the degree of optimism of the agent and the standard deviation of a bigger weight to the positive outcomes of the prospect with respect the prospect’s outcomes, 𝑝 is the objective probability assigned to the to an unbiased agent, and who sees in a larger variance an opportunity 𝑗 6 outcome 𝑥 in the prospect 𝑋, and 𝑥̄ is the reference point. to earn more from the risky opportunity. Conversely, the pessimist, In this kind of setting, the choice of a proper analytical expression represented by a value of the parameter 𝛾 ∈ [0, ), can be seen as an for the weighting function 𝜙(𝛾, 𝜎 ) is needed in order to apply the agent who assigns less weight to positive outcomes and who is scared model. We suggest the following of an increment in the variance. Finally, 𝛾 = represents a rational { ( ) } expected utility maximizer. Formally, by modelling these circumstances 1 1 1 1 − exp −𝜌 𝛾 − 𝜎 if ≤ 𝛾 < 1 ⎪ 2 2 𝑋 2 through the function 𝜙(𝛾, 𝜎 ), the subjective value of the prospect for an agent affected by an optimism/pessimism bias can be represented 𝜙(𝛾, 𝜎 ) = (3) [ { ( ) }] as 1 1 1 [ ] ⎪ 1 − exp −𝜌 𝛾 − 𝜎 1 2 2 if 0 < 𝛾 < 𝜎 <𝜎 2 2 𝑋 2 ⎩ 𝑋 𝑈(𝑋, 𝜎 , 𝛾) = 2 𝜙(𝛾, 𝜎 )E[𝑢(𝑥)] + [1 − 𝜙(𝛾, 𝜎 )]E[𝑢(𝑥)] (1) 𝑋 𝑋 + 𝑋 − where, by defining 𝜎 as the threshold variance beyond which a pes- where E[𝑢(𝑥)] and E[𝑢(𝑥)] are the subjective expected values of simistic agent will give up on the prospect faced, 1 is an indicator 2 2 + − 𝜎 <𝜎 respectively the gains and losses with respect to a reference point 𝑥̄ , and function which assumes the value one if the variance of the prospect’s 𝜙(𝛾, 𝜎 ), which assumes the interpretation of an optimism weighting outcomes is lower than the critical level 𝜎 and conversely is equal to 2 2 function. It determines the weight assigned to the gains (and thus to the zero when 𝜎 ≥ 𝜎 . 𝑋 ∗ losses) in the overall value function based on the degree of optimism Rocciolo et al. (2019) studied in detail how such preferences per- of the agent. form in terms of their descriptive power for many of the most ac- In order to sketch out how the model works, let us consider three knowledged ‘‘counter-examples’’ of the expected utility criterion. Their agents endowed with the same utility function 𝑢(𝑥) and level of absolute tests show in particular how the adjustment for optimism, characterized risk aversion 𝜌, and different degrees of optimism 𝛾 . In particular, let through the use of an optimism weighting function such as that in us assume that one of them is an optimist ( < 𝛾 ≤ 1), one a pessimist Eq. (3), can adapt expected utility theory in order to allow the latter to better describe the empirical evidence collected in a wide number (0 < 𝛾 ≤ ) and the last one is a pure rational expected utility of empirical studies, such as Allais (1953) and Kahneman and Tversky maximizer (𝛾 = ). With respect to a prospect 𝑋 faced, the three (1979). Moreover, they showed how the latter form is convenient, agents, while sharing the same utility function and risk aversion, might especially when applied in a CARA-Normal assumptions setting, in that end up with very different evaluations depending on the variance of the it allows the derivation of linear demand curves, as we will show in the outcome. As shown in Fig. 1, in fact, the bigger the outcome’s variance, next section. In this sense, our decision to make use of such an order of the more the optimist will assign a greater (lower) weight 𝜙(𝛾 , 𝜎 ) 1 𝑋 preferences finds justification in that improving the descriptive power to the prospect’s gains (losses), and the steeper (flatter) will be the of the expected utility criterion allows us to use the latter, which is adjusted utility function 𝑢 (𝑥, 𝜎 , 𝛾 ) (s)he employs in the evaluation of ∗ 𝑋 1 still the currently preferred framework in the asset pricing literature. the positive (negative) outcomes of the prospect. Conversely, the bigger As shown in the next section, this preference ordering also allows us to the outcome’s variance, the more the pessimist will assign a lower derive an asset pricing model expressed in the usual beta language. (greater) weight to the prospect’s gains (losses), the flatter (steeper) will be the adjusted utility function 𝑢 (𝑥, 𝜎 , 𝛾 ) that (s)he employs in the ∗ 𝑋 2 2.2. The alpha-neutral CAPM evaluation of the positive (negative) outcomes of the prospect. Thus, we have that, under such preferences and ceteris paribus, 𝑈(𝑋, 𝜎 , 𝛾 ) > 𝑋 1 As in the classic CAPM, let us consider as a basic framework an 𝑈(𝑋, 𝜎 , 𝛾 ) > 𝑈(𝑋, 𝜎 , 𝛾 ) if the prospect is risky, i.e. 𝜎 > 0, and 𝑋 3 𝑋 2 𝑋 economy free of taxes and transaction costs, characterized by 𝑛 risk- 𝑈(𝑋, 𝜎 , 𝛾 ) = 𝑈(𝑋, 𝜎 , 𝛾 ) = 𝑈(𝑋, 𝜎 , 𝛾 ) in the case of a risk-free 𝑋 1 𝑋 3 𝑋 2 averse utility maximizing agents, 𝑁 risky assets, each characterized opportunity, i.e. 𝜎 = 0. by a normally distributed gross return 𝑅 , and a risk-free asset with The strength of this representation evidently lies in being a mere 𝐹 8 an exogenously determined gross risk-free return 𝑅 . The market is adjustment applicable to a wide range of existing models in the field. always in equilibrium and each agent 𝑖 can invest any fraction of At the same time, it is able to reconcile one of the most widely his/her capital in either the risk-free asset or any of the risky assets acknowledged features in the decision-making literature – evidence traded in the market, and can freely borrow and lend funds at the gross that individuals make use of weighted probabilities (Kahneman & risk-free return 𝑅 . All 𝑛 agents are assumed to be price-takers and plan to trade over the same time horizon at prices that are determined as a consequence of the equilibrium condition. In addition, let us make the A possible issue that arises from this definition of optimism is that one following further assumptions: may suspect optimistic agents to be risk lovers. The authors analyse this possibility at length and show that an optimistic, risk-averse agent will not Assumption 1. All 𝑛 agents have the same information and beliefs be a risk seeker unless a highly skewed prospect is considered. Here, the term rational is interpreted in the sense of VNM expected about the objective joint probability distribution of the returns of all utility theory, as a characteristic of agents displaying preferences that do individual stocks not violate expected utility theory. In this paper, we consider the degree of agents’ optimism to be the sole source of non-rationality. Optimistic and pessimistic agents are not (fully) rational in the sense that they do not conform The inverse of the gross return and the gross risk-free return define to the coherence paradigms of expected utility theory (EUT) and they display respectively the stochastic discount factor 1∕𝑅 , specific to the asset 𝑗 and preference orderings that typically violate the latter. the risk-free discount factor 1∕𝑅 . 3 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 1 1 1 Fig. 1. The left-hand plot represents the optimism weighting 𝜙(𝛾, 𝜎 ) as a function of the prospect’s variance 𝜎 for the degrees of optimism < 𝛾 ≤ 1, 0 < 𝛾 ≤ and 𝛾 = . 𝑋 𝑋 1 2 3 2 2 2 The right-hand plot represents the different distortions in the utility functions of the three agents, according to their degrees of optimisms and the optimism weighting function 𝜙(𝛾, 𝜎 ). F 𝐹 Assumption 2. Every agent 𝑖 is equipped with an optimism-adjusted where 𝜅(𝛾 ) = 1 − 2𝛾 , R = 1𝑅 and 𝜮 is the 𝑁× 𝑁 covariance matrix 𝑖 𝑖 R negative exponential utility function of the type 𝑢 (𝑥, 𝜎 , 𝛾 ) = −2[1 − of risky asset gross returns. Under Assumptions 1 and 2, every agent holds ∗ 𝑋 𝑖 𝜙(𝛾, 𝜎 )] exp{−𝜌 𝑥}, where the function 𝜙(𝛾, 𝜎 ) takes the form in (3), a portfolio characterized by different combinations, according to his/her 𝑋 𝑖 𝑋 and where the parameters 𝜎 , 𝜌 and 𝛾 are respectively the standard risk aversion and degree of optimism, of the risk-free asset and the market 𝑋 𝑖 𝑖 deviation of the prospect 𝑋, the absolute risk aversion coefficient, and portfolio so that, at an aggregate level and for each asset 𝑗 traded in the the degree of optimism of the agent 𝑖. Risk aversion and agents’ degree market, the following relationship holds of optimism are assumed constant over time 𝐹 𝑀 E[𝑅 ] − 𝑅 = (𝜌 + 𝜅(𝛾))COV (𝑅 , 𝑅 ) (8) 𝑗 𝑗 The second assumption represents the actual breaking point with where 𝜌 and 𝛾 are aggregate measures of the agent’s absolute risk aversion ‘‘rational" asset pricing theory through the introduction of a behavioural and degree of optimism respectively, and where 𝑅 is the gross return on element in the evaluation of the assets, represented by the agent’s the market portfolio. degree of optimism. Being the unique difference with respect to the standard assumption set used in deriving the traditional CAPM, the The first result in Eq. (7) is the optimal individual demand schedule, asset pricing model we are going to derive makes a comparison with expressed in the usual hyperbolic form introduced in Grossman (1976), similar models in the literature an easy task. In particular, this greatly 9 and which can be found in many other studies, generalized for the case facilitates the study of where and how the original formulation of the in which 𝑁 risky assets are traded in the market, and adjusted for the CAPM fails and how it can be fixed simply by considering agents in the behavioural bias implicit in the order of preferences used. As for the market as they actually are, i.e. not (fully) rational. Since we are using second result, Eq. (8) again represents the usual expression that ties essentially the same framework as the traditional CAPM, the derivation the risky security excess returns to risk attitudes, adjusted through the of what follows traces the standard CARA-Normal procedure widely agents’ aggregate degree of optimism. discussed in, amongst others, Cochrane (2009). The term 𝜅(𝛾) = 1 − 2𝛾 ∈ [−1, 1], contained in both equations (7) Let us start by considering the problem from the point of view of a and (8), represents a quantification of the distance from rationality that single agent 𝑖 characterized, at time 𝑡 − 1, by an initial level of wealth characterizes typical agents who act in the economy. In particular, the 𝑊 (𝑡 − 1) that can be split how (s)he prefers between the risk-free and 𝑖 term 𝜅(𝛾) in Eq. (7) identifies the mitigation, in the case that the agent risky securities traded in the market, in order to maximize the utility of 𝑖 is an optimist, or the enhancement, in the case in which (s)he is a final level of wealth 𝑊 (𝑡). Let 𝑥 and x be respectively the amount of 𝑖 𝑖 pessimist, on the total impact that the asset’s risk has on the demand his(her) initial wealth invested in the risk-free asset and the 𝑁×1 vector function. of the amounts invested in the risky securities. His(her) maximization Starting from the result in Eq. (8), the pricing equation can be problem is given by rewritten in terms of the more commonly used beta language. Since, in fact, Eq. (8) holds for every agent 𝑖 and every asset 𝑗, it also holds arg max 𝑈(𝑊 (𝑡)) = E[−2[1 − 𝜙(𝛾, 𝜎 )] exp(−𝜌 𝑊 (𝑡))] (4) 𝑖 𝑊 𝑖 𝑖 𝑊 for the market portfolio. In particular, we have in this case that 𝑀 𝐹 2 subject to the following budget constraint E[𝑅 ] − 𝑅 = (𝜌 + 𝜅(𝛾))𝜎 ′ 𝑓 x 1 + 𝑥 = 𝑊 (𝑡 − 1) (5) 𝑖 𝑖 and thus, 𝑀 𝐹 where the agent’s final level of wealth is given by E[𝑅 ] − 𝑅 𝜌 + 𝜅(𝛾) = (9) ′ 𝐹 𝑊 (𝑡) = x R + 𝑥 𝑅 (6) 𝑖 𝑖 By plugging this last result into Eq. (8) and by defining the system- 𝑗 𝑀 R is the 𝑁 × 1 vector of gross risky returns and 1 an 𝑁 × 1 vector of COV(𝑅 ,𝑅 ) atic risk component beta in a conventional way as 𝛽 = , we 𝑗 2 ones. end up with Proposition 1. Under Assumption 2, the 𝑁 ×1 vector of individual optimal 𝐹 𝑀 𝐹 E[𝑅 ] − 𝑅 = (E[𝑅 ] − 𝑅 )𝛽 , (10) 𝑗 𝑗 demand schedules for the risky assets traded in the market, which solves the optimization problem in (4) subject to (5), is given by The most recent works include Cochrane (2009), Mendel and Shleifer E[R] − R −1 x = 𝜮 (7) (2012) and Banerjee and Green (2015). (𝜌 + 𝜅(𝛾 )) 𝑖 𝑖 4 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 which is a pricing equation consistent with the original representation traditional market 𝛽, and instead captured by the new factor. Notice of the CAPM, with the difference that the systematic risk beta reflects, that we have deliberately left the intercepts 𝛼 in Eq. (12) in accordance in this case, both the agents’ risk aversion and their degree of optimism. with the idea of mispricing of the traditional version of the model as as- This last expression, which is the fruit of pure algebraic manipula- sumed in Conjecture 2. If the traditional CAPM completely explains the tion, is not so innocuous as it might first appear. By recalling that in the covariance between the asset considered and the market, the intercepts original derivation of the CAPM only the risk aversion 𝜌 is taken into 𝛼 as well as the coefficient 𝜅(𝛾) should not be distinguishable from zero account in determining the market price of risk, it generates a clash since the latter would constitute an unnecessary explanatory variable between the model just derived, in which the behavioural bias is taken in the regression of the excess returns against the market risk premium, into account as well, and the traditional CAPM. To better understand since all of the co-movement between the asset and the market would this point, let us consider two types of asset pricing model, both with be fully captured by the alpha-neutral betas, 𝛽 , which would in this representation as in Eq. (10), which focus on two different conjectures case be equivalent to the market 𝛽 of the traditional CAPM. of the market risk premium. Conversely, in the case in which the model’s 𝛼 estimate is sig- Assume that the market is not uniquely composed of fully rational nificantly distinguishable from zero, and if, as we have conjectured, expected utility maximizers, i.e., 𝜅(𝛾) ≠ 0. the pricing errors are fully generated by behavioural biases, we should expect for every 𝛼 a model estimate 𝜅(𝛾) such that the net intercepts Conjecture 1. The market price of risk reflects not only the aggregate 𝑗 𝛿 = 𝛼 + 𝜅(𝛾)COV(𝑅 , 𝑅 ) are jointly indistinguishable from zero. degree of risk aversion but also the aggregate degree of agents’ optimism, 𝑗 𝑗 𝑗 In this sense, the model is an ‘‘alpha-neutral’’ version of the CAPM, i.e., Eq. (9) holds in that the new factor, which exists because of the presence of a Conjecture 2. The market price of risk reflects only agents’ aggregate mispricing according to Conjecture 2, does not enter in the asset pricing risk aversion without taking into consideration the potential presence of a equation as an explanatory variable for expected returns. Rather, it behavioural bias in their decisions, resulting in the traditional version of the appears only as a counterbalance to the assumed misprice, which, if CAPM, it works well, ends up ‘‘neutralizing’’ it. Moreover, if that is the case, 𝑀 𝐹 such a result is consistent with the intuition behind the optimism- E[𝑅 ] − 𝑅 𝜌 = (11) 2 based order of preferences employed. According to Eq. (12) and the definition of the factor 𝜅(𝛾) = 1 − 2𝛾 , in fact, in the presence of a The two conjectures are clearly not compatible simultaneously in positive unexplained excess return, the CAPM holds only if 𝜅(𝛾) < 0 that they give rise to different expressions for the unitary market’s in such a way that the net intercepts are nullified, and thus if agents risk premium. It is immediately clear that the only possible case in are on average optimistic about returns on the asset under study. The which the two expressions are equivalent is when 𝜅(𝛾) = 0, i.e., all contrary evidently applies in the case of negative alphas where we will agents in the market are purely rational expected utility maximizers. have, on average, pessimistic traders with regard to the asset under As a result, we have that, under Conjecture 1 in which the model takes consideration. Finally, by using the definition of net intercepts 𝛿 as account of agents’ behavioural biases in formulating asset prices, the above, the model in Eq. (12) can be rewritten as representation in Eq. (10) holds for every agent 𝑖 and for every security 𝑗 traded in the market. Thus, under Conjecture 1, prices determined 𝐹 𝑀 𝐹 ∗ E[𝑅 ] − 𝑅 = 𝛿 + (E[𝑅 ] − 𝑅 )𝛽 (13) 𝑗 𝑗 by the market and the model coincide. The same is evidently not true in the case of Conjecture 2, under which there will exist a misprice Eq. (13) tells the same story but from a different perspective. 𝛼 between the market and the model, given by the fact that we are The main difference with respect to the previous representation in imposing a model which assumes rational agents (as the CAPM does) Eq. (12) is in that the absorption of the intercepts by the behavioural on the prices of assets which are traded by agents who are not rational. component 𝜅(𝛾)COV(𝑅 , 𝑅 ) is made explicit here, so that the ex- In particular, we have the following different result. pression recalls the traditional CAPM representation under conditions of non-full rationality and explicitly in a market where agents suffer Proposition 2. Let 𝛼 be the misprice of asset 𝑗 as a consequence of from optimism/pessimism biases. In this sense, Eq. (13) defines a the assumption in Conjecture 2. Given the asset pricing model expressed by unique equilibrium characterized by an augmented security market line Eq. (8), under Conjecture 2 in which the model does not take into account (SML*), which will, in general, be steeper with respect to the traditional agents’ behavioural biases in formulating asset prices, Eq. (10) becomes SML defined by the traditional CAPM in Eq. (10). In fact, this change 𝐹 𝑀 𝐹 ∗ 𝑀 E[𝑅 ] − 𝑅 = 𝛼 + (E[𝑅 ] − 𝑅 )𝛽 + 𝜅(𝛾)COV (𝑅 , 𝑅 ) (12) 𝑗 𝑗 𝑗 in the measurement of the intercept inevitably generates a change in the measurement of the systematic risk beta, which will result in a which we will refer to from now on as an Alpha-Neutral CAPM, and where ∗ ‘‘purified’’, behaviourally driven part of the movement in the market 𝛽 is a measure of the systematic risk of asset 𝑗, which, as in the traditional which at the same time impacts positively on the slope of the SML. CAPM and according to Conjecture 2, reflects only agents’ risk aversion. In a comparison between the traditional CAPM in Eq. (10) and the Consistent with the name that we give to the model, we will refer to 𝛽 as Alpha-Neutral CAPM in Eqs. (12) and (13) we will refer to 𝛽 and 𝛽 alpha-neutral betas in what follows. as, respectively, traditional betas and alpha-neutral betas. A few comments are necessary on this last proposition. First, the In order to outline the intuition behind the model, let us consider a expression in Eq. (12) has to be interpreted as a single-factor asset simplified version of our economy in which only three assets named 𝐴, pricing model since, given our assumption-setting, we are still in an 𝐵 and 𝐶 are traded. Let us suppose that the cross-sectional errors from economy in which assets’ prices are determined only according to the 0 0 0 the traditional CAPM are 𝛼 > 0, 𝛼 > 0 and 𝛼 < 0 respectively for the 𝐴 𝐵 𝐶 systematic risk of assets. The new element 𝜅(𝛾)COV(𝑅 , 𝑅 ) is actually three assets. Let us then imagine running the regression in Eq. (12) and a direct consequence of the fact that we are considering a mispricing of finding the result that, consistent with the results previously obtained the traditional CAPM due to the non-fully rational behaviour of agents and with our Alpha-Neutral CAPM, the regressions on the assets 𝐴 and in the economy. In this sense, the factor 𝜅(𝛾) merely quantifies how 𝐶 generate pricing errors 𝛼 > 0 and 𝛼 < 0 respectively and, consistent much of the cross-sectional pricing error produced by the traditional 𝐴 𝐶 with these, the behavioural adjustments 𝜅 < 0 and 𝜅 > 0. Conversely, CAPM is explained by the behavioural component 𝜅(𝛾)COV(𝑅 , 𝑅 ). 𝐴 𝐶 let us suppose that the asset 𝐵 lies perfectly on the regression plane with This can be seen in the model as the portion of the covariance between the risky asset considered and the market left unexplained by the 𝛼 = 0 and 𝜅 = 0. 𝐵 𝐵 5 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 Fig. 3. The figure represents, with respect to the example considered with three assets, Fig. 2. The figure represents a hypothetical regression plane of the average excess the hypothetical security market lines respectively of the traditional CAPM (SML) and the Alpha-Neutral CAPM (SML*). returns on the three assets with respect to the betas of the latter and their associated behavioural bias factor 𝜅(𝛾). from the intersections of 10 portfolios of stocks sorted on size deciles Fig. 2 represents the situation for the three assets. Consistent with and three groups of 10 portfolios in which the stocks have been inde- the situation depicted, we have that the model describes well the excess pendently sorted with respect to their BTM ratio, investment (INV) and returns of the assets 𝐴 and 𝐶 if respectively the segment 𝐴 − 𝐴 is operating profitability (OP) deciles. Consistent with Fama and French equal to 0 − 𝜅 and 𝐶 − 𝐶 is equal to 𝜅 − 0. Regarding asset 𝐵, we 𝐴 𝐶 (1993, 1996a, 2015), the latter portfolios have been constructed at have instead that the model does not help in explaining the abnormal the end of each June using NYSE breakpoints and considering in the return 𝛼 predicted by the traditional CAPM in that the asset lies, in construction all NYSE, AMEX, and NASDAQ stocks for which returns equilibrium, on the plane with a coefficient 𝜅 equal to zero. As we and book values are available respectively on CRSP and COMPUSTAT. will show in the next section, this situation is quite rare, at least in the Table 1 shows the monthly average excess returns for the portfolios dataset that we employ. considered in Sample (a). It is easy to recognize the typical patterns Assuming that the latter conditions on the behavioural factors of the in the excess returns of the portfolios pointed out by Fama and French three assets are satisfied, Fig. 3 shows the traditional SML in Eq. (10) (1993, 1996a, 2015). The size effect, which is typically used to refer and the augmented SML* in Eq. (13) for the example we consider with to the phenomenon characterized by a fall in the average returns three assets. According to the previous results, assets A and C that were from small stocks to big stocks is persistent in each panel of data showing respectively positive and negative pricing errors under the analysed; exceptions are the first deciles of all three of the other firms’ traditional CAPM, result in equilibrium on the new SML* defined by the characteristics involved in the sorts — i.e., the BTM-Low (panel B), Alpha-Neutral model. In particular, as mentioned above, the augmented OP-Low (panel C) and INV-Low (panel D). SML* will, in general, be steeper than the traditional SML and the betas Panel B of Table 1 documents the value effect — i.e., the tendency associated with the assets’ return reduced since, as argued above, the of average returns to increase for higher values of the BTM ratio. This behavioural factor that we have included in the model also deadens relationship shows up clearly in each row of the panel and, consistent the spurious component present in the betas when agents are not fully with Fama and French (1993, 1996a, 2015), its effect is stronger for rational. small size portfolios. Panels C and Panel D of Table 1 instead provide evidence of the 3. The playing field so called profitability effect (Fama & French, 2015; Novy-Marx, 2013) and the investment effect (Aharoni, Grundy, & Zeng, 2013; Fama & 3.1. Data description French, 2015) respectively. In particular, we observe that average returns typically increase for stocks of firms with higher operating Our empirical tests concern two main datasets: (a) average returns profitability (Panel C) and decrease for stocks of firms that invest more from Kenneth French’s data library on 336 portfolios typically used (Panel D). in the literature to describe patterns in expected stock returns, and For Sample (b), we have considered monthly and daily excess (b) the average returns on portfolios that are considered to mimic the returns with respect to the one-month Treasury bill rate on the portfolio patterns in the portfolios in (a), plus the covariances between returns of all sample stocks, which can be considered a proxy for the market to each of the assets in (a) and the proxy for the market portfolio. portfolio, and the monthly returns on the portfolios typically used in For both samples, the period considered is July 1963–December 2016, order to mimic the risk factors acknowledged in the literature, repre- and the excess returns are observed at both a monthly and a daily sented by (i) size, (ii) value, (iii) momentum, (iv) operating profitability frequency, where the former have been used in order to perform the and (v) investment. The manner in which the latter portfolios have been main tests of the model, while the latter are employed only to compute constructed is described in detail in Fama and French (1993, 1996a) the covariances that will be used as explanatory variables as in Eq. (12). for portfolios (i) and (ii), Carhart (1997) for (iii), and Fama and French Sample (a) has been constructed by considering excess returns with (2015) for (iv) and (v). In what follows, we provide a brief summary. respect to the one-month U.S. Treasury Bill rate on 36 one-way sorted Portfolios (i) and (ii), named SMB (small minus big) and HML (high portfolios (18 portfolios with stocks sorted on size quantiles and 18 minus low), are constructed as the differences between, respectively, portfolios with stocks sorted on book-to-market (BTM) ratio quantiles) the average returns on three small-stock value-weighted portfolios and on three groups of 100 two-way sorted portfolios, which result and three big-stock value-weighted portfolios in the former case and 6 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 Table 1 Average monthly portfolio returns. Average monthly percent returns for portfolios formed on size, book-to-market ratio, size and book-to-market ratio, size and operating profitability and size and investment. The returns are in excess of the one-month Treasury bill rate. Period: July 1963–December 2016, 642 months. Panel A: 18 Size Portfolios + 18 Book-to-Market Portfolios Small 0.79 Size-10 0.76 BTM-Low 1.02 BTM-10 0.45 Size-2 0.73 Size-11 0.82 BTM-2 0.50 BTM-11 0.55 Size-3 0.52 Size-12 0.75 BTM-3 0.60 BTM-12 0.58 Size-4 0.77 Size-13 0.77 BTM-4 0.84 BTM-13 0.56 Size-5 0.78 Size-14 0.72 BTM-5 0.48 BTM-14 0.59 Size-6 0.74 Size-15 0.72 BTM-6 0.56 BTM-15 0.70 Size-7 0.69 Size-16 0.68 BTM-7 0.64 BTM-16 0.63 Size-8 0.50 Size-17 0.62 BTM-8 0.68 BTM-17 0.75 Size-9 0.78 Big 0.48 BTM-9 0.90 BTM-High 0.88 Panel B: 100 Size X Book-to-Market Portfolios BTM-Low BTM-2 BTM-3 BTM-4 BTM-5 BTM-6 BTM-7 BTM-8 BTM-9 BTM-High Small N/A N/A 0.16 0.92 0.74 0.90 0.97 1.07 1.13 1.12 Size-2 0.25 0.78 0.82 0.74 0.85 0.71 1.04 0.98 1.07 1.16 Size-3 0.44 0.58 0.69 0.80 0.81 0.88 1.00 0.95 1.16 0.68 Size-4 0.28 0.53 0.57 0.81 0.55 0.78 0.77 0.96 1.10 0.64 Size-5 0.42 0.65 0.77 1.18 0.74 0.89 0.92 0.97 1.05 1.02 Size-6 0.41 0.61 0.75 0.72 0.68 0.75 0.93 0.83 1.01 1.15 Size-7 0.70 0.55 0.63 0.69 0.63 0.77 0.90 0.84 0.88 0.87 Size-8 0.60 0.55 0.65 0.55 0.88 0.65 0.82 0.82 0.82 0.80 Size-9 0.50 0.54 0.60 0.67 0.72 0.67 0.73 0.61 1.20 0.90 Big 0.49 0.55 0.54 0.51 0.48 0.61 0.32 0.93 1.03 0.86 Panel C: 100 Size X Operating Profitability Portfolios OP-Low OP-2 OP-3 OP-4 OP-5 OP-6 OP-7 OP-8 OP-9 OP-High Small 0.44 0.85 1.02 0.99 1.02 0.91 1.13 0.99 1.03 0.76 Size-2 0.44 0.72 0.92 0.84 0.83 0.69 1.01 0.78 0.84 0.90 Size-3 0.57 0.88 0.93 0.86 0.75 1.05 0.78 0.77 0.93 0.99 Size-4 0.32 0.81 0.65 0.73 0.78 0.85 0.68 0.91 0.85 1.08 Size-5 0.37 0.70 0.86 0.82 0.93 0.85 0.76 0.73 1.00 0.97 Size-6 0.43 0.72 0.74 0.70 0.53 0.69 0.81 0.76 0.79 0.97 Size-7 0.41 0.47 0.88 0.82 0.65 0.71 0.62 0.69 0.82 0.83 Size-8 0.65 0.52 0.50 0.62 0.56 0.66 0.77 0.66 0.81 0.81 Size-9 1.05 0.62 0.46 0.61 0.69 0.49 0.62 0.52 0.70 0.61 Big N/A 0.27 0.30 0.36 0.44 0.45 0.42 0.55 0.61 0.51 Panel D: 100 Size X Investment Portfolios INV-Low INV-2 INV-3 INV-4 INV-5 INV-6 INV-7 INV-8 INV-9 INV-High Small 0.89 1.07 1.07 1.08 1.03 0.90 0.94 0.83 0.75 0.14 Size-2 0.84 0.98 0.90 0.78 1.11 0.91 0.93 0.83 0.63 0.23 Size-3 0.90 0.92 1.06 0.79 0.92 1.18 0.94 1.10 0.81 0.28 Size-4 0.97 0.83 0.89 0.78 0.99 0.70 0.91 0.77 0.64 0.31 Size-5 1.01 1.05 0.92 0.93 1.00 0.85 1.00 0.90 0.70 0.36 Size-6 0.75 0.84 0.84 0.94 0.74 0.76 0.69 0.78 0.76 0.31 Size-7 0.45 0.87 0.67 0.81 0.78 0.77 0.91 0.73 0.66 0.56 Size-8 1.02 0.75 0.66 0.78 0.67 0.76 0.68 0.72 0.76 0.26 Size-9 1.05 0.69 0.60 0.64 0.71 0.78 0.61 0.50 0.46 0.42 Big 1.02 0.69 0.56 0.50 0.49 0.43 0.55 0.42 0.54 0.35 the average returns on two high BTM stock value-weighted portfolios employed at this point do not affect the final result that is the principal objective of this paper. and two low BTM stock value-weighted portfolios in the latter case. Table 2 reports summary statistics for the monthly average returns Portfolio (iii), named UMD (up minus down), is computed by consid- on the portfolios that proxy for the risk factors. The extra three years ering the difference between the average returns on two high prior of data with respect to the sample used in Fama and French (2015) do (winner) stock value-weighted portfolios and two low prior (losers) not significantly change the picture regarding the descriptive statistics stock value-weighted portfolios. of the risk factors. The only relevant change can be found with respect Finally, portfolios (iv) and (v), named RMW (robust minus weak) to SMB, which results in an average value of six basis points less and and CMA (conservative minus aggressive), are determined as the dif- just 1.86 standard errors from zero. With respect to the remainder, we ferences, respectively, between the average returns on two robust can still find a negative correlation between the value, profitability, and operating profitability stock value-weighted portfolios and the average investment factors, and the market and size factors. An extremely high returns on two weak operating profitability stock value-weighted port- correlation between CMA and HML is still present, as well as evidence folios; and between the average returns on two conservative investment of non-correlation between RMW and HML. stock value-weighted portfolios and two aggressive investment stock To complete Sample (b), we have determined the covariances be- tween the daily excess returns of all the portfolios in (a) and the daily value-weighted portfolios. returns on the market portfolio for each month. Formally, for each Regarding portfolios (i), (ii), (iv), and (v), Fama and French (2015) month of 𝑚 days and by indicating with 𝑅 and 𝑅 the return on consider different methods of construction that differ from the 2 × 3 𝑙 𝑙 ̄𝑗 portfolio 𝑗 and on the market portfolio for the 𝑙th day and with 𝑅 sorts used in Fama and French (1993, 1996a). Although they find ̄𝑀 and 𝑅 the respective monthly averages, we have that interesting insights from the different ways of constructing the risk factors, in this paper we focus our attention just on the standard 𝑀 ̄ ̄𝑗 𝑀 𝜎 = (𝑅 − 𝑅 )(𝑅 − 𝑅 ) (14) 𝑗,𝑀 𝑙 𝑙 construction since they show in their paper that different procedures 𝑙=1 7 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 Table 2 Descriptive statistics for the risk factors. M F Summary statistics for the portfolios proxy for the risk factors: R -R are the monthly returns on the market portfolio proxy (portfolio of all the sample stocks) minus the one-month Treasury bill. SMB, HML, UMD, RMW and CMA are the value weighted monthly returns on the portfolios proxy for, respectively, size, value, momentum, operating profitability and investment. Panel A of the table shows the average returns and the standard deviations for each of the considered portfolios and the t-ratios. Panel B reports the correlations between each factor and the other five. Period: July 1963–December 2016, 642 months. Panel A: Averages, standard deviations and 𝑡-statistics for risk factors M F R -R SMB HML UMD RMW CMA Mean 0.510 0.227 0.373 0.664 0.242 0.310 Standard Deviation 4.424 3.087 2.819 4.228 2.234 2.007 𝑡-statistic 2.922 1.863 3.350 3.980 2.742 3.910 Panel B: Correlations between risk factors M F R -R SMB HML UMD RMW CMA M F R -R 1.000 0.295 −0.258 −0.132 −0.233 −0.384 SMB 0.295 1.000 −0.204 0.002 −0.404 −0.169 HML −0.258 −0.204 1.000 −0.187 0.074 0.691 UMD −0.132 0.002 −0.187 1.000 0.109 −0.013 RMW −0.233 −0.404 0.074 0.109 1.000 −0.037 CMA −0.384 −0.169 0.691 −0.013 −0.037 1.000 3.2. Estimation method model expressed as in Eq. (13) can be used to perform a test of the null hypothesis H ∶ 𝑑 = 0, ∀𝑗 ∈ [1, 𝑁] against the alternative hypothesis 0 𝑗 In order to test the performance of the Alpha-Neutral CAPM in H ∶ ∃𝑑 ≠ 0, 𝑖 ∈ [1, 𝑁] where 𝑁 is the number of portfolios considered. 1 𝑖 Eqs. (12) and (13), we have made use of a two-step procedure that The test statistic is given by extends the usual time series testing approach for the purpose of mak- ( )( )( ) ′ −1 d 𝜮 d 𝑇 𝑇 − 𝑁 − 𝐿 ing the latter suitable to test our model. The employment of this kind GRS = ∼ 𝐹 (17) 𝑁,𝑇 −𝑁−𝐿 −1 𝑁 𝑇 − 𝐿 − 1 ′ 1 + E[f] 𝜴 E[f] of testing approach is unusual in this context in that the behavioural component in Eq. (15) is not a traded asset and therefore, in general, where 𝑇 is the number of observations, 𝐿 is the number of factors a cross-sectional approach is usually favourable. Notwithstanding this, included in the regressions, d is the 𝑁 × 1 vector of estimated net the particular kind of setting in which the Alpha-Neutral CAPM is intercepts from the time series regressions, E[f] is the 𝐿 × 1 vector of conceived allows us the use of the GRS test provided by Gibbons factor averages, and 𝜮 and Ω are respectively the unbiased 𝑁 × 𝑁 et al. (1989) as in a normal setting with traded assets, without any covariance matrix of time series regression residuals and the 𝐿 × 𝐿 consequences for the test’s power or interpretation. In fact, we have matrix of covariances between the factors f employed. the following result which we demonstrate in the paper’s appendix. In both steps, we analyse the performance of the model in describing the excess returns of the portfolios considered against the performance Proposition 3. Given the Alpha-Neutral CAPM, the average net cross- of the other most accredited asset pricing models. In particular, we sectional pricing errors E[𝛿 ] coincide with the average time-series net consider the following alternatives to our model: intercepts E[𝑑 ]. The same does not apply to the standard cross sectional 𝑗 The traditional CAPM (Lintner, 1965; Mossin, 1966; Sharpe, 1964) pricing errors 𝛼 , which will in general be different on average from the 𝐹 𝑀 𝐹 𝑅 − 𝑅 = 𝑎 + 𝑏 (𝑅 − 𝑅 ) + 𝑒 (18) 𝑗 𝑗 𝑗,𝑡 time series intercepts 𝑎 given that the behavioural component is not a traded 𝑡 𝑡 𝑡 𝑡 security. The Fama–French three-factor model (Fama & French, 1993) More specifically, the test will be structured in the following way: At 𝑗 𝐹 𝑀 𝐹 𝑅 − 𝑅 = 𝑎 + 𝑏 (𝑅 − 𝑅 ) + 𝑠 SMB + ℎ HML + 𝑒 (19) 𝑡 𝑡 𝑗 𝑗 𝑡 𝑡 𝑗 𝑡 𝑗 𝑡 𝑗,𝑡 the first step, for each portfolio 𝑗 in Sample (a), we run the following 5- year rolling window time series regression of the type in Eq. (12), with The Carhart four-factor model (Carhart, 1997) the purpose of estimating the alpha-neutral betas and the behavioural 𝑗 𝐹 𝑀 𝐹 𝑅 − 𝑅 = 𝑎 + 𝑏 (𝑅 − 𝑅 ) + 𝑠 SMB + ℎ HML + 𝑢 UMD + 𝑒 (20) 𝑡 𝑡 𝑗 𝑗 𝑡 𝑡 𝑗 𝑡 𝑗 𝑡 𝑗 𝑡 𝑗,𝑡 factor 𝜅(𝛾), which represents the key element of our extension And the Fama–French five-factor model (Fama & French, 2015) 𝑗 𝐹 ∗ 𝑀 𝐹 𝑗 𝑀 𝑅 − 𝑅 = 𝑎 + 𝑏 (𝑅 − 𝑅 ) + 𝑘 𝜎(𝑅 , 𝑅 ) + 𝑒 (15) 𝑡 𝑡 𝑗 𝑗 𝑡 𝑡 𝑗 𝑡 𝑡 𝑗,𝑡 𝑗 𝐹 𝑀 𝐹 𝑅 −𝑅 = 𝑎 +𝑏 (𝑅 −𝑅 )+𝑠 SMB +ℎ HML +𝑟 RMW +𝑐 CMA +𝑒 where 𝑎 , 𝑏 , 𝑘 and 𝑒 are respectively the intercepts, the slopes for the 𝑡 𝑡 𝑗 𝑗 𝑡 𝑡 𝑗 𝑡 𝑗 𝑡 𝑗 𝑡 𝑗 𝑡 𝑗,𝑡 𝑗 𝑗 𝑗,𝑡 market risk factor, the behavioural factors which quantify the portions (21) of the covariances left unexplained by the market and explained by As a robustness check, we also run a performance test by con- agents’ non-rational behaviour, and the regressions’ residuals. Then in the second step, we consider the restriction characterized sidering the latter three models augmented for the behavioural bias measured by 𝜅. by the definition of the model’s net intercepts 𝑑 = 𝑎 + 𝑘 𝜎(𝑅 , 𝑅 ) as 𝑗 𝑗 𝑗 𝑡 𝑡 in the specification of the model given in Eq. (13). The restricted model will be 4. Results 𝐹 ∗ 𝑀 𝐹 𝑅 − 𝑅 = 𝑑 + 𝑏 (𝑅 − 𝑅 ) + 𝑒 (16) 𝑗 𝑗,𝑡 𝑡 𝑡 𝑗 𝑡 𝑡 4.1. Model performance summary which is not different from a CAPM adjusted for the hypothesis of agents’ limited rationality. We now turn to the main empirical results. As widely discussed in If the traditional CAPM still works after adjusting for the limited the paper, our main target is to test the extent to which the Alpha- rationality of agents in the market, we should find that all net intercepts Neutral CAPM is able to explain the excess returns of portfolios of are jointly indistinguishable from zero. In order to test this hypothesis, stocks, and to examine a comparison of the performance of our model we have made use of the GRS statistic which, when applied to the against those of the Fama,-French and Carhart multifactor models. We 8 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 test the performance of the models by looking both at the time series the net intercepts’ distribution. Specifically, a skewness close to zero regression-generated intercepts and at different measures of the overall and a low kurtosis are good news since that would imply that the explanatory power of the models involving the cross-sectional pricing pricing errors will be distributed homogeneously on the equilibrium errors. hyperplane and with a low frequency of values far from zero. Table 3 reports the GRS statistics of Gibbons et al. (1989) and the If, in general, the information contained in the descriptive statistics relative p-values, which test whether the models’ net intercepts with of the distribution of net intercepts is helpful in the interpretation of respect to the behavioural bias, 𝐴| 𝑑 | , are jointly statistically equal to the GRS test, it is also true that it is not sufficient to fully describe the zero – obviously, for models which do not consider the behavioural results. The Alpha-Neutral CAPM has a distribution of pricing errors adjustment in the pricing equation, the net intercepts will coincide clearly improved with respect to the traditional models for the 18 Size with the alphas – for the Alpha-Neutral CAPM and the seven alternative portfolios (Panel A), the 18 BTM portfolios (Panel B) and 100 Size X models considered. For every set of portfolios examined, the GRS test BTM portfolios (Panel C), with a skewness index that goes from −0.7 easily rejects the traditional CAPM along with all of the multifactor to 0.4 and kurtosis from 1.7 to 3.6. The same is not true for the 100 models that are not adjusted for the behavioural bias. Size X OP portfolios (Panel C) and 100 Size X INV portfolios (Panel D) Conversely, the test never rejects the Alpha-Neutral CAPM, a result in which the statistics seem to contradict the GRS test result, showing a that is strongly robust across all samples as documented by the high pricing error distribution for the Alpha-Neutral CAPM which is clearly level of the p-values (from 0.197 for the BTM portfolios to 0.995 for outperformed by the traditional model and, in particular, by the FF the Size X BTM portfolios). The conclusions from the Alpha-Neutral five-factor model, which is instead rejected by the formal test. variations of the multifactor models are less obvious: the results from the augmented FF three-factor model are not robust for the size (Panel 4.1.1. Size portfolios A) and BTM portfolios (Panel B) with GRS statistic p-values respectively The CAPM, along with the FF three-factor and the Carhart four- equal to 0.115 and 0.125, while the augmented FF five-factor model is factor models, are all easily rejected by the GRS test with p-values close clearly rejected for the same portfolios with p-values equal to 0.045 and to zero. The Alpha-Neutral version of the latter instead easily passes the 0.005 respectively. On the contrary, with regard to the two-way sorted test with p-values from 0.11 for the augmented three-factor model to portfolios in panels C, D, and E, the two models cannot be rejected. In 0.4 for the Alpha-Neutral CAPM. The traditional and the augmented any case, it is interesting to observe that all the augmented multifactor five-factor model share p-values around the threshold values and thus models are, in terms of their GRS statistic, systematically outperformed the asset pricing test is inconclusive in these cases. by the Alpha-Neutral CAPM, except for the 100 Size X BTM portfolios The average net intercept 𝐴| 𝑑 | produced by the Alpha-Neutral (Panel C), where the best performance is achieved by the behaviourally CAPM and the behaviourally augmented models are close in magnitude augmented Carhart four-factor model. to the traditional multifactor models, which also share similar values Table 3 reports for each model and panel of data, along with the for the descriptive statistics of the intercepts. Specifically, almost all GRS test, the estimated average absolute intercepts 𝐴| 𝑎 | , the estimated models share a slightly skewed and platykurtic distribution of pricing average absolute slope for the behavioural factor 𝐴| 𝜅 | , the percentage errors. An interesting exception is represented by the high kurtosis of sign reversals between the latter two, and the average absolute net displayed by the five-factor model (4.196), which identifies a higher intercepts 𝐴| 𝑑 | , along with some descriptive statistics which character- frequency of values far from zero that is coherent with a rejection of ize the empirical distribution of the latter: the maximum and minimum the GRS test. The best possible distribution is achieved for this sample values for the net estimated intercepts, their standard deviation 𝜎(𝑑 ), by the Alpha-Neutral variation of the FF three-factor model with a the skewness Sk(𝑑 ) and the kurtosis Ku(𝑑 ). skewness index equal to −0.265 and a kurtosis of just 1.952, a result 𝑗 𝑗 The intercepts 𝑎 generated by the Alpha-Neutral variation of the that is in contradiction with the rejection of the GRS test. models are always greater than those generated by the traditional models. However, by representing the pricing errors of the regressions’ 4.1.2. BTM portfolios hyperplanes that consider the behavioural biases as independent vari- For the 18 BTM portfolios, the test easily rejects the FF three-factor ables, their magnitude is not relevant when testing model performance model, the Carhart four factor model and the augmented five-factor in that, as discussed in the previous sections, the tests are conducted model. With p-values from 0.1 and 0.2, the test is not able to reject the in terms of the restriction applied to the models that the net intercepts Alpha-Neutral variation of the CAPM, the three-factor model and the equal zero. Moreover, the finding of higher standard intercepts in this four-factor model. Again, the test is inconclusive for the five-factor asset setting is not necessarily bad news in that it is simply a consequence of pricing model. The average net intercept 𝐴| 𝑑 | values for the Alpha- estimating an asset pricing model that makes use of a non-traded asset, Neutral CAPM are considerably higher than those of the traditional as explained in Proposition 3. multifactor models and in particular show a magnitude similar to those The behavioural coefficients 𝑘 show up as always statistically of the traditional CAPM. The maximum value assumed by the net significant in each sample. Consistent with the intuition of the model intercepts is equal to 0.32, which is again close to the 0.36 of the tradi- introduced, the behavioural adjustments generated by the products of tional CAPM and considerably greater than the maximum net intercept the latter with the associated covariances display signs that are inverted generated by the traditional multifactor models. However, the lower with respect to the intercepts 𝑎 in almost every portfolio analysed (the skewness (−0.2) and kurtosis (1.7) with respect to the other competing figure runs from 83% for the 18 BTM portfolios in Panel B to 100% for traditional models might justify the non-rejection of the GRS test. The the 18 size portfolios in Panel A). best possible distribution is achieved this time by the traditional FF With respect to the Alpha-Neutral CAPM, for every sample consid- three-factor model with a skewness value tending towards a normal ered, the average net intercept 𝐴| 𝑑 | is always significantly reduced (0.06) and a kurtosis of just 1.861. in magnitude by the presence of the behavioural component with respect to the traditional CAPM in which the latter is not considered. 4.1.3. Size-BTM portfolios This reduction is, however, never sufficient to generate net intercepts The GRS test does not reject the null hypothesis that the net inter- which are on average lower than the traditional multifactor model cepts are jointly equal to zero for all of the Alpha-Neutral variations alphas. Nevertheless, although highly emphasized in the literature, the of the traditional models considered and conversely, it easily rejects magnitude of the absolute average intercept is definitely not, on its the latter with p-values tending to zero. Again, the average absolute own, an unquestionable measure of the performance of an asset pricing net intercepts for the Alpha-Neutral CAPM are lower than those of the model, as highlighted by, among others, Barillas and Shanken (2016). traditional CAPM and higher than those of the traditional multifactor In fact, it is highly informative to also look at the higher moments of models. The descriptive statistics give strength to the non-rejection of 9 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 Table 3 Model performance summary. Summary statistics for performance tests of the Alpha-Neutral CAPM (𝛼-CAPM) against the performance of the traditional CAPM (equation 18), the Fama and French 3 Factor Model (equation 19), the Carhart 4 Factor Model (equation 20), the Fama and French 5 Factor Model (equation 21) and all the behavioural augmented versions of the latter three models. Sample: monthly excess returns on 18 Size portfolios (Panel A), 18 Book-to-Market portfolios (panel B), 100 Size and Book-to-Market portfolios (Panel C), 100 Size and Operating Profitability Portfolios (Panel D) and 100 Size and Investment Portfolios. The table shows, for each model and panel of data, the estimated average absolute intercepts 𝐴| 𝑎 | , the estimated average absolute slope for the behavioural factor 𝐴| 𝑘 | , the 𝑗 𝑗 percentage of sign reversals between the latter two, the average absolute net intercepts 𝐴| 𝑑 | , the standard deviation, the (absolute) minimum and the maximum estimated values, the skewness and the kurtosis of the distribution of the latter, the GRS statistic which test if the net intercepts are jointly equal to zero and the relative p -values. Period: July 1963–December 2016, 642 months. 𝐴| 𝑎 | 𝐴| 𝑘 | s.r. 𝐴| 𝑑 | 𝜎(𝑑 ) min(𝑑 ) max(𝑑 ) Sk(𝑑 ) Ku(𝑑 ) GRS p(GRS) 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 Panel A: 18 Size Portfolios CAPM 0.127 0.072 0.030 0.234 −0.939 3.079 1.968 0.010 Fama French 3 Factor 0.033 0.044 0.103 0.042 −0.743 2.376 2.333 0.001 Carhart 4 Factor 0.038 0.047 0.097 0.042 −0.891 2.267 2.571 0.000 Fama French 5 Factor 0.023 0.029 0.053 0.076 −0.408 4.196 1.857 0.017 𝛼-CAPM 0.477 0.019 1.0 0.034 0.046 0.064 0.102 0.400 2.460 1.047 0.404 𝛼-Fama French 3 Factor 0.061 0.002 0.8 0.037 0.043 0.097 0.044 −0.265 1.952 1.421 0.115 𝛼-Carhart 4 Factor 0.051 0.002 0.8 0.034 0.042 0.090 0.046 −0.488 2.005 1.354 0.148 𝛼-Fama French 5 Factor 0.054 0.002 0.7 0.039 0.039 0.080 0.062 0.246 2.345 1.644 0.045 Panel B: 18 Book-to-Market Portfolios CAPM 0.160 0.151 0.137 0.358 0.019 2.018 1.770 0.025 Fama French 3 Factor 0.069 0.083 0.140 0.133 0.058 1.861 2.263 0.002 Carhart 4 Factor 0.064 0.079 0.103 0.137 0.310 1.768 2.116 0.005 Fama French 5 Factor 0.050 0.064 0.083 0.159 0.523 2.612 1.680 0.038 𝛼-CAPM 0.268 0.005 0.8 0.153 0.138 0.106 0.320 −0.198 1.710 1.275 0.197 𝛼-Fama French 3 Factor 0.103 0.004 0.7 0.074 0.094 0.145 0.193 0.454 2.361 1.398 0.125 𝛼-Carhart 4 Factor 0.095 0.004 0.8 0.065 0.082 0.135 0.186 0.271 2.516 1.286 0.190 𝛼-Fama French 5 Factor 0.089 0.005 0.8 0.095 0.146 0.133 0.522 2.143 8.279 2.115 0.005 Panel C: 100 Size X Book-to-Market Portfolios CAPM 0.263 0.259 0.601 0.721 −0.515 3.445 2.248 0.000 Fama French 3 Factor 0.122 0.182 0.752 0.382 −1.468 6.934 2.176 0.000 Carhart 4 Factor 0.122 0.179 0.781 0.390 −1.676 7.570 2.057 0.000 Fama French 5 Factor 0.116 0.156 0.644 0.384 −0.763 5.278 2.418 0.000 𝛼-CAPM 0.512 0.017 0.9 0.214 0.252 0.708 0.575 −0.700 3.638 0.656 0.995 𝛼-Fama French 3 Factor 0.177 0.006 0.8 0.128 0.173 0.660 0.364 −0.802 4.653 0.624 0.998 𝛼-Carhart 4 Factor 0.164 0.005 0.8 0.125 0.164 0.659 0.365 −0.976 5.443 0.568 1.000 𝛼-Fama French 5 Factor 0.169 0.007 0.7 0.161 0.194 0.485 0.413 −0.030 2.394 0.988 0.517 Panel D: 100 Size X Operating Profitability Portfolios CAPM 0.232 0.207 0.410 0.600 −0.597 3.274 1.894 0.000 Fama French 3 Factor 0.142 0.183 0.515 0.349 −0.890 3.772 1.807 0.000 Carhart 4 Factor 0.150 0.189 0.535 0.371 −0.965 4.191 1.881 0.000 Fama French 5 Factor 0.110 0.138 0.258 0.455 0.511 3.282 1.596 0.001 𝛼-CAPM 0.520 0.017 0.9 0.168 0.200 0.662 0.407 −1.318 5.246 0.683 0.989 𝛼-Fama French 3 Factor 0.198 0.006 0.7 0.160 0.207 0.628 0.476 −0.737 3.835 0.671 0.992 𝛼-Carhart 4 Factor 0.203 0.005 0.7 0.166 0.210 0.627 0.523 −0.784 4.045 0.681 0.990 𝛼-Fama French 5 Factor 0.155 0.005 0.7 0.129 0.165 0.355 0.649 0.704 3.951 0.638 0.996 Panel E: 100 Size X Investment Portfolios CAPM 0.295 0.256 0.506 0.640 −0.991 3.824 2.620 0.000 Fama French 3 Factor 0.142 0.189 0.706 0.473 −1.310 6.201 2.454 0.000 Carhart 4 Factor 0.144 0.185 0.684 0.519 −1.209 6.144 2.319 0.000 Fama French 5 Factor 0.122 0.152 0.438 0.511 −0.101 4.280 2.083 0.000 𝛼-CAPM 0.547 0.017 0.9 0.232 0.264 0.702 0.507 −1.287 4.764 0.815 0.893 𝛼-Fama French 3 Factor 0.167 0.005 0.8 0.140 0.176 0.596 0.417 −0.925 4.773 0.707 0.982 𝛼-Carhart 4 Factor 0.166 0.005 0.8 0.141 0.173 0.555 0.457 −0.775 4.470 0.658 0.994 𝛼-Fama French 5 Factor 0.158 0.005 0.8 0.131 0.158 0.331 0.454 0.212 2.929 0.639 0.996 the Alpha-Neutral CAPM despite the higher magnitudes of the inter- for all of the Alpha-Neutral variations of the traditional models and cepts. The maximum is of a lower magnitude than for the traditional conversely, it easily rejects the latter with p-values tending to zero CAPM, while the minimum is lower in magnitude with respect to the FF for both samples. The results for the portfolios formed from stocks three-factor and the Carhart four-factor models. Skewness and kurtosis sorted on size and operating profitability and on size and investment are the lowest among the competing traditional multifactor models, are, however, the most controversial for the Alpha-Neutral CAPM. although, surprisingly, the values are higher than for the traditional Despite the clear non-rejection of the GRS test, the magnitude of the CAPM. The best performance in terms of the distribution of intercepts net average absolute intercepts, although inferior with respect to the is this time achieved by the augmented Fama and French five-factor traditional CAPM results, are again larger with respect to the traditional model with a skewness that tends towards the normal (−0.03) and multifactor models. Moreover, contradicting the results with respect to showing the only case of a platykurtic distribution amongst all the the previous samples, the distribution of net intercepts for the Alpha- competing models (Ku(𝑑 ) = 2.115). Neutral CAPM is in this case highly leptokurtic (Ku(𝑑 ) = 5.24 for Panel D and Ku(𝑑 ) = 4.76 for Panel E) and skewed (Sk(𝑑 ) = −1.32 for 𝑗 𝑗 4.1.4. Size-OP portfolios and size-INV portfolios Panel D and Ku(𝑑 )= −1.29 for Panel E), identifying a high frequency of extreme values with respect to the competing models. Thus, the As for the Size X BTM portfolios, the GRS test does not reject the null hypothesis that the net intercepts are jointly equal to zero descriptive statistics regarding the distribution of net intercepts are in 10 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 contradiction to the result of the formal test, showing the necessity to Following Fama and French (2016), since 𝛿 is constant, the cross- delve deeper in analysing the results obtained. sectional average over the expected value of 𝑑 is 2 2 2 2 2 2 E[𝑑 ] = E[𝛿 ] + E[𝜀 ] + E[𝜀 ] + 2E[𝜀 𝜀 ] = E[𝛿 ] + VAR[𝜀 ] (24) 𝑗,𝛼 𝑗,𝜅 𝑗 𝑗 𝑗,𝛼 𝑗,𝜅 𝑗 𝑗 4.2. Diagnostics where VAR[𝜀 ] is the variance of the net intercepts 𝛿 due to estimation The apparent clash between the information obtained from the GRS error which we estimate using the cross-sectional average standard 2 2 2 test and the net average absolute value of the estimated intercepts error of 𝑑 , 𝐴𝑠 (𝑑 ). The ratio 𝐴𝑠 (𝑑 )∕𝐴𝑑 thus measures the dispersion 𝑗 𝑗 𝑗 highlights an important question. A possible controversy that may arise of net intercept estimates due to estimation error. Along with the by observing the latter results concerns the extent to which the non- latter ratio, Table 4 also reports some metrics, introduced in Fama rejection of the GRS tests is due to chance rather than to an actual and French (2015, 2016), which estimate the proportion of the cross- section of expected returns left unexplained by the models. Let 𝑟 = contraction of the (true) magnitude of the net intercepts. 𝑅 − 𝑅, 𝑅 be the time series average excess return on the portfolio This issue can be addressed through a dissection of the GRS statistic 𝑗 𝑗 ′ −1 2 ̄ 𝑗 and 𝑅 is the cross-section average of 𝑅 , 𝐴| 𝑑 | ∕𝐴| 𝑟 | measures the into the unexplained ex-post squared Sharpe ratio 𝜃 = d 𝜮 d and 𝑗 𝑗 𝑗 𝑢 e 2 ′ −1 dispersion of average expected returns left unexplained by the models the factors’ Sharpe ratio 𝜃 = E[f] 𝜴 E[f] according to the economic 2 2 while 𝐴(𝑑 )∕𝐴(𝑟 ) is the variance of the cross-sectional expected returns interpretation of the GRS statistic given in Gibbons et al. (1989), in 𝑗 𝑗 of the portfolios left unexplained by the models. As pointed out by which they show the possibility of rewriting the latter as Fama and French (2016), high values of the latter two ratios are bad in ( )( )( 2 2 ) 𝜃 − 𝜃 𝑇 𝑇 − 𝑁 − 𝐿 𝑓 that this would suggest that the dispersion of intercepts is high relative GRS = (22) 𝑁 𝑇 − 𝐿 − 1 to the dispersion of test assets. Conversely, high values of the ratio 1 + 𝜃 2 2 𝐴𝑠 (𝑑 )∕𝐴𝑑 are good in that it would tell us that a higher proportion where 𝜃 is the Sharpe ratio of the ex post tangency portfolio spanned by of the dispersion is due to sampling error rather than to the dispersion the 𝑁 assets and the 𝐿 factors. According to this interpretation, the less of the true intercepts. is the relative distance between the ex post tangency portfolio Sharpe Except for the 18 Size portfolios (Panel A), the Alpha-Neutral CAPM ratio and the factor Sharpe ratio, the higher will be the unexplained displays dispersion coefficients that are always greater than one with Sharpe ratio 𝜃 , and thus the distance from intercepts that are thus more dispersed than the average returns. In the intercepts to zero. In a recent study, Barillas and Shanken terms of dispersion in particular, in this case, the best results are (2016) discuss this decomposition, showing that a comparison between achieved by the five-factor model, although some of the Alpha-Neutral competing models essentially relies on the magnitude of the factor variations of the multifactor models produce results that are at least Sharpe ratio while the test assets are shown as irrelevant unless one close to the three-factor model or to the five-factor model depending or more factors employed in the asset pricing model are not returns. 2 on the sample. Nevertheless, the ratio 𝐴𝑠 (𝑑 )∕𝐴𝑑 for the Alpha- Table 4 reports the decomposition of the GRS statistics into unex- Neutral CAPM has a minimum equal to 58% in Panel (B) and around 2 2 plained Sharpe ratios 𝜃 and Sharpe ratios of the factors 𝜃 along with 1 for the other four samples. Thus, there is strong evidence that the 𝑢 𝑓 Shanken’s 1987 efficiency ratio, 𝜌 = 𝜃 ∕𝜃. 𝑓 larger dispersion is due predominantly to estimation error in the net Ideally, if the portfolio given by the combination of factors is intercepts, which in particular is higher than for the traditional model, efficient, 𝜌 = 1. Consistent with the result in Proposition 3 and with according to Eqs. (23) and (24). the findings of Barillas and Shanken (2016), the unexplained Sharpe We conclude this section by reporting confidence intervals for the ratio is approximately the same for every model in each of the sam- true unexplained Sharpe ratio 𝜃 as suggested in Lewellen, Nagel, and ples considered. Thus, for each of the asset pricing models that have Shanken (2010). They show, in particular, that it is possible to find been considered, the actual explanatory power of the latter is wholly an exact confidence interval by representing the relative percentiles of represented by the factor Sharpe ratio 𝜃 . The Alpha-Neutral models the GRS statistic given by a non-centred Fisher F-distribution with non- always display values considerably higher than those of the traditional centrality parameter 𝑐 = 𝜃 ∕𝑁 , as a function of the unexplained Sharpe asset pricing models (at least eight times higher than the FF five-factor ratio, and by studying the intersection with the observed value of the model, which represents the best alternative amongst the traditional GRS statistic. In this last test, we focus only on a comparison between models). Notice also that the unexplained Sharpe ratios of the Alpha- the traditional model with the best performance, i.e. the Fama–French Neutral models, although remaining very close to those obtained from five-factor model, and the Alpha-Neutral CAPM which is the primary the traditional models, benefit from a consistent reduction in three out interest of this paper. of five of the samples. Fig. 4 represents the 5th, 50th, and 95th percentiles of random Consistent with the findings of Fama and French (2015), the five- extractions from a non-centred F-distribution with 𝑁 = 18 (Panel A) factor model always outperforms the three-factor model, but in terms and 𝑁 = 100 (Panel B), which constitute the theoretical distributions that we need to compare with the observed GRS statistics for the 18 of absolute efficiency, the combination of factors: MKT, SMB, HML, CMA and RMW, slightly exceeds 50% for the one-way sorted portfolios size and BTM portfolios (straight lines in Panel A) and the 100 size X BTM, size X OP and size X INV portfolios (straight lines in Panel (panel A and B) and 30% for the two-way sorted portfolios (panel B) respectively. In both panels, the blue lines are the GRS statistics C, D and E). Conversely, the Alpha-Neutral CAPM, along with all the produced by the Fama–French five-factor model while the red lines are adjusted multifactor models, display an efficiency coefficient of around produced by the Alpha-Neutral CAPM. Confidence intervals are formed 70% for every sample, which is surprisingly robust across the samples. taking the interceptions between the observed GRS statistics and the Another important point that is not always well addressed in the 95th percentiles for the left-hand side extreme and the 5th percentiles empirical literature is represented by the fact that, more important than for the right-hand side of the confidence intervals. the magnitude of the estimated intercepts themselves, is the proportion From Fig. 4 it can be immediately noticed that for each sample, the in the estimation represented by the real unknown pricing errors and confidence intervals for the FF five-factor model are always wider than the estimation errors which naturally arise from the application of the that for the Alpha Neutral model. More specifically in Panel A, the five- econometric technique employed. The estimated intercepts 𝑑 are in factor model shows intervals that are approximately [0,0.5] for both fact given by the true intercepts 𝛿 plus the sum of the estimation errors size and BTM portfolios while for the Alpha-Neutral model they are of the alphas, 𝜀 , and of the behavioural bias 𝜀 . 𝑗,𝛼 𝑗,𝜅 around [0,0.05]. The evidence for the 100 two-way sorted portfolios 𝑑 = 𝛿 + 𝜀 + 𝜀 (23) in Panel B is even stronger. Confidence intervals for the five-factor 𝑗 𝑗 𝑗,𝛼 𝑗,𝜅 11 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 Table 4 Test diagnostics. The table shows the summary statistics for the performance test’s diagnostics of the Alpha-Neutral CAPM (𝛼-CAPM) against the performance of the traditional CAPM (equation 18), the Fama and French three-factor model (equation 19), the Carhart four-factor model (equation 20), the Fama and French five-factor model (equation 21) and all the behavioural augmented versions of the latter three models. Sample: monthly excess returns on 18 Size portfolios (Panel A), 18 Book-to-Market portfolios (panel B), 100 Size and Book-to-Market portfolios (Panel C), 100 Size and Operating Profitability Portfolios (Panel D) and 100 Size and Investment Portfolios. The table shows, for each model and panel of data, the dispersion’s indexes of Fama and French (2015, 2016), the unexplained squared Sharpe ratio, the factor squared Sharpe ratio, Shanken’s 1987 index of efficiency. Period: July 1963–December 2016, 642 months. 𝐴| 𝑑 | 𝐴| 𝑑 | 𝐴𝑠 | 𝑑 | 𝑗 𝑗 𝑗 2 2 𝜃 𝜃 𝜌 2 2 𝑢 𝐴| 𝑟 | 𝐴| 𝑟 | 𝐴| 𝑑 | 𝑓 𝑗 𝑗 Panel A: 18 Size Portfolios CAPM 1.589 1.915 0.267 0.053 0.015 0.350 Fama French 3 Factor 0.417 0.192 0.979 0.064 0.049 0.467 Carhart 4 Factor 0.482 0.204 1.051 0.075 0.105 0.542 Fama French 5 Factor 0.288 0.087 0.977 0.054 0.100 0.577 𝛼-CAPM 0.431 0.202 1.006 0.051 0.868 0.804 𝛼-Fama French 3 Factor 0.468 0.205 0.884 0.071 0.898 0.781 𝛼-Carhart 4 Factor 0.427 0.176 1.003 0.072 1.026 0.790 𝛼-Fama French 5 Factor 0.494 0.205 0.911 0.083 0.911 0.769 Panel B: 18 Book-to-Market Portfolios CAPM 1.274 1.618 0.594 0.047 0.015 0.362 Fama French 3 Factor 0.546 0.279 1.055 0.062 0.049 0.470 Carhart 4 Factor 0.512 0.252 1.041 0.061 0.105 0.566 Fama French 5 Factor 0.397 0.177 0.997 0.049 0.100 0.590 𝛼-CAPM 1.215 1.403 0.576 0.060 0.788 0.784 𝛼-Fama French 3 Factor 0.585 0.358 1.035 0.067 0.822 0.778 𝛼-Carhart 4 Factor 0.517 0.285 0.989 0.067 0.977 0.793 𝛼-Fama French 5 Factor 0.755 0.889 1.009 0.103 0.848 0.742 Panel C: 100 Size X Book-to-Market Portfolios CAPM 1.484 2.094 0.287 0.379 0.015 0.167 Fama French 3 Factor 0.689 0.662 0.466 0.379 0.049 0.265 Carhart 4 Factor 0.691 0.643 0.460 0.377 0.105 0.345 Fama French 5 Factor 0.654 0.490 1.122 0.442 0.100 0.323 𝛼-CAPM 1.207 1.475 1.143 0.355 2.168 0.712 𝛼-Fama French 3 Factor 0.721 0.601 1.275 0.343 2.224 0.718 𝛼-Carhart 4 Factor 0.705 0.552 1.386 0.337 2.470 0.730 𝛼-Fama French 5 Factor 0.907 0.757 1.086 0.547 2.245 0.669 Panel D: 100 Size X Operating Profitability Portfolios CAPM 1.465 1.891 0.412 0.354 0.013 0.162 Fama French 3 Factor 0.894 0.883 0.400 0.348 0.046 0.266 Carhart 4 Factor 0.945 0.962 0.337 0.379 0.093 0.331 Fama French 5 Factor 0.696 0.570 0.607 0.326 0.106 0.363 𝛼-CAPM 1.061 1.179 1.764 0.397 2.047 0.694 𝛼-Fama French 3 Factor 1.009 1.135 0.840 0.401 2.133 0.698 𝛼-Carhart 4 Factor 1.049 1.203 0.756 0.425 2.271 0.698 𝛼-Fama French 5 Factor 0.811 0.779 1.150 0.388 2.184 0.703 Panel E: 100 Size X Investment Portfolios CAPM 1.713 2.288 0.252 0.489 0.013 0.142 Fama French 3 Factor 0.824 0.771 0.331 0.473 0.046 0.237 Carhart 4 Factor 0.840 0.763 0.311 0.467 0.093 0.308 Fama French 5 Factor 0.709 0.529 0.485 0.425 0.106 0.333 𝛼-CAPM 1.348 1.658 1.008 0.515 2.311 0.679 𝛼-Fama French 3 Factor 0.811 0.690 0.906 0.455 2.371 0.695 𝛼-Carhart 4 Factor 0.820 0.693 0.870 0.453 2.601 0.706 𝛼-Fama French 5 Factor 0.761 0.564 1.148 0.414 2.394 0.706 model go from [0.02,0.2] for the 100 size and investment portfolios others and thus, if it is redundant as an explanator of the test assets to [0.12,0.35] for the size and BTM portfolios. Conversely, the Alpha- considered. The nature of the model that we have introduced, the Neutral CAPM displays an unexplained Sharpe ratio confidence interval strength of our results and of the evidence about the contraction of of [0,0.02] for the size and investment portfolios and of [0,0.005] for the intercepts obtained in general compared with the three-factor and the other two samples. Thus, this test gives further confirmation to five-factor models (Fama & French, 1993, 1996a, 1996b, 2015, 2016) 2 2 the information contained by the ratio 𝐴𝑠 (𝑑 )∕𝐴𝑑 and in particular lead us to attempt to give an answer to an old controversy which usually characterizes the latter models, i.e. whether the ‘‘better’’ pricing to the intuition that much of the net intercepts generated constitutes attained by these model is rationally or irrationally driven (Fama & estimation error and that it most likely, at least for the sample that we have analysed, that the GRS test does not make an error in not rejecting French, 1993, 2017); Titman, Wei and Xie, 2013). the null. The intuition behind our spanning test is the same as for the general test of the performance of competing models. A factor is not redundant 4.3. Factor spanning test in the model if and only if the other factors considered in the regression are insufficient to price the latter correctly. Formally, by considering A common practice in the empirical asset pricing literature is to for instance a standard CAPM and a factor 𝑓 different from the market test whether a factor can be explained through a combination of the portfolio, the factor is important in order to explain average returns in 12 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 Fig. 4. The figure represents the sample distribution of the GRS statistic, with 𝑁 = 18 (Panel A) and 𝑁 = 100 (Panel B), and confidence intervals for the unexplained Sharpe ratios relative to the 336 test portfolios analysed. Sample period: July 1963–December 2016, 642 months. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) the test assets if, in a time-series regression of the type show highly significant estimated intercepts with p-values less than 0.01. Again consistent with Fama and French (2015, 2017), we find that 𝑀 𝐹 𝑓 = 𝑎 + 𝑏 (𝑅 − 𝑅 ) + 𝑒 (25) 𝑡 𝑓 𝑓 𝑓,𝑡 𝑡 𝑡 the value factor instead seems redundant with an estimated intercept of just 0.09 and a 𝑝-value of 0.27, while, different from them, the addition the intercepts 𝑎 are statistically different from zero. of the behavioural factor in the regression makes the profitability factor However, if the market’s mispricing of the factor is not due to the redundant as well with an intercept of 0.03 and an associated 𝑝-value presence of a real effect that is not caught by the market but instead of 0.06, although the result is less robust with respect to HML. because of the presence of a behavioural bias, exactly as occurred for By instead judging the explanatory power of the factors under the test assets, we will have a behavioural bias coefficient statistically the logic introduced by the Alpha-Neutral framework, the situation is different from zero and a representation of the model in Eq. (25) as completely reversed. With the exception of the market factor, all of the remaining factors are replicable with combinations of the others 𝑀 𝐹 𝑀 𝑀 𝐹 𝑓 = 𝑎 +𝑏 (𝑅 −𝑅 ) +𝑘 COV(𝑓 , 𝑅 ) +𝑒 = 𝑑 +𝑏 (𝑅 −𝑅 ) +𝑒 𝑡 𝑓 𝑓 𝑓 𝑡 𝑓,𝑡 𝑓 𝑓 𝑓,𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 within the Alpha-Neutral model, showing non-significant average net (26) intercepts with p-values that run from 0.25 for SMB to 0.84 for CMA. These results are, however, not decisive, especially for the momentum where now the condition in order for the factor to not be discarded is factor, in that much of the non-rejection of the t-test is due to substan- 𝑑 different from zero. tially increased standard errors obtained in the formation of the net In each regression, the behavioural factor COV(𝑓 , 𝑅 ) takes the 𝑡 intercepts. form of the covariances, estimated from the daily returns, between the The results shown in Panel B instead display more strength. In this market and the factor that has to be explained. Panel A of Table 5 shows case, we have used only the market and behavioural factors in order regressions in which six factors are used to explain the returns on the to explain the average monthly returns on each of the other factors. seventh. In terms of the estimated intercepts, our findings are similar to The results for the size factor are very interesting since, when all of those in Fama and French (2015, 2017). Judging each of the different the other factors are removed, we obtain estimated intercepts that are factors considered along with the market in terms of 𝑎 , almost all not significant with 𝑝-value 0.10. The behavioural bias represented seem to play a role in the explanation of the average returns of the by the coefficient 𝜅 has zero explanatory power, a result which 𝑆𝑀𝐵 test assets. In particular, the size, momentum and investment factors is consistent with the logic of the Alpha-Neutral CAPM in which a 13 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 behavioural bias exists in the case of mispricing of the traditional model market factor and the degree of optimism is in fact able to explain the represented by a pricing error different from zero. cross-section of the size, profitability and investment effects but not the Regarding the HML factor, our test clearly rejects the null hypoth- value or momentum effects, which seem instead to have a real impact esis that the value factor is redundant in an Alpha-Neutral framework on the average returns of the securities in the market. At the same time, which considers just optimism/pessimism as a departure from rational- it is also true that our representation of irrationality is limited in that ity. The intercepts ex ante and ex post netting are statistically different we are considering just one, albeit somehow encompassing, departure from zero, showing that the market factor alone cannot correctly price from rationality. Moreover, the results we have obtained do not render the HML factor and the latter is thus a necessary variable to include in the other factors studied in the literature outdated. We also have to the regressions to explain the average returns on the test asset. deal with the problem that the behavioural factor is not a return. In More controversial is the result concerning the momentum factor. this sense, the SMB, HML, UMD, RMW and CMA factors might remain The estimated intercepts are statistically different from zero with p- essential in order to construct a traded portfolio whose returns mimic values around 0.00 while, as for the previous cases, the net average the behavioural factor. intercepts from the t-test are equal to zero with a 𝑝-value of 0.34. Nevertheless, it is clear from the standard error that most of the non- CRediT authorship contribution statement rejection of the test is due to the magnitude of the latter, which is approximately five times the corresponding value associated with the Francesco Rocciolo: Conceptualization, Methodology, Software, Formal analysis, Writing – original draft. Andrea Gheno: Conceptual- estimated intercept. Thus we can easily see that the spanning test is not conclusive in this case. ization, Supervision, Writing – review & editing. Chris Brooks: Project The CMA and RMW factors show relevant results. The estimated administration, Conceptualization, Supervision, Writing – review & intercepts in this case are both statistically different from zero, showing editing. the presence of a consistent mispricing of the latter portfolios by the Appendix. Proofs of propositions market. However, different from the value factor, this mispricing seems in both cases to be behaviourally driven, given that the average net intercepts are statistically insignificant with p-values of 0.15 and 0.65 Proof of Proposition 1 respectively. An interesting insight is that the more a firm is charac- terized by a high level of operating profitability, the more the market Let us start by considering the variable 𝑊 (𝑡) in Eq. (6). If the vector is on average optimistic so that such a stock generates a misprice with of risky asset gross returns is distributed as a multivariate normal with a non-rational root. Conversely, and specifically for small firm stocks, mean E[R] and covariance matrix 𝜮 , we will have that the final level the larger the level of a firm’s investment, the less that firm will be of wealth 𝑊 (𝑡) will be normally distributed as well with moments: seen as stable by the market which will interpret a larger variance as a ′ F 𝐹 E[𝑊 (𝑡)] = x (E[R] − R ) + 𝑊 (𝑡 − 1)𝑅 (A) bad signal, and which would be reflected in investors being pessimistic 𝑖 𝑖 𝑖 about that stock. Thus, under this logic, profitability and investment effects have non-zero impacts on the average excess returns of the test VAR[𝑊 ] = x 𝜮 x (B) 𝑖 R 𝑖 assets not because of a real effect from these two variables on the stocks’ For a normally distributed variable 𝑊 (𝑡) and a constant 𝑎, the following returns, but because of behavioural biases generated among investors 𝑖 property holds regarding the firms’ investment decisions and the characteristics of their profitability. 1 E[exp(𝑎𝑊 (𝑡))] = exp(E[𝑎𝑊 (𝑡)] + VAR[𝑎𝑊 (𝑡)]) (C) 𝑖 𝑖 𝑖 5. Conclusions Applying this last expansion to Eq. (4) and by plugging in the results into Eqs. (A) and (B), we end up with the following objective function In this paper we have derived a capital asset pricing model in an [ ( )] ′ F 𝐹 ′ economy in which traders, consistent with recently developed theories −2[1−𝜙(𝛾 ,𝜮 )] exp −𝜌 (x (E[R]−R )+𝑊 (𝑡−1)𝑅 )+ x 𝜮 x (D) 𝑖 R 𝑖 𝑖 R 𝑖 𝑖 𝑖 in the decision-making literature, do not behave rationally in the sense of von Neumann–Morgenstern expected utility theory. In particular, we where, according to Eq. (3), have focused our attention on the inclusion of the agents’ degrees of { ( ) } ⎧ 1 1 1 optimism in the capital asset pricing model. In our view, this represents 1 − exp −𝜌 𝛾 − x 𝜮 x if ≤ 𝛾 < 1 𝑖 R 𝑖 ⎪ 𝑖 2 2 2 the most compelling departure from rationality and at the same time is 𝜙(𝛾,𝜮 ) = 0 if 0 < 𝛾 < ,𝜮 ≥ 𝜮 R ⎨ a crucial component in decision-making. The Alpha-Neutral CAPM that R ∗ { ( ) } 2 1 1 1 we derive provides an intuitive and analytically simple explanation of ⎪1 − exp −𝜌 𝛾 − x 𝜮 x if 0 < 𝛾 < ,𝜮 < 𝜮 𝑖 R 𝑖 R ∗ 2 2 2 the abnormal returns left unexplained by the Sharpe (1964), Lintner (1965) and Mossin (1966) traditional CAPM and by many of the cur- (E) rently most accredited multifactor models by attributing the presence As demonstrated by Rocciolo et al. (2019), the optimal demand of these ‘‘anomalies’’ to the limited rationality of traders. function x is a solution to the optimization problem in Eq. (D), which The results we present, both on the performances of competing will be the same for all three possible functional forms assumed by models in Table 3 and on the spanning tests in Table 5, are consistent 𝜙(𝛾,𝜮 ) as in Eq. (E). Thus, we can solve the problem just for the case with the idea that the SMB, CMA and RMW factors are not necessary in in which ≤ 𝛾 < 1. By plugging the explicit form of 𝜙(𝛾,𝜮 ) in Eq. (D), order to explain the average returns of the test assets. Conversely, the we can rewrite the latter as parsimonious representation of the Alpha-Neutral CAPM, comprising [ ( ( ) )] 𝜌 − 2𝛾 + 1 just one risk factor augmented by the behavioural bias, seems sufficient ′ F 𝐹 𝑖 𝑖 ′ − exp −𝜌 (x (E[R]−R )+𝑊 (𝑡−1)𝑅 )+𝜌 x 𝜮 x (F) 𝑖 𝑖 𝑖 R 𝑖 𝑖 𝑖 to explain the variation in average returns. 2 Does this mean that all factors considered in the literature are just This has a first order condition with respect to the demand x of: imperfect proxies for an effect that is purely behavioural and are thus not necessary? The answer to this question is not straightforward. First, from the spanning tests, not all of the factors are perfectly explicable in Such a portfolio could be constructed by taking the (normalized) weight terms of just the behavioural bias characterized in terms of the degree of the slope coefficients of the Fama–French and Carhart factors in a regression of optimism that we introduce in this paper. A combination of the where the dependent variable is the behavioural factor. 14 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 Table 5 Factor spanning test. Factor spanning test summary: MKT is the return on the market portfolio in excess of the one-month Treasury bill rate, SMB (small minus big) is the size factor, HML (high minus low) is the value factor, UMD (up minus down) is the momentum factor, CMA (conservative minus aggressive) is the investment factor and RMW (robust minus weak) is the profitability factor. 𝑎 and 𝐴(𝑑 ) are respectively the time series 𝑓 𝑓 regressions’ intercepts and the Alpha-Neutral net average net intercepts. Finally, Cov(f,MKT) is the covariance between the market and the dependent variable. Panel A shows the slopes of the regressions which use six factors to explain the return of the seventh. Panel B shows the slopes of the regressions which use just the market and the behavioural factor to explain the returns on SMB, HML, UMD, CMA and RMW. Period: July 1963–December 2016, 642 months. 𝑎 𝐴(𝑑 ) MKT SMB HML UMD CMA RMW Cov(f,MKT) 𝑅 𝑓 𝑓 Panel A: Spanning test involving multiple factors MKT Coefficient 1.48 1.85 0.20 −0.09 −0.16 −0.69 −0.28 −0.62 0.30 Standard Error 0.17 0.19 0.05 0.08 0.04 0.11 0.07 0.08 𝑡-Statistic (8.60) (9.77) (3.83) (1.16) (4.23) (6.50) (3.89) (7.70) p-values 0.00 0.00 0.00 0.00 0.25 0.00 0.00 0.00 SMB Coefficient 0.33 0.32 0.13 0.07 0.04 −0.13 −0.43 0.41 0.16 Standard Error 0.12 0.27 0.03 0.06 0.03 0.08 0.05 0.25 𝑡-Statistic (2.79) (1.15) (4.43) (1.18) (1.28) (1.64) (8.35) (1.67) p-values 0.01 0.25 0.00 0.24 0.20 0.10 0.00 0.10 HML Coefficient 0.09 0.09 0.0 0.04 −0.12 0.98 0.16 −0.10 0.52 Standard Error 0.08 0.15 0.0 0.03 0.02 0.04 0.04 0.13 𝑡-Statistic (1.09) (0.60) (0.18) (1.31) (6.70) (23.29) (4.24) (0.76) p-values 0.27 0.55 0.86 0.19 0.00 0.00 0.00 0.45 UMD Coefficient 0.73 0.78 −0.12 0.06 −0.52 0.40 0.26 −3.17 0.11 Standard Error 0.17 0.82 0.04 0.06 0.08 0.12 0.08 0.81 𝑡-Statistic (4.42) (0.94) (3.03) (1.12) (6.62) (3.43) (3.34) (3.93) p-values 0.00 0.34 0.00 0.26 0.00 0.00 0.00 0.00 CMA Coefficient 0.03 0.04 −0.09 −0.03 0.47 0.05 −0.15 −0.59 0.56 Standard Error 0.06 0.18 0.01 0.02 0.02 0.01 0.03 0.17 𝑡-Statistic (1.87) (0.20) (7.00) (1.61) (23.64) (3.70) (5.80) (3.41) p-values 0.06 0.84 0.00 0.11 0.00 0.00 0.00 0.00 RMW Coefficient 0.08 −0.07 −0.08 −0.23 0.18 0.09 −0.34 −1.39 0.23 Standard Error 0.08 0.24 0.02 0.03 0.04 0.02 0.06 0.23 𝑡-Statistic (2.61) (0.30) (4.17) (8.54) (4.45) (4.38) (5.96) (6.07) p-values 0.01 0.76 0.00 0.00 0.00 0.00 0.00 0.00 𝑎 𝐴(𝑑 ) MKT Cov(f,MKT) 𝑅 𝑓 𝑓 Panel B: Spanning test involving just the market and the behavioural bias SMB Coefficient 0.19 0.18 0.19 0.32 0.08 Standard Error 0.12 0.27 0.03 0.26 𝑡-Statistic (1.63) (0.67) (7.07) (1.24) p-values 0.10 0.50 0.00 0.21 HML Coefficient 0.42 0.41 −0.16 −0.30 0.07 Standard Error 0.11 0.15 0.02 0.17 𝑡-Statistic (3.84) (2.75) (6.60) (1.73) p-values 0.00 0.01 0.00 0.09 UMD Coefficient 0.74 0.79 −0.13 −3.26 0.04 Standard Error 0.16 0.82 0.04 0.83 𝑡-Statistic (4.52) (0.96) (3.40) (3.91) p-values 0.00 0.34 0.00 0.00 CMA Coefficient 0.31 0.27 −0.17 −0.58 0.15 Standard Error 0.08 0.18 0.02 0.24 𝑡-Statistic (3.89) (1.46) (9.81) (2.45) p-values 0.00 0.15 0.00 0.01 RMW Coefficient 0.21 0.11 −0.11 −1.11 0.08 Standard Error 0.09 0.24 0.02 0.24 𝑡-Statistic (2.44) (0.46) (5.70) (4.60) p-values 0.02 0.65 0.00 0.00 𝜌 [(E[R] − R ) − (𝜌 − 2𝛾 + 1)𝜮 x ] In Eq. (G), the exponential term is always positive so that the latter 𝑖 𝑖 𝑖 R 𝑖 F reduces to a concave programming problem which is solved by ′ 𝐹 × exp − 𝜌 ((x (E[R] − R ) + 𝑊 (𝑡 − 1)𝑅 ) + 𝑖 𝑖 ( ) ) 𝜌 − 2𝛾 + 1 E[R] − R 𝑖 𝑖 −1 x = 𝜮 (H) +𝜌 x 𝜮 x ) = 0 (G) 𝑖 R 𝑖 R (𝜌 + 𝜅(𝛾 )) 𝑖 𝑖 which is the first result of Proposition 1 embodied in Eq. (7). 15 F. Rocciolo et al. International Review of Financial Analysis 82 (2022) 102143 𝑗 𝑀 COV(𝑅 ,𝑅 ) With regard to the result in Eq. (8), we start by computing the and eventually, by defining beta in the usual way as 𝛽 = , aggregate demand for risky assets as we end up with ( ) 𝑛 𝑛 𝑛 ∑ ∑ ∑ 𝐹 𝑀 𝐹 ∗ 𝑀 E[R] − R −1 −1 F −1 E[𝑅 ] − 𝑅 = 𝛼 + (E[𝑅 ] − 𝑅 )𝛽 + 𝜅(𝛾)COV(𝑅 , 𝑅 ) (N) 𝑗 𝑗 𝑗 x = x = 𝜮 = 𝜮 (E[R] − R ) (𝜌 + 𝜅(𝛾 )) (I) 𝑗 𝑖 𝑖 𝑖 R R (𝜌 + 𝜅(𝛾 )) 𝑖 𝑖 𝑖=1 𝑖=1 𝑖=1 which is the result in Proposition 2. −1 −1 By defining (𝜌 +𝜅(𝛾)) = (𝜌 +𝜅(𝛾 )) , where 𝜌 and 𝜅(𝛾) might 𝑖 𝑖 𝑖=1 Proof of Proposition 3 be interpreted as aggregate measures of the absolute risk aversion and of the distance from rationality respectively, we have that, by inverting For the Alpha-Neutral CAPM in Eq. (12), the time series regression the first order condition in (I) is: E[R] − R = (𝜌 + 𝜅(𝛾))𝜮 x (J) 𝑗 𝑗 𝐹 ∗ 𝑀 𝑅 − 𝑅 = 𝑎 + 𝑏 MKT + 𝑘 𝜎(𝑅 , 𝑅 ) + 𝑒 (O) 𝑗 𝑡 𝑗 𝑗,𝑡 𝑡 𝑡 𝑗 𝑡 𝑡 where 𝜮 x is the 𝑁 × 1 vector of covariances between each asset’s 𝑀 𝐹 where MKT = (𝑅 − 𝑅 ). return 𝑅 and the return on the ‘‘comprehensive’’ portfolio obtained 𝑡 𝑡 𝑡 The correct asset pricing model is through the aggregation of all the individual portfolios held by the 𝑛 agents. 𝐹 ∗ E [𝑅 ] − 𝑅 = E [𝛿 ] + 𝑏 𝜆 (P) 𝑡 𝑗 𝑡 𝑗 MKT In fact, 𝜮 x is given by where 𝛿 are the pricing errors, theoretically equal to zero, and 𝜆 is ∑ 𝑗 MKT ⎡ ⎤⎡ ⎤ 𝜎 𝜎 … 𝜎 𝑥 12 1𝑁 𝑖,1 the market factor premium. 1 𝑖=1 ⎢ ⎥⎢ ⎥ 𝜎 𝜎 … 𝜎 21 2𝑁 𝑖,2 𝑖=1 Contrasting the correct model (P) with the expected value of the ⎢ 2 ⎥⎢ ⎥ ⎢ . . . . . . . . . . . ⎥⎢ ⋮ ⎥ time series regression in (O) we have ⎢ ⎥⎢ 𝑛 ⎥ 𝜎 𝜎 … 𝜎 𝑖,𝑁 ⎣ 𝑛1 𝑛2 ⎦⎣ 𝑖=1 ⎦ ∗ 𝑀 E [𝛿 ] = E [𝑎 ] + 𝑏 (E [MKT ] − 𝜆 ) + 𝑘 E [𝜎(𝑅 , 𝑅 )] (Q) ∑ ∑ ∑ 𝑡 𝑗 𝑡 𝑗 𝑡 𝑡 MKT 𝑗 𝑡 𝑛 𝑛 𝑛 𝑗 𝑡 𝑡 ⎡ ⎤ 𝑥 𝜎 + 𝑥 𝜎 + ⋯ 𝑥 𝜎 𝑖,1 𝑖,2 12 𝑖,𝑛 1𝑛 𝑖=1 1 𝑖=1 𝑖=1 ∑ ∑ ∑ ⎢ 𝑛 𝑛 𝑛 ⎥ The market factor is represented by the excess returns on the 𝑥 𝜎 + 𝑥 𝜎 + ⋯ 𝑥 𝜎 𝑖=1 𝑖,1 12 𝑖=1 𝑖,2 𝑖=1 𝑖,𝑛 2𝑛 ⎢ 2 ⎥ market portfolio so that E[MKT ] = 𝜆 , while, by definition, E [𝑎 ] + ⎢ ⋮ ⎥ 𝑡 MKT 𝑡 𝑗 ∑ ∑ ∑ ⎢ 𝑛 𝑛 𝑛 ⎥ 𝑀 𝑘 E [𝜎(𝑅 , 𝑅 )] = E [𝑑 ]. 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