The PAPM with Heterogeneous Preferences and Expectations

The Popularity Asset Pricing Model (PAPM) has similar assumptions to the Capital Asset Pricing Model (CAPM), but some different conclusions. In the CAPM, the expected excess return of each security is proportional to a single market systematic risk (beta), while in the PAPM, the expected excess return of each security is linearly related to multiple risk and non-risk characteristics (exposures). Relaxing the assumption of homogeneous forecasts and allowing for heterogeneous forecasts changes most of the CAPM’s conclusions but none of the PAPM’s conclusions other than introducing another reason that investors hold unique portfolios. With heterogeneous expectations, many investors believe they have skill, but some have more skill than others. These differences in forecasting skill, together with differences in risk and non-risk preferences lead to a variety of different types of market participants. We illustrate the impact of both heterogeneous expectations and popularity preferences on expected returns and portfolios through a series of highly stylized numerical examples. The PAPM with heterogeneous expectations can be viewed as an extension of the CAPM, but it may be more useful to view the CAPM as special case of the PAPM. By incorporating investor preferences and allowing for heterogeneous forecasts, the PAPM takes two major steps towards asset pricing in the real world. Electronic copy available at: https://ssrn.com/abstract=3451554 2 Introduction In Popularity: A Bridge Between Classical and Behavioral Finance, by Ibbotson, Idzorek, Kaplan, and Xiong (2018), henceforth IIKX, Chapter 5 presents a formal version of the Popularity Asset Pricing Model (PAPM) as a direct extension of the Capital Asset Pricing Model (CAPM), but introducing popularity preferences. The chapter starts by presenting the assumptions and conclusions of the CAPM, and then follows the same formal process in the development of the PAPM. An explicit desire was to start with the CAPM, and then to extend the CAPM to the PAPM to account for popularity preferences. As such, the PAPM of IIKX inherited most of the assumptions of the CAPM. One of those assumptions is that market participants share the same (homogeneous) forecasts or capital market assumptions. In this article we examine the implications of relaxing the assumption of homogeneous forecasts, arriving at a more generalized and realistic version of the PAPM in which rather than viewing the PAPM as an extension of the CAPM, the CAPM should be viewed as an elegant special case of the PAPM. Under the PAPM, some investors may knowingly and willingly give up some return in order to hold portfolios that reflect their unique preferences for certain characteristics, such as more liquidity, a preference for dividends, or a low carbon footprint. IIKX state, “[t]he investors with strong preferences might be called ‘the willing losers’. ”1 Aggregate investor preferences drive demand and thus asset prices and result in preference driven popularity premiums. The assumption of homogeneous forecasts allows for those without strong preferences to knowingly profit from those with strong preferences who knowingly sacrifice return for their preferences, but it does not allow for more skilled investors to profit from less skilled investors (beyond pure preferences). Heterogeneous forecasts give rise to additional types of market participants, beyond the knowing winner/losers caused by differing preferences. Market participants with more relative skills (with similar preferences) are expected to outperform those with less relative skill. Thus, heterogeneous expectations give us an additional reason to have winners/losers in the PAPM. We begin with a review of the assumptions and conclusions of the CAPM and PAPM with homogeneous expectations, and then examine the implications of heterogeneous forecasts. Next we provide a select literature review and contrast the PAPM with several important areas of asset pricing research prior to proceeding with a formal presentation of the CAPM and PAPM with heterogeneous expectations. Finally, we present an illustrative example in which we compare/contrast four models: the CAPM with homogeneous expectations, the CAPM with heterogeneous expectations, PAPM with homogeneous expectations, and the PAPM with heterogeneous expectations. 1 The phrase “willing loser” comes from a quote of Robert Arnott in Rostad (2013). Electronic copy available at: https://ssrn.com/abstract=3451554 3 CAPM and PAPM Assumptions and Conclusions with Homogeneous Forecasts With some rearranging relative to IIKX for the purposes of comparing assumptions, Exhibit 1 presents the assumptions of the CAPM and PAPM side-by-side. Sequentially, the first three assumptions are the same or nearly the same across the two models. The first assumption in Exhibit 1 is the key assumption that we will be examining: homogeneous forecasts. The PAPM is less restrictive and differs from the CAPM by assuming that security characteristics unrelated to risk matter and are important to investors with heterogeneous preferences that go beyond risk. Exhibit 1: Model Assumptions CAPM – Homogeneous PAPM – Homogeneous All investors have the same forecasts; that is, the same capital market assumptions (expected returns, standard deviations, and correlations). = All investors have the same forecasts; that is, the same capital market assumptions (expected returns, standard deviations, and correlations). All investors can borrow and lend at the same risk-free rate without limit. = All investors can borrow and lend at the same risk-free rate without limit. All investors use Mean-Variance Optimization (MVO) as described by Markowitz (1952, 1959, 1987) to select their portfolios. ≈ All investors use a generalized form of Mean-Variance Optimization (MVO) that linearly incorporates their preferences regarding security characteristics. Taxes, transaction costs, and other realworld considerations can be ignored. ≠ Investors have preferences regarding these characteristics in addition to their preferences regarding risk and expected return. N/A ≠ All investors agree on what the characteristics of the securities are. Moving to the conclusions, in Exhibit 2 we once again attempt to arrange what one might think of as corresponding model conclusions (even though the conclusions themselves are very different between the two models). Electronic copy available at: https://ssrn.com/abstract=3451554 4 Exhibit 2: Model Conclusions CAPM – Homogeneous PAPM – Homogeneous From among all possible portfolios of risky assets, the market portfolio (that is, the cap-weighted combination of all risky assets in the market) maximizes the Sharpe ratio; hence, it is on the efficient frontier. ≠ The market portfolio does not maximize the Sharpe ratio among all portfolios of risky assets. Each investor combines the market portfolio with long or short positions in the risk-free asset (cash). Hence, investors do not actually need to perform MVO to construct optimal portfolios. ≠ Each investor forms a customized portfolio of the risky assets that reflects his or her attitudes toward each security characteristic. This portfolio is combined with long or short positions in the risk-free asset. MVO is required to construct the overall investor-specific portfolio. The expected excess return of each security is proportional to its systematic risk with respect to the market portfolio (beta). ≠ The expected excess return of each security is a linear function of its beta and its popularity loadings, which measure the popularity of the security based on its characteristics relative to those of the betaadjusted market portfolio. Despite starting with very similar assumptions in Exhibit 1, we see in Exhibit 2 that the conclusions of the CAPM and PAPM are quite different from one another. The market portfolio is / is not the Sharpe maximizing portfolio; mean-variance optimization is not / is necessary to arrive at an investor’s ideal portfolio. In the CAPM, the expected excess return of each security is proportional to a single market systematic risk (beta), while in the PAPM, the expected excess return of each security is linearly related to multiple risk and non-risk characteristics (exposures). Rather, in the equation for the expected return of a given security, in addition to the beta term, there are terms consisting of the security’s popularity loadings multiplied by popularity premiums. The popularity premiums are aggregations of the preferences of the investors regarding the characteristics that investors care about besides risk. In this way, the market aggregates investor preferences in determining the influence of security characteristics on the expected returns and prices of the securities. The equation for expected return in the PAPM is the foundation for our primary question and model extension/generalization in this paper. What happens to the conclusions of the CAPM and PAPM if we relax the assumption that all investors share the same forecasts (while maintaining the other assumptions)? Electronic copy available at: https://ssrn.com/abstract=3451554 5 The quick answer is that the CAPM’s conclusions regarding individual investor portfolios do not hold while conclusions regarding security expected returns do on average. For the PAPM, the conclusions largely continue to hold, although investor portfolios differ from one another for an additional reason – heterogenous forecasts. As we shall see, it also becomes evident, that rather than viewing the PAPM as an extension of the CAPM as framed in IIKX, the PAPM with heterogeneous expectations is a more generalized, realistic asset pricing model that allows for unique characteristic preferences and that the CAPM with or without heterogeneous expectations represents more restrictive special cases of the more encompassing PAPM. The Implication of Heterogeneous Forecasts In a CAPM-world, if market participants all have different