Popularity
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Source: IDS.
5. The Popularity Asset Pricing Model
The capital asset pricing model (CAPM) has been the dominant model of expected returns for more than 50 years. Chapter 2 noted that, despite the distinction of the theory, subsequent empirical research has established the existence of various premiums and anomalies that violate the CAPM. Perhaps the model’s biggest strength—expressing investor preferences solely in terms of risk—is also its biggest limitation. Investors care about many characteristics having little to do with risk that we consider to be dimensions of popularity . These features include such asset characteristics as liquidity, taxability, scalability, divisibility, controllability, transparency, and the components of sustainability—namely, environmental, social, and governance (ESG) factors.
This chapter addresses nonrisk characteristics in a CAPM-like framework based on the concept of popularity that we discussed in Chapters 1 –3 with an equilibrium model that we call the “popularity asset pricing model” (PAPM). In Chapter 3 , we introduced the concept of a market that is “beyond efficient,” in which information irrelevant to “fair” value is reflected in security prices as a result of investors’ behavioral preferences. In such a market, we consider prices to be “biased” (as opposed to fair). The PAPM is a model in which markets are beyond efficient and prices are biased.
The idea of including security characteristics and investor attitudes toward them is not new. As we discussed in Chapter 4 , Ibbotson, Diermeier, and Siegel (1984) presented a sketch for an equilibrium model based on characteristics and investors’ attitudes toward them in their New Equilibrium Theory (NET).
In Chapter 1 , we presented a two-part taxonomy of security characteristics that could potentially affect security prices. The two sets of characteristics are classical and behavioral . Under the classical heading, we further classified characteristics as risks or frictional . Classical models, such as the CAPM and arbitrage pricing theory, take only the risk characteristics into account. NET extends classical models by taking into account the frictional characteristics, which include taxes, trading costs, and divisibility. The PAPM extends NET by including behavioral characteristics.
The PAPM is relevant in the context of an individual security as well as in an asset allocation context involving allocations to such assets as stocks, corporate and municipal bonds, real estate, and so forth. The characteristics modeled in the PAPM can also represent many of the psychological desires and preferences that are portrayed in behavioral finance—for example, prospect theory, affect, sentiment, and attentiveness. Furthermore, in equity markets, the characteristics function like factor premiums, such as value versus growth, momentum versus reversal, size, quality, liquidity, and even volatility (because, in some instances, investors might even prefer riskier assets). 31 The discovery of these premiums has led to the development of indexes that are investable as “smart” or “strategic” exchange-traded funds. Because in the PAPM such premiums are the result of popularity effects, we identify them with the popularity premiums that we discussed in previous chapters as a way of formally modeling them.
The existence of premiums raises these questions:
Why do they exist? The existence of premiums, other than the overall equity risk premium, appears to be the “free lunch” that, according to the CAPM, should not exist.
Which investors are on the opposite side? If some investors are systematically beating the market, then for the market to clear, there must be investors systematically falling behind the market. These investors are the ones Robert Arnott (quoted in Rostad 2013 ) calls “willing losers” or, by extension, unknowing losers.
To answer these questions, we need a model that
contains premiums and
allows for some investors to hold portfolios tilted toward the premiums (who thus outperform the market) and some investors to tilt away from these premiums (who thus underperform the market).
In Chapter 3 , we discussed what such a model might look like. In that chapter, we focused on characteristics of securities that we call “dimensions of popularity.” The idea is that each security characteristic can be ranked along a popularity continuum. For example, highly liquid stocks are regarded as popular, whereas illiquid stocks are regarded as unpopular. Investors who have a strong demand for popularity hold securities that rank high on the popularity scales and are willing to give up return to do so. Investors who do not demand popularity and who believe that they will be compensated for holding unpopular stocks hold securities that rank low on the popularity scales and can expect to earn superior returns.
The key to understanding equilibrium pricing in securities is to recognize that securities have both risk and nonrisk characteristics. In the PAPM, investors are risk averse and diversify, as in the CAPM, but they also vary in their preferences toward the other characteristics that securities embody. Securities supply the various characteristics, and investors demand them to varying degrees. As we discussed in Chapter 1 , supply does not change as quickly as demand. Thus, the characteristics and, ultimately, the securities are priced according to the weighted average of investor preferences. The investors are proportionally weighted by their wealth but inversely weighted by their risk aversion.
We present the PAPM here by formalizing the ideas that we discussed in Chapter 4 but applying them more broadly. We do this by extending the CAPM to include classical and behavioral security characteristics that different investors regard differently, both positively and negatively. This process leads to an equilibrium in which:
The expected excess return on each security is a linear function of its beta and its multiple popularity loadings, which measure the popularity of the security based on its characteristics relative to those of the beta-adjusted market portfolio.
Each investor holds a different portfolio based on his or her attitudes toward security characteristics.
Investor preferences determine the prices of the securities.
The remainder of this chapter is organized as follows. First, we review the CAPM in detail to set the stage for the PAPM. Then, building off the presentation of the CAPM, we present the PAPM in detail. Finally, we present a numerical example to illustrate the differences between the CAPM and the PAPM.
Review of the CAPM
The CAPM makes the following assumptions:
Taxes, transaction costs, and other real-world considerations can be ignored.
All investors use mean–variance optimization (MVO) as described by Markowitz (1952 , 1959 , 1987 ) to select their portfolios.
All investors have the same forecasts; that is, they use the same capital market assumptions (expected returns, standard deviations, and correlations) when constructing their portfolios.
All investors can borrow and lend at the same risk-free rate without limit.
From these assumptions, the following conclusions emerge:
From among all possible portfolios of risky assets, the market portfolio (i.e., the capitalization-weighted combination of all risky assets in the market) maximizes the Sharpe ratio (the expected return in excess of the risk-free rate per unit of total risk). Hence, it is on the efficient frontier.
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.
The expected excess return of each security is proportional to its systematic risk with respect to the market portfolio (beta).
To state Assumption 2 formally, let
n = the number of risky securities in the market
= the n -element vector of expected excess returns
Ψ = the n × n variance–covariance matrix of returns to the risky securities
= the n -element vector of investor i ’s allocations to the risky securities 32
λi = the risk aversion parameter of investor i
Then, investor i ’s MVO problem is to maximize utility by portfolio selection:
To state Conclusion 2 formally, let
m = the number of investors
w i = the fraction of wealth held by investor i ; that is,
Aggregating across investors provides the market level of risk aversion and the market portfolio: 33
and
(The M subscript indicates aggregation to the market level.) As we show in Appendix B , each investor holds the market portfolio in proportion to the ratio of his or her risk tolerance (the reciprocal of risk aversion) to the wealth-weighted average risk tolerance:
In the standard CAPM, the net supply of the risk-free asset (cash) is zero, so . Thus, Equation 5.4 states that if investor i is less risk averse than the average investor, he or she borrows at the risk-free rate and levers the market portfolio. Conversely, if investor i is more risk averse than the average investor, he or she holds a combination of the risk-free asset (cash) and the market portfolio.
Figure 5.1 illustrates Conclusions 1 and 2 graphically. It shows that in the CAPM, the market portfolio is on the MVO efficient frontier. Its location is the point of tangency between the capital market line and the efficient frontier. The capital market line is the line of tangency that emanates from the risk-free rate on the vertical axis. As Figure 5.1 shows, not only is the market portfolio on the capital market line but so are the portfolios of all investors. Investors who take more risk than the market portfolio have portfolios above it, indicating that they hold levered positions in it. Investors who take less risk than the market portfolio have portfolios below it, indicating that they hold delevered positions in it.
Figure 5.1. Equilibrium in the CAPM
To state Conclusion 3 formally, define the expected excess return on the market portfolio as
Define the variance of the market portfolio as
The familiar CAPM equation for expected excess returns can be written as
where
In other words, the expected excess return on each security is the product of its systematic risk with respect to the market portfolio (beta) and the expected excess return of the market portfolio.
Single-Period Valuation in the CAPM. So far, we have given the conventional presentation of the CAPM as a model of portfolio construction and expected return. The CAPM is also, however, a single-period valuation model (which is why it is called an “asset pricing” model). Let
v j = the current market value of security j
= the exogenous random end-of-period total value of security j
r f = the risk-free rate
The current value of each security j can then be written as
The expected end-of-period value of each security, , is, in a sense, the fundamental of the security that the market prices. 34 If all securities had the same systematic risk (beta), the denominator of Equation 5.9 would be the same for all securities and all market values would be proportional to this fundamental. But not all securities have the same systematic risk, so the market value of a security depends both on its fundamental and on its risk.
Equation 5.9 corresponds to the most common way that valuation is carried out—namely, by discounting the expected value of future cash flows (the numerator) by a risk-adjusted discount rate (the denominator). Another way to approach valuation, however, is to risk-adjust expected future flows (the fundamental) and then discount the risk-adjusted value by the risk-free rate. To demonstrate, let
= the random end-of-period value of the market as a whole
v M = the value of the market as whole
By definition,
Let denote the vector of random exogenous end-of-period total security values. Then, the distribution of constitutes the real economy. Denote the variance–covariance matrix of as Ω . The systematic risk of an individual end-of-period security value with respect to total economic output is the covariance of the economic output of j with that of all other economic output divided by the variance of total economic output:
The systematic risk of the value of economic output, γj , is related to the systematic risk of return, βj , as follows:
As we show in Appendix B , the value of security j can be expressed as:
Although Equation 5.13 yields the same results as Equation 5.9 , as we will show, the valuation equation that we derive for the PAPM is a generalization of Equation 5.9 .
The Popularity Asset Pricing Model
The PAPM is a generalization of the CAPM in which securities have characteristics other than risk and expected return that investors are concerned about. Its assumptions are as follows:
Each security has a bundle of characteristics.
Investors have preferences regarding these characteristics in addition to their preferences regarding risk and expected return.
All investors use a generalized form of MVO that incorporates their preferences regarding security characteristics.
All investors have the same forecasts; that is, they hold the same capital market assumptions (expected returns, standard deviations, and correlations).
All investors agree on what the characteristics of the securities are.
All investors can borrow and lend at the same risk-free rate without limit.
The conclusions of the PAPM are as follows:
The market portfolio does not maximize the Sharpe ratio among all portfolios of risk assets.
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. Portfolio optimization is required to find the overall investor-specific portfolio.
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 beta-adjusted market portfolio. The popularity loadings are multiplied by the popularity premiums, which are aggregations of the preferences of the investors regarding the characteristics. In this way, the market aggregates investor preferences in determining the influence of security characteristics on the expected returns and prices of the securities.
Note that the conclusions of the PAPM are nearly the exact opposite of those of the CAPM. Additionally, Conclusion 2 is much more consistent with observed investor portfolios.
Figure 5.2 illustrates Conclusions 1 and 2. The market portfolio is not on the Sharpe ratio–maximizing tangent line. Neither are the portfolios of Investors 1 and 2, as is the case in the CAPM. Investor 3’s portfolio, however, is on the tangent line. This investor is risk averse but has no other preferences for security characteristics and, therefore, holds an efficient portfolio. We present the specifics of this example in the next section.
To state Assumptions 2–4 formally, let
p = the number of characteristics (besides risk and expected excess return)
C = n × p matrix of characteristics of the securities (or asset classes)
= p -element vector of investor i ’s attitudes toward the characteristics
(The elements can be positive or negative.)
Figure 5.2. Equilibrium under the PAPM
Investor i ’s problem is the following:
which is the utility-maximization problem introduced in Chapter 4 to formalize the main idea of NET. Here, however, the nonrisk characteristics are not only the costs that rational investors care about, such as liquidity, but also characteristics that investors desire for irrational reasons, such as the popularity of the companies that issue the stock. In the NET framework, where all of the nonrisk characteristics are costs, the elements of are all negative. But, in the more general PAPM, the elements of that are on characteristics that the investor likes are positive.
This extension of MVO is similar to the formulation in Cooper, Evnine, Finkelman, Huntington, and Lynch (2016) . The main difference is in interpretation. In Cooper et al., the nonrisk characteristics are expected social-impact metrics (ESG factors), whereas in the PAPM, the nonrisk characteristics can include t
hese factors but can also include any number of other security characteristics that investors might care about.
Note that because the preferences for characteristics enter the utility function in parallel with expected returns, they should be in the same units. For example, if φ11 = 5%, Investor 1 would be indifferent between a 100% allocation to a security with exposure of 1.0 to characteristic 1 and a 5% increase in expected return.
The solution to the maximization problem shown in Equation 5.14 is
As we show in Appendix C , each investor’s portfolio can be expressed in terms of the market portfolio and the investor’s attitudes toward security characteristics:
where denotes the vector of the aggregation of investor attitudes toward the characteristics:
For reasons that will become apparent, we call the vector of popularity premiums .
Equation 5.16 shows how each investor’s portfolio differs from the market portfolio based on (1) the investor’s attitude toward risk and (2) his or her attitude toward security characteristics.
In Appendix C , we show that expected excess returns can be written as
where .
Equation 5.18 looks like a multifactor asset pricing model but with popularity premiums rather than risk premiums. Let