so we can write
Separating out the risk-free rate, we can write Equation 5.20 as an equation for the expected total return of security j :
Equation 5.21 is of the same form as the Popularity formula for expected return in Figure 3.3 that we postulated would hold in an equilibrium in which investors care about nonrisk characteristics. The main difference is that in the formula in Figure 3.3 , we have proxies for such characteristics as market cap (size) and market-based measures of value, whereas in Equation 5.21 we have the actual characteristics.
We call δjk security j ’s “popularity loading” on characteristic k . It is positive if security j ’s exposure to characteristic k is less than that of the beta-adjusted market portfolio and negative if the reverse is true. In this way, a popularity loading of a security is positive for a given characteristic if the security is unpopular with respect to the characteristic and negative if it is popular.
As a special case, the net attitude toward characteristics could be zero , so the CAPM equation for expected excess returns would still prevail and the market portfolio would be mean–variance efficient. Even in that case, however, each investor still tilts his or her portfolio toward the preferred characteristics and away from the ones disliked, as described by Equation 5.16 .
Valuation under the PAPM. Just as the equation for expected excess returns in the CAPM can be used to derive a one-period valuation formula (Equation 5.13 ), so Equation 5.18 can be used to derive a valuation formula for the PAPM.
In Appendix C , we derive the following PAPM valuation formula:
Equation 5.22 shows the respective roles that systematic risk (as measured by γj ) and nonrisk characteristics play in determining the market value of a security when the PAPM is used. As can be seen from the numerator in Equation 5.22 , systematic risk reduces the value of the security, much in the same way that it does in the CAPM (see Equation B.22 in Appendix B ). Systematic risk is part of a term deducted from the security’s fundamental, . In contrast, the nonrisk characteristics appear in the denominator where, multiplied by the popularity premiums, they form deductions from the risk-free rate. In this way, the market value of a security depends on both its risk and nonrisk characteristics.
A Numerical Example
To create Figure 5.1 and Figure 5.2 , we used an example with five securities and three investors. Table 5.1 presents the assumptions regarding the joint distribution of the end-of-period values of five securities. This is the model of the real economy in our example. To keep the example simple, we assume that there is one characteristic that investors care about, called popularity, which we also show in Table 5.1 .
Table 5.2 presents the assumptions regarding the investors. Note that Investor 3 has a popularity preference of zero; she is indifferent to whether a security is popular or not. This investor’s portfolio is on the tangent line in Figure 5.2 .
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Table 5.1. Assumptions about Popularity and the Real Economy for Five Hypothetical Securities
Security
Popularity
Expected Value ($)
Standard Deviation ($)
Correlation of End-of-Period Value with:
A
B
C
D
E
A
0.50
10
1.3
1.0
B
0.25
8
1.6
0.4
1.0
C
0.00
6
1.2
0.3
0.4
1.0
D
–0.25
4
1.0
0.2
0.3
0.4
1.0
E
–0.50
1
0.3
0.1
0.2
0.3
0.4
1.0
Table 5.2. Assumptions Regarding Investors
Description
Investor 1
Investor 2
Investor 3
Market
Fraction of market $ wealth (w j )
60%
30%
10%
100%
Risk aversion (λi )
4.00
4.00
1.50
3.43
Popularity preference (φi )a
5.0%
7.5%
0.0%
4.5%
a We scaled the popularity preferences in this example to be in the same units as expected returns.
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In addition to the assumptions presented in Tables 5.1 and 5.2 , we assume that the risk-free rate is 2%.
We solved the model under the assumptions of the CAPM (all popularity preferences set to zero) and under the assumptions of the PAPM by using the techniques described in Appendixes B and C , respectively. Table 5.3 and Table 5.4 present the results.
In Chapter 3 , we introduced the concept of a market that is “beyond efficient” in which prices are “biased,” as opposed to “fair” in an “efficient market.” With this numerical example of the CAPM (an efficient market model) and the PAPM (a model in which the market is beyond efficient), we can measure the pricing biases. In the last column of Table 5.3 , we show the percentage difference between the PAPM and CAPM value of each security and of the market as a whole. These percentages are the pricing biases. As expected, the prices of the popular securities are biased up and the prices of the unpopular securities are biased down relative to their fair values under the CAPM.
In Figure 5.3 , we show the popularity characteristic versus the pricing biases shown in the last column of Table 5.3 . Figure 5.3 reveals that not only is there a positive relationship between popularity and pricing bias but also the relationship is nearly linear.
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Table 5.3. Expected Returns and Valuations under the CAPM and the PAPM
Security
Expected Return
Value ($)
Pricing Bias
CAPM
PAPM
CAPM
PAPM
A
6.23%
3.85%
9.41
9.63
2.29%
B
10.33
9.03
7.25
7.34
1.19
C
9.62
9.54
5.47
5.48
0.07
D
10.25
11.38
3.63
3.59
–1.02
E
8.28
10.61
0.92
0.90
–2.10
Market
8.66
7.65
26.69
26.94
0.94
Table 5.4. Investor and Market Portfolios under the CAPM and the PAPM
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Figure 5.3. The Popularity Characteristic vs. Pricing Bias
These results reveal two striking features. First, from Table 5.3 , we see that. although values of the securities are similar in both models, the expected returns are quite different. The reason is that expected returns are highly sensitive to changes in value because of their inversely proportional relationship:
Because , the sensitivity is high. Figure 5.4 illustrates this sensitivity by showing how the expected return of Security E changes as the value of the security changes. (The curve appears to be linear rather than hyperbolic because the figure plots only a tiny part of the curve.)
The second striking feature is the radical change that Investor 3 undergoes in moving from the CAPM world to the PAPM world. As you can see from Table 5.2 , Investor 3 pays no attention to
the nonrisk characteristic and, therefore, focuses entirely on risk and expected return. For this reason, Investor 3’s portfolio is on the tangent line in Figure 5.2 . Because Investor 3 is much less risk averse than Investors 1 and 2, in the scenarios of both the CAPM and the PAPM, she takes on a lot of leverage. But the levered portfolios are quite different. In the CAPM, she levers the market portfolio. But in the PAPM, she holds a levered position in the Sharpe-ratio-maximizing portfolio, shown in Table 5.5 , which is short in the most popular security, Security A. As Table 5.4 shows, Security A is attractive to Investors 1 and 2 because of their preferences for popular stocks, but Investor 3, who has no preference regarding popularity, takes advantage of their preferences by shorting it. Thus, she follows a “risk arbitrage strategy” by going short the most popular security and going long the less popular ones. 35
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Figure 5.4. Relationship between the Value and Expected Return of Unpopular Security E
Table 5.5. The Portfolio with the Maximum Sharpe Ratio under the CAPM and the PAPM
Security/Statistic
CAPM
PAPM
A
35%
–11%
B
27%
37%
C
21%
34%
D
14%
26%
E
3%
13%
Expected return
8.66%
10.58%
Standard deviation
13.93%
19.03%
Sharpe ratio
0.48
0.45
This example illustrates how in the PAPM, where investors hold custom portfolios based on how much they care about (or do not care about) popularity characteristics, investors can be thought of as forming clienteles for the dimensions of popularity. For example, Investor 2, who has the strongest preference for the popularity characteristic, has 55% of his portfolio in the most popular security (Security A); Investor 3, who has no preference for the popularity characteristic, holds substantially more of the least popular security (Security E) than the other investors and the market as a whole.
Table 5.6. Expected Return Equations under the CAPM and the PAPM
Finally, Table 5.6 compares the equations for expected returns under the CAPM and under the PAPM. Note that the most popular security (A), has a negative popularity loading. Also note that the three least popular securities (C, D, and E) have positive popularity loadings, so these securities can be categorized as popular. In addition, note that under the PAPM, differences in popularity loadings lead to differences in expected returns larger than would be expected based only on the differences in betas. For example, Security E has a higher expected return than Security B (10.61% vs. 9.03%), even though Security E has a much lower beta than Security B (0.97 vs. 1.25). 36 This result is consistent with many empirical findings that differences in betas do not explain differences in returns, and it demonstrates how the theory of popularity can explain this phenomenon.
Conclusion
Equilibrium asset pricing models, such as the CAPM, predict that expected returns on securities will be linear functions of systematic risk factors. A large body of empirical evidence suggests, however, that premiums are related to characteristics not related to risk. Hence, we need a new model that takes nonrisk characteristics into account. In this chapter, we presented such a model, the popularity asset pricing model.
We formed the PAPM by extending the CAPM to include preferences for nonrisk security characteristics in investor objective functions. In the PAPM, an equilibrium emerges in which the expected excess return of each security is a linear function of its systematic risk with respect to the market portfolio (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. As we illustrated, differences in popularity loadings can cause differences in expected returns greater than would be expected if we consider only differences in betas. The coefficients on the popularity loadings—the popularity premiums—are the aggregated attitudes of investors toward the nonrisk security characteristics. Furthermore, the market portfolio does not maximize the Sharpe ratio; it is merely the aggregation across investors of each investor’s customized portfolio.
We illustrated that an investor who has only risk aversion preferences can benefit by loading up on the less popular securities. This investor can also use leverage and potentially short the most popular securities. But no risk-free arbitrage exists in the PAPM framework. The investor, or clientele of similar investors, can influence market prices but not remove the effect of popularity. The market is made up of all its participants, with each of them affecting prices and expected returns.
These conclusions have important practical implications. First, when estimating the equity cost of capital, adjustments need to be made for the characteristics of the security in question. Second, the conclusions imply that by focusing only on expected return and risk, an investor may be able to create portfolios that are more efficient than market-weighted indexes. Thus, one can profit by trading against investors who take into account security characteristics other than risk.
The approach that we took in constructing the PAPM can be extended to take into account other types of heterogeneity among investors. For example, investors could have heterogeneous views about the expected value of the real economic output associated with each security, much as in Lintner (1969) . In general, deriving the equilibrium of a model with investors who are heterogeneous in different respects should be possible.
Appendix B. Formal Presentation of the CAPM
Investor i ’s problem is as follows:
where
= 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
λi = the risk aversion parameter of investor i
From the first-order condition, we have
Solving for , we have
Let m be the number of investors and w i be the fraction of wealth held by investor i ; . Aggregating across investors, we have the market level of risk aversion and the market portfolio:
and
where the M subscript indicates the market.
Aggregating Equation B.3 across investors, we have
So,
From Equations B.3 and B.6 , we can see that each investor holds the market portfolio in proportion to the ratio of the wealth-weighted average risk aversion to his or her risk aversion:
In the standard CAPM, the net supply of the risk-free asset is zero, so . Therefore, Equation B.8 tells us that if investor i is less risk averse that 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, she or he holds a combination of the risk-free asset (cash) and the market portfolio.
Expected Excess Returns under the CAPM. The expected excess return on the market portfolio is
Hence, multiplying Equation B.7 through by yields
where . This is the variance of the market portfolio.
From Equation B.10 , it follows that
Substituting the right-hand side of Equation B.11 for λM in Equation B.7 and rearranging terms yields the familiar CAPM equation for expected excess returns:
where
Valuation under the CAPM. Because the CAPM is a one-period model, the value of each security j can be written as
where
v j = the total market value of security j
= the random exogenous end-of-period total value of security j
r f = the risk-free rate
Let be the random end-of-period value of the market as a whole and v M be the value of the market as whole. Then, by definition
and
The realized total return on security j is
Let denote the vector of random end-of-period total security values. The distribution of constitutes the real economy. Denote the variance–covariance matrix of as Ω . From the definition of Ω and Equation B.15 , it follows that the jq element of the variance–covariance matrix of returns, Ψ , can be written as:
So, the formula for βj can be rewritten as follows:
where
Substituting the final term in Equation B.20 for βj in Equation B.14 , rearranging terms, and simplifying yields the following equation for the total value of security j
The value of the market as a whole is
Substituting the right-hand side of Equation B.23 for v M in Equation B.22 yields:
Solving the CAPM. Equation B.24 states the values of the risky securities in terms of the underlying economic variables , the market risk premium (μM ), and the risk-free rate (r f ). From these values, we can derive all of the other variables in the CAPM by using the earlier equations. We take the risk-free rate as given; we still need to solve for the market risk premium.
The market risk premium depends on the market risk aversion. To see exactly how, first consider what Equation B.21 implies about the variance of return on the market portfolio, :
Substituting the right-hand side of Equation B.23 for v M in Equation B.25 yields
where
Substituting the right-hand side of Equation B.10 for in Equation B.26 and rearranging terms yields a quadratic equation for μM :
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