The relevant solution is as follows:
Figure B.1. Relationship between Market Risk Aversion and the Market Risk Premium under the CAPM
Equation B.29 is, in effect, the solution to the CAPM.
Figure B.1 is a plot showing the relationship between market risk aversion (λM ) and the market risk premium (μM ), taking the risk-free rate (r f ) and the parameters of the distribution of total market end-of-period value (κ) as given. The relationship is positive, so the higher the higher the level of market risk aversion, the higher the market risk premium.
Appendix C. Formal Presentation of the PAPM
Let
p = the number of popularity characteristics
C = n × p matrix of characteristics of the securities
= p -element vector of investor i ’s attitudes toward the characteristics (The elements can be positive or negative.)
Investor i ’s problem is as follows:
From the first-order condition, we have
The solution is
Aggregating Equation C.3 across investors, we have
where
For reasons that will become apparent, we call the vector of popularity premiums .
From Equations C.3 and C.4 , we derive an equation for the portfolio decision of each investor relative to the market portfolio:
Solving Equation C.4 for yields
Multiplying Equation C.7 through by yields
where . This is the vector of characteristics of the market portfolio.
From Equation C.8 , it follows that
Substituting the right-hand side of Equation C.9 for λM in Equation C.7 and rearranging terms yields the generalization of the CAPM equation for expected excess returns:
Equation C.10 looks like a multifactor asset pricing model but with the popularity premiums rather than risk premiums. Let
so we can write
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 popularity characteristics could be zero , so the CAPM equation for expected excess returns would still prevail and the market portfolio would be mean–variance efficient. But even in that case, each investor still tilts his or her portfolio toward the preferred characteristics and away from the disliked ones, as described by Equation C.6 .
Valuation in the PAPM. Just as the equation for expected excess returns in the CAPM can be used to derive a one-period valuation formula (Equation B.20 ), Equation C.10 can be used to derive a valuation formula in the PAPM.
To accomplish this derivation, first we write Equation C.10 for a single security as follows:
where is the vector formed from the j th row of C .
Equations B.17 and B.20 hold under the PAPM just as they do for the CAPM. From them, we have
where γj is as defined in Equation B.21 .
The value of security j can be written as
Substituting the right-hand side of Equation C.13 for μj in Equation C.15 , rearranging terms, and solving for v j yields the valuation equation presented as Equation 5.22 in the main text:
Solving the PAPM. Unlike the CAPM, the PAPM has no closed-form solution. Instead, to solve it, we need to solve a system of nonlinear equations.
Let denote the n -element vector-valued function that we are seeking to set to by finding the value of the vector of security values, , that does so. That is, we seek the solution to
Once the solution is found, all of the values of the variables of the model can be derived from the values of and the preceding equations in this appendix.
The values of are determined by making the following set of calculations:
and
6. New Empirical Evidence for Popularity
It is one thing to look in the past at the well-known premiums and anomalies in connection with a proposed asset pricing model and to find an after-the-fact explanation that seems to hold. It is another thing to see whether the predictions of a new model hold for security characteristics that have not been previously tested in the empirical asset pricing literature. In this chapter, we test the relationship between returns and various measures of characteristics that investors should nearly universally like or dislike. Broadly speaking, these measures can be divided into those that pertain to characteristics of companies and those that pertain to characteristics of the securities that they issue.
In the first section of this chapter, we analyze data on company characteristics that could make a company popular or unpopular. For each characteristic, we see if the stocks of the least popular companies outperform those of the most popular companies. We look at three characteristics that, to our knowledge, have not previously been examined in the literature: brand value, competitive advantage, and company reputation.
In the second section, we look at two stock-level characteristics: severe tail risk, as represented by low or negative coskewness with a market index (a nearly uniformly unpopular characteristic), and lottery-like payoffs (a nearly uniformly popular characteristic).
For each popularity measure, we consider four portfolios formed by dividing the universe of stocks into equally populated quartiles such that Quartile 1 contains the most popular stocks and Quartile 4 contains the least popular stocks. We also follow this convention, where possible, in Chapter 7 . Both here and in Chapter 7 , our empirical analysis is largely a comparison of the historical performance of the quartile portfolios formed using each popularity measure.
Popular Company Characteristics
Although a single mutually agreed-upon best measure of a security’s popularity does not exist, we have identified several previously unstudied characteristics that could serve as proxies for dimensions of popularity. These traits include the value of a brand, the degree to which a company is estimated to have a sustainable competitive advantage, and the reputation of the company. We use the following measures of these dimensions of popularity:
Interbrand’s annual “Best Global Brands” report—On an annual basis, Interbrand publishes a list of the 100 brands with the highest estimated “brand value.” We tested whether significant performance differences exist among the evolving top 100 brands.
Morningstar’s economic moat ratings—Morningstar’s equity analysts evaluate a number of factors related to a company’s relative sustainable competitive advantage (considered a moat to deter competition), including network effect, intangible assets, cost advantage, switching costs, and efficient scale. On the basis of this analysis, they classify each company as having (1) a wide moat, (2) a narrow moat, or (3) no moat. A sustainable competitive advantage is an example of a characteristic that investors would nearly uniformly agree is good.
Nielsen’s Harris Poll reputation quotient (see Harris Poll 2015 )—The Harris Poll reputation quotient measures the reputations of companies in the United States in which consumers rate corporations by 20 attributes that are categorized into six dimensions, which ultimately form the reputation quotient. We believe the reputation quotient aligns well with characteristics that investors seek and thus can serve as a proxy for a dimension of popularity. Our analysis is similar to that of Statman, Fisher, and Anginer (2008) , who studied Fortune ’s most admired companies, although the Fortune rankings are based on the opinions of senior executives and analysts rather than general consumers.
In the following sections, we present an analysis of the relationship between each of these measures of popularity and returns.
Brand Value. Interbrand was founded in 1974 and is one of the world’s largest branding consultancies. Starting in 2000, Interbrand began publishing an annual “Best Global Brands” report. 37 This report identifies and ranks the top global brands b
ased on a proprietary methodology for estimating the net present value (NPV) of a company’s earnings related to brand value. 38
Interbrand’s methodology combines financial, demand, and competitor analyses to estimate the NPV of earnings related to brand value. To be included in the Interbrand study, a company must meet a number of criteria; namely, 30% or more of revenue must come from outside the home region and the company must have a presence in at least three continents, have publicly available financials, and have the expectation of positive long-term economic profits.
To estimate the value of a brand, Interbrand starts by estimating the economic profit of the company in question. Economic profit is then multiplied by what Interbrand calls the “role of brand” measure, which attempts to identify the portion of the buying decision attributable to brand. According to the methodology description, the “Role of Brand reflects the portion of demand for a branded retailer that exceeds what the demand would be for the same offering if it were unbranded.” Multiplying the estimated economic value by the role of brand leads to what Interbrand calls “brand earnings.” To determine the brand value, brand earnings are discounted by a brand-specific discount rate, by which Interbrand evaluates brand strength along 10 dimensions. Although not a perfect measure of popularity, this measure of brand value captures the power of the brand and is thus strongly influenced by brand popularity.
Other researchers have not interpreted brand value as a proxy for popularity, but they have found links between brand value and stock returns, although not in the direction predicted by our popularity theory. For example, using 1,204 brand value estimates for 1991–1996, Barth, Clement, Foster, and Kasznik (1998) found brand value to be positively related to stock prices and returns. Madden, Fehle, and Fournier (2006) and Fehle, Fournier, Madden, and Shrider (2008) , after using the three-factor Fama–French model to adjust for risk plus a momentum factor, found that stocks associated with strong brands as measured by Interbrand statistically and economically outperformed.
These observations appear to be inconsistent with the popularity hypothesis, which predicts that stocks with strong brands are popular so they should underperform. The conclusions in Fehle et al. (2008) were drawn from a small sample size (only 111 stocks) and a short period that corresponded with the dot-com euphoria of 1994–2000; thus, the findings may not be robust over time. The August 1994 to December 2000 measurement period study by Fehle et al. could be considered the time of a unique inflating bubble, in which popular stocks became even more popular, with growth and large-capitalization stocks dramatically outperforming. Note also that Portfolio 3 in Table 2 of Fehle et al. was rebalanced annually and each year contained only the companies on the most recent Interbrand list. Hence, it is more relevant than Portfolios 1 and 2. The results for Portfolio 3 are not statistically significant, however, whereas the results for Portfolios 1 and 2 are. Thus, we find their conclusions unconvincing.
We took a different approach from Fehle et al. (2008) . Instead of grouping all stocks in the Interbrand list, we studied the cross-sectional performance differences among all stocks on that list. Our cross-sectional analyses allowed us to study the impact of changes in brand value over time, and we show that within the Interbrand list, the more popular stocks (higher brand values) underperform the less popular stocks (lower brand values).
Interbrand supplied us with a spreadsheet containing the top brands for each calendar year starting in 2000 and ending in 2017. Much of this information is publicly available, but it is tedious to consolidate in a usable form. 39 In the initial year, 2000, the list contained only 75 brands; the rest of the years contained 100 brands. Table 6.1 displays the first 50 brands, the brand ranks, and Interbrand’s estimated brand value (BV) for 2000 and 2017.
Some observations about the data in Table 6.1 are worth noting. First, the relationship between brands and stocks is not always one to one. To address this issue, in some cases, we mapped brand to publicly traded stocks. In other cases, where multiple brands (such as Volkswagen, Audi, and Porsche) are part of the same company (in this case, the Volkswagen Group), we combined the brand values under the parent company so we could eventually sort the group on the basis of the total brand value of an identified stock. Brands of privately held companies, such as IKEA, were excluded from the study.
The list of brands and stocks associated with them represent companies listed on the New York Stock Exchange (NYSE), Nasdaq Stock Market, and other international exchanges. For stocks from international stock exchanges with American depositary receipts (ADRs), we used the ADR. If an ADR was unavailable, we converted returns from the non-US exchange into US dollar–based returns. We carefully recorded mergers, acquisitions, and spin-offs. Some stocks ranked similarly throughout the study (examples are IBM and GE), whereas others changed dramatically. For example, Google was not even on the list in 2000 but had climbed to #2 by 2017; Nokia was #5 in 2000 but was no longer on the list by 2017. For the year 2000 rankings, we were able to link 51 of the 75 brands to a specific stock. Over time, the number of brands that we could link to unique stocks changed, with 79 brands linked to stocks in the final year of this analysis, 2017.
To study the impact of evolving popularity, starting prior to the first trading date in April of each year, based on the most recent BV ranking at the time (the rankings are released in late September or early October of the previous year), we linked brands to specific stocks. If multiple brands in the rankings were associated with a single stock, then prior to ranking the stocks, we summed the various brands belonging to the single company to arrive at an aggregate value for a given stock. We used those values as the basis of the ranking.
Table 6.1. Interbrand’s Annual Best Global Brands Top 50: 2000 and 2017
Note: FMCG stands for fast moving consumer goods.
We then divided the stocks into quartiles based on their associated brands. Each quartile contains the same number of constituents plus or minus one stock. We equally weighted the returns on the stocks within each quartile. If a company was acquired by another company, we removed the stock of the acquired company from the sample as of the month of the acquisition, which could cause the number of stocks in a quartile to temporarily be lower than the other quartiles. Quartile 1 consists of the 25% of stocks with the highest BV (51 possible stocks divided into quartiles resulted in approximately 13 stocks in 2000, and 79 stocks divided into quartiles resulted in approximately 20 stocks per quartile in 2017). Quartile 4 consists of the 25% of stocks with the lowest BV.
In this and the next section, we report portfolio performance as measured against both equally weighted and market cap–weighted benchmarks. We rebalanced equally weighted portfolios back to equal weights at the beginning of each month. Cap-weighted portfolio weights were based on market-cap values at the beginning of each month. Note that equally weighted portfolios tended to have better performance than cap-weighted portfolios because of the rebalancing premium. 40 Therefore, we used equally weighted portfolios as benchmarks for equally weighted portfolios in computing Jensen’s alpha (Jensen 1968 ) and used the Carhart four factors (Carhart 1997 ) to compute the Carhart alpha for cap-weighted portfolios because the Carhart four factors are cap weighted. 41
Table 6.2 presents summary statistics for the annually constituted (with monthly rebalancing back to equal weights) BV-based quartiles.
Focusing initially on annual geometric returns, we see when we move from left to right, from Quartile 4 (Q4) containing the stocks with the lowest BV (least popular stocks) to Quartile 1 (Q1) containing the stocks with the highest BV (most popular stocks) that the lower-BV quartiles monotonically outperformed the higher-BV quartiles. The same monotonic relationship holds for the Sharpe ratio: Q4 has a significantly higher Sharpe ratio than the other three quartiles. We found no consistent relationship for standard deviation across the quartiles.
Table 6.2. Summary Statistics of Equally Weighted Quartile Returns Based on Interbrand’s Global Brand Value (BV) Rankings, April 200
0–August 2017
Measure
Quartile 4
(lowest BV)
Quartile 3
Quartile 2
Quartile 1
(highest BV)
Geometric mean (%)
11.95
8.85
7.61
5.87
Arithmetic mean (%)
13.53
10.89
8.95
7.39
Standard deviation (%)
16.73
19.30
15.87
16.90
Sharpe ratio
0.705
0.476
0.459
0.340
Skewness
–0.556
–0.312
–0.076
–0.376
Jensen’s alpha (%)
3.50
–0.62
–0.32
–2.47
t -stat. of alpha
2.30
–0.44
–0.24
–2.04
The last two rows of Table 6.2 show Jensen’s annualized alpha and the corresponding t -statistic for each quartile. 42 We used the equally weighted portfolios for all stocks across the quartiles as the benchmark and ran simple regressions to get Jensen’s annualized alphas. Note the wide differences in Jensen’s alpha (and t -statistics) for the first and fourth quartiles.
Figure 6.1 shows the growth of a $1 investment in each of the quartiles on a logarithmic scale, which allows a comparison of return changes through time. Here, we see that the monotonic performance relationship captured by the summary statistics in Table 6.2 has not always held in the growth of $1 race.
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