Resurrecting the Capital Asset Pricing Model

Hammad Siddiqi

University of Queensland

This Version: October 2016

Abstract

The capital asset pricing model is generally considered incapable of explaining the stock market behavior. I show that this conclusion is premature, and incorporating a pragmatic approach to approximating optimal Bayesian inference in the model resurrects CAPM. The adjusted CAPM internalizes the well-known size, value, and momentum effects, high-alpha-of-low-beta-stocks, accruals, low volatility anomaly, stock-split anomaly, and reverse stock-split anomaly. The market equity premium is also larger with anchoring. Immediate applications of the adjusted CAPM include improved cost of equity calculation, and improved evaluation of managed portfolio performance.

Keywords: Size Premium, Value Premium, Stock Splits, Equity Premium Puzzle, Anchoring and Adjustment Heuristic, CAPM, Asset Pricing, Accrual Anomaly, Low Volatility Anomaly, Low-beta-high- alpha, Momentum Effect

JEL Classification: G12, G11, G02

Resurrecting the Capital Asset Pricing Model

It is fair to trace the birth of asset pricing theory to the capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965). Over five decades later, CAPM is still the most widely used model in practice. However, academic finance has largely moved away from CAPM due to the discovery of a large number of phenomena that are generally considered to be inconsistent with the model.1 Such empirical difficulties have led Fama and French (1992) (and others) to claim that the CAPM is dead.

In this article, I show that it is theoretically possible to internalize almost all of the prominent anomalies if a pragmatic approach to approximating optimal Bayesian inference is incorporated in the model. Hence, the conclusion that the CAPM is dead is premature.

The key principle on which financial decisions turn is the relationship between risk and return. How are beliefs about risks formed? In general, a large number of factors contribute to risk in any given situation. Consider any given firm. A large number of factors (such as industry structure, regulations, labor relations, technology, input supply constraints, business strategy, and investment policy) matter for the volatility of its stock. There are nearly 5000 firms listed in the New York stock exchange alone, so optimal Bayesian inference requiring an accurate assessment of all the relevant factors for every firm is intractable.

The problem becomes more manageable with the realization that many factors are sector-specific, that is, are common to every firm in the same sector. So, instead of re-inventing the wheel for every firm, a pragmatic approach is to leverage such commonalities by using the sector-specific factors to furnish an informative starting point and then attempting to make appropriate adjustments pertaining to individual firms in the same sector. This is the approach taken by brokerage houses, commercial banks, and investment banks, as they typically employ sector specialists who focus on a group of firms in the same sector.

To take a concrete example, suppose adverse weather conditions are affecting the supply of coffee beans. Clearly, this is a factor that matters for all coffee producers. A pragmatic approach is to analyze the impact of adverse weather conditions on the sector leader (about which information is readily available) and use it as an informative starting point for other firms.

1 Two prominent CAPM anomalies are size and value effects. Size effect is documented in Banz (1981), Reinganum (1983), Brown et al (1983), and more recently in Hou, Xue, and Zhang (2014), and Fama and French (2015) among many others. Value premium is documented in Fama and French (1998) among many others.

This informative starting points needs to be adjusted as other firms may have hedged differently in the futures market, may hold different levels of inventory, and have varying relations with suppliers among other factors. This pragmatic approach has the added advantage of making it likely that an analyst would at least be qualitatively correct in ranking the firms even if he gets the magnitudes wrong. Unsurprisingly, Da and Schaumburg (2011) find that equity analysts are consistently able to rank stocks correctly making them qualitatively or rank-order correct even though they are often quantitatively incorrect.

In this article, I study the implications of incorporating this pragmatic approach into CAPM. The surprising finding is that it gives rise to a unified theoretical explanation for almost all of the prominent anomalies concerning CAPM. Without asserting that anomalies are completely resolved, which requires a comprehensive empirical examination of a large number of phenomena, and so is beyond the scope of any single article, this article shows that the reliance on the pragmatic approach must be considered as an alternative explanation before rejecting CAPM. In this respect, the article paves the way for such empirical testing by explicitly deriving CAPM formula, adjusted for reliance on the pragmatic approach, form first principles.

In practice, CAPM is also widely used in calculating the cost of equity and in evaluating the performance of managed portfolios. This practice is considered problematic due to various anomalies facing CAPM (Fama and French 2004). As integrating the pragmatic approach with CAPM internalizes the anomalies, immediate applications of the adjusted CAPM are improved estimation of cost of equity and improved evaluation of portfolio performance.

This article is organized as follows. Section 1 discusses the importance of informative starting points in decision-making. Section 2 and 3 derive CAPM adjusted for reliance on informative starting points. Section 4 shows how various prominent anomalies are internalized. Section 5 discusses the improved calculation of cost and equity and improved evaluation of managed portfolios with the adjusted CAPM. Section 6 concludes.

1. Reliance on Informative Starting Points

Bayesian inference provides a unified framework for making decisions as new information arrives. However, outside of the simplest of situations, exact Bayesian inference is intractable, and must be approximated. Relying on informative starting points that are somewhat incorrect

3 and attempting to adjust them appropriately is a pragmatic solution, which is considered an optimal response of a Bayesian decision-maker facing finite computational resources (Lieder, Griffiths, & Goodman 2013). Evidence of the anchoring-and-adjustment process has been found in brain imaging studies demonstrating that we are hard-wired to rely on informative starting points and then attempting to adjust them (Quo, Zhao, and Luo 2008).

It is intractable for anyone to accurately measure all of the risk factors, which is what needs to be done, in order to make the optimal decision. A pragmatic approach is considering the factors that one can assess, and using them to furnish an informative starting point (anchor), and then attempting to adjust the starting point appropriately. Such anchoring-and-adjustment is a robust human decision-making heuristic. Describing the anchoring heuristic, Epley and Gilovich write (2001), “People may spontaneously anchor on information that readily comes to mind and adjust their response in a direction that seems appropriate, using what Tversky and Kahneman (1974) called the anchoring and adjustment heuristic. Although this heuristic is often helpful, the adjustments tend to be insufficient, leaving people’s final estimates biased towards the initial anchor value.” (Epley and Gilovich (2001) page. 1).

It is easy to provide several examples of such reliance on informative starring points. When respondents were asked which year George Washington became the first US President, most would start from the year the US became a country (in 1776). They would reason that it might have taken a few years after that to elect the first president so they add a few years to 1776 to work it out, coming to an answer of 1778 or 1779. George Washington actually became president in 1789, implying that, starting from 1776, adjustments are typically insufficient. Another example is the freezing temperature of vodka. People typically know that vodka freezes at a temperature below the freezing point of water (0 Celsius). So, when asked about the freezing point of vodka, they tend to start from the freezing point of water and then adjust downwards to form an estimate. However, their adjustments do not go far enough, and fall short of the right answer (which is -24 Celsius). The adjustments tend to stop once a plausible value is reached leaving the final answer biased towards the starting value (Epley and Gilovich 2006).

Hirshleifer (2001) considers anchoring-and-adjustment to be an “important part of psychology based dynamic asset pricing theory in its infancy” (p. 1535). Shiller (1999) argues that anchoring appears to be an important concept for financial markets. This argument has been supported quite strongly by recent empirical research on financial markets. Anchoring-and-adjustment has been found to matter for credit spreads that banks charge to firms (Douglas et al 2015), it matters in determining the price of target firms in mergers and acquisitions (Baker et al 2012), and it also

4 affects the earnings forecasts made by analysts in the stock markets (Cen et al 2013). Furthermore, Siddiqi (2015) shows that anchoring-and-adjustment provides a unified explanation for a number of key puzzles in options market.

Apart from evidence of anchoring-and-adjustment found via regression studies described above, financial market evidence of anchoring on an informative starting point and then attempting to adjust it appropriately has been reported in experimental studies as well. De Bondt (1993) is one of the first experimental studies demonstrating that people anchor on informative starting points and then attempt to adjust away from them appropriately while forming risk and return judgments. Siddiqi (2012) reports a series of experiments in which underlying stock volatility serves as the informative anchor for call option volatility.

Going back to the example of coffee producers, generally a large number of sector- specific factors matter for earnings volatility of firms. A pragmatic approach is to use these factors to furnish an informative starting point pertaining to the sector leader (about which information is readily available) and then attempt to make appropriate adjustments for other firms. This approach saves an analyst from re-inventing the wheel for every firm by leveraging the result for the most prominent firm (which he/she understands better and about which plenty of information is available) to other firms in the same sector.2 Sector leaders are special as a vast majority of analyst research time is devoted to them.

2 2퐴 If 휎퐿푡 is the sector leader earnings volatility, and 휎푠푡 is the estimated earnings volatility of another firm in the same sector, then for 퐴푡 > −1, one may write:

2퐴 2 휎푠푡 = 휎퐿푡(1 + 퐴푡) (1.1)

2 ̅ 휎푠푡 Define correct adjustment as 퐴푡 = 2 − 1. Anchoring bias implies that adjustments to the 휎퐿푡 2 ̅ 휎푠푡 starting point do not go far enough. That is, 퐴푡 = 푚퐴푡 = 푚 ( 2 − 1) where 0 ≤ 푚 ≤ 1. In 휎퐿푡 other words, 푚 is the fraction of the correct scaling-factor a decision-maker applies while adjusting away from the informative starting point. It follows that:

퐴 2 2 휎푠푡 = (1 − 푚)휎퐿푡 + 푚휎푠푡 (1.2)

The correct adjustment is a special case, which corresponds to 푚 = 1. Anchoring bias implies that 푚 < 1.

2 Discussions with several sector-specialists confirm this view.

2. CAPM with Informative Starting Points

A typical derivation of CAPM via utility maximization starts by considering an overlapping generations (OLG) economy with agents having identical beliefs. Each agent lives for two periods. Agents that are born at 푡 aim to maximize their utility of wealth at 푡 + 1. Their utility functions are identical and exhibit mean-variance preferences. They trade securities 푠 = 1, … . , 푆 푠 ∗푠 where security 푠 pays dividends 푑푡 and has 푛 shares outstanding, and invest the rest of their wealth in a risk-free asset that offers a rate of 푟퐹.

The market is described by a representative agent who maximizes:

훾 max 푛′{퐸 (푃 + 푑 ) − (1 + 푟 )푃 } − 푛′Ω 푛 푡 푡+1 푡+1 퐹 푡 2 푡

where 푃푡 is the vector of prices, Ω푡 is the variance-covariance matrix of 푃푡+1 + 푑푡+1, and 훾 is the risk-aversion parameter.

The maximization problem described above has three components about which beliefs must be formed. These elements are: 1) Vector of expected payoffs in the next period, 2 퐸푡(푃푡+1 + 푑푡+1). 2) Expected variances associated with these payoffs, 휎 (푃푡+1 + 푑푡+1), that is, the diagonal elements in the variance-covariance matrix Ω푡. 3) Expected covariances associated with these payoffs, 푐표푣(푃푖푡+1 + 푑푖푡+1, 푃푗푡+1 + 푑푗푡+1), which are the off-diagonal elements in

Ω푡.

How are judgments regarding 푃푡+1 + 푑푡+1 formed in practice? Empirical research as well as street wisdom suggests that stock price forecasts are typically made by estimating earnings per share and then multiplying earnings per share with an appropriate multiple which is often the 푃 price-to-earnings ( ⁄퐸) ratio specific to the industry. Asquith, Mikhail, & Au (2005) find that 99% of stock analysts sampled in their study rely on this method to make stock price forecasts.

Due to its popularity among practitioners3, this method is also taught in popular MBA texts on investment analysis.4

The method typically involves the following three steps:

1) Estimate future earnings of the company. Typically, sales are forecasted and an appropriate net profit margin is applied to arrive at the total earnings of the company.

2) Earning estimated in step 1 is divided by the total number of shares outstanding to arrive at earnings-per-share (퐸푃푆) estimate.

푃 3) A multiple (often the ⁄퐸 ratio specific to the industry) is applied to 퐸푃푆 in order to forecast the stock price. A fraction (often the payout ratio specific to the industry) is applied to 퐸푃푆 to forecast the dividend.

The above method, which is empirically found to be informative (see Brav and Lehavy (2003)), implies that the following holds in stock analysts’ judgment:

푃푡+1 + 푑푡+1 ∝ 퐸푃푆푡+1.

Introducing a constant of proportionality:

2 푃푡+1 + 푑푡+1 = 푐 ∙ 퐸푃푆푡+1. It follows that 퐸(푃푡+1 + 푑푡+1) = 푐 ∙ 퐸(퐸푃푆푡+1) and 휎 (푃푡+1 + 2 2 푑푡+1) = 푐 ∙ 휎 (퐸푃푆푡+1), where 푐 is industry or sector specific. For two companies in different sectors, the covariance of payoffs is: 휎12(푃1푡+1 + 푑1푡+1, 푃2푡+1 + 푑2푡+1 ) =

푐1푐2휎12(퐸푃푆1푡+1, 퐸푃푆2푡+1). For two companies in the same sector: 휎12(푃1푡+1 + 2 푑1푡+1, 푃2푡+1 + 푑2푡+1 ) = 푐 휎12(퐸푃푆1푡+1, 퐸푃푆2푡+1).

Hence, the expected earnings, their volatilities, and covariances drive judgments regarding the means, volatilities, and covariances of 푃푡+1 + 푑푡+1, which play a crucial role in the optimization exercise leading to CAPM.

3 A few example of market professionals using this method are: http://www.winninginvesting.com/how_to_set_target_prices.htm http://www.treetis.com/blog/2015/3/6/modeling-home-depot http://www.betterinvesting.org/Public/StartLearning/BI+Mag/Articles+Archives/1008Publicab.htm 4 One prominent example: Smart, Gitman, & Joehnk (2014)

As discussed earlier, many risk factors are common among firms in the same sector. So, a pragmatic approach used by sector-specialists is to analyse the impact on the sector-leader, which is relatively better understood, and then leverage this analysis for other firms in the same sector by attempting to make appropriate adjustments. Denote the earnings of the sector leader in 푗 푗 sector 푗 by 휋퐿, and the earning of another firm in the same sector by 휋푆 . Applying the pragmatic approach, earnings volatility of the sector-leader is an informative starting point to which a scaling-factor is applied. As discussed in section 1, anchoring bias implies that the scaling-factor has smaller magnitude than the correct value. Hence, a typical firm’s earnings volatility is inferred from (1.2) as follows:

퐴2 푗 2 푗 2 푗 휎 (휋푆 ) = (1 − 푚1)휎 (휋퐿) + 푚1휎 (휋푆 ) (2.1)

′ Denoting the number of shares outstanding of the typical firm by 푛푆, and the number of shares ′ outstanding of the sector leader by 푛퐿, (2.1) leads to:

퐴2 푗 ′2 퐴2 푗 휎 (휋푆 ) 푛퐿 2 푗 2 푗 휎 (퐸푃푆푆 ) = ′2 = (1 − 푚1) ∙ ′2 ∙ 휎 (퐸푃푆퐿 ) + 푚1휎 (퐸푃푆푆 ) (2.2) 푛푆 푛푆

As the firms belong to the same sector, multiplying (2.2) by 푐2 and substituting out 푐퐸푃푆 leads to:

′2 2퐴 푛퐿 2 2 휎 (푃푠푡+1 + 푑푠푡+1) = (1 − 푚1) ∙ ′2 ∙ 휎 (푃퐿푡+1 + 푑퐿푡+1) + 푚1휎 (푃푠푡+1 + 푑푠푡+1) (2.3) 푛푆

Following the same line of reasoning for expected payoffs, it follows:

′ 퐴 푛퐿 퐸 (푃푠푡+1 + 푑푠푡+1) = (1 − 푚2) ∙ ′ ∙ 퐸(푃퐿푡+1 + 푑퐿푡+1) + 푚2퐸(푃푠푡+1 + 푑푠푡+1) (2.4) 푛푆

The literature on financial decision making from Bayesian learning perspective tends to focus on volatility beliefs while assuming that payoff beliefs are correct (Weitzman 2007). This is because of the realization that errors in volatility beliefs are likely to be much larger due to task complexity, as volatility is an unobserved second order effect whereas payoff levels are directly observed. A robust finding from the anchoring literature is that the anchoring bias is directly proportional to task complexity (Blankenship et al 2008, Mueb and Proeger 2014, Meub, Proeger, & Bizer 2013, March and Ahn 2006). It follows that the anchoring bias must be larger for volatility beliefs than for payoff beliefs That is, 푚1 < 푚2 ≤ 1. Fo simplicity of exposition and without loss of generality, I set 푚2 = 1.

The variance-covariance matrix, Ω푡, includes variances as well as covariances. Assuming that they have similar complexity, one may allow the anchoring bias to matter equally for both 퐴 variances as well as covariances. To form the covariance judgment (휎푆1푆2) between a given typical firm (S1) and another typical firm (S2), the agent uses as the starting point, the covariance between the earnings of the sector-leader (L) in the same sector as the given typical stock and S2, denoted as 휎퐿푆2. Following the same steps as before:

′ 퐴 푛퐿 휎푆1푆2 = (1 − 푚) ′ 휎퐿푆2 + 푚휎푆1푆2 (2.5) 푛푆

The qualitative impact of anchoring in covariance judgments is identical to the impact of anchoring in variance judgments. Hence, for reasons of simplicity and clarity of exposition, only anchoring in variance judgments is modelled in the main body of this article. For completeness, appendix A presents an extended model that incorporates anchoring in covariance judgments as well.

Before solving the model, it is useful to compare sector-leader volatility with a median firm volatiltiy in the same sector. Table 1 shows the average earnings and volatility of earnings between 2011-2015 for two companies across 10 major sectors included in the S&P 500 index. In each sector, one company is the sector leader (by market capitalization) and second is the median company included in the index from the same sector. In every sector of the economy, the sector leader has higher average earning level, which is expected, as the asset base of the

9 sector leader is much larger when compared with an average sized firm in the same sector. Due to higher earnings levels, the sector leaders have larger earnings volatilities as well when compared with median firms in the same sector.5

3. Incorporating Informative Starting Points in CAPM

The previous section shows how the key elements relevant for the maximization problem of a representative agent may be influenced by relying on the sector-leader as the informative starting point. In this section, I discuss the implications of such reliance for CAPM. To keep the discussion as clear as possible, I discuss the implications in the increasing order of market complexity:

1) Only one sector in the economy with one sector-leader and one other firm.

2) Only one sector in the economy with one sector-leader and many normal/typical firms.

3) Many sectors in the economy with one sector-leader in each sector and many normal/typical firms in every sector.

3.1 CAPM with Informative Starting Points: The Simplest Case

Consider an overlapping-generations (OLG) economy in which agents are born each period and live for two periods. For simplicity, in the beginning, I assume that they trade in the stocks of two firms and invest in a risk-free asset. One firm is well-established with large payoffs (the sector-leader), and the second firm is a relative new-comer with much smaller payoffs (typical or normal firm). The next period payoff per share of the leader firm is denoted by 훿퐿푡+1 = 푃퐿푡+1 +

푑퐿푡+1 where 푃퐿푡+1 is the next period share price and 푑퐿푡+1 is the per share dividend of the large firm. Similarly, the next period payoff per share of the normal firm is defined by 훿푆푡+1 =

푃푆푡+1 + 푑푆푡+1. The risk-free rate of return is 푟퐹. At time 푡, each agent chooses a portfolio of stocks and the risk-free asset to maximize his utility of wealth at 푡 + 1. There are no transaction costs, taxes, or borrowing constraints.

5 If the domain is extended to include all firms in the same sector instead of only firms that are included in the S&P 500 index, then the difference between the median firm and the sector leader gets even starker.

The market dynamics are described by a representative agent who maximizes:

푛퐿{퐸푡(훿퐿푡+1) − (1 + 푟퐹)푃퐿푡} + 푛푆{퐸푡(훿푆푡+1) − (1 + 푟퐹)푃푆푡} 훾 − {푛2휎2 + 푛2휎2 + 2푛 푛 휎 } 2 퐿 퐿 푆 푆 퐿 푆 퐿푆

where 푛퐿, 푛푠, 푎푛푑 훾 denote the number of shares of the leader firm, the number of shares of the normal firm, and the risk aversion parameter respectively. Next period variances of the leader 2 2 firm and the normal firm payoffs per share are 휎퐿 = 푉푎푟(훿퐿푡+1) and 휎푆 = 푉푎푟(훿푆푡+1) respectively and 휎퐿푆 denotes their covariance.

The first order conditions of the maximization problem are:

2 퐸푡(훿퐿푡+1) − (1 + 푟퐹)푃퐿푡 − 훾푛퐿휎퐿 − 훾푛푆휎퐿푆 = 0 (3.1)

2 퐸푡(훿푆푡+1) − (1 + 푟퐹)푃푆푡 − 훾푛푆휎푆 − 훾푛퐿휎퐿푆 = 0 (3.2)

Solving (3.1) and (3.2) for prices yields:

2 퐸푡(훿퐿푡+1) − 훾푛퐿휎퐿 − 훾푛푆휎퐿푆 푃퐿푡 = 1 + 푟퐹

2 퐸푡(훿푆푡+1) − 훾푛푆휎푆 − 훾푛퐿휎퐿푆 푃푆푡 = 1 + 푟퐹

′ If the number of shares of the leader firm outstanding is 푛퐿, and the number of shares of the ′ normal firm outstanding is 푛푆, then the equilibrium prices are:

′ 2 ′ 퐸푡(훿퐿푡+1) − 훾푛퐿휎퐿 − 훾푛푆휎퐿푆 푃퐿푡 = (3.3) 1 + 푟퐹

′ 2 ′ 퐸푡(훿푆푡+1) − 훾푛푆휎푆 − 훾푛퐿휎퐿푆 푃푆푡 = (3.4) 1 + 푟퐹

Next, I show how reliance on informative starting points alters the above equilibrium. In accordance with the discussion in section 2, the representative agent’s judgment of normal stock’s payoff variance is modeled as in (2.3). With such anchoring-and-adjustment, the

11 equilibrium price of the normal firm falls; however, the equilibrium price of the leader firm remains unchanged.

Substituting from (2.3) into (3.4), the equilibrium price of the normal firm with anchoring is:

푃푆푡 푛 ′2 퐸 (훿 ) − 훾푛′ 푚휎2 − 훾푛′ (1 − 푚)휎2 퐿 − 훾푛′ 휎 푡 푆푡+1 푆 푆 푆 퐿 푛′2 퐿 퐿푆 = 푆 (3.5) 1 + 푟퐹

′ 2 Adding and subtracting 훾푛푆휎푆 from the numerator of the above equation and using ′ ′ ′ ′ 2 푐표푣(훿푆푡+1, 푛퐿훿퐿푡+1 + 푛푆훿푆푡+1) = 푛퐿휎퐿푆 + 푛푆휎푆 leads to:

푃푆푡 푛 ′2 퐸 (훿 ) − 훾 [푐표푣(훿 , 푛′ 훿 + 푛′ 훿 ) + 푛′ (1 − 푚) (휎2 퐿 − 휎2)] 푡 푆푡+1 푆푡+1 퐿 퐿푡+1 푆 푆푡+1 푆 퐿 푛′2 푆 = 푆 (3.6) 1 + 푟퐹

With correct adjustment, that is, with 푚 = 1, there is no anchoring bias and (3.6) reduces to (3.4).

Expressing (3.6) in terms of the expected stock return leads to:

′2 훾 ′ ′ ′ 2 푛퐿 2 퐸푡(푟푆) = 푟퐹 + [푐표푣(훿푆푡+1, 푛퐿훿퐿푡+1 + 푛푆훿푆푡+1) + 푛푆(1 − 푚) (휎퐿 ′2 − 휎푆 )] (3.7) 푃푆푡 푛푆

Anchoring does not change the share price of the leader firm. By re-arranging (3.3), the expected return expression for the stock price of the leader firm is obtained:

훾 ′ ′ 퐸푡(푟퐿) = 푟퐹 + [푐표푣(훿퐿푡+1, 푛퐿훿퐿푡+1 + 푛푆훿푆푡+1)] (3.8) 푃퐿푡

′ 푛푆푃푆푡 Expected return on the total market portfolio is obtained by multiplying (3.7) by ′ ′ 푛퐿푃퐿푡+푛푆푃푆푇

′ 푛퐿푃푆푡 and (3.8) by ′ ′ and adding them: 푛퐿푃퐿푡+푛푆푃푆푇

훾 ′ ′ 퐸푡[푟푀] = 푟퐹 + ′ ′ [푉푎푟(푛퐿훿퐿푡+1 + 푛푆훿푆푡+1) 푛퐿푃퐿푡 + 푛푆푃푆푡 ′2 ′2 2 푛퐿 2 + 푛푆 (1 − 푚) (휎퐿 ′2 − 휎푆 )] (3.9) 푛푆

Proposition 1 The expected return on the market portfolio with reliance on informative starting points is larger than the expected return on the market portfolio without such reliance.

Proof.

Follows directly from (3.9) by realizing that with starting points 푃푆푡 is smaller than what it would

′2 ′2 2 푛퐿 2 be without them, and the second term, 푛푆 (1 − 푚) (휎퐿 ′2 − 휎푆 ), which is positive with 푛푆 anchoring bias is equal to zero without the anchoring bias. This term is positive because the earnings volatility of the sector-leader is larger than the earnings volatility of the normal firm

′2 ′2 2 푛퐿 2 2 2 (Table 1 in section 2). To see this clearly: 푛푆 (1 − 푚) (휎퐿 ′2 − 휎푆 ) = 푐 (1 − 푚)(휎퐿 (휋퐿) − 푛푆 2 휎푆 (휋푆))

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From (3.9), one can obtain an expression for the risk aversion coefficient, 훾, as follows:

훾 ′ ′ (퐸푡[푟푀] − 푟퐹) ∙ (푛퐿푃퐿푡 + 푛푆푃푆푡) = ′2 (3.10) ′ ′ ′2 2 푛퐿 2 푉푎푟(푛퐿훿퐿푡+1 + 푛푆훿푆푡+1) + 푛푆 (1 − 푚) (휎퐿 ′2 − 휎푆 ) 푛푆

′ ′ Substituting (3.10) in (3.7) and (3.8) and using 푃푀푡 = 푛퐿푃퐿푡 + 푛푆푃푆푡 leads to:

퐸푡(푟푆) = 푟퐹 + 퐸푡[푟푀 − 푟퐹] ′2 ′ 2 푛퐿 2 푛푆(1 − 푚) (휎퐿 ′2 − 휎푆 ) 푛푆 퐶표푣(푟푆, 푟푀) + 푃푆푡푃푀푡 ∙ ′2 (3.11) ′2 2 푛퐿 2 푛푆 (1 − 푚) (휎퐿 ′2 − 휎푆 ) 푛푆 푉푎푟(푟푀) + 2 푃푀푡

퐸푡(푟퐿) = 푟퐹 + 퐸푡[푟푀 − 푟퐹]

퐶표푣(푟퐿, 푟푀) ∙ ′2 (3.12) ′2 2 푛퐿 2 푛푆 (1 − 푚) (휎퐿 ′2 − 휎푆 ) 푛푆 푉푎푟(푟푀) + 2 푃푀푡

Equations (3.11) and (3.12) are the expected return expressions for the normal stock and the leader stock respectively with anchoring. They give the expected return under the adjusted

CAPM (ACAPM). It is straightforward to see that substituting 푚 = 1 in (3.11) and (3.12) leads to the classic CAPM expressions. That is, without anchoring ACAPM converges to CAPM, with

퐶표푣(푟푠,푟푀) 퐶표푣(푟퐿,푟푀) beta being the only priced risk factor, 훽푆 = , and 훽퐿 = . 푉푎푟(푟푀) 푉푎푟(푟푀)

One can see the origins of various CAPM anomalies in (3.11). Compared to the classic

퐶표푣(푟푠,푟푀) CAPM where 훽푆 = , CAPM adjusted for reliance on the informative starting point of 푉푎푟(푟푀) ′2 ′ 2푛퐿 2 푛푆(1−푚)(휎퐿 ′2 −휎푆) 푛푆 퐶표푣(푟푆,푟푀)+ 푃푆푡푃푀푡 sector-leader leads to the following modified beta: ′2 . That is, there is ′2 2푛퐿 2 푛푆 (1−푚)(휎퐿 ′2 −휎푆) 푛푆 푉푎푟(푟푀)+ 2 푃푀푡 an additional term in the denominator as well as in the numerator. It is easy to see that if actual returns are determined in accordance with modified beta, and classic beta is used in a CAPM regression, then higher returns would be associated with low 푃푆푡(origin of value effect), and smaller size (as smaller size correlates with smaller price and smaller payoff volatility). Once can

퐶표푣(푟푠,푟푀) also see the origins of high-alpha-of-low-beta anomaly: low value of 훽푆 = typically 푉푎푟(푟푀) 2 2 corresponds to low value of 휎푆 , and low 휎푆 leads to higher alpha due to modified beta.

Proposition 2 shows that reliance on the informative starting point of sector-leader implies that the normal firm has a larger beta-adjusted return than the leader firm.

Proposition 2 The beta-adjusted excess return on the normal stock is larger than the beta-adjusted excess return on the leader stock.

Proof.

From (3.12):

퐸푡[푟퐿 − 푟퐹] < 퐸푡[푟푀 − 푟퐹] 퐶표푣(푟퐿, 푟푀) 푉푎푟(푟푀) and from (3.11)

퐸푡[푟푆 − 푟퐹] > 퐸푡[푟푀 − 푟퐹] 퐶표푣(푟푠, 푟푀) 푉푎푟(푟푀)

Hence, the beta-adjusted excess return on the normal stock is larger than the beta-adjusted excess return on the leader stock.

▄

In the next two sub-sections, the above results are generalized. In section 3.2, the results are generalized to include a large number of normal firms while keeping the number of leader firm at one. In section 3.3, the results are generalized to include a large number of leader firms as well (one leader firm in each sector).

3.2 Adjusted CAPM: One sector-leader and many normal firms

It is straightforward to extend the anchoring approach to a situation in which there are a large number of normal firms. Equation (3.6) remains unchanged. However, equation (3.9) changes slightly to the following:

푘 ′2 훾 ′2 2 푛퐿 2 퐸푡[푟푀] = 푟퐹 + [푉푎푟(훿푀푡+1) + ∑ 푛푆푖 (1 − 푚) (휎퐿 ′2 − 휎푆푖)] (3.13) 푃푀푡 푛 푖=1 푆푖 where 훿푀푡+1 is the payoff associated with the aggregate market portfolio in the next period, and 푘 is the number of normal firms in the market.

From (3.13), it follows that:

(퐸푡[푟푀] − 푟퐹) ∙ (푃푀푡) 훾 = ′2 (3.14) 푘 ′2 2 푛퐿 2 푉푎푟(훿푀푡+1) + ∑푖=1 푛푆푖 (1 − 푚) (휎퐿 ′2 − 휎푆푖) 푛푆푖

Substituting (3.14) in (3.7), the corresponding expression for a normal firm j’s expected return can be obtained: 퐸푡(푟푆푗) = 푟퐹 + 퐸푡[푟푀 − 푟퐹] ∙

′2 ′ 2푛퐿 2 푛푆푗(1−푚)(휎퐿 ′2 −휎푆푗) 푛푆푗 퐶표푣(푟 ,푟 )+ 푆푗 푀 푃 푃 푆푗푡 푀푡 (3.15) 푛 ′2 ′2( ) 2 퐿 2 푛푆푖 1−푚 (휎퐿 ′2 −휎푆푖) 푘 푛푆푖 푉푎푟(푟푀)+∑푖=1 2 푃푀푡

The corresponding expression for the leader firm is obtained by substituting (3.14) in (3.8):

퐶표푣(푟퐿, 푟푀) 퐸푡(푟퐿) = 푟퐹 + 퐸푡[푟푀 − 푟퐹] ∙ ′2 (3.16) ′2 2 푛퐿 2 푛푆푖 (1 − 푚) (휎퐿 ′2 − 휎푆푖) 푘 푛푆푖 푉푎푟(푟푀) + ∑푖=1 2 푃푀푡

(3.15) and (3.16) provide the expected return expressions corresponding to a situation in which there are a large number of normal firms and one leader firm. It is straightforward to check that in the absence of the anchoring bias, that is, when 푚 = 1, the model converges to the classic CAPM expressions of expected returns. In the next section, I generalize the model to include a large number of leader firms as well.

3.3 Adjusted CAPM: 푸 sectors with a sector-leader in each sector, and 풌 normal firms in every sector

I assume that there are 푄 sectors and every sector has one leader firm. I assume that the number of normal firms in every sector is 푘. That is, the total number of normal firms in the market is 푄 × 푘. As the total number of leader firms is 푄. The total number of all firms (both leader and normal) in the market is 푄 + (푄 × 푘).

Following a similar set of steps as in the previous two sections, the expected return expression for a normal firm 푗 in sector 푞 ∈ 푄 is given by:

퐸푡(푟푞푆푗) = 푟퐹 + 퐸푡[푟푀 − 푟퐹] ′2 ′ 2 푛푞퐿 2 푛푞푆푗(1 − 푚) (휎푞퐿 ′2 − 휎푞푆푗) 푛푞푆푗 퐶표푣(푟푞푆푗, 푟푀) + 푃푞푆푗푡푃푀푡 ∙ ′2 (3.17) ′2 2 푛푞퐿 2 푛푞푆푖(1 − 푚) (휎푞퐿 ′2 − 휎푞푆푖) 푄 푘 푛푞푆푖 푉푎푟(푟푀) + ∑푞=1 ∑푖=1 2 푃푀푡

The corresponding expression for the leader firm in sector 푞 is given by:

퐸푡(푟푞퐿) = 푟퐹 + 퐸푡[푟푀 − 푟퐹]

퐶표푣(푟푞퐿, 푟푀) ∙ ′2 (3.18) ′2 2 푛푞퐿 2 푛푞푆푖(1 − 푚) (휎푞퐿 ′2 − 휎푞푆푖) 푄 푘 푛푞푆푖 푉푎푟(푟푀) + ∑푞=1 ∑푖=1 2 푃푀푡

(3.17) and (3.18) are the relevant expected return expressions under the adjusted CAPM. As before, it is easy to see that, in the absence of the anchoring bias, that is, if 푚 = 1, adjusted CAPM expressions converge to the classic CAPM expressions.

Section 3.4 presents an alternate derivation of the adjusted CAPM by using portfolio expected returns and standard deviations.

3.4 Adjusted CAPM: A derivation based on portfolio expected return and standard deviation

Adjusted CAPM can also be derived by minimizing portfolio standard deviation for a given level of expected return. I illustrate the two firm case below with one leader firm, one normal firm, and a risk-free asset. Generalization to large number of normal and leader firms is straightforward. The portfolio standard deviation is:

2 2 2 2 휎0 = √푤퐿 휎푅퐿 + 푤푆 휎̂푅푆 + 2푤퐿푤푆퐶표푣(푟퐿, 푟푆)

2 where 푤퐿 and 푤푆 are portfolio weights of leader and normal stock respectively. 휎푅퐿is the 2 variance of leader stock’s return. 휎̂푅푆 is the judgment of the anchoring-prone investor regarding ′2 2 2푛퐿 푚휎푆 +(1−푚)휎퐿 ′2 2 푛푆 the return variance of normal stock. 휎̂푅푆 = 2 푃푆

The optimization problem of the representative agent is as follows:

퐿 = 휎0 + 휆(퐸(푟0) − 푤퐿퐸(푟퐿) − 푤푆퐸(푟푆) − (1 − 푤퐿 − 푤푆)푟퐹)

The first order conditions are:

1 2 (푤퐿휎푅퐿 + 푤푆퐶표푣(푟퐿, 푟푆)) = 휆(퐸(푟퐿) − 푟퐹) 휎0 17

′2 2 푛퐿 푤푆(1 − 푚)휎퐿 ′2 1 2 푛푆 푤푆푚휎푅푆 + 2 + 푤퐿퐶표푣(푟퐿, 푟푆) = 휆(퐸(푟푆) − 푟퐹) 휎0 푃푆 ( )

Multiply the first equation with 푤퐿, multiply the second equation with 푤푆, add them together and carry out a little algebraic manipulation to arrive at the following result at the point where

푤퐿 + 푤푆 = 1:

1 [퐸[푟푀] − 푟퐹]휎푀 = ′2 휆 2 2 푛퐿 2 푤푆 (1 − 푚) (휎퐿 ′2 − 휎푆 ) 2 푛푆 휎푀 + 2 푃푆 where 퐸[푟푀] and 휎푀 identify the portfolio expected return and the portfolio standard deviation 2 at the point 푤퐿 + 푤푆 = 1. That is, they correspond to the market portfolio. 휎푆 is the payoff variance of the normal stock. Note, if there is no anchoring bias, then the expression 1 [퐸[푟 ]−푟 ] corresponding to the classical CAPM is obtained: = 푀 퐹 . 휆 휎푀

′ 푛푠푃푠 As 푤푠 = , it follows that: 푃푀

1 [퐸[푟푀] − 푟퐹]휎푀 = ′2 휆 ′2 2 푛퐿 2 푛푠 (1 − 푚) (휎퐿 ′2 − 휎푆 ) 2 푛푆 휎푀 + 2 푃푀

1 Substituting in the first order conditions and carrying out a little algebraic manipulation leads 휆 to:

′2 ′ 2 푛퐿 2 푛푆(1 − 푚) (휎퐿 ′2 − 휎푆 ) 푛푆 퐶표푣(푟푆, 푟푀) + 푃푆푡푃푀푡 퐸푡(푟푆) = 푟퐹 + 퐸푡[푟푀 − 푟퐹] ∙ ′2 ′2 2 푛퐿 2 푛푆 (1 − 푚) (휎퐿 ′2 − 휎푆 ) 푛푆 푉푎푟(푟푀) + 2 푃푀푡

퐶표푣(푟퐿, 푟푀) 퐸푡(푟퐿) = 푟퐹 + 퐸푡[푟푀 − 푟퐹] ∙ ′2 ′2 2 푛퐿 2 푛푆 (1 − 푚) (휎퐿 ′2 − 휎푆 ) 푛푆 푉푎푟(푟푀) + 2 푃푀푡

The above two equations are identical to equations (3.11) and (3.12) respectively.

It is straightforward to generalize to the case of 푄 leader firms and 푄 × 푘 normal firms to obtain expressions identical to (3.17) and (3.18).

4. Adjusted CAPM and Asset-Return Anomalies

Finance theory predicts that risk-adjusted returns from all stocks must be equal to each other. The starting point for thinking about the relationship between risk and return is the Capital Asset Pricing Model (CAPM) developed in Sharpe (1964), and Lintner (1965). CAPM proposes that beta is the sole measure of priced risk. If CAPM is correct then the beta-adjusted returns from all stocks must be equal to each other. A large body of empirical evidence shows that beta-adjusted stock returns are not equal but vary systematically with factors such as “size” and “value”. Size premium means that small-cap stocks tend to earn higher beta-adjusted returns than large-cap stocks.6 Value premium means that value stocks tend to outperform growth stocks.7 Value stocks are those with high book-to-market value. They typically have stable dividend yields. Growth stocks have low book-to-market value and tend to reinvest a lot of their earnings. Value stocks are typically less volatile than growth stocks. Fama-French (FF) value and growth indices (monthly returns data from July 1963 to April 2002) show the following standard deviations of returns: FF small value: 19.20%, FF small growth: 24.60%, FF large value: 15.39%, and FF large growth: 16.65%. That is, among both small-cap and large-cap stocks, value stocks are less volatile than growth stocks.

Intuitively, the value premium is even more surprising than the size premium as it is plausible to argue that small size means higher risk (such as business cycle or liquidity risk) with size premium being compensation for higher risk; however, how can less volatility be riskier as the value premium seems to suggest?

The existence of size and value premiums has led to a growing body of research that attempts to explain them. In particular, there is the empirical asset pricing approach of Fama and French (1993) in which these factors are taken as proxies for risks with the assumption that all

6 Size effect is documented in Banz (1981), Reinganum (1983), Brown et al (1983), and more recently in Hou, Xue, and Zhang (2014), and Fama and French (2015) among many others. 7 Value premium is documented in Fama and French (1998) among many others.

19 risks are correctly priced.8 The task then falls to the asset pricing branch of theory to explain the sources of these risks. Fama and French (1997) argue that value firms are financially distressed, and the existence of the value premium is a compensation for bearing this risk. However, the empirical evidence is largely inconsistent with this view as distressed firms are found to have low returns rather than high returns (see Dichev (1998), Griffin and Lemmon (2002), and Campbell et al (2008) among others). Behavioral explanations of the value premium have also been put forward. In particular, De Bondt and Thaler (1987), Lakonishok et al (1994), and Haugen (1995) argue that traders overreact to news, and overprice “hot” stocks which tend to be growth firms, and underprice out of favor stocks which tend to be value firms. When this overreaction is eventually corrected, value premium arises.

Apart from size and value, there also exists, what is known as, the momentum effect in the stock market. Momentum effect refers to the tendency of “winning stocks” in recent past to continue to outperform “losing stocks” for an intermediate horizon in the future. Momentum effect has been found to be a robust phenomenon, and can be demonstrated with a number of related definitions of “winning” and “losing” stocks. Jegadeesh and Titman (1993) show that stock returns exhibit momentum behavior at intermediate horizons. A self-financing strategy that buys the top 10% and sells the bottom 10% of stocks ranked by returns during the past 6 months, and holds the positions for 6 months, produces profits of 1% per month. George and Hwang (2004) define “winning’ stocks as having price levels close to their 52-week high, and “losing” stocks as those with price levels that are farthest from their 52-week high, and show that a self-financing strategy that shorts “losing” stocks and buys “winning” stocks earns abnormal profits over an intermediate horizon (up to 12 months) consistent with the momentum effect.

Other well-documented anomalies include:

Low β High 휶

It is well known that the relation between univariate market β and average stock return is flatter than predicted by the CAPM of Sharpe (1964) and Lintner (1965) (see Black, Jensen, and Scholes (1972), Fama and MacBeth (1973), Frazzini and Pedersen (2014) among others). This is known as the low-beta-high-alpha anomaly.

8 Recently Fama and French (2015) argue that value premium is also captured by adding “investment” and “profitability” factors to size and beta factors.

Low Volatility Anomaly

Stocks with highly volatile returns tend to have low average returns irrespective of whether volatility is measured as the variance of daily returns or as the variance of the residuals from the FF three-factor model (see Baker, Bradley, & Wurgler (2011), Baker and Haugen (2012), and Ang et al (2006) among others).

Accruals

Sloan (1996) is the first paper to find that low returns are associated with high accruals. Accruals arise because accounting decisions cause book earnings to differ from cash earnings. The puzzle is that equity in firms that have a large accrual component of earnings performs worse than the equity in firms that have a lower accrual component.

Stock-Splits and Reverse Stock-Splits

Using data from 1975 to 1990, Ikenberry et al (1996) shows that stock-splits are associated with 8% positive abnormal returns after one year, and 16% abnormal returns over three years. Ikenberry et al (2003) uses data from 1990 to 1997 and confirms the earlier findings. Ohlson and Penman (1985) are the first to show that return volatility increases by about 30% following a stock split. Kim et al (2008) examine the long-run performance of 1600 firms with reverse stock- splits and reports negative abnormal returns. Koski (2007) shows a decrease in return volatility of 25% subsequent to a reverse stock-split.

4.1 Adjusted CAPM: A Unified Framework for Asset-Return Anomalies

Without asserting that the adjusted CAPM provides a complete explanation for the puzzles mentioned above, which requires a comprehensive empirical examination of a large number of anomalies, beyond the scope of any one article, I show that it provides a plausible unified explanation for the puzzles. I am not aware of any other approach, either traditional or behavioral, that provides such a unified explanation. Academic finance has largely given up on CAPM. The aim of this article is to demonstrate that this conclusion is premature and CAPM can internalize almost all of the prominent anomalies if a pragmatic approach of approximating optimal Bayesian inference is incorporated in the model. Hence, CAPM with the pragmatic approach must be considered as an alternative explanation.

4.1.1 ACAPM and the Size Effect

The size effect with adjusted CAPM can be easily seen after a little algebraic manipulation of (3.17). Beta-adjusted excess return on a normal firm’s stock is:

퐸(푟푞푆푗) − 푟퐹 퐵푒푡푎퐴푑푗푢푠푡푒푑 푒푥푐푒푠푠 푟푒푡푢푟푛 = 퐶표푣(푟푞푆푗, 푟푀) 푉푎푟(푟푀)

Beta-adjusted excess return on a normal firm’s stock can be written as:

퐵푒푡푎퐴푑푗푢푠푡푒푑 푒푥푐푒푠푠 푟푒푡푢푟푛 (푛표푟푚푎푙 푠푡표푐푘) = [ℎ] ∙ [1 + 𝑔]

푛′2 푛′2 ′ 2 푞퐿 2 ′ 2 푞퐿 2 푛푞푆푗(1−푚)(휎푞퐿 ′2 −휎푞푆푗) 푛푞푆푗(1−푚)(휎푞퐿 ′2 −휎푞푆푗) 푛푞푆푗 푛푞푆푗 where 𝑔 = = ′ 푃푞푆푗푡푃푀푡∙퐶표푣(푟푞푆푗,푟푀) 퐶표푣(푠푡표푐푘 푠 푝푎푦표푓푓,푚푎푟푘푒푡 푝푎푦표푓푓)

푉푎푟(푟푀)∙퐸푡[푟푀−푟퐹] and ℎ = ′2 = 푐표푛푠푡푎푛푡 푛푞퐿 푛′2 (1−푚)(휎2 −휎2 ) 푞푆푖 푞퐿푛′2 푞푆푖 푄 푘 푞푆푖 푉푎푟(푟푀)+∑푞=1 ∑푖=1 2 푃푀푡

In a given cross-section of stocks, ℎ is a constant. So, in order to make predictions about what happens to beta-adjusted return when size varies, we need to look at how 𝑔 =

′2 푛푞퐿 푛′ (1−푚)(휎2 −휎2 ) 푞푆푗 푞퐿푛′2 푞푆푗 푞푆푗 changes with size. 퐶표푣(푠푡표푐푘′푠 푝푎푦표푓푓,푚푎푟푘푒푡 푝푎푦표푓푓)

Proposition 3 (The Size Premium):

Beta-adjusted excess returns in adjusted CAPM fall as payoff size increases. In the classic CAPM, beta-adjusted excess returns do not vary with size and are always equal to the market risk premium.

Proof.

퐸(푟푞푆푗) − 푟퐹 퐵푒푡푎퐴푑푗푢푠푡푒푑 푒푥푐푒푠푠 푟푒푡푢푟푛 = 퐶표푣(푟푞푆푗, 푟푀) 푉푎푟(푟푀)

Substituting from (3.17) and re-arranging leads to:

퐵푒푡푎퐴푑푗푢푠푡푒푑 푒푥푐푒푠푠 푟푒푡푢푟푛

푉푎푟(푟푀) ∙ 퐸푡[푟푀 − 푟퐹] = 푛′2 (1 − 푚)(휎2 − 휎2 ) 푄 푘 푞푆푖 푞퐿 푞푆푖 푉푎푟(푟푀) + ∑푞=1 ∑푖=1 2 [ 푃푀푡 ] ′ 2 2 푛푞푆푗(1 − 푚)(휎푞퐿 − 휎푞푆푗) ∙ [1 + ] 푃푞푆푗푡푃푀푡 ∙ 퐶표푣(푟푞푆푗, 푟푀)

That is, beta-adjusted excess return can be written in the form:

퐵푒푡푎퐴푑푗푢푠푡푒푑 푒푥푐푒푠푠 푟푒푡푢푟푛 = [ℎ] ∙ [1 + 𝑔]

Clearly, as payoff variance and covariance with the market increase, 𝑔 falls. Firms with large asset bases (and total market capitalization, which typically proxies size) tend to have large earnings sizes and volatilities (see Table 1), which lead to large payoff volatilities and covariances. It follows that beta-adjusted excess returns fall as size increases when there is anchoring bias. In the absence of anchoring bias, that is, with 푚 = 1, it follows that 𝑔 = 0 and ℎ = 퐸[푟푀 − 푟퐹]. Hence, in the absence of anchoring, beta-adjusted excess return does not change with payoff covariance and payoff variance, and remains equal to the market risk premium. ▄

4.1.2 ACAPM and the Value Effect

A straightforward way of seeing the value premium is directly via the modified beta:

′2 ′ 2 푛푞퐿 2 푛푞푆푗(1 − 푚) (휎푞퐿 ′2 − 휎푞푆푗) 푛푞푆푗 퐶표푣(푟푞푆푗, 푟푀) + 푃푞푆푗푡푃푀푡 ′2 ′2 2 푛푞퐿 2 푛푞푆푖(1 − 푚) (휎푞퐿 ′2 − 휎푞푆푖) 푄 푘 푛푞푆푖 푉푎푟(푟푀) + ∑푞=1 ∑푖=1 2 푃푀푡

Value stock have low stock prices compared to fundamentals (book value of equity), whereas growth stocks have high stock prices. It follows that in a given cross-section, value stocks have higher modified beta when compared with growth stocks.

Next, I consider an equivalent alternative way of seeing how the value premium arises in adjusted CAPM. Value stocks have less volatile returns than growth stocks. Fama-French (FF) value and growth indices (monthly returns data from July 1963 to April 2002) show the following standard deviations of returns: FF small value: 19.20%, FF small growth: 24.60%, FF large value: 15.39%, and FF large growth: 16.65%. That is, among both small-cap and large-cap stocks, value stock returns are less volatile than growth stocks. By definition, value stocks have high book-to-market ratio when compared with growth stocks. It follows that, for a given book value of equity, value stocks have lower market prices when compared with growth stocks. Hence, the value stocks not only have smaller return volatility, but must also have smaller payoff volatility as return is payoff divided by price.

Proposition 4: (The Value Premium):

Beta-adjusted excess return on stocks with smaller payoff volatility is larger than the beta-adjusted excess returns on stocks with higher payoff volatility. In the classic CAPM, beta-adjusted returns do not vary with payoff volatility and are always equal to the market risk-premium.

Proof.

From the proof of proposition 3, we know that:

퐵푒푡푎퐴푑푗푢푠푡푒푑 푒푥푐푒푠푠 푟푒푡푢푟푛 = [ℎ] ∙ [1 + 𝑔]

′ 2 2 ′ 2 2 푛푞푆푗(1−푚)(휎푞퐿−휎푞푆푗) 푛푞푆푗(1−푚)(휎푞퐿−휎푞푆푗) where 𝑔 = = ′ 푃푞푆푗푡푃푀푡∙퐶표푣(푟푞푆푗,푟푀) 퐶표푣(푠푡표푐푘 푠 푝푎푦표푓푓,푚푎푟푘푒푡 푝푎푦표푓푓)

푉푎푟(푟 )∙퐸 [푟 −푟 ] and ℎ = 푀 푡 푀 퐹 = 푐표푛푠푡푎푛푡 푛′2 (1−푚)(휎2 −휎2 ) 푄 푘 푞푆푖 푞퐿 푞푆푖 푉푎푟(푟푀)+∑푞=1 ∑푖=1 2 푃푀푡

푛′ (1−푚)(휎2 −휎2 ) Note, that 𝑔 = 푞푆푗 푞퐿 푞푆푗 . As payoff volatility rises, 𝑔 falls because 퐶표푣(푠푡표푐푘′푠 푝푎푦표푓푓,푚푎푟푘푒푡 푝푎푦표푓푓) numerator falls and denominator rises. It follows that beta-adjusted excess return falls as payoff volatility rises. ▄

4.1.3 ACAPM and the Low-β-High-휶 Anomaly

Stocks with low betas tend to have low return volatilities and stocks with high betas tend to have high return volatilities (Baker et al 2011). Also, it is well known that earnings volatility is positively related to stock return volatility (see Beaver et al (1970), Graham, Campbell & Rajgopal (2005) among others). One can safely conclude that low beta stocks also have low earnings volatility, and high beta stocks have high earnings volatility.

From (3.17):

퐸푡(푟푞푆푗) = 푟퐹 + 퐸푡[푟푀 − 푟퐹] ′2 ′ 2 푛푞퐿 2 푛푞푆푗(1 − 푚) (휎푞퐿 ′2 − 휎푞푆푗) 푛푞푆푗 퐶표푣(푟푞푆푗, 푟푀) + 푃푞푆푗푡푃푀푡 ∙ ′2 ′2 2 푛푞퐿 2 푛푞푆푖(1 − 푚) (휎푞퐿 ′2 − 휎푞푆푖) 푄 푘 푛푞푆푖 푉푎푟(푟푀) + ∑푞=1 ∑푖=1 2 푃푀푡

As can be seen from above, in a given cross-section, stocks with higher value of the term

′2 푛푞퐿 푛′ (1−푚)(휎2 −휎2 ) 푞푆푗 푞퐿푛′2 푞푆푗 푞푆푗 outperform stocks with a lower value of the term on beta-adjusted basis. 푃푞푆푗푡푃푀푡 The above expression is larger for stocks with less volatile earnings. As low beta stocks tend to have low earnings volatility, the above term is higher for them. Hence, in adjusted CAPM, stocks with low CAPM betas must have high CAPM alphas leading to a flatter relationship between CAPM beta and average returns.

4.1.4. ACAPM and the Low Volatility Anomaly

As stocks with less volatile stock returns typically belong to firms with less volatile earnings (see Beaver et al (1970), Graham, Campbell & Rajgopal (2005) among others), the modified beta is larger for them. Hence, the explanation for the low volatility anomaly in adjusted CAPM is very similar to the explanation for low-beta-high-alpha anomaly.

4.1.5 ACAPM and the Accruals Anomaly

Sloan (1996) finds that low returns are associated with high accruals. Accruals arise because accounting decisions cause book earnings to differ from cash earnings. One expects that firms with large accrual component of earnings will have less persistent (more volatile) earnings when compared with firms with smaller accrual component of earnings. Sloan (1996) explicitly tests for this and finds strong support. The adjusted CAPM explanation is straightforward. As high accruals imply high earnings volatility, and high earnings volatility reduces the modified beta, average returns from high accrual firms must be lower. Note, that the explanation for accruals has the same underlying logic as the explanations for low-beta-high-alpha and low volatility anomalies. Hence, in the adjusted CAPM, these seemingly different anomalies are much the same phenomena.

4.1.6 ACAPM, Stock-Splits, and Reverse Stock-Splits

A stock-split increases the number of shares proportionally. In a 2-for-1 split, a person holding one share now holds two shares. In a 3-for-1 split, a person holding one share ends up with three shares and so on. A reverse stock-split is the exact opposite of a stock-split. Stock-splits and reverse stock-splits appear to be merely changes in denomination, that is, they seem to be accounting changes only that should not have any real impact on returns. With CAPM, stock- splits and reverse stock-splits do not change expected returns. To see this clearly, consider equation (3.4), which is reproduced below:

′ 2 ′ 퐸푡(훿푆푡+1) − 훾푛푆휎푆 − 훾푛퐿휎퐿푆 푃푆푡 = 1 + 푟퐹

A 2-for-1 split in the small firm’s stock divides the expected payoff by 2, divides the variance by 4 and covariance by 2, while multiplying the number of shares outstanding by 2. That is, a 2-for- 1 split leads to:

퐸 (훿 ) 2푛′ 휎2 훾푛′ 휎 푡 푆푡+1 − 훾 푆 푆 − 퐿 퐿푆 푆푝푙푖푡 2 4 2 푃푆푡 = 1 + 푟퐹

푃 It follows that 푃 푆푝푙푖푡 = 푆푡. 푆푡 2

That is, the price with split is exactly half of what the price would have been without the split. As both the expected payoff and the price are divided by two, there is no change in expected returns associated with a stock-split under CAPM. Similarly, there is no change in return variance as well. An equivalent conclusion follows for a reverse stock-split. For a reverse split, both the expected payoffs and the price increase by the same factor.

In adjusted CAPM, the price after the split is as follows:

퐸 (훿 ) 훾푛′ 휎 푡 푆푡+1 − 훾2푛′ 휎2퐴′ − 퐿 퐿푆 푆푝푙푖푡 2 푆 푆 2 푃푆푡 = (4.1) 1 + 푟퐹

2퐴′ where 휎푆 is the perceived payoff volatility after the split. A natural starting point for forming 2퐴′ 2퐴 judgment about 휎푆 is the perceived payoff volatility before the split, 휎푆 . The correct way of

휎2퐴 forming this additional judgment is to set: 휎2퐴′ = 푆 That is, starting from 휎2퐴, the investor 푆 4 푆

3휎2퐴 should subtract 푆 . Anchoring bias means that the adjustment falls short implying that 휎2퐴 > 4 푆

휎2퐴 푃 휎2퐴′ > 푆 . It follows that the stock price after the split is less than 푆푡 with anchoring. Hence, 푆 4 2 consistent with empirical findings, average return as well as volatility of return rise after the split.

With the 1-to-2 reverse stock-split, the anchoring influenced price after the reverse split is given by:

푅푒푣푒푟푠푒−푆푝푙푖푡 푃푆푡 ′ 푛푆 2퐴′ ′ 2퐸푡(훿푆푡+1) − 훾 휎푆 − 2훾푛퐿휎퐿푆 = 2 (4.2) 1 + 푟퐹

2퐴′ A natural starting point for forming judgment about 휎푆 is the perceived payoff volatility before 2퐴 2퐴′ the reverse-split, 휎푆 . The correct way of forming this additional judgment is to set: 휎푆 = 2퐴 2퐴 2퐴 4휎푆 . That is, starting from 휎푆 , the anchoring-prone investor should add 3휎푆 . Anchoring 2퐴 2퐴′ 2퐴 bias means that the adjustment falls short implying that 4휎푆 > 휎푆 > 휎푆 . It follows that the stock price after the reverse-split is more than 2푃푆푡 Hence, consistent with empirical findings, both the return as well as its volatility fall after the reverse-split.

4.1.7 Adjusted CAPM and the Momentum Effect

From (3.17), one can see that, in a given cross-section of stocks, low “m” stock do better than high “m” stocks on beta-adjusted basis. Plausibly, stocks that have shown superior performance recently, invite smaller adjustment from the sector-leader starting point, compared to stocks that have performed poorly. In this way, the momentum effect could arise in the adjusted CAPM. This explanation can be directly tested by calibrating the adjusted CAPM and testing for difference in “m” across superior vs inferior recent performers.

4.1.8 Adjusted CAPM and the High Equity Premium

Since its identification in Mehra and Prescott (1985), a large body of research has explored what is known in the literature as the equity premium puzzle. It refers to the fact that historical average return on equities (around 7%) is so large when compared with the historical average risk-free rate (around 1%) that it implies an implausibly large value of the risk aversion parameter. Mehra and Prescott (1985) estimate that a risk-aversion parameter of more than 30 is required, whereas a much smaller value of only about 1 seems reasonable.

If one uses CAPM to estimate risk-aversion then the following equation is appropriate:

훾 퐸푡[푟푀] = 푟퐹 + [푉푎푟(훿푀푡+1)] (4.3) 푃푀푡

One may recover the corresponding value of 훾, the risk-aversion parameter from (4.3) by substituting for all other variables in (4.3). The equity premium puzzle, when translated in the CAPM context, is that the recovered value of risk-aversion parameter is implausibly large.

With the adjusted CAPM, the expected return on the market portfolio is given by:

훾 퐸푡[푟푀] = 푟퐹 + [푉푎푟(훿푀푡+1) 푃푀푡

푄 푘 2′ 푛푞퐿 + ∑ ∑ 푛′2 (1 − 푚) (휎2 − 휎2 )] (4.4) 푞푆푖 푞퐿 푛2′ 푞푆푖 푞=1 푖=1 푞푠푖

A comparison of (4.4) and (4.3) shows that, with adjusted CAPM, a much smaller value of the risk-aversion parameter is required to justify the observed equity premium. Hence, adjusted CAPM is relevant for the equity premium puzzle.

5. Cost of Equity and Managed Portfolio Performance

Adjusted CAPM differs from classical CAPM in one key aspect. It replaces the classical beta,

퐶표푣(푟푠,푟푀) which is 훽푆 = , with the following modified beta: 푉푎푟(푟푀) ′2 ′ 2 푛푞퐿 2 푛푞푆푗(1 − 푚) (휎푞퐿 ′2 − 휎푞푆푗) 푛푞푆푗 퐶표푣(푟푞푆푗, 푟푀) + 푃푞푆푗푡푃푀푡 ′2 ′2 2 푛푞퐿 2 푛푞푆푖(1 − 푚) (휎푞퐿 ′2 − 휎푞푆푖) 푄 푘 푛푞푆푖 푉푎푟(푟푀) + ∑푞=1 ∑푖=1 2 푃푀푡

As the modified beta internalizes high-alpha-of-low-beta anomaly, and size and value effects, it is superior to classical beta for cost of equity calculation and portfolio performance evaluation. A three-step process can be used for estimating the modified beta: 1) Calculate classical beta and multiply with 푉푎푟(푟푀) to infer 퐶표푣(푟푠, 푟푀). 2) Insert other parameter values. The number of shares outstanding and market prices can be directly inserted. Time-series averages can be used for variances. 3) Finally, calibrate 푚 with historical data to estimate the modified beta.

The above exercise is simple enough to be carried out in an excel spreadsheet.

6. Conclusion

Academic finance generally considered CAPM to be an empirical failure. This article demonstrates that this conclusion is premature as it is possible to internalize almost all of the prominent anomalies in CAPM if a pragmatic approach to approximating optimal Bayesian inference is incorporated in the model. The article shows that the adjusted CAPM replaces the

퐶표푣(푟푠,푟푀) well-known expression for beta, which is, 훽푆 = , with a modified beta expression. 푉푎푟(푟푀) Almost all of the prominent anomalies can be seen in the modified beta. That is, if adjusted CAPM determines market prices and the classical CAPM is used to estimate beta, anomalies arise.

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Appendix A In this appendix, ACAPM is extended to allow for anchoring in covariance judgments as well. The basic case with three firms is discussed here. It is straightforward to extend it to a large number of blue-chips and normal firms. There is sector-leader firm, L, and two normal firms, S1 and S2. The agent needs to form judgments not only about variances of each normal firm, but also about their covariance, 휎푆1푆2. The judgment about variances of normal firms are formed in accordance with (2.3). The judgment about covariance of payoffs between normal firms is formed in accordance with (2.5), which is re-produced below:

′ 퐴 푛퐿 휎푆1푆2 = (1 − 푚) ′ 휎퐿푆2 + 푚휎푆1푆2 (퐴1) 푛푆 The equilibrium condition for S1 is:

′ 2퐴 ′ ′ 퐴 퐸푡(훿푆1푡+1) − 훾푛푆1휎푆 − 훾푛퐿휎퐿푆1 − 훾푛푆2휎푆1푆2 푃푆1푡 = (퐴2) 1 + 푟퐹 Substituting in (A2) from (2.3) and (A1) and simplifying leads to: 훾 퐸푡(푟푆1) = 푟퐹 + [푐표푣(훿푆1푡+1, 훿푀푡+1) 푃푆1푡 ′2 ′ ′ 2 푛퐿 2 ′ 푛퐿 + (1 − 푚) {푛푆1 (휎퐿 ′2 − 휎푆1) + 푛푆2 (휎퐿푆2 ′ − 휎푆1푆2)}] (퐴3) 푛푆1 푛푆1 From the equilibrium condition for S2:

훾 퐸푡(푟푆2) = 푟퐹 + [푐표푣(훿푆2푡+1, 훿푀푡+1) 푃푆2푡 ′2 ′ ′ 2 푛퐿 2 ′ 푛퐿 + (1 − 푚) {푛푆2 (휎퐿 ′2 − 휎푆2) + 푛푆1 (휎퐿푆2 ′ − 휎푆1푆2)}] (퐴4) 푛푆2 푛푆1 From the equilibrium condition for L: 훾 퐸푡(푟퐿) = 푟퐹 + [푐표푣(훿퐿푡+1, 훿푀푡+1)] (퐴5) 푃퐿

푛′ ∙푃 푛′ ∙푃 푛′ ∙푃 Multiply (A3) by 푆1 푆1, (A4) by 푆2 푆2, and (A5) by 퐿 퐿 and add to obtain the following 푃푀 푃푀 푃푀 ′ ′ ′ (푤ℎ푒푟푒 푃푀 = 푛푆1 ∙ 푃푆1 + 푛푆2 ∙ 푃푆2 + 푛퐿 ∙ 푃퐿): 훾 퐸푡(푟푀) = 푟퐹 + [푉푎푟(훿푀푡+1) 푃푀 ′2 ′2 ′2 2 푛퐿 2 ′2 2 푛퐿 2 + (1 − 푚) {푛푆1 (휎퐿 ′2 − 휎푆1) + 푛푆2 (휎퐿 ′2 − 휎푆2) 푛푆1 푛푆2 ′ ′ ′ 푛퐿 + 2푛푆1푛푆2 (휎퐿푆2 ′ − 휎푆1푆2)}] (퐴6) 푛푆1 It follows that:

퐸푡(푟푀 − 푟퐹)푃푀 훾 = ′2 ′ 2 ′2 2 푛퐿 2 ′ ′ 푛퐿 푉푎푟(훿푀푡+1) + (1 − 푚) {∑푖=1 푛푆푖 (휎퐿 ′2 − 휎푆푖) + 2푛푆1푛푆2 (휎퐿푆2 ′ − 휎푆1푆2)} 푛푆푖 푛푆1 Substituting the above in (A3) and (A4) leads to:

′2 ′ ′ 2푛퐿 2 ′ 푛퐿 (1−푚){푛푆1(휎퐿 ′2 −휎푆1)+푛푆2(휎퐿푆2 ′ −휎푆1푆2)} 푛푆1 푛푆1 퐶표푣(푟푆1,푟푀)+ 푃푆푡푃푀푡 퐸 (푟 ) = 푟 + 퐸 [푟 − 푟 ] ∙ ′ (퐴7) 푡 푆1 퐹 푡 푀 퐹 푛 ′2 푛 ( ) 2 ′2 2 퐿 2 ′ ′ 퐿 1−푚 {∑푖=1 푛푆푖(휎퐿 ′2 −휎푆푖)+2푛푆1푛푆2(휎퐿푆2 ′ −휎푆1푆2)} 푛푆푖 푛푆1 푉푎푟(푟푀)+ 2 푃푀푡

퐸푡(푟퐿) = 푟퐹 + 퐸푡[푟푀 − 푟퐹] 퐶표푣(푟퐿, 푟푀) ∙ ′2 ′ (퐴8) 2 ′2 2 푛퐿 2 ′ ′ 푛퐿 (1 − 푚) {∑푖=1 푛푆푖 (휎퐿 ′2 − 휎푆푖) + 2푛푆1푛푆2 (휎퐿푆2 ′ − 휎푆1푆2)} 푛푆푖 푛푆1 푉푎푟(푟푀) + 2 푃푀푡 It is immediately obvious that the qualitative impact of A7 and A8 is identical to the case when anchoring is only allowed in variance judgments. It is straightforward to extend A7 and A8 to include a large number of sectors (with a sector-leader in each sector) and a large number of normal firms.