Capital Asset Pricing Model

This paper presents a theory of equilibrium asset pricing that generalizes the recent work of Connor. The model extends Connor's results to more general sets of asset returns and consumer preferences, introduces production, and provides a framework for analyzing exact and approximate equilibrium asset pricing. The other major contribution of the paper is the introduction of geometric arguments that exploit the properties of induced prefer- ences over assets. This method of analyzing asset pricing provides an intuitively appealing way of analyzing equilibrium asset pricing theories. unifies several well-known asset-pricing models. This paper extends Connor's finite-asset model in a number of directions. These extensions are nontrivial, because they allow the theory to encompass asset-pricing models that were excluded by Connor's assumptions. The second major contribution of the paper is to introduce a method of argument that exploits the geometric properties of induced preferences over assets, and characterizes asset prices using these proper- ties. This technique provides an intuitively appealing way of analyzing asset pricing theories. (For a simple geometric exposition of the main ideas in this paper, see Milne (1987).) 0.2. To place this paper in the context of the literature, it is important to appreciate the development of asset-pricing models over the last two decades. The first-and perhaps the most famous-asset-pricing model is the Capital Asset Pricing Model (CAPM) developed by Sharpe (1964) and Lintner (1965). The model demonstrated a striking result: assuming (i) consumers characterize portfolios in terms of the means and variance of rates of return, (ii) there exists a risk-free asset, and (iii) asset equilibrium; then any asset's expected rate of return could be written as a linear function of the riskless interest rate, and the expected rate of return on the endowment of risky assets (the so-called market portfolio). The original CAPM result was derived from a one period model, but Merton (1973) showed that in a continuous time model where returns could be char- acterized by a diffusion process, the same mean-variance argument could be applied to obtain the CAPM formula. It was well-known that if consumers had von Neumann-Morgenstern prefer- ences, then either quadratic utility and/or multivariate normality of returns were

This chapter discusses theoretical foundations and empirical evidence for the most prominent asset pricing theory: the Capital Asset Pricing Model (CAPM). It represents a pricing model for risky assets. The CAPM has been extended to the multi– factor model (MFM) and arbitrage pricing theory (APT). The motivation of the latter has been to overcome the problems associated with the market portfolio in the CAPM by introducing a multi–factor approach. The subsequent chapter is kept simple and refers to the CAPM only. In some additional remarks crucial assumptions regarding investors’ preferences and stock return distributions are relaxed. Currently the debate whether exact pricing restrictions in the MFM and APT imply the return to the “good old CAPM” is still going on. By introducing state dependent relationships as well as general nonlinearities in a CAPM model one can try to unify the different views but we will not elaborate on such extensions. A more extensive treatment of static and dynamic portfolio theories is given in Part VI of the book.

Since Sharpe (1964) derived the CAPM, it has been the benchmark of asset pricing models and has been used to calculate the cost of equity capital and other asset pricing determinations for more than four decades. Many researchers have tried to relax the original assumptions and generalize the static CAPM. In addition, Merton (1973) and Black (1976) have generalized the static CAPM in terms of intertemporal CAPM. In this chapter, we survey the important alternative theoretical models of capital asset pricing and provide a complete review of the evolution of both static and intertemporal asset pricing models. We also discuss the interrelationships among these models and suggest several possible directions for future research. In addition, we review the asset pricing tests in terms of individual companies’ data instead of portfolio data. Our results might be used as a guideline for future theoretical and empirical research in capital asset pricing.

In this chapter we employ SD rules to prove that the CAPM is theoretically intact in a wide range of frameworks corresponding to the assumption about the distribution of returns and the length of the investment horizon. This is a surprising integration of the MV and SD paradigms, as these two paradigms represent two distinct branches of expected utility, each implying a different technique for portfolio investment selection. Each paradigm has its pros and cons. The obvious advantage of the MV approach is that it provides a simple and elegant method for determining the optimal diversification among risky assets which is necessary in establishing the Capital Asset Pricing Model (CAPM) (see Sharpe and Lintner). The main disadvantage of the MV paradigm is that it relies on the assumption of normal distribution of returns (or Elliptic distribution family, where the normal distribution belongs to this family), an assumption not needed for the employment of the SD rules. The normality assumption is obviously inappropriate for most assets traded in the stock market because asset prices cannot drop below zero (−100 % rate of return) whereas the normal distribution is unbounded. As the equilibrium risk-return relationship implied by the CAPM has very important theoretical and practical implications, it has been developed under other frameworks which do not assume normality. Levy has shown that technically the CAPM holds even if all possible mixes of distributions are log-normal (bounded from below by zero). However, this approach has also a drawback as with discrete time models, a new problem emerges: if x and y are log-normally distributed, a portfolio z, where z = α x + (l–α)y, will no longer distribute log-normally. Merton assumes continuous-time portfolio with infinite portfolio revisions and shows that under this assumption, the terminal wealth will be log-normally distributed and the CAPM will hold in each single instantaneous period. By employing the continuous portfolio revision, the nagging problem that the sum of log-normal distributions is not log-normal, which characterizes discrete models disappears. However, the disadvantage of the continuous time model is that the CAPM result breaks down if even minor transaction costs, no matter how small, are incorporated.

The CAPM: A Case of Elegance is for Tailors?

This thesis consists of three empirical studies on what drives stock market dynamics. The first empirical study explores the effect of crude oil price changes on the stock market returns of oil-exporting countries and oil-importing countries as well as those of a number of global stock indices. Using the Ordinary Least Squares (OLS) approach as well as the more robust Quantile Regression (QR) approach to explore the relationship between crude oil and stock market dynamics. The empirical findings suggest that the QR approach provides further insights compared to the OLS approach. For instance, the QR approach is able to identify specific quantiles where a significant relation exists. In particular crude oil price increases tend to have a negative impact on the stock market returns for some oil-exporting countries (such as Mexico, Iraq, Ecuador, and Venezuela) and a positive effect for other oil-exporting countries (such as Brazil and Algeria). However, the OLS approach suggests that these relationships are insignificant at the level of the mean. Overall, the empirical findings confirm that the QR approach can reveal more information about the relationship between crude oil price changes and stock market return across different quantiles of their distribution.The second study explores the extent to which implied volatility extracted from commodity markets and developed stock markets can predict the implied volatility of stock markets in BRICS countries. Using daily data from 2011 to 2016 and employing the newly developed Bayesian Graphical Vector Autoregressive (BGVAR) model of Ahelegbey et al. (2016) which does not suffer from over-parameterization and the identification problems associated with traditional VAR frameworks, this study finds that implied volatilities extracted from global and regional stock markets have a significant predictive power over the implied volatilities in BRICS stock markets. However, the predictive power of implied volatility from commodity markets are significant only in the case of South Africa.The third empirical study analyses the relationship between illiquidity and stock market returns in the G7 and BRICS countries. More specifically, this study explore the extent to which the Amihud (2002) illiquidity measure can improve the explanatory power of three commonly used asset pricing models, namely the Capital Asset Pricing Model (CAPM), the Fama-French three-factor model and the Carhart four-factor model. The empirical analysis is based on 15 years of monthly data on the returns of seven stock portfolios: 100 largest companies (Largest100), small value (S/V), small neutral (S/N), small growth (S/G) stocks, big value (B/V) stocks, big neutral (B/N) stocks, and big growth (B/G) stocks. The findings suggest that incorporating illiquidity as an additional factor results in a significant improvement in the explanatory power of these asset pricing models across several of the sample countries (8 countries in the case of the CAPM and Carhart four-factor model, and 6 countries in the case of the Fama-French three-factor model). For example, in the US adding illiquidity to the CAPM leads to an increase of the goodness of fit by 2.6% in the B/V portfolio, and for the Fama-French three-factor model the goodness of fit increases by up to 3% in all portfolios Moreover, the goodness of fit increases in all portfolios in the US by adding illiquidity to the Carhart four-factor model, with an up to 36% increase in the B/N portfolio.

The famous Capital Asset Pricing Model (CAPM), widely used in practice, takes into account only the business risk associated with investments in a specific company [not the entire market (or industry)]. In practice, most listing companies use debt financing and operate at a non-zero leverage level. This means that the financial risk associated with the use of debt financing, along with business risk, must be taken into account. The purpose of this paper is to simultaneously account for business and financial risk. We combined the CAPM theory and the Modigliani-Miller (MM) theory, which is the perpetual limit of the BFO (Brusov-Filatova-Orekhova) theory. The article shows that R. Hamada’s attempt to take into account both business and financial risks has proved unsustainable, and the formulas he obtained, widely used in practice, are incorrect. The paper outlines the correct formulae that made it possible to generalize CAPM for the first time, taking into account both business and financial risk. The application of the new CAPM 2.0 model to a number of companies is considered and the difference between the results obtained within the framework of CAPM 2.0 and CAPM is demonstrated. CAPM is one of the main models [along with APT (arbitrage pricing theory) and WACC] within the income approach to business valuation. This significantly increases the value of the developed CAPM 2.0 approach, which can significantly improve the accuracy of the assessment.

Finance models as metaphors

Introduction In deriving the Mean-Variance (M-V) rule and the Capital Asset Pricing Model (CAPM), one assumes explicitly or implicitly normal distributions of returns and risk aversion. With these assumptions, it emerges that the variance of the portfolio's return is the measure of the investment's risk. In Chapter 8, we saw that although normality of returns is statistically rejected, the economic loss from assuming normality when the distributions are actually not normal is relatively small. Moreover, in Chapter 4, we saw that even if the distributions of returns are not normal, the M-V rule can be employed as an approximation to expected utility, as long as risk aversion is assumed. Furthermore, it was shown empirically in Chapter 4 that the choices made by the M-V rule are almost fully correlated with the choices made by direct expected utility maximization for a wide set of risk-averse utility function. Considering the relatively small economic loss involved as a result of the normality assumption, in this chapter we assume that the M-V and the CAPM are derived based on this normality assumption, within the expected utility framework, rather than as an approximation to expected utility. However, even in this framework, as we saw in Chapters 9 and 10, the M-V and the CAPM are questionable for the following reasons: By the Prospect Theory (PT), people maximize a value function, which is defined on change of wealth (gains and losses) rather than on total wealth; by expected utility, the M-V rule and the CAPM are defined on total wealth. To derive the M-V rule and the CAPM, risk aversion is needed, whereas by PT, the value function has a risk-seeking segment. The risk- seeking segment of preference is steeper than the risk-aversion segment, implying loss aversion . Adopting the modified version of the PT, called Cumulative Prospect Theory (CPT), people employ cumulative decision weights . Thus, the cumulative distribution of returns is transformed, and in calculating the expected value (utility), the transformed distribution rather than the original distribution is employed. This seems to be a “deathblow” to the M-V rule and the CAPM because even if the original distribution of returns is normal, generally the transformed distribution is not. Moreover, although the normal distribution is symmetric, implying that skewness is equal to zero, the transformed distribution may be skewed, thus relying only on the mean, and variance may be misleading because there is ample evidence that investors like positive skewness and dislike negative skewness. For the preceding reasons, it would seem that the CPT paradigm and the M-V rule plus the CAPM paradigm cannot coexist. Moreover, the fact that Markowitz and Sharpe won the Noble Prize in Economics in 1990 for developing the M-V rule and the CAPM, and Kahneman won the Noble Prize in Economics in 2002 for introducing the PT, emphasizes the seeming contradiction between these two paradigms and makes it even more uncomfortable for researchers. If one adopts PT, it seems the M-V and the CAPM should be abandoned. If one employs the M-V and the CAPM, the important evidence regarding subjects’ choices, as clearly shown in PT experiments, are ignored.

Failure of Asset Pricing Models: Transaction Cost, Irrationality, or Missing Factors

Gerber and Shiu (1994) and Henne and Reichling (2006) have independently derived similar stochastic discount factor asset pricing principles in justifying their arguments regarding asset pricing. In this paper, we show that if we slightly modify the two principles, they reach the same asset pricing principle. To be speci fic, in deriving their pricing principles, both Gerber and Shiu (1994) and Henne and Reichling (2006) deal with the case of a single risk-free asset and a single risky asset for which Gerber and Shiu (1994) further assume a financial derivative. Here, we simply extend their logic to portfolios of the entire capital market, in line with the portfolio theory by Markowitz (1952, 1956, 1959), Sharpe (1964), Lintner (1965), and Mossin (1966), to make the two principles converge to each other. And based on the converged version of the pricing principles, we show that the capital asset pricing model by Sharpe (1964), Lintner (1965), and Mossin (1966), the capital asset pricing model with time-dependent beta by Henne and Reichling (2006), the arbitrage pricing theory by Ross (1976), the Gerber-Shiu model by Gerber and Shiu (1994) (a generalization of the Black-Scholes model by Black and Scholes (1973) and Merton (1973b)), and the Esscher insurance premium pricing principle by Buhlmann (1980) can all be derived under a unified pricing principle.

This paper derives an equilibrium capital asset pricing model (CAPM) in a market with trading constraints and asset price bubbles. The asset price processes are general semimartingales including Markov jump-diffusion processes as special cases, and the trading constraints considered include short sale restrictions, borrowing constraints, and margin requirements, among others. We derive a generalized intertertemporal CAPM and consumption CAPM for these markets. The implications for empirical testing are that additional systematic risk factors will exist in a market with trading constraints and price bubbles as contrasted with an otherwise equivalent unconstrained market with no price bubbles.

Since William Sharpe introduced the Capital Asset Pricing Model (CAPM) in the 1960s, it has profoundly influenced asset pricing, portfolio management, and corporate financial decision-making. This work employs a comprehensive literature review methodology, utilizing comparative and inductive analysis to elucidate the primary research trajectories and developmental patterns of the CAPM. The Capital Asset Pricing Model (CAPM) was formulated on the foundation of Markowitz's portfolio theory and introduced by William Sharpe. It has since had extensions, including the APT model, the three-factor model, and the four-factor model, along with the latest use of machine learning techniques, aligning capital price more closely with actual market conditions. The research conclusion indicates that, firstly, although the CAPM has certain limitations, its theoretical simplicity and logical rigor make it an important asset pricing tool for investors. Secondly, multi-factor models have become a new path for the development of the CAPM, and further increasing factors can enhance the model's explanatory power. Finally, behavioral finance provides a new perspective for understanding the limitations of the CAPM.

Rate distortion theory and the volatility of asset prices

This study re-evaluates the validity of the Capital Asset Pricing Model (CAPM) in the U.S. Consumer Discretionary sector over the turbulent 2015–2025 decade, a period marked by pandemics, inflationary surges, monetary tightening, and shifts in tariff policy. Adopting a two-pass regression framework with the Shanken (1992) correction to mitigate errors-in-variables bias, we analyze 20 representative S&P 500 firms across subsectors of the sector. Empirical results show the sector's average beta at 1.25, significant beta instability during structural shocks, and a flatter Security Market Line (SML) than CAPM's theoretical prediction. The market risk premium is positive but statistically insignificant after Shanken correction, and the low-beta anomaly is pronounced, with defensive stocks outperforming high-beta counterparts on a risk-adjusted basis. We find that non-market systematic factors—including tariff policy risk, interest rate duration, and operating leverage—undermine CAPM's explanatory power, alongside behavioral herding and speculative bubbles. Multifactor models (e.g., Fama-French Three-Factor) exhibit better performance but still struggle with tail-risk events. This research concludes that CAPM serves as a useful first-order risk approximation but is insufficient for the modern Consumer Discretionary sector. Dynamic, multifactor frameworks that incorporate policy and macroeconomic sensitivities are necessary for accurate asset pricing and investment decision-making in this volatile sector.