The Evolution of Capital Asset Pricing Models: Update and Extension

The specification of the stochastic process for stock prices is an important assumption underlying most equilibrium and option-pricing models in modern finance. This is particularly true of the continuous-time capital asset pricing models where it is assumed that asset prices follow diffusion processes with continuous sample paths (see Merton 1973a; Breeden 1979; Grossman and Shiller 1982). The extent to which the implications of these models generalize to discontinuous sample paths is an unanswered question in theoretical finance. However, the answer to this question is of more than just theoretical interest since accumulated empirical evidence appears to be inconsistent with the continuous sample path assumption (see Rosenberg 1972; Oldfield, Rogalski, and Jarrow 1977; Rosenfeld 1982). One purpose of this paper is to provide some answers to this question. We do this by extending Merton's (1973a) intertemporal asset pricing model, in the special case of a constant investment opportunity set, to include discontinuous sample paths for asset prices. The constant investment opportunity set assumption is imposed because we are interested in obtaining sufficient conditions under which an instantaneous capital asset pricing model (CAPM) results. The extension of our model to a stochastic opportunity set is discussed briefly in the text. This paper investigates an economy where stock prices follow a jump-diffusion process. The first part of the analysis derives a sufficient condition under which an instantaneous CAPM will characterize equilibrium expected returns. The sufficient condition is that the jump component of a stock's return must be unsystematic and diversifiable in the market portfolio. The second part of the paper is an empirical investigation of the satisfaction of this condition. The evidence is seen to be inconsistent with the satisfaction of this hypothesis, that is, the market portfolio appears to contain a jump component.

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 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.

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

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.

This paper explores the empirical research based on the asset pricing model in China's securities market under the COVID-19 epidemic. Capital asset pricing model, which are frequently employed in corporate finance and investment decision-making, serve as the foundation for price theory in financial markets. However, the research on asset pricing models in China is still in its infancy, and there is a lack of detailed and profound empirical studies. The COVID-19 epidemic causes some economic damage to China's securities market. Therefore, the empirical research in the era of the epidemic is necessary, and it can help investors find suitable investment portfolios. Using asset pricing models and related indicator formulas, the results of the study show that the risk of bond fonds, blend funds, and stock funds, in addition to monetary funds, increases with the arrival of COVID-19. Among them, blend funds and stock funds have better performances, but the operational risk of these three types of funds outweighs the return.

Two general approaches to the problem of valuing assets under uncertainty may be distinguished. The first approach relies on arbitrage arguments of one kind or another, while under the second approach equilibrium asset prices are obtained by equating endogenously determined asset demands to asset supplies, which are typically taken as exogenous. Examples of the former range from the static arbitrage arguments which underlie the Modigliani–Miller theorem to the dynamic arbitrage strategies which are the basis for the Option Pricing Model: such arbitrage based models can only yield the price of one asset relative to the prices of other assets. The Capital Asset Pricing Model (CAPM) is an example of an equilibrium model in which asset prices are related to the exogenous data, the tastes and endowments of investors although, as we shall see below, the CAPM is often presented as a relative pricing model.

Asset pricing theories provide an understanding of the risk factors that determine the price of assets. The identification of risk factors assists investors in seeking out profitable investment opportunities. The difficulty in observing how investors identify such opportunities and how they react to it has largely restricted the literature from determining the factors that matter to an investor. There is a paucity of empirical studies that give insight into the investment decision-making process of an investor. Some of the recent studies that seek to fill this gap have used mutual fund flows to infer which asset pricing model investors use. The fund flows are used as a measure of investors’ response for the identification of a positive net present value investment opportunity. These studies suggest that the Capital Asset Pricing Model (CAPM) is closest to the asset pricing model used by investors in the US market. Taking this literature forward, we enquire whether investors from Indian markets exhibit a similar pattern when making investment decisions. Using the fund flows to actively managed equity schemes, we have investigated the risk factors that matter for mutual fund investors in India for the period from April 2006 to December 2019. We use alternative performance measure and then evaluate the sensitivity of fund flows to each of the performance measures. Our results suggest that investors assess the performance of competing investment opportunities based on naïve measures. Further, though uncertainty has a negative impact on fund flow, the flow–performance relation is not sensitive to the level of uncertainty.

The paper discusses efficiency tests of two popular asset pricing models; Sharpe-Lintner (CAPM) and Fama-French 3 factor model using banking stock data at the Nairobi Securities Exchange.The purpose is to propose a more efficient model between the two based on end year returns for 10 banks over the study period 2010-2015.Design: Comparative analysis: Fitting data on stock returns, market returns, size and value factors to the two asset pricing models to find their statistical power measured by R 2 and significance of the coefficient alpha.Results: Fama-French 3 factor model has better fit (R 2 =77.5%) compared to CAPM (R 2 =57%) when the bank stock data is analysed and fitted into the two models.However value factor is irrelevant in the 3 factor model.The coefficient alpha is observed not to be different from zero signifying better efficiency for the FF model.Stock return is positively related with market risk(r=0.775,p=0.000) but negatively associated with size (r= -0.841, p=000). Originality:The paper makes unique contribution by validating asset pricing models using bank stock prices at the Nairobi Securities Exchange.It uses end year stock data (low frequency) which can then be compared with higher frequency data models.Lastly a shortened version of Fama-French (1993) is proposed for pricing bank stocks that captures only market risk and size effects as explanatory factors at the NSE.

We show that the external habit‐formation model economy of Campbell and Cochrane (1999) can explain why the Capital Asset Pricing Model (CAPM) and its extensions are betterapproximate asset pricing models than is the standard onsumption‐based model. The model economy produces time‐varying expected eturns, tracked by the dividend–price ratio. Portfolio‐based models capture some of this variation in state variables, which a state‐independent function of consumption cannot capture. Therefore, though the consumption‐based model and CAPM are both perfect conditional asset pricing models, the portfolio‐based models are better approximate unconditional asset pricing models.

In a competitive market, what should the rate of return on insurance companies’ equity be? What premiums should insurance companies charge? Traditionally, the answers to these questions have been based on actuarial and accounting concepts.1 More recently financial models of the insurance firm have been developed. Ferrari (1968) suggested a descriptive model which allowed an algebraic expression for the rate of return on equity as a function of the premiums charged to be derived. Combining this with the capital asset pricing model (CAPM) meant that an equilibrium value for the return on equity and the corresponding level of premiums could be found. This model is known as the insurance CAPM.2 The development of other asset pricing models in finance has also led to insurance counterparts. Thus the Arbitrage Pricing Theory, and option pricing models have been used to derive the return on equity and the level of premiums.3 Although the approaches based on asset pricing models have advantages compared to those based on traditional actuarial and accounting concepts, they are not ideal. One of the most important problems is that they are not well suited for finding premiums when an insurance company has multiple divisions. The difficulty is that it is not clear how earnings on reserves should be allocated among the various divisions of the firm.

The CAPM: A Case of Elegance is for Tailors?

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.

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.

Risk, the Pricing of Capital Assets, and the Evaluation of Investment Portfolios