Mean-variance Capital Asset Pricing Model Research Articles

Which measure of systematic risk should we use? An empirical study on systematical risk and Treynor measure using the economic index of riskiness and operational measure of riskiness

AbstractThis paper empirically studies the differences among the systematic risks of three asset pricing models, namely; the mean–variance capital asset pricing model (MV‐CAPM), AS‐CAPM and FH‐CAPM. The last two are derived by replacing variance with the Aumann‐Serrano (AS) index and the Foster‐Hart (FH) as the risk measure in MV‐CAPM. We use the Dow Jones Industrial Average (DJIA) index as a proxy for the market portfolio, and its component stocks to check if the systematic risks and the Treynor measures are different. The monthly return data from January 1997 to October 2017 are used for empirical estimations. The results show that the three systematic risks are highly correlated. Similarly, high correlation is also found for the three Treynor measures. It seems that even though they are derived under different risk measures, they produce almost the same systematic risk and performance measure for individual stocks. Therefore the findings of the present study suggest that any of the above measures can be used in empirical finance in the area of risk management. As this finding is different from those of other studies in the existing literature in this area, this study makes a contribution to the finance literature.

Computing equilibrium prices for a capital asset pricing model with heterogeneous beliefs and margin-requirement constraints

The mean-variance capital asset pricing model (CAPM) is a useful mathematical tool for studying a variety of financial problems. In contrast to existing work in the literature, which has primarily focused on deriving analytical solutions under restrictive assumptions, we propose a numerical algorithm for efficiently computing the set of equilibrium prices of a CAPM model with heterogeneous investors and arbitrary margin requirements. We present the mathematical formulation of the CAPM model, derive structural properties of the portfolio selection and excess demand functions, and establish the asymptotic convergence of the proposed algorithm under mild conditions. To illustrate the utility of the algorithm, we perform sensitivity analysis on a simple example to study the impact of marginal requirements and interest rates on the resulting equilibrium prices. Numerical studies are also carried out to compare the performance of the algorithm with that of two other popular methods, namely, the fixed point method and the brand-and-bound algorithm.

Downside risk analysis of returns on the Karachi Stock Exchange

Purpose – The purpose of this paper is to analyze the mean-variance capital asset pricing model (CAPM) and downside risk-based CAPM (DR-CAPM) developed by Bawa and Lindenberg (1977), Harlow and Rao (1989), and Estrada (2002) to assess which downside beta better explains expected stock returns. The paper also explores whether investors respond differently to stocks that co-vary with declining market than to those of co-vary with rising market. Design/methodology/approach – The paper uses monthly data of closing prices of stocks listed at the Karachi Stock Exchange (KSE). The data cover the period from January 2000 to December 2012. The standard, downside, and upside betas are estimated for different sub-periods,and then,their validity to quantify the risk premium is tested for subsequent sub-periods in a cross sectional regression framework. Though our empirical methodology is similar to that of Fama and MacBeth (1973) for testing the CAPM and the DR-CAPM, our approach to estimate the downside beta is different from earlier studies. In particular, we follow Estrada ' s (2002) suggestions and obtain the correct and unbiased estimation of the downside beta by running the time series regression through origin. The authors carry out the two-pass regression analysis using the generalized method of moment (GMM) in the first pass and the generalized least squares (GLS) estimation method in the second pass. Findings – The results indicate that the mean-variance CAPM shows a negative risk premium for monthly returns of selected stocks. However, the results for the DR-CAPM of Bawa and Lindenberg (1977) and Harlow and Rao (1989) provide evidence of a positive risk premium for the downside beta. In contrast, the DR-CAPM of Estrada (2002) shows a negative risk premium in some sub-periods while the positive premium in the others. By comparing the risk premium for both downside and upside risks in a single-equation framework, the authors show that the stocks that co-vary with a declining market are compensated with a positive premium for bearing the downside risk. Yet, the risk premium for stocks that are negatively correlated with declining market returns is negative for all the three-downside betas in all the examined sub-periods. Practical implications – The empirical findings of the paper are of great significance for investors for designing effective investment strategies. Specifically, the results help investors to identify an appropriate measure of risk and to construct well-diversified portfolio. The results are also useful for firm managers in capital budgeting decision-making process as they enable them to cost equities appropriately. The results also suggest that the risk-return relationship implied by mean-variance CAPM is negative and therefore this model is not suitable for gauging the risk associated with stocks traded in KSE. Yet, the authors show that DR-CAPM out performs in quantifying the risk premium. Originality/value – Unlike prior empirical studies, the authors follow Estrada’s (2002) suggestions where downside beta is calculated using regression through origin to find correct and unbiased beta. Departing from the existing literature the authors estimate three different versions of DR-CAPM along with the standard CAPM for comparison purpose. Finally, the authors apply sophisticated econometrics methods that help in lessening the problem of non-synchronous trading and the issue of non-normality of returns distribution.

An Empirical Analysis of Higher Moment Capital Asset Pricing Model for Bangladesh Stock Market

Capital Asset Pricing Model (CAPM) describes a relationship which is linear between expected return and risk of an asset. Within the contents of this paper, the higher moments of return distributions for companies listed in the Dhaka Stock Exchange (DSE) market have been inspected for the period of January 2005 to December 2009. The mean-variance CAPM model is extended by taking higher moments-Skewness and Kurtosis. Monthly stock returns from 80 non-financial companies, covering ten sectors (Engineering, Food & Allied, Fuel & Power, Textile, Pharmaceuticals, Services & Real Estate, Cement, Tannery, Ceramic and Miscellaneous) are studied in this research. From the empirical analysis, it is observed that the intercept term is significantly different from zero and insignificant relationship between beta and excess returns both in mean-variance CAPM and higher moment CAPM conditions. This means that the market excess returns provide no explanation for the asset rate of return, whether or not third and fourth moments are considered in the regression model. But, when the higher moments are introduced, the adjusted R-square increases 0.037 to 0.257. It is noticed that the risk premium for co-skewness risk is positive for the period 2005-2009, indicating that the co-skewness risk is compensated in the DSE market for the studied period. Also, the co-kurtosis risk is rewarded by the market. Thus, in describing risk-return relationship in emerging markets like Bangladesh stock market, the higher moment CAPM performs comparatively well.

Second-Order Stochastic Dominance, Reward-Risk Portfolio Selection, and the CAPM

AbstractStarting from the reward-risk model for portfolio selection introduced in De Giorgi (2005), we derive the reward-risk Capital Asset Pricing Model (CAPM) analogously to the classical mean-variance CAPM. In contrast to the mean-variance model, reward-risk portfolio selection arises from an axiomatic definition of reward and risk measures based on a few basic principles, including consistency with second-order stochastic dominance. With complete markets, we show that at any financial market equilibrium, reward-risk investors' optimal allocations are comonotonic and, therefore, our model reduces to a representative investor model. Moreover, the pricing kernel is an explicitly given, non-increasing function of the market portfolio return, reflecting the representative investor's risk attitude. Finally, an empirical application shows that the reward-risk CAPM captures the cross section of U.S. stock returns better than the mean-variance CAPM does.

A Mean-Variance Capital Asset Pricing Model for Long Short Equity Hedge Fund Portfolios

In this paper a single factor mean-variance Capital Asset Pricing model is derived for long short equity investment portfolios. This model is empirically tested using a statistical arbitrage momentum trading strategy based on Dow Jones 30 historical equity data from 1986 to 2002. This trading strategy commonly utilized in practice by hedge funds is typically classified as systematic statistical arbitrage within the long short equity style. The equilibrium two moment model developed has a desirable feature in that it is leverage invariant - a useful quality if used for long short hedge fund performance measurement where leverage may be unknown, non-constant and which may vary widely between various long short hedge fund portfolios making alpha estimation difficult. Alpha is modeled as a function of beta using the concept of entropy as interpreted by Shannon\cite{Shannon48}. The two moment equations derived constitute equilibrium mean-variance model and as such long only, short only and fractional beta long/short portfolios can be priced in the CAPM sense. The approach taken is to assume that during the life of the investment, the overall portfolio can be represented by the returns of the two sub-portfolios represented by long and short positions via separation with a corresponding correlation coefficient. It is shown that the expected portfolio return model agrees with the classical CAPM for expected portfolio return\cite{Sharpe64} in the theory of market equilibrium under conditions of risk. The same analytical technique used for the first moment model derivation is then used to derive the CAPM variance model. The validity of this variance model is tested at the 95 per cent confidence level by empirically simulating a variable fractional beta long short statistical arbitrage momentum trading strategy using Dow Jones 30 daily equity price data from 1988 to 2002. The mean-variance model is then developed further in leverage invariant equilibrium which leads to the formulation of an analytical dynamic alpha equation. As this function can have a maximum value as a function of beta, the usual calculus techniques are used to derive the analytical objective function which allows the determination of the optimal mean-variance long short beta portfolio. Minimum existence conditions are then derived which dictate the minimum pre-requisite performance expectations for the creation of a long/Short equity portfolio for domination of the long only market index in the mean-variance sense. Finally, since both the mean and variance expectation equations are derived, a stochastic differential equation is formulated and the Black-Scholes option price derived.

Second Order Stochastic Dominance, Reward-Risk Portfolio Selection and the CAPM

Starting from the reward-risk model for portfolio selection introduced in DeGiorgi (2005), we derive the reward-risk Capital Asset Pricing Model (CAPM) analogously to the classical mean-variance CAPM. In contrast to the mean-variance model, reward-risk portfolio selection arises from an axiomatic definition of reward and risk measures based on few basic principles, including consistency with second order stochastic dominance. With complete markets, we show that at any financial market equilibrium, reward-risk investors' optimal allocations are comonotonic and therefore our model reduces to a representative investor model. Moreover, the pricing kernel is an explicitly given, non-increasing function of the market portfolio return, reflecting the representative investor's risk attitude. Finally, an empirical application shows that the reward-risk CAPM better captures the cross-section of US stock returns than the mean-variance CAPM does.

Pareto optima in incomplete financial markets

This paper demonstrates the structure of the set of Pareto optima in exchange economy models where the choice sets are unbounded and satiation is possible. The main examples are found among asset market models with short-selling and incomplete markets, such as the mean–variance Capital Asset Pricing Model (CAPM) without a riskless asset.

Positive Prices in CAPM

Some equilibrium prices in CAPM may be negative because of nonmonotonicity of preferences. We identify several sets of sufficient conditions for prices to be positive. The central conditions impose bounds on the investors' risk aversion. These bounds do not need to hold globally but only in a relevant range of portfolios or combinations of mean and standard deviation. The relevant range is specified on the basis of exogenous parameters and variables, and it must contain any endogenously determined equilibrium. The bounds on risk aversion ensure that the preferences for assets are sufficiently well-behaved within the relevant range. THE TWO-PERIOD MEAN-VARIANCE capital asset pricing model (CAPM) has the peculiar property that preferences may not be monotone. The expected return to an investor's portfolio increases as he holds more and more shares of the assets, but so does the variance of return. It may be that at some point, the additional expected return gained from adding more shares to the portfolio is not sufficient to compensate for the increase in variance. If so, then the induced preferences for assets are not monotone. Nonmonotonicity of preferences in the mean-variance model is analyzed in Nielsen (1987). Because portfolio preferences are not necessarily monotone, equilibrium asset prices may be negative or zero. In fact, it is possible that the value of the market portfolio of risky assets is negative. This is demonstrated in Nielsen (1985, 1990a) and in two examples in the present paper. When the CAPM is used as a model of the prices of stocks, which in reality have limited liability, it is of course disturbing that it may predict negative prices. It can be seen as a symptom of the fact that mean-variance preferences are not always a good approximation to reality. This paper analyzes various conditions on the exogenous parameters which ensure that the preferences are sufficiently realistic to yield positive prices. The exogenous parameters in question are the expectations, variances and covariances of total returns, the initial allocations of the assets (endowments), and the investors' utility functions for mean and variance of return. The analysis applies to the classical version of the CAPM with a riskless asset, as developed by Sharpe (1964), Lintner (1965) and Mossin (1966), as well as the CAPM with risky assets only, which was introduced by Black (1972), and has been much used in empirical work under the name the zero-beta or two-factor CAPM.

Existence of equilibrium in CAPM

In the mean-variance capital asset pricing model (CAPM), non-monotonicity of preferences may lead to satiation, and short-selling as well as satiation may lead to non-existence of equilibrium. This paper identifies several sets of sufficient conditions for equilibrium to exist. The nature of the central conditions is to impose bounds on the investors' degree of risk aversion, as measured by the marginal rate of substitution between mean and standard deviation of return. These bounds do not need to hold globally but only in a relevant range of portfolios or combinations of mean and standard deviation.

Asset Market Equilibrium with Short-Selling

This paper presents simple conditions and a simple proof of the existence of equilibrium in asset markets where short-selling is allowed and satiation is possible. Unlike standard non-satiation assumptions, the one used here is weak enough to be reasonable in the mean-variance Capital Asset Pricing Model and in asset market models where investors maximize expected utility and where total returns to individual assets may be negative. This paper analyses the existence of equilibrium in exchange economy models where choice sets may be unbounded below and satiation is likely. There are two main examples of such models. One is an asset market model where investors maximize expected utility and the total return to individual assets may be negative with positive probability. The other is the mean-variance Capital Asset Pricing Model (CAPM), where investors do not necessarily maximize expected utility but instead maximize a function of only the mean and variance of return to their portfolios. In both of these models, the investors' choice sets will be unbounded if short-selling is allowed. To short-sell a share of an asset means to borrow it and sell it, promising to buy it back and return it to the lender at a later date. Formally, short-selling corresponds to holding a negative number of shares. If unlimited short-selling is allowed, then there is no limit to how large negative numbers of shares can be held, and therefore the choice sets are unbounded. Satiation can also occur both in the mean-variance model and in the expected-utility model (unless there is a riskless asset). In the expected-utility model, there may be satiation if the individual assets have negative returns with positive probability. As the investor gets more and more shares, the potential positive returns get larger and larger, but so do the potential negative returns. If the investor is very averse to large negative returns, then he may eventually be satiated and not want any more shares. In the mean-variance model, the expected return to an investor's portfolio increases as he holds more and more shares of the assets, but so does the variance of return. It may be that at some point, the additional expected return gained from adding more shares to the portfolio is not sufficient to compensate for the increase in variance. If so, then there will be satiation. Satiation in the mean-variance model iie analysed in Nielsen (1987, 1988). The expected-utility model with short-selling has been used in many specific analyses of portfolio selection and asset pricing. To name but one, Connor (1984) studies arbitrage pricing in a general equilibrium model where there are no short sales restrictions. The mean-variance CAPM is of course very prominent in the finance literature. Because the

Systematic Risk, Unsystematic Risk, and Property-Liability Rate Regulation

Several researchers suggest that property-liability rates be regulated using total variance risk measures. Others insist that only systematic risk measures are relevant. A normative model is proposed to reconcile these diverse approaches. Empirical analysis indicates that the model provides a higher risk premium and is reliable when compared to the mean-variance Capital Asset Pricing Model.

Uniqueness of Equilibrium in the Classical Capital Asset Pricing Model

General equilibrium in the classical two-period mean-variance capital asset pricing model is not unique. Corresponding to one single set of expectations, utility functions, and an initial wealth distribution, there may be several equilibria, and an asset may have different prices, expected rates of return, and betas in different equilibria. However, any equilibrium portfolio is sustained by a unique price system, and if investors have decreasing risk aversion, then any equilibrium allocation of the risky assets is sustained by a unique price system.

An Integration of CAPM with IS-LM Analysis into a Unified Theory of Aggregate Demand

Ever since James Tobin [23] explained the demand for money in a portfolio context economists have been aware of the relation between portfolio behavior and aggregate demand. On a most basic level, since the demand for money is at least partly dependent on portfolio behavior, the LM and aggregate demand schedules are also dependent. Nevertheless, formal derivations of aggregate demand schedules (e.g., from IS-LM schedules) have not as yet taken portfolio considerations fully into account.' Likewise, the vast literature on capital market theory emanating out of the Sharpe [22]-Lintner [18]-Mossin [19] mean-variance capital asset pricing model (CAPM) assumes that the level of aggregate demand (or income) is exogenously determined. Two of the most important literatures in macroeconomics, aggregate demand theory and capital market theory, have been largely developed along separate paths. The purpose of this paper is to unify the relationship between aggregate demand theory and capital market theory by explicitly incorporating the CAPM into an IS-LM framework.2 In this way we can examine the effects of shocks to the IS and LM schedules upon the returns to riskless and risky securities and the feedback effects upon aggregate demand. The influence of these shocks on aggregate demand (i.e., the monetary and autonomous spending multipliers) will be shown to depend on the amounts of security risk (beta coefficients) as well as upon investors' risk-return attitudes.3

Notes on Multiperiod Valuation and the Pricing of Options: A Comment

rates as somehow deeper and more significant than those that serve other purposes, whereas the two types of models differ mainly in terms of the variables assumed to be endogenous or exogenous. Bhattacharya believes that the assumptions of the SS model: exponential utility, multivariate normally distributed cash flows and non-stochastic future interest rates are mutually inconsistent. He also does not realize that, unlike most of the models he surveys, the SS model is fully consistent with the single period mean-variance Capital Asset Pricing Model (CAPM) i.e., has 'rationally expected' normal distributions of future stock prices. Another major advantage of the SS model is that it states firm values as functions of inter- and intratemporal covariances of cash flows analogous to the single period CAPM, while maintaining the essential features of the multiperiod valuation problem.' Given these lacunae in the Bhattacharya survey, we can only conclude that the relationship between different multiperiod models requires further clarification. We first deal with the substantive point about the consistency of the SS assumptions and then comment on the linkages between the assumptions and results of various multiperiod models.

Expected Monetary Changes and Relative Prices: A Look at Evidence from the Stock Market

At the heart of the natural rate theory is the assumption that fully perceived monetary growth is neutral with respect to relative price changes. Unfortunately, there have been few attempts to test this assumption directly. Because the belief that active contracyclical monetary policies are not superior to Friedman's constant growth rule rests upon the assumption that expected monetary changes have neutral relative price effects, more evidence is needed on whether such changes are, in fact, neutral. To this end, we test the hypothesis that only the unexpected components, and not the expected components, of monetary changes affect relative price changes by looking at the stock market. Testing is done on a modification of the well known mean-variance Capital Asset Pricing Model, the CAPM. Specifically, we modify the CAPM to allow for differential stock returns depending upon expected and unexpected monetary growth rates. Section II explains the basic model used to test for non-neutral relative price effects of monetary change. Section III deals with the empirical definitions of expected and unexpected money growth. Section IV discusses the basic data while Sections V and VI make neutrality tests of monetary growth using actual, expected, revisions in expected, and unexpected money growths. The last part summarizes the paper.

Mean-Variance Theory in Complete Markets

Two paradigms in the pricing of risky assets are the model of Arrow and Debreu and mean-variance pricing as embodied in the capital asset pricing model (CAPM). The former is primarily of theoretical interest. In fact, prior to the recent work of Banz and Miller (1978) and Breeden and Litzenberger (1978) there was little thought or hope of practical applications. On the other hand, the hegemony of the CAPM is due mostly to its apparent ease of applicability and, to a lesser extent, its empirical justification. In a competitive market with no taxes, transactions costs, or institutional constraints on portfolio holdings, the allocation of the available assets will be Pareto optimal. The end-of-period distribution of wealth (or the consumption goods), however, will typically be Pareto optimal only when there is available an Arrow-Debreu security for each contingency. Since, by definition, a Pareto superior reallocation is weakly preferred by all investors, a complete set of Arrow-Debreu markets should naturally arise provided they are not too costly to establish. The complete-markets model of Arrow and Debreu and the meanvariance capital asset pricing model (CAPM) are two paradigms of risky asset markets. This paper provides an integration of these two theories. We prove that the standard meanvariance separation theorem obtains in a complete market only if all investors have quadratic utility. In addition, the familiar CAPM pricing relation can hold for all assets in a complete market only if arbitrage opportunities exist. On the positive side, we show that the validity of this relation for the primary assets in a complete market is unaffected by the distribution of returns on the created financial assets.

Divergent Rates, Financial Restrictions and Relative Prices in Capital Market Equilibrium

The mean-variance capital asset pricing model (CAPM) of Sharpe and Lintner was extended by Brennan [3] to incorporate divergent borrowing and lending rates. He found that in equilibrium the security market line (SML) has the same structure as the SML under the single-rate CAPM of Sharpe and Lintner. That is, the expected return of a security or a portfolio remains linear in its systematic risk, with the intercept replaced by an equivalent risk-free return, which is an average of the divergent borrowing and lending rates weighted by the investors' taste parameters. The equivalent risk-free return is larger than the riskless lending rate and, hence, does not represent an inconsistency with the empirical findings by Friend and Blume [4] and by Black, Jensen and Scholes [1[ that the intercept of empirical SML estimated for the single-rate CAPM is larger than the riskless rate. Moreover, Brennan attempted to show that his construct can be extended to the extreme case where there are no riskless opportunities. The case of no riskless opportunities was of course investigated by Black [2], who generalized the CAPM and SML by inventing the concept of zero-beta port-folio to account for the same empirical problem encountered in the traditional SML tests of CAPM. Since the Sharpe-Lintner single-riskless-rate CAPM implies a perfect loan market, we may view the attempts by Black and Brennan as generalizing the CAPM by incorporating financial restrictions and loan market imperfections. Their primary motive, however, is empirical, i.e., to reconcile the results from the traditional SML tests with their generalized CAPM.