The systematic risk estimation models: A different perspective

There is a long-standing debate in the statistical, epidemiological, and econometric fields as to whether nonparametric estimation that uses machine learning in model fitting confers any meaningful advantage over simpler, parametric approaches in finite sample estimation of causal effects. We address the question: when estimating the effect of a treatment on an outcome, how much does the choice of nonparametric vs parametric estimation matter? Instead of answering this question with simulations that reflect a few chosen data scenarios, we propose a novel approach to compare estimators across a large number of data-generating mechanisms drawn from nonparametric models with semi-informative priors. We apply this proposed approach and compare the performance of two nonparametric estimators (Bayesian adaptive regression tree and a targeted minimum loss-based estimator) to two parametric estimators (a logistic regression-based plug-in estimator and a propensity score estimator) in terms of estimating the average treatment effect across thousands of data-generating mechanisms. We summarize performance in terms of bias, confidence interval coverage, and mean squared error. We find that the two nonparametric estimators can substantially reduce bias as compared to the two parametric estimators in large-sample settings characterized by interactions and nonlinearities while compromising very little in terms of performance even in simple, small-sample settings.

We consider the problem of choosing between two density estimates, a non-parametric estimate with the the standard properties of nonparametric estimates (universal consistency, robustness, but not extremely good rate of convergence) and a special estimate designed to perform well on a given set T of densities. The special estimate can often be thought of as a parametric estimate. The selection we propose is based upon the L1 distance oetween both estimates. Among otner things, we show how one should proceed to insure that the selected estimate matches the special estimate's rate on T, and that it matches the nonparametric estimate's rate off T

D UE to Sharpe (1964), Lintner (1965) and Mossin (1966), the Capital Asset Pricing Model (CAPM) has been employed to estimate systematic and performance measure and to predict the risk-return relationship. The predictive ability of this model has been examined by Friend and Blume (FB) (1973), Black, and Scholes (BJS) (1972) and Blume and Friend (BF) (1970). They have concluded that the empirical results obtained from the CAPM are significantly different from the ex ante expectation of this model. The effects of investment horizon on the estimate of the systematic were first investigated by (1969). Based upon the instantaneous systematic concept, he concluded that the logarithmic linear form of the CAPM can be used to eliminate the effects of time horizon on the estimated systematic risk; in other words, the basic specification for the CAPM is a Cobb-Douglas type functional form. Levy (1972) has shown that the assumption of a holding period that is different from the true investment horizon will lead to systematic bias of the performance measure index. Recently, Cheng and Deets (CD) (1973) have shown that the logarithmic linear form of the CAPM not only implies a linear relationship but produces an instantaneous risk, dependent upon the length of observed horizon.' In addition, they proposed a new instantaneous systematic entitled the Cheng-Deets instantaneous systematic risk to substitute for the Jensen instantaneous Neither nor CD has ever investigated the effects of finite investment horizon when market equilibrium is not instantaneous. The main purposes of this paper are to derive two alternative functional forms for the CAPM, which will explicitly include the investment horizon parameter, to improve the explanatory power of the CAPM, and to reduce the bias of the estimated systematic risk. In the second section the risk-return relationship is reexamined under the assumptions that true investment horizon is either observable or not observable. In the third section both likelihood ratio and constant elasticity of substitution (CES) function methods are proposed to derive a testable generalized CAPM in accordance with the assumption that all investors have identical investment horizons.2 In the fourth section models derived in the third section are related to Merton's (1970, 1973) continuous time models and Fama and Macbeth's (1973, 1974) empirical work, which supports the linearity of CAPM. In the fifth section, a set of sample data from the New York Stock Exchange (NYSE) during 19671972 will be employed to estimate the related parameters of the nonlinear CAPM being derived in this paper. In addition, the results obtained from nonlinear CAPM will be compared with those obtained from the linear CAPM. Finally, in the last section, the results of this paper will be summarized.

Determinants of Electric Utility Betas. One important aspect of utility regulation is the estimation of cost of equity capital of the firm. Several techniques have been used to estimate the cost of equity, including the discounted cash flow model and the capital asset pricing model (CAPM). CAPM has its foundations in modern portfolio theory and its application has generated a lot of controversy — both from academia and the professional world. Much of the problem in using CAPM in utility rate cases has centered on the issue of estimating the beta coefficient. Myers (1972) points out that problems exist in the following areas: measurement of beta; stability of beta; and incomplete description of risk and return by CAPM. There is evidence to believe that CAPM is still widely used be expert witnesses to explain risk‐return relationships in utility rate cases (Cooley, 1980).

The paper provides a multivariate testing of the Sharpe-Lintner and Black Capital Asset Pricing Model (CAPM) for the Hong Kong stock market. Adopting the multivariate approach has many advantages, such as elimination of the error in variables problem encountered in the univariate approach as well as provision of an efficiency gain in the parameter estimates. The estimation results reject both versions of CAPM. The source of rejection seems to come from the firm size effect as it is found that small firms (which also have small betas) have earned returns higher than the CAPM has predicted. Furthermore, the estimated return of the zero beta portfolio in the Black CAPM is found to be statistically insignificant.

The Correlations and Volatilities of Stock Returns: The CAPM Beta and the Fama-French Factors

Systematic risk (beta) is one of the most effective factors in predicting the appropriate required rate of returns of portfolios. By understanding the systematic risk of usual portfolios of various companies, investors will be able to consider financial investment more confidentially. The aim of this study was to examine the relationship between productivity indices (operating leverage, financial leverage, combined leverage) as independent variables and systematic risk (Beta) as dependent variables. To do so, 112 companies accepted in Tehran Stock Market were selected based on screening (systematic deletion) in a five-year- period, 2004 to 2008. The required data were gathered from basic financial statements, committee reports, and other available documents in Tehran Stock Market. Regression and Pearson correlation were applied to analyze the data. The results of the study revealed that there was no significant relationship between productivity indices and systematic risk. Some suggestions are provided regarding the topic of the research. Key words: Operating leverage, financial leverage, combined leverage, systematic risk (beta).

Estimating the cost of equity for the regulated energy and infrastructure sectors in India

Process capability indices (PCIs) are widely used to assess whether an in-control process meets manufacturing specifications. In most applications of classical PCIs, the process characteristic is assumed normally distributed. However, the normal distribution has been found inappropriate in various applications. In the literature, the percentile-based PCIs are widely used to deal with the nonnormal process. One problem associated with the percentile-based PCIs is that they do not provide a quantitative interpretation to the process capability. In this study, new PCIs that have a consistent quantification to the process capability for both normal and nonnormal processes are proposed. The proposed PCIs are generalizations of the classical normal PCIs in the sense that they are the same as the classical PCIs when the process characteristic follows a normal distribution, and they offer the same interpretation to the process capability as the classical PCIs when the process characteristic is nonnormal. We then discuss nonparametric and parametric estimation of the proposed PCIs. The nonparametric estimator is based on the kernel density estimation and confidence limits are obtained by the nonparametric bootstrap, while the parametric estimator is based on the maximum likelihood estimation and confidence limits are constructed by the method of generalized pivots. The proposed methodologies are demonstrated using a real example from a manufacturing factory.

Two general classes of nonparametric kernel estimators of the hazard function are introduced, which include both a 1-parameter estimator and a more complex 3-parameter estimator. In addition, employing the idea of cross-validation, the authors present a data-based algorithm for smoothing parameter selection. The article compares the data-based 1and 3-parameter estimators in a simulation experiment to the maximum likelihood estimator assuming the correct failure distribution and censoring mechanism. The 3-parameter estimator is found to perform well over a wide range of settings. On the average, the estimator recovers the shape of the underlying failure hazard and is competitive with the parametric estimator over a subset of the positive half line. Two examples illustrate possible uses of the nonparametric estimators.

A well known category of classification error rate estimators is so called parametric error rate estimators. These estimators are typically expressed as functions of the training sample size, the dimensionality of the observation vector and the Mahalanobis distance between the classes. However, all parametric classification error rate estimators are biased and the main source of this bias is the estimate of the Mahalanobis distance. In this paper we propose a new Mahalanobis distance estimation method that is designed for use in parametric classification error rate estimators. Experiments with real world and synthetic data sets show that new estimator helps to reduce the bias of the most common parametric classification error rate estimators. Additionally, non-parametric classification error rate estimators, such as resubstitution, repeated 10-fold cross-validation and leave-one-out are outperformed (in terms of root-mean-square error) by parametric estimators that use new estimates of the Mahalanobis distance.

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.

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

The current study scrutinizes the relationship between the cost of equity and corporate governance practices within companies listed on the Pakistan Stock Exchange (PSX).The Capital Asset Pricing Model (CAPM) is applied to determine the cost of equity. A Sys-GMM model was applied to a sample of 170 businesses listed on the Pakistan Stock Exchange (PSX) from 2014 to 2019 to establish a connection between the variables. Sys-GMM is effective in addressing endogeneity issues related to corporate governance variables. In addition, the Sys-GMM results were compared with pooled OLS and fixed effects. It was discovered that the subsequent approaches failed to address the endogeneity issue, consequently yielding biased results. The findings unveiled a negative nexus amid cost of equity and block ownership. The output indicated that board size plays a key role in reducing the cost of equity. Interestingly, audit quality is directly related to equity cost, however, female directors and leverage were not linked with the cost of equity. The findings will direct policymakers in expanding or adapting the scope and depth of corporate governance practices, which will certainly facilitate the production of the right corporate climate in the country.