ALTERNATIVE TESTS OF THE ZERO‐BETA CAPM

In this paper I develop an analytical Wald test of the zero‐beta capital asset pricing model (CAPM) in a simple iid (independent and identically distributed) setting and extend the Wald test to the generalized method of moments (GMM) framework that allows for a general form of serial correlation and conditional heteroskedasticity. The size and power of these tests, along with some existing tests, are investigated under normal errors and other alternative distributional specifications. The results show that, under alternative distributional assumptions for the error terms, the proposed Wald and GMM tests have reliable sizes for medium‐size samples, whereas the likelihood ratio test (LRT) rejects the efficiency too often, especially when the error terms significantly deviate from normality. However, the LRT is more powerful than both the Wald and GMM tests. JEL classification: C13, C53, G14.     

Risk Measurement Performance of Alternative Distribution Functions

This paper evaluates the performance of three extreme value distributions, i.e., generalized Pareto distribution (GPD), generalized extreme value distribution (GEV), and Box‐Cox‐GEV, and four skewed fat‐tailed distributions, i.e., skewed generalized error distribution (SGED), skewed generalized t (SGT), exponential generalized beta of the second kind (EGB2), and inverse hyperbolic sign (IHS) in estimating conditional and unconditional value at risk (VaR) thresholds. The results provide strong evidence that the SGT, EGB2, and IHS distributions perform as well as the more specialized extreme value distributions in modeling the tail behavior of portfolio returns. All three distributions produce similar VaR thresholds and perform better than the SGED and the normal distribution in approximating the extreme tails of the return distribution. The conditional coverage and the out‐of‐sample performance tests show that the actual VaR thresholds are time varying to a degree not captured by unconditional VaR measures. In light of the fact that VaR type measures are employed in many different types of financial and insurance applications including the determination of capital requirements, capital reserves, the setting of insurance deductibles, the setting of reinsurance cedance levels, as well as the estimation of expected claims and expected losses, these results are important to financial managers, actuaries, and insurance practitioners.     

On the distribution and conditional heteroscedasticity in Taiwan stock prices

This paper studies the distribution and conditional heteroscedasticity in stock returns on the Taiwan stock market. Apart from the normal distribution, in order to explain the leptokurtosis and skewness observed in the stock return distribution, we also examine the Student-t, the Poisson–normal, and the mixed-normal distributions, which are essentially a mixture of normal distributions, as conditional distributions in the stock return process. We also use the ARMA (1,1) model to adjust the serial correlation, and adopt the GJR–generalized autoregressive conditional heteroscedasticity (GARCH (1,1)) model to account for the conditional heterscedasticity in the return process. The empirical results show that the mixed–normal–GARCH model is the most probable specification for Taiwan stock returns. The results also show that skewness seems to be diversifiable through portfolio. Thus the normal–GARCH or the Student-t–GARCH model which involves symmetric conditional distribution may be a reasonable model to describe the stock portfolio return process¹.     

Financial market models with Lévy processes and time-varying volatility

Asset management and pricing models require the proper modeling of the return distribution of financial assets. While the return distribution used in the traditional theories of asset pricing and portfolio selection is the normal distribution, numerous studies that have investigated the empirical behavior of asset returns in financial markets throughout the world reject the hypothesis that asset return distributions are normally distribution. Alternative models for describing return distributions have been proposed since the 1960s, with the strongest empirical and theoretical support being provided for the family of stable distributions (with the normal distribution being a special case of this distribution). Since the turn of the century, specific forms of the stable distribution have been proposed and tested that better fit the observed behavior of historical return distributions. More specifically, subclasses of the tempered stable distribution have been proposed. In this paper, we propose one such subclass of the tempered stable distribution which we refer to as the “KR distribution”. We empirically test this distribution as well as two other recently proposed subclasses of the tempered stable distribution: the Carr–Geman–Madan–Yor (CGMY) distribution and the modified tempered stable (MTS) distribution. The advantage of the KR distribution over the other two distributions is that it has more flexible tail parameters. For these three subclasses of the tempered stable distribution, which are infinitely divisible and have exponential moments for some neighborhood of zero, we generate the exponential Lévy market models induced from them. We then construct a new GARCH model with the infinitely divisible distributed innovation and three subclasses of that GARCH model that incorporates three observed properties of asset returns: volatility clustering, fat tails, and skewness. We formulate the algorithm to find the risk-neutral return processes for those GARCH models using the “change of measure” for the tempered stable distributions. To compare the performance of those exponential Lévy models and the GARCH models, we report the results of the parameters estimated for the S&P 500 index and investigate the out-of-sample forecasting performance for those GARCH models for the S&P 500 option prices.     

Efficient Hedge Fund Strategy Allocations – Systematic Framework for Investors that Incorporates Higher Moments

In this paper, we provide a realistic framework that investors can use to optimize hedge fund portfolio strategy allocations. Our approach includes important aspects that investors need to address in the real world, such as how limited resources can restrict the ability to conduct multiple due diligences. Additionally, our approach is not based on a utility function, but on an easily quantifiable preference parameter, lambda. We need to account for higher moments of the return distribution within our optimization and approximate a best‐fit distribution. Thus we replace the empirical return distributions, which are often skewed or exhibit excess kurtosis, with two normal distributions. We then use the estimated return distributions in the strategy optimization. Our dataset comes from the Lipper TASS Hedge Fund Database and covers the June 1996‐December 2008 time period. Our results show in‐ and out‐of‐sample that our framework yields superior results to the Markowitz framework. It is also better able to manage regime switches, which tend to occur frequently during crises. Lastly, to test our results for stability, we use robustness tests, which allow for the time‐varying correlation structures of the strategies.     

Capital Asset Pricing Compatible with Observed Market Value Weights

We show that the set of expected return vectors, for which an observed portfolio is mean variance (MV) efficient, is a two-parameter family. We identify ten ways to specify the time series behavior of the two parameters; the result highlights a number of inconsistencies involved in MV modelling. For each of the cases, it permits the inference of the time series of expected return vectors, as well as all the other Capital Asset Pricing Model (CAPM) variables, compatible with a known covariance matrix and the observed time series of market value weights. The empirical work shows that there are substantial case-to-case differences in the time series of mean vectors and many of them are quite different from the constant mean vector envisioned in tests of the CAPM.     

Estimating and testing beta pricing models: Alternative methods and their performance in simulations

We conduct a simulation analysis of the Fama and MacBeth[1973. Risk, returns and equilibrium: empirical tests. Journal of Political Economy 71, 607–636.] two-pass procedure, as well as maximum likelihood (ML) and generalized method of moments estimators of cross-sectional expected return models. We also provide some new analytical results on computational issues, the relations between estimators, and asymptotic distributions under model misspecification. The generalized least squares estimator is often much more precise than the usual ordinary least squares (OLS) estimator, but it displays more bias as well. A “truncated” form of ML performs quite well overall in terms of bias and precision, but produces less reliable inferences than the OLS estimator.     

Investment decisions when utility depends on wealth and other attributes

The problem of optimal investment under a multivariate utility function allows for an investor to obtain utility not only from wealth, but other (possibly correlated) attributes. In this paper we implement multivariate mixtures of exponential (mixex) utility to address this problem. These utility functions allow for stochastic risk aversions to differing states of the world. We derive some new results for certainty equivalence in this context. By specifying different distributions for stochastic risk aversions, we are able to derive many known, plus several new utility functions, including models of conditional certainty equivalence and multivariate generalisations of HARA utility, which we call dependent HARA utility. Focusing on the case of asset returns and attributes being multivariate normal, we optimise the asset portfolio, and find that the optimal portfolio consists of the Markowitz portfolio and hedging portfolios. We provide an empirical illustration for an investor with a mixex utility function of wealth and sentiment.     

The Analytics of Performance Measurement Using a Security Market Line

Security market line (SML) analysis, while an important tool, has never been fully justified from a theoretical standpoint. Assuming symmetric information and an inefficient index, we show that SML analysis can be grossly misleading, since, in general, efficient and inefficient portfolios can plot above and below the SML. On a more positive note, if SML analysis uses the return on a marketed riskless asset for the zero-beta rate, efficient portfolios must plot above the SML. Nonetheless, arbitrarily inefficient portfolios also plot above the SML.     

Nonparametric rank tests for event studies

Because stock prices are not normally distributed, the power of nonparametric rank tests dominate parametric tests in event study analyses of abnormal returns on a single day. However, problems arise in the application of nonparametric tests to multiple day analyses of cumulative abnormal returns (CARs) that have caused researchers to normally rely upon parametric tests. In an effort to overcome this shortfall, this paper proposes a generalized rank (GRANK) testing procedure that can be used on both single day and cumulative abnormal returns. Asymptotic distributions of the associated test statistics are derived, and their empirical properties are studied with simulations of CRSP returns. The results show that the proposed GRANK procedure outperforms previous rank tests of CARs and is robust to abnormal return serial correlation and event-induced volatility. Moreover, the GRANK procedure exhibits superior empirical power relative to popular parametric tests.     

A Note: Natural Generalization of Black-Scholes in the Presence of Skewness,Using Stable Processes

The Lognormal price model is generalized to the class of Log-Stable Processes, a family which possesses self-similarity properties usually only associated with the Lognormal, but which, more generally, can model negatively skewed distributions of return. This generalization appears to explain several discrepancies between the Black-Scholes Model and observed market phenomena, such as the variation of implied volatility of option price with exercise price and term to expiry, and the nonzero probability of bankruptcy or ‘crash’. It will be argued that the class of maximally negatively skewed Stable distributions (a class which, paradoxically, contains the normal) may be utilized to produce models which imply these phenomena naturally.     

The empirical relationship between risk and return: evidence from the UK stock market

Previous studies reach no consensus on the relationship between risk and return using data from one market. We argue that the world market factor should not be ignored in assessing the risk-return relationship in a partially integrated market. Applying a bivariate generalized autoregressive conditional heteroscedasticity in mean (GARCH-M) model to the weekly stock index returns from the UK and the world market, we document a significant positive relationship between stock returns and the variance of returns in the UK stock market after controlling for the covariance of the UK and the world market return. In contrast, conventional univariate GARCH-M models typically fail to detect this relationship. Nonnested hypothesis tests supplemented with other commonly used model selection criteria unambiguously demonstrate that our bivariate GARCH-M model is more likely to be the true model for UK stock market returns than univariate GARCH-M models. Our results have implications for empirical assessments of the risk-return relationship, expected return estimation, and international diversification.     

ON VARIANCE AND LOWER PARTIAL MOMENT BETAS THE EQUIVALENCE OF SYSTEMATIC RISK MEASURES

The market equilibrium mean-lower partial moment (LPM) model may serve as an alternative to the traditional mean-variance (MV) CAPM for security analysis. It appears that the merits of the MV-CAPM vis-à-vis the mean-LPM model continue to be debated (see Nantell and Price, 1979; Price, Price and Nantell, 1982; and Homaifer and Gaddy, 1990). This paper demonstrates that from a theoretical perspective the two models are equivalent for general riskaverse investors. This modeling consistency is based on the fact that all return distributions which yield the mean-LPM equilibrium model must permit two-fund portfolio separation, and the well-known MV-CAPM is the direct pricing result of the two-fund portfolio separation. Hence any debate about the bias of systematic risk measures between the mean-LPM model and the MV-CAPM is meaningless. Nevertheless, although the economic equilibrium results of the models are identical, this does not indicate that the LPM portfolio theory is redundant or irrelevant. It may be that some behavioral perspective, sensitivity to downside risk is more appropriate than variance as a risk proxy.     

Skewness and leptokurtosis in GARCH-typed VaR estimation of petroleum and metal asset returns

This paper utilizes the most flexible skewed generalized t (SGT) distribution for describing petroleum and metal volatilities that are characterized by leptokurtosis and skewness in order to provide better approximations of the reality. The empirical results indicate that the forecasted Value-at-Risk (VaR) obtained using the SGT distribution provides the most accurate out-of-sample forecasts for both the petroleum and metal markets. With regard to the unconditional and conditional coverage tests, the SGT distribution produces the most appropriate VaR estimates in terms of the total number of rejections; this is followed by the nonparametric distribution, generalized error distribution (GED), and finally the normal distribution. Similarly, in the dynamic quantile test, the VaR estimates generated by the SGT and nonparametric distributions perform better than that generated by other distributions. Finally, in the superior predictive test, the SGT distribution has significantly lower capital requirements than the nonparametric distribution for most commodities.     

Modeling and forecasting expected shortfall with the generalized asymmetric Student-t and asymmetric exponential power distributions

Financial returns typically display heavy tails and some degree of skewness, and conditional variance models with these features often outperform more limited models. The difference in performance may be especially important in estimating quantities that depend on tail features, including risk measures such as the expected shortfall. Here, using recent generalizations of the asymmetric Student-t and exponential power distributions to allow separate parameters to control skewness and the thickness of each tail, we fit daily financial return volatility and forecast expected shortfall for the S&P 500 index and a number of individual company stocks; the generalized distributions are used for the standardized innovations in a nonlinear, asymmetric GARCH-type model. The results provide evidence for the usefulness of the general distributions in improving fit and prediction of downside market risk of financial assets. Constrained versions, corresponding with distributions used in the previous literature, are also estimated in order to provide a comparison of the performance of different conditional distributions.     

The spirit of capitalism among the income classes

This paper tests the consumption-based capital asset model within the context of the spirit of capitalism. The spirit of capitalism asserts that consumers gain utility not just from consumption of goods and services, but also from the social status obtained from wealth. We examine two asset pricing models developed by Bakshi and Chen (1996) that employ wealth in the utility function, for households sorted by income quintiles. In the first model, households obtain utility from both consumption and the social status that comes from their own wealth. In the second model, households gain utility from both consumption and the social status obtained from their own wealth relative to the wealth of other peer households. Our results indicate that both models are inconsistent with the data regardless of income. However, using cointegration methods as a diagnostic tool, we find that the data are “loosely” consistent with the spirit of capitalism, at least for the upper income quintiles.     

Variance and Lower Partial Moment Measures of Systematic Risk: Some Analytical and Empirical Results

As a measure of systematic risk, the lower partial moment measure requires fewer restrictive assumptions than does the variance measure. However, the latter enjoys far wider usage than the former, perhaps because of its familiarity and the fact that two measures of systematic risk are equivalent when return distributions are normal. This paper shows analytically that there are systematic differences in the two risk measures when return distributions are lognormal. Results of empirical tests show that there are indeed systematic differences in measured values of the two risk measures for securities with above average and with below average systematic risk.     

Diversification and generalized tracking errors for correlated non‐normal returns

Abstract

The probability distribution for the relative return of a portfolio constructed from a subset n of the assets from a benchmark, consisting of N assets whose returns are multivariate normal, is completely characterized by its tracking error. However, if the benchmark asset returns are not multivariate normal then higher moments of the probability distribution for the portfolio's relative return are not related to its tracking error. We discuss the convergence of generalized tracking error measures as the size of the subset of benchmark assets increases. Assuming that the joint probability distribution for the returns of the assets is symmetric under their permutations we show that increasing n makes these generalized tracking errors small (even though n « N). For n » 1 the probability distribution for the portfolio's relative return is approximately symmetric and strongly peaked about the origin. The results of this paper generalize the conclusions of Dynkin et al (Dynkin L, Hyman J and Konstantinovsky V 2002 Sufficient Diversification in Credit Portfolios (Lehman Brothers Fixed Income Research)) to more general underlying asset distributions.     

Measuring changes in credit risk: The case of CDS event studies

This paper examines the size and power of test statistics designed to detect abnormal changes in credit risk as measured by credit default swap (CDS) spreads. We follow a simulation approach to examine the statistical properties of normal and abnormal CDS spread changes and assess the performance of normal return models and test statistics. Using daily CDS data, we find parametric test statistics to be generally inferior to non-parametric tests, with the rank test performing best. A CDS factor model based on factors identified in the empirical literature is generally well specified and more powerful in detecting abnormal performance than some of the classical normal return models. Finally, we examine abnormal CDS announcement spread changes around issuer's rating downgrades to demonstrate the effect of different CDS spread change measures and normal return models on event study results.