Covariance complexity and rates of return on assets

This paper considers the estimation of the expected rate of return on a set of risky assets. The approach to estimation focuses on the covariance matrix for the returns. The structure in the covariance matrix determines shared information which is useful in estimating the mean return for each asset. An empirical Bayes estimator is developed using the covariance structure of the returns distribution. The estimator is an improvement on the maximum likelihood and Bayes–Stein estimators in terms of mean squared error. The effect of reduced estimation error on accumulated wealth is analyzed for the portfolio choice model with constant relative risk aversion utility.     

Time‐Varying Risk–Return Trade‐off in the Stock Market

We uncover a strong comovement of the stock market risk–return trade‐off with the consumption–wealth ratio (CAY). The finding reflects time‐varying investment opportunities rather than countercyclical aggregate relative risk aversion. Specifically, the partial risk–return trade‐off is positive and constant when we control for CAY as a proxy for investment opportunities. Moreover, conditional market variance scaled by CAY is negatively priced in the cross‐section of stock returns. Our results are consistent with a limited stock market participation model, in which shareholders require an illiquidity premium that increases with CAY, in addition to the risk premium that is proportional to conditional market variance.     

Models for stock returns

Historically, the normal variance model has been used to describe stock return distributions. This model is based on taking the conditional stock return distribution to be normal with its variance itself being a random variable. The form of the actual stock return distribution will depend on the distribution for the variance. In practice, the distributions chosen for the variance appear to be very limited. In this note, we derive a comprehensive collection of formulas for the actual stock return distribution, covering some sixteen flexible families. The corresponding estimation procedures are derived by the method of moments and the method of maximum likelihood. We feel that this work could serve as a useful reference and lead to improved modelling with respect to stock market returns.     

Non-parametric estimation of historical volatility

Evolving volatility is a dominant feature observed in most financial time series and a key parameter used in option pricing and many other financial risk analyses. A number of methods for non-parametric scale estimation are reviewed and assessed with regard to the stylized features of financial time series. A new non-parametric procedure for estimating historical volatility is proposed based on local maximum likelihood estimation for the t-distribution. The performance of this procedure is assessed using simulated and real price data and is found to be the best among estimators we consider. We propose that it replaces the moving variance historical volatility estimator.     

There is a risk-return trade-off after all

This paper studies the intertemporal relation between the conditional mean and the conditional variance of the aggregate stock market return. We introduce a new estimator that forecasts monthly variance with past daily squared returns, the mixed data sampling (or MIDAS) approach. Using MIDAS, we find a significantly positive relation between risk and return in the stock market. This finding is robust in subsamples, to asymmetric specifications of the variance process and to controlling for variables associated with the business cycle. We compare the MIDAS results with tests of the intertemporal capital asset pricing model based on alternative conditional variance specifications and explain the conflicting results in the literature. Finally, we offer new insights about the dynamics of conditional variance.     

Asset pricing implications of benchmarking: a two-factor CAPM

The paper considers the equilibrium effects of an institutional investor whose performance is benchmarked to an index. In a partial equilibrium setting, the objective of the institutional investor is modelled as the maximization of expected utility (an increasing and concave function, in order to accommodate risk aversion) of final wealth minus a benchmark. In equilibrium this optimal strategy gives rise to the two-beta CAPM: together with the market beta a new risk-factor (termed active management risk) is brought into the analysis. This new beta is defined as the normalized (to the benchmark's variance) covariance between the asset excess return and the excess return of the market over the benchmark index. The empirical test supports the model's predictions. The cross-section return on the active management risk is positive and significant, especially after 1990, when institutional investors became the representative agent of the market.     

A study of diffusion processes for foreign exchange rates

The nature of the stochastic process generating the time path of foreign exchange rates plays an important role in dynamic theories of international financial economics. An important consideration in this stochastic process is the relationship between currency return variances and exchange rate levels. Using five years of daily data separated into quarterly intervals, this study demonstrates that currency return variances depend on exchange rate levels and the dependency is unstable intertemporally. Thus, these empirical results contradict the assumption of stable log-normal distribution and the more general assumption of constant elasticities of variances. For elasticity coefficients ordinary least squares estimates are compared to maximum likelihood estimates; the maximum likelihood estimator clearly is superior. Implications of these results for models of foreign exchange rates are discussed.     

Extracting Information from Options Markets: Smiles, State–Price Densities and Risk Aversion

In this paper, recent techniques of estimating implied information from derivatives markets are presented and applied empirically to the French derivatives market. We determine nonparametric implied volatility functions, state–price densities and historical densities from a high–frequency CAC 40 stock index option dataset. Moreover, we construct an estimator of the risk aversion function implied by the joint observation of the cross–section of option prices and time–series of underlying asset value. We report a decreasing implied volatility curve with the moneyness of the option. The estimated relative risk aversion functions are positive and globally consistent with the decreasing relative risk aversion assumption.     

Bayesian realized-GARCH models for financial tail risk forecasting incorporating the two-sided Weibull distribution

The realized-GARCH framework is extended to incorporate the two-sided Weibull distribution, for the purpose of volatility and tail risk forecasting in a financial time series. Further, the realized range, as a competitor for realized variance or daily returns, is employed as the realized measure in the realized-GARCH framework. Sub-sampling and scaling methods are applied to both the realized range and realized variance, to help deal with inherent micro-structure noise and inefficiency. A Bayesian Markov Chain Monte Carlo (MCMC) method is adapted and employed for estimation and forecasting, while various MCMC efficiency and convergence measures are employed to assess the validity of the method. In addition, the properties of the MCMC estimator are assessed and compared with maximum likelihood, via a simulation study. Compared to a range of well-known parametric GARCH and realized-GARCH models, tail risk forecasting results across seven market indices, as well as two individual assets, clearly favour the proposed realized-GARCH model incorporating the two-sided Weibull distribution; especially those employing the sub-sampled realized variance and sub-sampled realized range.     

A dynamic equilibrium model for U-shaped pricing kernels

This paper proposes a dynamic equilibrium model that can provide a unified explanation for the stylized facts observed in stock index markets such as the fat tails of the risk-neutral return distribution relative to the physical distribution, negative expected returns on deep OTM call options and negative realized variance risk premiums. In particular, we focus on the U-shaped pricing kernel against the stock index return, which is closely related to the negative call returns. We assume that the stock index return follows a time-changed Lévy process and that a representative investor has power utility over the aggregate consumption that forms a linear regression of the stock index return and its stochastic activity rate. This model offers a macroeconomic interpretation of the stylized facts from the perspective of the sensitivity of the activity rate and stock index return on aggregate consumption as well as the investor’s risk aversion.     

Zero-intelligence realized variance estimation

Given a time series of intra-day tick-by-tick price data, how can realized variance be estimated? The obvious estimator—the sum of squared returns between trades—is biased by microstructure effects such as bid–ask bounce and so in the past, practitioners were advised to drop most of the data and sample at most every five minutes or so. Recently, however, numerous alternative estimators have been developed that make more efficient use of the available data and improve substantially over those based on sparsely sampled returns. Yet, from a practical viewpoint, the choice of which particular estimator to use is not a trivial one because the study of their relative merits has primarily focused on the speed of convergence to their asymptotic distributions, which in itself is not necessarily a reliable guide to finite sample performance (especially when the assumptions on the price or noise process are violated). In this paper we compare a comprehensive set of nineteen realized variance estimators using simulated data from an artificial “zero-intelligence” market that has been shown to mimic some key properties of actual markets. In evaluating the competing estimators, we concentrate on efficiency but also pay attention to implementation, practicality, and robustness. One of our key findings is that for scenarios frequently encountered in practice, the best variance estimator is not always the one suggested by theory. In fact, an ad hoc implementation of a subsampling estimator, realized kernel, or maximum likelihood realized variance, delivers the best overall result. We make firm practical recommendations on choosing and implementing a realized variance estimator, as well as data sampling.     

On the Distributional Conditions for a Consumption-oriented Three Moment CAPM

In this paper, we develop sufficient conditions on probability distributions for a three moment (mean, variance, and skewness) consumption-oriented capital asset pricing model (CAPM) to price correctly a subset of assets. The assumptions that individuals in an allocationally efficient capital market have identical probability beliefs and monotone increasing strictly concave utility functions displaying nonincreasing absolute risk aversion imply an aggregate preference function that exhibits preference for expected return, aversion to variance of return, and preference for positive skewness. For otherwise arbitrary preferences, we show that quadratic characteristic lines are sufficient for a subset of assets to be priced according to a three moment consumption-oriented CAPM.     

Risk premia: Exact solutions vs. log-linear approximations

We derive exact expressions for the risk premia for general distributions in a Lucas economy and show that the errors when using log-linear approximations can be economically significant when the shocks are nonnormal. Assuming growth rates are Normal Inverse Gaussian (NIG) and fitting the distribution to the data used in Mehra and Prescott (1985), the coefficient of relative risk aversion required to match the equity premium is more than halved compared to the finding in their article. We also consider a standard long-run risk model and, by comparing our exact solutions to the log-linear approximations, we show that the approximation errors are substantial, especially for high levels of risk aversion.     

An Assessment of Empirical Model Performance When Financial Market Transactions are Observed at Different Data Frequencies: An Application to East Asian Exchange Rates

This paper compares the performance of alternative models of east Asian exchange rates at different data frequencies. Selected models employ different specifications of the conditional variance and the conditional error distribution. Conditional variance specifications include: homoscedasticity, GARCH, LGARCH, and EGARCH. Conditional error distribution specifications include normal and Student t. The best exchange rate model specification is clearly conditional on data frequency. Higher frequency (daily, weekly) data commonly exhibit characteristics that demand more sophisticated estimation methods than analysts commonly employ. These characteristics generally vanish at lower (monthly, quarterly) frequencies. Overall we find significant benefit from accommodating heteroscedasticity and leptokurtic properties of the conditional distribution as data frequency increases. Using a likelihood ratio test we compare the relative gain from addressing heteroscedasticity (through use of GARCH models) versus accommodation of leptokurtosis. This comparison suggests that the gains from correct specification of the conditional distribution dominate those obtained from addressing problems of heteroscedasticity.     

The role of autoregressive conditional skewness and kurtosis in the estimation of conditional VaR

This paper investigates the role of high-order moments in the estimation of conditional value at risk (VaR). We use the skewed generalized t distribution (SGT) with time-varying parameters to provide an accurate characterization of the tails of the standardized return distribution. We allow the high-order moments of the SGT density to depend on the past information set, and hence relax the conventional assumption in conditional VaR calculation that the distribution of standardized returns is iid. The maximum likelihood estimates show that the time-varying conditional volatility, skewness, tail-thickness, and peakedness parameters of the SGT density are statistically significant. The in-sample and out-of-sample performance results indicate that the conditional SGT-GARCH approach with autoregressive conditional skewness and kurtosis provides very accurate and robust estimates of the actual VaR thresholds.     

The bias of tests for a risk premium in forward exchange rates

The pure expectations theory of unbiased forward exchange rates predicts that the slope coefficient in a regression of the change in the spot rate on the difference between the current forward and spot rates should equal unity. In the recent empirical work by Fama, the estimates of this coefficient turn out to be negative in all regressions for nine major industrialized nations. This paper demonstrates that under the expectations theory, the sampling distribution of the regression estimator of this coefficient is upward-biased relative to unity and strongly skewed to the right. The likelihood of negative values is essentially zero. Thus, the estimator is biased in a direction opposite to what is observed. Since the observed estimates lie far out in the thin left-hand tail of the estimator's sampling distribution, the evidence against the hypothesis of unbiased forward rates is much stronger than previously believed.     

On Mixed and Compound Mixed Poisson Distributions

Recursive formulae are derived for the evaluation of the t-th order cumulative distribution function and the t-th order tail probability of compound mixed Poisson distributions in the case where the derivative of the logarithm of the mixing density can be written as a ratio of polynomials. Also, some general results are derived for the evaluation of the t-th order moments of stop-loss transforms. The recursions can be applied for the exact evaluation of the probability function, distribution function, tail probability and stop-loss premium of compound mixed Poisson distributions and the corresponding mixed Poisson distributions. Several examples are also presented.     

Stock market expectations and risk aversion of individual investors

We study the relationship between stock market return expectations and risk aversion of individuals and test whether the joint effects arising from the interaction of these two variables affect investment decisions. Using data from the Dutch National Bank Household Survey, we find that higher risk aversion is associated with lower stock market expectations. We identify significant and negative effects on the probability that individuals invest in stocks arising from the interaction between stock market expectations and risk aversion. These effects are in addition to a significant and positive impact from stock market return expectations as well as a significant and negative effect from risk aversion separately. However, once individuals participate in the stock market, their stock market expectations alone remain significant in determining their portfolio allocation decisions.     

Currency Hedging Using the Mean-Gini Framework

The mean-Gini framework has been suggested as a robust alternative to the portfolio approach to futures hedging given its optimality under general distributional conditions. However, calculation of the Gini hedge ratio requires estimation of the underlying price distribution. We estimate minimum-Gini hedge ratios using two widely-used estimation procedures, the empirical distribution function method and the kernel method, for three emerging market and three developed market currencies. We find that these methods yield different Gini hedge ratios. These differences increase with risk aversion and are statistically significant for all developed market currencies but only one emerging market currency. In-sample analyses show that the empirical distribution function method is more effective at risk reduction than the kernel method for developed market currencies, whereas the kernel method is superior for emerging market currencies. Post-sample analyses strengthen the superiority of the empirical distribution function method for developed market and, in several cases, for emerging market currencies.JEL Classification: F31, G15     

Mean-Gini,Portfolio Theory,and the Pricing of Risky Assets

This paper presents the mean-Gini (MG) approach to analyze risky prospects and construct optimum portfolios. The proposed method has the simplicity of a mean-variance model and the main features of stochastic dominance efficiency. Since mean-Gini is consistent with investor behavior under uncertainty for a wide class of probability distributions, Gini's mean difference is shown to be more adequate than the variance for evaluating the variability of a prospect. The MG approach is then applied to capital markets and the security valuation theorem is derived as a general relationship between average return and risk. This is further extended to include a degree of risk aversion that can be estimated from capital market data. The analysis is concluded with the concentration ratio to allow for the classification of different securities with respect to their relative riskiness.