Stochastic Discount Factor Models Research Articles

Financial crises, the decoupling–recoupling hypothesis, and the risk premium on the Greek stock index futures market

Our results shed light on the contribution of local and regional factors to the risk premium on the Greek stock index futures market. Building upon the stochastic discount factor model, we estimate a multivariate exponential GARCH-in-mean model to uncover the risk premium on the FTSE/ASE-20 stock index futures traded in the ADEX, Greece. We provide evidence that the risk premium was mainly driven by its regional (European) component before the recent financial crisis. We also report that the local (Greek) component has become more important for the risk premium on the Greek stock index futures market after the recent debt crisis in Greece. Importantly, our results suggest that the decoupling–recoupling hypothesis, according to which the recent financial crisis has strengthened international financial links, does not apply to the risk premium on the stock index futures market. Rather, we report evidence consistent with a recoupling–decoupling hypothesis.

The Realised Recursive Portfolio Performance of Linear Stochastic Discount Factor Models

The Realised Recursive Portfolio Performance of Linear Stochastic Discount Factor Models

Arbitrage and the Evaluation of Linear Factor Models in UK Stock Returns

AbstractI examine the impact of the no arbitrage restriction on the estimation and evaluation of linear factor models in UK stock returns. The no arbitrage restriction reduces volatility and eliminates most of the negative values of the fitted stochastic discount factor models. All of the factor models are rejected and there are significant differences in the pricing performance between models under the no arbitrage restriction. The no arbitrage restriction can have a significant impact on both the parameter estimates and pricing errors for some models.

The business cycle and the equity risk premium in real time

Building on the stochastic discount factor model, we estimated a multivariate exponential GARCH-in-mean model to analyze the link between the business cycle and the equity risk premium in the United States. In order to measure the business cycle, we used revised and real-time monthly data on industrial production for the period from 1965 to 2008. The main result of our empirical analysis is that estimates of the equity risk premium based on real-time macroeconomic data may significantly differ from estimates of the equity risk premium based on revised macroeconomic data.

Time Variations in the Equity Premium: Is It Habit Formation or Loss Aversion?

The large spread between returns and risk-free rates (the equity premium puzzle) has been the subject of intense debate. Two main families of models claim to solve this puzzle: habit-formation models and loss-aversion models. The goal of this paper is to assess empirically which of them best fits the observed excess returns. In order to do this, I first show how to express both models as linear stochastic discount factor models. This gives explicit and testable constraints for the excess return variations. I then compare the theoretical evolution generated by these models with the observed path. I find that the constant relative risk aversion model and the external habit model are not likely to correspond to the observed data. The loss aversion model and the internal habit model could fit the observed excess return evolution.

A general framework for the derivation of asset price bounds: an application to stochastic volatility option models

We present a generalization of Cochrane and Saa-Requejo’s good-deal bounds which allows to include in a flexible way the implications of a given stochastic discount factor model. Furthermore, a useful application to stochastic volatility models of option pricing is provided where closed-form solutions for the bounds are obtained. A calibration exercise demonstrates that our benchmark good-deal pricing results in much tighter bounds. Finally, a discussion of methodological and economic issues is also provided.

Evaluation of linear asset pricing models by implied portfolio performance

We present a theoretical perspective that motivates the use of the Generalized Least Squares R-Square, prominently advocated by Lewellen et al. [Lewellen, J., Nagel, S., Shanken, J., forthcoming. A skeptical appraisal of asset-pricing tests. Journal of Financial Economics], as an evaluation measure for multivariate linear asset pricing models. Adapting results from Shanken [Shanken, J., 1985. Multivariate tests of the zero-beta CAPM. Journal of Financial Economics 14, 327–348] and Kandel and Stambaugh [Kandel, S., Stambaugh, R.F., 1995. Portfolio inefficiency and the cross-section of expected returns. Journal of Finance 50, 157–184], we provide various interpretations and a graphical account in mean-variance space of this measure, facilitating a better understanding of its properties. We furthermore relate it to another leading evaluation metric, the HJ-distance of Hansen and Jagannathan [Hansen, L.P., Jagannathan, R., 1997. Assessing specification errors in stochastic discount factor models. Journal of Finance 52, 557–590]. Additionally, we present a comparison between these evaluation measures using mean-variance mathematics in risk-return space, and we provide a simple formula for calculating both model evaluation measures that involves only the parameters of the mean-variance asset and factor frontiers.

A General Framework for the Derivation of Asset Price Bounds: An Application to Stochastic Volatility Option Models

We present a generalization of Cochrane and Saa-Requejo’s good-deal bounds which allows to include in a flexible way the implications of a given stochastic discount factor model. Furthermore, a useful application to stochastic volatility models of option pricing is provided where closed-form solutions for the bounds are obtained. A calibration exercise demonstrates that our benchmark good-deal pricing results in much tighter bounds. Finally, a discussion of methodological and economic issues is also provided.

Improvement in finite sample properties of the Hansen–Jagannathan distance test

Jagannathan and Wang [Jagannthan, R., and Wang, Z., “The conditional CAPM and the cross-section of expected returns.” Journal of Finance, 51 (1996), 3–53] derive the asymptotic distribution of the Hansen–Jagannathan distance (HJ-distance) proposed by Hansen and Jagannathan [Hansen, L.P., and Jagannathan, R., Assessing specific errors in stochastic discount factor models." Journal of Finance, 52 (1997), 557–590], and develop a specification test of asset pricing models based on the HJ-distance. While the HJ-distance has several desirable properties, Ahn and Gadarowski [Ahn, S.C., and Gadarowski, C., “Small sample properties of the GMM specification test based on the Hansen–Jagannathan distance.” Journal of Empirical Finance, 11 (2004), 109–132] find that the specification test based on the HJ-distance overrejects correct models too severely in commonly used sample size to provide a valid test. This paper proposes to improve the finite sample properties of the HJ-distance test by applying the shrinkage method [Ledoit, O., and Wolf, M., “Improved estimation of the covariance matrix of stock returns with an application to portfolio selection.” Journal of Empirical Finance, 10 (2003), 603–621] to compute its weighting matrix. The proposed method improves the finite sample performance of the HJ-distance test significantly.

Historical simulation approach to the estimation of stochastic discount factor models

We propose an approach to the estimation of the parameters of stochastic discount factor (SDF) models which is based on the idea that the next period joint distribution of the variables in a SDF and asset returns can be well approximated by their joint historical distribution. The estimates of the SDF parameters may therefore be found as the values of the parameters at which the mean of the historical distribution of the product of the SDF with an asset return equals one. Each time period, the estimates are updated using the most recent periods of data and hence can change over time. This method can be viewed as an alternative to the approaches that specify a particular functional form relating the SDF parameters to proxies for the state of the world.

Macroeconomic Sources of Risk in the Term Structure

We develop a new way of modeling time variation in term premia, based on the stochastic discount factor model of asset pricing. The joint distribution of excess U.S. bond returns of different maturity and the observable fundamental macroeconomic factors is modeled using multivariate GARCH with conditional covariances in the mean to capture the term premia. By testing the assumption of no arbitrage we derive a specification test of our model. We estimate the contribution made to the term premia at different maturities through real and nominal sources of risk. From the estimated term premia we recover the term structure of interest rates and examine how it varies through time. Finally, we examine whether the reported failures of the rational expectations hypothesis can be attributed to an omitted time‐varying term premium.

Performance evaluation considering the coskewness

PurposeThe paper aims to examine the performance of Spanish mutual funds between 1999 and 2003.Design/methodology/approachThe methodolgy uses the stochastic discount factor (SDF) framework across a variety of models developed in the recent asset pricing literature. This approach is a fairly recent innovation in the evaluation of investment performance.FindingsThe present work complements the research of Farnworth et al. and Fletcher and Forbes, adding a new issue to the SDF, the third co‐moment of asset returns. Recent asset pricing studies show the relevance of the component of an asset's skewness related to the market portfolio's skewness, the coskewness, and how it helps to explain the time‐variation of ex‐ante market risk premiums. It is found that the effects of adding coskewness to evaluate the performance is significant even when factors based on size, book‐to‐market and momentum are included.Practical implicationsThe omission of a coskewness factor may lead to erroneous evaluations of a fund's performance, and therefore, issues such as the persistence of performance should be revised.Originality/valueThis paper explores, for the first time, the effects of incorporating a coskewness factor in the analysis of investment performance, both in an unconditional and a conditional framework using SDF models.

Econometric specification of stochastic discount factor models

Abstract We consider the problem of derivative pricing when the stochastic discount factors are exponential-affine functions of underlying state variable. In particular we discuss the conditionally Gaussian framework and introduce semi-parametric pricing methods for models with path dependent drift and volatility. This approach is also applied to more complicated frameworks, such as pricing of a derivative written on an index, when the interest rate is stochastic.

Specification tests of international asset pricing models

This study evaluates the cross-sectional pricing performances of several international asset pricing models. The comparison metric is the Hansen and Jagannathan [Hansen, L., Jagannathan, R., 1997. Assessing specification errors in stochastic discount factor models. Journal of Finance 52, 557–590] distance, and the models are required to price size and book-to-market portfolios from the US, the UK and Japan. When betas and risk premiums are constant over business cycles, none of the models can pass the specification test. By allowing time-varying betas and risk premiums in conditional models, most models can pass the specification test. This is because these models capture the assets' different sensitivities to the time-varying risk premiums, which explain most of the book-to-market return spread. The Fama–French factors are redundant in conditional models. Finally, exchange risk premiums account for a significant part of the excess returns on international assets, and the conditional International CAPM with exchange risk performs the best. The market integration hypothesis is also supported.

Testing Parameters in GMM Without Assuming that They Are Identified

We propose a Generalized Method of Moments (GMM) Lagrange multiplier statistic, i.e. the K-statistic, that uses the Jacobian at the evaluated parameter value instead of the expected Jacobian. To obtain its limit behavior, we use a novel assumption that brings GMM closer to maximum likelihood and which is easily satisfied. The usual asymptotic x2 distribution of the K-statistic then holds under a wider set of circumstances, like weak and many instrument asymptotics and combinations thereof, than the standard full rank case for the Jacobian. The behavior of the K-statistic can be spurious around inflexion points and the maximum of the objective function since the moment conditions are then not satisfied. Combinations of the K-statistic with statistics that test the validity of the moment equations overcome the spurious behavior. We conduct a power comparison to test for the risk aversion parameter in a stochastic discount factor model and construct its confidence set for observed consumption growth and asset return series.

The Asymmetric Effect of the Business Cycle on the Relation Between Stock Market Returns and their Volatility

We examine the relation between US stock market returns and the US business cycle for the period 1960 - 2003 using a new methodology that allows us to estimate a time-varying equity premium. We identify two channels in the transmission mechanism. One is through the mean of stock returns via the equity risk premium, and the other is through the volatility of returns. We confirm previous findings based on simple correlation analysis that the relation is asymmetric with downturns in the business cycle having a greater negative impact on stock returns than the positive effect of upturns. We also obtain a new result, that demand and supply shocks affect stock returns differently. Our model of the relation between returns and their volatility is derived from the stochastic discount factor model of asset pricing which encompasses CAPM, consumption CAPM and Merton's (1973) inter-temporal CAPM. It is implemented using a multi-variate GARCH-in-mean model with a time-varying conditional heteroskedasticity and correlation structure.

An examination of alternative CAPM-based models in UK stock returns

We evaluate the performance of unconditional and conditional versions of seven stochastic discount factor models in UK stock returns between January 1975 and December 2001. We find that the conditional four-moment capital asset pricing model (CAPM) has the best performance among the models we consider in terms of the lowest [Hansen, L.P., Jagannathan, R., 1997. Assessing specification errors in stochastic discount factor models. Journal of Finance 52, 591–607] distance measure and explaining the time-series predictability of industry portfolio excess returns. Conditional models also do a better job than unconditional models. However we find that the superior performance of the conditional four-moment CAPM, and conditional models in general, arises in part due to overfitting the data.

An empirical examination of UK emerging market unit trust performance

We examine the unconditional performance of UK unit trusts with emerging market equity objectives between January 1993 and December 2003 using 13 candidate models of the stochastic discount factor. No one stochastic discount factor model dominates the others in pricing passive trading strategies or detecting significant superior performance. There is no evidence of significant superior performance by trusts. The average (median) trust does not earn sufficient returns to cover expenses and trading costs. These findings are robust across all the stochastic discount factor models.

Macroeconomic Sources of Risk in the Term Structure

In this paper we develop a new way of modelling time variation in term premia. This is based on the stochastic discount factor model of asset pricing with observable macroeconomic factors. The joint distribution of excess holding period US bond returns of different maturity and the fundamental macroeconomic factors is modelled using multivariate GARCH with conditional covariances in the mean to capture the term premia. We show how by testing the assumption of no arbitrage we can derive a specification test of our model. We estimate the contribution made to the term premia at different maturities by real and nominal macroeconomic sources of risk. From the estimated term premia we recover the term structure of interest rates and examine how it varies through time. Finally, we examine whether the large number of reported failures of the rational expectations hypothesis of the term structure can be attributed to an omitted time-varying term premium.

Generalized Reduced Rank Tests Using the Singular Value Decomposition

We propose a novel statistic to test the rank of a matrix. The rank statistic overcomes deficiencies of existing rank statistics, like: Necessity of a Kronecker covariance matrix for the canonical correlation rank statistic of Anderson (1951), sensitivity to the ordering of the variables for the LDU rank statistic of Cragg and Donald (1996) and Gill and Lewbel (1992), a limiting distribution that is not a standard chi-squared distribution for the rank statistic of Robin and Smith (2000) and usage of numerical optimization for the objective function statistic of Cragg and Donald (1997). The new rank statistic consists of a quadratic form of a (orthogonal) transformation of the smallest singular values of a unrestricted estimate of the matrix of interest. The quadratic form is taken with respect to the inverse of a unrestricted covariance matrix that can be estimated using a heteroscedasticity autocorrelation consistent estimator. The rank statistic has a standard chi-squared limiting distribution. In case of a Kronecker covariance matrix, the rank statistic simplifies to the canonical correlation rank statistic. In the non-stationary cointegration case, the limiting distribution of the rank statistic is identical to that of the Johansen trace statistic. We apply the rank statistic to test for the rank of a matrix that governs the identification of the parameters in the stochastic discount factor model of Jagannathan and Wang (1996). The rank statistic shows that non-identification of the parameters can not be rejected. We further use the stochastic discount factor model to illustrate the validity of the limiting distribution and to conduct a power comparison.