Specification tests of asset pricing models using excess returns

In this paper, we discuss the impact of different formulations of asset pricing models on the outcome of specification tests that are performed using excess returns. We point out that the popular way of specifying the stochastic discount factor (SDF) as a linear function of the factors is problematic because (1) the specification test statistic is not invariant to an affine transformation of the factors, and (2) the SDFs of competing models can have very different means. In contrast, an alternative specification that defines the SDF as a linear function of the de-meaned factors is free from these two problems and is more appropriate for model comparison. In addition, we suggest that a modification of the traditional Hansen–Jagannathan distance (HJ-distance) is needed when we use the de-meaned factors. The modified HJ-distance uses the inverse of the covariance matrix (instead of the second moment matrix) of excess returns as the weighting matrix to aggregate pricing errors. Asymptotic distributions of the modified HJ-distance and of the traditional HJ-distance based on the de-meaned SDF under correctly specified and misspecified models are provided. Finally, we propose a simple methodology for computing the standard errors of the estimated SDF parameters that are robust to model misspecification. We show that failure to take model misspecification into account is likely to understate the standard errors of the estimates of the SDF parameters and lead us to erroneously conclude that certain factors are priced.     

Why not use SDF rather than beta models in performance measurement?

This paper analyzes performance measurement based on stochastic discount factors, compared to beta models traditionally used in computing funds’ (Jensen) alphas. From a theoretical point of view, standard alphas suffer from several limitations. Our paper addresses this issue from an empirical point of view using a sample of Swiss mutual funds from 2000 to 2011. Our results suggest that the key for a “fair” comparison between stochastic discount function (SDF) and beta models is the specification of the set of primitive assets used to calibrate the SDF function. Once this is established, the size of (absolute) performance differences considerably decreases between the two model families. However, there are sizeable performance deviations in the cross-section of funds if conditioning information is incorporated in the tests, up to some 20 basis points per month, or about 2.3 % per year. In almost all cases, the SDF-alphas are lower than the standard (Jensen) alphas. In absolute terms, the average SDF-based underperformance of the funds is way larger than the average total expense ratio (TER) of the funds, both in a conditional and unconditional setting.     

A Critique of the Stochastic Discount Factor Methodology

In this paper, we point out that the widely used stochastic discount factor (SDF) methodology ignores a fully specified model for asset returns. As a result, it suffers from two potential problems when asset returns follow a linear factor model. The first problem is that the risk premium estimate from the SDF methodology is unreliable. The second problem is that the specification test under the SDF methodology has very low power in detecting misspecified models. Traditional methodologies typically incorporate a fully specified model for asset returns, and they can perform substantially better than the SDF methodology.     

Market-wide illiquidity and the distribution of non-parametric stochastic discount factors

Employing out-of-sample non-parametric estimation techniques, we show that market-wide liquidity risk matters for asset pricing independently of the specific functional form of the stochastic discount factor (SDF) and, therefore, of the asset pricing model specification. Market-wide illiquidity significantly affects the distribution of the SDF. Specifically, it boosts up the volatility of the SDF, causing minor effects on higher moments of its distribution. This finding is robust to the use of different sets of test assets in the estimation of the SDF, including equity and corporate bond portfolios, and the use of a high-dimensional data estimation procedure.     

Asset pricing with Second-Order Esscher Transforms

The purpose of the paper is to introduce, in a discrete-time no-arbitrage pricing context, a bridge between the historical and the risk-neutral state vector dynamics which is wider than the one implied by a classical exponential-affine stochastic discount factor (SDF) and to preserve, at the same time, the tractability and flexibility of the associated asset pricing model. This goal is achieved by introducing the notion of exponential-quadratic SDF or, equivalently, the notion of Second-Order Esscher Transform. The log-pricing kernel is specified as a quadratic function of the factor and the associated sources of risk are priced by means of possibly non-linear stochastic first-order and second-order risk-correction coefficients. Focusing on security market models, this approach is developed in the multivariate conditionally Gaussian framework and its usefulness is testified by the specification and calibration of what we name the Second-Order GARCH Option Pricing Model. The associated European Call option pricing formula generates a rich family of implied volatility smiles and skews able to match the typically observed ones.     

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.     

Conditional performance evaluation for German equity mutual funds

We investigate the conditional performance of a sample of German equity mutual funds over the period from 1994 to 2003 using both the beta-pricing approach and the stochastic discount factor (SDF) framework. On average, mutual funds cannot generate excess returns relative to their benchmark that are large enough to cover their total expenses. Compared to unconditional alphas, fund performance sharply deteriorates when we measure conditional alphas. Given that stock returns are to some extent predictable based on publicly available information, conditional performance evaluation raises the benchmark for active fund managers because it gives them no credit for exploiting readily available information. Underperformance is more pronounced in the SDF framework than in beta-pricing models. The fund performance measures derived from alternative model specifications differ depending on the number of primitive assets taken to calibrate the SDF as well as the number of instrument variables used to scale assets and/or factors.     

开放经济下随机折现因子对经济周期与经济波动的解释能力

本文通过研究随机折现因子(SDF)与经济周期以及经济波动的关系,旨在探索金融市场与宏观经济的内在联系。我们构建了一个开放经济定价模型(OEAP model),将汇率、通货膨胀率、国内消费以及市场收益率纳入统一的框架内,探讨SDF对经济周期及经济波动的解释能力。基于模型的估计结果表明在开放经济的模型假设下SDF具有显著的反周期特点并且SDF的波动性方差可以作为衡量经济波动一个很好的指标。另外,模型的模拟结果表明,相对于封闭经济假设下的Epstein-Zin模型,OEAP模型对消费具有更好的拟合效果。这说明OEAP模型对SDF具有更准确的估计。     

Semi-Parametric Specification Tests for Discrete Probability Models

Loss functions play an important role in analyzing insurance portfolios. A fundamental issue in the study of loss functions involves the selection of probability models for claim frequencies. In this article, we propose a semi‐parametric approach based on the generalized method of moments (GMM) to solve the specification problems concerning claim frequency distributions. The GMM‐based testing procedure provides a general framework that encompasses many specification problems of interest in actuarial applications. As an alternative approach to the Pearson χ² and other goodness‐of‐fit tests, it is easy to implement and should be of practical use in applications involving selecting and validating probability models with complex characteristics.     

The coskewness puzzle

We propose a novel approach to testing non-linear stochastic discount factor (SDF) specifications that arise in rational representative investor models. Our approach does not require overly-restrictive assumptions about the shape of investors’ preferences, typically imposed by the extant literature, and is based instead on restrictions that rule out “good deals”, i.e., arbitrage opportunities as well as unduly large Sharpe ratios. We apply this framework to test the empirical admissibility of 3 and 4-moment versions of the CAPM. We find that, while coskewness and cokurtosis risk help price a number of stock strategies and portfolios, including static strategies based on a fine industry-level diversification, momentum strategies and portfolios managed on the basis of available information, the CAPM and its 3 and 4-moment versions cannot provide an exhaustive account of observed asset returns.     

Learning by Failing: A Simple VaR Buffer

We study in this article the problem of model risk in VaR computations and document a procedure for correcting the bias due to specification and estimation errors. This practical method consists of “learning from model mistakes”, since it dynamically relies on an adjustment of the VaR estimates – based on a back‐testing framework – such as the frequency of past VaR exceptions always matches the expected probability. We finally show that integrating the model risk into the VaR computations implies a substantial minimum correction to the order of 10–40% of VaR levels.     

Martingalized historical approach for option pricing

In a discrete time option pricing framework, we compare the empirical performance of two pricing methodologies, namely the affine stochastic discount factor (SDF) and the empirical martingale correction methodologies. Using a CAC 40 options dataset, the differences are found to be small: the higher order moment correction involved in the SDF approach may not be that essential to reduce option pricing errors. This paper puts into evidence the fact that an appropriate modelling under the historical measure associated with an adequate correction (that we call here a “martingale correction”) permits to provide option prices which are close to market ones.     

Formal Analysis of Segregation of Duties (SoD) in Accounting: A Computational Approach

This study examines a computational framework for segregation of duties (SoD) in the design as well as implementation of accounting systems. The framework consists of a model of workflows in accounting systems based on workflow graphs, a partial order model of roles performed by the actors in the accounting system, and a specification of SoD rules. We develop a set of algorithms for four SoD rules that can be used in the enforcement of SoD. For the SoD rule that precludes task type conflicts, our results show that while compliance verification can be carried out efficiently, finding an SoD compliant assignment of tasks is computationally intractable. For those situations, we present an integer linear programming (ILP) formulation for finding compliant assignments using public domain ILP solvers. For the remaining three SoD rules, we demonstrate efficient ways of testing compliance for a given assignment as well as finding compliant assignments.     

Individual Demand for Local Public Schooling: Evidence from Swedish Survey Data

In this paper we investigate the demand for local public school expenditures in Sweden using survey data, a method previously never applied to Swedish data. We compare our results to those of earlier US studies, where the same method is used in a different institutional setup. Estimating a linear demand specification, we find that demand is inelastic with respect to income and taxprice, much in line with previous Swedish findings in a median voter framework. Estimation of a log-linear demand specification indicates that the elasticities of demand for schooling are higher in Sweden than in the US. Testing the hypothesis that municipal employees tend to have a higher demand for public spending than others, we conclude that income, as well as taxprice and grants, enters the demand function differently for the two groups of employees.     

Testing the CAPM with Time-Varying Risks and Returns

This paper draws on Engle's autoregressive conditionally heteroskedastic modeling strategy to formulate a conditional CAPM with time-varying risk and expected returns. The model is estimated by generalized method of moments. A CAPM that allows mean excess returns to shift in January survives generalized method of moments specification tests for a number of omitted variables. However, a residual dividend yield component is found to remain in the excess returns of smaller firms. We find significant monthly and quarterly components in the risk premia and beta estimates.     

A Comparison of the Efficacy of Liquidity,Momentum, Size and Book‐to‐Market Value Factors in Equity Pricing on a Heterogeneous Sample: Evidence from Asia

This paper compares the size and book‐to‐market value factors of Fama and French (1993) alongside Momentum of Jagadeesh and Titman ( 1993 ) with two Liu ( 2006 ) liquidity factors formed from 1 year rebalancing and 1 month rebalancing respectively. A heterogeneous and comprehensive sample of the top blue chip stocks of all national Asian equity markets with further differentiation undertaken between sub samples formed for Japan only and Asia excluding Japan for period January 2000 to August 2014. Our empirical results suggest that multifactor time invariant pricing models based on augmented capital asset pricing model (CAPM) framework are ineffective in explaining the cross section of stock returns in the presence of significant inter and intra‐market segmentation. However an alternative model specification based on a time varying parameter specification and using same sets of factors yields significant enhancements in explaining cross section of stock returns across universe. We find that momentum factor largely lacks significance while a time varying two factor model, based on CAPM plus liquidity factor, is optimal. The liquidity factor being that of Liu (2006) and annually rebalanced. Our findings are important for investment managers seeking appropriate factors and modelling techniques to hedge against risks as well as firm's financial managers seeking to reduce costs of equity capital.     

Does tax convexity matter for risk? A dynamic study of tax asymmetry and equity beta

The purpose of this study is to explore the effect of tax convexity on firms’ market risk, where tax convexity measures the progressivity of firms’ tax function. We examine the relation between equity beta and tax convexity based on a standard contingent-claims model, in which firms face nonlinear tax schedules. We verify that in the presence of default and growth options, the effect of tax convexity on beta is significant and depends on several countervailing forces. Tax convexity has a direct, positive effect on beta, as well as two indirect countereffects through default and growth options. The overall effect is ambiguous and quantitatively significant. As asymmetric tax schedules are used in most countries, assuming a linear tax schedule in the valuation framework may misestimate beta and thus fail to assess risk accurately. Our theoretical model shows that tax convexity should be taken into consideration when estimating equity beta.     

Macroeconomic sources of foreign exchange risk in new EU members

We address the issue of foreign exchange risk and its macroeconomic determinants in several new EU members. We derive the observable macroeconomic factors—consumption and inflation—using the stochastic discount factor (SDF) approach. The joint distribution of excess returns in the foreign exchange market and the factors are modeled using a multivariate GARCH-in-mean specification. Our findings show that both real and nominal factors play important roles in explaining the variability of the foreign exchange risk premium. Both types of factors should be included in monetary general equilibrium models employed to study excess returns. To contribute to the further stability of domestic currencies, the new EU members should strive to implement stabilization policies aimed at achieving nominal as well as real convergence with the core EU members.     

Shrinking the cross-section

We construct a robust stochastic discount factor (SDF) summarizing the joint explanatory power of a large number of cross-sectional stock return predictors. Our method achieves robust out-of-sample performance in this high-dimensional setting by imposing an economically motivated prior on SDF coefficients that shrinks contributions of low-variance principal components of the candidate characteristics-based factors. We find that characteristics-sparse SDFs formed from a few such factors—e.g., the four- or five-factor models in the recent literature—cannot adequately summarize the cross-section of expected stock returns. However, an SDF formed from a small number of principal components performs well.