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.     

An Interpretation of SDF Based Performance Measures

This note discusses stochastic discount factor (SDF) measuresof mutual fund performance. It shows that the most common SDFperformance measures can be interpreted as Jensen's "alphas".JEL Classification Numbers: G11, G12, G23     

Long-term factorization in Heath–Jarrow–Morton models

The long-term factorization decomposes the stochastic discount factor (SDF) into discounting at the rate of return on the long bond and a martingale that defines a long-term forward measure. We establish sufficient conditions for existence of the long-term factorization in HJM models. A condition on the forward rate volatility ensures existence of the long bond volatility. This yields existence of the long bond and convergence of \(T\)-forward measures to the long forward measure. It contrasts with the familiar risk-neutral factorization that decomposes the SDF into discounting at the short rate and a martingale defining the risk-neutral measure.     

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.     

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.     

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。     

Empirical Evaluation of Asset–Pricing Models: A Comparison of the SDF and Beta Methods

The stochastic discount factor (SDF) method provides a unified general framework for econometric analysis of asset–pricing models. There have been concerns that, compared to the classical beta method, the generality of the SDF method comes at the cost of efficiency in parameter estimation and power in specification tests. We establish the correct framework for comparing the two methods and show that the SDF method is as efficient as the beta method for estimating risk premiums. Also, the specification test based on the SDF method is as powerful as the one based on the beta method.     

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.     

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.     

Risk‐Adjusting the Returns to Venture Capital

We adapt stochastic discount factor (SDF) valuation methods for venture capital (VC) performance evaluation. Our approach generalizes the popular Public Market Equivalent (PME) method and allows statistical inference in the presence of cross‐sectionally dependent, skewed VC payoffs. We relax SDF restrictions implicit in the PME so that the SDF can accurately reflect risk‐free rates and returns of public equity markets during the sample period. This generalized PME yields substantially different abnormal performance estimates for VC funds and start‐up investments, especially in times of strongly rising public equity markets and for investments with betas far from one.     

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.     

Nonparametric estimation and testing of stochastic discount factor

This paper attempts to estimate stochastic discount factor (SDF) proxies nonparametrically using the conditional Hansen–Jagannathan distance. Nonparametric estimation can not only avoid misspecification when dealing with nonlinearity in the model but also provide more precise information about the local properties of the estimators. Empirical studies show that our method performs better than the alternative parametric polynomial models, and furthermore, we find that the return on aggregate wealth can sufficiently explain the SDF proxies when one deals with nonlinearity appropriately.     

Asset Pricing at the Millennium

This paper surveys the field of asset pricing. The emphasis is on the interplay between theory and empirical work and on the trade-off between risk and return. Modern research seeks to understand the behavior of the stochastic discount factor (SDF) that prices all assets in the economy. The behavior of the term structure of real interest rates restricts the conditional mean of the SDF, whereas patterns of risk premia restrict its conditional volatility and factor structure. Stylized facts about interest rates, aggregate stock prices, and cross-sectional patterns in stock returns have stimulated new research on optimal portfolio choice, intertemporal equilibrium models, and behavioral finance.     

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.     

Estimation and Evaluation of Conditional Asset Pricing Models

We find that several recently proposed consumption‐based models of stock returns, when evaluated using an optimal set of managed portfolios and the associated model‐implied conditional moment restrictions, fail to capture key features of risk premiums in equity markets. To arrive at these conclusions, we construct an optimal Generalized Method of Moments (GMM) estimator for models in which the stochastic discount factor (SDF) is a conditionally affine function of a set of priced risk factors, and we show that there is an optimal choice of managed portfolios to use in testing a null model against a proposed alternative generalized SDF.     

Financial Intermediaries and the Cross‐Section of Asset Returns

Financial intermediaries trade frequently in many markets using sophisticated models. Their marginal value of wealth should therefore provide a more informative stochastic discount factor (SDF) than that of a representative consumer. Guided by theory, we use shocks to the leverage of securities broker‐dealers to construct an intermediary SDF. Intuitively, deteriorating funding conditions are associated with deleveraging and high marginal value of wealth. Our single‐factor model prices size, book‐to‐market, momentum, and bond portfolios with an R² of 77% and an average annual pricing error of 1%—performing as well as standard multifactor benchmarks designed to price these assets.     

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.     

External balances,trade flows and financial conditions

This paper studies how changing expectations concerning future trade and financial conditions are reflected in international external positions. In the absence of Ponzi schemes and arbitrage opportunities, the net foreign asset position of any country must, as a matter of theory, equal the expected present discounted value of future trade deficits, discounted at the cumulated world stochastic discount factor (SDF) that prices all freely traded financial assets. I study the forecasting implications of this theoretical link in 12 countries (Australia, Canada, China, France, Germany, India, Italy, Japan, South Korea, Thailand, The United States and The United Kingdom) between 1970 and 2011. I find that variations in the external positions of most countries reflect changing expectations about trade conditions far into the future. I also find the changing forecasts for the future path of the world SDF are reflected in the dynamics of the U.S. external position.     

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.