Can Stochastic Discount Factor Models Explain the Cross Section of Equity Returns?

The stochastic discount factor model provides a general framework for pricing assets. By specifying the discount factor suitably it encompasses most of the theories currently in use, including CAPM and consumption CAPM. The SDF model has been based on the use of single and multiple factors, and on latent and observed factors. In most situations, and especially for the term structure, single factor models are inappropriate, whilst latent variables require the somewhat arbitrary specification of generating processes and are difficult to interpret. In this paper we survey the principal different implementations of the SDF model for bonds, equity and FOREX and propose a new approach. This is based on the use of multiple factors that are observable and modelling the joint distribution of excess returns and the factors using a multi–variate GARCH–in–mean process. We argue that in general single equation and VAR models, although widely used in empirical finance, are inappropriate as they do not satisfy the no–arbitrage condition. Since risk premia arise from conditional covariation between the returns and the factors, both a multi–variate context and having conditional covariances in the conditional mean process, is essential. We explain how apparent exceptions, such as the CIR and Vasicek models, in fact meet this requirement — but at a price. We explain our new approach, discuss how it might be implemented and present some empirical evidence, mainly from our own researches. Partly, to enable comparisons to be made, the survey also includes evidence from recent empirical work using more traditional approaches.

I. INTRODUCTION The consumption-based capital asset pricing model (CCAPM) has become the dominant paradigm in asset pricing theory because of its intuitive characterization of asset risk. That is, unlike expectations-based asset pricing theories (e.g. the present value model of stock prices) which inherently assume risk neutrality, the CCAPM is predicated upon risk-averse agents attempting to achieve a smooth consumption profile through the trading of assets. From this perspective, the risk of an asset is captured by the covariance of the asset's return with agents' marginal utility of consumption, or, more broadly, agents' intertemporal marginal rate of substitution of wealth. With this characterization, the CCAPM extends the results of the standard CAPM, i.e. the risk of an asset is reflected in the covariance of the asset's return with the market portfolio, to a general equilibrium setting. However, this intuitively appealing theory has not found much empirical support. Perhaps the most famous demonstration of inconsistency between the theory and data was Rajnish Mehra and Edward Prescott's [1985] analysis of the equity premium, i.e. the difference between the average return on equity and that on government bonds. Using calibration methods rather than a formal statistical approach, they showed that a standard CCAPM severely underpredicts the size of the risk premium on equity. The important implication of this finding is that the basic CCAPM incorrectly models the covariance structure between the return on equity and asset holders' intertemporal marginal rate of substitution. Economists' response to this empirical rejection has been to either identify new sources of risk (e.g., that implied by a particular monetary environment as in Labadie [1989]) or define alternative preferences toward risk (e.g., the use of non-expected utility theory as presented in Weil [1989] and Epstein and Zin [1989]). The potential for this research was illustrated recently by George Constantinides's [1990] resolution of the equity premium puzzle in a model that included habit formation in consumption. By relaxing the standard CCAPM's assumption of time-separable preferences, Constantinides demonstrated that a relatively smooth consumption profile and large risk premium on equity are consistent with equilibrium in a complete markets, rational-expectations framework.(1) I use this insight to study another risk premium puzzle; specifically the behavior of the nominal term premium. The existence of a nominal term premium puzzle was demonstrated in papers by Backus, Gregory, and Zin [1989] and Salyer [1990] in which the term structure implications within a Lucas [1982] type cash-in-advance (CIA) economy were studied. While some equilibrium characteristics of this model were consistent with the data, both papers demonstrated that the model predicts the wrong sign of the term premium: the theory implies the term premium should be negative while, based on quarterly data for yields of three- and six-month U.S. Treasury Bills over the period 1959.1-1989.4, the observed term premium is positive. Similar to the equity premium puzzle as posed by Mehra and Prescott [1985], the prediction of a negative average term premium signals a critical discrepancy between the risk associated with nominal bonds as perceived by market participants and as captured by the models. Moreover, this term premium puzzle is in some sense a stronger rejection of the underlying asset pricing model than that embodied by the equity premium puzzle. To see this, recall that within the consumption-based asset pricing model, the sign of the risk premium on any asset depends on the sign of the covariance between the marginal utility of consumption and that asset's return. Applying this reasoning to the case of the equity premium suggests that, while the discrepancy in size between the observed equity premium and that generated within the model is troubling, the model's prediction of a positive equity premium implies that the basic payout structure of equity within the model is consistent with that in the data (e. …

The consumption-based capital asset pricing model: International evidence

I incorporate household debt and delinquency decisions into a standard model of lifecycle consumption-saving-investment. I also impose a punishment to the delinquent behavior by assuming that the percentage of endowment available is a linear function of the default decision. Theoretically such additional investor decisions are playing a relevant role in terms of completing markets. In practice, it enables me to derive an extended system of Euler equations which does not alter consumption-based fundamental asset pricing equation. It imposes the pricing kernel to account jointly for two additional first-order conditions. I perform empirical exercises aiming to price equity premium in United States from 1987:1 to 2018:1. I find significant elasticity of intertemporal substitution in consumption of the representative agent ranging from 0.24 to 0.55 and risk aversion from 1.82 to 3.51. This approach is also useful to account for the cross-section behavior of domestic assets. I can also use this framework to draw bounds for the household decisions on loan and delinquency and to propose a new rule of thumb relating preferences parameters and credit variables.

Stochastic Discount Factor, Asset Pricing, and Dependence of Random Variables in Discrete Period

Can stochastic discount factor models explain the cross-section of equity returns?

Transaction costs, liquidity risk, and the CCAPM

This paper proposes a novel robustness framework for studying the unit elasticity of intertemporal substitution (EIS) assumption based on the Perron-Frobenius sieve estimation model by Christensen, 2017. The sieve nonparametric decomposition is a central model that connects key strands of the long run risk literature and recovers the stochastic discount factor (SDF) under the unit EIS assumption. I generate various economies based on Epstein–Zin preferences to simulate scenarios where the EIS deviates from unity. Then, I study the main estimation mechanism of the decomposition as well as the time discount factor and the risk aversion parameter estimation surface. The results demonstrate the robustness of estimating the average yield, change of measure, and preference parameters but also reveal an “absorption effect” arising from the unit EIS assumption. The findings highlight that asset pricing models assuming a unit EIS produce distorted parameter estimates, caution researchers about the potential under- or over-estimation of risk aversion, and provide insight into trends of misestimation when interpreting the results. I also identify an additional source of failure from a consumption component, which demonstrates a more general limit of the consumption-based capital asset pricing model and the structure used to estimate relevant preference parameters.

Hedge Fund Performance Evaluation: A Stochastic Discount Factor Approach

Management Insights

In this paper we develop alternative ways to compare asset pricing models when it is understood that their implied stochastic discount factors do not price all portfolios correctly. Unlike comparisons based on x2 statistics associated with null hypothesis that models are correct, our measures of model performance do not reward variability of discount factor proxies. One of our measures is designed to exploit fully the implications of arbitrage-free pricing of derivative claims. We demonstrate empirically the usefulness of methods in assessing some alternative stochastic factor models that have been proposed in asset pricing literature.

In this article we develop alternative ways to compare asset pricing models when it is understood that their implied stochastic discount factors do not price all portfolios correctly. Unlike comparisons based on statistics associated with null hypotheses that models are correct, our measures of model performance do not reward variability of discount factor proxies. One of our measures is designed to exploit fully the implications of arbitrage‐free pricing of derivative claims. We demonstrate empirically the usefulness of our methods in assessing some alternative stochastic factor models that have been proposed in asset pricing literature.

In this dissertation, I investigate three related topics on asset pricing: the consumption-based asset pricing under long-run risks and fat tails, the pricing of VIX (CBOE Volatility Index) options and the market price of risk embedded in stock returns and stock options. These three topics are fully explored in Chapter II through IV. Chapter V summarizes the main conclusions. In Chapter II, I explore the effects of fat tails on the equilibrium implications of the long run risks model of asset pricing by introducing innovations with dampened power law to consumption and dividends growth processes. I estimate the structural parameters of the proposed model by maximum likelihood. I find that the stochastic volatility model with fat tails can, without resorting to high risk aversion, generate implied risk premium, expected risk free rate and their volatilities comparable to the magnitudes observed in data. In Chapter III, I examine the pricing performance of VIX option models. The contention that simpler-is-better is supported by the empirical evidence using actual VIX option market data. I find that no model has small pricing errors over the entire range of strike prices and times to expiration. In general, Whaley’s Black-like option model produces the best overall results, supporting the simpler-is-better contention. However, the Whaley model does under/overprice out-of-the-money call/put VIX options, which is contrary to the behavior of stock index option pricing models. In Chapter IV, I explore risk pricing through a model of time-changed Lévy processes based on the joint evidence from individual stock options and underlying stocks. I specify a pricing kernel that prices idiosyncratic and systematic risks. This approach to examining risk premia on stocks deviates from existing studies. The empirical results show that the market pays positive premia for idiosyncratic and market jump-diffusion risk, and idiosyncratic volatility risk. However, there is no consensus on the premium for market volatility risk. It can be positive or negative. The positive premium on idiosyncratic risk runs contrary to the implications of traditional capital asset pricing theory.

Arbitrage, Good Deals and Stochastic Discount Factor Restrictions on Multi-Factor Models With Coskewness

This paper introduces a consumption-based capital asset pricing model (CCAPM) that combines undiversifiable income shocks and external habit formation. Using US state-level data, the paper provides realistic estimates for preference parameters when the external habit of the state investors is based on the consumption of the four Census regions. The model also implies four asset pricing factors: the cross-sectional means of consumption growth and habit growth (capturing national systematic risk) and the cross-sectional variances of consumption growth and habit growth (capturing regional systematic risk). This four-factor model has greater power in explaining expected returns than the CCAPM described in Breeden (1979).