Stochastic Discount Factor Models Research Articles

Machine learning private equity returns

In this paper, we use two machine learning techniques to learn the aggregated return time series of complete private capital fund segments. First, we propose Stochastic Discount Factor (SDF) model combination to determine the public factor exposure of private equity. Here, we describe our theoretical motivation to favor model combination over model selection. This entails that we apply simple coefficient averaging to obtain multivariate SDF models that mimic the factor exposure of all major private capital fund types. As a second step, we suggest componentwise L2 boosting to estimate the error-term time series associated with our factor models. The simple addition of the public factor model returns and the error terms then yields the total return time series. These return time series can be applied for proper integrated public and private risk management or benchmarking.

Testing the boundaries of applicability of standard Stochastic Discount Factor models

We provide a joint non-parametric test to gather insights on the boundaries of applicability of Stochastic Discount Factor (SDF) models. We find that a non-trivial class of models cannot price the U.S. stock market equally weighted portfolio, implying non-monotonic SDFs, especially over the last 50/60 years in (recessionary) periods characterized by higher market volatility. Stocks responsible for this rejection mostly belong to the smallest NYSE market cap decile, are characterized by high idiosyncratic risk, and typically cannot be priced via SDF models where the aggregate level of risk aversion is bigger then 9 or 10. Excluding these stocks increases the ability to explain the cross-section of returns without impairing the ability to span the mean–variance frontier.

The Risk of Expected Utility Under Parameter Uncertainty

We derive analytical expressions for the risk of an investor’s expected utility under parameter uncertainty. In particular, our analysis focuses on characterizing the out-of-sample utility variance of three portfolios: the classic mean-variance portfolio, the minimum-variance portfolio, and a shrinkage portfolio that combines both. We then use our analytical expressions to study a robustness measure that balances out-of-sample utility mean and volatility. We show that neither the sample mean-variance portfolio nor the sample minimum-variance portfolio exhibits maximal robustness individually, and one needs to combine both to optimize portfolio robustness. Accordingly, we introduce a robust shrinkage portfolio that delivers an optimal tradeoff between out-of-sample utility mean and volatility and is more resilient to estimation errors. Our results highlight the importance of considering out-of-sample performance risk in designing and evaluating investment strategies and stochastic discount factor models. This paper was accepted by Kay Giesecke, finance. Funding: N. Lassance gratefully acknowledges financial support by the Fonds de la Recherche Scientifique [Grant J.0115.22]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.00178 .

The reduced-rank beta in linear stochastic discount factor models

In a linear stochastic discount factor model, failure of the full-rank conditions affects the standard statistical inference of coefficients. We propose a novel risk measurement, the reduced-rank beta, which is the risk sensitivity to the effective part of factors for the full-rank covariance matrix. Our reduced-rank beta is a generalisation of the standard beta when the full-rank condition is not satisfied. By considering the Fama–French five-factor (FF5) model for the US equity market, the failure of the full-rank condition is found to affect beta estimates. We demonstrate the reduced-rank beta has important empirical implications for model reductions and anomaly explanations.

Moneyness, Underlying Asset Volatility, and the Cross-Section of Option Returns

Abstract We study the effect of an asset’s volatility on the expected returns of European options on the asset. Deriving predictions from a stochastic discount factor model, we show that the effect depends on whether variations in the asset’s volatility are driven by systematic or idiosyncratic volatility. While idiosyncratic-volatility-induced variations only affect the option elasticity, systematic-volatility-induced variations also oppositely affect the expected return of the asset. Since the expected asset return (elasticity) effect dominates for options with more linear (non-linear) payoffs, systematic volatility prices sufficiently in-the-money (out-of-the-money) options with the opposite (same) sign as idiosyncratic volatility. Using single-stock calls as test assets, double-sorted portfolios and Fama–MacBeth (1973) regressions broadly support the model’s predictions.

Evaluating the performance of U.S. international equity closed-end funds

This study examines whether clientele effects are important in the evaluation of the performance of U.S. international equity closed-end funds (CEF) using the best clientele (BC) performance measure of Chretien and Kammoun (2017), and alternative stochastic discount factor models based on global factor models. The study finds that clientele effects are important when evaluating the performance of international CEFs, as there are significant differences between the BC performance and performance using the global factor models. International CEF provide significant superior performance using the BC measure and neutral performance with the global factor models.

Asset Pricing with Panel Trees Under Global Split Criteria

We introduce a class of interpretable tree-based models (P-Trees) for analyzing panel data, with iterative and global (instead of recursive and local) splitting criteria to avoid overfitting and improve model performance. We apply P-Tree to generate a stochastic discount factor model and test assets for cross-sectional asset pricing. Unlike other tree algorithms, P-Trees accommodate imbalanced panels of asset returns and grow under the no-arbitrage condition. P-Trees also graphically capture nonlinearity and interaction effects and accommodate regime-switching and interactions between macroeconomic states and firm characteristics. For example, P-Tree identifies inflation as the most important macro predictor with regime-switching in U.S. equity data. Based on multiple pricing, prediction, and investment metrics, we find that (boosted or time-series) P-Trees outperform standard factor models and PCA latent factor models. An equally-weighted portfolio for five factors generated by P-Trees delivers an excess alpha of 1.09% against the Fama-French 3-factor benchmark, producing an annualized Sharpe ratio of 1.98 out-of-sample. Data-driven cutpoints in P-Trees reveal that long-run reversal, volume volatility, and industry-adjusted market equity drive cross-sectional return variations, consistent with variable importance analysis using random forests.

Generalized aggregation of misspecified models: With an application to asset pricing

We propose a generalized aggregation approach for model averaging. The entropy-based optimal criterion is a natural choice for aggregating information from many “globally” misspecified models as it adapts better to the underlying model uncertainty and obtains more robust approximations. Unlike almost all other approaches in the existing literature, we do not require a “reference model,” or a true data generation process contained in the set of models — neither implicitly nor in otherwise popular limiting forms. This shift in paradigm prioritizes stochastic optimization and aggregation of information about outcomes over parameter estimation of an optimally selected model. Stochastic optimization is based on a risk function of aggregators across models that satisfies oracle inequalities. Our generalized aggregators relax the common perfect substitutability of the candidate models, implicit in linear averaging and pooling. The aggregation weights are data-driven and obtained from a proper (Hellinger) distance measure. The empirical results illustrate the performance and economic significance of the aggregation approach in the context of stochastic discount factor models and inflation forecasting.

Comparing Asset Pricing Models by the Conditional Hansen-Jagannathan Distance*

AbstractWe compare nonnested parametric specifications of the stochastic discount factor (SDF) using the conditional Hansen–Jagannathan (HJ-) distance. This distance measures the discrepancy between a parametric model-implied SDF and the admissible SDF’s satisfying all the conditional (dynamic) no-arbitrage restrictions, instead of just few unconditional no-arbitrage restrictions for managed portfolios chosen through the instrument selection. We estimate the conditional HJ-distance by a generalized method of moments estimator and establish its large sample properties for model selection purposes. We compare empirically several SDF models including multifactor beta pricing specifications and some recently proposed SDF models that are conditionally linear in consumption growth.

Growth in Average Firm Size of U.S. Industrial Portfolios and the Cross-Section of Expected Returns

This paper shows that growth in average firm size in U.S. industrial portfolios predicts future growth in average firm size. Moreover, the payoffs of industrial portfolios sorted by growth in average firm size in the previous period increase linearly as we move from lowest to highest growth in average firm size. The spread between highest and lowest growth in average firm size is economically large and cannot be explained by exposures to standard risk factors or the asset growth effect (Cooper, Gulen, and Schill, 2008). Principal component analysis reveals that this growth in average firm size effect is linked to the first principal component. Moreover, stochastic discount factor model analysis shows that the spread is marginal useful for pricing the cross section of U.S. industrial portfolios.

Portfolio performance of linear SDF models: an out-of-sample assessment

We evaluate linear stochastic discount factor models using an ex-post portfolio metric: the realized out-of-sample Sharpe ratio of mean–variance portfolios backed by alternative linear factor models. Using a sample of monthly US portfolio returns spanning the period 1968–2016, we find evidence that multifactor linear models have better empirical properties than the CAPM, not only when the cross-section of expected returns is evaluated in-sample, but also when they are used to inform one-month ahead portfolio selection. When we compare portfolios associated to multifactor models with mean–variance decisions implied by the single-factor CAPM, we document statistically significant differences in Sharpe ratios of up to 10 percent. Linear multifactor models that provide the best in-sample fit also yield the highest realized Sharpe ratios.

Portfolio Performance of Linear SDF Models: An Out-of-Sample Assessment

We evaluate linear stochastic discount factor models using an ex-post portfolio metric: the realized out-of-sample Sharpe ratio of mean-variance portfolios backed by alternative linear factor models. Using a sample of monthly US portfolio returns spanning the period 1968-2016, we find evidence that multifactor linear models have better empirical properties than the CAPM, not only when the cross-section of expected returns is evaluated in-sample, but also when they are used to inform one-month ahead portfolio selection. When we compare portfolios associated to multifactor models with mean-variance decisions implied by the single-factor CAPM, we document statistically significant differences in Sharpe ratios of up to 10 percent. Linear multifactor models that provide the best in-sample fit also yield the highest realized Sharpe ratios.

Estimating stochastic discount factor models with hidden regimes: Applications to commodity pricing

We develop new likelihood-based methods to estimate factor-based Stochastic Discount Factors (SDF) that may accommodate Hidden Markov dynamics in the factor loadings. We use these methods to investigate whether it is possible to find a SDF that jointly prices the cross-section of eight U.S. portfolios of stocks, Treasuries, corporate bonds, and commodities. In particular, we test a range of possible different specification of the SDF, including single-state and Hidden Markov models and compare their statistical and pricing performances. In addition, we assess whether and to which extent a selection of these models replicates the observed moments of the return series, and especially correlations. We report that regime-switching models clearly outperform single-state ones both in term of statistical and pricing accuracy. However, while a four-state model is selected by the information criteria, a two-state three-factor full Vector Autoregression model outperforms the others as far as the pricing accuracy is concerned.

Corruption, Carry Trades, and the Cross Section of Currency Returns

This is the first paper to explore the effects of perceived corruption on the FX market. It finds that the currencies of countries perceived to suffer from high levels of corruption generate statistically significantly lower returns than the currencies of countries perceived to have low levels of corruption. Moreover, the portfolio spread is highly correlated with NBER recessions and U.S. consumption growth of nondurable goods. Interestingly, stochastic discount factor model analysis reveals that the portfolio spread is useful for pricing the cross section of currency returns, even when controlling for standard FX risk factors.

Moneyness, Total, Systematic, and Idiosyncratic Volatility, and the Cross-Section of European Option Returns

We use a stochastic discount factor model to show that the expected returns of European options are not unambiguously positively or negatively related to their underlying asset’s volatility, a conclusion strikingly different from those derived in the prior literature. Higher idiosyncratic volatility always lowers (raises) expected call (put) returns through increasing the chance of non-zero option payoffs in high (low) stochastic discount factor states. Higher systematic volatility can raise or lower the expected call or put return since it produces both the above option-payoff effect, but also an oppositely-signed effect arising through a higher underlying asset risk. The strike price modulates the strengths of the oppositely-signed effects and thus determines the signs of both the effects of systematic volatility and of total volatility on expected option returns. Single-stock call option data strongly confirm the testable implications of our European option pricing theory.

Can No-Arbitrage SDF Models with Regime Shifts Explain the Correlations between Commodity, Stock, and Bond Returns?

We investigate whether it is possible to find a Stochastic Discount Factor (SDF) that jointly prices the cross-section of eight U.S. portfolios of stocks, Treasuries, corporate bonds, and commodities and replicates their observed moments, and especially correlations. We use the first three principal components extracted from a set of 112 U.S. macro variables as pricing factors. We also introduce commodity-based factors in the SDF, motivated by commodity pricing theories and we use them either separately or in conjunction with the macro-based ones. We report that it is not impossible to find a set of macroeconomic factors able to jointly price the cross section of stocks, bonds, and commodities; however, this task is accomplished by a small set of commodity-based factors. Observed correlations for commodities are perfectly matched by a parsimonious, single state, diagonal factor VAR model where only three commodity-based factors enter the SDF. While introducing regimes does not improve the performance in all cases, they could be beneficial to replicating correlations among commodity and bond portfolios.

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

We propose a multivariate test based on no-arbitrage conditions under the stochastic discount factor approach, which compares cross-sectional variation in equity returns to the cross-sectional variation in their conditional covariance with the discount factors. Using the multivariate generalized heteroskedasticity in mean model to estimate the 25 portfolios formed on size and book-to-market ratio, together each with its own arbitrage condition, we find that the no-arbitrage test rejects the consumption-based capital asset pricing model (C-CAPM). Although the conditional covariances of returns with consumption exhibit negative variation across size, they do not vary across the book-to-market ratio. Thus, the C-CAPM can capture size effect, but not value effect. Allowing the coefficients on the consumption covariances to be different largely improves the fit of the C-CAPM, however. The value effect appears to be associated with book-to-market ratio as well as size. Book-to-market ratio separately does not generate information about average returns that cannot be explained by the C-CAPM. One possible explanation for this extra dimension of risk is the investment growth prospect of firms. Low book-to-market ratio firms may be expected to have higher rates of growth while small firms may also be expected to behave similarly.

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

We propose a multivariate test of the capital asset pricing model (C-CAPM) of the cross-sectional variation in equity returns in which we compare cross-sectional variation in equity returns to the cross-sectional variation in their conditional covariance with stochastic discount factors. We use a multivariate generalized heteroskedasticity in mean model to estimate 25 portfolios that are formed on size and the book-to-market ratio. Each portfolio is allowed to have its own no-arbitrage condition. We find that although the conditional covariances of returns with consumption exhibit negative variation across size, they do not vary across the book-to-market ratio. Thus, C-CAPM can capture the size effect, but not the value effect. The fit is, however, improved by allowing the coefficients on the consumption covariances to be different. The value effect appears to be associated with the book-to-market ratio as well as size. On its own the book-to-market ratio does not generate additional information about average returns to C-CAPM. A possible explanation for these findings is that both small and low book-to-market ratio firms are expected to have higher rates of growth.

Identification and Inference in Linear Stochastic Discount Factor Models with Excess Returns

When excess returns are used to estimate linear stochastic discount factor (SDF) models, researchers often adopt a normalization of the SDF that sets its mean to 1, or one that sets its intercept to 1. These normalizations are often treated as equivalent, but they are subtly different both in population, and in finite samples. Standard asymptotic inference relies on rank conditions that differ across the two normalizations, and which can fail to differing degrees. I first establish that failure of the rank conditions is a genuine concern for many well-known SDF models in the literature. I also describe how failure of the rank conditions can affect inference, both in population and in finite samples. I propose using tests of the rank conditions not only as a diagnostic device, but also for model reduction. I show that this model reduction procedure has desirable properties in a Monte-Carlo experiment with a calibrated model.

On the Social Discount Rate of Climate Change Economics: A Stochastic Discount Factor Model Perspective

On the Social Discount Rate of Climate Change Economics: A Stochastic Discount Factor Model Perspective