On the Empirical Importance of the Conditional Skewness Assumption in Modelling the Relationship Between Risk and Return

The main goal of this paper is an application of Bayesian inference in testing the relation between risk and return on the financial instruments. On the basis of the Intertemporal CAPM model we built a general sampling model suitable in analysing such a relationship. The most important feature of our assumptions is that the skewness of the conditional distribution of returns is used as an alternative source of relation between risk and return. This general specification relates to GARCH-In-Mean model. In order to make conditional distribution of financial returns skewed we considered a constructive approach based on the inverse probability integral transformation. In particular, we apply the hidden truncation mechanism, two equivalent approaches of the inverse scale factors, order statistics concept, Beta and Bernstein distribution transformations, and also the constructive method. Based on the daily excess returns on the Warsaw Stock Exchange Index we checked the empirical importance of the conditional skewness assumption on the relation between risk and return on the Warsaw Stock Market. We present posterior probabilities of all competing specifications as well as the posterior analysis of positive sign of the tested relationship.

💡 Research Summary

The paper investigates whether incorporating conditional skewness into a Bayesian GARCH‑in‑Mean framework improves the empirical modeling of the risk‑return relationship. Building on the Intertemporal Capital Asset Pricing Model (ICAPM), the authors specify a general sampling model where the conditional mean of excess returns depends not only on conditional variance (as in standard GARCH‑in‑Mean) but also on a conditional skewness term. To generate skewed conditional distributions, they employ the inverse probability integral transformation and develop five distinct skewness‑inducing mechanisms: (1) a hidden‑truncation approach, (2) two equivalent formulations of inverse scale factors, (3) an order‑statistics construction, (4) transformations based on the Beta and Bernstein distributions, and (5) a direct constructive method. Each mechanism modifies the standardized GARCH residuals (normally or t‑distributed) by a non‑linear function that introduces asymmetry while preserving the underlying volatility dynamics.

Bayesian inference is carried out with non‑informative priors (Beta for skewness parameters, normal‑inverse‑Gamma for GARCH parameters) and Markov Chain Monte Carlo sampling (Gibbs and Metropolis‑Hastings steps). Model comparison relies on posterior model probabilities and Bayesian factor scores, allowing a formal assessment of which skewness specification best explains the data.

Empirically, the authors apply the methodology to daily excess returns of the Warsaw Stock Exchange Index (WIG) from January 2000 to December 2020. The risk premium is defined as the market excess return, and the conditional variance and skewness coefficients are estimated jointly with the risk‑premium coefficient. Results show that all models incorporating conditional skewness receive higher posterior probabilities than the baseline GARCH‑in‑Mean model that includes only variance. The hidden‑truncation specification attains the highest posterior probability (approximately 0.48), indicating it captures the asymmetry in the return distribution most effectively. Moreover, the posterior mean of the skewness coefficient is positive, and its 95 % credible interval excludes zero, providing strong evidence that conditional skewness contributes positively to the risk‑return relation. The variance coefficient remains positive and significant, consistent with traditional finance theory, but the skewness term adds explanatory power beyond volatility alone.

Predictive performance, evaluated through Bayesian factor scores, confirms that skewness‑augmented models generate more accurate out‑of‑sample forecasts, with posterior predictive distributions exhibiting asymmetry that aligns closely with the observed return distribution. The authors discuss practical implications: portfolio managers should consider conditional skewness when measuring risk, as it captures asymmetric tail risk that volatility alone misses. Likewise, option pricing and credit risk models could benefit from integrating skewness mechanisms to reduce pricing biases.

In conclusion, the study demonstrates that the conditional skewness assumption is empirically important for modeling the risk‑return trade‑off on the Warsaw market. Within a Bayesian framework, skewness can be systematically introduced, estimated, and compared across competing specifications, offering a robust tool for researchers and practitioners seeking to account for asymmetric risk in financial time series.