An approach to measuring the relation between risk and return. Bayesian analysis for WIG data

The main goal of this paper is an application of Bayesian inference in testing the relation between risk and return of the financial time series. On the basis of the Intertemporal CAl’M model, proposed by Merton (1973), we built a general sampling model suitable in analysing such relationship. The most important feature of our model assumptions is that the possible skewness of conditional distribution of returns is used as an alternative source of relation between risk and return. Thus, pure statistical feature of the sampling model is equipped with economic interpretation. This general specification relates to GARCH-In-Mean model proposed by Osiewalski and Pipień (2000). 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 approaches based on the inverse scale factors in the positive and the negative orthant, order statistics concept, Beta distribution transformation, Bernstein density transformation and the method recently proposed by Ferreira and Steel (2006). Based on the daily excess returns of WIG index we checked the total impact of conditional skewness assumption on the relation between return and risk on the Warsaw Stock Market. Posterior inference about skewness mechanisms confirmed positive and decisively significant relationship between expected return and risk. The greatest data support, as measured by the posterior probability value, receives model with conditional skewness based on the Beta distribution transform ation with two free parameters.