# Research

### Published Papers

**VIRBICKAITE, A, ****C. FREY AND D. MACEDO (2020): "****Bayesian sequential stock return prediction through copulas****," ****The Journal of Economic Asymmetries****.**

In this paper we perform density prediction for the equity returns in a non-linear manner by employing a copula-based approach. The use of asymmetric copulas allows to model asymmetric predictive densities and non-linear dependencies between equity returns and some predictor variable. In our proposed approach, the copula parameter and the marginals are estimated simultaneously by using Sequential Monte Carlo techniques. We apply proposed models to daily log returns of 20 assets traded at the NYSE. Among other findings, we show that in terms of predictive log Bayes Factors the asymmetric copula is preferred by more assets than the symmetric copula, advocating the use of non-linear models. Also, dividend yield is a better predictor variable than the lagged returns overall, but this result is reversed if we consider a volatile period only. These results have major implications for the investors when making portfolio decisions or measuring tail risk.

We propose a Bayesian shrinkage approach for vector autoregressions that uses survey forecasts as additional non-sample information. In particular, we augment the vector of dependent variables by their survey nowcasts and claim that each variable of the VAR and its nowcast are likely to depend in a similar way on the lagged dependent variables. The idea is that this additional information will help us pin down the model coefficients. We find that the forecasts obtained from a VAR fitted by our new shrinkage approach typically yield smaller mean squared forecast errors than the forecasts obtained from a range of benchmark methods.

### Working Papers

**Alkafri, N. AND C. Frey (2020): "Shrinkage estimation for Risk-Parity Portfolio," Working Paper.**

We investigate the impact of shrinkage estimation techniques for the moments of asset returns in risk-parity portfolios. In contrast to mean-variance portfolios, the risk contribution of individual assets in risk-parity portfolios is fixed a priori. This additional information is commonly found to stabilize empirical portfolio weights in time. We show that the marginal risk-budget for each portfolio asset indeed serves as a natural shrinkage target and hence provide a new perspective on risk-parity portfolios. In an extensive empirical application for multi-asset portfolios, we compare the various shrinkage strategies to popular risk-parity approaches from the literature and find that the former show better out-of-sample performance based on various performance criteria. These Results also turn out to be particularly attractive for high-dimensional portfolios.

**FREY, C. AND W. POHLMEIER (2015): "Bayesian Shrinkage of Portfolio Weights," Working Paper, Department of Economics, University of Konstanz.**** (Revise and Resubmit Journal of Empirical Finance)**

We propose a novel regression approach for optimizing portfolios by means of Bayesian regularization techniques. In particular, we represent the weight deviations of the global minimum variance portfolio from a reference portfolio (e.g. the naive 1/N portfolio) as regression coefficients and apply different shrinkage techniques in order to stabilize the portfolios against estimation errors. Modeling the optimal portfolio weights through Bayesian priors avoids estimating the moments of the asset return distribution and substantially reduces the dimensionality of the problem. We compare the proposed Bayesian shrinkage strategies to popular frequentist approaches and find that the former show better out-of-sample performance based on various criteria and especially for larger portfolio dimensions.

**FREY, C. AND W. POHLMEIER (2016): "On the Posterior Distribution of Portfolio Weights," Working Paper, Department of Economics, University of Konstanz.**

We propose a novel regression approach for optimizing portfolios by means of Bayesian regularization techniques. In particular, we represent the weight deviations of the global minimum variance portfolio from a reference portfolio (e.g. the naive 1/N portfolio) as regression coefficients and apply different shrinkage techniques in order to stabilize the portfolios against estimation errors. Modeling the optimal portfolio weights through Bayesian priors avoids estimating the moments of the asset return distribution and substantially reduces the dimensionality of the problem. We compare the proposed Bayesian shrinkage strategies to popular frequentist approaches and find that the former show better out-of-sample performance based on various criteria and especially for larger portfolio dimensions.

### Work in Progress Papers

**"Using Analysts' Forecasts for Stock Predictions - An Entropic Tilting Approach".**

In this paper, we combine predictive density forecasts for US stock returns from Bayesian vector autoregressions with financial analysts' forecasts via entropic tilting. In particular, we modify the predictive density of the asset returns to match moment conditions that are formed based on average analysts' forecasts. The advantage of this approach is that we can combine model-based time-series information with external, forward-looking information in a parsimonious way using closed-form solutions. Using a model with time-varying coefficients and stochastic volatility, we show that tilting the predictive asset return distribution towards the mean and variance of target price implied expected returns increase prediction accuracy for both point and density forecasts.

**"Adjusting Risk Models to Minimize Capital Requirements," joint work with Sebastian Bayer.**

The paper is concerned with minimizing the latest Basel daily capital requirements (CR) of market risk through optimally adjusted Value-at-Risk (VaR) forecasts. While statistical optimal VaR forecasts minimize the tick loss, we propose a correction mechanism for the VaR that directly minimizes the CR loss function. In particular, we shrink the conditional GARCH volatility forecast towards the unconditional variance in order to reduce the VaR forecast in absolute values during calm periods and to increase it in crisis periods. The idea is to avoid high capital costs through too conservative forecasts in the former case, or due to an excessive number of violations in the latter one. We motivate our approach in a simulation study and demonstrate that the true parameters of the volatility process do not minimize the VaR with respect to the CR. Further in an empirical application, we show that our proposed strategy significantly reduces daily capital charges without violating the Basel penalty limits.