Publikationen

This dissertation is composed of three research papers that I have been working on during my doctoral studies in the "Doctoral Programme in Quantitative Economics and Finance" at the University of Konstanz. The three papers are empirical studies on time series analysis and empirical macroeconomics. The first article analyses the effects of fiscal policy shocks in the United States of America (U.S.). It uses additional data on the domestic economic activities that enter the usual vector autoregressive model (VAR) in the form of augmented factors. In the second paper a similar factor augmented vector autoregressive model (FAVAR) is used to assess the transmission mechanisms of monetary policy in four newly joined European Union (EU) member states. In the last article the Euro Area (EA) Gross Domestic Product (GDP) is "nowcasted" using a novel approach that combines wavelet analysis with the bridge equations model.

The financial crisis has shown that quantitative asset allocation models and risk management models have not been sufficiently understood. Since the seminal paper of Markowitz (1952), the academic portfolio management literature has relied on sample estimates of return moments, which have led to extreme and unstable portfolio weights. In the 1990s, Britten-Jones (1999) and others demonstrated the poor empirical performance of the standard Markowitz model. As particularly the mean is difficult to estimate, literature often relies on the minimum variance portfolio (for a sensitivity analysis to changes to the mean see Best and Grauer, 1991a). Still, the estimation error in the covariance matrix is large (Chan et al., 1999). To reduce the estimation error in the covariance matrix, Ledoit and Wolf (2004) suggest a shrinkage approach.

Several regularization procedures have been proposed. Jagannathan and Ma (2003) advise short-selling restrictions. The approach is generalized by Brodie et al. (2009) who penalize short positions. They introduce a Lasso penalty on the norm of the portfolio weights. Stabilizing portfolio weights by shrinking them directly towards some predefined target is studied by Frahm and Memmel (2010). Despite numerous efforts, DeMiguel et al. (2009b) show that naive portfolio strategies such as the equally weighted portfolio are difficult to outperform. Clearly, it is not yet fully understood under which circumstances portfolio optimization fails. This thesis advocates and contributes to the development of alternatives and extensions to current portfolio optimization procedures. One possibility to reduce estimation risk is the combination of different asset allocation models. Model combination in a forecasting context is analyzed in Chapter 1, while the transfer to asset allocation is presented in Chapter 2. The combined model turns out to have a stable performance and does not (strongly) suffer from misspecification of the individual models. Other possibilities are alternative asset allocation strategies or improvements upon current portfolio optimization procedures. Chapter 3 advocates a portfolio strategy, called Minimax, which does not suffer from estimation risk in the returns’ moments. We show that the Minimax can be considered as an alternative to Markowitz portfolio optimization. An extension to standard portfolio optimization is given in Chapter 4. We introduce a penalty term for the norm of portfolio weights to prevent extreme asset allocations. The penalty is shown to improve the performance of standard asset allocation strategies.

Chapter 1 is a joint paper with Fabian Krüger and analyzes the performance of model combination. The performance of a probabilistic forecast model is commonly measured by strictly proper scoring rules (Gneiting and Raftery, 2007). It has been shown that some popular scoring rules are concave functions of the forecast. By Jensen’s inequality it holds that the average score is necessarily smaller (i.e. worse) than the score of the average forecast. This feature is often related to the good empirical performance of forecast combination, compared to the individual forecasts. The success of forecast combination is partly a consequence of the forecast evaluation methodology. We generalize the literature by showing that (smooth and strictly proper) scoring rules cannot be entirely convex, may be entirely concave, and are at least locally concave around the true probability. The finding implies that if forecast predictions are sufficiently close to the true probability, the performance of the average forecast is at least as good as the average performance of the individual models. Concavity depends on the true probability, which is not known in practice. For a given set of forecasts, we suggest to derive a range of true probabilities for which concavity holds. As an example, we analyze the prediction of US recessions based on a Probit and a survey of experts. We find that the range of probabilities under which the (spherical) score is concave is typically much larger than the interval defined by the two forecasts. Further, we find that the ex-post better model and the combined model significantly outperform the ex-post worse model. However, the ex-post better model and the combined model are statistically indistinguishable. We conclude that model combination is rewarding for most scenarios and scoring rules.

Chapter 2 transfers the insights gained in Chapter 1 to portfolio optimization. Under a (strictly) concave performance function, the average model is better than the average performance of the individual models. We find that many performance measures used in asset allocation are concave, while others are concave under certain conditions. So far, the literature was hunting for one single ”true” asset allocation model. We run a large empirical study to analyze the average model of several well-known asset allocation models. We use five models of three model classes applied to six different data sets and evaluate them by five different performance measures. We find that no single model constantly outperforms the others. The ranking of the models depends on the performance measure as well as on the chosen data set. Even for a certain performance measure and a certain data set, the ranking of the models strongly changes over time. The finding confirms that in calm periods sophisticated models outperform naive models; in rough periods data-ignorant models outperform sophisticated models. In the situation of changing model ranking and concave performance measures, the average model has to perform well by definition. The theoretical conclusion can be affirmed by our empirical study. The average model performs almost as good as the ex-post best model.

Chapter 3 is a joint paper with Steffen Schaarschmidt proposing a portfolio strategy called Minimax. The strategy deviates from classical risk measures and defines risk in terms of the worst case scenario. Common symmetric risk measures have undesirable properties. First, large positive returns should not be considered as risk. Second, rare extreme losses might get too little attention. In view of the recent economic turmoil, investors may prefer our conservative risk measure. Additionally, our approach circumvents the estimation of unstable means as well as the estimation of a large covariance matrix. A typical investor might be a corporation, a pension fund, or a bank which has to implement daily risk management. The target is to minimize daily investment losses. A second type of investor is an investor who is facing mark-to-market accounting. He aims to minimize the margin calls as the portfolio falls short of a certain level. The Minimax strategy is a ”pessimistic” trading strategy, as it chooses the portfolio weights such that the portfolio payoff is maximized for the minimum outcome. Our contribution is to show that the Minimax portfolio is implementable for a multi asset investor. In our empirical study, we use US stock, bond, real estate and commodity indexes to construct portfolios with yearly holding periods. We compare the performance of the Minimax portfolio to the performance of the asset allocation of a typical pension fund. Additionally, we use alternative asset allocation strategies, such as the equally weighted portfolio, the Minimum Variance portfolio and the (short-selling restricted) Mean Variance portfolio. We find that the Minimax portfolio performs well compared to the benchmarks considered.

Chapter 4 is a joint paper with Prof. Dr. Pohlmeier considering a recent regularization approach for portfolio weights. To stabilize portfolio weights, Jagannathan and Ma (2003) find that the no-shortsale constraint works well in portfolio optimization. Still, the approach is too restrictive, as moderate short positions can enhance the performance of the portfolio (Fan, 2010). Brodie et al. (2009) introduce a L1 norm restriction for the portfolio weights. In the statistics literature, the type of restriction was introduced as the ”Lasso” by Tibshirani (1996). The restriction can be interpreted as a penalization of the portfolio’s short positions. Naturally, the question arises how to determine the optimal penalty level for short-selling. Our contribution is to introduce a rule-of-thumb for the penalty level. The resulting ”Lasso portfolio” is easy to implement and asymptotically optimal. It performs well in a simulation
study. In an empirical study, we find that the Lasso portfolio outperforms the no short-sale Mean Variance portfolio, the unrestricted Mean Variance portfolio as well as various alternative strategies proposed by literature.

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The mean-variance approach was first proposed by Markowitz (1952), and laid the foundation of the modern portfolio theory. Despite its theoretical appeal, the practical implementation of optimized portfolios is strongly restricted by the fact that the two inputs, the means and the covariance matrix of asset returns, are unknown and have to be estimated by available historical information. Due to the estimation risk inherited from inputs, desired properties of estimated optimal portfolios are dramatically degraded. This problem has been addressed by empirical research and is well known by both practitioners and academics for many years. However, only quite recently, some studies such as Kan & Zhou (2007) and Frahm & Memmel (2010) tried to provide analytical insights into the real-world portfolio choice problems which help us to understand key aspects of the empirical portfolios and to find the possible way to improve the portfolio performance. This dissertation is a collection of three stand-alone papers and contributes to the recent literature by taking some important issues into account such as the investor’s risk preference, non-negativity constraints as well as the presence of structural breaks.

The first chapter analyzes the estimation risk of efficient portfolio selection. We use the concept of certainty equivalent as the basis for a well-defined statistical loss function and a monetary measure to assess estimation risk. For given risk preferences we provide analytical results for different sources of estimation risk such as sample size, dimension of the portfolio choice problem and correlation structure of the return process. Our results show that theoretically sub-optimal portfolio choice strategies turn out to be superior once estimation risk is taken into account. Since estimation risk crucially depends on risk preferences, the choice of the estimator for a given portfolio strategy becomes endogenous depending on sample size, number of assets and properties of the return process. We show that a shrinkage approach accounting for estimation risk is generally superior to simple theoretically subopti- mal strategies. Moreover, focusing on just one source of estimation risk, e.g. risk reduction in covariance estimation, can lead to suboptimal portfolios.
Imposing portfolio constraints is one of the most effective ways to improve plug-in estimates of mean-variance portfolios. Jagannathan & Ma (2003) show that the non-negativity constraint in construction of the global minimum variance portfolio has a shrinkage interpretation and could improve the portfolio performance even if the constraint is wrong in population. The second chapter generalizes the theoretical result of Jagannathan & Ma (2003) to the efficient portfolio case where the investor’s risk preference plays a crucial role in portfolio construction. We show that imposing the non-negativity constraint on efficient portfolios is equivalent to using a modified covariance matrix which depends on asset expected returns and the risk preferences of investors. We conduct a simulation study with realistic inputs to demonstrate the trade-off between the theoretical and empirical losses of the constrained portfolio with respect to the investor’s risk preferences. In addition, different constrained and unconstrained portfolio strategies are compared in both simulation and empirical studies. We find that conservative but unconstrained portfolio strategies proposed by recent studies could outperform constrained portfolios even in the small sample case where the mean and the covariance matrix are estimated with large estimation errors.

The third chapter of the thesis takes possible structural breaks into account and analyzes the estimation risk of different mean-variance portfolio strategies with and without the adding-up constraint. Building upon as idea from Pesaran & Timmermann (2007), we provide an analytical comparison of empirical portfolios estimated by including pre-break data with pure post-break portfolio strategies. It is shown that portfolios incorporating pre-break information can be dominating with respect to their certainty equivalents and the dominance relationship is consistent for dif- ferent risk aversion levels. Although the theoretical result is obtained under the assumption that there is only a unique structural break whose date is known, our approach combining portfolios estimated from pre- and post-break data can be easily generalized to the multiple break case with unknown break points. In addition, un- der the normality assumption, we provide an unbiased way to estimate the difference of certainty equivalents between combined portfolios and pure post-break portfolios which allows us to identify the benefit of using pre-break information in portfolio construction.