## G11 Portfolio Choice; Investment Decisions

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#### Keywords

- Portfoliomanagement (2)
- structured products (2)
- Anlageverhalten (1)
- Cluster-Analyse (1)
- Diversifikation (1)
- Einzelinvestor (1)
- Erwarteter Nutzen (1)
- Kurtosis (1)
- Mixture-Model (1)
- Optionen (1)

The classic Capital Asset Pricing Model and the portfolio theory suggest that investors hold the market portfolio to diversify idiosyncratic risks. The theory predicts that expected return of assets is positive and that reacts linearly on the overall market. However, in reality, we observe that investors often do not have perfectly diversified portfolios. Empirical studies find that new factors influence the deviation from the theoretical optimal investment. In the first part of this work (Chapter 2) we study such an example, namely the influence of maximum daily returns on subsequent returns. Here we follow ideas of Bali et al. (2011). The goal is to find cross-sectional relations between extremely positive returns and expected average returns. We take account a larger number of markets worldwide. Bali et al. (2011) report with respect to the U.S. market a robust negative relation between MAX (the maximum daily return) and the expected return in the subsequent time. We extent substantially their database to a number of other countries, and also take more recent data into account (until end of 2009). From that we conclude that the relation between MAX and expected returns is not consistent in all countries. Moreover, we test the robustness of the results of Bali et al. (2011) in two time-periods using the same data from CRSP. The results show that the effect of extremely positive returns is not stable over time. Indeed we find a negative cross-sectional relation between the extremely positive returns and the average returns for the first half of the time series, however, we do not find significant effects for the second half. The main results of this chapter serve as a basis for an unpublished working paper Yuan and Rieger (2014b). While in Chapter 2 we have studied factors that prevent optimal diversification, we consider in Chapter 3 and 4 situations where the optimal structure of diversification was previously unknown, namely diversification of options (or structured financial products). Financial derivatives are important additional investment form with respect to diversification. Not only common call and put options, but also structured products enable investors to pursue a multitude of investment strategies to improve the risk-return profile. Since derivatives become more and more important, diversification of portfolios with dimension of derivatives is of particularly practical relevance. We investigate the optimal diversification strategies in connection with underlying stocks for classical rational investors with constant relative risk aversion (CRRA). In particular, we apply Monte Carlo method based on the Black-Scholes model and the Heston model for stochastic volatility to model the stock market processes and the pricing of the derivatives. Afterwards, we compare the benchmark portfolio which consists of derivatives on single assets with derivatives on the index of these assets. First we compute the utility improvement of an investment in the risk-free assets and plain-vanilla options for CRRA investors in various scenarios. Furthermore, we extend our analysis to several kinds of structured products, in particular capital protected notes (CPNs), discount certificates (DCs) and bonus certificates (BCs). We find that the decision of an investor between these two diversification strategies leads to remarkable differences. The difference in the utility improvement is influenced by risk-preferences of investors, stock prices and the properties of the derivatives in the portfolio. The results will be presented in Chapter 3 and are the basis for a yet unpublished working paper Yuan and Rieger (2014a). To check furthermore whether underlyings of structured products influence decisions of investors, we discuss explicitly the utility gain of a stock-based product and an index-based product for an investor whose preferences are described by cumulative prospect theory (CPT) (Chapter 4, compare to Yuan (2014)). The goal is that to investigate the dependence of structured products on their underlying where we put emphasis on the difference between index-products and single-stock-products, in particular with respect to loss-aversion and mental accounting. We consider capital protected notes and discount certificates as examples, and model the stock prices and the index of these stocks via Monte Carlo simulations in the Black-Scholes framework. The results point out that market conditions, particularly the expected returns and volatility of the stocks play a crucial role in determining the preferences of investors for stock-based CPNs and index-based CPNs. A median CPT investor prefers the index-based CPNs if the expected return is higher and the volatility is lower, while he prefers the stock-based CPNs in the other situation. We also show that index-based DCs are robustly more attractive as compared to stock-based DCs for CPT investors.

This dissertation includes three research articles on the portfolio risks of private investors. In the first article, we analyze a large data set of private banking portfolios in Switzerland of a major bank with the unique feature that parts of the portfolios were managed by the bank, and parts were advisory portfolios. To correct the heterogeneity of individual investors, we apply a mixture model and a cluster analysis. Our results suggest that there is indeed a substantial group of advised individual investors that outperform the bank managed portfolios, at least after fees. However, a simple passive strategy that invests in the MSCI World and a risk-free asset significantly outperforms both the better advisory and the bank managed portfolios. The new regulation of the EU for financial products (UCITS IV) prescribes Value at Risk (VaR) as the benchmark for assessing the risk of structured products. The second article discusses the limitations of this approach and shows that, in theory, the expected return of structured products can be unbounded while the VaR requirement for the lowest risk class can still be satisfied. Real-life examples of large returns within the lowest risk class are then provided. The results demonstrate that the new regulation could lead to new seemingly safe products that hide large risks. Behavioral investors who choose products based only on their official risk classes and their expected returns will, therefore, invest into suboptimal products. To overcome these limitations, we suggest a new risk-return measure for financial products based on the martingale measure that could erase such loopholes. Under the mean-VaR framework, the third article discusses the impacts of the underlying's first four moments on the structured product. By expanding the expected return and the VaR of a structured product with its underlying moments, it is possible to investigate each moment's impact on them, simultaneously. Results are tested by Monte Carlo simulation and historical simulation. The findings show that for the majority of structured products, underlyings with large positive skewness are preferred. The preferences for variance and for kurtosis are ambiguous.