Abgeschlossene
Statistische Methoden zur Überwachung des Lageverhaltens von multivariaten Zeitreihen (November 2004 - Juli 2007)
Inhalt: In unserem Vorhaben setzten wir uns mit einem Thema aus der Sequentialanalyse auseinander. Man findet zwar in der Literatur, vor allem in der Ökonometrie, viele Beiträge über Strukturveränderungen bei Zeitreihen (z. B. Broemeling und Tsurumi (1987), Maddala und Kim (2002), Greene (2003)). Dabei wird in den meisten Fällen eine retrospektive Vorgehensweise gewählt. Dies bedeutet, dass die Stichprobe bereits vollständig vorliegt und im nachhinein untersucht wird, ob eine Veränderung in der vorgegebenen Stichprobe aufgetreten ist. Häufig unterstellt man auch Vorinformationen über die Position der Veränderung (Bayes--Ansatz). Das Ziel ist stets die Einführung eines Tests auf Veränderung. Dies ist eine vollkommen andere Vorgehensweise als die von uns gewählte. Wir unterstellten, dass die Daten sequentiell erhoben werden. Liegt eine neue Beobachtung vor, so wurde überprüft, ob sie mit dem Sollprozess noch verträglich ist. Falls nicht, endet das Verfahren. Falls ja, wird die nächste Beobachtung gezogen. An die Stelle von Tests traten sogenannte Kontrollkarten.
Zusammenfassung: In diesem Projekt wurden mehrere Überwachungsverfahren für das Lageverhalten multivariater Zeitreihen hergeleitet. Die Vorgehensweise ist sequentiell. Dies erlaubt Veränderungen im Prozessverlauf schneller zu erkennen. Es stellt einen entscheidenden Vorteil in Bezug auf die vielfach praktizierten retrospektiven Methoden dar. Damit können Veränderungen wesentlich schneller erkannt werden. Das stochastische Verhalten der eingeführten Kontrollkarten wurde eingehend analysiert. Es wurden Aussagen über die Richtungsinvarianz der Karten gemacht und das Verhalten der Verfahren wurde mittels stochastischer Ungleichungen analysiert. Ferner wurden diese Verfahren mittels unterschiedlicher Gütekriterien miteinander verglichen. Es konnten Aussagen über die Wahl der Designparameter und zur Anwendung der verschiedenen Verfahren gemacht werden. Es wurden ebenfalls Kontrollverfahren hergeleitet, die auf einer Vorinformation über die Lageveränderung basieren. Im Prinzip muss bekannt sein, in welchem Unterraum die Veränderung liegt.
Die Anwendungsmöglichkeiten der erzielten Ergebnisse sind vielfältig. In den Arbeiten wurden vor allem Anwendungen in der Finanzwirtschaft betrachtet, allerdings können sie ebenfalls in den Ingenieurwissenschaften, den Naturwissenschaften, der Medizin und in anderen Bereichen der Wirtschaftswissenschaften verwendet werden.
Sequenzielle Überwachung von optimalen Portfoliogewichten - gemeinsames Projekt mit Prof. Dr. Vasyl Golosnoy, Universität Bochum (November 2005 - Juli 2009)
Inhalt: Dieses Projekt setzt sich mit der sequenziellen Überwachung der Gewichte eines Portfolios auseinander. Es wird unterstellt, dass ein Anleger zu einem gewissen Zeitpunkt sein Portfolio nach gewissen Optimalitätskriterien (z. B. Minimum-Varianz-Portfolio) zusammengestellt hat. Da sich das Renditeverhalten von Aktien im Zeitverlauf ändern kann, stellt sich die Frage, ob diese Konstellation zu einem späteren Zeitpunkt immer noch optimal ist oder nicht.
Im Rahmen dieses Projektes sollen statistische Methoden eingeführt werden, die Veränderungen in der optimalen Gewichtung frühzeitig erkennen. Hierzu wird auf eine Methode zurückgegriffen, die insbesondere in den Ingenieurwissenschaften verbreitet ist, sogenannte Kontrollkarten. Es sollen Kontrollschemata für die Portfoliogewichte eingeführt werden und es soll kritisch überprüft werden, inwieweit diese Verfahren für Investoren vorteilhaft sind.
Ziel: Das Ziel des Projektes ist die Herleitung von statistischen Überwachungsmethoden für optimale Portfoliogewichte. Es soll ein neuer Ansatz zur Konstruktion von Kontrollkarten verfolgt werden. Dieser wird motiviert durch eine Approximation an die geschätzten Portfoliogewichte. Durch die stärkere Berücksichtigung des lokalen Verhaltens erhält man eine alternative Kenngröße, die nicht mehr die Defizite der Differenzkarten besitzt und außerdem auch nicht die starken Abhängigkeiten der geschätzten Gewichte. Aus diesen Gründen scheint diese Vorgehensweise sehr Erfolg versprechend zu sein. Die bisherigen Karten sind Verfahren für das Lageverhalten der geschätzten Portfoliogewichte. Da sich in der Praxis eine Veränderung ebenfalls in der Kovarianzmatrix der geschätzten Gewichte widerspiegelt, soll dies ebenfalls bei der Konstruktion der Karten berücksichtigt werden.
Ferner wurden ausschließlich Kontrollstatistiken vom EWMA-Typ betrachtet. Es ist beabsichtigt, Kontrollverfahren basierend auf kumulierten Summen (CUSUM-Ansatz) einzuführen. Hierzu ist geplant, die Karten von Crosier (1988), Pignatiello und Runger (1990) und Ngai und Zhang (2001) für das Lageverhalten von unabhängigen multivariaten Beobachtungen auf die Portfolioüberwachung zu erweitern.
Sämtliche Verfahren sollen miteinander verglichen werden. Dies geschieht durch die Betrachtung verschiedener Gütekriterien für Kontrollkarten. Überdies soll im Rahmen einer empirischen Studie analysiert werden, inwieweit ein Kontrollkartenansatz für einen Investor von Nutzen ist. Zunächst sollen diese Ansätze zur Überwachung der Gewichte des globalen Minimum-Varianz-Portfolios verwendet werden. Zur Berechnung dieser Gewichte geht nur die Kovarianzmatrix, aber nicht der Erwartungsvektor ein.
Im weiteren Projektverlauf sollen andere Optimalitätskriterien zur Portfolioauswahl, vor allem die "mean-variance" Zielfunktion, untersucht werden. Dabei tritt das zusätzliche Problem auf, dass Veränderungen in der erwarteten Renditestruktur die Portfolioselektion beeinflussen. Es ist beabsichtigt, Kontrollkarten für die Portfoliogewichte derartiger Nutzenfunktionen herzuleiten.
Im letzten Teil des Projekts werden die optimalen Gewichte, die aus einem multiperiodischen Portfolioproblem stammen, überwacht. Dabei wird unterstellt, dass die Anzahl der Veränderungen in der Renditeverteilung durch einen Poisson-Prozess beschrieben werden kann. Es sollen Verfahren zur Überwachung der Gewichte in Abhängigkeit vom Zeithorizont des Investors hergeleitet werden.
Sequentielle Überwachungsmethoden für das Risikoverhalten komplexer Prozesse (März 2008 - Dezember 2010)
Inhalt: Dieses Projekt setzt sich mit der Herleitung von statistischen Überwachungsmethoden für das Risikoverhalten komplexer Prozesse auseinander. Hierzu wird ein sequenzieller Ansatz gewählt. Dies bedeutet, dass nach dem Eingang einer neuen Informationseinheit sofort überprüft wird, ob Abweichungen vom unterstellten Verlauf auftreten. Damit hat der Anwender die Möglichkeit schneller auf Veränderungen zu reagieren. Es wird unterstellt, dass sich der ungestörte Prozess durch einen multivariaten stationären Prozess modellieren lässt. Zur Messung des Risikoverhaltens werden unterschiedliche Kenngrößen wie z. B. die unbedingte und die bedingte Kovarianzmatrix des Prozesses herangezogen. Zum Monitoring dieser Kenngrößen werden neue Kontrollkarten hergeleitet. Es sollen Aussagen über die stochastischen Eigenschaften dieser Überwachungsverfahren gemacht werden. Insbesondere sollen die eingeführten Methoden eingehend miteinander verglichen werden. Hierzu wird das Kontrolldesign jedes Verfahrens derartig bestimmt, dass im Falle keiner Veränderung die Karten dasselbe Signalverhalten besitzen. Zur Bewertung der Überwachungsmethoden im Falle einer Veränderung wird als Gütemaß die maximale Verzögerung herangezogen.
Ziel: Das Ziel des Projekts ist die Herleitung von sequenziellen Überwachungsmethoden für das Risikoverhalten eines multivariaten Prozesses. Der ungestörte Prozess wird entweder durch einen stationären multivariaten Gaußschen Prozess oder einen stationären multivariaten nicht-linearen Prozess modelliert. Das Risikoverhalten wird durch unterschiedliche Kenngrößen wie etwa der unbedingten Kovarianzmatrix bzw. der bedingten Kovarianzmatrix beschrieben. Es sollen Kontrollverfahren für diese Kenngrößen hergeleitet werden. Dabei liegt der Schwerpunkt auf der Betrachtung von Kontrollstatistiken, die auf kumulierten Summen basieren. Die stochastischen Eigenschaften der eingeführten Überwachungsmethoden sollen eingehend analysiert werden. Insbesondere sollen die Verfahren miteinander verglichen werden. Als Gütekriterium wird hierzu die maximale durchschnittliche Verzögerung herangezogen. In einem zweiten Schritt soll der Einfluss der Parameterschätzung diskutiert werden. Es wird üblicherweise davon ausgegangen, dass die Parameter des ungestörten Prozesses bekannt sind. In der Praxis ist diese Anforderung allerdings selten erfüllt. Damit stellt sich die Frage, wie die Parameterschätzung die Kontrollschranken und damit auch die Güte der Überwachungsverfahren beeinflusst. Diese Fragestellung soll für unterschiedliche Schätzmethoden in Abhängigkeit vom Stichprobenumfang analysiert werden.
Statistical Analysis of Portfolio Characteristics for Different Risk Measures (September 2010 - voraussichtlich August 2013)
Inhalt: In the pioneering work of Markowitz (1952) an optimal portfolio is obtained by minimizingthe portfolio variance for a given value of the portfolio return. Although in the meantimemany other approaches for constructing an optimal portfolio have been introduced, the meanvarianceanalysis of Markowitz (1952) is still the most popular method in practice. For along time one of the crucial assumptions for the derivation of the optimal portfolio weightswas that the parameters of the underlying return process are known. It was recommended toestimate these quantities by historical data. Recently several authors started to analyze thisproblem from a statistical point of view. Various estimators and tests of optimal portfolioweights and portfolio characteristics have been proposed and compared with each other(e.g., Okhrin and Schmid (2006), Bodnar and Schmid (2008a/b, 2009)). In these papers thevariance is chosen as a risk measure of the portfolio. In the last years, however, it has beenshown in several papers that the variance is not a good risk measure and other measuresshould be favored (e.g., Artzner et al. (1999)). The aim of this project is to consider theportfolio selection problem by using more suitable risk measures. Estimators and tests forthe corresponding optimal portfolio weights and portfolio characteristics will be derived.
Ziel: In 1952 Markowitz invented the mean-variance analysis which provides an easy and intelligent solution of the optimal portfolio selection problem. Following this approach the weights of the optimal portfolio, i.e. the parts of the investor’s wealth invested into the selected assets, are obtained by minimizing the variance of the portfolio for a given level of the expected return. Because of its simplicity, the mean-variance analysis has become very popular within practitioners and researches on the financial sector. As a result, Markowitz received the Nobel Memorial Prize in economic sciences in 1990. Depending on the chosen level of the expected return, different optimal portfolios are obtained. All these portfolios lie in a set in the mean-variance space, the so-called efficient frontier, and they possess the property that a larger value of the expected return corresponds to a larger value of the risk. Hence, it is impossible to increase the profit of the portfoliots of the investor’s wealth invested into the selected assets, are obtained by minimizing the variance of the portfolio for a given level of the expected return. Because of its simplicity, the mean-variance analysis has become very popular within practitioners and researches on the financial sector. As a result, Markowitz received the Nobel Memorial Prize in economic sciences in 1990. Depending on the chosen level of the expected return, different optimal portfolios are obtained. All these portfolios lie in a set in the mean-variance space, the so-called efficient frontier, and they possess the property that a larger value of the expected return correspondsto a larger value of the risk. Hence, it is impossible to increase the profit of the portfolio without increasing its risk. Merton (1972) studied this point in detail. He showed that the dependence between the expected return and the risk is non-linear. He derived the equation of the efficient frontier which is a parabola in the mean-variance space and a hyperbola in the mean-standard eviation space. Several other approaches have been proposed in financial literature to construct an optimal portfolio like, e.g., the Sharpe ratio (Sharpe (1966, 1994), Lo (2002), Memmel (2003), Hooker and Xiang (2007), Ledoit and Wolf (2008)) and the Treynor ratio (Treynor (1965)). The maximization of the expected quadratic utility (EU) function provides an alternative procedure (e.g., Ingersoll (1987)). Kroll et al. (1984) reported that the mean-variance portfolio has a maximum utility function or at least a near optimum expected utility. In these cases the risk of the portfolio is mainly measured by its standard deviation and the solution presents a tradeoff between the expected return and the standard deviation. The portfolio allocation problem with higher moments has been considered among others by Samuelson (1970), Lai (1991), Chunhachinda et al. (1997), Prakash et al. (2003), Jondeau and Rockinger (2006). A new class of risk measures was proposed by Artzner et al. (1999), the so-called coherent measures. These approaches make use of the knowledge of the parameters of the asset return distribution. Consequently, the derived optimal weights and portfolio characteristics like, e.g., the expected portfolio return and the portfolio variance, depend on these parameters. In practice the unknown parameters are estimated by suitable estimators. In most cases the estimated optimal portfolio weights and the estimated characteristics of the optimal portfolios are obtained by plugging the sample estimators of the mean vector and the covariance matrix into the corresponding formulas instead of the unknown parameters. This means that we are always working with estimators of the optimal portfolio weights and estimators of the portfolio characteristics. This fact was nearly completely ignored in literature over a long time and there are only a few papers dealing with this problem in the last century. Assuming the asset returns to be independently and multivariate normally distributed Jobson and Korkie (1980) studied the distributional properties of the estimated weights resulting from the Sharpe ratio, while Jobson and Korkie (1981) showed that the estimated Sharpe ratio is asymptotically normally distributed. A more detailed analysis of the estimated portfolio weights started with Okhrin and Schmid (2006). They derived the first two moments of the estimated weights of the mean-variance portfolio and the maximum expected quadratic utility portfolio assuming independent and multivariate normally distributed returns. Moreover, they proved that the first moment does not exist for the estimated weights of the maximum Sharpe ratio portfolio. Schmid and Zabolotskyy (2008) extended this result. They showed that there is no unbiased estimator for the weights of the Sharpe optimal portfolio. The asymptotic behavior of the estimated portfolio weights obtained by the EU approach for a stationary return process is analyzed in Okhrin and Schmid (2006) as well. Another important point is to characterize the performance of the underlying portfolio. The problem of testing the efficiency of a portfolio has been discussed in a large number of studies. In the absence of a riskless asset Gibbons (1982), Kandel (1984), Shanken (1985) and Stambaugh (1982) have analyzed multivariate testing procedures for the mean-variance efficiency of a portfolio. Jobson and Korkie (1989) and Gibbons et al. (1989) derived exact F-tests for testing the efficiency of a given portfolio. Britten-Jones (1999) presented an F-test for the efficiency of a portfolio with respect to the portfolio weights which is based on a single linear regression. More recently, Bodnar and Schmid (2008a) and Bodnar (2009) proposed tests for the weights of the global minimum variance portfolio and, moreover, they derived the distribution of the test statistics for finite sample case. Further tests introduced for the portfolio analysis are the well-known spanning and intersection tests given in Huberman and Kandel (1987), Jobson and Korkie (1989), and Kan and Zhou (2008). Control charts for monitoring the global minimum variance portfolio weights are given by Golosnoy and Schmid (2007) and Bodnar (2007). Bodnar (2009) derived sequential monitoring procedures for the tangency portfolio weights. The distributional properties of the sample efficient frontier were studied among others by Jobson and Korkie (1980), Jobson (1991), Kan and Smith (2008), and Bodnar and Schmid (2008b, 2009). While Jobson and Korkie (1980) considered the asymptotic behavior of the estimated parameters of the efficient frontier, recent results of Kan and Zhou (2007), Kan and Smith (2008), and Bodnar and Schmid (2008b) dealt with the behavior for a finite sample size. Jobson (1991) constructed an asymptotic confidence region and Bodnar and Schmid (2009) provided a confidence set for finite samples. While in most papers the parameters of the return process are estimated by the sample counterparts, shrinkage estimators have been applied by Jorion (1986), Frost and Savarino (1986), Kashima (2005), Golosnoy and Okhrin (2007), Kan and Zhou (2007), Okhrin and Schmid (2007, 2008) among others in order to reduce the variance of the estimated optimal portfolio weights. Ledoit and Wolf (2003, 2004) suggested an improved estimator of the covariance matrix for the portfolio selection problem. Moreover, Okhrin and Schmid (2007) provided an overview and a comparison of several estimation procedures. In most of the above mentioned papers, the results are obtained assuming the asset returns to be independently and normally distributed. However, this assumption is not realistic if data of the asset returns of higher frequency is available, like daily or more frequent returns, or if data are taken from emerging markets. It appear that the asset returns are dependent and non-normally distributed. Bodnar et al. (2009) derived the distributional properties of the estimated efficient frontier in the case of a dependent gaussian process, while Bodnar and Zabolotskyy (2008, 2009) obtained the asymptotic distributions of the estimated weights and the estimated characteristics of the optimal portfolios for conditionally heteroscedastic models of the asset returns. In nearly all of these papers the variance is used as a measure of the portfolio risk. Recently, several authors postulated desirable properties on risk measures. This procedure led to the introduction of coherent risk measures (e.g., Artzner et al. (1999)). The subject of this project is to discuss the problem of portfolio selection by using more suitable risk measures than the variance. We want to introduce estimators and tests for the corresponding optimal portfolio weights and portfolio characteristics.
Bayesian Estimation of the Multi-Period Optimal Portfolio Weights and Risk Measures (Sept. 2014 - September 2018)
Inhalt: The project consists of two main parts. In the first part of the project we are planning to derive analytical and/or recursive solutions of the multi-period portfolio choice problem for an quadratic utility as well as for an exponential utility. The second part of the project is more applied. In this part the methods of the Bayesian statistics will be used for deriving the posterior distributions of the estimated weights and the corresponding risk measures of multi-period optimal portfolios.
In the mathematical part of our project we plan further research of optimization procedures used in the multi-period portfolio choice problems. First, we contribute to the existing literature by deriving the closed-form (recursive) solution of the dynamic portfolio choice problem with and without a riskless asset under rather weak assumptions. The only conditions imposed on the distributions of the asset returns will be the existence of the conditional mean vectors and of the conditional covariance matrices. No assumptions about the correlation structure between different time points or about the distribution of the asset returns, like normality, will be used. The suggested method can be applied for both stationary and non-stationary stochastic models. The results will be obtained assuming that the investor makes his decision on the basis of the quadratic utility function (Li and Ng (2000), Leippold et al. (2004), Brandt and Santa-Clara (2006), Bodnar and Schmid (2008b, 2009), Celikyurt and Özekici (2007)). This is one of the most commonly used procedures since the paper of Tobin (1958) where it is shown that the Bernoulli principle is satisfied for the mean-variance solution only if one of the following two conditions is valid: the asset returns are normally distributed, which is rarely the case in application, or the utility function is quadratic. On the other hand, the quadratic utility function is usually considered as a good approximation of the other utility functions (cf. Brandt et al. (2005)).
In an empirical study we are going to apply the obtained results to real data and to compare the performance of the suggested strategies with existing multi-period portfolio allocation methods. Second, we plan to extend the result of the paper by Canakoglu and Özekici (2009) who solved the portfolio choice problem for an exponential utility assuming that the stochastic market follows a discrete time Markov chain and all parameters of the asset returns, i.e., mean vector and covariance matrix, depend only on the current state of the stochastic market and not on the previous states. Another paper closely related to this part of the project is written by Soyer and Tanyeri (2006) where a Bayesian computational approach with the exponential utility was presented. The authors write that the solution of the multi-period portfolio choice problem with the exponential utility under the assumption of normality ”...cannot be evaluated in closed form and the optimal portfolio cannot be obtained analytically”.
Furthermore, we note that the application of the exponential utility function is more plausible than the use of the quadratic utility since the first one is monotonically decreasing. That is why the exponential utility function is commonly used in portfolio selection theory. Moreover, the optimization of the expected exponential utility function leads to the well known mean-variance utility maximization problem and consequently its solution lays on the mean-variance efficient frontier. We intend to derive an analytical solution of the multiperiod portfolio choice problem with the exponential utility function under the assumption that the asset returns depend on certain predictable variables. The joint process consists of the asset returns and the predictable variables and it is assumed to follow a vector autoregressive (VAR) process. This approach is very popular in finance and it is often used for modeling the asset returns (see, e.g., Campbell (1991, 1996), Barberis (2000), Avramov (2002), Brandt (2010)).
Third, we plan to consider partial cases of the obtained general solutions in detail. We expect that the expressions of general solutions in case of the quadratic utility could be quite complicate and cannot be evaluated for an arbitrary model of the asset returns. For this reason we will suggest a reasonable approximation for the weights and evaluate its accuracy. Furthermore, under the additional assumption of independence we are going to show that at each time point the optimal multi-period portfolio weights obtained by maximizing the quadratic utility function can be presented in a similar way as the optimal single-period portfolio weights. A very remarkable result is expected to be obtained for the portfolio selection problem based on the tangency portfolio. Assuming independent returns it will be proved that the solution of the multi-period problem and the solutions of the simple-period problems are the same. Similarly, we deal with the multi-period portfolio choice problem for an exponential utility function under the assumption that the asset returns are independent. We show that in this case the results can be obtained as a partial case of Canakoglu and Özekici (2009) where the stochastic market was presented by a discrete time Markov chain. In that case the asset returns depend on the present state of the market and not on the previous ones which implies the independence of the asset return over time.
In the statistical part of the project we are planning to do research in three directions too. In the first direction we will start with assigning of informative priors to the parameters of the datagenerating process. The aim will be to derive the posterior distributions of the optimal multi-period portfolio weights obtained by maximizing a quadratic utility function as well as an exponential utility. We also plan to obtain the posteriors for the characteristics of the multi-period optimal portfolios, like the expected return and the variance. It is noted that assigning an informative prior for the parameters of the asset return distribution has already been used in the application of Bayesian statistics to portfolio theory. For instance, Pastor (2000) suggested a class of priors that incorporates an investor’s beliefs on an asset pricing model. Tu and Zhou (2010) dealt with this point in detail in the single period case. They proposed a way how to allow Bayesian priors to reflect the economic beliefs. In this part of the project, the results of Tu and Zhou (2010) will be extended to the multi-period portfolio choice problems where the information is updated sequentially.
On the other hand, the assignment of non-informative priors (Laplace (1812), Jeffreys (1946, 1961)) are preferable in statistical literature. It is argued that the prior has to reflect no (or only vague) information and it should have only a minor impact on the posterior distribution. In 1979 Bernardo suggested a new approach for constructing a non-informative prior for one group of parameters in the Bayesian statistics. The idea behind this approach is based on maximizing the distance between the calculated prior and the corresponding resulting posterior, i.e. the best choice of the non-informative prior is those that has a smallest impact on the posteriors. The Shannon’s mutual information, also known as the Kullback-Leibler divergence distance, was used in that paper as a distance between the prior. We will deal with the determination of non-informative priors for the multi-period portfolio choice problems in the second direction. We plan to derive noninformative priors for the weights of optimal portfolios obtained as solutions of the multi-period portfolio choice problems for a quadratic utility as well as for an exponential utility. Using these priors the posteriors for the optimal portfolio weights and for the corresponding risk measures will be obtained.
Third, we plan to compare the posterior distributions calculated under assigning informative as well as non-informative priors. Using these posteriors we will derive credible intervals for the optimal portfolio weights and the corresponding risk measures as well as calculate the coverage probabilities of these intervals. Because we deal with the multi-period portfolio choice problems, several credible intervals should be compared simultaneously. It requires a definition of a suitable performance measure which is the next goal of the project in this direction. Finally, all the derived theoretical results will be evaluated for real and simulated data. The first research assistant (3=4 position) will participate mainly in the theoretical research of the mathematical part of the project, while the second research assistant (full position) in the theoretical research of the statistical part. Note that both parts of the project require much programming for simulation studies and empirical examples which will be performed by these two researches. Refer also to Section 4 for further details.
Ziele: The objective of this project is to make a series of theoretical and empirical contributions in two major fields. The first field deals with the derivation of the analytical and/or recursive expressions for the solutions of multi-period portfolio choice problem for several utility functions. First, we will consider an investor who based his investment strategy on the quadratic utility function and derive (recursive) closed-form expressions for the optimal portfolio weights. Second, we will deal with the multi-period portfolio choice problem based on maximizing the exponential utility function which appears to be more plausible in portfolio theory than the application of the quadratic utility since it is monotonically decreasing. Third, the general solutions can be simplified to several important partial cases by, for example, assuming that the asset returns are independently distributed or the asset returns depend on predictable variables and that the joint random process of the asset returns and the predictable variables follow a vector autoregressive process. The later assumption is usually applicable when a high-dimensional portfolio is of interest (see, e.g. Brandt (2010)). The second field of contribution is statistically oriented. Here, the methods of Bayesian statistics will be applied to derive the distributional properties of the estimated multi-period optimal portfolios and their risk measures. First, the information from the pre-investment period will be used for the determination of informative prior. Second, we will use the theory of Berger and Bernardo (1992) for deriving a noninformative prior for optimal portfolio weights and risk measures. In both case, the posterior distributions will be obtained and will be used for constructing of the credible intervals for the quantities of interest. Third, we will apply the obtained theoretical results to real data in an empirical study and compare the performance of the suggested strategies with existing multi-period portfolio allocation methods.
Wishart Processes in Statistics and Economics (September 2011 - August 2013)
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Ziel: