# Projects supported by the German Research Society (DFG)

**Bayesian Estimation of the Multi-Period Optimal Portfolio Weights and Risk Measures (Sept. 2014 - Sept. 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.