Simulation of Diversified Portfolios in a Continuous Financial Market

In this paper we analyze the simulated behavior of diversified portfolios in a continuous financial market. In particular, we focus on equally weighted portfolios. We illustrate that these well diversified portfolios constitute good proxies of the growth optimal portfolio. The multi-asset market models considered include the Black-Scholes model, the Heston model, the ARCH diffusion model, the geometric Ornstein-Uhlenbeck volatility model and the multi-currency minimal market model. The choice of these models was motivated by the fact that they can be simulated almost exactly and, therefore, very accurately also over longer periods of time. Finally, we provide examples, which demonstrate the robustness of the diversification phenomenon when approximating the growth optimal portfolio of a market by an equal value weighted portfolio. Significant out performance of the market capitalization weighted portfolio by the equal value weighted portfolio can be observed for models.growth optimal portfolio; diversification Theorem; diversified portfolios; equally weighted portfolio; exact simulation

Simulation of Diversified Portfolios in a Continuous Financial Market

The paper analyzes the simulated long-term behavior of well diversified portfolios in continuous financial markets. It focuses on the equi-weighted index and the market portfolio. The paper illustrates that the equally weighted portfolio constitutes a good proxy of the growth optimal portfolio, which maximizes expected logarithmic utility. The multi-asset market models considered include the Black-Scholes model, the Heston model, the ARCH diffusion model, the geometric Ornstein-Uhlenbeck volatility model and a multi-asset version of the minimal market model. All these models are simulated exactly or almost exactly over an extremely long period of time to analyze the long term growth of the respective portfolios. The paper illustrates the robustness of the diversification phenomenon when approximating the growth optimal portfolio by the equi-weighted index. Significant outperformance in the long run of the market capitalization weighted portfolio by the equi-weighted index is documented for different market models. Under the multi-asset minimal market model the equi-weighted index outperforms remarkably the market portfolio. In this case the benchmarked market portfolio is a strict supermartingale, whereas the benchmarked equi-weighted index is a martingale. Equal value weighting overcomes the strict supermartingale property that the benchmarked market portfolio inherits from its strict supermartingale constituents under this model.growth optimal portfolio; diversification theorem; diversified portfolios; market portfolio; equi-weighted index; almost exact simulation; minimal market model

Exact Scenario Simulation for Selected Multi-dimensional Stochastic Processes

Accurate scenario simulation methods for solutions of multi-dimensional stochastic differential equations find application in stochastic analysis, the statistics of stochastic processes and many other areas, for instance, in finance. They have been playing a crucial role as standard models in various areas and dominate often the communication and thinking in a particular field of application, even that they may be too simple for more advanced tasks. Various discrete time simulation methods have been developed over the years. However, the simulation of solutions of some stochastic differential equations can be problematic due to systematic errors and numerical instabilities. Therefore, it is valuable to identify multi-dimensional stochastic differential equations with solutions that can be simulated exactly. This avoids several of the theoretical and practical problems encountered by those simulation methods that use discrete time approximations. This paper provides a survey of methods for the exact simulation of paths of some multi-dimensional solutions of stochastic differential equations including Ornstein-Uhlenbeck, square root, squared Bessel, Wishart and Levy type processes.exact scenario simulation; multi-dimensional stochastic differential equations; multi-dimensional Ornstein-Uhlenbeck process; multi-dimensional square root process; multi-dimensional squared Bessel process; Wishart process; multi-dimensional Levy process

Quasi-exact Approximation of Hidden Markov Chain Filters

This paper studies the application of exact simulation methods for multi-dimensional multiplicative noise stochastic differential equations to filtering. Stochastic differential equations with multiplicative noise naturally occur as Zakai equation in hidden Markov chain filtering. The paper proposes a quasi-exact approximation method for hidden Markov chain filters, which can be applied when discrete time approximations, such as the Euler scheme, may fail in practice.stochastic differential equations; Zakai equation; quasi-exact approximation; hidden Markov chain filtering

Modeling diversified equity indices

University of Technology, Sydney. Faculty of Science.The objective of this thesis is to study and model the dynamics of aggregate wealth, that is, the dynamics of the market capitalization weighted world stock index in different currency denominations. In order for the considered model to be valid over long time periods, it turns out that it needs to be formulated in a general financial modeling framework, the benchmark approach. In order to visualize and test the proposed aggregate wealth dynamics, exact and almost exact simulation techniques for multi-dimensional stochastic processes have been developed. Moreover, the model specification is preceded by a detailed study of the distribution of log-returns of world stock indices in different currency denominations. Various types of world equity indices are constructed and systematically studied, in particular, equi-weighted indices. When the number of constituents is increasing and the given investment universe is well securitized the Naive Diversification Theorem states that a sequence of equi-weighted indices approximates the growth optimal portfolio, which is also the numéraire portfolio. Finally, by conjecturing for the normalized world stock index the dynamics of a time transformed square root process, and by establishing a list of stylized empirical facts, a two-component index model has been proposed. This model is very parsimonious and driven only by the non-diversifiable risk of the market. Via almost exact simulation this model is shown to reflect well all listed empirical stylized facts and is difficult to falsify

Empirical Evidence on Student-t Log-Returns of Diversified World Stock Indices

The aim of this paper is to document some empirical facts related to log-returns of diversified world stock indices when these are denominated in different currencies. Motivated by earlier results, we have obtained the estimated distribution of log-returns for a range of world stock indices over long observation periods. We expand previous studies by applying the maximum likelihood ratio test to the large class of generalized hyperbolic distributions, and investigate the log-returns of a variety of diversified world stock indices in different currency denominations. This identifies the Student-t distribution with about four degrees of freedom as the typical estimated log-return distribution of such indices. Owing to the observed high levels of significance, this result can be interpreted as a stylized empirical fact.diversified world stock index; growth optimal portfolio; log-return distribution; Student-t distribution; generalized hyperbolic distribution; likelihood ratio test

Using Dynamic Copulae for Modeling Dependency in Currency Denominations of a Diversifed World Stock Index

The aim of this paper is to model the dependencya mong log-returns when security account prices are expressed in units of a well diversified world stock index. The paper uses the equi-weighted index EWI104s, calculated as the average of 104 world industry sector indices. The log-returns of its denominations in different currencies appear to be Student-t distributed with about four degrees of freedom. Motivated by these findings, the dependency in log-returns of currency denominations of the EWI104s is modeled using time-varying copulae, aiming to identify the best fitting copula family. The Student-t copula turns generally out to be superior to e.g. the Gaussian copula, where the dependence structure relates to the multivariate normal distribution. It is shown that merely changing the distributional assumption for the log-returns of the marginals from normal to Student-t leads to a significantly better fit. Furthermore, the Student-t copula with Student-t marginals is able to better capture dependent extreme values than the other models considered. Finally, the paper applies copulae to the estimation of the Value-at-Risk and the expected shortfall of a portfolio, constructed of savings accounts of different currencies. The proposed copula-based approach allows to split market risk into general and specific market risk, as defied in regulatory documents. The paper demonstrates that the approach performs clearly better than the Risk Metrics approach.diversified world stock index; Student-t distribution; time-varying copula; Value-at-Risk; expected shortfall

Composite materials used in the manufacture of cars

Bakalárska práca pojednáva o použití jednotlivých druhov kompozitov v odvetviach automobilového priemyslu. V prvej časti je zameraná na úvod do problematiky kompozitných materiálov, spôsoby výroby, ich výhody a nevýhody. Druhá časť práce sa sústredí na konkrétne aplikácie v automobilovom priemysle. Vo finálnej časti je zhodnotená budúcnosť a celkový pohľad na kompozitné materiály v oblasti automobilov.Bachelor’s thesis deals with various types of composites used in the spheres of automotive industry. In the first section, it’s focused on the introduction to issues of composite materials, methods of production, their advantages and disadvantages. The second part of thesis focuses on specific applications in the automotive industry. In the final part there is a future evaluation and an overall view of the composite materials in the automotive field.

Approximating the numéraire portfolio by naive diversification

Estimation theory has shown, owing to the limited estimation window available for real asset data, that the sample-based Markowitz mean-variance approach produces unreliable weights that fluctuate substantially over time. This article proposes an alternate approach to portfolio optimization, being the use of naive diversification to approximate the numéraire portfolio (NP). The NP is the strictly positive portfolio that, when used as benchmark, makes all benchmarked non-negative portfolios either mean decreasing or trendless. Furthermore, it maximizes expected logarithmic utility and outperforms any other strictly positive portfolio in the long run. The article proves for a well-securitized market that the naive equal value-weighted portfolio converges to the NP when the number of constituents tends to infinity. This result is model independent and, therefore, very robust. The systematic construction of diversified stock indices by naive diversification from real data is demonstrated. Even when taking transaction costs into account, these indices significantly outperform the corresponding market capitalization- weighted indices in the long run, indicating empirically their asymptotic proximity to the NP. Finally, in the time of financial crisis, a large equi-weighted fund carrying the investments of major pension funds and insurance companies would provide important liquidity. It would not only dampen the drawdown of a crisis, but would also moderate the excesses of an asset price bubble. © 2012 Macmillan Publishers Ltd