Bitcoin is an unregulated digital currency originally introduced in 2008 without legal tender status. Based on a decentralized peer-to-peer network to confirm transactions and generate a limited amount of new bitcoins, it functions without the backing of a central bank or any other monitoring authority. In recent years, Bitcoin has seen increasing media coverage and trading volume, as well as major capital gains and losses in a high volatility environment. Interestingly, an analysis of Bitcoin returns shows remarkably low correlations with traditional investment assets such as other currencies, stocks, bonds or commodities such as gold or oil. In this paper, we shed light on the impact an investment in Bitcoin can have on an already well-diversified investment portfolio. Due to the non-normal nature of Bitcoin returns, we do not propose the classic mean-variance approach, but adopt a Conditional Value-at-Risk framework that does not require asset returns to be normally distributed. Our results indicate that Bitcoin should be included in optimal portfolios. Even though an investment in Bitcoin increases the CVaR of a portfolio, this additional risk is overcompensated by high returns leading to better return-risk ratios.

<p>The advances in three related areas of state-space modeling, sequential Bayesian learning, and decision analysis are addressed, with the statistical challenges of scalability and associated dynamic sparsity. The key theme that ties the three areas is Bayesian model emulation: solving challenging analysis/computational problems using creative model emulators. This idea defines theoretical and applied advances in non-linear, non-Gaussian state-space modeling, dynamic sparsity, decision analysis and statistical computation, across linked contexts of multivariate time series and dynamic networks studies. Examples and applications in financial time series and portfolio analysis, macroeconomics and internet studies from computational advertising demonstrate the utility of the core methodological innovations.</p><p>Chapter 1 summarizes the three areas/problems and the key idea of emulating in those areas. Chapter 2 discusses the sequential analysis of latent threshold models with use of emulating models that allows for analytical filtering to enhance the efficiency of posterior sampling. Chapter 3 examines the emulator model in decision analysis, or the synthetic model, that is equivalent to the loss function in the original minimization problem, and shows its performance in the context of sequential portfolio optimization. Chapter 4 describes the method for modeling the steaming data of counts observed on a large network that relies on emulating the whole, dependent network model by independent, conjugate sub-models customized to each set of flow. Chapter 5 reviews those advances and makes the concluding remarks.</p> / Dissertation

Title: Optimization and stress tests Author: Diana Fašungová Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Jitka Dupačová, DrSc., Department of Probability and Mathematical Statistics Abstract: In the thesis we apply contamination technique on a portfolio optimiza- tion problem using minimization of risk measure CVaR. The problem is considered from a risk manager point of view. We stress correlation structure of data and of revenues using appropriately chosen data for this kind of problem and for ge- nerated stress scenarios. From behaviour of CVaR with regard to contamination bounds, we formulate recommendations for the risk manager optimizing his port- folio. The recommendations are interpreted for both types of stress scenarios. In the end, limitations of the model and possible ways of improvement are discussed. Keywords: contamination bounds, stress tests, portfolio optimization, risk mana- gement

This thesis' topic is stochastic programming, in particular with regard to portfolio optimization and heavy tailed data. The first part of the thesis mentions the most common types of problems associated with stochastic programming. The second part focuses on solving the stochastic programming problems via the SAA method, especially on the condition of data with heavy tailed distributions. In the final part, the theory is applied to the portfolio optimization problem and the thesis concludes with a numerical study programmed in R based on data collected from Google Finance.

In this thesis, a portfolio optimization with integer variables which influ- ence optimal assets allocation, is studied. Measures of risk are defined and the cor- responding mean-risk models are derived. Two methods are used to develop robust models involving uncertainty in probability distribution: the worst-case analyses and contamination. The uncertainty in values of scenarios and in their probabili- ties of the discrete probability distribution is assumed separately followed by their combination. These models are applied to stock market data with using optimization software GAMS.

In this thesis, a portfolio optimization with integer variables which influence optimal assets allocation in domestic as well as in international environment, is studied. At the beginning with basic terms, assets and portfolio background, incentives of portfolio creation, fields of portfolio application and portfolio management is dealt. Following the characteristics of assets and portfolios (expected return, risk, liquidity), which are used by investors to value their properties, are introduced. Next the mean-risk models are derived for the measures of risk - variance, Value at Risk, Conditional Value at Risk and prepared for a practical application. Heuristics implemented in Matlab and standard algorithms of software GAMS are used for solving problems of the portfolio optimization. At the end optimization methods are applied on real financial data and an outputs are compared.

In order to evaluate and manage portfolio risk, we separated this project into three sections. In the first section we constructed a portfolio with 15 different stocks and six options with different strategies. The portfolio was implemented in Interactive Brokers and rebalanced weekly through five holding periods. In the second section we modeled the loss distribution of the whole portfolio with normal and student-t distributions, we computed the Value-at-Risk and expected shortfall in detail for the portfolio loss in each holding week, and then we evaluated differences between the normal and student-t distributions. In the third section we applied the ARMA(1,1)-GARCH(1,1) model to simulate our assets and compared the polynomial tails with Gaussian and t-distribution innovations.

This report is for MA575: Market and Credit Risk Models and Management, given by Professor Marcel Blais. In this project, three different methods for estimating Value at Risk (VaR) and Expected Shortfall (ES) are used, examined, and compared to gain insightful information about the strength and weakness of each method. In the first part of this project, a portfolio of underlying assets and vanilla options were formed in an Interactive Broker paper trading account. Value at Risk was calculated and updated weekly to measure the risk of the entire portfolio. In the second part of this project, Value at Risk was calculated using semi-parametric model. Then the weekly losses of the stock portfolio and the daily losses of the entire portfolio were both fitted into ARMA(1,1)-GARCH(1,1), and the estimated parameters were used to find their conditional value at risks (CVaR) and the conditional expected shortfalls (CES).

In order to evaluate and manage portfolio risk, we separated this project into three sections. In the first section we constructed a portfolio with 15 different stocks and six options with different strategies. The portfolio was implemented in Interactive Brokers and rebalanced weekly through five holding periods. In the second section we modeled the loss distribution of the whole portfolio with normal and student-t distributions, we computed the Value-at-Risk and expected shortfall in detail for the portfolio loss in each holding week, and then we evaluated differences between the normal and student-t distributions. In the third section we applied the ARMA(1,1)-GARCH(1,1) model to simulate our assets and compared the polynomial tails with Gaussian and t-distribution innovations.

Esta tese é dedicada ao estudo de modelos de otimização de carteiras de investimento multi-período. Daremos ênfase a um modelo com restrições intermediárias formulado como um problema de controle ótimo e resolvido utilizando técnicas de programação dinâmica. Serão tratados aspectos teóricos e práticos desta classe de problemas. Primeiramente faremos uma revisão das principais hipóteses dos modelos de otimização de carteiras e o caso uni-período. Analisaremos a seguir as generalizações para o caso multi-período, onde os modelos utilizam apenas restrições para o valor esperado e/ou para a variância da carteira no instante final do período analisado. Apresentaremos então o principal resultado proposto neste trabalho onde consideramos o problema de seleção ótima de ativos multi-período no qual podemos incorporar ao modelo restrições intermediárias para o valor esperado e variância da carteira durante o período de análise. A grande vantagem desta técnica é permitir o controle do valor esperado e/ou da variância da carteira ao longo de todo o horizonte de análise. Faremos uma comparação o entre as formulações apresentadas e realizaremos experimentos numéricos com o modelo proposta nesta tese. Os principais resultados originais desta tese encontram-se no Capítulo 5. No Capítulo 6 apresentamos as simulações numéricas realizadas com o modelo proposto. / The subject of this thesis is the study of multi-period portfolio optimization problems. We focus on a model with intermediate constraints formulated as an optimal control problem and solved by using dynamic programming techniques. Both theoretical and practical issues are addressed. Firstly we will analyze the main hypothesis of portfolio optimization models and the single period case. Then we will present the generalization for the multi-period case, where the models use only constraints for the expected value and variance at the final period. The main result proposed in this work considers the multi-period portfolio selection problem with intermediate constraints on the expected value and variance of the portfolio taken into account in the optimization problem. The main advantage of this technique is that it is possible to control the intermediate expected value or variance of the portfolio during the time horizon considered. Comparison between the presented formulations and numerical experiments of the proposed model will be exposed. The main original results of this thesis can be found in Chapter 5. In Chapter 6 we present numerical simulations with the proposed model.