Finance a řízení rizik

Stochastic interest rates modeling, Diploma thesis, MFF UK. This thesis studies different stochastic models of interest rates. Theoretical part describes short-rate models, HJM framework and LIBOR Market model. It focuses in detail on widely known short-rate models, i.e. Vašícek, Hull-White and Ho-Lee model, and on LIBOR Market model. This part ends by valuation of interest rate options and model calibration to real data. Analytical part analyses valuation of real non-standard interest rate derivative using different models. Part of this derivative valuation is comparison among models in terms of general valuation and also in terms of capturing the dynamics of interest rates. The aim of this thesis is to describe different stochastic models of interest rates and mainly to compare them with each other.

KMV model in terms of the Czech capital market, Diploma thesis, University of Economics. The thesis is focused on the options of quantifying credit risk by using the concept of the KMV model. The introduction outlines the basic approaches to measuring credit risk. In the following chapters is specified the nature of KMV model with the focus on its application in the Czech capital market. Self-calibration of the KMV model is made in this part. The analytical part related to the quantification of credit risk using the KMV model is implemented on selected companies which are traded on the Prague Stock Exchange. The results obtained are consequently confronted with the official rating degrees of agency Moody's.

Credit derivatives market, Diploma thesis, University of Economics. The main objective of this thesis is to acquaint the reader with the main types of credit derivatives, with the methods of functioning and with main valuation principles. The theoretical part focus on description of credit derivatives market developement with more detailed description of subjcts, who are operating on credit derivatives market. The analysis fosus on how the financial crises influenced these subjects and their credit derivatives portfolios. New suggested regulation changes are also described. As a conclusion is the estimation of the regulation changes on the credit derivatives market size.

A Comparison of EVT and Standard VaR Estimations, University of Economics, Working Paper. In this paper, Extreme value theory (EVT) is applied in estimating low quantiles of P/L distribution and the results are compared to common VaR methodologies. The fundamental theory behind EVT is built, and peaks-over-threshold method is used for modeling the tail of the distribution of losses with Generalized Pareto Distribution (GPD). The different VaR methods are then compared using backtesting procedures. Practical issues such as time varying volatility of returns, and multivariate time series (portfolio of financial instruments) are covered.

A Two-Factor Model for PD and LGD Correlation, University of Economics, Working Paper. The paper proposes a two-factor model to capture retail portfolio probability of default (PD) and loss given default (LGD) parameters and in particular their mutual correlation. We argue that the standard one-factor models standing behind the Basel II formula and used by a number of studies cannot capture well the correlation between PD and LGD on a large (asymptotic) portfolio. Parameters of the proposed model are estimated using the Markov Chain Monte Carlo (MCMC) method on a sample of real banking data. The results confirm positive stand-alone PD and LGD correlations and indicate a positive mutual PD x LGD correlation. The estimated Bayesian MCMC distributions of the parameters show that the stand alone correlations are strongly significant with a lower significance of the mutual correlation probably due to a too short observed time period.

Credit Risk Management and Modeling, Textbook. The main goal of this textbook is to provide an overview of credit risk management, as well as modeling approaches from the point of view of a classical commercial bank, savings institution, finance company, or other financial market players. It is targeted mainly on current or future credit risk managers and analysts working in the credit risk departments of those institutions. It will be also useful to those overseeing, or working on the loan underwriting side, as well as to middle and top managers. Our point of view will be usually that of a universal bank, since risk management organization, and the techniques for the other institutions can be described as an appropriate subset of the universal financial institutions. The book can be also useful for corporate risk managers, where risk management organization and modeling techniques are, however, normally significantly simpler compared to those of a bank.

Portfolio Credit Risk Modeling, Diploma thesis, University of Economics. This thesis focuses on state-of-the-art credit models largely implemented by banks into their banking risk-assessment and complementary valuation system frameworks. Reader is provided in general with both theoretical and applied (practical) approaches that are giving a clear notion how selected portfolio models perform in real-world environment. Our study comprises CreditMetrics, CreditRisk+ and KMV model. In the first part of the thesis, our intention is to clarify theoretically main features, modeling principles and moreover we also suggest hypotheses about strengths/drawbacks of every scrutinized model. Subsequently, in the applied part we test the models in a lab-environment but with real-world market data. Noticeable stress is also put on model calibration. This enables us to con_rm/reject the assumptions we made in the theoretical part.

Presentation form the Breakfest with SAS held on 28th April, 2010. After an overview of the key risk management principles and tools the presentation focuses on challenging tasks related to Basel II implementation, in particular LGD, EAD, and correlation estimates. The last part then looks on the latest BCBS proposals leading to a Basel III regulatory package trying to learn a lesson from the recent financial crisis.

Analysis of guaranteed investment funds, Diploma thesis, University of Economics. This thesis focuses on guaranteed investment funds, which have become very popular among investors in the Czech Republic in recent years. The reason for this popularity is the conservativeness of a typical domestic investor, who appreciates the lower bound for the value of his investment. Guaranteed funds characteristically have a complex structure and valuation of their profitability based solely on intuition is therefore impossible. This analysis tries to provide an answer to the question if investing in these funds is reasonable. A large part of the thesis is dedicated to the option theory and option valuation methods, including the famous Black-Scholes formula, as guaranteed investment funds have the characteristics of an option. Thanks to the complicated structure of these products, the analysis itself is done by Monte Carlo simulation.

Survival Analysis in LGD Modeling. IES Working Paper. The paper proposes an application of the survival time analysis methodology to estimations of the Loss Given Default (LGD) parameter. The main advantage of the survival analysis approach compared to classical regression methods is that it allows exploiting partial recovery data. The model is also modified in order to improve performance of the appropriate goodness of fit measures. The empirical testing shows that the Cox proportional model applied to LGD modeling performs better than the linear and logistic regressions. In addition a significant improvement is achieved with the modified “pseudo” Cox LGD model.

Analysis and Comparison of Different Value at Risk Models for Nonlinear Portfolio. Diploma Thesis, MFF UK. The thesis describes Value-at-Risk (VaR) and Expected Shortfall (ES) models for measuring market risk. Parametric method, Monte Carlo simulation, and Historical simulation (HS) are presented. The second part of the thesis analyzes Extreme Value Theory (EVT). The fundamental theory behind EVT is built, and peaks-over-threshold (POT) method is introduced. The POT method is then used for modelling the tail of the distribution of losses with Generalized Pareto Distribution (GPD), and is simultaneously illustrated on VaR and ES calculations for PX Index. Practical issues such as multiple day horizon, conditional volatility of returns, and backtesting are also discussed. Subsequently, the application of parametric method, HS and EVT is demonstrated on a sample nonlinear portfolio designed in Mathematica and the results are discussed.

Estimating LGD Correlation. University of Economics, Working Paper. The paper proposes a new method to estimate correlation of account level Basle II Loss Given Default (LGD). The correlation determines the probability distribution of portfolio level LGD in the context of a copula model which is used to stress the LGD parameter as well as to estimate the LGD discount rate and other parameters. Given historical LGD observations we apply the maximum likelihood method to find the best estimated correlation parameter. The method is applied and analyzed on a real large data set of unsecured retail account level LGDs and the corresponding monthly series of the average LGDs. The correlation estimate comes relatively close to the PD regulatory correlation. It is also tested for stability using the bootstrapping method and used in an efficient formula to estimate ex ante one-year stressed LGD, i.e. one-year LGD quantiles on any reasonable probability level.

Loss, Default, and Loss Given Default Modeling. IES WP 9/2009. The paper follows previous study of capital requirement sensitivity to the definition of default (Witzany, 2008). We study the phenomenon in the context of single-factor models where default and loss given default are driven by one systemic factor and by one or more idiosyncratic factors. In this theoretical framework we propose and analyze a relatively simple remedy of the problem requiring that the LGD parameter is estimated as a quantile of possible losses given default on the same probability level as used to stress probability of default (PD).

Basle II Capital Requirement Sensitivity to the Definition of Default. Risk management. Prague : Oeconomica, 2008. The paper is motivated by a disturbing observation according to which the outcome of the regulatory formula significantly depends on the definition of default used to measure the probability of default (PD) and the loss given default (LGD) parameters. Basel II regulatory capital should estimate with certain probability level unexpected credit losses on banking portfolios and so it should not depend on a particular definition of default that does not change real historical and expected losses. The paper provides an explanation of the phenomenon based on the Merton default model and test it using a Monte Carlo simulation. Finally we perform a structural model based simulation to test the hypothesis according to which scoring functions developed with a soft definition of default provide weaker predictive power than the ones developed with a hard definition of default.

Step by Step Credit Risk Model Construction. Bachelor thesis, MFF UK. The aim of the present work is to outline a principle of scoring models construction. Describtion of the logistic regression method, its parameters estimation and their significance testing. On the ground of odds ratio variables it defines the Independence model as an estimate of the conditional odds of client’s ability to pay. It generalizes this model by adding individual weights to groups and categories of clients characteristic. Using this way it comes to the WOE model and Full logistic model. This work also studies the way of measuring the diversification power of the models by the Lorenz curve and Somer’s d statistics as an estimate of the Gini coefficient. It applies the described methods to the practical scoring model construction. On a real data there are compared suitability and diversification power of the introduced models.

Estimations of Market and Credit Value at Risk. Bachelor thesis, MFF UK. This present work studies statistical estimations of market and credit risk by measure of risk called Value at Risk. This work describes the defintion of Value at Risk, estimations of Value at Risk for market risk by the variance and covariance method, by historical simu- lation method, by Monte Carlo simulation method and estimations of Value at Risk for credit risk by the most widely known methods CreditMetrics, CreditRisk+ and KMV. This part ends by historical development and cal- culation of capital adequacy. The analytical part of the work analyses main advantages and disadvantages of Value at Risk on the example of portfolio compact of exchange-traded funds. The aim of this work is to describe Value at Risk as a whole, describe its advantages and analyse disadvantages.

Speech on a doctoral seminar Stochastic modelling in economics and finance, MFF UK. The topic is searching for an investment portfolio with a maximal expected investor's utility, exponential utility and relative entropy.

Valuation of Convexity Related Derivatives. IES WP 4/2006. We will investigate valuation of derivatives with payoff defined as a nonlinear though close to linear function of tradable underlying assets. Derivatives involving Libor or swap rates in arrears, i.e. rates paid in a wrong time, are a typical example. It is generally tempting to replace the future unknown interest rates with the forward rates. We will show rigorously that indeed this is not possible in the case of Libor or swap rates in arrears. We will get a precise fully analytical formula applicable to a wide class of convexity related derivatives. We will illustrate the techniques and different results on a case study of a real life controversial exotic swap.

Markowitz model. Term exercise on the subject Optimization II with applications in finance, MFF UK, 2006. Use of Markowitz model (1952) to find optimal equity portfolio in terms of maximizing profit and minimizing risk (i.e. the role of multicriteria optimization). Use of that model on real data of the Czech stock market. Portfolio optimization itself is carried out for five different sets of restrictive conditions.

Maintaining price stability in the Czech Rep. and the euro area. Essay on the subject European Economic Policies, IES FSV UK. This work aims to summarize the relatively short history of inflation targeting in the Czech Republic and describes the mechanism, including the so-called inflation outlook, which is its key attribute. Then it analyzes the role of fiscal policy in the context of targeting inflation. The last chapter is devoted to maintaining price stability in the euro area, which is performed under the supervision of the European Central Bank.

Financial analysis Spolana, a.s. Seminar work on the subject Introduction on financial markets, IES FSV UK. Seminar work deals with financial analysis of company Spolana, a.s. whose shares are traded on the Prague Stock Exchange. In the first part characterizes the company analyzes the internal and external company environment and SWOT analysis is carried out. Financial analysis includes analysis of financial ratios based on the balance sheet and P&L statement then describes methods of valuation. Finally the work contains company valuation and comparison of used valuating methods.

Seigniorage in Continuous Time. Article published in Politická ekonomie magazine, 2004. The government is able to acquire real goods through printing money. Because government does not create wealth through printing money, this revenue, the seigniorage, is at the expense of the public, as the purchasing power of monetary units decreases because of the issue of new money. The authors use the model of auctions to which the public comes with their money and the government with the newly issued money. The value of goods acquired by the government in such an auction equals the newly printed money divided by the sum of the newly printed money and the money spent by the public. Upon this auction model, the authors develop the formula for seigniorage in continuous time. The seigniorage calculated in this way is lower than the seigniorage calculated upon the assumption of discrete changes in economic variables.

Use of Mathematica in the evaluation of financial derivatives. Bachelor thesis, MFF UK. In the theoretical part the work deals with the definition, classification and use of financial derivatives in general terms. For the valuation of European options is described Black-Scholes formula and binomial trees model for American options. The practical part are valuation models using system Mathematica and individual models were compared among themselves, including the problem of convergence of each numerical procedure.

Derivative products in domestic market – legislation, use, options. Essay on the subject Banking, IES FSV UK. This work briefly desribes the principle, practical use and classification of financial derivatives. It also deals with the legislative definition of financial derivatives and their real potential application in the domestic market (in force in 2003).