ALGEBRA MATRICIALE PDF

Calculate the trace or the sum of terms on the main diagonal of a matrix. Invert a square invertible matrix or find the pseudoinverse of a non-square matrix. Compute the characteristic polynomial of a matrix:. Perform various operations, such as conjugate transposition, on matrices.

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Fornire agli studenti conoscenze di base nelle tecniche per la costruzione di modelli probabilistici e nelle tecniche di analisi statistica inferenziale comunemente utilizzate in ambito finanziario per descrivere ed analizzare processi di valutazione, prendere decisioni tra alternative di investimento e controllare il rischio di mercato.

Illustrare tali tecniche con esempi tratti dalla pratica finanziaria. Esame esclusivamente in forma scritta, a libri chiusi, uguale per frequentanti e non frequentanti. Tutte le prove passate sono disponibili in rete con relative soluzioni. Provide the students with basic techniques for probabilistic modelling and statistical inference commonly applied in the field of finance in order to describe and analyze valuation processes, choose between investments and control market risk.

The techniques presented in the course are illustrated with examples drawn from actual financial practice. Written, closed books, exam, Identical for both participants and non participants in the course. No partial exam.

All past exams are available in the Internet together with solutions. Courses offered in Academic Programs a. Course a. Classi: 15 I sem. Materiali didattici classi 15 Programmi d'aula classi Introduzione ai problemi statistici in finanza. Dati e loro trasformazioni. Modelli probabilistici per la distribuzione di rendimenti: Gaussiana o non Gaussiana? Dipendenza o indipendenza? Nozioni di dipendenza. Indici di dipendenza. Modelli fattoriali in finanza.

Il modello lineare. Inferenza nel modello lineare. Il problema della previsione. Lettura dei risultati di un modello lineare. La Style Analysis e la valutazione della performance di un fondo gestito. Metodi per la stima della matrice delle varianze e covarianze. Il metodo delle componenti principali. Applicazioni al risk management.

Metodi bayesiani e selezione del portafogli. Il modello di Black e Litterman. Ruppert, Statistics and Finance , Springer Prove d'esame passate e relative soluzioni. Una scelta di articoli a cura del docente e disponibili in rete tra i quali: Fisher, Statman, A Behavioral framework for time diversification, Litterman Winkelmann , Estimating covariance matrices, Sharpe , Asset allocation: management style and performance measurement, Variabile aleatoria, funzione di distribuzione, modelli di funzione di distribuzione Bernoulli, Binomiale, Geometrica, Poisson, Gaussiana, Esponenziale negativa, Chi quadro, T di Student.

Momenti, quantili ed altri valori di sintesi. Vettore aleatorio a n dimensioni, distribuzioni congiunte, marginali e condizionate, momenti misti. Stima puntuale, e stima per intervalli. Concetti di non distorsione, efficienza e consistenza. Test di ipotesi statistiche. Algebra delle matrici: concetto di matrice e vettore, operazioni fondamentali somme, prodotti, trasposta, ecc.

Forme quadratiche e loro classificazione. Autovalori e autovettori di una matrice semi definita positiva. Classes: 16 I sem. Click here to see the ILOs of the course Knowledge and understanding At the end of the course student will be able to do: Know the statistical and econometrics tools of quantitative finance and risk management topics, in particular techniques for stochastic modeling and inferential statistical analysis, as applied in the field of finance for valuation and management of single positions and of portfolios Applying knowledge and understanding At the end of the course student will be able to do: Make use of acquired knowledge related to statistics and econometrics in quantitative finance and risk management topics, in order to apply techniques of probabilistic modelling and inferential analysis to investment and risk management decisions.

An introduction to statistical problems in finance. Data and data transforms. Returns, different definitions and aggregation properties w. Probability models for return distributions: gaussian or non gaussian? Dependence or independence? Univariate problems: risk premium and its estimation, volatility estimation.

VaR estimation, confidence intervals for the VaR. Multivariate problems. Matrix algebra and Statistics. Concepts of dependence. Measures of dependence. Factor models in finance. The linear model. Inference for the linear model. The least squares method and its properties under several hypothetical settings.

How to read the results of a linear model. Style analysis and its use for fund management performance evaluation.

Estimations methods for the covariance matrix. Principal components method. Its applications to risk management. The Markowitz model, its main properties and its limits in applications. Bayesian methods and portfolio selection. The Black and Litterman model. Note: while dealing with points 1 to 4, the basic concepts of probability and statistics required as a prerequisite of the course shall be recalled and re-examine.

Click here to see the assessment methods Written exam s with open ended questions. Detailed Description of Assessment Methods. Handouts available on e-learning Excel and Matlab examples. Past exams: questions and answers. A selection of papers available on e-learning: Fisher, Statman , A Behavioral framework for time diversification , Litterman Winkelmann , Estimating covariance matrices , See the detailed program of the course. Probability: definition of event, algebra of events, definition of probability, conditional probability.

Basic results: probability of a non disjoint union, decomposition of the probability of an event into conditional and marginal probability, Bayes theorem. Function of a random variable. Moments, quantiles and other summaries of the properties of a distribution.

Two dimensional random vector, conditional distributions, conditional expectation and conditional variance: definitions and properties. N dimensional random Vector: joint, marginal and conditional distributions, mixed moments. Random sequence, convergence in Law and in square mean. Statistical inference: Sample and sample functions, sampling variability.

Point estimate, interval estimate. Unbiased, efficient and consistent estimates. Method of moments and maximum likelihood method. Testing statistical hypothesis.

Multiple regressor linear model. Matrix algebra: Concept of matrix and vector, basic operations matrix sums, products, transpose etc.

Quadratic forms and their classification. Eigenvalues and eigenvectors of a semi positive definite matrix and the spectral theorem. In this section Courses offered in Academic Programs a. Se continui a navigare sul sito, accetti espressamente il loro utilizzo.

APART PC1000R PDF

Matrix norm

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ESCUPIRE SOBRE VUESTRA TUMBA PDF

20191 - FINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE - MODULE 1

In mathematics , a matrix norm is a vector norm in a vector space whose elements vectors are matrices of given dimensions. A matrix norm that satisfies this additional property is called a submultiplicative norm [1] in some books, the terminology matrix norm is used only for those norms which are submultiplicative [2]. For symmetric or hermitian A , we have equality in 1 for the 2-norm, since in this case the 2-norm is precisely the spectral radius of A. For an arbitrary matrix, we may not have equality for any norm; a counterexample being given by. In any case, for square matrices we have the spectral radius formula :. This inequality can be derived from the fact that the trace of a matrix is equal to the sum of its eigenvalues. It can be shown to be equivalent to the above definitions using the Cauchy—Schwarz inequality.

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