Portfolio optimization is a fundamental topic in financial mathematics, allowing to obtain portfolios that represent a good compromise between return and risk. Risk can be estimated through several risk measures, among which variance, semivariance, mean absolute deviation, semi-mean absolute deviation and Conditional Value-at-Risk. In this thesis, for each of the previously mentioned risk measures, we face two classical problems in the field of portfolio optimization: constructing the efficient frontier and determining the optimal risky portfolio through Sharpe ratio maximization. Variance and semivariance lead to quadratic programming problems, whereas mean absolute deviation, semi-mean absolute deviation and Conditional Value-at-Risk lead to linear programming problems. Computational experiments are carried out using historical data from companies belonging to the FTSE MIB, the benchmark stock market index for the Borsa Italiana. For each of the considered risk measures, we discuss the composition of efficient portfolios, focusing on the minimum risk portfolio and the optimal risky portfolio, and stressing how the investment should be diversified among stocks of companies classified in different industries. Moreover, a sliding window approach is employed to evaluate the evolution of an initial wealth on a test period of one year. Both with five and three year window length, the strategies involving the optimal risky portfolios obtained with each of the considered risk measures lead to an upward trend in the wealth evolution. In particular, with a five year window length, these strategies outperform the uniform one, that tracks the FTSE MIB, in the periods in which the value of the FTSE MIB drops.
L'ottimizzazione di portafoglio è un aspetto fondamentale della matematica finanziaria, che consente di ottenere portafogli che rappresentino un buon compromesso tra rendimento e rischio. Quest'ultimo può essere stimato tramite diverse misure di rischio, tra cui varianza, semivarianza, mean absolute deviation, semi-mean absolute deviation e Conditional Value-at-Risk. In questa tesi affrontiamo due problemi tipici nell'ambito dell'ottimizzazione di portafoglio: costruire la frontiera efficiente e determinare l'optimal risky portfolio attraverso la massimizzazione dell'indice di Sharpe. Per entrambi, presentiamo varie formulazioni corrispondenti alle diverse misure di rischio precedentemente citate: l'uso di varianza e semivarianza produce problemi di programmazione quadratica, mentre l'impiego di mean absolute deviation, semi-mean absolute deviation e Conditional Value-at-Risk determina problemi di programmazione lineare. I modelli proposti sono applicati a dati storici di alcune delle società che compongono il FTSE MIB, l'indice azionario di riferimento di Borsa Italiana. Per ognuna delle misure di rischio considerate, analizziamo la composizione dei portafogli efficienti, in particolare del portafoglio a rischio minimo e dell'optimal risky portfolio; si evidenzia come l'investimento debba essere diversificato tra azioni di società appartenenti a vari settori. Inoltre, utilizziamo un approccio sliding window (con una finestra di cinque o tre anni) per valutare l'evoluzione di una ricchezza iniziale su un periodo di un anno. Le strategie basate sugli optimal risky portfolio ottenuti con ciascuna delle misure di rischio considerate determinano una crescita della ricchezza iniziale. In particolare, con una finestra di cinque anni, queste strategie hanno prestazioni migliori rispetto a quella ottenuta diversificando l'investimento in maniera omogenea, nei periodi in cui il valore del FTSE MIB scende.
Portfolio optimization
Brambilla, Letizia
2021/2022
Abstract
Portfolio optimization is a fundamental topic in financial mathematics, allowing to obtain portfolios that represent a good compromise between return and risk. Risk can be estimated through several risk measures, among which variance, semivariance, mean absolute deviation, semi-mean absolute deviation and Conditional Value-at-Risk. In this thesis, for each of the previously mentioned risk measures, we face two classical problems in the field of portfolio optimization: constructing the efficient frontier and determining the optimal risky portfolio through Sharpe ratio maximization. Variance and semivariance lead to quadratic programming problems, whereas mean absolute deviation, semi-mean absolute deviation and Conditional Value-at-Risk lead to linear programming problems. Computational experiments are carried out using historical data from companies belonging to the FTSE MIB, the benchmark stock market index for the Borsa Italiana. For each of the considered risk measures, we discuss the composition of efficient portfolios, focusing on the minimum risk portfolio and the optimal risky portfolio, and stressing how the investment should be diversified among stocks of companies classified in different industries. Moreover, a sliding window approach is employed to evaluate the evolution of an initial wealth on a test period of one year. Both with five and three year window length, the strategies involving the optimal risky portfolios obtained with each of the considered risk measures lead to an upward trend in the wealth evolution. In particular, with a five year window length, these strategies outperform the uniform one, that tracks the FTSE MIB, in the periods in which the value of the FTSE MIB drops.File | Dimensione | Formato | |
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https://hdl.handle.net/10589/188917