Contributions published at Finance and Insurance (Cosimo Munari)

Value-at-Risk (VaR) is a widely used statistical measure of financial risk. It provides an estimate of the maximum expected loss at a given confidence level over a specified time horizon. VaR models can be based on different statistical approaches, including Parametric models (Standard Normal, Weighted Standard Normal Increasing and Decreasing) and NonParametric models (Historical, Weighted Historical Increasing and Decreasing). However, there is no consensus on which approach is the best for predicting financial risk, and the choice of model can have a significant impact on the accuracy and robustness of VaR estimates. The goal of this study is to compare the performance of two different VaR models in forecasting the Value at Risk of different indices from the three asset classes of Equities, Commodities, and Fixed Income. The models to be tested are the Standard Normal and Historical, and for both we will analyze the three cases of Equally Weighted, Weighted Increasing and Weighted Decreasing. The primary objective of this study is to identify the model that provides the most accurate and robust predictions of risk for the three different indices. To achieve this objective, we will calculate VaR using the two different models for three different indices from three different asset classes. For each of the two models, we will apply different weights to the sample observations, equal, increasing and decreasing, so that each time different importance will be attributed to more dated or recent data. We will then backtest the VaR estimates using the Traffic Light Approach from the Basel II regulation, which is a supervisory tool used by regulators to assess the operational risk management practices of banks. We will classify the VaR estimates into three categories based on their performance: green, yellow, and red. The green category represents VaR estimates that perform well, the yellow category represents VaR estimates that need improvement, and the red category represents VaR estimates that are not acceptable. We will compare the performance of the VaR models based on the number of green lights achieved during the backtesting process. We will also analyze the results to determine which VaR model is the most robust and accurate for predicting the risk of different indices. Moreover, we will compare the results obtained utilizing 2-years data with the ones obtained utilizing 10-years data in order to add robustness to the findings. The expected findings of this study are that one of the VaR models will perform better than the others in terms of accuracy and robustness. We also expect to observe that for different asset classes the best performing model will vary, showing how one or another model best suits the different characteristics of each. The findings of this study will contribute to the existing literature on VaR modeling and model selection. In conclusion, this study aims to provide insights into the performance of different VaR models and their suitability for predicting financial risk. The findings of this study will be of interest to risk managers, investors, and regulators who use VaR as a tool for measuring and managing financial risk.

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We establish general "collapse to the mean" principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our results also apply beyond the convex setting. We illustrate this by providing a complete account of the "collapse to the mean" for quasiconvex functionals. In the special cases of consistent risk measures and Choquet integrals, we can even dispense with quasiconvexity. In addition, we relate the "collapse to the mean" to the study of solutions of a broad class of optimisation problems with law-invariant objectives that appear in mathematical finance, insurance, and economics. We show that the corresponding quantile formulations studied in the literature are sometimes illegitimate and require further analysis.

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We introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum amount of capital that has to be raised and injected into a financial position X to ensure that Expected Shortfall ESp (X) does not exceed a pre-specified threshold g(p) for every probability level p ∈ [0, 1]. Through the choice of the benchmark risk profile g one can tailor the risk assessment to the specific application of interest. We devote special attention to the study of risk profiles defined by the Expected Shortfall of a benchmark random loss, in which case our risk measures are intimately linked to second-order stochastic dominance.

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We discuss when law-invariant convex functionals "collapse to the mean". More precisely, we show that, in a large class of spaces of random variables and under mild semicontinuity assumptions, the expectation functional is, up to an affine transformation, the only law-invariant convex functional that is linear along the direction of a nonconstant random variable with nonzero expectation. This extends results obtained in the literature in a bounded setting and under additional assumptions on the functionals. We illustrate the implications of our general results for pricing rules and risk measures.

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