Even earlier, Leavens ( 1945) published a paper describing the benefits of diversification. Value-at-risk metrics first emerged in finance during the 1980s, but they were preceded by various other PMMRs, including Markowitz’s ( 1952) variance of simple return. They will also illustrate a framework for thinking about value-at-risk measures (and, more generally, measures of PMMRs), which we shall formalize in Section 1.8. These will introduce basic concepts and standard notation. Let’s consider some examples of risk measures. 14.6 Example: Backtesting a One-Day 95% EUR Value-at-Risk Measure.14.5 Backtesting With Independence Tests.14.4 Backtesting With Distribution Tests.11.6 Shortcomings of Historical Simulation.11.5 Flawed Arguments for Historical Simulation.11.3 Calculating Value-at-Risk With Historical Simulation.11.2 Generating Realizations Directly From Historical Market Data.10.4 Monte Carlo Transformation Procedures.10.3 Quadratic Transformation Procedures.7.4 Unconditional Leptokurtosis and Conditional Heteroskedasticity.5.7 Breaking the Curse of Dimensionality.5.6 Implementing Pseudorandom Number Generators.5.5 Testing Pseudorandom Number Generators.4.8 White Noise, Moving-Average and Autoregressive Processes.3.17 Quantiles of Quadratic Polynomials of Joint-Normal Random Vectors.3.13 Quadratic Polynomials of Joint-Normal Random Vectors.3.8 Bernoulli and Binomial Distributions.3.5 Linear Polynomials of Random Vectors.2.3 Gradient & Gradient-Hessian Approx.1.6 Other Applications of Value-at-Risk.If I have any questions regarding my rights or privacy, I can contact the email address provided. If we are unable to resolve any issue, you will provide an independent service to arbitrate a resolution. I also understand that if there is a dispute regarding my personal data, I can contact someone who is responsible for handling data-related concerns. Likewise, if I properly request to be forgotten, you will delete the data you have for me, or make it inaccessible. I understand that if I revoke my consent, you may be unable to provide contracted products or services to me, and I can not hold you responsible for that. If I revoke my consent you will stop collecting or processing my personal data. I also understand that I can revoke my consent and that I have the right to be forgotten. You have agreed to provide me with a record of my personal data in a readable format. I understand that I have the right to request access annually to any personal data you have obtained or collected regarding me. I understand that in some cases it may be required to use cookies or similar tracking to provide those products or services. I understand that certain data, including personal data, must be collected or processed in order for you to provide any products or services I have requested or contracted for. I have been fully informed and consent to the collection and use of my personal data for any purpose in connection with the software, products and/or services. I have read and accept any EULA, Terms and Conditions, Acceptable Use Policy, and/or Data Processing Addendum which has been provided to me in connection with the software, products and/or services. I confirm that I am at least 16 years of age or older
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