The investment horizon is H = 5 days, and that is also the block length for the bootstrapped data. In addition, we assume a linear correlation between asset returns of ρ = 0.3. In particular, we assume three DGPs: following standard reasoning, we model prices to be lognormally distributed with P t = P t − 1 e r t where r t ∼ N ( μ = 0.05 / 250 σ 2 = 0.25 2 / 250 ). We start with generating the sample of observed prices for our purposes, we are not taking a historical time series as is, but simulating data to ensure that we can have a validation sample that comes from the same data generating process (DGP). Here we rule out these situations and focus on the remaining two: we make assumptions about the distribution that might or might not be suitable, but validation data follow the same rules as the observed data on which the estimations and optimizations are based. Perfect foresight is highly desirable but unachievable in the real world working with inappropriate data is anything but desirable but sometimes hard to avoid and recognizable only with hindsight. Finally, we might miss that there is some structural break and that the data we fit our model on is not representative of the actual investment period. 15 Thirdly, if parametric assumptions are abandoned and order statistics are used to estimate the empirical VaR, then important aspects in the overall distribution might be ignored. In addition, the assumed distribution might not be appropriate for the investigated asset, which adds another source of errors that is typically more systematic. In the absence of perfect foresight, these parameters have to be estimated, which introduces a first source of errors. Enrico Schumann, in Numerical Methods and Optimization in Finance (Second Edition), 2019 14.4.2 Setting up experimentsĮstimating VaR is less of a problem if the distribution of the asset returns or the value itself follows some parametric distribution with known parameters. Consider the usual assumption of a normal distribution: 6.4 standard deviations below the mean represent a “one day since the big bang” quantile the FTSE knows five such days in the 25-year horizon of October 1985 to September 2010 alone. At the same time, parametric distributions often seem to work reasonably well for the mass of the distribution, but not so well for the tails. This risk measure focuses on rare events and, by definition, there are only few historical observations to calibrate models on. Another reason is that it is sanctioned by regulators in the Basel II and Basel III accords.Īlthough conceptually rather simple, estimating the VaR is often challenging. One main reason for its popularity is that it is very intuitive, and its numerical values are easier to interpret than other risk measures such as variance or the omega. VaR has gained considerable importance as a main risk measure. Value-at-Risk (VaR) is the maximum loss that one will not exceed with a certain probability α within a given time horizon. Enrico Schumann, in Numerical Methods and Optimization in Finance (Second Edition), 2019 9.2.1 Basic concepts
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