Quest for the VaR: Historical bootstrap method
This particular thesis will address a practical problem which econometric theory is not very fond of: a small sample size.
Our goal is to obtain a distribution and Value-at-Risk (Jorion, 2006) of the behaviour of our future portfolio: in the medium- or long-run. This is not a straightforward task because of a few problems with our data. The first problem has to do with the fact that it is not possible to use very old data; economic shocks from 1934 cannot be used as the economy at that time is very different from the economy nowadays. Therefore it is necessary to find a method that can provide us with enough plausible yearly or quarterly scenarios to find a distribution of the performance of our portfolio. The second problem is that our current sample does not consist of enough scenarios to be representative. Therefore, we introduced a new tool to estimate both the distribution and Value-at-Risk of longer time periods in the future based on the bootstrap method (Efron, 1979) and Historical Simulation (Pritsker, 2006). We refer to this method as the Historical Bootstrap Simulation. The general idea of the new method is to use data from at least a period of one month to obtain medium/long-term shock scenarios. We have to weigh the consequences of choosing monthly data over weekly or daily data. Choosing smaller time intervals gives us a larger data set, however, modeling data of smaller time intervals is more complicated. There will be more temporary shocks and more dependence, whilst monthly data is expected to be on the verge of having no serial correlation, but will still provide us with a representable sample size. Our method is interested in an unconventional statistic and a derived value (the VaR), therefore it pushes us away from the theoretical research that statisticians and econometricians are interested in. The difference of the research done in this thesis and research done in most of the bootstrapping literature is the addition of an ’in between’ step. We want to construct the scenarios from our data – constructing yearly or quarterly data from monthly data of risk drivers – before mapping the risk drivers onto the portfolio performance in R. Our statistic will therefore not be derived from resampling actual historical scenarios, but from generated scenarios.