# LO 2.2: Describe historical simulation using non-parametric density estimation.

LO 2.2: Describe historical simulation using non-parametric density estimation.
The clear advantage of the traditional historical simulation approach is its simplicity. One obvious drawback, however, is that the discreteness of the data does not allow for estimation of VaRs between data points. If there were 100 historical observations, then it is straightforward to estimate VaR at the 95% or the 96% confidence levels, and so on. However, this method is unable to incorporate a confidence level of 95.5%, for example. More generally, with n observations, the historical simulation method only allows for n different confidence levels.
One of the advantages of non-parametric density estimation is that the underlying distribution is free from restrictive assumptions. Therefore, the existing data points can be used to smooth the data points to allow for VaR calculation at all confidence levels. The simplest adjustment is to connect the midpoints between successive histogram bars in the original data sets distribution. See Figure 1 for an illustration of this surrogate density function. Notice that by connecting the midpoints, the lower bar receives area from the upper bar, which loses an equal amount of area. In total, no area is lost, only displaced, so we still have a probability distribution function, just with a modified shape. The shaded area in Figure 1 represents a possible confidence interval, which can be utilized regardless of the size of the data set. The major improvement of this non-parametric approach over the traditional historical simulation approach is that VaR can now be calculated for a continuum of points in the data set.
Figure 1: Surrogate Density Function
Distribution
Tail
Following this logic, one can see that the linear adjustment is a simple solution to the interval problem. A more complicated adjustment would involve connecting curves, rather than lines, between successive bars to better capture the characteristics of the data.
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Topic 2 Cross Reference to GARP Assigned Reading – Dowd, Chapter 4
W e i g h t e d H i s t o r i c a l S i m u l a t i o n A p p r o a c h e s