ELT Simulation Methodology¶
Basic Resources¶
- Simulations of ELTs require two resources:
ELT data uploaded to S3, which specifies the severity distribution / secondary uncertainty parameters
A LossSetELT template, which specifies the frequency and seasonality distributions.
Secondary Uncertainty Sampling Methodology for Beta Distributions¶
This is the loss sampling approach used when the ELT data specifies the beta distribution for severity sampling.
A loss must be generated for each event, utilizing its distributional parameters. For this process, two assumptions are made:
\(L \sim Beta(\alpha,\beta)\)
The quantile presented from the YET corresponds to the 100% correlated quantile to sample from.
Where \(L\) is the loss distribution and the “100% correlated” refers to the correlation between loss sets. Given the above, to sample any loss, a beta distribution must be fitted to each event record. To do so the mean, \(\mu\), and standard deviation, \(\sigma\), of the distribution are computed as follows:
\(\mu = \frac{Mean}{ExpValue}\)
\(\sigma = \frac{StandardDeviationI+StandardDeviationC}{ExpValue}\)
Note that all beta distributions are assumed to be unimodal and parameterization is solvable using the method of moments which enforces that \(\sigma^2 < \mu(1 - \mu)\). Given these assumptions, a bounded \(\sigma\) is computed such that:
\(\sigma_{b} = min(\sigma, (\mu(1 - \mu)^{1/2} = \epsilon))\)
where \(\epsilon\) is the machine error for floating point values.
Given the above, the \(\alpha\) and \(\beta\) parameters for the Beta distribution are computed using a closed form method as follows:
\(\alpha = \mu\left(\frac{\mu(1-\mu)}{\sigma_{b}^2}-1\right)\)
\(\beta = (1 - \mu)\left(\frac{\mu(1-\mu)}{\sigma_{b}^2}-1\right)\)
Once the beta distribution above has been computed, the sampling quantile for the distribution must be obtained. The correlation of severity of an event across all loss sets is not assumed to be 1.0 as the severity for different exposure may be different. Since the standard deviation is split into two components, it is possible to obtain the assumed level of correlation between exposures in loss sets. This correlation coefficient can be computed by:
\(r = \frac{StandardDeviationC}{StandardDeviationI+StandardDeviationC}\)
Using the correlation coefficient, \(r\), a correlated uniform random quantile is produced using Cholesky Decomposition. This ensures that the quantile used for a loss set for a given event maintains a correlation of \(r\) with the quantile present in the YET. This new quantile is computed as follows:
\(\rho = 2 * sin\left(r\frac{\pi}{6}\right)\)
\(\tau = \left(1 - \rho^2\right)^{1/2}\)
\({q}' = F_{N}\left(Q_{N}(q)\rho+Q_{N}(x)\tau\right)\)
where \({q}'\) is the quantile used for sampling a loss, \(F_{N}\) denotes the cumulative density function of a standard normal distribution, \(Q_{N}\) denotes the quantile function of a standard normal distribution and \(x\) is a randomly generated value such that \(X \sim U(0,1)\).
Given the fitted Beta distribution and the correlated quantile, \({q}'\), a loss is sampled via:
\(l = Q_{Beta(\alpha,\beta)}({q}')*ExpValue\)
Where \(Q_{Beta(\alpha,\beta)}\) is the quantile function for the \(Beta(\alpha, \beta)\) distribution.
In cases where the standard deviations sum to zero (i.e. no uncertainty parameters were provided, or both standard deviation values were provided as zero) the Mean of the distribution will be returned.