# Solvency Analytics News

## In a nutshell

SolvencyAnalytics’ Fast Fourier Transform (FFT) based algorithm reduces the SST target capital’s calculation time by a factor of several thousands. This is relevant for various practical applications.

The target capital of the standard SST model (Delta-Gamma model) is typically calculated using Monte Carlo simulations.
However, sometimes the speed and precision of this approach is unsatisfactory. We see at least three applications
where both speed and high precision are crucial.

1. For portfolio optimisation the target capital has to be calculated many times for the algorithm to
converge towards an optimal solution. Running an MC simulation repeatedly several thousands of times is often not
feasible at a satisfactory level of precision.
2. Portfolio simulation tools require a fast and precise implementation of the SST as users expect
simulation results in a fraction of a second rather than after minutes.
3. When screening a broad asset universe and assessing each e.g. bond’s impact on the overall SST ratio, we need to recalculate
the target capital thousands of times. Considering that the Barclays Global Aggregate Bond Index consists of 10 thousand bonds,
the importance a fast solution becomes even more evident.

### Speed improvement of the SST Target Capital Calculation

Target capital in the standard SST market model is typically calculated by Monte Carlo simulation

However, for Monte Caro simulations there is a trade-off between computational time and precision:
some applications need fast and exact calculation of the target capital

SST target capital calculation by a Fast Fourier Transform based algorithm:

• Exact results (avoiding the statistical error of Monte Carlo simulations)
• Exceptionally fast calculation of target capital
• Increases flexibility for portfolio optimisation
• Details of the algorithm described in the technical paper

## Mathematical Foundation

The change of the risk bearing capital during a ‘normal year’ given by $\Delta RBC = y = \delta x + 1/2 x’ \Gamma x$
is a sum of non-centrally chi-square distributed random variables.

$y$’s density is a multi-dimensional convolution of chi-square densities:
$$f(y) = \int \sum_{k=1}^n \left(\frac{dx_k}{\sqrt{2\pi}}e^{-\frac{1}{2}x_k^2} \right) \delta \left(\frac{1}{2} \sum_{k=1}^n d_k (x_k – \tilde\mu_k)^2+bx_0 – y\right).$$

Computation becomes much faster if we use the Fourier transform of $f$:
$$\hat f(t) = e^{-\frac{1}{2}t^2 b^2}\prod_{k=1}^n \frac{1}{\sqrt{1-it\tilde d_k}}\left(\frac{1}{2}\tilde \mu_k^2 \frac{i t \tilde d_k}{1-it\tilde d_k} \right),$$

since the computationally intensive convolution becomes a fast multiplication when working with Fourier transforms.

Further steps are required to include ‘rare scenarios’ defined under the SST – details are shown in our technical paper of the FFT implementation.