Proxy techniques have grown in popularity in recent years, with insurers increasingly using them to model the full distribution of their balance sheet assets and liabilities to support capital calculations. So why use them?
Proxy techniques have grown in popularity in recent years, with insurers increasingly using them to model the full distribution
of their balance sheet assets and liabilities to support capital calculations. So why use them?
The advent of these techniques has been partly due to changes in the regulatory landscape (for example, Solvency II in Europe) but also a greater desire by senior management to better understand their risk exposure and feed this into their decision-making processes.
Asset liability management (ALM) systems are designed to calculate accurate balance sheet values and as such are granular and complex. As a result, they were not originally designed for real-world stochastic modelling, particularly for complex liabilities with options and guarantees which require a nested stochastic approach. The large computational requirement and long run times that would be required to support such calculations mean that many insurers have turned to proxy techniques, which aim to emulate the results from the ALM system without the associated run times.
Proxy techniques have developed rapidly in recent years. Initially firms developed replicating portfolios to model their liabilities. However, for more complex liabilities it can prove difficult to find assets with the same risk characteristics and typically they don't deal with non-market risks, limiting the usefulness of the technique.
As a result, curve fitting and, more recently, the Least Squares Monte Carlo (LSMC) technique, originally developed by Longstaff and Schwartz as a method for valuing American options, have gained in popularity. These techniques use more general mathematical functions rather than a replicating portfolio-to-model liabilities (or assets). The process for generating
these functions is to:
? Identify risk factors to be modelled for the business
? Generate real-world 'outer' fitting points
? Value the liability (or asset) at the fitting points using the ALM system
? Use linear regression techniques to fit a function to the results for the fitting points.
These proxy techniques have the advantage that they provide significant runtime reductions over the ALM systems, in particular where a nested stochastic approach would be required. By their very nature, care needs to be taken with proxy techniques, and we now look at some of the practical considerations.
Which risks are important?
A key part of the curve fitting or LSMC approach is deciding which risk factors (market and non-market) to include in the proxy function. It is easy to overlook this stage of the process but the choice of risk factors can have a significant impact on the effectiveness of the fit and there can be considerable expert judgment required.
The asset or liability value being modelled will be a function of a large number of variables and including them all in the proxy function is not usually practical, so an exercise to reduce the dimensionality of these risks will usually be required (for example, using principal components to represent the term structure of a nominal yield curve). A pitfall to be avoided is the inclusion of highly correlated risk factors that can cause problems with the fit.
Choosing the fitting points
A lot of focus can be on the regression process for fitting the proxy function but generation of fitting points is a critical part of the fitting process and can be split into two areas:
? Generating real-world 'outer' fitting points
? Generating associated market consistent 'inner' scenarios.
The real-world 'outer' fitting points can either be user defined (for example user defines 100 points for curve fitting) or automated using algorithms to generate fitting points within a multi-dimensional space for a given set of parameters.
Liabilities with options or guarantees are typically valued using Monte Carlo simulation. Therefore, for each real-world 'outer' fitting point there is also a requirement to generate an associated set of market consistent 'inner' scenarios, and herein lies the key difference between curve fitting and LSMC approaches. Curve fitting uses a small set of 'accurate' fitting points whereas LSMC uses a large number of 'inaccurate' fitting points. For example, let us consider a firm with a fitting scenario budget of 200,000 scenarios:
? For curve fitting the firm might typically choose 100 'outer' fitting points 'accurately' valued using 2,000 inner scenarios (Figure 1).
? For LSMC the firm could choose 100,000 'outer' fitting points but they are 'inaccurately' valued using only two corresponding inner scenarios (Figure 2).
LSMC requires less expert judgment than curve fitting for generating fitting points and can often generate better fits for a given scenario budget due to the larger number of fitting points spanning the multi-dimensional space. One practical issue with LSMC relates to modelling non-market risks in the ALM models. While the use of economic scenario generators (ESGs) has meant that it is common for the value of economic inputs to vary between scenarios, insurance risks have usually been constant across all scenarios included in a model run. One solution is a hybrid approach of LSMC and curve fitting, whereby the market risk factors span the multi-dimensional space (for example, LSMC) but are divided into a small number of tranches, with each tranche taking a different value for each of the non-market risk factors (for example, curve fitting).
A key requirement for LSMC is being able to recalibrate the market consistent ESG for each real world fitting point. Ideally, the market consistent 'inner' scenarios should be recalibrated, by either stressing the ESG parameters or recalibrating to stressed economic variables.
Fitting the proxy function
The most common approach to function fitting is least squares regression with polynomial basis functions. The process can be split into two key stages:
? Identifying the functional form
? Solving the coefficients for the terms of the functional form.
In the early days of curve fitting and LSMC, the identification of the functional form was a very manual and iterative process as users tried to identify which terms (combinations of risk factors) in the polynomial were important to the fit. This is an extremely time-consuming process and does not guarantee an optimal fit.
These days there are specific software solutions available with algorithms that can automate the process of identifying the optimal functional form and solving for the coefficients using matrix mathematics. This does not mean there is no need for manual intervention as, even with the use of software, the fitting process can still be an iterative process, particularly the first time.
Practically, there may be some constraints on the goodness of fit. For example, if the scenario budget for generating fitting points is relatively small then it may be difficult to achieve a good fit. Also, if a large number of risk factors are being used for an individual proxy function, the regression and optimisation problem can become computationally very complex.
Are they fit for purpose?
Proxy functions can provide very good fits for assets and liabilities, but firms need to be able to validate that they are fit for purpose. The simplest way to assess the accuracy of the proxy function is to generate a set of independent validation scenarios and check the proxy function values against 'accurate' ALM values (Figure 3). The results can be used to assess accuracy using analysis of the min/max errors and error distribution. In addition, it can be useful to include confidence intervals in this analysis to provide insight into the sampling error associated with the fitting points.
Other outputs and metrics can be used to assess the fit and robustness of the regression process. The R-squared metric gives insight into the relative fit of different proxy functions for the same data. For LSMC it can be useful to analyse the residuals associated with the fitting points. Analysing the distribution of the residuals can highlight whether there may be issues such as bias or heteroscedasticity.
