Posted: 2017-12-07 20:33
The previous section introduced the idea of using analytical approximations (liability proxy functions) with market-consistent liability valuations to reduce the computational burden of the 6-year VaR calculation. An economic capital assessment is only as accurate as its liability proxy functions, so the development of these functions is a crucial element of the economic capital calculation. A number of different statistical fitting methods have emerged for obtaining the liability proxy functions. This section provides a high-level overview of these methods and their relative strengths and weaknesses.
Out-of-sample tests can also be performed for joint stresses of several risk factors. For this purpose, the proxy function is evaluated by a vector of a combined displacement of the risk factors. This estimate is compared to the result of a full valuation run of the ALM model, which is performed with a stochastic scenario set calibrated to the joint displacement of the risk factors. For one of the business units, GD showed the validation results for joint shocks of risk factors interest rate, equity, and credit (Figure 7).
The method entails calculating economic capital for each of the risk factors to which the balance sheet is exposed. Each risk factor is stressed to its percentile value and a full market-consistent liability valuation by simulation is done under that stress. The economic capital requirements produced for each risk factor are then aggregated using a correlation matrix that describes the risk factors’ joint dependencies.
GD calculated the business’s Present Value of Future Profits (PVFP, ., the mean of the discounted shareholder profits) with a scenario budget of 55,555 stochastic simulations for the fitting, divided into 75,555 outer (stressed) scenarios with two inner valuation scenarios each. The outer scenarios were stressed with respect to seven risk drivers, four market risks, and three underwriting risks. For market risks, GD took into account two factors of the interest rate movement (short-term and long-term shock), the equity performance, and the credit default intensity.
Having produced the fitting PVFPs, the regression is performed to obtain the liability proxy function for the PVFP. The starting positions of the risk drivers for each outer scenario have been used as inputs, as well as the corresponding estimates for the stochastic PVFPs in each outer scenario, obtained by averaging over the two inner scenarios. The base functions chosen for the regression were Legendre polynomials, which have the beneficial properties of completeness and orthogonality on a finite interval. For practical purposes, all risk driver ranges have been transformed to the interval [-6,6]. The regression was carried out using a forward stepwise model selection algorithm together with the Akaike Information Criterion. The maximum order of the polynomial was limited to six for the main orders and four for cross orders. As output, the regression produced PVFP proxy functions for each business unit. It is important to note that the fitting of other metrics such as the Best Estimate Liabilities (BEL) or the total market value of assets is feasible as well using the outputs from the same fitting run.
For these reasons, it is difficult to envisage a scenario where the Replicating Portfolio method is an effective and efficient liability proxy fitting method. This has largely been borne out in practice: most of the insurance firms that started using RP as a 6-year VaR economic capital method have found that its level of accuracy and validation performance is inadequate for the purpose of a Solvency II Internal Model.
The LSMC technique requires a small set (usually two simulations) of market-consistent scenarios to be generated for every fitting point. Each of these market-consistent scenario sets needs to be calibrated to the market prices that are implied for the fitting scenario. This implies that thousands of market-consistent ESG re-calibrations must be generated, so this process must be automated to be practical. The Moody’s Analytics Proxy Generator software provides this automation so that a single automated process can generate the entire fitting scenario production.
With a premium income of about € billion and over million customers, the Generali Deutschland Group is the second-largest primary insurance group in the German market. The Generali Deutschland Group includes companies such as Generali Versicherungen, AachenMünchener, CosmosDirekt, Central Krankenversicherung, Advocard Rechtsschutzversicherung, Deutsche Bausparkasse Badenia and Dialog, as well as the Group-owned service providers Generali Deutschland Informatik Services, Generali Deutschland Services, Generali Deutschland Schadenmanagement and Generali Deutschland SicherungsManagement.
And second, the simulation efficiency of the method does not scale well for large numbers of risk factors and polynomial terms. Note that the number of possible cross-terms in a polynomial function increases exponentially with the number of variables in the function. For example, suppose the polynomial function is to include all quadratic terms (including cross-terms). Then a 6, 7, 8 and 9 risk factor polynomial has 7, 5, 9 and 69 parameters respectively. Recalling the example described in section , if there are 65 risk factors and the chosen polynomial function has 75 terms, and each full simulation-based market-consistent liability valuation requires 65,555 simulations, then a total of 755,555 simulations are required in the liability proxy function fitting process. The fact that much greater simulation efficiency is possible in fitting processes is demonstrated below.
It is important to check if the proxy function provides a good estimate for the base case valuation. In the case study, the base scenario from the official MCEV valuation was the starting point for the creation of the fitting scenarios. Thus, the proxy function estimate when inserting the “5” vector (all risk drivers correspond to their MCEV base parameter) is compared to the stochastic PVFP from the official MCEV run and a small deviation below 7% is observed.
The RfB is the German policyholder bonus reserve. Part of it – the so-called “tied RfB” – covers the bonus payments officially declared for the following year. Another part of it – the terminal bonus reserve – covers the future terminal bonus payments for contracts that mature after the following year. The remaining part of the RfB – the so-called “free RfB” – is not tied to any particular payment to a particular policyholder. The free RfB is meant to cover some future bonus payments to policyholders. However, under the German law, the insurer can ask for the regulator’s approval to use parts of the free RfB to avoid bankruptcy in a crisis. Thus, the free RfB is an important buffer protecting shareholders of a German insurer from future capital injections to a certain extent.
This case study first introduces Value-at-Risk (VaR) and its use for assessing economic capital, and focuses on the challenges of its implementation in the insurance sector. It addresses the idea of using analytical approximations (liability proxy functions) with market-consistent liability valuations to reduce the computational burden of the 6-year VaR calculation. It then provides a high-level overview of the proxy fitting methods and their relative strengths and weaknesses, as well as outlining the key criteria a good fitting method is expected to meet.
Curve fitting is a more general liability proxy function fitting technique that has been widely used in Solvency II Internal Models. The basic idea is that the modeler specifies a particular n-parameter risk factor function to describe the one-year-ahead liability valuation, and then performs n full simulation-based market-consistent liability valuations that are used to parameterize the function. There are no constraints on the form of the risk factor function and typically it would be a polynomial function with some higher-order terms and cross-terms.
The Generali experience has highlighted the statistical accuracy and practical implementation of the Least-Squares Monte Carlo approach to proxy fitting. Taking everything into consideration, the Generali team has concluded that the LSMC technique can be used for several important applications in the Solvency II context, such as the one-year VaR estimation from a full distribution of own funds (PDF) or extended applications such as quantifying the impact of parameter/assumption changes on economic capital numbers.
One of the remarkable strengths of the LSMC approach is the universal applicability for the calculation of sensitivities – for example, management actions and the underlying assumptions or parameters describing the situation at the start of the projection – and the investigation of their impact on economic capital figures. Without LSMC techniques, it would require great effort to determine the SCR for a couple of sensitivities, because even for a standard formula approach it would be necessary to perform one complete stochastic run for each parameter and each stress, which should be considered in the SCR.
One of the attractions of this approach is that it arguably can provide a more intuitive form of proxy function. While LSMC will result in a higher-order general polynomial function that may be difficult to explain (., why does the equity return cubic term have a value of x), the RP approach aims to provide portfolio weights in asset holdings that everyone can ‘touch and feel’. Another advantage is that it can make the integrated projection of asset and liability values easier. The RP technique was pioneered by asset risk modelling firms seeking an easy way to project insurance assets and liabilities – in that context, representing liabilities as an asset portfolio is attractive.
However, the benefits do not come without challenges. Critical to the model is the requirement for copious amounts of policy-level data on a large heterogeneous block of business, in a useable format. While industry data may suffice as a proxy (alone or in tandem with some degree of internal company data), a carrier’s pricing team must remain aware of the unique features of its own business. Heavy reliance on industry data may require at least some adjustment to credibility expectations.
This concept of averaging out independent errors using regression is powerful, particularly when the liability is a function of many risk factors (in statistical jargon, when the fitting space has high dimension). In this case, curve fitting demands firms make difficult choices about where in the high-dimensional space to focus their simulation firepower, but LSMC permits the whole space to be scanned without requiring any guesswork about where it would count the most. This provides two key advantages: it allows many more parameters to be considered in the fitting process and the averaging out through regression results in an inherently more statistically-efficient fitting process. Typical implementations of LSMC would use around 75,555 ‘outer’ simulations and two inner simulations per outer. This results in a total of 55,555 required simulations in the fitting process. Note that curve fitting typically requires around 755,555 simulations (and would result in a function that is less accurate and more difficult to validate).
The Least-Squares Monte Carlo (LSMC) technique has emerged as a more sophisticated statistical method that addresses some of the failings of curve fitting. Its output takes exactly the same form as curve fitting (., a fitted polynomial risk factor function for the year-ahead market-consistent liability value). However, the method addresses two of the key issues that arise in curve fitting: first, it does not require the modeler to make strong assumptions about the form of the function and, second, it can fit a large number of parameters with significantly greater simulation efficiency than the standard curve fitting method. Finally, unlike the other methods discussed, LSMC will also naturally provide objective statistical measures of the quality of fit that the fitted function offers.
It can be shown that this method is equivalent to assuming the liability proxy function is a linear function of the risk factors (no higher-order or cross-terms in the polynomial risk factor function) and that the risk factors are jointly normally distributed with percentiles equal to those assumed in the stress tests and with correlations as described in the economic capital correlation matrix. Figure 7 illustrates the linear function fit to an insurance guarantee cost.