Solvency Capital Research Articles

Longevity Risk Measurement of Life Annuity Products

This paper captures and measures the longevity risk generated by an annuity product. The longevity risk is materialized by the uncertain level of the future liability compared to the initially foretasted or expected value. Herein we compute the solvency capital (SC) of an insurer selling such a product within a single risk setting for three different life annuity products. Within the Solvency II framework, we capture the mortality of policyholders by the mean of the Hull–White model. Using the numerical analysis, we identify the product that requires the most SC from an insurer and the most profitable product for a shareholder. For policyholders we identify the cheapest product by computing the premiums and the most profitable product by computing the benefit levels. We further study how sensitive the SC is with respect to some significant parameters.

Capital allocation and RORAC optimization under solvency 2 standard formula

Solvency II Directive 2009/138/EC requires an insurance and reinsurance undertakings assessment of a Solvency Capital Requirement by means of the so-called “Standard Formula” or by means of partial or full internal models. Focusing on the first approach, the bottom-up aggregation formula proposed by the regulator allows for a capital reduction due to the diversification effect, according to the typical subadditivity property of risk measures. However, once the overall capital has been assessed no specific allocation formula is provided or required in order to evaluate the contribution of each risk source on the overall Solvency Capital Requirement. The aim of this paper is twofold. First, we provide a closed formula for capital allocation fully compliant with the Solvency II Capital Requirement assessed by means of the Standard Formula. The solution enables a top-down approach to assess the allocated Solvency Capital Requirement among the risks considered in the Solvency II multilevel aggregation scheme; we demonstrate that the allocation formula adopted is consistent with the Euler allocation principle. Second, a solution is found as a result of an optimum capital allocation problem based on a Return On Risk Adjusted Capital measure; we establish the equivalence between the Return On Risk Adjusted Capital optimization, when the risk adjusted capital is calculated according to the Standard Formula, and the Markowitz mean-variance optimization.

Time Consistent Solvency Capital Requirement Under Solvency II.

In this paper, we analyse the current dynamic definitions of the Solvency Capital Requirement (SCR) and we propose a new dynamic and time consistent formulation of the SCR, that is compliant with the Solvency II directive. In case of a single liability cash-flow at maturity, Devolder and Lebegue (2017) analyse the time-consistent dynamic formulation of the SCR using iterated risk measures. Firstly, we extend the iterated-SCR (ISCR) formulation to the case of multiple liability cash-flows. However, the iterated formulation of the SCR presents some drawbacks. Devolder and Lebegue (2017) show that using iterated risk measures, the ISCR becomes quite expensive for long term liabilities. Moreover, we show that in case of multiple liability cash-flows, the ISCR is not completely compliant with the Solvency II directive. In fact, ISCR does not answer to a fundamental regulator request: the capital to be held by insurance company to meet obligations over the following year. Then, starting from the static definition in Christiansen and Niemeyer (2014), we propose a dynamic version of the SCR, that is time consistent and encompasses the drawbacks of the iterated formulation. Our proposed dynamic definition of the SCR, called additive-SCR (ASCR), indicates the expected total capital requirement for the period from time zero to the liabilities maturity. Moreover, the ASCR can be decomposed in the annual expected-SCRs (ESCR), which represent the expected regulatory capital for the next year, not only at time zero but also at the futures payment dates. Hence, we build a term structure of expected annual SCRs.

El reaseguro en el riesgo de mortalidad en Solvencia II

The aim of this paper is to analyze the mitigating effect that, for an insurance company, the solvency capital requirement for the mortality risk of the life underwriting risk module has. In our model, the solvency capital required to the insurance company is computed through an internal model based in the Monte Carlo’s simulation method. That capital is obtained as the value in risk at 99,5% of the difference between the present value of assets minus the present value of liabilities corresponding to two consecutive years, where the reinsurance company policy is taken into account for its calculation. The forms of reinsurance considered in the analysis are the quota share, the surplus and the stop-loss. Finally, and for the case of the quota share, a sensitivity analysis of the required solvency capital of the company with respect to the ratio retention is carried out. Furthermore, and for the case of the surplus and stop-loss, a sensitivity analysis with respect to the priority is also made.

Application of Bayesian penalized spline regression for internal modeling in life insurance

Solvency 2 requires insurance companies to compute a Best-Estimate of their Liabilities (BEL) as well as a Solvency Capital Requirement (SCR). Life insurance companies being in the business of selling participating contracts with financial guarantees have to rely on a Monte-Carlo approach to appropriately value their BEL, which is the source of a first Monte-Carlo statistical error. In addition, several insurance companies rely on a (partial) internal model to derive their SCR. In this context, insurance companies rely again on a Monte-Carlo approach to value their SCR, which is the source of a second Monte-Carlo statistical error. These later computations require evaluating the BEL several thousand times, which is not possible in practice, since one single Monte-Carlo BEL evaluation can be a computational burden. The BEL has therefore to be approximated by an analytic proxy function, which introduces an additional source of numerical approximation error. In this paper, we show how these three sources of error (statistical and numerical) are intrinsically related to one another. We show that to obtain the best possible SCR accuracy, the computing power invested in assessing the Monte-Carlo SCR should be directly related to that invested in computing the Monte-Carlo BEL. Interestingly, and to achieve these results, we introduce a novel proxy method, which is highly practical, modular, smooth and naturally relates the approximation errors to the Monte-Carlo statistical errors. Furthermore, our approach allows insurance companies to naturally and transparently start reporting confidence levels on their prudential reporting, which is not disclosed so far by insurance companies and would be a relevant information within solvency disclosures for the industry.

Planned Solvency III Regulation: Should It Be Adopted Outside the European Union?

Abstract Several countries outside the European Union consider adopting its solvency regulation for their insurance industries. However, Solvency I and (to a lesser extent) Solvency II were found to run the risk of inducing more rather than less risk-taking by insurers (Zweifel, Peter. 2014. “Solvency Regulation of Insurers: A Regulatory Failure?” Journal of Insurance Issues 37 (2): 135–157.). Companies are led to neglect parameters that link them to developments in the capital market when determining their endogenous perceived efficiency frontier (EPEF), causing it to become steeper. Given homothetic risk preferences, senior management is predicted to opt for increased rather than reduced volatility. By way of contrast, if modeled after Basel III for banks, planned Solvency III will ask insurers to take developments in the capital market into account in their formulation of business strategies designed to ensure solvency (Principle 5 of Basel III). In addition, the stipulated decrease in their leverage ratio is shown to reduce the slope of the EPEF for insurers with little solvency capital. Contrary to its predecessors, Solvency III is therefore predicted to make insurers take on less risk, which argues for its for adoption beyond the European Union if properly implemented.

An Investigation of Machine Learning Approaches in the Solvency II Valuation Framework

The insurance regulatory regime introduced in the European Union by the "Solvency II" Directive 2009/138, that has become applicable on January 1, 2016, is aimed to safeguard policyholders and beneficiaries by requiring insurance undertakings to hold own funds able to cover losses, in excess to the expected ones, at the 99.5% confidence level, over a one-year period. In order to assess risks and evaluate the regulatory Solvency Capital Requirement undertakings should compute the probability distribution of the Net Asset Value - i.e., value of assets minus value of liabilities - over a one-year period, with a financially inspired market consistent approach. In life insurance, given the peculiarities of the contracts, the valuation of the Net Asset Value distribution requires a nested Monte Carlo simulation, which is extremely time consuming. Machine learning techniques are considered a promising candidate to reduce the computational burden of nested simulations. This work investigates the potential of well-established methods, such as Deep Learning Networks and Support Vector Regressors, when applied to the valuation of the Solvency Capital Requirement of participating life insurance polices, by empirically assessing their effectiveness and by comparing their efficiency and accuracy, also w.r.t. the "traditional" Least Squares Monte Carlo technique. The work aims also to contribute to the global process of renewal of the European insurance industry, where Solvency II has made the board of directors fully responsible for the choice of evaluation techniques and algorithmic processes, under the periodic monitoring of national supervisory authorities.

Calibration of VaR models with overlapping data

Abstract Under the European Union’s Solvency II regulations, insurance firms are required to use a one-year VaR (Value at Risk) approach. This involves a one-year projection of the balance sheet and requires sufficient capital to be solvent in 99.5% of outcomes. The Solvency II Internal Model risk calibrations require annual changes in market indices/term structure for the estimation of risk distribution for each of the Internal Model risk drivers. This presents a significant challenge for calibrators in terms of: Robustness of the calibration that is relevant to the current market regimes and at the same time able to represent the historically observed worst crisis; Stability of the calibration model year on year with arrival of new information. The above points need careful consideration to avoid credibility issues with the Solvency Capital Requirement (SCR) calculation, in that the results are subject to high levels of uncertainty. For market risks, common industry practice to compensate for the limited number of historic annual data points is to use overlapping annual changes. Overlapping changes are dependent on each other, and this dependence can cause issues in estimation, statistical testing, and communication of uncertainty levels around risk calibrations. This paper discusses the issues with the use of overlapping data when producing risk calibrations for an Internal Model. A comparison of the overlapping data approach with the alternative non-overlapping data approach is presented. A comparison is made of the bias and mean squared error of the first four cumulants under four different statistical models. For some statistical models it is found that overlapping data can be used with bias corrections to obtain similarly unbiased results as non-overlapping data, but with significantly lower mean squared errors. For more complex statistical models (e.g. GARCH) it is found that published bias corrections for non-overlapping and overlapping datasets do not result in unbiased cumulant estimates and/or lead to increased variance of the process. In order to test the goodness of fit of probability distributions to the datasets, it is common to use statistical tests. Most of these tests do not function when using overlapping data, as overlapping data breach the independence assumption underlying most statistical tests. We present and test an adjustment to one of the statistical tests (the Kolmogorov Smirnov goodness-of-fit test) to allow for overlapping data. Finally, we explore the methods of converting “high”-frequency (e.g. monthly data) to “low”-frequency data (e.g. annual data). This is an alternative methodology to using overlapping data, and the approach of fitting a statistical model to monthly data and then using the monthly model aggregated over 12 time steps to model annual returns is explored. There are a number of methods available for this approach. We explore two of the widely used approaches for aggregating the time series.

A Full and Synthetic Model for Asset-Liability Management in Life Insurance, and Analysis of the SCR With the Standard Formula

The aim of this paper is to introduce a synthetic ALM model that catches the main specificity of life insurance contracts. First, it keeps track of both market and book values to apply the regulatory profit sharing rule. Second, it introduces a determination of the crediting rate to policyholders that is close to the practice and is a trade-off between the regulatory rate, a competitor rate and the available profits. Third, it considers an investment in bonds that enables to match a part of the cash outflow due to surrenders, while avoiding to store the trading history. We use this model to evaluate the Solvency Capital Requirement (SCR) with the standard formula, and show that the choice of the interest rate model is important to get a meaningful model after the regulatory shocks on the interest rate. We discuss the different values of the SCR modules first in a framework with moderate interest rates using the shocks of the present legislation, and then we consider a low interest framework with the latest recommandation of the EIOPA on the shocks. In both cases, we illustrate the importance of matching cash-flows and its impact on the SCR.

Between ℙ and ℚ: The ℙℚ Measure for Pricing in Asset Liability Management

Insurance companies issue guarantees that need to be valued according to the market expectations. By calibrating option pricing models to the available implied volatility surfaces, one deals with the so-called risk-neutral measure Q , which can be used to generate market consistent values for these guarantees. For asset liability management, insurers also need future values of these guarantees. Next to that, new regulations require insurance companies to value their positions on a one-year horizon. As the option prices at t = 1 are unknown, it is common practice to assume that the parameters of these option pricing models are constant, i.e., the calibrated parameters from time t = 0 are also used to value the guarantees at t = 1 . However, it is well-known that the parameters are not constant and may depend on the state of the market which evolves under the real-world measure P . In this paper, we propose improved regression models that, given a set of market variables such as the VIX index and risk-free interest rates, estimate the calibrated parameters. When the market variables are included in a real-world simulation, one is able to assess the calibrated parameters (and consequently the implied volatility surface) in line with the simulated state of the market. By performing a regression, we are able to predict out-of-sample implied volatility surfaces accurately. Moreover, the impact on the Solvency Capital Requirement has been evaluated for different points in time. The impact depends on the initial state of the market and may vary between −46% and +52%.

One-Year Change Methodologies for Fixed-Sum Insurance Contracts

We study the dynamics of the one-year change in P&C insurance reserves estimation by analyzing the process that leads to the ultimate risk in the case of “fixed-sum” insurance contracts. The random variable ultimately is supposed to follow a binomial distribution. We compute explicitly various quantities of interest, in particular the Solvency Capital Requirement for one year change and the Risk Margin, using the characteristics of the underlying model. We then compare them with the same figures calculated with existing risk estimation methods. In particular, our study shows that standard methods (Merz–Wüthrich) can lead to materially incorrect results if the assumptions are not fulfilled. This is due to a multiplicative error assumption behind the standard methods, whereas our example has an additive error propagation as often happens in practice.

An option pricing approach for measuring Solvency Capital Requirements in Insurance Industry

Solvency capital requirements indicated by Solvency II against longevity risk involve distortions and inconsistencies caused by the invariance of the longevity shock compared to the age and time assumed by the regulatory model. To overcome the problem we introduce a temporal structure of the time mortality volatility which is included as a driver of longevity shock, by modeling a rolling window affine stochastic model.We then derive the longevity shock as a function of mortality rate time volatility evolution and time, according to a Black and Scholes environment. The original approach shows that the suggested internal model is able to reflect the risk profile of a specific undertaker by allowing consequently the level of own funds it needs and removing the effects of the invariance longevity shock.Numerical illustrations are provided in different settings and they highlight the consistency of the new approach.

Defining principles of a robust insurance solvency regime

The article deals with the fundamental features of a solvency regime: valuation and risk modelling. The paper argues that for solvency testing purposes the valuation method of choice is market consistent valuation. Statutory valuation based on historical costs is at best irrelevant and possibly misleading in the case of a compulsory winding up of a company and of forced sales of assets. As far as risk modelling is concerned, it is argued that stress tests and scenarios have an important role to play. They can complement or replace stochastic risk models. They have in particular the advantage of enabling an effective involvement of senior managers and Board members in the company’s quantitative risk management. It is argued that supervisors could use two co-existing models for solvency testing purposes: a probabilistic model and a set of scenarios based Solvency Capital Requirements. The paper draws on the experience of the EU regulatory authorities, CEIOPS and EIOPA, with the development of Solvency II. It also draws on the experience of FINMA, the Swiss supervisory authority, with the implementation of the Swiss Solvency Test. Finally, it takes into account the lessons of the Japanese life insurance crisis from 1997 to 2001.

Solvency capital requirement and the claims development result

AbstractIn this paper, four variants of calculating the Solvency Capital Requirement for long-tail liabilities satisfying Solvency II regulations are discussed. The merits of each metric are related to the stated objectives of Solvency II. Assumptions made in the calculations are assessed for suitability for the determination of an appropriate level of Solvency Capital. We show that two methods for calculating Solvency Capital provide insufficient capital to restore the Economic Balance Sheet in the event of distress. The standard formula referencing the Claims Development Result is shown to be too conservative when models are correctly specified.

The impact of insurance premium taxation

In many countries insurance premiums are subject to an insurance premium tax that replaces the common value-added tax (VAT) used for most products and services. Insurance companies cannot deduct VAT payed on inputs from premium tax; also corporate buyers of insurance cannot deduct premium tax payments from VAT on their outputs. Such deductions would be allowed, if insurance premiums were subject to VAT instead of insurance tax. In the current paper, we investigate the impact of the premium tax on insurance companies, insurance holders and government revenues from multiple perspectives. We explicitly compare tax systems with premium tax and tax systems that allow deductions. We find that the competitiveness of corporate buyers of insurance, the ruin probabilities of insurance firms and their solvency capital are hardly affected by the tax system. In contrast, the tax system has a significant influence on the cost of insurance, insurance demand, government revenues and the profitability of insurance firms.

Operational Choices for Risk Aggregation in Insurance: PSDization and SCR Sensitivity

This work addresses crucial questions about the robustness of the PSDization process for applications in insurance. PSDization refers to the process that forces a matrix to become positive semidefinite. For companies using copulas to aggregate risks in their internal model, PSDization occurs when working with correlation matrices to compute the Solvency Capital Requirement (SCR). We examine how classical operational choices concerning the modelling of risk dependence impacts the SCR during PSDization. These operations refer to the permutations of risks (or business lines) in the correlation matrix, the addition of a new risk, and the introduction of confidence weights given to the correlation coefficients. The use of genetic algorithms shows that theoretically neutral transformations of the correlation matrix can surprisingly lead to significant sensitivities of the SCR (up to 6%). This highlights the need for a very strong internal control around the PSDization step.

Analyzing the Impact of Time Horizon, Volatility and Profit Margins on Solvency Capital: Proposing a New Model for the Global Regulation of the Insurance Industry

The European Solvency II regime requires a solvency capital covering risks with a given shortfall probability of 1/200=0.5% on a one-year time horizon, which is extremely short compared to the contractual terms in traditional life insurances, as well as the settlement periods of several decades in some casualty branches. This approach undermines the importance of a high return margin and, given a risk-averse approach to management, may lead to an overall riskier business strategy in the long run. In light of this, we cannot help but ask whether such a short time horizon is capable of providing a meaningful guideline for a sustainable business and risk management that has a long-term perspective. In response to this question, we present a new model for assessing the evolution of the equity of an insurance company and calculating the probability that the initial equity of an insurance company will be depleted during a given time period. This model demonstrates that insolvencies mostly do not occur in the first year. Therefore, if one only considers a one-year window, as is the case under Solvency II, the risk will be underestimated. Even more serious is that the business will be managed too cautiously without aiming for a suitably high profit margin, which significantly reduces the risk only in the long term.

An Assessment of the Market Risk Solvency Capital Requirement Simplifications for Insurance Undertakings

The Solvency II regulatory framework has been implemented as of January 1st, 2016 and among other things it introduced economic risk-based capital requirements across all EU Member States for the first time, applicable for insurance and reinsurance undertakings. Similar to Basel II whose scope is banks, the Solvency II directive provides a new regime based on three pillars for insurers and reinsurers: 1) pillar 1: harmonized valuation and risk based capital requirements, 2) pillar 2: harmonized governance and risk management requirements and 3) pillar 3: harmonized supervisory reporting and public disclosure. The Solvency Capital Requirement (SCR) should correspond to the Value-at-Risk of the basic own funds of an insurance or reinsurance undertaking subject to a confidence level of 99.5% over a one-year period. The Solvency II directive provides a range of methods to calculate the SCR. This allows insurance or reinsurance undertakings to choose a method that is proportionate to the nature, scale and complexity of the risk that is measured. In order to calculate the SCR, an insurance undertaking can use a fully internal model, the standard formula and a partial internal model, the standard formula with undertaking-specific parameters, the standard formula as it is or a simplification. When introducing a simplification, the SCR estimate could deviate from the calculation without the simplification. A simplification could lead in important/crucial information missing from the SCR calculation. In some occasions the SCR is overestimated and in some others it is underestimated. It is therefore of interest to find the range of this deviation, potential bounds—if any and the effect it can have on the required capital. In this paper we attempt to measure this deviation for simplifications pertaining to the interest rate risk for insurance companies.

Solvency II reporting: How to interpret funds’ aggregate solvency capital requirement figures

Depending on the current risk exposure of an insurance company, the impact of buying an additional unit of a fund on an insurer’s overall Solvency II capital charges, i.e., the Solvency Capital Requirement (SCR), will differ. We call this impact the fund’s SCR contribution and show in which boundaries it lies if only the fund’s aggregate sub-SCR figures are known but not the risk exposures of the insurance company buying the fund. The upper bound of this range, the worst-case SCR contribution, can be used as a conservative measure to assess funds’ Solvency II risk contributions or to assign them to different Solvency II risk categories. We believe that providing funds’ worst-case SCR contributions can be useful information to insurance companies when screening from a broad investment universe.

Calculation of Solvency Capital Requirements for Non-life Underwriting Risk Using Generalized Linear Models

The paper presents various GLM models using individual rating factors to calculate the solvency capital requirements for non-life underwriting risk in insurance. First, we consider the potential heterogeneity of claim frequency and the occurrence of large claims in the models. Second, we analyse how the distribution of frequency and severity varies depending on the modelling approach and examine how they are projected into SCR estimates according to the Solvency II Directive. In addition, we show that neglecting of large claims is as consequential as neglecting the heterogeneity of claim frequency. The claim frequency and severity are managed using generalized linear models, that is, negative-binomial and gamma regression. However, the different individual probabilities of large claims are represented by the binomial model and the large claim severity is managed using generalized Pareto distribution. The results are obtained and compared using the simulation of frequency-severity of an actual insurance portfolio.