Risk aggregation in Solvency II through recursive log-normals

It is argued that the accuracy of risk aggregation in Solvency II can be improved by updating skewness recursively. A simple scheme based on the log-normal distribution is developed and shown to be superior to the standard formula and to adjustments of the Cornish–Fisher type. The method handles tail-dependence if a simple Monte Carlo step is included. A hierarchical Clayton copula is constructed and used to confirm the accuracy of the log-normal approximation and to demonstrate the importance of including tail-dependence. Arguably a log-normal scheme makes the logic in Solvency II consistent, but many other distributions might be used as vehicle, a topic that may deserve further study.     

Ein zweidimensionales Durationskonzept

The German proposal for a Solvency II-compatible standard model for the life insurance branch calculates the risk capital that is necessary for a sufficient risk capitalisation of the company at hand. This capital is called ‘‘target capital’’ or Solvency Capital Requirement (SCR for short). For this to achieve it is applied the book value of the actuarial reserve onto the well-known market value formula getting the market value (or present value) by means of the classical duration concept as a global approach (cf. the documentation of the standard model of the GDV p. 26). This formula takes into account the impact of the interest rate but leaves aside all the other actuarial assumptions. In particular, the influence of the biometrical assumptions is not considered. This is at least one reason, why this ansatz is – at the time being – no more compatible with the Solvency II requirements and thus does no more satisfy its own entitlements. In the work at hand it is proposed and worked out a concept that overcomes this drawback. The result is a formula with the help of which the present value of the actuarial liabilities is calculated from their book value in fact by taking into account the interest rate as well as the biometrical assumptions. It is to be remarked that the proposed two-dimensional duration concept may be developed completely along the lines given by the classical one-dimensional analogue leaving some arbitraries only on determining the biometrical gauge, i.e., the mapping of the vector that represents the formula of the active lives remaining onto its average value. For this to achieve one has to consider the underlying business in force. The superordinate relevance of such a two-dimensional ansatz lies in the fact that the developments of the project Solvency II during the last months have shown that its success depends crucially on the availability of efficient and well-elaborated approximation procedures.     

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.     

Longevity risk,cost of capital and hedging for life insurers under Solvency II

The cost of capital is an important factor determining the premiums charged by life insurers issuing life annuities. This capital cost can be reduced by hedging longevity risk with longevity swaps, a form of reinsurance. We assess the costs of longevity risk management using indemnity based longevity swaps compared to costs of holding capital under Solvency II. We show that, using a reasonable market price of longevity risk, the market cost of hedging longevity risk for earlier ages is lower than the cost of capital required under Solvency II. Longevity swaps covering higher ages, around 90 and above, have higher market hedging costs than the saving in the cost of regulatory capital. The Solvency II capital regulations for longevity risk generates an incentive for life insurers to hold longevity tail risk on their own balance sheets, rather than transferring this to the reinsurance or the capital markets. This aspect of the Solvency II capital requirements is not well understood and raises important policy issues for the management of longevity risk.     

A generic model for spouse’s pensions with a view towards the calculation of liabilities

We introduce a generic model for spouse’s pensions. The generic model allows for the modeling of various types of spouse’s pensions with payments commencing at the death of the insured. We derive abstract formulas for cashflows and liabilities corresponding to common types of spouse’s pensions. In particular, we show that our generic model allows for simple modeling of longevity improvements, enabling the calculation of the Solvency II capital requirements related to longevity risk for spouse’s pensions.     

Deterministic shock vs. stochastic value-at-risk — an analysis of the Solvency II standard model approach to longevity risk

In general, the capital requirement under Solvency II is determined as the 99.5% Value-at-Risk of the Available Capital. In the standard model’s longevity risk module, this Value-at-Risk is approximated by the change in Net Asset Value due to a pre-specified longevity shock which assumes a 25% reduction of mortality rates for all ages. We analyze the adequacy of this shock by comparing the resulting capital requirement to the Value-at-Risk based on a stochastic mortality model. This comparison reveals structural shortcomings of the 25% shock and therefore, we propose a modified longevity shock for the Solvency II standard model. We also discuss the properties of different Risk Margin approximations and find that they can yield significantly different values. Moreover, we explain how the Risk Margin may relate to market prices for longevity risk and, based on this relation, we comment on the calibration of the cost of capital rate and make inferences on prices for longevity derivatives.     

Impact of insurance for operational risk: Is it worthwhile to insure or be insured for severe losses?

Under the Basel II standards, the Operational Risk (OpRisk) advanced measurement approach allows a provision for reduction of capital as a result of insurance mitigation of up to 20%. This paper studies different insurance policies in the context of capital reduction for a range of extreme loss models and insurance policy scenarios in a multi-period, multiple risk setting. A Loss Distributional Approach (LDA) for modeling of the annual loss process, involving homogeneous compound Poisson processes for the annual losses, with heavy-tailed severity models comprised of α-stable severities is considered. There has been little analysis of such models to date and it is believed insurance models will play more of a role in OpRisk mitigation and capital reduction in future. The first question of interest is when would it be equitable for a bank or financial institution to purchase insurance for heavy-tailed OpRisk losses under different insurance policy scenarios? The second question pertains to Solvency II and addresses quantification of insurer capital for such operational risk scenarios. Considering fundamental insurance policies available, in several two risk scenarios, we can provide both analytic results and extensive simulation studies of insurance mitigation for important basic policies, the intention being to address questions related to VaR reduction under Basel II, SCR under Solvency II and fair insurance premiums in OpRisk for different extreme loss scenarios. In the process we provide closed-form solutions for the distribution of loss processes and claims processes in an LDA structure as well as closed-form analytic solutions for the Expected Shortfall, SCR and MCR under Basel II and Solvency II. We also provide closed-form analytic solutions for the annual loss distribution of multiple risks including insurance mitigation.     

Intensity-based framework for surrender modeling in life insurance

In this paper, we propose an intensity-based framework for surrender modeling. We model the surrender decision under the assumption of stochastic intensity and use, for comparative purposes, the affine models of Vasicek and Cox–Ingersoll–Ross for deriving closed-form solutions of the policyholder’s probability of surrendering the policy. The introduction of a closed-form solution is an innovative aspect of the model we propose. We evaluate the impact of dynamic policyholders’ behavior modeling the dependence between interest rates and surrendering (affine dependence) with the assumption that mortality rates are independent of interest rates and surrendering. Finally, using experience-based decrement tables for both surrendering and mortality, we explain the calibration procedure for deriving our model’s parameters and report numerical results in terms of best estimate of liabilities for life insurance under Solvency II.     

Minimum standards for investment performance: A new perspective on non-life insurer solvency

The aim of this paper is to develop an alternative approach for assessing an insurer’s solvency as a proposal for a standard model for Solvency II. Instead of deriving minimum capital requirements–as is done in solvency regulation–our model provides company-specific minimum standards for risk and return of investment performance, given the distribution structure of liabilities and a predefined safety level. The idea behind this approach is that in a situation of weak solvency, an insurer’s asset allocation can be adjusted much more easily in the short term than can, for example, claims cost distributions, operating expenses, or equity capital. Hence, instead of using separate models for capital regulation and solvency regulation–as is typically done in most insurance markets–our single model will reduce the complexity and costs for insurers as well as for regulators. In this paper, we first develop the model framework and second test its applicability using data from a German non-life insurer.     

Biometric worst-case scenarios for multi-state life insurance policies

It is common actuarial practice to calculate premiums and reserves under a set of biometric assumptions that represent a worst-case scenario for the insurer. The new solvency regime of the European Union (Solvency II) also uses worst-case scenarios for the calculation of solvency capital requirements for life insurance business. Surprisingly, the actuarial literature so far offers no exact method for the construction of biometric scenarios that let premiums and reserves be always on the safe side with respect to a given confidence band for the biometric second-order basis. The present paper partly fills this gap by introducing a general method that allows one to construct such scenarios for homogenous portfolios of life insurance policies. The results are especially informative for life insurance policies with mixed character (e.g. survival and occurrence character). Two examples are given that illustrate the new method, demonstrate its usefulness for the calculation of premiums and reserves, and show how the new approach could improve the calculation of biometric solvency reserves for Solvency II.     

From deterministic to stochastic surrender risk models: Impact of correlation crises on economic capital

In this paper we raise the matter of considering a stochastic model of the surrender rate instead of the classical S-shaped deterministic curve (in function of the spread), still used in almost all insurance companies. For extreme scenarios, due to the lack of data, it could be tempting to assume that surrenders are conditionally independent with respect to a S-curve disturbance. However, we explain why this conditional independence between policyholders decisions, which has the advantage to be the simplest assumption, looks particularly maladaptive when the spread increases. Indeed the correlation between policyholders decisions is most likely to increase in this situation. We suggest and develop a simple model which integrates those phenomena. With stochastic orders it is possible to compare it to the conditional independence approach qualitatively. In a partially internal Solvency II model, we quantify the impact of the correlation phenomenon on a real life portfolio for a global risk management strategy.     

Multi-year non-life insurance risk of dependent lines of business in the multivariate additive loss reserving model

We derive analytical estimators of non-life insurance risk in multi-year view for the multivariate additive loss reserving model. Thereby we jointly assess reserve and premium risks of multiple years for portfolios of possibly dependent lines of business in one integrated approach. By extending existing formulae for the univariate additive model to the multivariate case, risk estimators for the aggregated portfolio now include the inherent dependencies among all lines of business. The resulting risk evaluation over one-year and general multi-year horizons is fundamental to regulatory reporting (e.g. the ORSA process in Solvency II) and risk-based business planning of non-life insurers with multiple lines of business. A case study illustrates the fruitful application of our formulae and reproduces previous findings for the special case of ultimo view.     

Dependent interest and transition rates in life insurance

For market consistent life insurance liabilities modelled with a multi-state Markov chain, it is of importance to consider the interest and transition rates as stochastic processes, for example in order to consider hedging possibilities of the risks, and for risk measurement. In the literature, this is usually done with an assumption of independence between the interest and transition rates. In this paper, it is shown how to valuate life insurance liabilities using affine processes for modelling dependent interest and transition rates. This approach leads to the introduction of so-called dependent forward rates. We propose a specific model for surrender modelling, and within this model the dependent forward rates are calculated, and the market value and the Solvency II capital requirement are examined for a simple savings contract.     

Managing longevity and disability risks in life annuities with long term care

The aim of the paper is twofold. Firstly, it develops a model for risk assessment in a portfolio of life annuities with long term care benefits. These products are usually represented by a Markovian Multi-State model and are affected by both longevity and disability risks. Here, a stochastic projection model is proposed in order to represent the future evolution of mortality and disability transition intensities. Data from the Italian National Institute of Social Security (INPS) and from Human Mortality Database (HMD) are used to estimate the model parameters. Secondly, it investigates the solvency in a portfolio of enhanced pensions. To this aim a risk model based on the portfolio risk reserve is proposed and different rules to calculate solvency capital requirements for life underwriting risk are examined. Such rules are then compared with the standard formula proposed by the Solvency II project.     

Capital allocation based on the Tail Covariance Premium Adjusted

The current Solvency II process makes risk capital allocation to different business lines more and more important. This paper considers two business lines with the exponential loss distributions linked by a Farlie–Gumbel–Morgenstern (FGM) copula, modelling the dependence between them. As an allocation principle we use the Tail Covariance Premium Adjusted and obtain expressions for the allocation to the two business lines.     

On the lifetime and one-year views of reserve risk,with application to IFRS 17 and Solvency II risk margins

This paper brings together analytic and simulation-based approaches to reserve risk in general (P&C) insurance, applied to the traditional actuarial view of risk over the lifetime of the liabilities and to the one-year view of Solvency II. It also connects the lifetime and one-year views of risk. The framework of the model in Mack (1993) is used throughout, although the results have wider applicability.The advantages of a simulation-based approach are highlighted, giving a full predictive distribution, which is used to estimate risk margins under Solvency II and risk adjustments under IFRS 17. We discuss methods for obtaining capital requirements in a cost-of-capital risk margin, and methods for estimating risk adjustments using risk measures applied to a simulated distribution of the outstanding liabilities over their lifetime.     

Issues with the Smith–Wilson method

We analyse various features of the Smith–Wilson method used for discounting under the EU regulation Solvency II, with special attention to hedging. In particular, we show that all key rate duration hedges of liabilities beyond the Last Liquid Point will be peculiar. Moreover, we show that there is a connection between the occurrence of negative discount factors and singularities in the convergence criterion used to calibrate the model. The main tool used for analysing hedges is a novel stochastic representation of the Smith–Wilson method.     

Modelling of insurers’ rating determinants. An application of machine learning techniques and statistical models

In recent years, financial regulations such as Basel II and Solvency II have highlighted the utility of credit risk assessments through internal rating systems, particularly for estimating the probability of default (PD) of credit exposures.     

On market value margins and cost of capital

The valuation of insurance liabilities plays a central role in the design of any solvency framework. We investigate the notion of “fair value of liabilities” at a conceptional level and compare several implementations which are currently discussed in the Solvency II project. Our focus is on the cost of capital approach based on market information. In particular, we discuss the applicability of arguments borrowed from financial mathematics.     

Risk aggregation in non-life insurance: Standard models vs. internal models

Standard models for capital requirements restrict the correlation between risk factors to the linear measure and disregard undertaking-specific parameters. We consider an alternative framework for risk aggregation in non-life insurance using vine copulas that allow non-linear dependence and are estimated with undertaking-specific parameters. We empirically compare our alternative risk model with three regulatory standard models (Korean risk-based capital, Solvency II, Swiss Solvency Test) and show that the standard models lead to more than 50% higher capital requirements on average. Half of the overestimation results from the uniform parameter selection imposed by regulations and the other half comes from the linear correlation assumption. The differences might distort competition when both standard models and internal risk models are used in a single market.