The Risk Margin under Solvency II is determined as the cost of holding capital over the lifetime of liabilities, whereby future costs are discounted to the valuation date at risk-free rates. An implicit assumption of the current method is that the Risk Margin should allow for new capital to be raised after the occurrence of losses no larger than required capital. Using “Cost of Capital” as general valuation method, various approaches are discussed, giving rise to several alternative calculation methods of the Risk Margin. A comparison is made with the adjustment proposed by EIOPA in 2020 and also an approach is explored where future capital raisings are treated as contingent commitments. Each of the approaches discussed can be justified on its own merits in the context of Solvency II legislation, and leads to substantially different results for liabilities with long durations. Therefore, a more precise specification of the function of the Risk Margin and underlying assumptions is desirable.

There is a lack of publicly available information covering the practices insurers employ to manage their exposure to reinsurance recapture risk. A working party was set-up to shed light on the different approaches insurers use to mitigate this complicated to manage risk. This report is intended to form part of a publicly available information repository that market practitioners can refer to and reflect on as best practice evolves and develops.

For life insurers in the United Kingdom (UK), the risk margin is one of the most controversial aspects of the Solvency II regime which came into force in 2016.

The risk margin is the difference between the technical provisions and the best estimate liabilities. The technical provisions are intended to be market-consistent, and so are defined as the amount required to be paid to transfer the business to another undertaking. In practice, the technical provisions cannot be directly calculated, and so the risk margin must be determined using a proxy method; the method chosen for Solvency II is known as the cost-of-capital method.

Following the implementation of Solvency II, the risk margin came under considerable criticism for being too large and too sensitive to interest rate movements. These criticisms are particularly valid for annuity business in the UK – such business is of great significance to the system for retirement provision. A further criticism is that mitigation of the impact of the risk margin has led to an increase in reinsurance of longevity risks, particularly to overseas reinsurers.

This criticism has led to political interest, and the risk margin was a major element of the Treasury Committee inquiry into EU Insurance Regulation.

The working party was set up in response to this criticism. Our brief is to consider both the overall purpose of the risk margin for life insurers and solutions to the current problems, having regard to the possibility of post-Brexit flexibility.

We have concluded that a risk margin in some form is necessary, although its size depends on the level of security desired, and so is primarily a political question.

We have reviewed possible alternatives to the current risk margin, both within the existing cost-of-capital methodology and considering a wide range of alternatives.

We believe that requirements for the risk margin will depend on future circumstances, in particular relating to Brexit, and we have identified a number of possible changes to methodology which should be considered, depending on circumstances.

Delong et al. (2018) presented a theory of fair (market-consistent and actuarial) valuation of insurance liability cash-flow streams in continuous time. In this paper, we investigate in detail two practical applications of our theory of fair valuation. In the first example, we consider the fair valuation of a terminal benefit which is contingent on correlated tradeable and non-tradeable financial risks. In the second example, we consider a portfolio of unit-linked contracts contingent on a non-tradeable insurance and a tradeable financial risk. We derive partial differential equations (PDEs) which characterize the continuous-time fair valuation operators in these two examples and we find explicit solutions to these PDEs. The fair values of the liabilities are decomposed into the best estimate of the liability and a risk margin. The arbitrage-free representations of the fair values of the liabilities are derived and the dynamic hedging strategies associated with the continuous-time fair valuation operators are also established. Detailed interpretations of the results, which should be useful both for researchers and practitioners, are provided.

In 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.

Following the EU Gender Directive, that obliges insurance companies to charge the same premium to policyholders of different genders, we address the issue of calculating solvency capital requirements (SCRs) for pure endowments and annuities issued to mixed portfolios. The main theoretical result is that, if the unisex fairness principle is adopted for the unisex premium, the SCR at issuing time of the mixed portfolio calculated with unisex survival probabilities is greater than the sum of the SCRs of the gender-based subportfolios. Numerical results show that for pure endowments the gap between the two is negligible, but for lifetime annuities the gap can be as high as 3–4%. We also analyze some conservative pricing procedures that deviate from the unisex fairness principle, and find that they lead to SCRs that are lower than the sum of the gender-based SCRs because the policyholders are overcharged at issuing time.

Solvency II is currently one of the most sophisticated insurance regulatory regimes in the world. It is built around the principles of market consistency and embedding strong risk management and governance within insurance companies. For business with long-term guarantees, the original basis produced outcomes that were unacceptable to the member states. The original design was amended through Omnibus II. The working party has looked back at the outcome of the final regulation and comments on how well Solvency II has fared, principally from a UK perspective, relative to its initial goals of improved consumer protection, harmonisation, effective risk management and financial stability. We review Pillar 1’s market consistent valuation (including the risk margin and transitional measures) as well as the capital requirements (including internal models). We look at the impact this has on asset and liability management, pro-cyclicality and product design. We look at Pillars 2 and 3 in respect of the Own Risk and Solvency Assessment, liquidity and disclosure. Finally, we stand back and look at harmonisation and the implications of Brexit. In summary we conclude that Solvency II represents a huge improvement over Solvency I although it has not fully achieved the goals it aspired to. There are acknowledged shortfalls and imperfections where adjustments to Solvency II are likely. There remain other concerns around pro-cyclicality, and the appropriateness of market consistency is still open to criticism. It is hoped that the paper and the discussion that goes with it provide an insight into where Solvency II has taken European Insurance regulation and the directions in which it could evolve.

This paper is concerned with dependency between business segments in the non-life insurance industry. When considering the business of an insurance company at the aggregate level, dependence structures can have a major impact in several areas of Enterprise Risk Management, such as in claims reserving and capital modelling. The accurate estimation of the diversification benefits related to the dependence structures between lines of business (LoBs) is crucial for (i) capital efficiency, as one should avoid holding unnecessarily high levels of capital, and (ii) solvency of the insurance company, as an underestimation, on the other hand, may lead to insufficient capitalisation and safety. There seems to be a great deal of preconception as to how dependent insurance claims should be. Often, presence of dependence is taken as a given and rarely discussed or challenged, perhaps because of the lack of extensive datasets to be publicly analysed. In this paper, we take a different approach, and consider how much correlation some real datasets actually display (the Meyers–Shi dataset from the USA, and the AUSI dataset from Australia). We develop a simple theoretical framework that enables us to explain how and why correlations can be illusory (and what we mean by that). We show with some real examples that, sometimes, most (if not all) of the correlation can be “explained” by an appropriate methodology. Two major conclusions stem from our analysis.

1. 1. In any attempt to measure cross-LoB correlations, careful modelling of the data needs to be the order of the day. The exercise will not be well served by rough modelling, such as the use of simple chain ladders, and may indeed result in the prescription of excessive risk margins and/or capital margins.

2. 2. Such empirical evidence as examined in the paper reveals cross-LoB correlations that vary only in the range zero to very modest. There is little evidence in favour of the high correlation assumed in some jurisdictions. The evidence suggests that these assumptions derived from either poor modelling or a misconception of the cross-LoB dependencies relevant to the purpose to which they are applied.

We develop quantile functions from regression models in order to derive risk margin and to evaluate capital in non-life insurance applications. By utilizing the entire range of conditional quantile functions, especially higher quantile levels, we detail how quantile regression is capable of providing an accurate estimation of risk margin and an overview of implied capital based on the historical volatility of a general insurers loss portfolio. Two modeling frameworks are considered based around parametric and non-parametric regression models which we develop specifically in this insurance setting. In the parametric framework, quantile functions are derived using several distributions including the flexible generalized beta (GB2) distribution family, asymmetric Laplace (AL) distribution and power-Pareto (PP) distribution. In these parametric model based quantile regressions, we detail two basic formulations. The first involves embedding the quantile regression loss function from the nonparameteric setting into the argument of the kernel of a parametric data likelihood model, this is well known to naturally lead to the AL parametric model case. The second formulation we utilize in the parametric setting adopts an alternative quantile regression formulation in which we assume a structural expression for the regression trend and volatility functions which act to modify a base quantile function in order to produce the conditional data quantile function. This second approach allows a range of flexible parametric models to be considered with different tail behaviors. We demonstrate how to perform estimation of the resulting parametric models under a Bayesian regression framework. To achieve this, we design Markov chain Monte Carlo (MCMC) sampling strategies for the resulting Bayesian posterior quantile regression models. In the non-parametric framework, we construct quantile functions by minimizing an asymmetrically weighted loss function and estimate the parameters under the AL proxy distribution to resemble the minimization process. This quantile regression model is contrasted to the parametric AL mean regression model and both are expressed as a scale mixture of uniform distributions to facilitate efficient implementation. The models are extended to adopt dynamic mean, variance and skewness and applied to analyze two real loss reserve data sets to perform inference and discuss interesting features of quantile regression for risk margin calculations.

This paper brings together the work of the GI Solvency II Technical Provisions working party. The working party was formed in 2009 for the primary purpose of raising awareness of Solvency II and the impact it would have on the work that reserving actuaries do. Over the years, the working party’s focus has shifted to exploring and promoting discussion of the many practical issues raised by the requirements and to promoting best practice. To this end, we have developed, presented and discussed many of the ideas contained in this paper at events and forums. However, the size of the subject means that at no one event have we managed to cover all of the areas that the reserving actuary needs to be aware of. This paper brings together our thinking in one place for the first time. We hope experienced practitioners will find it thought provoking, and a useful reference tool. For new practitioners, we hope it helps to get you up-to-speed quickly. Good luck!

It is essential for insurance regulation to have a clear picture of the risk measures that are used. We compare different mathematical interpretations of the Solvency Capital Requirement (SCR) definition from Solvency II that can be found in the literature. We introduce a mathematical modeling framework that enables us to make a mathematically rigorous comparison. The paper shows similarities, differences, and properties such as convergence of the different SCR interpretations. Moreover, we generalize the SCR definition to future points in time based on a generalization of the value at risk. This allows for a sound definition of the Risk Margin. Our study helps to make the Solvency II insurance regulation more consistent.

This paper investigates market-consistent valuation of insurance liabilities in the context of Solvency II among others and to some extent IFRS 4. We propose an explicit and consistent framework for the valuation of insurance liabilities which incorporates the Solvency II approach as a special case.

The proposed framework is based on replication over multiple (one-year) time periods by a periodically updated portfolio of assets with reliable market prices, allowing for “limited liability” in the sense that the replication can in general not always be continued. The asset portfolio consists of two parts: (1) assets whose market price defines the value of the insurance liabilities, and (2) capital funds used to cover risk which cannot be replicated. The capital funds give rise to capital costs; the main exogenous input in the framework is the condition on when the investment of the capital funds is acceptable.

We investigate existence of the value and show that the exact calculation of the value has to be done recursively backwards in time, starting at the end of the lifetime of the insurance liabilities. We derive upper bounds on the value and, for the special case of replication by risk-free one-year zero-coupon bonds, explicit recursive formulas for calculating the value. In the paper, we only partially consider the question of the uniqueness of the value.

Valuation in Solvency II and IFRS 4 is based on representing the value as a sum of a “best estimate” and a “risk margin”. In our framework, it turns out that this split is not natural. Nonetheless, we show that a split can be constructed as a simplification, and that it provides an upper bound on the value under suitable conditions. We illustrate the general results by explicitly calculating the value for a simple example.

Under new solvency regulations, general insurance companies need to calculate a risk margin to cover possible shortfalls in their liability runoff. A popular approach for the calculation of the risk margin is the so-called cost-of-capital approach. A comprehensive cost-of-capital approach involves the consideration of multiperiod risk measures. Because multiperiod risk measures are rather complex mathematical objects, various proxies are used to estimate this risk margin. Of course, the use of proxies and the study of their quality raises many questions, see IAA position paper [8]. In the present paper we provide a first discourse on multiperiod solvency considerations for a general insurance liability runoff. Within a chain ladder framework, we derive analytic formulas for the risk margin which allow to compare the comprehensive approach to the different proxies used in practice. Moreover, a case study investigates and answers questions raised in [8].