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.     

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.     

Insurance valuation: A computable multi-period cost-of-capital approach

We present an approach to market-consistent multi-period valuation of insurance liability cash flows based on a two-stage valuation procedure. First, a portfolio of traded financial instrument aimed at replicating the liability cash flow is fixed. Then the residual cash flow is managed by repeated one-period replication using only cash funds. The latter part takes capital requirements and costs into account, as well as limited liability and risk averseness of capital providers. The cost-of-capital margin is the value of the residual cash flow. We set up a general framework for the cost-of-capital margin and relate it to dynamic risk measurement. Moreover, we present explicit formulas and properties of the cost-of-capital margin under further assumptions on the model for the liability cash flow and on the conditional risk measures and utility functions. Finally, we highlight computational aspects of the cost-of-capital margin, and related quantities, in terms of an example from life insurance.     

Efficient approximations for numbers of survivors in the Lee–Carter model

In portfolios of life annuity contracts, the payments made by an annuity provider (an insurance company or a pension fund) are driven by the random number of survivors. This paper aims to provide accurate approximations for the present value of the payments made by the annuity provider. These approximations account not only for systematic longevity risk but also for the diversifiable fluctuations around the unknown life table. They provide the practitioner with a useful tool avoiding the problem of simulations within simulations in, for instance, Solvency 2 calculations, valid whatever the size of the portfolio.     

On the modelling of nested risk-neutral stochastic processes with applications in insurance

We propose a modelling framework for risk-neutral stochastic processes nested in a real-world stochastic process. The framework is important for insurers that deal with the valuation of embedded options and in particular at future points in time. We make use of the class of State Space Hidden Markov models for modelling the joint behaviour of the parameters of a risk-neutral model and the dynamics of option market instruments. This modelling concept enables us to perform non-linear estimation, forecasting and robust calibration. The proposed method is applied to the Heston model for which we find highly satisfactory results. We use the estimated Heston model to compute the required capital of an insurance company under Solvency II and we find large differences compared to a basic calibration method.     

Valuation and hedging of participating life-insurance policies under management discretion

The valuation and hedging of participating life insurance policies, also known as with-profits policies, is considered. Such policies can be seen as European path-dependent contingent claims whose underlying security is the investment portfolio of the insurance company that sold the policy. The fair valuation of these policies is studied under the assumption that the insurance company has the right to modify the investment strategy of the underlying portfolio at any time. Furthermore, it is assumed that the issuer of the policy does not setup a separate portfolio to hedge the risk associated with the policy. Instead, the issuer will use its discretion about the investment strategy of the underlying portfolio to hedge shortfall risks. In that sense, the insurer’s investment portfolio serves simultaneously as the underlying security and as the hedge portfolio. This means that the hedging problem can not be separated from the valuation problem. We investigate the relationship between risk-neutral valuation and hedging of these policies in complete and incomplete financial markets.     

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.     

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.     

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.     

Fair valuation of insurance contracts under Lévy process specifications

The valuation of options embedded in insurance contracts using concepts from financial mathematics (in particular, from option pricing theory), typically referred to as fair valuation, has recently attracted considerable interest in academia as well as among practitioners. The aim of this article is to investigate the valuation of participating and unit-linked life insurance contracts, which are characterized by embedded rate guarantees and bonus distribution rules. In contrast to the existing literature, our approach models the dynamics of the reference portfolio by means of an exponential Lévy process. Our analysis sheds light on the impact of the dynamics of the reference portfolio on the fair contract value for several popular types of insurance policies. Moreover, it helps to assess the potential risk arising from misspecification of the stochastic process driving the reference portfolio.     

Importance sampling for integrated market and credit portfolio models

A sophisticated approach for computing the total economic capital needed for various stochastically dependent risk types is the bottom-up approach. In this approach, usually, market and credit risks of financial instruments are modeled simultaneously. As integrating market risk factors into standard credit portfolio models increases the computational burden of calculating risk measures, it is analyzed to which extent importance sampling techniques previously developed either for pure market portfolio models or for pure credit portfolio models can be successfully applied to integrated market and credit portfolio models. Specific problems which arise in this context are discussed. The effectiveness of these techniques is tested by numerical experiments for linear and non-linear portfolios.     

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.     

Risk-minimization for life insurance liabilities with basis risk

In this paper we study the hedging of typical life insurance payment processes in a general setting by means of the well-known risk-minimization approach. We find the optimal risk-minimizing strategy in a financial market where we allow for investments in a hedging instrument based on a longevity index, representing the systematic mortality risk. Thereby we take into account and model the basis risk that arises due to the fact that the insurance company cannot perfectly hedge its exposure by investing in a hedging instrument that is based on the longevity index, not on the insurance portfolio itself. We also provide a detailed example within the context of unit-linked life insurance products where the dependency between the index and the insurance portfolio is described by means of an affine mean-reverting diffusion process with stochastic drift.     

Das Vorhersagerisiko der Sterblichkeitsentwicklung – Kann es durch eine geeignete Portfoliozusammensetzung minimiert werden?

Annuities as well as term insurance create risks for the insurance companies due to changes in mortality/longevity – especially in low-interest phases. For the past decades an increase in life expectancy was observed. In this article, we examine whether an insurance company can minimise the longevity risk by means of an appropriate composition of its portfolio. We use stochastic interest rates and mortality trends. For annuities and term insurance different mortality trends are used. Based on an example we show the impact of the portfolio composition on the longevity risk. The results prove that a deliberate portfolio composition can significantly reduce the longevity risk for the insurance company.     

Valuation and risk assessment of participating life insurance in the presence of credit risk

In participating life insurance, management decisions regarding the asset composition can substantially impact the value of a policy from the policyholders’ perspective as well as the insurer’s risk situation. Due to the long-term guarantees often embedded in these contracts, life insurers typically invest a considerable portion of their capital in long-term assets such as corporate and government bonds. Besides interest rate risk, the value of these bond investments is thus particularly influenced by credit risk. Thus, the aim of this paper is to examine the impact of market risk associated with the asset composition on fair valuation and risk assessment with focus on credit risk and its interaction with equity risk and interest rate risk. Our analysis emphasizes that the consideration of credit risk associated with bonds has a strong impact on the fair valuation and risk measurement in the context of participating life insurance contracts, even in case of higher grade bond exposures.     

非风险中性定价意义下的欧式期权定价公式

用较简单的数学方法 ,推导出了非风险中性定价意义下的股票欧式期权定价公式 ,该公式在风险中性意义下包含了原始的 Black-Scholes公式 .     

Ein allgemeines Modell zur Analyse und Bewertung von Guaranteed Minimum Benefits in Fondspolicen

Variable Annuities with embedded guarantees are very popular in the US-market. There exists a great variety of products with both, guaranteed minimum death benefits (GMDB) and guaranteed minimum living benefits (GMLB). Although several approaches for pricing some of the corresponding guarantees have been proposed in the academic literature, there is no general framework in which the existing variety of such guarantees can be priced consistently. The present paper fills this gap by introducing a model, which permits a consistent and extensive analysis of all types of guarantees currently offered within Variable Annuity contracts. Aside from a valuation assuming that the policyholder follows a given strategy with respect to surrender and withdrawals, we are able to price the contract under optimal policyholder behavior. Using both, Monte-Carlo methods and a generalization of a finite mesh discretization approach proposed by Tanskanen and Lukkarinen (2004), we find that some guarantees are overpriced, whereas others, e.g. guaranteed annuities within guaranteed minimum income benefits (GMIB), are offered significantly below their risk-neutral value. We identify a variety of ‘‘new risks’’ associated with such products.     

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.     

From regulatory life tables to stochastic mortality projections: The exponential decline model

Often in actuarial practice, mortality projections are obtained by letting age-specific death rates decline exponentially at their own rate. Many life tables used for annuity pricing are built in this way. The present paper adopts this point of view and proposes a simple and powerful mortality projection model in line with this elementary approach, based on the recently studied mortality improvement rates. Two main applications are considered. First, as most reference life tables produced by regulators are deterministic by nature, they can be made stochastic by superposing random departures from the assumed age-specific trend, with a volatility calibrated on market or portfolio data. This allows the actuary to account for the systematic longevity risk in solvency calculations. Second, the model can be fitted to historical data and used to produce longevity forecasts. A number of conservative and tractable approximations are derived to provide the actuary with reasonably accurate approximations for various relevant quantities, available at limited computational cost. Besides applications to stochastic mortality projection models, we also derive useful properties involving supermodular, directionally convex and stop-loss orders.     

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.