Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility

We consider the pricing of long-dated insurance contracts under stochastic interest rates and stochastic volatility. In particular, we focus on the valuation of insurance options with long-term equity or foreign exchange exposures. Our modeling framework extends the stochastic volatility model of Schöbel and Zhu (1999) by including stochastic interest rates. Moreover, we allow all driving model factors to be instantaneously correlated with each other, i.e. we allow for a general correlation structure between the instantaneous interest rates, the volatilities and the underlying stock returns. As insurance products often incorporate long-term exposures, they are typically more sensitive to changes in the interest rates, volatility and currencies. Therefore, having the flexibility to correlate the underlying asset price with both the stochastic volatility and the stochastic interest rates, yields a realistic model which is of practical importance for the pricing and hedging of such long-term contracts. We show that European options, typically used for the calibration of the model to market prices, and forward starting options can be priced efficiently and in closed-form by means of Fourier inversion techniques. We extensively discuss the numerical implementation of these pricing formulas, allowing for a fast and accurate valuation of European and forward starting options. The model will be especially useful for the pricing and risk management of insurance contracts and other exotic derivatives involving long-term maturities.     

Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility

We consider the pricing of long-dated insurance contracts under stochastic interest rates and stochastic volatility. In particular, we focus on the valuation of insurance options with long-term equity or foreign exchange exposures. Our modeling framework extends the stochastic volatility model of Schöbel and Zhu (1999) by including stochastic interest rates. Moreover, we allow all driving model factors to be instantaneously correlated with each other, i.e. we allow for a general correlation structure between the instantaneous interest rates, the volatilities and the underlying stock returns. As insurance products often incorporate long-term exposures, they are typically more sensitive to changes in the interest rates, volatility and currencies. Therefore, having the flexibility to correlate the underlying asset price with both the stochastic volatility and the stochastic interest rates, yields a realistic model which is of practical importance for the pricing and hedging of such long-term contracts. We show that European options, typically used for the calibration of the model to market prices, and forward starting options can be priced efficiently and in closed-form by means of Fourier inversion techniques. We extensively discuss the numerical implementation of these pricing formulas, allowing for a fast and accurate valuation of European and forward starting options. The model will be especially useful for the pricing and risk management of insurance contracts and other exotic derivatives involving long-term maturities.     

Evaluation of insurance products with guarantee in incomplete markets

Life insurance products are usually equipped with minimum guarantee and bonus provision options. The pricing of such claims is of vital importance for the insurance industry. Risk management, strategic asset allocation, and product design depend on the correct evaluation of the written options. Also regulators are interested in such issues since they have to be aware of the possible scenarios that the overall industry will face. Pricing techniques based on the Black & Scholes paradigm are often used, however, the hypotheses underneath this model are rarely met.To overcome Black & Scholes limitations, we develop a stochastic programming model to determine the fair price of the minimum guarantee and bonus provision options. We show that such a model covers the most relevant sources of incompleteness accounted in the financial and insurance literature. We provide extensive empirical analyses to highlight the effect of incompleteness on the fair value of the option, and show how the whole framework can be used as a valuable normative tool for insurance companies and regulators.     

Calibrating affine stochastic mortality models using term assurance premiums

In this paper, we focus on the calibration of affine stochastic mortality models using term assurance premiums. We view term assurance contracts as a “swap” in which policyholders exchange cash flows (premiums vs. benefits) with an insurer analogous to a generic interest rate swap or credit default swap. Using a simple bootstrapping procedure, we derive the term structure of mortality rates from a stream of contract quotes with different maturities. This term structure is used to calibrate the parameters of affine stochastic mortality models where the survival probability is expressed in closed form. The Vasicek, Cox-Ingersoll-Ross, and jump-extended Vasicek models are considered for fitting the survival probabilities term structure. An evaluation of the performance of these models is provided with respect to premiums of three Italian insurance companies.     

Valuation of guaranteed annuity options using a stochastic volatility model for equity prices

Guaranteed annuity options are options providing the right to convert a policyholder’s accumulated funds to a life annuity at a fixed rate when the policy matures. These options were a common feature in UK retirement savings contracts issued in the 1970’s and 1980’s when interest rates were high, but caused problems for insurers as the interest rates began to fall in the 1990’s. Currently, these options are frequently sold in the US and Japan as part of variable annuity products. The last decade the literature on pricing and risk management of these options evolved. Until now, for pricing these options generally a geometric Brownian motion for equity prices is assumed. However, given the long maturities of the insurance contracts a stochastic volatility model for equity prices would be more suitable. In this paper explicit expressions are derived for prices of guaranteed annuity options assuming stochastic volatility for equity prices and either a 1-factor or 2-factor Gaussian interest rate model. The results indicate that the impact of ignoring stochastic volatility can be significant.     

One-year Value-at-Risk for longevity and mortality

Upcoming new regulation on regulatory required solvency capital for insurers will be predominantly based on a one-year Value-at-Risk measure. This measure aims at covering the risk of the variation in the projection year as well as the risk of changes in the best estimate projection for future years. This paper addresses the issue how to determine this Value-at-Risk for longevity and mortality risk. Naturally, this requires stochastic mortality rates. In the past decennium, a vast literature on stochastic mortality models has been developed. However, very few of them are suitable for determining the one-year Value-at-Risk. This requires a model for mortality trends instead of mortality rates. Therefore, we will introduce a stochastic mortality trend model that fits this purpose. The model is transparent, easy to interpret and based on well known concepts in stochastic mortality modeling. Additionally, we introduce an approximation method based on duration and convexity concepts to apply the stochastic mortality rates to specific insurance portfolios.     

Risk-neutral valuation of participating life insurance contracts in a stochastic interest rate environment

Over the last years, the valuation of life insurance contracts using concepts from financial mathematics has become a popular research area for actuaries as well as financial economists. In particular, several methods have been proposed of how to model and price participating policies, which are characterized by an annual interest rate guarantee and some bonus distribution rules. However, despite the long terms of life insurance products, most valuation models allowing for sophisticated bonus distribution rules and the inclusion of frequently offered options assume a simple Black–Scholes setup and, more specifically, deterministic or even constant interest rates.We present a framework in which participating life insurance contracts including predominant kinds of guarantees and options can be valuated and analyzed in a stochastic interest rate environment. In particular, the different option elements can be priced and analyzed separately. We use Monte Carlo and discretization methods to derive the respective values.The sensitivity of the contract and guarantee values with respect to multiple parameters is studied using the bonus distribution schemes as introduced in [Bauer, D., Kiesel, R., Kling, A., Ruß, J., 2006. Risk-neutral valuation of participating life insurance contracts. Insurance: Math. Econom. 39, 171–183]. Surprisingly, even though the value of the contract as a whole is only moderately affected by the stochasticity of the short rate of interest, the value of the different embedded options is altered considerably in comparison to the value under constant interest rates. Furthermore, using a simplified asset portfolio and empirical parameter estimations, we show that the proportion of stock within the insurer’s asset portfolio substantially affects the value of the contract.     

随机利率条件下最优投资消费与寿险购买策略

随着我国利率市场化的深入发展, 利率的随机波动对投资者的最优投资消费策略将产生重要影响. 与此同时, 随着我国寿险市场的渐趋完善, 寿险购买也越来越受到投资者的重视, 投资者的最优策略也将发生改变. 现研究由 Vasicek 模型来刻画的随机利率条件下最优投资消费与寿险购买策略. 投资者的目标在于选择最优投资消费与寿险购买策略使期望效用最大化. 通过运用 Legendre 转换方法求出最优投资消费与寿险购买的显性解. 通过数值分析的方法, 实证分析相关变量的变化对投资者最优投资与寿险购买策略的影响.     

Stochastic portfolio specific mortality and the quantification of mortality basis risk

In the last decade a vast literature on stochastic mortality models has been developed. However, these models are often not directly applicable to insurance portfolios because:

(a) For insurers and pension funds it is more relevant to model mortality rates measured in insured amounts instead of measured in the number of policies.

(b) Often there is not enough insurance portfolio specific mortality data available to fit such stochastic mortality models reliably.

Therefore, in this paper a stochastic model is proposed for portfolio specific mortality experience. Combining this stochastic process with a stochastic country population mortality process leads to stochastic portfolio specific mortality rates, measured in insured amounts. The proposed stochastic process is applied to two insurance portfolios, and the impact on the Value at Risk for longevity risk is quantified. Furthermore, the model can be used to quantify the basis risk that remains when hedging portfolio specific mortality risk with instruments of which the payoff depends on population mortality rates.     

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.     

A Hybrid Model for Pricing and Hedging of Long-dated Bonds

Abstract

Long-dated fixed income securities play an important role in asset-liability management, in life insurance and in annuity businesses. This paper applies the benchmark approach, where the growth optimal portfolio (GOP) is employed as numéraire together with the real-world probability measure for pricing and hedging of long-dated bonds. It employs a time-dependent constant elasticity of variance model for the discounted GOP and takes stochastic interest rate risk into account. This results in a hybrid framework that models the stochastic dynamics of the GOP and the short rate simultaneously. We estimate and compare a variety of continuous-time models for short-term interest rates using non-parametric kernel-based estimation. The hybrid models remain highly tractable and fit reasonably well the observed dynamics of proxies of the GOP and interest rates. Our results involve closed-form expressions for bond prices and hedge ratios. Across all models under consideration we find that the hybrid model with the 3/2 dynamics for the interest rate provides the best fit to the data with respect to lowest prices and least expensive hedges.     

Zinsmodelle für Versicherungen – Diskussion der Anforderungen und Vergleich der Modelle von Hull-White und Cairns

Applications of term structure models in insurance mathematics are characterized by the usage of interest rates of various times to maturity, by the long lifetimes of insurance products and typically by long-term guarantees over the lifetime of the product such as a guaranteed minimal return. For term structure models in insurance, the main requirement is their ability to realistically implement these aspects over long time periods. Due to the lack of observable market data for most products, analysis of term structure models for insurance purposes is difficult. As a consequence, one first has to rely on the evaluation of the theoretical properties of term structure models with respect to their applicability in insurance and second one has to rely on comparisons to certain benchmark models. This in turn implies significant problems regarding a comparable implementation of models, in particular considering calibration and modeling of interest rate shocks.     

Valuation of contingent claims with mortality and interest rate risks

We consider the pricing of life insurance contracts under stochastic mortality and interest rates assumed not independent of each other. Employing the method of change of measure together with the Bayes’ rule for conditional expectations, solution expressions for the value of common contracts are obtained. A demonstration of how to apply our proposed stochastic modelling approach to value survival and death benefits is provided. Using the Human Mortality Database and UK interest rates, we illustrate that the dependence between interest rate and mortality dynamics has considerable impact in the value of even a simple survival benefit.     

A linear algebraic method for pricing temporary life annuities and insurance policies

We recast the valuation of annuities and life insurance contracts under mortality and interest rates, both of which are stochastic, as a problem of solving a system of linear equations with random perturbations. A sequence of uniform approximations is developed which allows for fast and accurate computation of expected values. Our reformulation of the valuation problem provides a general framework which can be employed to find insurance premiums and annuity values covering a wide class of stochastic models for mortality and interest rate processes. The proposed approach provides a computationally efficient alternative to Monte Carlo based valuation in pricing mortality-linked contingent claims.     

Valuing inflation-linked death benefits under a stochastic volatility framework

In this paper we construct a framework to price the inflation-linked derivatives with the stochastic inflation rate, the stochastic interest rate, and stochastic risky assets with stochastic volatility. Because of the popularity of the guaranteed minimum death benefit (GMDB) in insurance market, we mainly study two types of GMDBs: the inflation guarantee and the combination guarantee. We consider the guaranteed minimum death benefit as an European option with a random maturity date, the closed-form pricing formulas for the GMDBs are derived by Fourier-based method. Moreover, we give an elaborate sensitivity analysis to explain economical behaviors of our models. The numerical results show that the death benefit of inflation guarantee is slightly overpriced in constant volatility of stock situation.     

The role of the dependence between mortality and interest rates when pricing Guaranteed Annuity Options

In this paper we investigate the consequences on the pricing of insurance contingent claims when we relax the typical independence assumption made in the actuarial literature between mortality risk and interest rate risk. Starting from the Gaussian approach of Liu et al. (2014), we consider some multifactor models for the mortality and interest rates based on more general affine models which remain positive and we derive pricing formulas for insurance contracts like Guaranteed Annuity Options (GAOs). In a Wishart affine model, which allows for a non-trivial dependence between the mortality and the interest rates, we go far beyond the results found in the Gaussian case by Liu et al. (2014), where the value of these insurance contracts can be explained only in terms of the initial pairwise linear correlation.     

Money,prices and interest rates in a non-aggregate stochastic general equilibrium model

This paper explores the relationships between money, prices, uncertainty and interest rates in a stochastic general equilibrium model. Taking a non-aggregate pure exchange economy with time and uncertainty as the starting point, money is introduced as a means to keep track of past transactions of goods and insurance services and as an instrument to settle debts. As a result, in this stochastic general equilibrium model the desire to hold money arises from the demand of goods and services, Arrow-Debreu securities, and assets. Since these sources of demand for money are strongly related to the economy output, the economy degree of uncertainty, and the interest rates, this paper provides not only an alternative framework to the traditional keynesian analysis of the liquidity preference, but also an extension of the cash-in-advance models for introducing money in a general equilibrium model.     

多利率下的寿险费率测算模型

为了能够在多利率条件下测算人寿保险的费率,本文建立了一个线性规划模型.根据该模型,能够合理安排保费资金的投资期限以达到最大的保险利益,从而为费率和红利的测算提供了依据.列出了两个典型寿险产品的计算数据,结果表明,寿险费率的测算主要取决于长期利率.对于储蓄型寿险,资金的运用应该以长期投资为主,分红水平可以由长期利率与预定利率之差来确定.     

Pricing and hedging GLWB in the Heston and in the Black–Scholes with stochastic interest rate models

Valuing Guaranteed Lifelong Withdrawal Benefit (GLWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Forsyth and Vetzal (2014) the Black and Scholes framework seems to be inappropriate for such a long maturity products. They propose to use a regime switching model. Alternatively, we propose here to use a stochastic volatility model (Heston model) and a Black–Scholes model with stochastic interest rate (Hull–White model). For this purpose we present four numerical methods for pricing GLWB variables annuities: a hybrid tree-finite difference method and a Hybrid Monte Carlo method, an ADI finite difference scheme, and a Standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GLWB contract, and to calculate the Greeks used in hedging. Both constant withdrawal and optimal withdrawal (including lapsation) strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions.