Modelling the fair value of annuities contracts: the impact of interest rate risk and mortality risk

The purpose of this paper is to analyze the problem of the fair valuation of annuities contracts. The market consistent valuation of these products requires a pricing framework which includes the two main sources of risk affecting the value of the annuity, i.e. interest rate risk and mortality risk. As the IASB has not set any specific guidelines as to which models are the most appropriate for these risks, in this note we consider a range of different models calibrated with historical data. We calculate the fair value of the annuity as a portfolio of zero coupon bonds, each with maturity set equal to the date of the annuity payments; the weights in the portfolio are given by the survival probabilities. Moreover, we focus on the additional information provided by stochastic simulations in order to define a suitable risk margin. The nature of the risk margin is one of the main key issues concerning the IASB and Solvency project

Pricing of guaranteed annuity conversion options.

In this note we introduce a theoretical model for the pricing and valuation of guaranteed annuity conversion options associated with certain deferred annuity pension-type contracts in the UK. The valuation approach is based on the similarity between the payoff structure of the contract and a call option written on a coupon-bearing bond. The model makes use of a one-factor Heath-Jarrow-Morton framework for the term structure of interest rates. Numerical results are investigated and the sensitivity of the price of the option to changes in the key parameters is also analyzed

The fair valuation problem of guaranteed annuity options: The stochastic mortality environment case

In this paper, we extend the analysis of the behaviour of pension contracts with guaranteed annuity conversion options (as presented in Ballotta and Haberman [Insurance: Math. Econ. 33 (2003) 87]) to the case in which mortality risk is incorporated via a stochastic model for the evolution over time of the underlying hazard rates. The pricing framework makes also use of a Black–Scholes/Heath–Jarrow–Morton economy in order to obtain an analytical solution to the fair valuation problem of the liabilities implied by these particular pension policies. The solution is not in closed form, and therefore, we resort to Monte Carlo simulation. Numerical results are investigated and the sensitivity of the price of the option to changes in the key parameters from the financial and mortality models is also analyzed

Investment Strategies and Risk Management for Participating Life Insurance Contracts

This paper proposes an asset allocation strategy for the risk management of the broad category of participating life insurance policies. The nature of the liability implied by these contracts allows us to treat them as options written on the reference portfolio backing the policy; consequently, the valuation approach is based on classical contingent claim theory. This leads to the identification of additional safety loadings against the risk of default implied by these contracts, and the setting up of suitable investment strategies aimed at minimizing this risk. The impact on the solvency requirements for the capital of the insurer of the proposed asset allocation strategy is analyzed by means of Monte Carlo techniques. Stress testing is considered as well with respect to the key risk factors of the model, such as the equity volatility and the market interest rate. The numerical analysis shows that, for the specific policy design considered in this paper, a suitable choice of the participation rate combined with the proposed investment strategy minimizes the overall default risk of the insurance company, both in terms of probability of default and expected severi

The IASB Insurance Project for life insurance contracts: Impact on reserving methods and solvency requirements

In this communication, we review the fair value-based accounting framework promoted by the IASB Insurance Project for the case of a life insurance company. In particular, for the case of a simple participating contract with minimum guarantee, we show that the fair valuation process allows for the identification of a suitable safety loading to hedge against default risk; furthermore, we show that, when compared with the “traditional” accounting system based on the construction of mathematical reserves, the fair value approach offers a sounder reporting framework in terms of covering of the liability, implementation costs, volatility of assets and liabilities and solvency capital requirements

The impact of longevity and investment risk on a portfolio of life insurance liabilities

In this paper we assess the joint impact of biometric and financial risk on the market valuation of life insurance liabilities. We consider a stylized, contingent claim based model of a life insurance company issuing participating contracts and subject to default risk, as pioneered by Briys and de Varenne (Geneva Pap Risk Insur Theory 19(1):53–72, 1994, J Risk Insur 64(4):673–694, 1997), and build on their model by explicitly introducing biometric risk and its components, namely diversifiable and systematic risk. The contracts considered include pure endowments, deferred whole life annuities and guaranteed annuity options. Our results stress the predominance of systematic over diversifiable risk in determining fair participation rates. We investigate the interaction of contract design, market regimes and mortality assumptions, and show that, particularly for lifelong benefits, the choice of the participation rate must be very conservative if longevity improvements are foreseeable

Temporal Lobe Spikes Affect Distant Intrinsic Connectivity Networks

Objective: To evaluate local and distant blood oxygen level dependent (BOLD) signal changes related to interictal epileptiform discharges (IED) in drug-resistant temporal lobe epilepsy (TLE). Methods: Thirty-three TLE patients undergoing EEG–functional Magnetic Resonance Imaging (fMRI) as part of the presurgical workup were consecutively enrolled. First, a single-subject spike-related analysis was performed: (a) to verify the BOLD concordance with the presumed Epileptogenic Zone (EZ); and (b) to investigate the Intrinsic Connectivity Networks (ICN) involvement. Then, a group analysis was performed to search for common BOLD changes in TLE. Results: Interictal epileptiform discharges were recorded in 25 patients and in 19 (58%), a BOLD response was obtained at the single-subject level. In 42% of the cases, BOLD changes were observed in the temporal lobe, although only one patient had a pure concordant finding, with a single fMRI cluster overlapping (and limited to) the EZ identified by anatomo-electro-clinical correlations. In the remaining 58% of the cases, BOLD responses were localized outside the temporal lobe and the presumed EZ. In every patient, with a spike-related fMRI map, at least one ICN appeared to be involved. Four main ICNs were preferentially involved, namely, motor, visual, auditory/motor speech, and the default mode network. At the single-subject level, EEG–fMRI proved to have high specificity (above 65%) in detecting engagement of an ICN and the corresponding ictal/postictal symptom, and good positive predictive value (above 67%) in all networks except the visual one. Finally, in the group analysis of BOLD changes related to IED revealed common activations at the right precentral gyrus, supplementary motor area, and middle cingulate gyrus. Significance: Interictal temporal spikes affect several distant extra-temporal areas, and specifically the motor/premotor cortex. EEG–fMRI in patients with TLE eligible for surgery is recommended not for strictly localizing purposes rather it might be useful to investigate ICNs alterations at the single-subject level

The valuation of guaranteed lifelong withdrawal benefit options in variable annuity contracts and the impact of mortality risk

n light of the growing importance of the variable annuities market, in this paper we introduce a theoretical model for the pricing and valuation of guaranteed lifelong withdrawal benefit (GLWB) options embedded in variable annuity products. As the name suggests, this option offers a lifelong withdrawal guarantee; therefore, there is no limit on the total amount that is withdrawn over the term of the policy because if the account value becomes zero while the insured is still alive, he or she continues to receive the guaranteed amount annually until death. Any remaining account value at the time of death is paid to the beneficiary as a death benefit. We offer a specific framework to value the GLWB option in a market-consistent manner under the hypothesis of a static withdrawal strategy, according to which the withdrawal amount is always equal to the guaranteed amount. The valuation approach is based on the decomposition of the product into living and death benefits. The model makes use of the standard no-arbitrage models of mathematical finance, which extend the Black-Scholes framework to insurance contracts, assuming the fund follows a geometric Brownian motion and the insurance fee is paid, on an ongoing basis, as a proportion of the assets. We develop a sensitivity analysis, which shows how the value of the product varies with the key parameters, including the age of the policyholder at the inception of the contract, the guaranteed rate, the risk-free rate, and the fund volatility. We calculate the fair fee, using Monte Carlo simulations under different scenarios. We give special attention to the impact of mortality risk on the value of the option, using a flexible model of mortality dynamics, which allows for the possible perturbations by mortality shock of the standard mortality tables used by practitioners. Moreover, we evaluate the introduction of roll-up and step-up options and the effect of the decision to delay withdrawing. Empirical analyses are performed, and numerical results are provided

Efficient pricing of ratchet equity-indexed annuities in a variance-gamma economy

In this paper we propose a new method for approximating the price of arithmetic Asian options in a Variance-Gamma (VG) economy, which is then applied to the problem of pricing equityindexed annuity contracts. The proposed procedure is an extension to the case of a VG-based model of the moment-matching method developed by Turnbull and Wakeman and Levy for the pricing of this class of path-dependent options in the traditional Black-Scholes setting. The accuracy of the approximation is analyzed against RQMC estimates for the case of ratchet equityindexed annuities with index averaging