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Session V-2 Insurance (18/12/2009 à 08h30)

D. Gabay

Impact of Stochastic Interest Rates and Stochastic Volatility on Variable Annuities

Auteurs : Eric Benhamou (Pricing Partners); Pierre Gauthier (Pricing Partners)

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Intervenants : Pierre Gauthier (Pricing Partners)

Rapporteurs : Daniel Gabay

With the success of variable annuities, insurance companies are piling up large risks in terms of both equity and fixed income assets. These risks should be properly modeled as the resulting dynamic hedging strategy is very sensitive to the modeling assumptions. The current literature has been largely focusing on simple variations around Black-Scholes model with basic interest rates term structure models. However, in a more realistic world, one should account for both Stochastic Volatility and Stochastic Interest rates. In this paper, we examine the combine effect of a Heston-type model for the underlying asset with a HJM affine stochastic interest rates model and apply it to the pricing of GMxB (GMIB, GMDB, GMAB and GMWB). We see that stochastic volatility and stochastic interest rates have an impact on the resulting fair value of the contract and the resulting fair fee as well as mainly on the vega hedge. Interestingly, using a stochastic volatility model leads to scenarios with high level of volatility for long maturities resulting in a higher contract value and a resulting fair fee. We also see that the impact of stochastic interet rates and volatility is more pronounced on the vega hedge than on the delta hedge.

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Calculating capital requirements for longevity risk in life insurance products using an internal model in line with Solvency II

Auteurs : Ralph Stevens (CentER and Netspar, Tilburg University); Anja De Waegenaere (Department of Econometrics & OR and Netspar, Tilburg University); Bertrand Melenberg (Department of Econometrics & OR and Netspar, Tilburg University)

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Intervenants : Ralph Stevens (CentER and Netspar, Tilburg University)

Rapporteurs : Rivo Randriarivony

Over the last decades, significant improvements in life expectancy have been observed in most Western countries. More importantly, there is considerable uncertainty regarding the future development of life expectancy. This uncertainty imposes significant risk on pension providers and life insurance companies, and is referred to as longevity risk. The new Solvency II regulation requires that insurers and pension funds hold a reserve capital in order to limit the probability of underfunding in a one year horizon to 0.5%, taking into account the impact of longevity risk on funding ratio volatility. In this paper we develop a methodology to determine reserve requirements for longevity risk in life insurance products. We consider the case where, as suggested in Solvency II, the risk premium for longevity risk is determined by the Cost of Capital approach. Because longevity risk arises from uncertainty in future
survivor probabilities, the capital reserve depends on the probability distribution of future mortality rates. The literature has devoted considerable attention to the development of statistical models to forecast future mortality improvements. However, using such statistical models to determine solvency requirements can be highly time-consuming. The goal of the paper is twofold. First, we propose a computationally tractable approach that yields an accurate approximation for the required solvency capital for di®erent portfolios of life insurance products, in case mortality rates are forecasted by means of the Lee and Carter (1992) model. Second, we quantify the effects of a number of simplified approaches, as suggested in the Solvency II proposal, on the level of the required solvency capital.

Valuation of Equity-Linked Life Insurance Contracts with Flexible Guarantees in a Non Gaussian Economy

Auteurs : François Quittard-Pinon (Université de Lyon and EMLyon Business School); Rivo Randrianarivony (EMLyon Business School)

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Intervenants : Rivo Randrianarivony (EMLyon Business School)

Rapporteurs : Ralph Stevens

In this paper, we focus on the pricing of a particular life insurance contract where the conditional payoff to the policyholder is the maximum of two risky assets. The first one has larger expected returns but is riskier while the second one is less risky but still can earn more
than an investment in a risk-free asset. Of course this payoff can be seen as the result of an investment in the first asset and a long position in an exchange option. The latter was priced under Gaussian assumptions by Margrabe (1978). To take kurtosis into account the underlying dynamics have to be changed. In this paper, we suggest modelling the underlying dynamics of the second asset by a simple diffusion, i.e. a geometric Brownian motion with a low volatility while the riskier asset follows a jump diffusion. More precisely, this process has a Brownian component and a compound Poisson one, where jump size is driven by a double exponential distribution. This stochastic process introduced by Kou (2002) is easy to manage and proves to be a versatile tool. To price our life insurance contract, we use a generalized Fourier transform and obtain the solution numerically. As far as we know, this is the first paper to use this approach. This methodology proves to be very efficient both with respect to accuracy and to computational time. We also consider a contrat with a fixed guarantee and price it while taking into account stochastic volatility and jumps. We incorporate mortality using a classical Makeham law.

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Hedging (Co)Variance Risk with Variance Swaps

Auteurs : José Da Fonseca (Ecole Supérieure d'Ingénieurs Léonard de Vinci); Florian Ielpo (Pictet & Cie Asset Management); Martino Grasselli (University of Padua & Ecole Superieure d'Ingenieurs Leonard de Vinci)

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Intervenants : Martino Grasselli (University of Padua & Ecole Superieure d'Ingenieurs Leonard de Vinci)

Rapporteurs : Pierre Gauthier

In this paper we measure the impact of variance and covariance risks in financial markets. In an asset allocation framework with stochastic (co)variances, we consider the possibility to invest not only in the risky assets but also in the variance swaps associated that are non redundant derivatives which span the volatility as well as the co-volatility risks. We provide explicit solutions for the portfolio optimization problem in both the incomplete and completed market cases. We use the ratio between the initial wealths leading to the same expected utility in the two market cases as a criterion in order to measure the impact of (co)variance risk. Using real data on major indexes and this criterion, we find that the impact of (co)variance risk on the optimal strategy is huge. We especially discuss the sensitivity of the criterion proposed to measure (co)variance risk with respect to the volatility of volatility parameter and it is found to be huge. This is consistent with the single asset empirical literature and the fast development of variance and covariance-based derivative products.

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