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Killing the Law of Large Numbers: Mortality Risk Premiums and the Sharpe Ratio
 The Journal of Risk and Insurance
, 2006
"... authors would like to acknowledge helpful comments and feedback from the participants at this event, as well as the conference organizer and JRI editor, Richard MacMinn. We provide an overview of how the classical law of large numbers breaks down when pricing lifecontingent claims under stochastic ..."
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authors would like to acknowledge helpful comments and feedback from the participants at this event, as well as the conference organizer and JRI editor, Richard MacMinn. We provide an overview of how the classical law of large numbers breaks down when pricing lifecontingent claims under stochastic as opposed to deterministic mortality (probability, hazard) rates. In a stylized situation we derive the limiting perpolicy risk and show that it goes to a nonzero constant, which is in contrast to the classical situation when the underlying mortality decrements are known with certainty. We decompose the standard deviation per policy into systematic and nonsystematic components, akin to the analysis of individual stock (equity) risk in a Markowitz portfolio framework. We then draw upon the nancial analogy of the Sharpe Ratio to develop a premium pricing methodology under aggregate mortality risk. Our paper is presented in discrete time and is a companion to the continuoustime formulation developed in Milevsky, Promislow and Young (2005). 1
Financial valuation of mortality risk via the instantaneous Sharpe ratio. Working paper
, 2005
"... Abstract: We develop a theory for pricing nondiversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortalitycontingent claim requires compensation for this risk in the form of a prespecified instantaneous Sharpe ratio. We prove that our ensuing va ..."
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Abstract: We develop a theory for pricing nondiversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortalitycontingent claim requires compensation for this risk in the form of a prespecified instantaneous Sharpe ratio. We prove that our ensuing valuation formula satisfies a number of desirable properties. For example, we show that it is subadditive in the number of contracts sold. A key result is that if the hazard rate is stochastic, then the riskadjusted survival probability is greater than the physical survival probability, even as the number of contracts approaches infinity.
Valuation of Mortality Risk via the Instantaneous Sharpe Ratio: Applications to Life Annuities
, 2008
"... Applications to Life Annuities Abstract: We develop a theory for valuing nondiversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortalitycontingent claim requires compensation for this risk in the form of a prespecified instantaneous Sharpe rati ..."
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Applications to Life Annuities Abstract: We develop a theory for valuing nondiversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortalitycontingent claim requires compensation for this risk in the form of a prespecified instantaneous Sharpe ratio. We apply our method to value life annuities. One result of our paper is that the value of the life annuity is identical to the upper good deal bound of Cochrane and SaáRequejo (2000) and of Björk and Slinko (2006) applied to our setting. A second result of our paper is that the value per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting value as an expectation with respect to an equivalent martingale measure (as in BlanchetScalliet, El Karoui, and Martellini (2005)), and from this representation, one can interpret the instantaneous Sharpe ratio as an annuity market’s price of mortality risk.
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, bu ..."
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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 U.S. 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 closed form expressions are derived for prices of guaranteed annuity options assuming stochastic volatility for equity prices and either a 1factor or 2factor Gaussian interest rate model. The results indicate that the impact of ignoring stochastic volatility can be significant.
Valuation of guaranteed annuity options in affine term structure models
"... We propose three analytic approximation methods for numerical valuation of the guaranteed annuity options in deferred annuity pension policies. The approximation methods include the stochastic duration approach, Edgeworth expansion and analytic approximation in affine diffusions. The payoff structur ..."
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We propose three analytic approximation methods for numerical valuation of the guaranteed annuity options in deferred annuity pension policies. The approximation methods include the stochastic duration approach, Edgeworth expansion and analytic approximation in affine diffusions. The payoff structure in the annuity policies is similar to a quanto call option written on a coupon bearing bond. To circumvent the limitations of the onefactor interest rate model, we model the interest rate dynamics by a twofactor affine interest rate term structure model. The numerical accuracy and computational efficiency of these approximation methods are analyzed. We also investigate the value sensitivity of the guaranteed annuity option with respect to different parameters in the pricing model.
On the regulatorinsurerinteraction in a structural model
, 2007
"... Abstract: In this paper we provide a new insight of the previous work of Grosen and Jørgensen [2002]. More precisely, firstly, we investigate the impact of regulatory authorities’ rules on the fair value of company’s liabilities and assets. We study how to choose regulation intervention levels in or ..."
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Abstract: In this paper we provide a new insight of the previous work of Grosen and Jørgensen [2002]. More precisely, firstly, we investigate the impact of regulatory authorities’ rules on the fair value of company’s liabilities and assets. We study how to choose regulation intervention levels in order to control the shortfall probability of the issuing company. The fact that the regulation rule depends on the assets ’ volatility implies that “fixed volatility rule” becomes completely useless whenever the insurance company follows a dynamic investment strategy. Therefore, secondly, we study the interaction between the regulator who determines the regulation rule and the insurance company’s risk management. Following the recent work of Ballotta, Haberman and Wang [2005] and the guidelines of the IASB, we develop an analysis of the model error when the insurance company is informed of the regulation rules and trades according to a certain discrete risk management hedging strategy instead of staying passive until the contract’s maturity.
FAST SIMULATION OF EQUITYLINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION
"... In this paper, we consider equitylinked life insurance contracts that give their holder the possibility to surrender their policy before maturity. Such contracts can be valued using simulation methods proposed for the pricing of American options, but the mortality risk must also be taken into acc ..."
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In this paper, we consider equitylinked life insurance contracts that give their holder the possibility to surrender their policy before maturity. Such contracts can be valued using simulation methods proposed for the pricing of American options, but the mortality risk must also be taken into account when pricing such contracts. Here, we use the leastsquares Monte Carlo approach of Longstaff and Schwartz coupled with quasiMonte Carlo sampling and a control variate in order to construct efficient estimators for the value of such contracts. We also show how to incorporate the mortality risk into these pricing algorithms without explicitly simulating it. 1
A note on utility based pricing and asymptotic risk diversification
, 2011
"... In principle, liabilities combining both insurancial risks (e.g. mortality/longevity, crop yield,...) and pure financial risks cannot be priced neither by applying the usual actuarial principles of diversification, nor by arbitragefree replication arguments. Still, it has been often proposed in th ..."
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In principle, liabilities combining both insurancial risks (e.g. mortality/longevity, crop yield,...) and pure financial risks cannot be priced neither by applying the usual actuarial principles of diversification, nor by arbitragefree replication arguments. Still, it has been often proposed in the literature to combine these two approaches by suggesting to hedge a pure financial payoff computed by taking the mean under the historical/objective probability measure on the part of the risk that can be diversified. Not surprisingly, simple examples show that this approach is typically inconsistent for risk adverse agents. We show that it can nevertheless be recovered asymptotically when the number of sold claims goes to infinity and the absolute risk aversion of the agent goes to zero simultaneously. This follows from a general convergence result on utility indifference prices which is valid for both complete and incomplete financial markets. In particular, if the underlying financial market is complete, the limit price corresponds to the hedging cost of the mean payoff. If the financial market is incomplete but the agent behaves asymptotically as an exponential utility maximizer with vanishing risk aversion, we show that the utility indifference price converges to the expectation of the discounted payoff under the minimal entropy martingale measure.
Mortality Derivatives: Valuation and Hedging of the RuinContingent Life Annuity (RCLA)
, 2009
"... This paper analyzes a type of mortality contingentclaim called a ruincontingent life annuity (RCLA). This product fuses together a type of equity put option with a personal longevity call option. The annuitant’s (i.e. long position) payoff from a generic RCLA is $1 of income per year for life, aki ..."
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This paper analyzes a type of mortality contingentclaim called a ruincontingent life annuity (RCLA). This product fuses together a type of equity put option with a personal longevity call option. The annuitant’s (i.e. long position) payoff from a generic RCLA is $1 of income per year for life, akin to a defined benefit pension, but deferred until a prespecified financial diffusion process hits zero. We derive the PDE and relevant boundary conditions satisfied by the RCLA value (i.e. the hedging cost) assuming a complete market where No Arbitrage is possible. We then describe some efficient numerical techniques and provide estimates of a typical RCLA under a variety of realistic parameters. The motivation for studying the RCLA is that it is now embedded in approximately $800 billion worth of U.S. variable annuity (VA) policies which have recently attracted scrutiny from analysts and regulators. 1
Individual Welfare Gains from Deferred LifeAnnuities
"... Individual welfare gains from deferred lifeannuities under stochastic LeeCarter mortality SFB 649 discussion paper, No. 2009,022 Provided in Cooperation with: ..."
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Individual welfare gains from deferred lifeannuities under stochastic LeeCarter mortality SFB 649 discussion paper, No. 2009,022 Provided in Cooperation with: