Indifference Pricing of a GLWB Option in Variable Annuities

We investigate the valuation problem of variable annuities with guaranteed lifelong/lifetime withdrawal benefit (GLWB) options, which give the policyholder the right to withdraw a specified amount as long as he or she lives, regardless of the performance of the investment. We assume the static approach that the policyholder’s withdrawal rate is a constant throughout the life of the contract. We apply the principle of equivalent utility to find the indifference price for a variable annuity with a GLWB contract with an equity-indexed death benefit. Using an exponential utility function, Hamilton-Jacobi-Bellman (HJB) type partial differential equations (PDEs) are derived for the pricing functions. We first assume the mortality is deterministic, and the pricing PDE is solved numerically using a finite difference method. The effects of various parameters are investigated, including the age at inception of the policyholder, withdrawal rate, risk-free rate, and volatility of the underlying asset. We also consider a roll-up option and analyze the effect of delaying the start of the withdrawals. Another pricing PDE is derived with a stochastic mortality, when the force of mortality is modeled with a stochastic differential equation. A finite difference method is used again to solve the pricing PDE numerically, and the sensitivities of the GLWB contracts with respect to the withdrawal rate and the risk-free rate are explored.     

GMWB for life an analysis of lifelong withdrawal guarantees

In this paper, we analyze the latest guarantee feature in the variable annuities market: guaranteed minimum withdrawal benefits for life (GMWB for life) which are also called guaranteed lifelong withdrawal benefits (GLWB). This option gives the client the right to deduct a certain amount annually from the policy??s account value until death??even if a unit-linked account value drops to zero. We show how such products can be analyzed within a general framework presented in Bauer et al. (ASTIN Bull. 38(2):621?C651, 2008). We price the embedded guarantee for different product designs and parameters under both, deterministic and optimal client behavior.     

The valuation of GMWB variable annuities under alternative fund distributions and policyholder behaviours

In this paper, we present a dynamic programming algorithm for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) under a general Lévy processes framework. The GMWB gives the policyholder the right to make periodical withdrawals from her policy account even when the value of this account is exhausted. Typically, the total amount guaranteed for withdrawals coincides with her initial investment, providing then a protection against downside market risk. At each withdrawal date, the policyholder has to decide whether, and how much, to withdraw, or to surrender the contract. We show how different policyholder’s withdrawal behaviours can be modelled. We perform a sensitivity analysis comparing the numerical results obtained for different contractual and market parameters, policyholder behaviours and different types of Lévy processes.     

Pricing of guaranteed minimum withdrawal benefits in variable annuities under stochastic volatility,stochastic interest rates and stochastic mortality via the componentwise splitting method

This paper values guaranteed minimum withdrawal benefit (GMWB) riders embedded in variable annuities assuming that the underlying fund dynamics evolve under the influence of stochastic interest rates, stochastic volatility, stochastic mortality and equity risk. The valuation problem is formulated as a partial differential equation (PDE) which is solved numerically by employing the operator splitting method. Sensitivity analysis of the fair guarantee fee is performed with respect to various model parameters. We find that (i) the fair insurance fee charged by the product provider is an increasing function of the withdrawal rate; (ii) the GMWB price is higher when stochastic interest rates and volatility are incorporated in the model, compared to the case of static interest rates and volatility; (iii) the GMWB price behaves non-monotonically with changing volatility of variance parameter; (iv) the fair fee increases with increasing volatility of interest rates parameter, and increasing correlation between the underlying fund and the interest rates; (v) the fair fee increases when the speed of mean-reversion of stochastic volatility or the average long-term volatility increases; (vi) the GMWB fee decreases when the speed of mean-reversion of stochastic interest rates or the average long-term interest rates increase. We investigate both static and dynamic (optimal) policyholder's withdrawal behaviours; we present the optimal withdrawal schedule as a function of the withdrawal account and the investment account for varying volatility and interest rates. When incorporating stochastic mortality, we find that its impact on the fair guarantee fee is rather small. Our results demonstrate the importance of correct quantification of risks embedded in GMWBs and provide guidance to product providers on optimal hedging of various risks associated with the contract.     

保险市场逆向选择的信号传递博弈研究

在保险市场中,投保人比保险人更了解自己的风险状况,保险双方之间的这种信息不对称难以避免地会产生逆向选择问题,于是在保险人混同定价的情形下,低风险投保人要承受过高的费率而受损,高风险投保人因保险成本过低而削弱控制风险的激励,导致整个市场资源配置低效甚至因逆向选择螺旋而崩溃。通过引入信号传递机制来实现保险市场的分离定价,从...     

Willow tree algorithms for pricing Guaranteed Minimum Withdrawal Benefits under jump-diffusion and CEV models

This paper presents the willow tree algorithms for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB), where the underlying fund dynamics evolve under the Merton jump-diffusion process or constant-elasticity-of-variance (CEV) process. The GMWB rider gives the policyholder the right to make periodic withdrawals from his policy account throughout the life of the contract. The dynamic nature of the withdrawal policy allows the policyholder to decide how much to withdraw on each withdrawal date, or even to surrender the contract. For numerical valuation of the GMWB rider, we use willow tree algorithms that adopt more effective placement of the lattice nodes based on better fitting of the underlying fund price distribution. When compared with other numerical algorithms, like the finite difference method and fast Fourier transform method, the willow tree algorithms compute GMWB prices with significantly less computational time to achieve a similar level of numerical accuracy. The design of our pricing algorithm also includes an efficient search method for the optimal dynamic withdrawal policies. We perform sensitivity analysis of various model parameters on the prices and fair participating fees of the GMWB riders. We also examine the effectiveness of delta hedging when the fund dynamics exhibit various jump levels.     

What if Variable Annuity Policyholders With Guaranteed Lifelong Withdrawal Benefit Were Rational?

This article examines the lapse risk inherent to the guaranteed lifelong withdrawal benefit option embedded in a variable annuity product valuated from a pure derivatives perspective, that is, as a Bermudian option given to the policyholder. We assume rational behavior and quantify the potential impact of the lapse risk, defined as the difference between no lapse and optimal lapsing. We develop a sensitivity analysis that shows how the value of the product varies with the key parameters, and calculate the fair fee using Monte Carlo simulations. Empirical analyses are performed and numerical results are provided.     

Risk based capital for guaranteed minimum withdrawal benefit

The guaranteed minimum withdrawal benefit (GMWB), which is sold as a rider to variable annuity contracts, guarantees the return of total purchase payment regardless of the performance of the underlying investment funds. The valuation of GMWB has been extensively covered in the previous literature, but a more challenging problem is the computation of the risk based capital for risk management and regulatory reasons. One needs to find the tail distribution of the profit–loss function, which differs from its expected payoff required for pricing the GMWB contract. GMWB has embedded two option-like features: Management fees are proportional to the current value of the policyholder’s account which results in an average price of the account. Thus the contract resembles an Asian option. However, the fees are charged only up to the time of the account hitting zero which resembles a barrier option payoff. Thus the GMWB is mathematically more complicated than Asian or barrier options traded on the financial markets. To the authors’ best knowledge, this is the first paper in the literature to formulate and analyse profit–loss distribution using PDE methods of such a product with intricate option-like features. Our approach is much more efficient than the current market practice of rather intensive and expensive Monte Carlo simulations due to the lack of samples for extreme cases.     

Cheaper by the bundle: The interaction of frictions and option exercise in variable annuities

Typical Variable Annuity products combine complex baseline contracts at substantial fees with optional guarantees. We argue this product design aligns with the benefits of bundling to the provider, to the extent that the baseline option features can reduce total replication value. This is possible due to market frictions, and particularly taxation rules, affecting policyholder exercise behavior. We demonstrate the relevance of this mechanism in the context of popular withdrawal guarantees, both theoretically and empirically. Specifically, we show that in the presence of personal taxes, adding on a common death benefit at baseline may decrease the total contract value to the provider.     

Generic pricing of FX,inflation and stock options under stochastic interest rates and stochastic volatility

We consider the pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility, for which we use a generic multi-currency framework. We allow for a general correlation structure between the drivers of the volatility, the inflation index, the domestic (nominal) and the foreign (real) rates. Having the flexibility to correlate the underlying FX/inflation/stock index with both stochastic volatility and stochastic interest rates yields a realistic model that is of practical importance for the pricing and hedging of options with a long-term exposure. We derive explicit valuation formulas for various securities, such as vanilla call/put options, forward starting options, inflation-indexed swaps and inflation caps/floors. These vanilla derivatives can be valued in closed form under Schöbel and Zhu [Eur. Finance Rev., 1999, 4, 23–46] stochastic volatility, whereas we devise an (Monte Carlo) approximation in the form of a very effective control variate for the general Heston [Rev. Financial Stud., 1993, 6, 327–343] model. Finally, we investigate the quality of this approximation numerically and consider a calibration example to FX and inflation market data.     

保单持有人退保行为对变额年金利益保证风险对冲的影响研究

给出保单持有人退保行为影响下的变额年金定价模型和对冲模型,当保单持有人分别采取无退保、固定退保和动态退保三种行为策略时,基于包含最低身故利益保证、最低满期利益保证和最低提取利益保证的三种不同的变额年金,运用蒙特卡罗模拟测试出保单持有人采取不同的退保策略对不同利益保证的变额年金风险对冲有着显著的不同影响。     

Immediate Annuity Pricing in the Presence of Unobserved Heterogeneity

Abstract

One of the acknowledged difficulties with pricing immediate annuities is that underwriting the annuitantis life is the exception rather than the rule. In the absence of underwriting, the price paid for a life-contingent annuity is the same for all sales at a given age. This exposes the market (insurance company and potential policyholder alike) to antiselection. The insurance company worries that only the healthiest people choose a life-contingent annuity and therefore adjust mortality accordingly. The potential policyholders worry that they are not being compensated for their relatively poor health and choose not to purchase what would otherwise be a very beneficial product.

This paper develops a model of underlying, unobserved health. Health is a state variable that follows a first-order Markov process. An individual reaches the state “death” either by accident from any health state or by progressively declining health state. Health state is one-dimensional, in the sense that health can either “improve” or “deteriorate” by moving farther from or close to the “death” state, respectively. The probability of death in a given year is a function of health state, not of age. Therefore, in this model a person is exactly as old as he or she feels.

I first demonstrate that a multistate, ageless Markov model can match the mortality patterns in the common annuity mortality tables. The model is extended to consider several types of mortality improvements: permanent through decreasing probability of deteriorating health, temporary through improved distribution of initial health state, and plateau through the effects of past health improvements.

I then construct an economic model of optimal policyholder behavior, assuming that the policyholder either knows his or her health state or has some limited information. the value of mortality risk transfer through purchasing a life-contingent annuity is estimated for each health state under various risk-aversion parameters. Given the economic model for optimal purchasing of annuities, the value of underwriting (limited information about policyholder health state) is demonstrated.     

Improving risk classification and ratemaking using mixture-of-experts models with random effects

In the underwriting and pricing of nonlife insurance products, it is essential for the insurer to utilize both policyholder information and claim history to ensure profitability and proper risk management. In this paper, we apply a flexible regression model with random effects, called the Mixed Logit-weighted Reduced Mixture-of-Experts, which leverages both policyholder information and their claim history, to categorize policyholders into groups with similar risk profiles, and to determine a premium that accurately captures the unobserved risks. Estimates of model parameters and the posterior distribution of random effects can be obtained by a stochastic variational algorithm, which is numerically efficient and scalable to large insurance portfolios. Our proposed framework is shown to outperform the classical benchmark models (Logistic and Lognormal GL(M)M) in terms of goodness-of-fit to data, while offering intuitive and interpretable characterization of policyholders' risk profiles to adequately reflect their claim history.     

Asset pricing under information with stochastic volatility

Based on a general specification of the asset specific pricing kernel, we develop a pricing model using an information process with stochastic volatility. We derive analytical asset and option pricing formulas. The asset prices in this rational expectations model exhibit crash-like, strong downward movements. The resulting option pricing formula is consistent with the strong negative skewness and high levels of kurtosis observed in empirical studies. Furthermore, we determine credit spreads in a simple structural model.      

Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results

Stochastic volatility (SV) and local stochastic volatility (LSV) processes can be used to model the evolution of various financial variables such as FX rates, stock prices and so on. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes. Many issues remain, though, including the efficacy of the standard alternating direction implicit (ADI) numerical methods for solving SV and LSV pricing problems. In general, the amount of required computations for these methods is very substantial. In this paper, we address some of these issues and propose a viable alternative to the standard ADI methods based on Galerkin-Ritz ideas. We also discuss various approaches to solving the corresponding pricing problems in a semi-analytical fashion. We use the fact that in the zero correlation case some of the pricing problems can be solved analytically, and develop a closed-form series expansion in powers of correlation. We perform a thorough benchmarking of various numerical solutions by using analytical and semi-analytical solutions derived in the paper.     

The option CAPM and the performance of hedge funds

We evaluate the investment performance of hedge funds using an asset pricing model that is characterized by a piecewise-linear stochastic discount factor, and which we estimate using the generalized method of moments by minimizing the Hansen–Jagannathan distance. Our results show that, once non-linearities and public information are taken into account, there is only evidence of positive performance for the overall hedge fund index, equity-market neutral strategy and the global macro strategy.     

Speculative trading and stock returns: A stochastic dominance analysis of the Chinese A-share market

The pricing of A-shares in China has long puzzled financial economists. This paper applies recent tests of stochastic dominance (SD) to examine whether differences in the return distributions of A- and B-shares in China are consistent with market efficiency. As SD is nonparametric, market efficiency can be examined without the joint test problem arising from misspecifications in the asset pricing benchmark. Our results show A-shares have second-order dominated B-shares from 1996 to 2005. This dominance was most significant during the market segmentation period, but has continued, albeit to a lesser extent even after the B-share market was opened to local investors in 2001. Our results are robust to using residual returns from an international asset pricing model instead of raw returns. We conclude that the superior performance of A-shares cannot be attributed to risk. The results are more likely due to a return bias caused by intense speculation among retail individuals under limited arbitrage.     

动态资本资产定价理论评述

本文主要讨论了动态资产定价理论的产生和发展.默顿和布里登使用贝尔曼开创的动态规划方法和伊藤随机分析技术,重新考察在由随机过程驱动的不确定环境下,个人如何连续地做出消费/投资决策,使得终身效用最大化.无须单期框架中的严格假定,他们也获得了连续时间跨期资源配置的一般均衡模型--时际资产定价模型(ICAPM)以及消费资产定价模型(CCAPM).这些工作开启了连续时间金融方法论的新时代.     

Self-Annuitization and Ruin in Retirement

Abstract

At retirement, most individuals face a choice between voluntary annuitization and discretionary management of assets with systematic withdrawals for consumption purposes. Annuitization–buying a life annuity from an insurance company–assures a lifelong consumption stream that cannot be outlived, but it is at the expense of a complete loss of liquidity. On the other hand, discretionary management and consumption from assets–self-annuitization–preserves flexibility but with the distinct risk that a constant standard of living will not be maintainable.

In this paper we compute the lifetime and eventual probability of ruin (PoR) for an individual who wishes to consume a fixed periodic amount–a self-constructed annuity–from an initial endowment invested in a portfolio earning a stochastic (lognormal) rate of return. The lifetime PoR is the probability that net wealth will hit zero prior to a stochastic date of death. The eventual PoR is the probability that net wealth will ever hit zero for an infinitely lived individual.

We demonstrate that the probability of ruin can be represented as the probability that the stochastic present value (SPV) of consumption is greater than the initial investable wealth. The lifetime and eventual probabilities of ruin are then obtained by evaluating one minus the cumulative density function of the SPV at the initial wealth level. In that eventual case, we offer a precise analytical solution because the SPV is known to be a reciprocal gamma distribution. For the lifetime case, using the Gompertz law of mortality, we provide two approximations. Both involve “moment matching” techniques that are motivated by results in Arithmetic Asian option pricing theory. We verify the accuracy of these approximations using Monte Carlo simulations. Finally, a numerical case study is provided using Canadian mortality and capital market parameters. It appears that the lifetime probability of ruin–for a consumption rate that is equal to the life annuity payout–is at its lowest with a well-diversified portfolio.