Optimal hedging strategies for multi-period guarantees in the presence of transaction costs: A stochastic programming approach

Multi-period guarantees are often embedded in life insurance contracts. In this paper we consider the problem of hedging these multi-period guarantees in the presence of transaction costs. We derive the hedging strategies for the cheapest hedge portfolio for a multi-period guarantee that with certainty makes the insurance company able to meet the obligations from the insurance policies it has issued. We find that by imposing transaction costs, the insurance company reduces the rebalancing of the hedge portfolio. The cost of establishing the hedge portfolio also increases as the transaction cost increases. For the multi-period guarantee there is a rather large rebalancing of the hedge portfolio as we go from one period to the next. By introducing transaction costs we find the size of this rebalancing to be reduced. Transaction costs may therefore be one possible explanation for why we do not see the insurance companies performing a large rebalancing of their investment portfolio at the end of each year.     

Hedging Large Portfolios of Options in Discrete Time*

The problem studied is that of hedging a portfolio of options in discrete time where underlying security prices are driven by a combination of idiosyncratic and systematic risk factors. It is shown that despite the market incompleteness introduced by the discrete time assumption, large portfolios of options have a unique price and can be hedged without risk. The nature of the hedge portfolio in the limit of large portfolio size is substantially different from its continuous time counterpart. Instead of linearly hedging the total risk of each option separately, the correct portfolio hedge in discrete time eliminates linear as well as second and higher order exposures to the systematic risk factors only. The idiosyncratic risks need not be hedged, but disappear through diversification. Hedging portfolios of options in discrete time thus entails a trade‐off between dynamic and cross‐sectional hedging errors. Some computations are provided on the outcome of this trade‐off in a discrete‐time Black–Scholes world.     

Risk-minimization for life insurance liabilities with basis risk

In this paper we study the hedging of typical life insurance payment processes in a general setting by means of the well-known risk-minimization approach. We find the optimal risk-minimizing strategy in a financial market where we allow for investments in a hedging instrument based on a longevity index, representing the systematic mortality risk. Thereby we take into account and model the basis risk that arises due to the fact that the insurance company cannot perfectly hedge its exposure by investing in a hedging instrument that is based on the longevity index, not on the insurance portfolio itself. We also provide a detailed example within the context of unit-linked life insurance products where the dependency between the index and the insurance portfolio is described by means of an affine mean-reverting diffusion process with stochastic drift.     

Quantile hedging for equity-linked contracts

We consider an equity-linked contract whose payoff depends on the lifetime of the policy holder and the stock price. We provide the best strategy for an insurance company assuming limited capital for the hedging. The main idea of the proof consists in reducing the construction of such strategies for a given claim to a problem of superhedging for a modified claim, which is the solution to a static optimization problem of the Neyman-Pearson type. This modified claim is given via some sets constructed in an iterative way. Some numerical examples are also given.     

Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies

The present paper investigates the net loss of a life insurance company issuing equity-linked pure endowments in the case of periodic premiums. Due to the untradability of the insurance risk which affects both the in- and outflow side of the company, the issued insurance claims cannot be hedged perfectly. Furthermore, we consider an additional source of incompleteness caused by trading restrictions, because in reality the hedging of the contingent claims is more likely to occur at discrete times. Based on Møller [Møller, T., 1998. Risk-minimizing hedging strategies for unit-linked life insurance contracts. Astin Bull. 28, 17–47], we particularly examine the situation, where the company applies a time-discretized risk-minimizing hedging strategy. Through an illustrative example, we observe numerically that only a relatively small reduction in ruin probabilities is achieved with the use of the discretized originally risk-minimizing strategy because of the accumulated extra duplication errors caused by discretizing. However, the simulated results are highly improved if the hedging model instead of the hedging strategy is discretized. For this purpose, Møller’s [Møller, T., 2001. Hedging equity-linked life insurance contracts. North Amer. Actuarial J. 5 (2), 79–95] discrete-time (binomial) risk-minimizing strategy is adopted.     

Pricing and hedging of variable annuities with state-dependent fees

We investigate the problem of pricing and hedging variable annuity contracts for which the fee deducted from the policyholder’s account depends on the account value. It is believed that state-dependent fees are beneficial to policyholders and insurers since they reduce policyholders’ incentives to lapse the policies and match the costs incurred by policyholders with the pay-offs received from embedded guarantees. We consider an incomplete financial market which consists of two risky assets modelled with a two-dimensional Lévy process. One of the assets is a security which can be traded by the insurer, and the second asset is a security which is the underlying fund for the variable annuity contract. In our model we derive an equation from which the fee for the guaranteed benefit can be calculated and we characterize a strategy which allows the insurer to hedge the benefit. To solve the pricing and hedging problem in an incomplete financial market we apply a quadratic objective.     

Fair valuation of insurance contracts under Lévy process specifications

The valuation of options embedded in insurance contracts using concepts from financial mathematics (in particular, from option pricing theory), typically referred to as fair valuation, has recently attracted considerable interest in academia as well as among practitioners. The aim of this article is to investigate the valuation of participating and unit-linked life insurance contracts, which are characterized by embedded rate guarantees and bonus distribution rules. In contrast to the existing literature, our approach models the dynamics of the reference portfolio by means of an exponential Lévy process. Our analysis sheds light on the impact of the dynamics of the reference portfolio on the fair contract value for several popular types of insurance policies. Moreover, it helps to assess the potential risk arising from misspecification of the stochastic process driving the reference portfolio.     

Corporate valuation,capital structure and risk management: A stochastic DCF approach

In this article we discuss a general stochastic framework for designing corporate investment, financing and risk management strategies for financially constrained firms. The strategy entailing the highest benefits for shareholders is considered to be the optimal strategy. This paper focuses on a simulation of present value distributions of the capital positions of a company, explicitly taking into account the risk of fluctuations in future cash flow as well as the risk of insolvency. The present value distribution of equity is used as a central instrument for evaluation of shareholder benefits. Expected present values are also computed. The investment and financing policy of the company pursued at the time of the valuation is reflected in certain global model parameters, which themselves influence the future profit distribution policy of the company. The main parameters are the extent of debt, the annual debt funding requirements, the average earnings power of the company – expressed as an expected annual return on total capital – and the risk of annual earnings – expressed as the standard deviation of the annual return on total capital. An explicit illustration of the volatility risk and default risk seems not only to be a suitable way of illustrating the impact of capital structure on corporate value. Such an depiction may also provide answers to the question of the link between hedging and enterprise value. This paper highlights the fact that investment, finance and hedging strategies should go hand in hand.     

实物期权定价与风险对冲方法研究

实物期权定价面临的一个主要问题是其基本资产不可交易问题，在这种情况下，通常的解决办法是在市场中寻找一个与该基本资产最为相关的可交易资产，利用可交易资产的价格信息来对特定实物期权进行定价和风险对冲．利用随机动态规划法，本文得到基本资产不可交易时实物期权的最优风险对冲策略，在一定条件下还可以得到近似定价．     

The role of index bonds in universal currency hedging

This study examines the demand for index bonds and their role in hedging risky asset returns against currency risks in a complete market where equity is not hedged against inflation risk. Avellaneda's uncertain volatility model with non-constant coefficients to describe equity price variation, forward price variation, index bond price variation and rate of inflation, together with Merton's intertemporal portfolio choice model, are utilized to enable an investor to choose an optimal portfolio consisting of equity, nominal bonds and index bonds when the rate of inflation is uncertain. A hedge ratio is universal if investors in different countries hedge against currency risk to the same extent. Three universal hedge ratios (UHRs) are defined with respect to the investor's total demand for index bonds, hedging risky asset returns (i.e. equity and nominal bonds) against currency risk, which are not held for hedging purposes. These UHRs are hedge positions in foreign index bond portfolios, stated as a fraction of the national market portfolio. At equilibrium all the three UHRs are comparable to Black's corrected equilibrium hedging ratio. The Cameron-Martin-Girsanov theorem is applied to show that the Radon-Nikodym derivative given under a P -martingale, the investor's exchange rate (product of the two currencies) is a martingale. Therefore the investors can agree on a common hedging strategy to trade exchange rate risk irrespective of investor nationality. This makes the choice of the measurement currency irrelevant and the hedge ratio universal without affecting their values.     

Polynomial diffusion models for life insurance liabilities

In this paper we study the pricing and hedging problem of a portfolio of life insurance products under the benchmark approach, where the reference market is modelled as driven by a state variable following a polynomial diffusion on a compact state space. Such a model can be used to guarantee not only the positivity of the OIS short rate and the mortality intensity, but also the possibility of approximating both pricing formula and hedging strategy of a large class of life insurance products by explicit formulas.     

Optimal investment strategies for participating contracts

Participating contracts are popular insurance policies, in which the payoff to a policyholder is linked to the performance of a portfolio managed by the insurer. We consider the portfolio selection problem of an insurer that offers participating contracts and has an S-shaped utility function. Applying the martingale approach, closed-form solutions are obtained. The resulting optimal strategies are compared with portfolio insurance hedging strategies (CPPI and OBPI). We also study numerical solutions of the portfolio selection problem with constraints on the portfolio weights.     

Hedging mortality/longevity risks of insurance portfolios for life insurer/annuity provider and financial intermediary

In this paper, we propose two risk hedge schemes in which a life insurer (an annuity provider) can transfer mortality (longevity) risk of a portfolio of life (annuity) exposures to a financial intermediary by paying the hedging premium of a mortality-linked security. The optimal units of the mortality-linked security which maximize hedge effectiveness for a life insurer (an annuity provider) can be derived as closed-form formulas under the risk hedge schemes. Numerical illustrations show that the risk hedge schemes can significantly hedge the downside risk of loss due to mortality (longevity) risk for the life insurer (annuity provider) under some stochastic mortality models. Besides, finding an optimal weight of a portfolio of life and annuity business, the financial intermediary can reduce the sensitivity to mortality rates but the model risk; a security loading may be imposed on the hedge premium for a higher probability of gain to compensate the financial intermediary for the inevitable model risk.     

Hedging of unit-linked life insurance contracts with unobservable mortality hazard rate via local risk-minimization

In this paper we investigate the local risk-minimization approach for a combined financial-insurance model where there are restrictions on the information available to the insurance company. In particular we assume that, at any time, the insurance company may observe the number of deaths from a specific portfolio of insured individuals but not the mortality hazard rate. We consider a financial market driven by a general semimartingale and we aim to hedge unit-linked life insurance contracts via the local risk-minimization approach under partial information. The Föllmer–Schweizer decomposition of the insurance claim and explicit formulas for the optimal strategy for pure endowment and term insurance contracts are provided in terms of the projection of the survival process on the information flow. Moreover, in a Markovian framework, this leads to a filtering problem with point process observations.     

Capital Asset Pricing Model (CAPM) with drawdown measure

The notion of drawdown is central to active portfolio management. Conditional Drawdown-at-Risk (CDaR) is defined as the average of a specified percentage of the largest drawdowns over an investment horizon and includes maximum and average drawdowns as particular cases. The necessary optimality conditions for a portfolio optimization problem with CDaR yield the capital asset pricing model (CAPM) stated in both single and multiple sample-path settings. The drawdown beta in the CAPM has a simple interpretation and is evaluated for hedge fund indices from the HFRX database in the single sample-path setting. Drawdown alpha is introduced similarly to the alpha in the classical CAPM and is evaluated for the same hedge fund indices. Both drawdown beta and drawdown alpha are used to prioritize hedge fund strategies and to identify instruments for hedging against market drawdowns.     

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.     

Analysis of survivorship life insurance portfolios with stochastic rates of return

A general portfolio of survivorship life insurance contracts is studied in a stochastic rate of return environment with a dependent mortality model. Two methods are used to derive the first two moments of the prospective loss random variable. The first one is based on the individual loss random variables while the second one studies annual stochastic cash flows. The distribution function of the present value of future losses at a given valuation time is derived. For illustrative purposes, an AR(1) process is used to model the stochastic rates of return, and the future lifetimes of a couple are assumed to follow a copula model. The effects of the mortality dependence, the portfolio size and the policy type, as well as the impact of investment strategies on the riskiness of portfolios of survivorship life insurance policies are analyzed by means of moments and probability distributions.     

Cramér-Lundberg型下保险公司的最优投资和混合再保

假定保险公司既可以投资在风险资产上,同时又允许混合再保险.用经典的Cramér-Lundberg模型来近似保险公司的盈余过程,考虑了在破产概率最小限制下保险公司的最优投资和再保策略满足的HJB方程,证明了解的存在性和最优性,并对最优策略下的破产概率进行了近似估计.     

随机利率及随机收益保证下的投资策略

考虑随机利率环境及随机收益保证下基金经理的投资组合问题。利用鞅方法,得到了最优投资策略的显性解。结论表明,最优投资策略包括三个部分：投机策略、利率套期保值策略以及随机收益保证的复制策略,且该最优策略等价于将一部分资金投资于确保终端时刻获得最低收益的基准组合,而剩余资金则依照无保证情况下的最优策略进行投资。     

A risk-based model for the valuation of pension insurance

In the US, defined benefit plans are insured by the Pension Benefit Guaranty Corporation (PBGC). Taking account of the fact that the PBGC covers only the residual deficits of the pension fund the sponsoring company is unable to cover and that the plans can be prematurely terminated, we consider a model that accounts for the joint dynamics of the pension fund’s and sponsoring firm’s assets in order to effectively determine the risk-based pension premium for the insurance provided by the PBGC. We obtain a closed-form pricing formula for this risk-based premium. Its magnitude depends highly on the investment portfolio of the pension fund and of the sponsoring company as well as the correlation between these two portfolios.