Optimal Portfolio Choice with Annuitization

We study the optimal consumption and portfolio choice problem over an individual's life-cycle taking into account annuity risk at retirement. Optimally, the investor allocates wealth at retirement to nominal, inflation-linked, and variable annuities and conditions this choice on the state of the economy.We also consider the case in which there are, either for behavioral or institutional reasons, limitations in the types of annuities that are available at retirement.Subsequently, we determine how the investor optimally anticipates annuitization before retirement.We find that i) using information on term structure variables and risk premia significantly improves the optimal annuity choice, ii) restricting the annuity menu to nominal or inflation-linked annuities is costly for both conservative and more aggressive investors, and iii) adjustments in the optimal investment strategy before retirement induced by the annuity demand due to inflation risk and time-varying risk premia are economically significant.This holds as well for sub-optimal annuity choices.The adjustment to hedge real interest rate risk is negligible.We estimate that the welfare costs of not taking these three factors into account at retirement are 9% for an individual with an average risk aversion ( = 5).Not hedging annuity risk before retirement causes an additional welfare costs between 1% and 13%, depending on the annuitization strategy implemented at retirement.optimal life-cycle portfolio choice;annuity risk

Efficient Hedging and Pricing of Equity-Linked Life Insurance Contracts on Several Risky Assets

The authors use the efficient hedging methodology for optimal pricing and hedging of equitylinked life insurance contracts whose payoff depends on the performance of several risky assets. In particular, they consider a policy that pays the maximum of the values of n risky assets at some maturity date T , provided that the policyholder survives to T . Such contracts incorporate financial risk, which stems from the uncertainty about future prices of the underlying financial assets, and insurance risk, which arises from the policyholder's mortality. The authors show how efficient hedging can be used to minimize expected losses from imperfect hedging under a particular risk preference of the hedger. They also prove a probabilistic result, which allows one to calculate analytic pricing formulas for equity-linked payoffs with n risky assets. To illustrate its use, explicit formulas are derived for optimal prices and expected hedging losses for payoffs with two risky assets. Numerical examples highlighting the implications of efficient hedging for the management of financial and insurance risks of equity-linked life insurance policies are also provided.Financial markets;

Managing Food Industry Business and Financial Risks with Commodity-Linked Credit Instruments

This paper reviews the use and structure of commodity-linked credit instruments. It is argued that in the absence of contingent markets food firms face increasing financial risk reduced investment, and limited access to debt markets. One strategy is to issue commodity-linked credit whose payment structure is linked to the price of an underlying commodity. In some cases, a commodity-linked bond (CLB) can be structured to provide an incentive to investors by sharing in profit gains. If the goal is to hedge financial risks, CLB's can also be constructed that reduce the loan principle or coupons depending on price movements.Agribusiness, Risk and Uncertainty,

Real World Pricing of Long Term Contracts

Long dated contingent claims are relevant in insurance, pension fund management and derivative pricing. This paper proposes a paradigm shift in the valuation of long term contracts, away from classical no-arbitrage pricing towards pricing under the real world probability measure. In contrast to risk neutral pricing, the long term excess return of the equity market, known as the equity premium, is taken into account. Further, instead of the savings account, the numeraire portfolio isused, as the fundamental unit of value in the analysis. The numeraire portfolio is the strictly positive, tradable portfolio that when used as benchmark makes all benchmarked non negative portfolios supermartingales, which means intuitively that these are downward trending or at least trendless. Furthermore, the benchmarked real world price of a benchmarked claimis defined to be its real world conditional expectation. This yields the minimal possible price for its hedgable part and minimizes the variance of the benchmarked hedge error. The pooled total benchmarked replication error of a large insurance company or bank essentially vanishes due to diversification. Interestingly, in long terml iability and asset valuation, real world pricing can lead to significantly lower prices than suggested by classical no-arbitragea rguments. Moreover, since the existence of some equivalent risk neutral probability measure is no longer required, a wider and more realistic modeling framework is available for exploration. Classical actuarial and risk neutral pricing emerge as special cases of real world pricing.long term pricing; real world pricing; risk neutral pricing; numeraire portfolio; law of the minimal price; strong arbitrage; hedges imulation; diversification; liquidity premium

A unified pricing of variable annuity guarantees under the optimal stochastic control framework

In this paper, we review pricing of variable annuity living and death guarantees offered to retail investors in many countries. Investors purchase these products to take advantage of market growth and protect savings. We present pricing of these products via an optimal stochastic control framework, and review the existing numerical methods. For numerical valuation of these contracts, we develop a direct integration method based on Gauss-Hermite quadrature with a one-dimensional cubic spline for calculation of the expected contract value, and a bi-cubic spline interpolation for applying the jump conditions across the contract cashflow event times. This method is very efficient when compared to the partial differential equation methods if the transition density (or its moments) of the risky asset underlying the contract is known in closed form between the event times. We also present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a benchmark for practitioners and researchers developing pricing of variable annuity guarantees.Comment: Keywords: variable annuity, guaranteed living and death benefits, guaranteed minimum accumulation benefit, optimal stochastic control, direct integration metho

Loss Analysis of a Life Insurance Company Applying Discrete-time Risk-minimizing Hedging Strategies

In this paper, we consider the net loss of a life insurance company issuing identical equity-linked pure endowment contracts in the case of periodic premiums. Under this construction, financial risks as well as the mortality risk are included. Based on Møller (1998), we particularly investigate the situation where the company applies a time-discretized risk-minimizing hedging strategy, i.e., a trading restriction is imposed on a continuous-time risk-minimizing strategy. Therefore, the considered model is incomplete where the incompleteness results not only from the mortality risk but also from the trading restrictions. Through an illustrative example, it is observed from the simulations that a substantial reduction in the ruin probability is achieved by using the time-discretized risk-minimizing strategy. However, as the hedging frequency is set higher, this advantage almost disappears, because a higher frequency leads to more hedging errors which constitute a vital part of the hedger’s net loss. In order to improve the simulated results, another type of discrete-time risk-minimizing strategy is taken into consideration. It is obtained by discretizing the hedging model instead of the hedging strategy. For this purpose, Møller’s (2001) discrete-time (binomial) risk-minimizing strategy is adopted. For both strategies, a number of sensitivity analyses are carried out, e.g. how the ruin probability changes with the fair combination of the minimum interest rate guarantee and the participation rate.Net Loss, Discrete-time Risk-minimizing Hedging Strategies, Pure Endowment Equity-linked Life Insurance

Optimal Portfolio Choice with Annuitization

We study the optimal consumption and portfolio choice problem over an individual's life-cycle taking into account annuity risk at retirement. Optimally, the investor allocates wealth at retirement to nominal, inflation-linked, and variable annuities and conditions this choice on the state of the economy.We also consider the case in which there are, either for behavioral or institutional reasons, limitations in the types of annuities that are available at retirement.Subsequently, we determine how the investor optimally anticipates annuitization before retirement.We find that i) using information on term structure variables and risk premia significantly improves the optimal annuity choice, ii) restricting the annuity menu to nominal or inflation-linked annuities is costly for both conservative and more aggressive investors, and iii) adjustments in the optimal investment strategy before retirement induced by the annuity demand due to inflation risk and time-varying risk premia are economically significant.This holds as well for sub-optimal annuity choices.The adjustment to hedge real interest rate risk is negligible.We estimate that the welfare costs of not taking these three factors into account at retirement are 9% for an individual with an average risk aversion ( = 5).Not hedging annuity risk before retirement causes an additional welfare costs between 1% and 13%, depending on the annuitization strategy implemented at retirement

Worst-Case Valuation of Equity-Linked Products Using Risk-Minimizing Strategies

The global market for life insurance products has been stable over the years. However, equity-linked products which form about �fifteen percent of the total life insurance market has experienced a decline in premiums written. The impact of model risk when hedging these investment guarantees has been found to be significant�. We propose a framework to determine the worst case value of an equity-linked product through partial hedging using quantile and conditional value-at-risk measures. The model integrates both the mortality and the financial risk associated with these products to estimate the value as well as the hedging strategy. We rely on robust optimization techniques for the worst case hedging strategy. To demonstrate the versatility of the framework, we present numerical examples of point-to-point equity-indexed annuities in multinomial lattice dynamics