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.     

Portfolios with fuzzy returns: Selection strategies based on semi-infinite programming

This paper provides new models for portfolio selection in which the returns on securities are considered fuzzy numbers rather than random variables. The investor's problem is to find the portfolio that minimizes the risk of achieving a return that is not less than the return of a riskless asset. The corresponding optimal portfolio is derived using semi-infinite programming in a soft framework. The return on each asset and their membership functions are described using historical data. The investment risk is approximated by mean intervals which evaluate the downside risk for a given fuzzy portfolio. This approach is illustrated with a numerical example.     

The mean-absolute deviation portfolio selection problem with interval-valued returns

In real-world investments, one may care more about the future earnings than the current earnings of the assets. This paper discusses the uncertain portfolio selection problem where the asset returns are represented by interval data. Since the parameters are interval valued, the gain of returns is interval valued as well. According to the concept of the mean-absolute deviation function, we construct a pair of two-level mathematical programming models to calculate the lower and upper bounds of the investment return of the portfolio selection problem. Using the duality theorem and applying the variable transformation technique, the pair of two-level mathematical programs is transformed into a conventional one-level mathematical program. Solving the pair of mathematical programs produces the interval of the portfolio return of the problem. The calculated results conform to an essential idea in finance and economics that the greater the amount of risk that an investor is willing to take on the greater the potential return.     

Indifference pricing of a life insurance portfolio with risky asset driven by a shot-noise process

In this paper, we investigate the pricing problem for a portfolio of life insurance contracts where the life contingent payments are equity-linked depending on the performance of a risky stock or index. The shot-noise effects are incorporated in the modeling of stock prices, implying that sudden jumps in the stock price are allowed, but their effects may gradually decline over time. The contracts are priced using the principle of equivalent utility. Under the assumption of exponential utility, we find the optimal investment strategy and show that the indifference premium solves a non-linear partial integro-differential equation (PIDE). The Feynman–Kač form solutions are derived for two special cases of the PIDE. We further discuss the problem for the asymptotic shot-noise process, and find the probabilistic representation of the indifference premium. We also provide some numerical examples and analyze parameter sensitivities for the results obtained in this paper.     

Managing a portfolio of long term service agreements

Long-term service agreements (LTSAs) for the maintenance of capital-intensive equipments such as gas turbines and aircraft engines are gaining wide acceptance. A typical LTSA contract spans over a period of around 10 years making a manufacturer fully responsible for maintaining the customer equipment. In this paper, we address the management of a portfolio of such contracts from the manufacturer’s perspective. The goal is to meet all the service requirements imposed by the contracts while minimizing total cost incurred. We develop a deterministic integer programming model to generate the optimal maintenance schedules that minimize the total portfolio cost. We then propose two heuristic algorithms for the problem.     

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.     

Pricing of long dated equity-linked life insurance contracts

This article adopts an approach to pricing of equity-linked life insurance contracts, which only requires the existence of the numéraire portfolio. An equity-linked life insurance contract is equivalent to a sum of the guaranteed amount and the value of an option on the equity index with some mortality risk attached. The numéraire portfolio equals the growth optimal portfolio and is used as numéraire or benchmark, where the real-world probability measure is taken as pricing measure. To obtain tractable solutions the short rate is modelled as a quadratic form of some Gaussian factor processes. Furthermore, the dynamics of the mortality rate is modelled as a threshold life table. The dynamics of the discounted equity market index or benchmark is modelled by a time transformed squared Bessel process. The equity-linked life insurance contracts are evaluated analytically.     

Additive portfolio improvement and utility-efficient payoffs

How can individual financial contracts be improved in an additive manner, such that any portfolio comprising improved contracts is at least as attractive as the portfolio of original contracts? We show that any additive procedure that improves contracts for all expected utility maximizers is a conditional expectation operator. Improved contracts are also attractive under robust Savage preferences. Furthermore, we generalize Bondarenko’s definition of ‘statistical arbitrage’ and show that the improved contracts do not admit this kind of arbitrage.     

Life insurance in a portfolio context

The ownership of life insurance may be modeled as a portfolio problem in which the return on the life insurance contract is negatively correlated with the return on a claim to future wage income. The mean-variance model developed in the paper uses such a framework to express the optimal amount of insurance in terms of two components: the expected value of the wage claim and the risk/return characteristics of the insurance contract. The model thus offers an appealing way to formulate the life insurance problem in a portfolio context. Implications of the model for the functioning of a life insurance market are examined and the existence of accidental death contracts is explained.     

Valuation of large variable annuity portfolios under nested simulation: A functional data approach

A variable annuity (VA) is equity-linked annuity product that has rapidly grown in popularity around the world in recent years. Research up to date on VA largely focuses on the valuation of guarantees embedded in a single VA contract. However, methods developed for individual VA contracts based on option pricing theory cannot be extended to large VA portfolios. Insurance companies currently use nested simulation to valuate guarantees for VA portfolios but efficient valuation under nested simulation for a large VA portfolio has been a real challenge. The computation in nested simulation is highly intensive and often prohibitive. In this paper, we propose a novel approach that combines a clustering technique with a functional data analysis technique to address the issue. We create a highly non-homogeneous synthetic VA portfolio of 100,000 contracts and use it to estimate the dollar Delta of the portfolio at each time step of outer loop scenarios under the nested simulation framework over a period of 25 years. Our test results show that the proposed approach performs well in terms of accuracy and efficiency.     

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.     

Anticipative portfolio optimization under constraints and a higher interest rate for borrowing

We study a stochastic control problem to maximize expected utility from terminal and/or consumption. The novel feature of our work is that the portfolio is allowed to anticipate the future with constraints and a higher interest rate for borrowing. The investor possesses information about the terminal values of the components of the Brownian motion, possibly distorted by ‘noise’. We use the technique from the so-called enlargement of filtrations, to model our problem. General existence results are established for optimal portfolio and consumption strategies. Equivalent conditions for optimality are obtained, and explicit solutions leading to feedback formulae are derived for special utility functions and for deterministic coefficients.     

On pricing and reserving with-profits life insurance contracts

As a first approximation, asset and liability management issues faced by life insurance companies originate from the sale of with-profits contracts. These contracts are bond-type products with several rate guarantees and other interestsensitive embedded options. Benefits paid out to policyholders mostly depend on the investment performance of a given asset portfolio in which premiums are invested. Thus, guarantees and options granted to policyholders may become effective when the investment performance of the asset portfolio is poor. Issuing a with-profits contract is therefore not equivalent to issuing plain-vanilla debt. The purpose of this paper is to value with-profits liabilities in a consistent option-pricing framework and to develop efficient asset or liability strategies to manage profitability and variability of shareholder value.     

Optimal trading and shipping of agricultural commodities

We develop and implement a model for a profit maximizing firm that provides an intermediation service between commodity producers and commodity end-users. We are motivated by the grain intermediation business at Los Grobo—one of the largest commodity-trading firms in South America. Producers and end-users are distributed over a realistic spatial network, and trade with the firm through contracts for delivery of grain during the marketing season. The firm owns spatially distributed storage facilities, and begins the marketing season with a portfolio of prearranged purchase and sale contracts with upstream and downstream counterparts. The firm aims to maximize profits while satisfying all previous commitments, possibly through the execution of new transactions. Under realistic constraints for capacities, network structure and shipping costs, we identify the optimal trading, storing and shipping policy for the firm as the solution of a profit-maximizing optimization problem, encoded as a minimum cost flow problem in a time-expanded network that captures both geography and time. We perform extensive numerical examples and show significant efficiency gains derived from the joint planning of logistics and trading.     

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.     

Robust portfolio selection involving options under a “ marginal+joint ” ellipsoidal uncertainty set

In typical robust portfolio selection problems, one mainly finds portfolios with the worst-case return under a given uncertainty set, in which asset returns can be realized. A too large uncertainty set will lead to a too conservative robust portfolio. However, if the given uncertainty set is not large enough, the realized returns of resulting portfolios will be outside of the uncertainty set when an extreme event such as market crash or a large shock of asset returns occurs. The goal of this paper is to propose robust portfolio selection models under so-called “ marginal+joint” ellipsoidal uncertainty set and to test the performance of the proposed models. A robust portfolio selection model under a “marginal + joint” ellipsoidal uncertainty set is proposed at first. The model has the advantages of models under the separable uncertainty set and the joint ellipsoidal uncertainty set, and relaxes the requirements on the uncertainty set. Then, one more robust portfolio selection model with option protection is presented by combining options into the proposed robust portfolio selection model. Convex programming approximations with second-order cone and linear matrix inequalities constraints to both models are derived. The proposed robust portfolio selection model with options can hedge risks and generates robust portfolios with well wealth growth rate when an extreme event occurs. Tests on real data of the Chinese stock market and simulated options confirm the property of both the models. Test results show that (1) under the “ marginal+joint” uncertainty set, the wealth growth rate and diversification of robust portfolios generated from the first proposed robust portfolio model (without options) are better and greater than those generated from Goldfarb and Iyengar’s model, and (2) the robust portfolio selection model with options outperforms the robust portfolio selection model without options when some extreme event occurs.     

A portfolio approach to managing procurement risk using multi-stage stochastic programming

Procurement is a critical supply chain management function that is susceptible to risk, due mainly to uncertain customer demand and purchase price volatility. A procurement approach in the form of a portfolio that incorporates the common procurement means is proposed. Such means include long-term contracts, spot procurements and option-based supply contracts. The objective is to explore possible synergies among the various procurement means, and so be able to produce optimal or near optimal results in profit while mitigating risk. The implementation of the portfolio approach is based on a multi-stage stochastic programming model in which replenishment decisions are made at various stages along a time horizon, with replenishment quantities being determined by simultaneously considering the stochastic demand and the price volatility of the spot market. The model attempts to minimise the risk exposure of procurement decisions measured as conditional value-at-risk. Numerical experiments to test the effectiveness of the proposed model are performed using demand data from a large air conditioner manufacturer in China and price volatility data from the Shanghai steel market. The results indicate that the proposed model can fairly reliably outperform other approaches, especially when either the demand and/or prices exhibit significant variability.     

Dynamic hybrid products with guarantees—An optimal portfolio framework

Dynamic hybrid products are pension products that consist of a dynamic combination of classical with profits participating life insurance contracts (or a bank account) and fund savings plans. To put such products in an optimal utility framework, we derive an optimal combination of a money market account, a CPPI-style fund and a free fund in continuous trading via transforming the original investment problem into a conventional portfolio problem in the presence of a guarantee condition. By this, we obtain (semi-) explicit forms of the dynamic weights for the different ingredients of a dynamic hybrid product.     

均衡的再保险市场

本文讨论了再保险市场达到均衡状态的定义、性质以及与 Pareto最优的关系 .     

A dynamic stochastic programming model for international portfolio management

We develop a multi-stage stochastic programming model for international portfolio management in a dynamic setting. We model uncertainty in asset prices and exchange rates in terms of scenario trees that reflect the empirical distributions implied by market data. The model takes a holistic view of the problem. It considers portfolio rebalancing decisions over multiple periods in accordance with the contingencies of the scenario tree. The solution jointly determines capital allocations to international markets, the selection of assets within each market, and appropriate currency hedging levels. We investigate the performance of alternative hedging strategies through extensive numerical tests with real market data. We show that appropriate selection of currency forward contracts materially reduces risk in international portfolios. We further find that multi-stage models consistently outperform single-stage models. Our results demonstrate that the stochastic programming framework provides a flexible and effective decision support tool for international portfolio management.