The design of equity-indexed annuities

There is a rich variety of tailored investment products available to the retail investor in every developed economy. These contracts combine upside participation in bull markets with downside protection in bear markets. Examples include equity-linked contracts and other types of structured products. This paper analyzes these contracts from the investor’s perspective rather than the issuer’s using concepts and tools from financial economics. We analyze and critique their current design and examine their valuation from the investor’s perspective. We propose a generalization of the conventional design that has some interesting features. The generalized contract specifications are obtained by assuming that the investor wishes to maximize end of period expected utility of wealth subject to certain constraints. The first constraint is a guaranteed minimum rate of return which is a common feature of conventional contracts. The second constraint is new. It provides the investor with the opportunity to outperform a benchmark portfolio with some probability. We present the explicit form of the optimal contract assuming both constraints apply and we illustrate the nature of the solution using specific examples. The paper focusses on equity-indexed annuities as a representative type of such contracts but our approach is applicable to other types of equity-linked contracts and structured products.     

Fuzzy portfolio selection model with real features and different decision behaviors

In the ever changing financial markets, investor’s decision behaviors may change from time to time. In this paper, we consider the effect of investor’s different decision behaviors on portfolio selection in fuzzy environment. We present a possibilistic mean-semivariance model for fuzzy portfolio selection by considering some real investment features including proportional transaction cost, fixed transaction cost, cardinality constraint, investment threshold constraints, decision dependency constraints and minimum transaction lots. To describe investor’s different decision behaviors, we characterize the return rates on securities by LR fuzzy numbers with different shape parameters in the left- and right-hand reference functions. Then, we design a novel hybrid differential evolution algorithm to solve the proposed model. Finally, we provide a numerical example to illustrate the application of our model and the effectiveness of the designed algorithm.     

Robust Management and Pricing of Liquefied Natural Gas Contracts with Cancelation Options

Liquefied Natural Gas contracts offer cancelation options that make their pricing difficult, especially if many gas storages need to be taken into account. We develop a valuation mechanism from the buyer’s perspective, a large gas company whose main interest in these contracts is to provide to clients a reliable supply of gas. The approach combines valuation with hedging, taking into account that price-risk is driven by international markets, while volume-risk depends on local weather and is stage-wise dependent. The methodology is based on setting risk-averse stochastic mixed 0-1 programs, for different contract configurations. These difficult problems are solved with light computational effort, thanks to a robust rolling-horizon approach. The resulting pricing mechanism not only shows how a specific set of contracts will impact the company business, but also provides the manager with alternative contract configurations to counter-propose to the contract seller.     

The Optimal Interaction between a Hedge Fund Manager and Investor

ABSTRACT

This study explores hedge funds from the perspective of investors and the motivation behind their investments. We model a typical hedge fund contract between an investor and a manager, which includes the manager’s special reward scheme, i.e., partial ownership, incentives and early closure conditions. We present a continuous stochastic control problem for the manager’s wealth on a hedge fund comprising one risky asset and one riskless bond as a basis to calculate the investors’ wealth. Then we derive partial differential equations (PDEs) for each agent and numerically obtain the unique viscosity solution for these problems. Our model shows that the manager’s incentives are very high and therefore investors are not receiving profit compared to a riskless investment. We investigate a new type of hedge fund contract where the investor has the option to deposit additional money to the fund at half maturity time. Results show that investors’ inflow increases proportionally with the expected rate of return of the risky asset, but even in low rates of return, investors inflow money to keep the fund open. Finally, comparing the contracts with and without the option, we spot that investors are sometimes better off without the option to inflow money, thus creating a negative value of the option.     

Contract design and supply chain coordination in the electricity industry

In this article we propose a model of the supply chain in electricity markets with multiple generators and retailers and considering several market structures. We analyze how market design interacts with the different types of contract and market structure to affect the coordination between the different firms and the performance of the supply chain as a whole. We compare the implications on supply chain coordination and on the players’ profitability of two different market structures: a pool based market vs. bilateral contracts, taking into consideration the relationship between futures and spot markets. Furthermore, we analyze the use of contracts for differences and two-part-tariffs as tools for supply chain coordination. We have concluded that there are multiple equilibria in the supply chain contracts and structure and that the two-part tariff is the best contract to reduce double marginalization and increase efficiency in the management of the supply chain.     

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.     

A principal–agent problem with heterogeneous demand distributions for a carbon capture and storage system

Mechanism design problems optimize contract offerings from a principal to different types of agents who have private information about their demands for a product or a service. We study the implications of uncertainty in agents’ demands on the principal’s contracts. Specifically, we consider the setting where agents’ demands follow heterogeneous distributions and the principal offers a menu of contracts stipulating quantities and transfer payments for each demand distribution. We present analytical solutions for the special case when there are two distributions each taking two discrete values, as well as a method for deriving analytical solutions from numerical solutions. We describe one application of the model in carbon capture and storage systems to demonstrate various types of optimal solutions and to obtain managerial insights.     

Equity-linked pension schemes with guarantees

This paper analyses the relationship between the level of a return guarantee in an equity-linked pension scheme and the proportion of an investor’s contribution needed to finance this guarantee. Three types of schemes are considered: investment guarantee, contribution guarantee and surplus participation. The evaluation of each scheme involves pricing an Asian option, for which relatively tight upper and lower bounds can be calculated in a numerically efficient manner.We find a negative (and for two contract specifications also concave) relationship between the participation in the surplus return of the investment strategy and the guarantee level in terms of a minimum rate of return. Furthermore, the introduction of the possibility of early termination of the contract (e.g. due to the death of the investor) has no qualitative and very little quantitative impact on this relationship.     

Optimal procurement strategies for online spot markets

Spot markets have emerged for a broad range of commodities, and companies have started to use them in addition to their traditional, long-term procurement contracts (forward contracts). In comparison to forward contracts, spot markets offer products at essentially negligible lead time, but typically command a higher expected price for this added flexibility while also exhibiting substantial price uncertainty. In our research, we analyze the resulting procurement challenge and quantify the benefits of using spot markets from a supply chain perspective. We develop and solve mathematical models that determine the optimal order quantity to purchase via forward contracts and the optimal quantity to purchase via spot markets. We analyze the most general situation where commodities can be both bought and sold via a spot market and derive closed-form results for this case. We compare the obtained results to the reference scenario of pure contract sourcing and we include results for situations where the use of spot markets is restricted to either buying or selling only. Our approaches can be used by decision makers to determine optimal procurement strategies based on key parameters such as, demand and spot price volatilities, correlation between demand and spot prices, and risk aversion. The results of our analysis demonstrate that significant profit improvements can be achieved if a moderate fraction of the commodity demand is procured via spot markets. The results also show that companies who use spot markets can offer a higher expected service level, but that they might experience a higher variability in profits than companies who do not use spot markets. We illustrate our analytical results with numerical examples throughout the paper.     

Equilibrium contract selection strategy in chain-to-chain competition with demand uncertainty

This paper investigates the equilibrium contract selection problem for the dominant suppliers in two competing supply chains with stochastic and price-sensitive demand. The two suppliers, acting as the Stackelberg leaders, produce substitutable products and distribute them through each exclusive retailer, and can provide either a consignment contract or a wholesale-price contract. The equilibrium behaviours of the suppliers and retailers are investigated in three different scenarios: (1) the consignment contract scenario; (2) the wholesale-price contract scenario; and (3) the hybrid contract scenario. We prove that the equilibrium contracting strategy is of the threshold type: when the cost-share rates of the two retailers are above certain thresholds, both suppliers select consignment contracts; when the cost-share rates of the two retailers are lower than certain thresholds, both suppliers select wholesale-price contracts; when one retailer’s cost-share rate is above a certain threshold and the other is lower than a certain threshold, the supplier with large retailer’s cost-share rate selects the consignment contract and the other supplier with small retailer’s cost-share rate selects the wholesale-price contract. Furthermore, these thresholds depend on price sensitivities.     

Optimizing product service system by franchise fee contracts under information asymmetry

This paper studies the problem of how to effectively provide product service system (PSS) in a service-oriented manufacturing supply chain under asymmetric private demand information. The PSS in the supply chain is operated heterogeneously and complementarily, in which the manufacturer provides the product while the retailer who possesses private demand information is responsible for adding the necessary value-added service on the basic product. We address the issue of how different contracts affect the decisions and profitability of the supply chain members. Three types of contracts are developed to help supply chain partners to make decisions and enhance the supply chain’s efficiency. The first is the franchise fee (FF) contract, under which the manufacturer provides a two-part tariff contract (wholesale price and franchise fee) to influence the retailer’s decision and to detect her private demand information. The second is the franchise fee with service requirement (FFS) contract, under which the manufacturer specifies the service level required in addition to the two-part tariff contract terms. The third is the franchise fee with centralized service requirement (FFCS) contract, which is similar to the FFS contract but that the service level specified by the manufacturer is the system optimal solution. Our analytical results show that all three contracts enable the manufacturer to detect the retailer’s private demand information, with the FFCS contract achieving the greatest channel profit. Finally, numerical examples are presented, and sensitivity analysis of service level and profit are conducted to compare the performance of the three contracts under different settings. The paper provides managerial guidelines for the manufacturer in contract offering under different conditions.     

Consignment contracts with retail competition

Consignment contracts have been widely employed in many industries. Under such contracts, items are sold at a retailer’s but the supplier retains the full ownership of the inventory until purchased by consumers; the supplier collects payment from the retailer based on actual units sold. We investigate how competition among retailers influences the supply chain decisions and profits under different consignment arrangements, namely a consignment price contract and a consignment contract with revenue share. First, we investigate how these two consignment contracts and a price only contract compare from the perspective of each supply chain partner. We find that the retailers benefit more from a consignment price contract than from a consignment contract with revenue share or a price only contract, regardless of the level of retailer differentiation. The supplier’s most beneficial contact, however, critically depends upon the level of retailer differentiation: a consignment contract with revenue share is preferable for the supplier if retailer differentiation is strong; otherwise a consignment price contract is preferable. Second, we study how retailer differentiation affects the profits of all supply chain partners. We find that less retailer differentiation improves the supplier’s profit for both types of consignment contract. Moreover, less retailer differentiation improves profits of the retailers in a consignment price contract, but not necessarily in a consignment contract with revenue share.     

Entry and exit of service providers under cost uncertainty: a real options approach

From a real options perspective, this paper examines a service provider's entry and exit decisions toward two types of service outsourcing contracts under service transaction cost uncertainties. Specifically, for a service contract with a flexible duration, the service provider has an option to terminate the contract at any time point by paying a pre-determined exit penalty. For a contract with a fixed-duration, the service provider is obligated to deliver services for a pre-determined period of time. Under this framework, this study seeks to derive the transaction cost conditions that trigger the service provider’s exercise of entry and exit options. Furthermore, via analytical and numerical examinations, this study also uncovers how service transaction cost uncertainty and other business factors (eg, exit penalty and contract duration) influence the service provider’s entry and exit decisions as well as the choice of contract type (ie, fixed-duration versus flexible-duration).     

Buyer-supplier currency exchange rate flexibility contracts in global supply chains

This paper analyzes a decentralized global supply chain under a newsvendor setting, where a supplier delivers a certain quantity of a single product to a buyer in accordance with the terms of a mutually agreed upon contract. This contract is signed prior to the delivery of the product and subsequent payment, thus, exposing the supply chain to the risk of currency exchange rate fluctuations. We propose two types of currency exchange rate flexibility contracts to explore the characteristics of exchange rate risk mitigation policies for the buyer and the supplier. Furthermore, we investigate the effects of the contract structures on the optimal order quantity, as well as the expected profits of both supply chain members. Our results show that the optimal order quantity of the buyer decreases when the wholesale price is uncertain due to exchange rate volatility. Also, both our proposed contracts tend to improve the expected profits of both the buyer and the supplier, when the payment is made in the supplier’s currency, indicating the desirability of adopting such contractual agreements from the perspective of both parties. On the other hand, when the payment is made in the buyer’s currency, our suggested contracts do not yield such win-win scenarios. Finally, we examine the effectiveness of availing the services of a local vendor, which is capable of satisfying any demand in excess of the quantity ordered from the foreign source with short notice, in order to mitigate the risks associated with an overseas order.     

Contracting with an urgent supplier under cost information asymmetry

We investigate a contract setting problem faced by a manufacturer who can procure major modules from an overseas supplier, as well as a local supplier. The overseas supplier is prime and offers quality products, whereas the local supplier is viewed only as a backup, and its products are inferior in quality. As the local supplier needs to put in additional effort to fulfill the urgent orders, it is difficult for the manufacturer to estimate this urgent supplier’s production cost. This asymmetric cost information becomes an obstacle for the manufacturer in managing the urgent supplier. In this paper, we study two types of contingent contracts. One is the common price-only contract, and the other is a contract menu consisting of a transfer payment and a lead time quotation. We construct a Stackelberg game model and evaluate how the involvement of an urgent supplier with private cost information affects performances of the prime supplier and the manufacturer in different scenarios (with or without the urgent supplier, under different contingent contracts). We also conduct numerical experiments to show how the parameters of the contracts affect profits of the manufacturer.     

Pricing and hedging equity-linked life insurance contracts beyond the classical paradigm: The principle of equivalent forward preferences

By applying the principle of equivalent forward preferences, this paper revisits the pricing and hedging problems for equity-linked life insurance contracts. The equity-linked contingent claim depends on, not only the future lifetime of the policyholder, but also the performance of the reference portfolio in the financial market for the segregated account of the policyholder. For both zero volatility and non-zero volatility forward utility preferences, prices and hedging strategies of the contract are represented by solutions of random horizon backward stochastic differential equations. Numerical illustration is provided for the zero volatility case. The derived prices and hedging strategies are also compared with classical results in the literature.     

Dominant strategy mechanisms for contract auctions with risk aversion and moral hazard

Within the class dominant strategy incentive compatible mechanisms, we show that there exists an optimal contracting mechanism for the principal for a version of the incomplete information principal-agent problem in which several agents compete for a contract and the principal selects an agent via a contract auction. In our auction model, we assume that the principal and the agents are risk averse, and we allow for uncountably many agent types. We also assume that the principal's probability measure over type profiles in such that correlation between agent's types is possible. Thus, we do not require that agents' types be independently distributed. Finally, we impose limited liability constraints upon the set of contracts. Due to the nature of the individual rationality and incentive compatibility constraints, the existence problem is nonstandard and novel existence arguments are required. We prove existence using a measurable selection result and a new notion of compactness called K-compactness.     

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.     

Utility indifference pricing and hedging for structured contracts in energy markets

In this paper we study the pricing and hedging of structured products in energy markets, such as swing and virtual gas storage, using the exponential utility indifference pricing approach in a general incomplete multivariate market model driven by finitely many stochastic factors. The buyer of such contracts is allowed to trade in the forward market in order to hedge the risk of his position. We fully characterize the buyer’s utility indifference price of a given product in terms of continuous viscosity solutions of suitable nonlinear PDEs. This gives a way to identify reasonable candidates for the optimal exercise strategy for the structured product as well as for the corresponding hedging strategy. Moreover, in a model with two correlated assets, one traded and one nontraded, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Finally, numerical results are provided.     

Ein allgemeines Modell zur Analyse und Bewertung von Guaranteed Minimum Benefits in Fondspolicen

Variable Annuities with embedded guarantees are very popular in the US-market. There exists a great variety of products with both, guaranteed minimum death benefits (GMDB) and guaranteed minimum living benefits (GMLB). Although several approaches for pricing some of the corresponding guarantees have been proposed in the academic literature, there is no general framework in which the existing variety of such guarantees can be priced consistently. The present paper fills this gap by introducing a model, which permits a consistent and extensive analysis of all types of guarantees currently offered within Variable Annuity contracts. Aside from a valuation assuming that the policyholder follows a given strategy with respect to surrender and withdrawals, we are able to price the contract under optimal policyholder behavior. Using both, Monte-Carlo methods and a generalization of a finite mesh discretization approach proposed by Tanskanen and Lukkarinen (2004), we find that some guarantees are overpriced, whereas others, e.g. guaranteed annuities within guaranteed minimum income benefits (GMIB), are offered significantly below their risk-neutral value. We identify a variety of ‘‘new risks’’ associated with such products.