Single-period Model Research Articles

Competitive strategy in remanufacturing and the effects of government subsidy

In this paper, we consider a single-period model comprised of an original manufacturer (OM) who produces only new products and a remanufacturer who collects used products from consumers and produces remanufactured products. The OM and the remanufacturer compete in the product market. We examine the effects of government subsidy as a means to promote remanufacturing activity. In particularly, we consider three subsidy options: subsidy to remanufacturer, subsidy to consumers, and subsidy shared by remanufacturer and consumers. We find that the introduction of government subsidy on remanufacturer or consumers always increases remanufacturing activity. We also find that subsidy to remanufacturer is the best subsidy option, because subsidy to remanufacturer results in lower price of remanufactured products, thus leading to higher consumer surplus.

Coordinating a three level supply chain under generalized parametric interval-valued distribution of uncertain demand

In uncertain supply chain management problem, the optimal order quantity often depends heavily on the distribution of uncertain demand. When the exact possibility distribution is unavailable, it is required to develop a novel method to characterize the uncertain demand and deal with the corresponding supply chain coordination problem. This paper addresses the coordination problem for a three level supply chain in a single period model, where the uncertain demand is characterized by generalized parametric interval-valued possibility distribution. We define the lambda selection of uncertain demand, and discuss its parametric possibility distribution and mean value. In addition, we construct L–S measure by parametric possibility distribution of the lambda selection, and use it to define the L–S integral of uncertain profits under different scenarios. Under the risk-neutral criterion, we demonstrate that the mean supply chain profit in centralized decision is greater than the total mean supply chain profit in decentralized decision. Then a three level supply chain with revenue-sharing contract and return policy is studied and the analytical expressions of the optimal order quantity for different members are derived. Under the variable parametric possibility distribution of demand, we provide the sufficient conditions to ensure the three level supply chain can be fully coordinated and show the total mean profit of the channel can be allocated with any specified ratios among the members. Finally, we provide some managerial implications in a practical supply chain coordination problem. The computational results demonstrate the efficiency of the proposed parametric credibilistic optimization method.

Disclosure Dynamics and Investor Learning

This paper examines whether investor learning about profitability (i.e., the mean of earnings distribution) leads to persistence in disclosure decisions. A repeated single-period model shows that persistent investor beliefs about profitability lead to persistent disclosure decisions. Using earnings forecast data, I structurally estimate the model and perform several counterfactual analyses. I find that, when investors are assumed to know profitability, the persistence of management forecast decisions significantly declines by 17% to 27%, and 24% of firms would have disclosed differently, resulting in 3.9% net change in the amount of information (i.e., posterior variance) provided to the capital market. Collectively, the results indicate the importance of learning profitability in understanding disclosure decisions and the capital market consequences of disclosures.

Robust Security Design

We consider the optimal contract between an entrepreneur and investors in a single-period model when both parties have limited liability, are risk-neutral toward cash flow risk, and are ambiguity-averse. Ambiguity aversion is modeled by multiplier preferences for robustness toward model uncertainty, as in Hansen and Sargent (2001). Efficient ambiguity-sharing implies that the first-best contract consists of either convertible debt or levered equity. As is customary, in the second-best contract, moral hazard is alleviated by giving more cash to investors in low cash flow states. Under many settings in our model, the optimal security has an equity-like component in high cash flow states, providing a contrast to the results in Innes (1990).

Pricing of excess inventory on Groupon

We consider the problem faced by a business that is considering using the Groupon platform to sell excess inventory. We discuss how demand functions can be derived using management knowledge. Then, using a single period model where excess inventory is exogenous, we show that the decision to use Groupon and the price to set on that channel depend on two parameters: the relative price sensitivity of Groupon customers as compared to the retailer's regular customers and the relative size of the Groupon market as compared to the regular market. Under a two-period model, when initial inventory is a decision, we show optimal inventory quantities. Our two-period model suggests that managers may plan on using Groupon and order inventory accordingly. We discuss the implications on third party channels as well as retail managers.

Why does bank screening matter? Private information and publicly traded securities

Purpose The purpose of this paper is to use an equilibrium model to identify the public and private informational requirements for equilibrium pricing and shows that unless these informational requirements are met, skin-in-the-game policies will not be fully effective against moral hazard for banks with relatively large market share. Selling securitizations with recourse can be. Design/methodology/approach The single-period model shows equilibrium prices depend on both public and private information, the latter produced as banks screen loans. If bank has a sufficiently large market share, it can profit by omitting the screening unless investors can detect the change. The author derives the profit function for not screening, shows that a skin-in-the-game policy cannot fully offset its incentives, and proposes a sale with recourse policy that can. Findings To value securitizations correctly, investors require both publicly and privately available information. If investors cannot monitor banks closely, correct pricing can be frustrated by profit maximization incentives, since banks with large market shares can profit from not screening. Skin-in-the-game policies cannot fully offset these incentives. Research limitations/implications The equilibrium model identifies the public and private informational requirements for equilibrium pricing and shows that unless these informational requirements are met, skin-in-the-game policies will not be fully effective for banks with relatively large market share. Selling securitizations with recourse can be more fully effective. Practical implications If it is difficult for investors to obtain private information, skin-in-the-game policies are not provide fully effective remedies against moral hazard. Sales with recourse policies offer promise because they are easy for investors to understand and difficult to evade. Social implications Trading on the basis of private information can create perverse incentives, and appropriate corrective policies can help offset them. Originality/value The general equilibrium methodology, the findings of incentives to avoid screening, the flaws with skin-in-the-game policies, and the proposal for sale with recourse are all new.

Single- and Multi-Period Portfolio Optimization with Cone Constraints and Discrete Decisions

Portfolio optimization literature has come quite far in the decades since the first publication, and many modern models are formulated using second-order cone constraints and take discrete decisions into consideration. In this study we consider both single-period and multi-period portfolio optimization problems based on the Markowitz (1952) mean/variance framework, where there is a trade-off between expected return and the risk that the investor may be willing to take on. Our model is aggregated from current literature. In this model, we have included transaction costs, conditional value-at-risk (CVaR) constraints, diversification-by-sector constraints and buy-in-thresholds. Our numerical experiments are conducted on portfolios drawn from 20 to 400 different stocks available from the S&P 500 for the single period-model. The multi-period portfolio optimization model is obtained using a binary scenario tree that is constructed with monthly returns of the closing price of the stocks from the S&P 500. We solve these models with a MATLAB based Mixed Integer Linear and Nonlinear Optimizer (MILANO). We provide substantial improvement in runtimes using warmstarts in both branch-and-bound and outer approximation algorithms.

Dynamic stochastic cooperative reinsurance strategy in a\\ continuous time model

The problem of the expected utility maximization in a reinsurance market for a single period model is well understood under a general setting. This paper studies the problem in a continuous time model. For reciprocal type of reinsurance strategies and exponential utility functions, with the concept of the Pareto optimality in game theory, we obtain the characterization of the core of cooperative reinsurance games and show that the core, which contains all of the Pareto optimal cooperative reinsurance strategies, is non-empty. Examples are provided to illustrate that the core is non-empty under the given situations and the Pareto optimal cooperative reinsurance strategy is shown to be a proportional reinsurance strategy when there are only two insurance companies considered.

Shock Diffusion: Does Inter-Sectoral Network Structure Matter?

This paper introduces the concept of diffusion of shocks in a macroeconomic network consisting of inter-sectoral production linkages. I show that if sectors have different reaction horizons it would lead to diffusion of shocks through the network over time which prevents the inter-sectoral linkages to form the feedback loop structure essential to generate aggregate volatility. This result is different from other recent papers which have single period model with contemporaneous production linkages between different sectors thus generating sectoral shock amplification as one sector reacts to another contemporaneously resulting in bigger aggregate fluctuations. In contrast if sectors have different production horizons due to varying complexity of their production process or supply chain, it would break down the feedback architecture present in single period models. I further show that if the diffusion rate is varied for different sectors, the contribution of network structure to aggregate volatility can be insignificant. Also, it is no longer sufficient to characterize this contribution of inter-sectoral production network to aggregate volatility by just looking at input-output matrix or its summary statistics like degree distribution. The paper thus highlights the stark difference between the study of financial and inter-sectoral production networks because of the possibility of contemporaneous amplification and hence cascades in the case of financial networks. In the end, I propose lead time indicator as a possible proxy for measuring differential sectoral diffusion rates.

Market Value, Time, and Risk

(Revised 4/29/15, with minor edits by Craig William French)Abstract by Craig William FrenchThis paper reprints a slightly edited version of Jack L. Treynor's 1961 CAPM manuscript, which has previously been unavailable to the public. The author's facsimile of the original was obtained thanks to the kind generosity of Mr. and Mrs. Treynor. Edits in the present version, which differ from the original, include minor typographical corrections and minor notation differences for some variables in the formulae. Pagination is as in the original. In 1958, Jack Treynor was employed by Arthur D. Little. That summer he took a three-week vacation to Evergreen, Colorado, during which he produced forty-four pages of mathematical notes on capital asset pricing and capital budgeting. Over the next two years, Treynor refined his notes into what is in all likelihood the first CAPM. Treynor gave a copy of this early model to John Lintner at Harvard in 1960. While in business school at Harvard from 1953 through 1955, Mr. Treynor had taken nearly every finance course offered, and though he signed up for Lintner’s economics course he was forced to cancel due to a schedule conflict. In 1960, John Lintner was the only economist he knew even slightly. Treynor refined his 1960 model into the 45-page “Market Value, Time, and Risk” [the present paper]. This paper, Treynor (1961), develops the CAPM using the concept of experiment space to quantify risk and risk relations. Without his knowledge or encouragement, one of Mr. Treynor’s colleagues sent the draft to Merton Miller in 1961, after Miller had moved to the University of Chicago from Carnegie Institute of Technology. Miller sent the paper to Franco Modigliani at MIT in the spring of 1962, and Modigliani invited Treynor to embark on a program of graduate work at MIT under his supervision. Treynor did so during the 1962-1963 academic year; in addition to Modigliani’s course, he took Bob Bishop’s price theory and Ed Kuh’s econometrics courses, among others. By the fall of 1962, Treynor had consolidated the first part of Treynor (1961), on the single-period model, into “Toward a Theory of Market Value of Risky Assets,” and presented it to the MIT finance faculty.A more complete description of the development of the Treynor CAPM may be found in French, Craig W., The Treynor Capital Asset Pricing Model. Journal of Investment Management, Vol. 1, No. 2, pp. 60-72, 2003. Available at SSRN: http://ssrn.com/abstract=447580.A reprint of Treynor's 1962 paper, “Toward a Theory of Market Value of Risky Assets,” may be found in French, Craig W., Jack Treynor's 'Toward a Theory of Market Value of Risky Assets' (December 28, 2002). Available at SSRN: http://ssrn.com/abstract=628187.

Single-period inventory model for one-level assembly system with stochastic lead times and demand

Replenishment planning of an assembly system with one type of finished product assembled from different types of components is considered. The components are procured from diverse external suppliers to satisfy finished product demand. It is supposed that the component lead times and finished product demand are random discrete variables. The assembly company must determine what are the best quantities of components and when is the right time to order. The objective is to minimise the total cost which is composed of holding component costs, tardiness penalties, lost sales and surplus item costs for finished products. A single-period analytical model is proposed. Several properties of the objective function are proven. They are used to develop a Branch and Bound algorithm. Numerical tests for the algorithm are presented. Five heuristics based on Newsvendor model for lead time and demand are proposed and compared with the Branch and Bound algorithm. These tests show that the suggested Branch and Bound algorithm can solve large size problems within a short time. The proposed heuristics but one are not competitive with the Branch and Bound algorithm. The truncated version of Branch and Bound gives better results. The model suggested is better adapted to actual contract assembler environments, more realistic and can better approximate real-life industrial situations. The proposed exact algorithm provides optimal solutions for all discrete distributions of probabilities of lead times and demand. A new general approach to design such discrete optimisation algorithms is presented.

Inventory Planning for a Modular Product Family

This paper is motivated by observing that an increasing number of firms are offering modular products assembled with multiple option choices for the consumer. Starting with the PC offerings by Dell which allowed (and still allows) users to configure their product by choosing among multiple choices for each option, the current market place seems to have evolved to a make‐to‐stock scenario where Apple offers its IPAD series with multiple models each with a unique storage size, color, and wireless chip technology. The focus of our work is on determining the optimal stocking level of modular end‐products. Our analysis is based on a benchmark model with the aim of maximizing expected profit subject to an aggregate fill rate constraint as well as variant‐specific individual fill rates under a make‐to‐stock setting. To further assess the robustness of our finding, we consider the extensions of correlated market preferences over options, price‐dependent demand, and alternative probability distributions for characterizing uncertainty in market preferences or aggregate demand. Finally we also show how to extend the single period model into a multiple‐period setting. Through extensive computational analysis, we find that more precise estimates of market preferences for various modular options constitute extremely valuable information that goes beyond the usefulness of forecasts of aggregate market demand. From a practical perspective, this might be indicative of another classic marketing‐operations trade‐off. Offering more options for consumers would be preferred by marketing managers since this would reach more consumers and hence, enhance product sales. On the other hand, the ability to obtaining greater forecast accuracy would decline when the number of options increase. Hence, from an operational perspective, it would be preferred to limit option choices (so that better forecasts can be obtained) since this would lead to lower stocking costs and hence, higher profits.

The Time Contradiction in Asset Pricing and Asset Management

Investors live in a multi-period, volatile world and base their decisions on theories of asset pricing, and asset allocation, often derived from a single period model. They make assumptions about asset returns and volatilities and use optimizers to set their long term allocations, and often naively rebalance back to these allocations, hire managers etc. Even if they have perfect foresight in predicting asset returns, are they making decisions based on noise or truly useful information? Alternatively, what confidence can one have in these estimates over typical investment horizons of 5-10 years? Or how much time does one need to establish a successful investment outcome as opposed to just experiencing noise? A simple formula designed to examine whether asset managers are lucky or skillful can be used to examine these questions. This analysis suggests that the time window to have high confidence in the efficacy of the approach utilized by most investors is much greater than the typical horizon of these approaches – this is the time contradiction in investments. Alternatively, for a given investment horizon, this methodology suggests the degree of confidence an investor might have in their investment inputs and allocation decisions. Living in a volatile world implies that this confidence is low for short time windows. This time contradiction between required time horizon and that used by investors has interesting implications not only for the investment models and approaches used, including the examination of funded versus unfunded programs, but also for how investors are compared to peers and compensated.

Dynamic acquisition pricing policy under uncertain remanufactured-product demand

Purpose– The purpose of this paper is to investigate the dynamic acquisition pricing strategy for collecting used products (also known as cores or returns) in a finite planning horizon. In particular, this paper studies a cost-minimization model in which a firm offers acquisition price that impacts the quantity of the returns, and remanufactures the used product to satisfy the customer demand.Design/methodology/approach– This paper uses multi-period stochastic dynamic programming theory to model a remanufacturing system that faces the random demand for remanufactured products. The number of the returns at each period is uncertain and increases linearly with the acquisition price offered.Findings– The study shows that when the uncertainty of demand for remanufactured products increases, the remanufacturer should hold a higher core stock level to minimize the expected total cost and thus a higher acquisition price is needed to attract returns. However, given demand uncertainty, the optimal price decreases in the initial core stock level in each period. It also indicates that the optimal acquisition price increases in the variance of the returns, but decreases in the mean of the returns.Practical implications– The findings suggest that a remanufacturer could reduce the expected total cost by adjusting the acquisition price according to the number of returns periodically.Originality/value– Introducing the impact of supply uncertainty on the acquisition price of used products, this paper uses a multi-period dynamic model, instead of single period model in previous studies, to examine the remanufacturer’s dynamic acquisition pricing policy.

A model for a large investor trading at market indifference prices. I: Single-period case

We develop a single-period model for a large economic agent who trades with market makers at their utility indifference prices. A key role is played by a pair of conjugate saddle functions associated with the description of Pareto optimal allocations in terms of the utility function of a representative market maker.

On the risk-averse procurement strategy under unreliable supply

This study investigates an effective procurement/inventory strategy for a risk-averse retailer facing unreliable supply and stochastic demand. By using an increasing and concave utility function to describe risk aversion, we construct a basic newsvendor (single-period) model and its multi-period extension. Both models are found to have unique solutions, as the optimized expected utility is strictly concave in initial inventory level. As a result, there is a unique optimal order quantity for the effective control of supply risk. For the single-period model, the optimal order quantity is derived in its analytical form. We then show by numerical analysis that the value of the optimized expected utility is a function of the initial inventory level when the retailer is risk-averse, becomes less sensitive to initial inventory level when the degree of risk aversion decreases, and is insensitive for the risk-neutral case. This finding suggests that in our setting the inventory holding matters only when the retailer is risk-averse. For the multi-period model, we propose a solution procedure using backward induction since a direct extension of the single-period solution is impossible. We also conduct a sensitivity analysis of demand and supply with the aim of giving some managerial suggestions for demand risk control and supplier selection.

The Value of Information in Container Transport

Planning the transport of maritime containers from the seaport to final inland destinations is challenged by uncertainties regarding the time the container is released for further transport and the transit time from the port to its final destination. This paper assesses the value of information in container transport in terms of efficiency and reliability. The analysis uses a stylized single period model where a decision maker allocates released containers to two transport modes (slow, low price, no flexible departure times versus fast, high price, flexible departure times) and plans the departure time of the inflexible mode. We construct Pareto frontiers and the corresponding Pareto optimal decisions under various information scenarios and show that the Pareto frontiers move in a favorable direction when the level of information increases. The mathematical results are explained and illustrated by means of a numerical example involving barge transport. We also perform a sensitivity analysis and study the impact of erroneous information on efficiency and reliability based on a numerical analysis.

An optimal hybrid emission control system in a multiple compliance period model

We characterize an optimal hybrid policy for pollutant emissions that includes a permit price ceiling and emission cap in a multi-period model. Previous studies have primarily focused on a hybrid policy of emission regulation that included an emission cap and permit price ceiling in a single period model, we extend this literature by developing such an optimal hybrid model in a multi-period framework where banking and borrowing of emission permits is allowed. In our model, we compare the case of a regulator who sets the emission cap and price ceiling to be consistent with a long run emission objective with the case of a regulator who occasionally is motivated to deviate from the optimal long run regulatory policy in order to correct for unexpected but exceptionally high emissions. Using a discrete dynamic programming model with stochastic emissions, we show that the hybrid model gives the regulator a degree of freedom in making an optimal price and quantity choice.

A Study of a Single-Period Model for the Parallel Fleet Replacement Problem

This paper presents a new approach for the parallel fleet replacement problem. The problem involves determining an optimal replacement schedule for a fleet of vehicles that results in minimal total cost of owning and operating the fleet. Four main aspects including utilization patterns, O&M cost patterns, salvage value patterns and replacement policies are considered. The proposed approach is to use a single-period integer programming model instead of the multiple-period model that requires much higher computing power and data requirement. The approach is tested on a small numerical example in eight scenarios. Promising results of off-optimal percentage are obtained.

On a dynamic adaptation of The Distribution Builder approach to investment decisions

Sharpe et al. proposed the idea of having an expected utility maximizer choose a probability distribution for future wealth as an input to her investment problem instead of a utility function. They developed a computer program, called The Distribution Builder, as one way to elicit such a distribution. In a single-period model, they then showed how this desired distribution for terminal wealth can be used to infer the investor's risk preferences. We adapt their idea, namely that a risk-averse investor can choose a desired distribution for future wealth as an alternative input attribute for investment decisions, to continuous time. In a variety of scenarios, we show how the investor's desired distribution combines with her initial wealth and market-related input to determine the feasibility of her distribution, her implied risk preferences, and her optimal policies throughout her investment horizon. We then provide several examples.