Financial Derivatives in Corporate Risk Management

Mulong Wang, B.S. Dissertation Presented to the Faculty of the Graduate School of the University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin August 2001 UMI Number: 3036610 ________________________________________________________ UMI Microform 3036610 Copyright 2002 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ____________________________________________________________ ProQuest Information and Learning Company 300 North Zeeb Road PO Box 1346 Ann Arbor, MI 48106-1346 Dedicated to Helen, My wife Acknowledgements I want to thank all those kind people who advised or helped my dissertation research. I am greatly indebted to my supervisors, Drs. Patrick Brockett and Richard MacMinn. This dissertation would not have been finished without their kind guidance, discussion, and encouragements. It has been an extremely valuable experience to study and work under their supervision. I also would like to thank my dear wife, Helen. With her consistent encouragements and help, I can dedicate to this research. Drs. Jonathan Bard, Douglas Morrice and Thomas Sager provided numerous comments on this dissertation. Their service on my dissertation committee is greatly appreciated. Colleagues in Center of Risk Management and Insurance and Center of Management and Operations for Logistics also provided a lot of help in my PhD student career. Their discussions inspired a lot of this dissertation research and other research projects. v Financial Derivatives in Corporate Risk Management Publication No. Mulong Wang, Ph.D. The University of Texas at Austin, 2001 Supervisors: Patrick L. Brockett and Richard D. MacMinn This dissertation addresses how the weather derivative hedges the corporate risk, how to price the indexed derivative as an exotic derivative instrument, and the implications of basis risk embedded in the weather derivative. The traditional one-dimension financial market framework is expanded to include the weather index. Under this expanded framework, the stock market values of the unhedged and hedged firms are studied first. This provides the base to investigate the pricing formula for weather derivative under the expanded framework. It is found that both financial and actuarial approaches are integrated vi to price the weather derivative. A positive risk management paradigm must provide the criteria to choose the optimal hedging instrument(s) for separable risks. This dissertation provides the criteria to choose optimal hedging contract set to hedge the weather risk, under different corporate leverage levels. It has been found that weather derivative outperforms the traditional commodity forward in most of the scenarios. When corporate leverage levels increase, the positive role of the weather derivative or the commodity forward diminishes. Basis risk arises by introducing the standard weather index, and providing the industry-standard payment when the weather derivative is exercised. The implication of basis risk is investigated under the same expanded framework. It is found that in most of the scenarios, basis risk is innocuous. vii TABLE OF CONTENTS Chapter I. Introduction…………………………………………………1 1.1 Background……………………………………………….2 1.2 Literature on Risk and Risk Management………….7 1.3 The Frontier of Risk Management ………………….11 1.4 Weather Risk and Weather Derivatives ……………13 Chapter II. Weather Derivative and Its Valuation……………….17 2.1 Introduction…………………………………………………17 2.2 Basic Model …………………………………………………28 2.3 Valuation of Weather Derivative………………………..38 Chapter III. Weather Derivative and Commodity Forward……..48 3.1 Scenario sets…………………………………………………49 3.2 Optimal Hedging……...…………………………………….58 Chapter IV. Basis Risk and Its Implications……………………….86 4.1 Introduction………………………………………………….86 4.2 Implications of Basis Risk…………………………………89 viii Appendix I. Figures ………………………………………………………106 Appendix II. Extensions of Principle of Increasing Uncertainty…125 Reference…………………………………………………………………….135 Vita……………………………………………………………………………142 ix Chapter I Introduction This dissertation addresses how the weather derivative hedges the corporate risk, how to price the indexed derivative as an exotic derivative instrument, and the implications of basis risk. These topics are summarized in an expanded uncertainty model. Under this framework, different hedging instruments for studying the optimal hedging portfolios are compared. In economics and finance literature, risk has been a subject of interest and study in many fields including management science, decision science, and psychology. With the new risks being continuously discovered, innovative strategies and tools were created to manage them or transfer them. In this chapter, the background of risk creation, identification and the importance of risk management are discussed first. In the second chapter, the economic and financial literature on risk and risk management, most of which concentrates on the mechanism of financial derivatives to hedge the risk, or insurance contract to transfer the individual risk or corporate risk, is reviewed. The third chapter studies the new frontier of risk management strategies and tools. In the last part, the weather 1 derivative is introduced as an example of novel risk management tool. The focus of this dissertation is to study how the weather derivative hedges the specific risk and its positive effects on creating value for the hedged firms. The valuation of the indexed derivative was investigated and the comparison of different hedging instruments in the corporate finance framework was made, to investigate the optimal hedging strategies under the presence of separable risks. This dissertation investigated the pricing problem of the weather derivative. In addition, the comparison of the weather derivative and the commodity forward under different corporate leverage scenarios was made. Finally, the implications of basis risk were discussed. 1.1 Background In general, risk is the uncertainty in the future, and has been traditionally separated into two categories: pure risk or speculative risk. A pure risk is a chance of loss or no loss, and a speculative risk is characterized as a chance of loss or gain. An example of pure risk is catastrophe risk, such as an earthquake, flood or hurricane. Gambling is an example of speculative risk, which may yield a gain or 2 loss in the end. In the economics and finance literature, the definitions of pure risk and speculative risk differentiate insurance from finance. Much of the insurance literature has concentrated on the management of pure risk and much of the finance literature on risk management has concentrated on the management of speculative risk. However, recent applications of risk management have blurred the line between these fields. For example, the Catastrophe (CAT) Bond, created to hedge the catastrophic event risk, is an example of securitization, seen mostly in the mortgage market. However, the separation between insurance and finance paradigms was blurred by the creation of the CAT bond. Like investing in traditional bonds, i.e., Treasury bonds or corporate bonds, investors bear the speculative risk. The cedent, paying the premium to the safe trust, obtains the protection as traditional insurance provides. Therefore, CAT bond is not only an investment vehicle for the investors, but also a hedging vehicle for the cedents, which represents a vehicle combining both pure risk and speculative risk. With the discovery of new risks, new hedging instruments will be developed to further blur the line of pure risk and speculative risk, making the distinction between insurance and finance even more 3 ambiguous. Nowadays, the risk-management process is becoming an increasingly important financial area for virtually all corporations. With the dramatically growing costs of losses from different risk sources, business firms can gain a competitive cost advantage through the development of a set of cost-effective and efficient risk- management strategies. The advantages of a well-managed risk management program include not only a lower total loss cost and an improved business bottom line, but also an increased predictability of future losses and cost, which ensures greater budget control and reduced ambiguity for future net revenue stream. A less risky corporate operation and payoff preserve more value for the risk-averse investors and therefore are preferred. In addition, a good risk management program will immunize the corporation from the sharp loss in the lower tail of the loss distribution curve [c.f. Stulz (1996)]. Another advantage for an efficient risk management program may be a lower cost of employment, since workers are more willing to be employed in a financially stable firm with a lower wage than work in a risky firm. In summary, an efficient risk management program will not only reduce the level of losses incurred by a firm, but it will also 4 help the firm improve its financial performance and employee morale. From a corporate finance perspective, risk management is becoming a more important field. For instance, during 1990, U.S. domestic firms spent more than $6001 billion on projects related to risk management, such as product liability, workers’ compensation, employee health, dental, disability and pension benefits, and other insurance- related products. Research has disclosed that firms typically pay approximately 40% of their payroll costs on risk management activities with health care and employee benefits alone, which account to approximately 26% of the cash flow of a typical firm. For instance, General Motors Corp. spent more than $4 billion on health care costs for its employees and is now attempting to cut the costs by transferring some of it to employees. Recently, large jury verdicts awarding compensations to customers or employees of business have skyrocketed, making risk management even more important for businesses. These types of costs of uncontrolled risk can, and have, bankrupted corporations while clearly focusing other corporations on the importance of managing and controlling their own 5 1 See George Rejad’s Principle of Risk Management and Insurance. risks. An integral part of managing risk is to transfer the financial impacts of such risks to another party, if possible, or if the transfer is not too expensive. Historically, a common mechanism for such transfers is to use the insurance industry to retain these transferred risks, by assuming the insurance companies are risk neutral. Theoretically, the law of large numbers, which states that as the number of independent and identically distributed risks becomes arbitrarily large, the standard deviation of the average loss distribution converges toward zero, makes the insurance industry viable and profitable. In fact, apart from government regulation, insurance may still be the single most important mechanism for regulating risk and safety in society, though many alternative risk- transferring vehicles have been introduced and implemented recently. Risk management and insurance costs and protection are also important for individual consumers. Individuals traditionally purchase automobile, homeowners, and other insurance to stabilize their financial conditions, and prepare for uncontrollable risks such as accidents, death or illness. 6 1.2 Literature on Risk and Risk Management In the economics and finance literature, risk and risk management have drawn considerable attention. One of the major instruments to manage risk is insurance. At the corporate level, the role and impact of using insurance to transfer risk have been studied extensively [c.f. MacMinn (1987a); Mayers and Smith (1987); MacMinn (1989); Garven and MacMinn (1993), MacMinn (1999)]. Insurance is an efficient instrument to manage the pure risk [c.f. MacMinn (1999)], and preserves corporate value when there is a probability of insolvency. At the individual level, the demand for insurance in an investor’s optimal portfolio was also investigated [c.f. Doherty (1981); Doherty (1984); Doherty and Schlesinger (1983a); Doherty and Schlesinger (1983b); Mayers and Smith (1982); Mayers and Smith (1983)]. In general, insurance has been demonstrated to be an indispensable risk-management instrument. It was found that insurance should be included in the optimal investment portfolio when pure risk is present. The role that insurance plays in financial markets can be explained by distinguishing between risks. The finance paradigm 7 characterizes risk as systematic or non-systematic, or equivalently, diversifiable or non-diversifiable. The insurance paradigm, however, classifies risks as either pure or speculative. The different definition of risk separated finance from insurance literatures. For a long time, finance and insurance were separated because of this line, and the convergence of hedging by financial instruments and insurance contracts was largely ignored. The main purpose of risk management is to preserve or create values by selecting optimal contract sets. For firms, corporate value is preserved by the inclusion of hedging instruments in the financial contracts. For individual investors, more expected utility is created by the inclusion of insurance and/or other hedging contracts in the investment portfolio. Before the demise of the Glass-Steagall Act in the 1990s, insurance and finance industries were treated separately. Recent mergers, particularly the merger between Citicorp and Travelers, represent the convergence of finance and insurance in practice. These new acts and mergers changed the conditions and more risk management instruments are combined with insurance and financial techniques. It is not hard to follow the convergence trend. The main 8 function of financial markets is to efficiently allocate the risk and facilitate the redistribution of risk. The development of financial markets has been benefited from the better understanding of separable risks. When the risk becomes more transparent, it facilitates the redistribution process in the financial market. Insurance contract is a subset of financial contracts. If the combination of insurance and financial hedging contracts creates more value for the firm, it is preferred. The recent convergence in the finance and insurance industries enhances the main function of risk management, that is, by efficiently combining risk management techniques, more value is created. The development of catastrophe bond is a good example of the convergence of financial and insurance techniques. By tapping the pure catastrophic risk into the financial market, a better allocation of risk and resource is achieved between firms and investors. Consequently, more value is generated. Therefore, there is a need to extend the current economics and finance paradigms to better understand how the financial instruments can hedge the risk and allow an efficient allocation of risk bearing and resources. Based on the extended paradigms, further studies on how to combine the different hedging instruments to reach 9 the optimal allocation of risks and resources can then be made. A robust model to incorporate and compare different hedging instruments has not yet been developed in the literature. A better understanding of risks and risk comparisons is needed, though a perspective for comparing risks has been provided [c.f., Rothschild and Stiglitz (1970)]. The process of risk being generated and valued must also be well understood to compare different hedging instruments. The optimal hedging contract set can then be chosen to preserve the most value. The recent developments of risk management instruments and financial markets reveal the continued separation of risks so that each can be redistributed at the least cost to society. To understand this behavior, more fundamental notions and understanding of risks are required. A robust model is needed to allow the risks to be valued separately and so the choices for an optimal hedging contract set can be made. 10 1.3 The Frontier of Risk Management With the convergence of finance and insurance, many new risk management tools were developed. Alternative Risk Transfer (ART) is one subset of them. With the rapid identification and separation of new risks, many of the ART solutions are tailored to specific client problems and offer integrated risk management solutions, for instance, a multi-year or multi-trigger cover. ART solutions focus on increasing the efficiency of the risk transfer, broadening the coverage of insurable risks and tapping the capital markets for additional capacity. These risk management solutions make it easier for a company to efficiently manage those risks from which it gains no comparative advantage in managing itself. The development of ART is consistent with the objective of an efficient risk- management solution by allowing institutional clients to allocate more capital to their core businesses, thereby generating higher returns. Over the last few years, ART solutions have expanded rapidly. The initial focus is on captives. Such solutions will allow companies to deal with high-frequency risks in a more cost-efficient way than through traditional industrial insurance. Since 1980s, captives have been increasingly used as a financing instrument for some low- 11 frequent, but high-severity risks which could not be placed in the traditional ways. Despite the continuous erosion of tax benefits and the persistently low premium rates in the traditional market during the 1990s, captives are set to come into their own as holistic risk management tools. The potential for the sustained growth of new risk management strategies and tools is considerable. Captive is one example. Other new instruments have been created and implemented. Attempts to transfer insurable risks directly to capital market investors have received special attention over the past a few years. This can be through the securitization of risks in the form of insurance bonds or via derivative transactions. In this way, the policyholder obtains additional capacity without incurring any credit risk, in particular, for the catastrophe losses, while investors are able to further diversify their investment portfolios with the invention of new instruments such as CAT bonds. The rapid development of risk management instruments must be based on a widely acceptable economic and/or financial mechanism for the instruments to efficiently hedge the risks. For instance, weather derivatives may be a good example to hedge the 12 weather-related risks. To tap the capacity of financial markets, a widely understandable pricing formula and the positive role of adding weather derivative into the optimal contract set of the firm must be investigated. In addition, the advantages and disadvantages of using those new hedging instruments must also be understood for them to be widely implemented and traded. 1.4 Weather Risk and Weather Derivatives As different as they may be, the business and production processes of utility companies, theme parks, fashion houses, ice- cream manufacturers, building companies and sports goods manufacturers, all have one thing in common: their business success is highly dependent on prevailing weather conditions. Nearly everyone talks about the weather but few do anything about it. Virtually all sectors of the economy are directly or indirectly subject to the influence of the weather in some form or other. For instance, daily beer consumption can be increased if the temperature rises. Weather risks are important to the energy and power supply industry because their product price is highly sensitive to the spot weather conditions. The agriculture industry is another example. Production level is highly 13 correlated with weather conditions, which affects profit. Unusual weather patterns have increasingly prompted companies whose results are affected by the prevailing weather conditions to seek protections against effects of this kind. An unusually warm winter may cut the revenue figure of a natural gas company, but an unusually cold winter may sharply increase the demand of natural gas. Hedging weather risks not only will yield a more predictable revenue stream, but also increase the shareholder’s value. The energy and power sectors may benefit most from the new tools to hedge weather-related risks, with other hedging instruments including weather derivatives in the firm’s portfolio will stabilize the net revenue stream and reduce the loss of revenue in the lower tail of the weather- risk distribution. A weather derivative is an exotic derivative because the underlying asset is not negotiable or traded. It is based on the weather index, such as Cool Degree Days or Heating Degree Days, which influence the volume of the goods, particularly the energy products, traded in the market. Therefore, the major effect of weather derivatives is to hedge the volume risk, instead of the price risk. For 14 instance, a warmer-than-average winter season may drive the natural gas demand lower, thus a weather put2 option will be in the money and will provide compensations to the option buyer, since the weather index (HDD in this case) is below the strike value. The notion behind a weather hedge is that the results of weather-sensitive sectors can be subject to great volatility, even if prices remain unchanged, due to a change in demand or volume. An important, but unresolved issue is a unanimous pricing formula for the weather derivative. This prevents the weather derivative from being traded efficiently in the market. The underlying asset is not traded and the traditional Black-Sholes model cannot be applied. In addition, the assumption of the log-normal distribution of the weather index, such as CDD or HDD, is difficult to verify due to the lack of a robust empirical test and consideration of seasonal change. To solve this issue, the dynamism of the weather (trends) and the actuarial weather forecast information must be included in the pricing model to best reflect the value of weather derivatives. In general, risk is a commodity that may be produced, exchanged or preserved. The way in which businesses and society 15 2 A put option will be in the money when the index is below the strike value. In the assess, control and transfer risk has been examined extensively and with the invention of new risk management tools being studied continuously. This dissertation is an attempt to examine how weather derivatives are incorporated into the corporate contract sets, how to price the indexed derivatives, and the implications of the basis risk. The Appendix is an expansion of the uncertainty model [c.f., MacMinn and Holtmann (1983)], including more stochastic inputs, to examine whether the widely cited Principle of Increasing Uncertainty (PIU) may still hold in this extended model. Chapter two discusses the inclusion of the weather derivative into the firm’s hedging contract set and valuation of the weather derivative. Chapter three delineates several scenarios based on corporate-leverage levels. For each scenario, the impacts of different hedging strategies on the corporate value are studied. In chapter four, implications for the corporate payoff and corporate value are studied when the basis risk is present. 16 warmer winter, the HDD index will be lower, since less heating is needed. Chapter II Weather Derivatives and Its Valuation 2.1 Introduction 2.1.1 Weather Risk and The Creation of Weather Derivatives In recent years, several unique conditions, e.g., unpredictable weather conditions that affect corporate revenue, combined with the need to manage risks produced one of the novel financial products of the last decade: weather derivative. The historical circumstances that accelerated the development of the trading market include: unusual weather conditions in different regions, and change of the demand for weather-associated commodities, causing possible price risks. These extreme weather patterns exposed the high level of weather-related risk embedded in the operations of many companies that, as a result, affected both their revenues and net earnings. For example, an unusually cool summer will decrease the demand for ice cream or electricity consumption because of the lesser use of air conditioning, and an unusually cold winter will increase the demand for natural gas 17 for heating purpose as in the past winter of 2000. In fact, virtually every sector of the economy is affected by the weather to one extent or another. Energy products may be the most sensitive to unusual weather conditions. Weather remains the single largest variable in the energy spot-market price. According to research by Koch Industrial and Utility Services [c.f., Richter (1998)], a 10 percent colder-than-normal temperature in summer can decrease the natural gas spot price by 15 percent. If average heating-season temperatures rise 1.43 degrees Fahrenheit above normal, the drop in demand for natural gas will outweigh the increase in demand that occurs naturally each year to satisfy economic growth, forcing prices down overall. Meanwhile, electric utilities also exhibit heightened weather sensitivity in the summer, and hydroelectric utilities are affected not only by temperature anomalies, but also by snow and rain anomalies. Recent data for the unit price of natural gas over the past few years3 clearly shows a strong negative correlation between the temperature and the 18 3 These data were kindly provided by Enron Capital. spot price for natural gas in Henry Hub4 (see Figure A in Appendix II). To measure the correlation between weather conditions and earnings, it is necessary to have a generally acceptable way to interpret the weather conditions in different seasons, particularly the summer and winter. In general, weather conditions are interpreted by the introduction of a weather index in different seasons. A widely accepted weather index is Cooling-Degree-Days (CDD) in the summer or Heating-Degree-Days (HDD) in the winter. The daily HDD may be defined as the maximum of zero and the difference between 65 degrees Fahrenheit and the daily average temperature, where the daily average temperature is the average of the maximum and minimum temperatures (midrange) recorded at a designated reporting station during a 24-hour period beginning at midnight. That is, if we let the daily average temperature be iT , the daily number of “heating degrees” is i iHDD max(65 T ,0)= − and the accumulated “heating- 19 4 Henry Hub is one of the major marketplace that Enron Capital uses to trace the energy commodity price in different seasons. It provides research data to illustrate degree-days” (HDD) over one month (30 days) period ending at date t is 30 t t i 1 i 1 X HDD − + = = ∑ 5. Hence, the larger the HDD, the colder the winter, and more heating may be needed. Similarly, in the summer season, the daily number of “cooling degrees” is i iCDD max(T 65,0)= − and the accumulated “cooling-degree-days” (CDD) over a 30-day period ending at date t is 30 t t i 1 i 1 X CDD − + = = ∑ . It follows that a cool summer may have an overall low CDD numerical value. With broad applicability ranging from electric utilities to long- underwear companies to theme parks, the weather-derivative market has the potential to exceed the $200-billion electricity market, according to Enron Capital & Trade Inc. The growing weather derivative market may help control its effects on businesses. Through October 2000, there are more than a dozen companies actively engaged in the transactions of weather derivatives, making market capitalization about $2 billion6. The weather derivative is designed to manage the weather risk, a risk that does not have an immediate and 20 the relationship between energy commodity price and temperature. 5 Therefore, the accumulated HDD by day 5 will be the sum of 5 4 1HDD ,HDD ,...,HDD . 6 Provided by Swiss Reinsurance Co. direct impact on other risks. It is usually written on an index of the weather conditions, such as CDDs or HDDs, and can take the form of calls, puts, swaps, caps, collars, or floors. The weather-derivative market is in many ways a perfect area for the development of risk- management products. The need for protection is universal and the information about weather conditions is widely available. By efficiently targeting volumetric risk in situations in which price-based derivatives have previously fallen short, weather derivatives provide the ability to combine hedges on weather-related risks, such as temperature or precipitation, with a more typical price-based hedge on energy commodities, such as natural gas or electricity tied to a weather index. A simple._. interpretation of weather options is provided in the following table: 21 Table 1 Option Type Hedge For Exercised While Option Value HDD Call Unusually Cold Winter HDD>Strike Value f(HDD-Strike Value)* HDD Put Unusually Warm Winter HDD<Strike Value f(Strike Value-HDD)* CDD Call Unusually Hot Summer CDD>Strike Value f(CDD-Strike Value)* CDD Put Unusually Cool Summer CDD<Strike Value f(Strike Value-CDD)* (*The option value will be dependent on the difference of actual weather index and strike values, i.e., it is a function of this difference.) A more complicated hedging structure may be constructed by a combination of the above option types, and varies according to customer needs. One of the earliest examples of a weather-derivative deal can be dated back to July 1996, when the power marketing group of Aquila Energy structured a weather hedge for Consolidated Edison Co.’s Megawatt Hour Store in New York City. The transaction was based on the Cooling-Degree-Days (CDD) for the month of August 1996 in New York City’s Central Park. In this contract, the embedded risk was that in a cooler-than-normal month spot-market prices for power would be lower than the fixed price at which Consolidated Edison Co. had purchased the commodity. In addition, a cooler summer will cut the electric sales for its Megawatt Store, causing revenue shortfalls. By giving it a rebate for a cooler-than-normal 22 summer, Consolidated Edison Co. had the opportunity to recoup the lost opportunities in the spot market. The detailed option structure was: if accumulated CDDs in August 1996 were from 0 to 10% below the expected 320 as weather stations did, the company received no discount to the power price, but if the accumulated CDDs were 11 to 20% below normal (320 in this case), Consolidated Edison Co. received a $16,000 discount in total. From 20 to 30% below normal, the discount value will be increased to $32,000, and if the total CDDs are more than 30% below 320, the maximum discount in total was $48,000. In fact, August did have some muggy and cooler days, which reduced the total CDDs and provided protections for Consolidated Edison Co., which was what it wanted. One of the major obstacles to the universal acceptance of weather derivatives is that its valuation stays unresolved. This is because the underlying “asset” (weather index as HDD or CDD) is untradeable. This untradeable index separates the weather derivative from other traditional hedging instrument, because there is no basis to price it as a financial derivative. The underlying “asset” of the weather derivative is based on data such as temperature, which influences the trading volume of other goods. Usually the weather 23 index, i.e., the underlying “asset” in the weather derivative, can be interpreted as CDD, i.e., cooling-degree-days, or HDD, heating-degree- days. This challenging valuation problem must be resolved to tap the weather derivative to the greater financial market. Two competing paradigms exist: one is based on the application of actuarial techniques, and the other is based on the Black-Sholes approach to the pricing of derivatives. The unique valuation challenges of weather derivatives may lead to the emergence of a hybrid approach and a new theory of valuation. A weather hedge is important because it can stabilize the forecast for the future revenue, or income stream, for the weather- sensitive sectors, thus a well-predictable business can result in increased shareholder value7. The objective of risk management is to preserve or create values for the firm. If the revenue of the firm is sensitive to the weather conditions, for instance, an energy firm, hedging with weather derivatives will recoup part or all the revenue loss due to negative weather conditions. When there is a risky leverage, the insolvency probability may be reduced or eliminated by 24 7 As illustrated by the following sections, a hedged firm generally will have less risky production than unhedged one. By taking a less risky position, corporate value is increased. hedging with weather derivatives. Therefore, the bond value is preserved, and more corporate value is achieved. For individual investors, investing in the energy firms hedged with weather derivative reduces the overall risk in the portfolio. Assume that the investors are risk-averse, a portfolio with the same expected return but less risk will be preferred, since more expected utility is achieved, i.e., more value is created for the investors. The development of weather derivatives also represents one of the recent trends toward the convergence of insurance and finance [c.f., MacMinn (2000)]. By efficiently tapping capital markets, this convergence will transfer the risk to a much larger capital capacity pool. The creation of weather derivatives also challenges another fundamental difference in the way that insurance and financial industries solve the problem of risk management. In general, the property/casualty insurance industry uses the principle of diversification, pooling many uncorrelated risks and charging a premium based on actuarial probabilities of occurrence of different risks and their correlations. Financial derivatives in risk management are based on the option pricing and hedging algorithms, which were originally developed in the 1970s. Which paradigm(s) should be 25 selected to price the weather derivatives still remains unclear. This paper is an attempt to price it in an expanded economic model, by integrating both the actuarial and financial approaches. 2.1.2 Literatures on Economics and Finance One of the fundamental research topics on the weather derivative is how to price this new hedging instrument. Although the structure of the weather derivative and the mechanism of how it hedges the weather risk have been extensively studied [c.f., Leggio and Lien (2000), Muller and Grandi (2000), Richter (1998)], little research has been done to provide an explicit explanation of how to price the weather derivative. In addition, most of the literature does not present a model where the weather-related risk was incorporated into a robust economic framework, as an explicit source of uncertainty. Economic and finance literature on the theory of the firm has concentrated on the uncertainty arising either from the demand of its product or from the firm’s technology [c.f., Leland (1972), MacMinn and Holtmann (1983)]. In these studies, firms had a single source of risk: a one- dimension uncertainty framework. None of the models had included 26 multiple sources of risk. In fact, one unique characteristic of weather risk is its independence with most other risks. Consequently, it may be considered as a second source of risk, in addition to other existing risk in the corporate finance literature. Another implication of using the weather derivative is from the basis risk. Though basis risk has been extensively interpreted recently [c.f., MacMinn (2000)], few studies quantified basis risk according to the corporate finance perspective. It is widely known that the basis risk arises from using a standard industry index. By using the standard index, some of the problems often hampering risk management, such as moral hazard and adverse selection, are eliminated. Basis risk may also arise when a standard index is used. Some of the most recent empirical work on corporate hedging behavior [c.f., Haushalter (2000)] revealed that firms would prefer to use hedging instruments with little basis risk. An explicit explanation in a theoretical framework has not yet been provided and the implications of basis risk for corporate hedging remains unclear. An economic model to explicitly incorporate weather risk into the firm’s payoff structure was presented in this chapter. Several important questions arising from using the weather derivatives were 27 addressed. For example, how to quantify the weather risk in an incomplete financial market; whether the incorporation of weather derivatives in the firm’s contract set adds value; whether the role of weather derivatives can be duplicated by other hedging instruments such as swaps or forwards, and if so, an alternative pricing formula may be provided for the weather derivative based on the no arbitrage principle; what is the optimal contract set to hedge the weather; what are the positive and negative impacts by using different set of hedging instruments? Chapter four discusses further whether the basis risk should be hedged or retained under different conditions. 2.2 Basic Model In the standard finance literature, there exists an economic state set = ωeZ [0, ] , where the economic state ∈e ez Z spans the space ω+! 1 in a complete financial market without the inclusion of weather risk. There are two dates in the model for which the economy is operated, now and then, i.e., =t 0 and 1. All of the uncertainties will be resolved at time =t 1. All financial and operation decisions are made now while all payoffs on those decisions will be received then. In the standard complete financial market model, an economic state may 28 be interpreted as an index of economic conditions and it is assumed that there are as many stock contracts, i.e., the basis stock, as there are states of nature, and each basis stock contract pays one dollar in a particular state and zero otherwise. A corporate stock may then be interpreted as a portfolio of these basis stock contracts. It follows that in a complete financial market, a corporate stock is a speculative risk. The economy is composed of individual investors, who make their investment portfolios to maximize the utility, and corporations where there are managers, who make operation and financial decisions on behalf of the shareholders. One simple interpretation of the managers’ objective is to maximize the current shareholders’ market value. With the inclusion of weather risk, the otherwise complete financial market becomes incomplete. In the economy constructed with weather risk, the state space is expanded and the weather index wz is incorporated. For simplicity, throughout this dissertation, the weather index is assumed as heating-degree-days (HDD), and a natural gas utility firm is the object firm being studied. Therefore, “then” defined in this model may be interpreted as a winter season. With the inclusion of the weather index into the original model, 29 the state space is interpreted now as ∈ ×e w e w(z ,z ) Z Z . The variable ez is interpreted as an index of economic conditions and = ωeZ [0, ] is the set of these index numbers. The variable wz represents a weather state and = υwZ [0, ]8 is the set of these states. Throughout this dissertation, the weather index wz represents different HDDs in the winter season. The weather risk is a speculative risk, i.e., weather- related loss could be positive or negative. If it is positive, it means the natural gas utility suffers a loss because of the weather condition, i.e., the HDDs are below the neutral level. If it is negative, it means the firm gains from the weather conditions, i.e., the HDDs are above the neutral level. The corporate net payoff is represented as e w(q,z ,z )Π , including the impacts from economic and weather conditions. The firm’s payoff function is Π = −e wP(z ,z )q c(q) , where P is the unit price of the firm’s product then, and c(q) is the cost function for a production level of q . If the cost function is assumed convex, the corporate payoff is concave 30 8 This is the set of HDD values for the observation period. Based on the definition of HDD in the first part (Introduction) of this chapter, the sum of the daily HDD number over the observation period must be at least zero. in production level. It is a competitive firm facing with price uncertainty. The firm’s payoff function is satisfied with the principle of increasing uncertainty (see Appendix). Let +wZ and − wZ be the weather states that bring the positive and negative loss to the firm respectively, i.e., +wZ represents lower HDDs and −wZ represents higher HDDs at time t 1= . Let = o 0 w wZ {z } be the neutral weather state, i.e., the actual HDDs coincide with the forecast of weather stations. Table 2 presents the components of this model. Table 2 eZ Economic index set [0, ]ω wZ Weather index set [0, ]υ + wZ Weather index set with positive loss − wZ Weather index set with negative “loss” 0 wz Neutral weather index ez Individual economic index wz Individual weather index9 waz HDD index used in weather derivative k wz Strike value selected by firm Π Firm’s payoff function q Firm’s production level For the firm studied in this dissertation, a HDD put option is 31 used to hedge an unusually warm winter. If kwz is the strike HDD value, and the measurement site in the weather derivative contract has a HDD numerical value as waz over the observation period10, the payoff of the weather derivative is = −kwa w waG(z ) I[max (z z ,0)] , where − k w wamax (z z ,0) is the exercise value of weather derivative contract. − k w waI[max (z z ,0)] is the industry standard dollar amount11 of this exercise value. By using the industry standard dollar amount, and measuring on the benchmark site12 instead of the exposure site, the moral hazard and adverse selection problems are eliminated but the basis risk13 is introduced. 32 9 In this dissertation, the weather index is the numerical sum of HDDs over the observation period. 10 The weather risk exposure site, i.e., hedged site, may not be the same site where the HDD in weather derivative contract was measured. The uncertainty of this difference may be minimal, though. First, the utility firm is usually local, and a local measurement site is available at airport or within the city. Even if it is not close to the exposure site, there must be a strong HDD correlation between the measurement site and exposure site, because both are within the same local city. Second, the HDD difference may be minimized over the observation period if the exposure and measurement sites are close and the period is long enough. Usually the observation period in weather derivative contract is at least 30 days. Therefore, the uncertainty of the HDD difference is negligible. A detailed discussion can be found in Chapter IV. 11 An industry standard rebate eliminates the moral hazard problem. A plausible way is to categorize firms into small, medium, and large operation size based on the revenue history, and rebates to the hedged firm based on its category, the expected economic condition and the ex post HDD value. 12 A benchmark site is usually highly reliable and has a long history of recording. Usually it is sponsored by the government. Examples of benchmark measurement sites are airports, central parks etc. 13 A detailed discussion of basis risk can be found in Chapter IV. If the firm selects weather hedging and we let = e wz (z ,z ) , the payoff becomes Π + = Π + − wa k w wa (q,z) G(z ) (q,z) I[max(z z ,0)] (0.1) If the weather derivative is exercised, the corporate payoff is Π + −ke w w wa(q,z ,z ) I(z z ) (0.2) With the absence of basis risk14, the firm would have recouped its revenue loss due to negative weather conditions, by exercising the weather derivative. Contrasted with the industry standard loss − k w waI(z z ), the firm’s actual loss experience is = Π − Πke w e wL (q,z ,z ) (q,z ,z ). In the absence of basis risk, ≡I L 15. The value of the basis risk is −I L . If the weather derivative is in the money, a swap may be used to eliminate the basis risk. The firm exchanges the HDD put with the real loss experience less the swap price p, i.e., −L p . If the swap price is less than the negative of basis risk, i.e., < −p L I, it would be preferred. 33 14 Basis risk is the difference between the payoff of weather derivatives and the actual loss experienced by the firm due to negative weather conditions. The implications of basis risk will be discussed in Chapter IV. For simplicity, it is assumed that ≡w waz z and ≡I L in this chapter and the following Chapter III, i.e., basis risk is assumed absent. Let B be the corporate leverage level. For an unhedged firm, the net payoff then is Π − B . For a hedged firm, the net payoff then is Π − +B W , where W is the rebate provided by the weather derivative, i.e., = wW G(z ). With a risky leverage, firm has a probability of insolvency. When the firm becomes insolvent, stock value is zero and bondholder becomes the creditor of the firm16. To study the stock and bond value of the firm, an iso-earning curve in the space of ×e wZ Z is interpreted as below. To better analyze the firm’s economic behavior, the firm’s payoff function Π is assumed to be well defined. That is, it is continuously twice differentiable at each of its arguments ez and wz (Continuity and Differentiability), and strictly increasing at each of its arguments ez and wz , i.e., ∂Π > ∂ e 0 z and ∂Π > ∂ w 0 z (Monotonicity). In addition, for each given production level q , the iso-earning curve Π = c in the plane of ×e wZ Z is strictly rotund (convexity), i.e., if both 34 15 I is the payoff of the weather derivative, and L is the loss actually experienced by the firm due to negative weather conditions. = 1 1 1 e wz (z ,z ) and = 2 2 2 e wz (z ,z ) generate the same payoff c, any weighted average of 1z and 2z must generate more payoff. It is noted that for different level of earning, the iso-earning curves may have different shapes in the plane of ×e wZ Z , though all strictly convex. For a given leverage level B, the iso-earning curve17 may be interpreted by a smooth and continuous convex curve as in Figure 118. To better illustrate the figure, the notations are defined in Table 3. 35 16 It is assumed that bondholder is the only creditor for the firm if it is insolvent. 17 In Figure 1, the iso-earning curve Π = B is also the insolvency boundary line for an unhedged firm. 18 It is noted that ∂ = − Π Π < ∂ e w e w z 0 z , and ∂ > ∂ 2 e 2 w z 0 z by PIU, where Π = ∂Π ∂w wz and Π = ∂Π ∂e ez . Table 3 Figure 1 (See Appendix II) displays the iso-earning curve then, after the production decision q was determined19. For each given q , the firm’s payoff Π is determined by = e wz (z ,z ) then. Principle of Increasing Uncertainty gives the decreasing shape of each iso-earning curve Π = c under all possible production decision q . Each 36 19 Figure 1 is an iso-earning curve then with given production level q . A well maintained technology is implicitly assumed, and firm manager knows the location 1Θ The sets when the weather put is exercised, i.e., the sets × < ke w w wZ {z : z z } 2Θ The sets when weather put is not exercised, i.e., the sets × ≥ ke w w wZ {z : z z } 1A Solvent states with negative weather condition, without using weather derivative, i.e., the sets × Π > ∩ Θe w e w 1{z z : (z ,z ) B} 2A Solvent states with neutral or positive weather condition, i.e., the sets × Π > ∩ Θe w e w 2{z z : (z ,z ) B} 1aB Insolvent states difference b/w using weather derivative and without using weather derivative, under negative weather conditions, i.e., the sets × Π < < Π + ∩ Θe w e w e w 1{z z : (z ,z ) B (z ,z ) W} 1bB Insolvent states with using weather derivative, under negative weather conditions, i.e., the sets × Π + < ∩ Θe w e w 1{z z : (z ,z ) W B} 2B Insolvent states with neutral or positive weather condition, i.e., the sets × Π < ∩ Θe w e w 2{z z : (z ,z ) B} combination of e w(z ,z ) in the black convex curve represents the same corporate payoff Π = B 20. The more corporate leverage level is, the more upward the iso-earning curve has to move. It is because a better economic condition and/or a more positive weather condition are/is needed to keep solvent at a higher leverage level. It is noted that different iso-earning curves under given production decision q may have different shapes (elasticity), though all convex and decreasing. The function format of Π e w(z ,z ) determines the shape21. For unhedged firm, the iso-earning curve (insolvency boundary) is Π = − =e w e w(z ,z ) P(z ,z )q c(q) B 22. For hedged firm, the iso-earning curve (insolvency boundary) is Π + = Π Π =ke w e wW max[ (z ,z ), (z ,z )] B23. In Figure 1, ∪1 2A A denotes the combination of economic and weather states where the firm is solvent with a risky debt, without using the weather derivative; ∪1 2B B denotes the condition in which 37 of production surface and his location on production surface with control of input choice. Firm manager also knows the cost of each input choice. 20 It is noted that B was determined now, and payoff Π is realized then. 21 Let the iso-earning curve be =e wz g(z ) in Fig 1, different iso-earning curve yields different function g such that Π =w w(g(z ), z ) B . In fact, =e B wz g (z ) may be used to better interpret that function g is dependent on B. 22 Shown as the black curve in Fig 1. 23 Shown as the red curve in Fig 1. With the absence of basis risk, payoff of the weather derivative recoups the loss due to negative weather conditions for the firm, i.e., = − = = Π − Πk kw w e w e wW I(z z ) L (q,z ,z ) (q,z ,z ) when it is exercised. firm is insolvent without using the weather derivative. Θ1 represents the combination of the states on which the weather derivative contract can be written. If weather derivative is used, the insolvency boundary line is dropped to the red line shown in Figure 124, and the insolvency states can be interpreted by 1b 2B B∪ . Therefore, by using weather derivatives, the insolvency probability is reduced. 2.3 Valuation of Weather Derivative One of the fundamental problems on weather derivative is its valuation. With a non-tradable underlying “asset”, the traditional financial valuation technique for derivatives cannot be applied. Actuarial technique, which is widely used as an insurance pricing paradigm, may not well hold to price weather derivative, either. This section is an attempt to price the weather derivative on the extended basic model. In our economy, there are two agents in the market: real risk- 38 24 It is because within Θ1 , i.e., weather derivative is exercised since < kw wz z , the firm’s payoff becomes Π + = Π ke wW (z ,z ) . At the iso-earning curve as well as the insolvency boundary line Π + =e w(z ,z ) W B for the hedged firm, there must be a averse individual agents and the fiduciary agents for the corporations25. The economy has two independent risks, which can be characterized by weather states ∈w wz Z , and economic states ∈e ez Z . The state set is expanded to ∈ ×e w e w(z ,z ) Z Z . The standard finance literature assumes the existence of basis stock in a complete market. The inclusion of weather risk makes the market incomplete. If the firm does not hedge the weather risk, the risk-averse investors will find a way to hedge it. Consider purchasing put and call option portfolios in the following way: Investors can go long in corporate stock and put options, and go short in the call options at the same time. If the exercise price in both puts and calls is fixed at E , the put has a value as − Πmax[0,E ], and the call has a value as Π −max[0, E]. The portfolio has a value as Π + − Π − Π − =max[0,E ] max[0, E] E . For each given wz , the firm’s payoff may be interpreted by the various straight lines (black ones) in Figure 2 (See Appendix II)26. For each given ez , 39 unique = ke B wz g (z ) to define the insolvency boundary line as the red line in Fig 1. This is due to the monotonicity of the firm’s payoff function Π . 25 It is not necessary to introduce another fiduciary agent in the economy. However, it makes the model explicitly clear with the introduction of fiduciary agent. 26 It is due to the monotonicity of Π , i.e., Π >e 0 . * w ez (z ) is defined as the risk-adjusted weather state, i.e., Π = Π∫ w * e w w e w wZ (z ,z ) f(z ) (z ,z )dZ 27. The straight line (red one28) in Figure 2 displays the payoff function Π *e w e[z ,z (z )] . If the option strike price is set as = Π = Π*e w eE (z ,z ) E( z ) for each economic index ez , it follows by Jensen’s inequality29 that, this hedging portfolio is preferred by risk-averse investors. This hedging behavior allows a simple interpretation of the stock value for the unhedged firm. Let ep(z ) be the basis stock price now which will pay one dollar then in state ez and zero otherwise and assume such a basis stock exists30. The hedged31 payoff on corporate stock on the contingency of economic index ez is Π = Π * e w e(z ,z ) E( z ). 40 27 In this definition, wf(z ) is the density function. * wz is an economic risk-adjusted weather index. Although wz and ez are independent, * wz is generally dependent on ez , i.e., for different ez , * wz may be different. 28 For simplicity, Π *e w e[z ,z (z )] is displayed as a straight red line. Since * wz is generally dependent on ez , Π * e w e[z ,z (z )]may be a curved line instead of a straight line. 29 Jensen’s inequality indicates that for risk-averse utility functions, E[u(c)] u[E(c)]< . In our framework, it means Π < Πe eE[u( z )] u[E( z )] . 30 This assumption holds because of the independence of ez and wz . 31 If the firm is unhedged, risk-averse investors prefer to hedge the weather risk on their own accounts. This is because all individual investors are assumed risk-averse in the financial market. By holding the portfolios discussed earlier, investors gain more utility. Since there are only real agents, i.e., individual investors, and fictitious agents, i.e., fiduciaries for the firms, in the market, the payoff on corporate stock becomes Π = Π*e w e(z ,z ) E( z ) when all risk-averse investors hold the same portfolio. The unhedged firm’s stock value is then interpreted as Θ = Π = Π = Π ∫ ∫ ∫ ∫∫ e e w u * e e w eZ e w e w w eZ Z w e e w e w S p(z ) (z ,z )dz p(z )[ f(z ) (z ,z )dz ]dz f(z )p(z ) (z ,z )dz dz (0.3) These results are summarized in the following proposition. Proposition 1: Assume the independence of the weather index and the economic index32, an unhedged firm has a stock value as Θ ω υ = Π = Π ∫∫ ∫ ∫ u w e e w e w w e e w e w0 0 S f(z )p(z ) (z ,z )dz dz f(z )p(z ) (z ,z )dz dz 41 32 Though weather conditions may have an impact on the economic index, the impact may be negligible. Two reasons may explain this: First, a certain weather condition may have negative impacts on some firms, but may be positive on other segments of the economy. Second, economic conditions may be more affected by other factors, such as notional GDP, financial markets, unemployment rates etc, making them less dependent on weather conditions. Proof: See above results. QED Fiduciary for the firm can also hedge on the corporate account by using weather derivatives. Assume that the strike index value in the weather derivative is set at kwz , the hedged firm’s payoff is υ υ = − + Π + Π − Π = − + Π Π ∫ ∫ ∫ ∫ e e h k k e w e w e w e w w eZ 0 k k e w e w e w w eZ 0 S P p(z ){ f(z )[ (z ,z ) max(0, (z ,z ) (z ,z ))]dz }dz P p(z ){ f(z )max[ (z ,z ), (z ,z )]dz }dz where kP is the price of weather put with a strike value as kwz . The no arbitrage principle shows that the unhedged firm has the same stock value as the hedged firm. Equivalently, it sets the value of weather derivative as υ ω υ ω υ Θ = Π + Π − Π − Π = Π − Π = ∫ ∫ ∫ ∫ ∫ ∫ ∫∫ e 1 k k e w w w w w eZ 0 w e e w e w0 0 k w e w w e w0 0 w e w w e P p(z ){ f(z )[ (z ) max(0, (z ) (z ))]dz }dz f(z )p(z ) (z ,z )dz dz f(z )p(z )max[0, (z ) (z )]dz dz f(z )p(z )G(z )dz dz (0.4) 42 That is, the price of weather derivatives must be equal to the risk-adjusted future payoff of the derivative. The above pricing formula implicitly combines the actuarial technique33 and financial paradigm34. This valuation stands in the economic model constructed here. Other hedging strategies exist. Suppose the firm goes long in a set of put options with exercise price at Π 0e w(z ,z ) at each economic state ez , where 0 wz is the ‘neutral” weather state defined in Table 2. The payoff for that set of put options is Π − Π0e w e wmax[ (z ,z ) (z ,z ),0] on the contingency of economic state ez . It follows that the price for the set of put options is Θ Π − Π∫∫ 0w e e w e w e wf(z )p(z )max[ (z ,z ) (z ,z ),0]dz dz . If the strike value in the weather derivative is also set at 0wz , it duplicates the set of put options. Therefore, there is an efficient gain by the inclusion of weather derivative in the financial market. These results are summarized as the following proposition. 43 33 The actuarial technique is implicitly included in the probability distribution of the weather index, i.e., the density function as wf(z ) . 34 The financial paradigm is implicitly included in the price of basis stock for each economic index, i.e., ep(z ) . Proposition 2: For a weather derivative with strike value at kwz , its price is interpreted as ω ._.free scenario, and makes forward contract more preferable. Scenario 2 If the firm’s leverage level is Π ≤ ≤ Πq oe w(z ,0) B (0,z ), the iso- earning curve for corporate payoff function may be displayed as the Figure 13 (See Appendix II). 96 For the first component of basis risk, it follows that ∂ < ∂ V 0 c , the same as shown in Scenario 1. The implication of second component of basis risk is not unambiguous. The presence of basis risk does not make weather derivative less preferable compared with forward in this case. This is because weather derivative still provides a positive payoff to the hedged firm under negative weather conditions, which is irrelevant with the presence or absence of basis risk. Forward contracts generally cannot reduce the insolvency probability unambiguously as weather derivative can in this scenario. Therefore, the presence of basis risk may diminish part of the positive role that an otherwise basis risk-free weather derivative does, it does not reverse to such a level that makes weather derivative less preferable. When there is probability of insolvency with the presence of basis risk, hedging with weather derivative is still optimal, but hedging with forward is not optimal. If weather derivative has been used, hedging with additional forward is not optimal. 97 Scenario 3 In Scenario 3, firm’s leverage level B increases to o f o w e w(0,z ) B (z ,z )Π < ≤ Π , where f ez is implicitly defined as f o e wf P(z ,z )= . It may be interpreted by Figure 14 (See Appendix II) with the presence of basis risk. For the first component of basis risk, it follows that ∂ < ∂ V 0 c , the same as shown in Scenario 1 and 2. The implication of second component of basis risk is not unambiguous though. With similar arguments as did in Scenario 2, the presence of basis risk does not make weather derivative less preferable compared with forward in this case. The presence of basis risk may reduce part of the positive role that an otherwise basis risk-free weather derivative does, it does not reverse to such a level that makes weather derivative less preferable. Addition of forward contract adds additional value to the firm, which follows from υ = ∂ ∂ ∂Π = > ∂ ∂ ∂∫ ∫ w hw k(z ) e w e wq q 0 0 V q p(z )f(z ) dz dz 0 n n q . . Therefore, hedging with weather derivative is still optimal, but hedging with forward is not optimal. If weather derivative has been used, hedging with additional forward further increases the firm’s 98 value. Scenario 4 In this Scenario, the corporate leverage level is increased to f o o e w w(z ,z ) B ( ,z )Π < ≤ Π ω . The presence of basis risk may be illustrated by Figure 15 (See Appendix II). For the first component of basis risk, it follows that ∂ < ∂ V 0 c , the same as shown in Scenario 1. The implication of second component of basis risk is not unambiguous though. With similar arguments as did in previous scenarios, the presence of basis risk does not make weather derivative less preferable compared with forward in this case. The presence of basis risk may reduce part of the positive role that an otherwise basis risk-free weather derivative does, it does not reverse to such a level that makes weather derivative less preferable. If forward is already used, addition of weather derivative adds additional value to the firm, which follows from υ = ∂ ∂ ∂Π = > ∂ ∂ ∂∫ ∫ w hf h(z ) e w e wk kq q 0 0 w w V q p(z )f(z ) dz dz 0 z z q . . While there is basis risk, hedging with weather derivative is still optimal compared with forward. Hedging with forward is not optimal. 99 If forward has been used, hedging with additional weather derivative will further improve the corporate value, but may not outperform the single use of weather derivative as a hedging instrument. The presence of basis risk may reduce or increase the positive role that weather derivative provides in hedging, however, the weather derivative still reduces the insolvency probability unambiguously. Scenario 5 In this scenario, the corporate leverage level is increased to o P w e( ,z ) B (z , )Π ω < ≤ Π υ 68, where P ez is implicitly defined as P eP(z , ) fυ = . The inclusion of basis risk was interpreted by Figure 16 (See Appendix II). When the firm’s leverage level comes to this high, weather derivative with a strike value =k 0w wz z cannot reduce the insolvency probability, and no firm’s value is preserved. The presence of basis risk may reduce the insolvency probability, if c is negative enough. It follows from ∂ < ∂ V 0 c . However, this implication is not unambiguous. Forward contract alone neither provides positive role in preserving 100 corporate value. While there is probability of insolvency, hedging with weather derivative is not optimal69, neither is hedging with forward. If forward has been used, hedging with additional weather derivative will not improve the corporate value, neither the opposite case, under the presence of basis risk. Scenario 6 In this scenario, the corporate leverage level is increased to P e(z , ) B ( , )Π υ < ≤ Π ω υ , where P ez is implicitly defined as P eP(z , ) fυ = . The presence of basis risk may be interpreted by Figure 17 (See Appendix II). When the firm’s leverage level comes to this high, weather derivative with a strike value =k 0w wz z cannot reduce the insolvency probability, and no firm’s value is preserved. The presence of basis risk may reduce the insolvency probability, if c is negative enough. It follows from ∂ < ∂ V 0 c . However, this implication is not unambiguous. 101 68 While this may not be guaranteed to hold, the exceptions will be discussed in the following scenarios. The presence of basis risk cannot generally provide positive incentives for firm’s managers to preserve more value70. Forward contract provides negative role in preserving corporate value, and more insolvency probability is introduced. In Scenario 5, the corporate leverage level is set as o P w e( , z ) B (z , )Π ω < ≤ Π υ , which may not hold always. One exception is as the following Figure 18 (See Appendix II). It is noted that forward contract plays a negative role, and weather derivative does provide less insolvency possibility. The presence of basis risk may reduce or increase the insolvency probability that weather derivative already reduced, but cannot eliminate the positive role of weather derivative. That is, with the presence of basis risk, weather derivative still provides a rebate under negative weather conditions, which reduces the insolvency probability as shown in Figure 18. If weather derivative is coupled with a forward, it is noted that 102 69 Unless the strike value of weather derivative is increased to >k 0w wz z . hw w e w e wq q 0 0 k(z ) e w e w0 0 V q p(z )f(z ) dz dz n n q q p(z )f(z ) dz dz n q 0 υ ω = υ ∂ ∂ ∂Π = ∂ ∂ ∂ ∂ ∂Π = ∂ ∂ < ∫ ∫ ∫ ∫ (1.3) Therefore, addition of forward will diminish the positive role that weather derivative already provides, and not preferable. Weather derivative alone provides the optimal hedging contract set. The following table summarizes the optimal contract set for hedging the weather risk, and achieve the most efficient allocation of resource, with the presence of basis risk. Table 9: Impacts of hedging portfolio with the presence of basis risk Scenario Forward Weather Derivative Both included 1 Positive (Best) Positive Positive 2 Neutral Positive Positive 3 Neutral Positive Positive (Best) 4 Neutral Positive Neutral 5 Neutral Neutral Neutral 6 Negative Neutral Negative 6b Negative Positive Neutral In general, the presence of basis risk does not make the weather derivative less desirable, except the scenario 1. It is quite intuitive. By 103 70 When the corporate leverage level comes to this high level, the rebate provided by weather derivative under negative weather conditions cannot reduce the insolvency probability, and preserve corporate value. providing a rebate to the firm under negative weather conditions, weather derivative still outperforms forward in most of the scenarios. The presence of basis risk makes the rebate more or less than the firm actually desires, but this impact cannot change the basic function of weather derivative. With the presence of basis risk, weather derivative still provides a more holistic hedging in most of the scenarios, yet the rebate is more uncertain, i.e., more or less than the firm actually desires. While basis risk seems innocuous in most of the scenarios, it does hamper the positive hedging role provided by weather derivative, for instance, in Scenario 1. The corporate value preserved by weather derivative may be reduced in scenario 1, therefore makes it less desirable. This is consistent with some recent empirical research which demonstrates a strong incentive for the firm manager to select a hedging portfolio with little basis risk [c.f.. Haushalter (2000)], because Scenario 1 represents a low corporate leverage level, which may be seen in many firms. In the presence of basis risk, this chapter has demonstrated that it is optimal to hedge against the basis loss, and the less the 104 basis difference is, the better the hedging instrument will be preferred by the risk-averse firm managers. The results are consistent with recent empirical studies in the oil and gas industries, which document strong incentives to hedge for large companies and companies which have little basis risk. 105 Figure A : G as Price v.s. Season A ppendix I Figures H enryH ub G as Price v.s. Tim e 0 2 4 6 8 10 12 14 16 18 8/31/1995 11/30/1995 2/29/1996 5/31/1996 8/31/1996 11/30/1996 2/28/1997 5/31/1997 8/31/1997 11/30/1997 2/28/1998 5/31/1998 8/31/1998 11/30/1998 2/28/1999 5/31/1999 8/31/1999 11/30/1999 2/29/2000 5/31/2000 8/31/2000 11/30/2000 2/28/2001 Tim e Points Price Index H igh Price Low Price 106 ez 0 + wZ wZ − o wz υ ω 1A 2A 1aB 2B Θ1 Θ2 Π − = Π + − = B 0 ( W B 0) Π + − = = k 0 w w W B 0 (z z ) Π − >B 0 Π Π Figure 1 Displays the a as for which firm is in the sol or insolvent states Π − =B 0 1bB 1B >k 0w wz z <k 0w wz z 107 wzventre ez e(z , )Π υ e(z ,0)Π Π *e w e[z ,z (z )] Π Figure 2: Corporate payoff structure 10 8 P f= B 0Π − = B n(f P) 0Π − + − = ez wz 109 Figure 3: Compare different iso-earning curves in the surface ez ω =P f Π − =B 0=n q Π − + =B W 0 Π − + − =B n(f P) 0 wz υ Figure 4: Compare different iso-earning curves 110 P f= B 0Π − = B n(f P) 0Π − + − =ez wz υ ω o wz n q= n 0= W B 0Π + − = 0 n q< < A 111 Figure 5: Compare different iso-earning curves P f= B 0Π − = B n(f P) 0Π − + − =e z ω o wz W B 0Π + − = B W n(f P) 0Π − + + − = 112 Figure 6: Compare different iso-earning curves wz υ P f= B 0Π − = B n(f P) 0Π − + − =e z wz υ ω o wz W B 0Π + − = B W n(f P)Π − + + − 113 Figure 7: Compare different iso-earning curves P f= B 0Π − = B n(f P) 0Π − + − =e z wz υ ω o wz 114 Figure 8: Compare different iso-earning curves o w wz z= B 0Π − = B n(f P) 0Π − + − =e z ω o wz 115 Figure 9: Compare different iso-ea o w wz z=P f= wz υ rning curves B W 0Π − + = B n(f P) 0Π − + − =e z ω o wz B W n(f P) 0Π − + + − = B 0Π − = 116 Figure 10: Compare different iso-ea o w wz z=P f= wz υ rning curves ez 0 + wZ wZ − 0 wz υ ω 1A 2A 2B Θ1 Θ2 Π − Π + B 0 ( W Π + − = = k 0 w w W B 0 (z z ) Π − >B 0 Π Π Figure 11 Displays the insolvency boundaries when basis risk is present Π − =B 0 >c 0 <c 0 =c 0 117 =wz − =B 0) ez ω =P f Π − =B 0=n q Π − + =B W 0 Π − + − =B n(f P) 0 wz υ 0 wz Figure 12: Compare different iso-earning curves with basis risk 118 P f= B 0Π − = B n(f P) 0Π − + − =ez wz υ ω o wz n q= n 0= W B 0Π + − = 0 n q< < A 119 Figure 13: Compare different iso-earning curves with basis risk P f= B 0Π − = B n(f P) 0Π − + − =e z wz υ ω o wz W B 0Π + − = B W n(f P) 0Π − + + − = 120 Figure 14: Compare different iso-earning curves P f= B 0Π − = B n(f P) 0Π − + − =e z ω o wz W B 0Π + − = B W n(f P)Π − + + − 121 Figure 15: Compare different iso-earning curves wz υ P f= B 0Π − = B n(f P) 0Π − + − =e z wz υ ω o wz 122 Figure 16: Compare different iso-earning curves o w wz z= B 0Π − = B n(f P) 0Π − + − =e z ω o wz 123 Figure 17: Compare different iso-ea o w wz z=P f= wz υ rning curves P f= B W 0Π − + = B n(f P) 0Π − + − =e z wz υ ω o wz B W n(f P) 0Π − + + − = B 0Π − = 124 Figure 18: Compare different iso-earning curves o w wz z= Appendix II Extensions of Principle of Increasing Uncertainty The risk aversion nature of the firm has been studied in several articles. It is widely believed that a risk adverse entrepreneur will select the optimal level of output where the expected unit price is the sum of marginal cost and the marginal risk premium. Both the demand uncertainty and supply uncertainty of firm have been extensively studied in the past literature. One of the representative works on the production decision given demand uncertainty is Leland (1972), where the principle of increasing uncertainty given demand uncertainty has been introduced and discussed. MacMinn and Holtmann complemented and generalized this work on the theory of firm by investigating the effect of technological uncertainty on the entrepreneur’s decision of optimal production level. Both models generate the same optimality condition, namely the expected marginal productivity equals the factor price, the marginal cost plus the marginal risk premium. MacMinn and Holtmann’s work further concluded that the sign of the marginal risk premium was not always 125 positive, as assumed in most of the literature, and it was dependent on the structure of the firm’s technology function. With different technology functions, the firm’s optimal decision would be different. In MacMinn and Holtmann’s work, the single risk averse71 nature of the entrepreneur was not enough to yield one of the general results in the theory of the firm: the optimal production level for a risk averse entrepreneur was always less than that of a risk neutral entrepreneur, unless the Principle of Increasing Uncertainty (PIU) also holds. The model presented in their work is a single owner- entrepreneur who makes the production decision, where the notion of technological uncertainty may be interpreted in distinct ways. Nonetheless, the technology is not well specified, due to the uncertainty in their model, which yields the uncertainty in the production and profit functions, and an increase of technological uncertainty will increase the risk of production and therefore, the profit distribution. In the economic and finance literature, the technological uncertainty is specified only by one non-negative continuous random variable. The Principle of Increasing (Decreasing) Uncertainty was 126 derived under this single stochastic input framework. This may not be always the case for the firm. For instance, for a firm that produces energy products like electricity or natural gas, its optimal production decision will be generally impacted by several stochastic variables, such as economic conditions, weather conditions, and credit conditions of energy marketers. Some of those variables will be independent. For example, it is possible that economic conditions and weather conditions may not be possibly dependent with each other, and others will be dependent. For instance, the general economic conditions may have an impact on the credit conditions of the energy marketers. With the introduction of new risk hedging products from the development of financial engineering, such as weather derivatives, it is necessary to have a more generalized model, which includes multiple stochastic variables to better interpret the firm’s technological uncertainty. In this chapter, an expanded model was conducted in which the entrepreneur selects one input and shows what impact technological uncertainty has on the firm’s scale of operation. Differing from the existing literature, this model includes two stochastic variables, both 127 71 Risk averse nature mentioned here implicitly defined the nature of utility function of which inject the risk into the firm’s technology. To introduce the technological uncertainty, let 1Z , 2Z be two non-negative continuous random variables. Let iΨ be their distribution functions and iψ be their density functions, where i 1,2= . The mean and variance of these two variables are iz i i i0 z d (z ) ∞ µ = Ψ∫ and i i 2 2 z i z i0 (z ) d ∞ σ = − µ Ψ∫ Let the firm’s production function be 1 2F(x,z ,z ) where x is the quantity of the nonstochastic input. Therefore, the firm’s output is a random variable 1 2Y F(x,z ,z )= . Let the marginal productivity of nonstochastic input be positive, i.e., x FF 0 x ∂ = > ∂ and let the production function be concave in x , i.e., xxF 0< . The firm’s profit is a random variable 1 2 1 2(x,z ,z ) F(x,z ,z ) c(x)Π = − , where c(x) is the cost function. An increase of riskiness in either 1Z or 2Z or a combination of 1 2(Z ,Z ) will have an impact on the riskiness of both the production function Y and profit function Π . However, whether the impact 128 for the firm’s entrepreneur. increases or decreases the riskiness of the production and profit functions is not clear in this extended model with more than one stochastic input variable. The purpose of this chapter is to extend and generalize the theory of the firm given more than one stochastic variable in the firm’s production function, in particular, to study how well the PIU or PDU principle may hold in this extended model. Assume that the firm’s objective is to maximize the expected utility of profit. Let u : →! ! be firm’s utility function and u'' 0 u'≤ < . Let 1 2(z ,z )ψ be the joint density function for the distribution of 1 2(z ,z ). The expected utility may be interpreted as H : →! ! by 1 2 1 2 1 20 0H(x) u[ (x,z ,z )] (z ,z )dz dz ∞ ∞ = Π ψ∫ ∫ If the cost function is linear, i.e., c(x) wx= , it may be noted that H is a well-defined concave function. To interpret it, simply note that 2 xx xx x 1 2 1 20 0 H [u' u''( ) ] (z ,z )dz dz 0 ∞ ∞ = Π + Π ψ < ∫ ∫ The first order condition yields that x x 1 2 1 2 1 20 0H u'( )[F (x,z ,z ) c '(x)] (z ,z )dz dz 0 ∞ ∞ = Π − ψ = ∫ ∫ 129 The FOC may also be interpreted as x 1 2 x 1 2E[F (x,z ,z )] c '(x) [ Cov(u'( ),F (x,z ,z ))/E(u')]= + − Π The second term of RHS is the marginal risk premium in this extended model with two stochastic variables. It follows that if the marginal risk premium is positive (negative), the expected marginal value product will be greater (less) than the marginal cost product. It is clear that a single risk averse nature of the utility function cannot determine the sign of marginal risk premium. It is also noted that the sign of the covariance term is dependent on the relationship between the profit function Π and the marginal value product function xF , since E(u') 0> due to the risk averse nature of the entrepreneur’s utility function. To show the effect on MRP of the sign of the covariance term of the above first order condition, all the possibilities were listed below in Table A1 and Table A2. 130 MRP 1z F 1xz F 2z F 2xz F 1 2(z z ) F 1 2x(z z ) F + + + + + + + ? + + + + + - ? + + + + - + + + + + + - - ? + + + - + + ? + + + - + - ? + + + - - + ? + + + - - - ? + + - + + + ? + + - + + - ? + + - + - + ? + + - + - - + + + - - + + ? + + - - + - ? + + - - - + + + + - - - - ? + - + + + + ? + - + + + - ? + - + + - + ? + - + + - - ? + - + - + + - + - + - + - - + - + - - + ? + - + - - - ? + - - + + + - + - - + + - - + - - + - + ? + - - + - - ? + - - - + + ? + - - - + - ? + - - - - + ? + - - - - - (Table A1) *MRP refers to the sign of Marginal Risk Premium 131 MRP 1z F 1xz F 2z F 2xz F 1 2(z z ) F 1 2x(z z ) F ? - + + + + + ? - + + + + - ? - + + + - + ? - + + + - - ? - + + - + + - - + + - + - - - + + - - + ? - + + - - - ? - + - + + + - - + - + + - - - + - + - + ? - + - + - - ? - + - - + + ? - + - - + - ? - + - - - + ? - + - - - - + - - + + + + ? - - + + + - ? - - + + - + + - - + + - - ? - - + - + + ? - - + - + - ? - - + - - + ? - - + - - - ? - - - + + + ? - - - + + - ? - - - + - + ? - - - + - - + - - - - + + ? - - - - + - ? - - - - - + + - - - - - - (Table A2) *MRP refers to the sign of Marginal Risk Premium 132 It should be noted that the sign of marginal risk premium depends on the sign of the covariance term, which comes from the relationship between Π and xF . A sufficient condition for a positive marginal risk premium is that 1 1z xz F 0Π > AND 2 2z xz F 0Π > AND 1 2 1 2z z x(z z ) F 0Π > , since ii zz F (i 1,2)= Π = and 1 2 1 2z z z z F = Π . Table 1 gives all of the possible 62 64= conditions, of which 16 possibilities will yield a clear sign of marginal risk premium, i.e., either positive or negative MRP. The other 48 possibilities will not unambiguously yield the sign of the marginal risk premium, therefore, will not give a clear conclusion on technological uncertainty. It should also be noted that the PIU condition concluded here, i.e., 1 1z xz F 0Π > , 2 2z xz F 0Π > and 1 2 1 2z z x(z z ) F 0Π > does not require the independence of stochastic inputs 1z and 2z , therefore may apply to the general uncertainty scenarios. From the first order condition, it is clear that a positive marginal risk premium mandates the nature of risk aversion and the PIU assumptions. It follows that a risk averse entrepreneur would make less production under the uncertainty scenario other than the certainty scenario. The more risk averse the entrepreneur is, the less 133 the production level would be. The proof (not shown) follows from the definition of riskiness by Rothschild and Stiglitz (1970) and the work by MacMinn and Holtmann (1983). 134 Reference Arrow, K. J. (1965). Aspects of the Theory of Risk Bearing, Helsinki: Johnsonin Saatie. Arrow, K. J. (1974a). Essays in the Theory of Risk Bearing, North Holland. Arrow, K. J. (1974b). “Optimal Insurance and Generalized Deductibles.” Scandinavian Actuarial Journal 1-42. Binswanger, M. and H. Wealth (1993). “Wealth, Weather Risk and the Composition and Profitability of Agricultural Investments.” Economics Journal 103 (416): 56-78. Bowers, J. and G. Mould (1994). “Weather Risk in Offshore Projects.” Journal of the Operational Research Society 45 (4): 409-18. Brockett, P. L. and Y. Kahane (1992). “Risk, Return, Skewness and Preference.” Management Science 851-66. Brockett, P. L. and J. Garven (1999). “ A Reexamination of the relationship Between Utility Functions, Risk Preference, and Moment Orderings.” Geneva Papers on Risk and Insurance 135 D’Arcy, S. and N.A. Doherty (1988). “Financial Theory of Pricing Property-Liability Insurance Contracts”, 1988, Philadelphia: S.S. Huebner Foundation. D’Arcy, S. and J.R. Garven (1990). “Property-Liability Insurance Pricing Models: An Empirical Evaluation”, Journal of Risk and Insurance, July 1990, pp.391-430 Doherty, N.A. (1981). “A Portfolio Theory of Insurance Capacity”, Journal of Risk and Insurance, 1981, pp. 405-420 Doherty, N.A.(1984). “Portfolio Efficient buying strategies”, Journal of Risk and Insurance, 1984, pp. 205-224 Doherty, N. and J.R. Garven (1986). “Price Regulation in Property- Liability insurance: A Contingent Claim Approach”, Journal of Finance, 1986, pp.1031-1050 Doherty, N. and H. Schlesinger (1983a). “Optimal Insurance in Incomplete Markets”, Journal of Political Economy, 1983a, pp.1045-1054. Doherty, N. and H. Schlesinger (1983b). “The Optimal Deductible for an Insurance Policy when Initial Wealth is Random”, Journal of Business, 56, 1983b, pp.555-565. 136 Froot, K. A. (1999). “The Evolving Market for Catastrophic Event Risk.” Risk Management and Insurance Review Volume II (3): 1- 28. Green, R. C. (1984). “Investment Incentives, Debt and Warrants.” Journal of Financial Economics 13: 115-36. Haushalter, G. (2000). “Financing Policy, Basis Risk, and Corporate Hedging: Evidence from Oil and Gas Producers.” Journal of Finance 55 (1): 107-52. Jensen, M. and C. Smith (1985). “Stockholder, Manager, and Creditor Interests: Applications of Agency Theory.” Recent Advances in Corporate Finance. E. Altman and M. Subrahmanayam, Richard D. Irwin. Jensen, M. and W. Meckling (1976). “Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Structure.” Journal of Financial Economics 3: 305-60. Leggio, K. and D. Lien (2000). “Hedging Gas Bills with Weather Derivatives.” University of Kansas. 137 Leland, H. E. (1970). “Theory of the Firm Facing Uncertain Demand.” Institute for Mathematical Studies in the Social Sciences, Stanford University, Tech. Report No. 24. Jan. 1970. Leland, H. E. (1972a). “Theory of the Firm Facing Uncertain Demand.” American Economic Review 62: 278-91. Leland, H. E. (1972b). “On the Existence of Optimal Policies under Uncertainty.” Journal of Economic Theory 4: 35-44. Leland, H. E. (1978). “Information, Managerial Choice and Stockholder Unanimity.” Review of Economic Studies 45 (3): 527-43. MacMinn, R. D. and A, Holtmann (1983). “Technological Uncertainty and the Theory of the Firm.” Southern Economic Journal 50: 120-36. MacMinn, R. D. (1987a). “Forward Markets, Stock Markets, and the Theory of the Firm.” Journal of Finance 42(5): 1167-85. MacMinn, R. D. (1987b). “ Insurance and Corporate Risk Management.” Journal of Risk and Insurance 54(4): 658-77. MacMinn, R. (1989) “Limited Liability, Corporate Objectives and Management Decisions” Working Paper, University of Texas. 138 MacMinn, R. D. and F. H. Page (1991). “Stock Options and the Corporate Objective Function.” Working Paper, University of Texas at Austin. MacMinn, R. D. (1993a). “On the Risk Shifting Problem and Convertible Bonds.” Advances in Quantitative Analysis of Finance and Accounting. MacMinn, R. D. and J. Garven (1993b). “The Under-investment Problem, Bond Covenants and Insurance.” Journal of Risk and Insurance 60(4): 635-45. MacMinn, R. D. (1999). “On Corporate Risk Management and Insurance.” Working Paper, University of Texas at Austin. MacMinn, R. D. (2000). “Risk and Choice: A Perspective on the Integration of Finance and Insurance.” Risk Management and Insurance Review. Mayers, D. and C. Smith (1982). “On the Corporate Demand for Insurance.” Journal of Business 55: 281-96. Mayers, D. and C. W. Smith Jr. (1983). “The Interdependence of Individual Portfolio Decisions and the Demand for Insurance”, Journal of Political Economy, April 1983, pp.304-11 139 Mayers, D. and C. W. Smith Jr. (1994). “Managerial discretion, regulation, and stock insurer ownership structure”, Journal of Risk and Insurance, December 1994, pp.638-55 Modigliani, F. and M. H. Miller (1958). “The Cost of Capital, Corporation Finance and the Theory of Investment.” American Economic Review Vol. XLVIII (3): 261-97. Muller A. and M. Grandi (2000). “Weather Derivatives: A Risk Management Tool for Weather-sensitive Industries.” Geneva Papers on Risk and Insurance 25 (2): 273-87. Pratt, J. W. (1964). “Risk Aversion in the Small and in the Large.” Econometrica 32: 122-36. Richter, R. (1998) “Temperature, price and profit: Managing weather risk.” Public Utilities Fortnightly 136(16):40-49. Rothschild, M. and J. E. Stiglitz (1970). “Increasing Risk: I. A Definition.” Journal of Economic Theory 2: 225-43. Schlesinger, H. (1983a). “Optimal Insurance in Incomplete Markets.” Journal of Political Economy 1045-54. Schlesinger, H. (1983b). “The Optimal Deductible for an Insurance Policy when Initial Wealth is Random.” Journal of Business 56: 555-65. 140 Smith, C. W. and R. M. Stulz (1985). “The Determinants of Firms’ Hedging Policies.” Journal of Financial and Quantitative Analysis 20 (4): 391-405. Stulz, R. M. (1984). “Optimal Hedging Policies.” Journal of Financial and Quantitative Analysis 19 (2): 127-40. Stulz, R. M. (1996). “Rethinking Risk Management.” Journal of Applied Corporate Finance (Fall): 8-24. Zolkos, R. (2000). “Forecast calls for growth in weather risk securitization.” Business Insurance 34 (5): 3-22. 141 Vita Mulong Wang was born in the County of Funing, Jiangsu Province of China on April 5, 1976, the youngest son of Dongmei Mao and Minxuan Wang. After completing his work in Suzhou high school, he entered the University of Science and Technology of China in 1991 and received his B.S. in 1996. He came to the University of Texas at Austin in 1997 as a Ph.D. student in the Department of Management Science and Information Systems. Permanent Address: Room 20-103, GuiHua XinChun, Suzhou, Jiangsu, China, 215006 This dissertation was typed by the author. 142 ._.