On Corporate Risk Management and Insurance
Revised March 2000
On Corporate Risk Management and Insurance
Insurance contracts provide the corporation with an instrument to manage risk and create value. Insurance is designed to manage pure risk; a pure risk only yields a loss unlike speculative risks that may yield a gain or loss. Received theory does not provide the necessary distinction between pure and speculative risks that would allow the role of insurance to be investigated. Pure and speculative risks are modeled here as independent random variables. The role that insurance plays in determining an optimal capital structure and otherwise managing risk is investigated. This analysis is a generalization of a classic financial market model and it shows that earlier results such as the use of insurance to control the riskshifting problem continue to hold. This role, however, can be duplicated by a variety of other instruments including convertible bonds and futures. In an attempt to provide a distinction the analysis is extended. It provides a new tax result in which the corporation creates value by substituting a safe tax shelter for a risky tax shelter; this is accomplished by issuing debt to purchase insurance so that the random deduction due to the pure loss is replaced by a known deduction due to the debt. Hence, the analysis reveals a role that insurance can play in developing an optimal capital structure. Finally, the analysis provides the basis for comparing the use of financial futures with that of insurance in managing risk. If the probability of insolvency exceeds the probability of a gain on the futures contract and the risk management choices are mutually exclusive then the corporation prefers insurance to futures in managing risk; under similar conditions without the mutually exclusive condition, the corporation prefers to use both contracts. Hence, the analysis provides a means of comparing the risk management instruments and exposits one case in which insurance and futures are complements.
One of the primary purposes of a financial market system is to facilitate the allocation of risk bearing. A competitive financial market system can generate an efficient allocation of risk bearing and resources. Insurance contracts compose a subset of the contracts traded in a financial market system. Like other financial contracts, the purpose of an insurance contract is to affect a risk transfer that results in a redistribution of the risk. Like other financial contracts, the insurance contract does generate an allocation of risk bearing among the market agents but received financial market theory provides no paradigm that adequately explains the existence of the insurance mechanism.^{[1]} Indeed, a generalization of the 1958 ModiglianiMiller theorem shows that corporate insurance is irrelevant, e.g., (MacMinn 1987). That theorem, however, also shows that financial leverage is irrelevant and implies that hedging with futures or forward contracts is irrelevant. Therefore, there is a need to modify the current finance paradigm^{[2]} so that the existence and function of the insurance mechanism can be analyzed. The current paradigm is modified here and the question of an optimal mix of financial contracts is addressed in a market that includes bond, stock, insurance, and futures contracts. The insurance contract plays an integral role in the determination of the corporation’s optimal nexus of contracts; in fact, the analysis shows that the corporation has an incentive to insure property risks and to do so by levering. Hence, insurance plays a role in generating an optimal capital structure. The analysis also shows that the firm does not have an incentive to use futures contracts in isolation since a full hedge does not capture all the value that can be captured with insurance or, more generally, a combination of futures and insurance.
A number of authors have considered corporate insurance (Cummins 1976; Mayers and Smith 1982; Main 1983; Shapiro and Titman 1985; MacMinn 1987; Mayers and Smith 1987; MacMinn 1989; Garven and MacMinn 1993) This literature [RM1]provides a number of motivations for corporate insurance but it does not provide a sufficiently clear role for the insurance contract to play in risk management that is different from that of other financial instruments!^{[3]} A positive risk management theory must provide some guidance in determining which financial instrument to use and when. This paper is an attempt at determining the role that insurance plays in a financial market setting and at specifying some circumstances that lead management to use insurance as well as other instruments in managing corporate risk.
The role that insurance plays in financial markets can be appreciated by distinguishing between risks. The finance paradigm categorizes the notion of risk as systematic or nonsystematic, or equivalently, as diversifiable or nondiversifiable. One might also characterize the risks in the finance paradigm as generic and specific. The insurance paradigm, however, classifies risks as either speculative or pure (Magee 1961; Bickelhaupt 1974). A speculative risk is characterized by a random payoff that may be either positive or negative; an investor may make a capital gain or a capital loss on most financial contracts and so the finance paradigm might be described as only classifying risks as speculative. A pure risk is characterized by a random payoff that is nonpositive with probability one. The losses covered by insurance contracts are of this variety.
This analysis extends the scope of the existing literature on financial markets by introducing and exploring the notion of pure[RM2] risks as well as speculative risks and by viewing insurance contracts as just one means of transferring risk in an integrated financial market setting. The liability system determines an initial allocation of pure risk bearing. The financial markets allow that allocation to be altered via trading among risk averse agents. Other things being equal,^{[4]} investors should be able to diversify the pure corporate risks. This, in turn, should allow the firm's financial instruments to be valued. What is more, given the other things being equal assumption, this analysis yields a generalized version of the 1958 ModiglianiMiller theorem, which shows that the value of the insured firm equals that of the uninsured firm. This result follows because of a no arbitrage result that shows that the premiums on the insurance contracts must equal the value of the portfolios that investors would have to form to diversify the pure risks.
Once the other things being equal assumption is relaxed, it is possible to show that the insurance contract is preferred to the investors' diversification on personal accounts. While investors can hedge the corporate pure risks, that hedging behavior does not affect the corporation's probability of events such as bankruptcy. An appropriately structured insurance contract can change that probability of bankruptcy and so alter the incentives associated with the corporate investment and operating decisions. Therefore, the insurance contract can change decisions and corporate values by altering bankruptcy events. By reducing the risk of bankruptcy, the insurance contracts motivate corporate decisions that generate an efficient allocation of resources and risk bearing.
Since the current literature does not contain a financial market model that incorporates and uses pure and speculative risks, it has not been possible to address a number of questions that arise naturally in this setting. For example, are financial futures contracts and insurance contracts substitutes or complements? Both contracts are instruments that can be used to hedge risk. The futures contract is designed for speculative risks while the insurance contract is designed for pure risk. The analysis here shows that there are some cases in which the two contract types are complements rather than substitutes. The futures contract only affects the earnings distribution but the joint use of insurance and futures can reduce financial distress and so alleviate the adverse incentive problems. This is one step in the direct of integrating the notions of risk management in insurance and finance.
In the next section, the notions of pure and speculative risks are introduced in a financial market economy. A random process different from that of the speculative risks characterizes the pure risk. The analysis shows how investors or corporations can hedge these risks and how the pure risks are valued. In section three, the model is structured so that the firm makes financing, insurance, and futures decisions, and then subsequently makes a production decision. The analysis shows those cases in which insurance and futures can be used to change incentives and increase corporate value. The last section contains the concluding remarks.
Consider an economy operating between the dates t = 0 and 1, referred to subsequently as now and then, respectively. All decisions are made now and all payoffs on those decisions are received then. The economy is composed of real and fictitious agents. The real agents are risk averse investors while the fictitious agents are corporations. Investors make portfolio decisions on personal account to maximize expected utility subject to a budget constraint. The fictitious agents act on behalf of their principals, i.e., the investors who are shareholders.^{[6]}
Let X ´ Z be the state space for the economy. All risks are functions from X ´ Z to the real numbers. State x in X represents an index of economic conditions and X is the set of these index numbers. The state z represents an accident state and Z is the set of these states. The speculative risk is the random variable P: X ® R and the pure risk is a random variable L: Z ® R. The corporate payoff is G = P  L.
Suppose the financial markets are competitive. A basis stock in this economy is a promise to pay one dollar then if state x is observed and zero otherwise; let p(x) be the price of the basis stock that pays one dollar if state x occurs and zero otherwise. There are as many basis stocks as there are states in X.
The following summarizes the notation used in the development of the model:
economic state variable 

set of economic states 

accident state variable 

set of accident states 

P 
speculative risk; 
pure risk 

G 
random corporate payoff then; G = P  L 
p(x) 
basis stock price now 
P(x) 

i 
insurance premium now 
S^{u} 
stock value of uninsured firm now 
S^{i} 
stock value of insured firm now 
This generalization of the financial model introduces a new valuation problem. Even if investors purchasing stock in the corporation know that a particular economic state will occur then, the corporate payoff is still uncertain until the accident state has been resolved. This is shown in the accompanying figure. If the corporation does not hedge the pure risk, risk averse investors will seek a means of hedging it. The construction of a hedge may be sketched as follows: To hedge the pure risk, consider the construction of a portfolio of put and call options. Let E be the random exercise price of an option written on the corporate payoff G. Suppose an investor goes long in corporate stock and the put and short in the call. Let the payoff on the put option be max{0, E  G} and the payoff on the call be max{0, G  E}. Such a portfolio of options allows the investor to hedge the pure risk since the payoff on the portfolio is G + max{0, E  G}  max{0, G  E}_{ }= E. If the exercise price on the put and the call is set at the expected value of the corporate payoff given x, i.e., E(x) = E{G  x} = P  EL, for each x^{[7]} then Jensen's Inequality may be used to show that all risk averse investors prefer to hedge the pure risk. An equivalent hedge may be constructed by writing options on the pure risk. To see this first observe that
Similarly
Hence, the call constructed here is equivalent to a put with an exercise price of EL written on the pure risk; similarly the put constructed here is equivalent to a call with an exercise price of EL written on the pure risk.^{[8]} Therefore, the hedge can also be constructed by going long in the stock and call and short in the put since
This hedging behavior enables a simple statement of the stock market value of the corporation now. Note that p(x) is the price now of a basis stock that pays one dollar in state x and zero otherwise; such a basis stock exists for each x. Let P(x) denote the sum of these basis stock prices over the set {t Î X  t £ x}. Since the hedged payoff on the corporate stock is E{G  x} = P(q, x)  EL, it may be shown that the stock value of the uninsured firm is S^{u} where
^{[9]} (1)
The analysis will also show that if it is possible to fully insure the loss, then the stock market value of the insured firm is
_{ }_{(2)}
A no arbitrage argument shows that the stock value of the uninsured firm equals that of the insured firm. Equivalently, the insurance premium must equal the present value of the expected loss. Therefore, the two representations of stock value are equal and establish a generalization of the 1958 ModiglianiMiller theorem (Modigliani and Miller 1958) since it shows that, ceteris paribus, the introduction of pure risk does not alter the earlier generalizations in the literature which showed that the value of the uninsured firm equals that of the insured firm.
A no arbitrage argument establishes the result that the value of the insurance premium equals the present value of the expected loss, i.e., i = p EL, where p is the sum of the basis stock prices, or equivalently, the discount factor for a safe asset.
max{0, P  G} = max{0, P  (P  L)} = max{0, L}. (3)
Hence, the payoff on a put with a random strike price P is zero if no loss occurs and the loss if it does occur. Also observe that (3) indicates that the contract could have simply been written on the accident index. Therefore, there is an efficiency gain due to the existence of insurance markets. If there is a deductible then the insurance takes the form max{0, L  d} where d is the deductible; the insurance premium becomes
^{[10]} (4)
This sketch of the analysis provides the base case. A relaxation of the ceteris paribus assumptions implicit in the 1958 ModiglianiMiller theorem will allow a characterization of the conditions under which insurance plays a positive role in alleviating conflict of interest problems and aligning incentives to achieve an efficient allocation of risks.
The corporate decisions are modeled in this section. The firm is assumed to make a sequence of decisions.^{[11]} The first set of decisions determines the corporate risk, or equivalently, the contract set that the firm uses to raise capital and control risk. The firm can use any mix of debt, equity, and insurance contracts to raise capital and control the speculative and pure risks; the futures contract is added to the mix in a subsequent section. The second set of decisions determines the corporate operations. The two sets of decisions are not independent. This analysis focuses on the interdependencies that can occur given a generic form of the risk shifting, equivalently, moral hazard problem, that the firm faces in making operating decisions.
q 
production level 
P(q, x) 
speculative risk; 
pure risk; L(0) = 0; L(1) = L 

q 
accident probability; P{L = L} 
G(q, x, z) 
random corporate payoff then; G = P  L 
b 
promised payment on a zero coupon bond then 
p(x) 
basis stock price now 
P(x) 

p 

d 

i(d) 
insurance premium now 
A risk shifting problem exists when it is possible for the firm to increase the stock value and decrease the bond value by increasing the risk of the corporate operations (Jensen and Meckling 1976; Mayers and Smith 1982; Jensen and Smith 1985). In the absence of a solution to the riskshifting problem, the stockholders bear the agency cost. In a setting with no pure risks, (Green 1984) and (MacMinn 1989) showed that a properly structured convertible bond could be used to solve the problem. (MacMinn 1987) showed that an insurance contract could also be used to solve the riskshifting problem. The analysis here differs from the earlier literature by allowing the pure and speculative risks to be generated by different random processes.
Here it is assumed that the two sets of decisions are made now. In the first step, the firm raises I dollars to invest through bond or equity issues and selects its insurance coverage. In the second step, the firm selects a production level q; that decision, like the others, is made prior to knowing the payoff P(q, x) and prior to knowing whether or not a loss occurs. The payoff P is defined as P(q, x) = R(q, x)  c(q), where R(q, x) is the random revenue then and c(q) is the cost function. Suppose the payoff is concave in the production level q. The structure of this payoff is consistent with that of a competitive firm facing price uncertainty (Baron 1970; Sandmo 1971) or that of a monopolist facing demand uncertainty (Leland 1972); the payoff satisfies the Principle of Increasing Uncertainty, i.e., PIU, (Leland 1972). It has been shown (MacMinn and Holtmann 1983) that the PIU implies that after adjusting for a change in the expected payoff, this principle guarantees that an increase in the production increases risk in the RothschildStiglitz sense, (Rothschild and Stiglitz 1970).
Suppose the firm does not use insurance to manage the pure risk and make the following assumption:
Assumption: The loss is large enough to cause insolvency if the accident occurs but not otherwise.
This assumption is made for convenience rather than necessity. Without this assumption the specification of a financial distress event becomes more cumbersome without adding insight. The assumption does not, of course, hold for all debt levels and it is relaxed in the section on futures.
Let d Î X be the boundary of the distress event and let it be implicitly defined by the condition P(q, d)  b  L = 0. Bankruptcy requires the loss event in addition to the distress event, i.e., {1} ´ {x Î X  0 < x < d }. Then the equity payoff is max{0, P(q, x)  L  b}. It follows that the equity payoff is zero if the loss occurs and x £ d while it is P(q, x) – b > 0 if the loss does not occur and x £ d. Similarly, the equity payoff is P(q, x)  b  L if the loss occurs and x > d while it is P(q, x)  b if the loss does not occur and x > d. Hence, the hedged equity payoff is (1  q) (P  b) for states x £ d and (1  q) (P  b) + q (P  b  L) = P  b  q L for states x > d. The stock market value [RM3]is
(5)
The firm selects the production now to maximize current shareholder value. Observe that the production level is greater with distress risk, i.e., q^{l} > q^{u}. This is a variant of the riskshifting problem. This result is summarized in the following theorem.
Theorem [RM4]1. If the principle of increasing uncertainty holds and there is a positive probability of insolvency then, ceteris paribus, the production decision of the levered firm is greater than that of the unlevered firm, i.e., q^{l} > q^{u}.
The following theorem characterizes the optimal insurance level, i.e. deductible, when all else is equal.
Theorem 2. Ceteris paribus, insuring is optimal if there is a positive probability of insolvency.
This theorem shows that the firm has an incentive to insure in order to reduce distress risk. The distress risk can be reduced, ceteris paribus, by reducing the deductible, equivalently, increasing insurance coverage. Current shareholders receive the additional value created by reducing what would otherwise be an agency cost.
The introduction of taxes has traditionally been important in providing an explanation for an optimal capital structure in the small or in the large. The corporate tax motivates the use of debt and helps explain the optimal use of that contract, e.g., (Miller and Modigliani 1963; Miller 1977; DeAngelo and Masulis 1980; Miller 1988). Taxes also help explain the use of insurance and other hedging instruments, e.g., (Mayers and Smith 1982; Main 1983). The results in the literature depend on a convex tax liability function.^{[12]}
A new tax result is introduced here. Convexity does not play a role in the result reported here. The corporate tax rate is treated as a constant in the remaining analysis. In addition, neither a positive probability of insolvency nor a positive probability that earnings do not cover deductions is necessary to establish the result; either positive probability would provide the convexity necessary for the standard result in the literature. Hence, convexity is not the source of the tax result. Rather, the result depends on the substitution of a safe tax shelter for a risky tax shelter; this is accomplished by issuing debt to purchase insurance so that the random deduction due to the pure loss is replaced by a known deduction due to the debt. This result continues to hold even if the probability of distress is zero.
To modify the model in the previous sections, suppose the tax liability of the corporation is T = t max{0, P  b  L}; this assumes that the principle and interest are deductible.^{[13]} The equity payoff is P  b  L  T and so the stock value is
(6)
The corporate value becomes
Let the corporate objective be G º V  I  i, as before, but where corporate value is now specified by (7).
The manager, acting in the interests of current shareholders, makes the finance and insurance decisions to maximize the objective function F. The first order condition for a bond issue is
The second equality in (8) follows due to the stage two first order condition for an optimal output. The first two terms on the right hand side of (8) represent the marginal value of the debt tax shelter while the last term on the right hand side represents the marginal agency cost of the bond issue. Equation (8) implies the result that the firm issues bonds and pushes the bond issue to the point at which the marginal value of the tax shelter equals the marginal agency cost. Hence, ceteris paribus, (8) implies a risky debt issue.
The manager also makes an insurance decision to maximize current shareholder value. The first order condition is
The second equality follows due to the stage two first order condition for an optimal output. The first term on the right hand side of (9) represents the marginal value of the tax shelter while the second term on the right hand side represents the marginal agency cost. (9) implies the result that the firm increases its deductible, equivalently, reduces its insurance to the point at which the marginal value of the tax shelter equals the marginal agency cost. (9) does not yield a conclusion like the bond issue because setting the deductible to zero does not eliminate the distress risk; the contrary is more nearly true.
Despite the limitations in interpreting the first order condition in (9), it is possible to demonstrate a demand for insurance in this version of the model. It is possible for the firm to increase its leverage with a bond issue and counter the increase in the agency cost by simultaneously increasing its insurance coverage. A one to one tradeoff in the size of the bond issue and the size of the deductible suffices to eliminate the agency cost at the margin and to increase the value of the tax shelter. Hence, there is a tax driven demand for insurance. The result is summarized in the following theorem.
Theorem 3. The corporate tax suffices to generate a demand for insurance.
Sketch of Proof. Suppose that for every dollar increase in leverage, the firm reduces the deductible by a dollar. Then the firm can generate an increase in value. Letting v = (v_{1}, v_{2}) = (1,  1) and D_{v}G denote the derivative of the objective function in the direction v, observe that
(10)
QED
Theorem three demonstrates a strong motivation to insure. The loss or deductible represents a risky tax shelter while the bond represents a certain tax shelter; note that t is the marginal value of the tax shelter and (1  q) is the probability of no property loss and so no loss shelter; hence (1  q) is the portion of the marginal value acquired by moving from the risky tax shelter to the safe tax shelter. Hence, the theorem shows that it is optimal to replace a risky with a safe tax shelter since the latter is more valuable. The theorem also shows that value can be increased as long as there are uninsured losses.
An insurance contract is just one of many contracts that the firm can use in managing corporate risk. It is also possible to use forward and futures contracts. A positive theory of risk management must provide some rationale for the use of each type of instrument. To the extent that both insurance and futures contracts can be used to hedge risk, it is natural to assert that the two contracts are substitutes. If they are perfect substitutes then it is only necessary to use one. It is possible to frame an argument that says that the firm will be indifferent to the use of either contractual form.^{[14]} The analysis here shows that when insurance does not yield undiluted incentives, other mechanisms can complement the risk transfer capabilities of the insurance contract. The analysis shows that a corporate risk management scheme exists that provides the right incentives and increases value; it also shows the circumstances that motivate the combined use of insurance and futures to hedge risk. To motivate the joint use of insurance and futures, consider the form of the futures contract and the comparison of corporate values.
Consider a futures contract. Let F denote the unit payoff on a stock index futures contract and let f denote the futures price. Suppose the unit payoff is increasing in state x. The payoff on the futures position is j (f  F(x)), where j is the position taken by the firm in futures. The payoff on the futures contract is shown in figure two. The payoff depicted is sometimes referred to as a short position in the futures contract.
One possible comparison that is instructive is the difference in corporate values of an insured firm and a hedged firm; the hedged firm here denotes the uninsured firm that takes a position in futures. In the insured case, suppose that full insurance is selected. In the hedged case, also suppose that the firm selects a fully hedged position. If the probability of distress is greater than the probability of a gain on the futures contracts then the value of the insured firm is greater than that of the hedged firm. If this assertion holds then it provides motivation for the use of insurance.
The payoffs for the firm hedging in financial futures are shown in the following figure. The equity payoffs are (1 – t) (P  b + j (f  F)) and (1 – t) max{0, P  b  d + j (f  F)} in the event of no accident and accident, respectively; the payoffs are shown in the figure; the dashed lines represent the equity payoffs given an increased hedge in futures. The bond payoff is b with probability 1  q. Similarly, the bond payoff is max{0, P  L + j (f  F)} if x < d with probability q and b if x > d with probability q. The payoffs rotate about their values at h and so the probability of insolvency increases in the event of an accident if the probability of distress exceeds that of a gain on futures, or equivalently
The inequality in (11) says that the probability of a gain on the futures contract is less than the probability of financial distress in the event of an accident. It follows that hedging cannot increase the left tail of the payoff enough to compensate for the accident loss. What is more, hedging in this case increases the probability of distress, or equivalently the probability of default. It follows that the payoff goes to zero in the event of an accident as the firm increases its hedge in financial futures. In this case the corporate payoff of the hedged firm becomes (1 – t) (P + j (f  F)) + t b in the event of no accident and zero otherwise.^{[15]} Given full insurance, the corporate payoff of the insured firm becomes (1 – t) P + t b in either the accident or no accident events. It may be noted that hedging in futures does not change value given no accident but it reduces value given an accident. This observation motivates the following comparison in corporate values. Letting V^{h} and V^{i} denote the hedged and insured corporate values, respectively, it follows that the difference in corporate values is
(12)
Hence the insured value is greater than the hedged value. Insurance is preferred if the difference in corporate value exceeds the insurance premium paid now, equivalently
Next, consider the conditions that lead to a joint use of futures and insurance in managing risk. The corporate payoff is (1  t) max{0, (P  L + max{0, L  d} + j (f  F))} + t b. Theorem three showed that the firm has an incentive to fully insure its property risk and equation (8) showed that the firm has an incentive to issue risky bonds. Now suppose assumption one is violated as shown in figure four. There is a positive probability of distress in the accident and no accident events. The state g is implicitly defined by the condition P(q, g) – b = 0 and represents the boundary of the default event given no accident while d represents the boundary of the distress event as previously defined. If the firm fully insures the pure risk in this case a positive probability of default remains. With a deductible of zero, the corporate payoff function becomes (1  t) max{0, (P  b + j (f  F))}. The equity payoffs are shown in figure four; the lower payoff merges with the upper payoff given full insurance and the upper payoff rotates clockwise through the intersection at state h as shown by the dashed line in the figure. The objective now is to show that the firm can increase current shareholder value by hedging in futures contracts.
The positive probability of default even with full insurance leaves the firm with a riskshifting problem now and so the shareholders must bear the remaining agency cost of any debt issue. The stock market value of a fully insured firm is
(14)
where g is the boundary of the default event. The condition for an optimal production decision is
This first order condition yields a production decision greater than that of an insured unlevered firm, i.e., q^{l} > q^{u} similar to theorem one. The next lemma shows that hedging in futures reduces the output decision.
Lemma 3. Let the state h be implicitly defined by F(h) = f, and let h > g for the unhedged firm. The production decision is decreasing in the size of the hedge.
Proof. The optimal output is a decreasing function of the size of the hedge since (15) implies
The inequality in (16) follows since ¶P/¶q is negative at g by the PIU and ¶g/¶j is negative for g < h. QED
Lemma three shows that the futures contracts decrease the output decision. It follows because the corporate hedge reduces the agency costs of the leverage; hedging moves the firm closer to a socially efficient production level, or equivalently, to a production level that maximizes value for all corporate stakeholders.
While the lemma shows that the production level is reduced, it does not establish the incentive to hedge. The manager makes the finance, insurance, and futures decisions to maximize corporate value. The bond and corporate values are
(17)
and
(18)
respectively. Now G(b, d, j) = V(b, d, j) – I  i.
The following theorem shows that forward contracts can be used in conjunction with insurance contracts to increase value and control risk.
a.
b.
then futures contracts form part of the insured firm’s optimal contract set.
Proof. By theorem 3, the firm selects full insurance. Hence, it suffices to show that the derivative of G with respect to j is positive at j = 0. Using Leibnitz’s [RM5]rule and noting that the derivatives with respect to the limit g vanish, we have
(19)
Evaluating corporate value at the optimal production level yields
The inequality in (20) follows because the risk adjusted net present value of the futures contract is zero, i.e.,
,
because the optimal output decreases with the size of the hedge j, and because the PIU yields a negative marginal payoff in the event of insolvency, i.e., ¶P/¶q < 0 for all x Î (0, g]. QED
This theorem demonstrates a case in which the joint use of insurance and financial futures is optimal. By insuring the pure risk, the firm increases the value of its tax shelter; just insuring, however, does not reduce or eliminate the riskshifting problem. The firm can continue to increase value by hedging with financial futures; this will reduce the riskshifting problem and so create value that can be captured by the current shareholders.
Although the finance and insurance disciplines are related, the language in each of the literatures is often quite different. The theory of insurance markets is often considered in isolation, i.e., without reference to other financial markets. The theory of financial markets is also typically considered without pure risks. This research is an attempt to provide a partial synthesis of finance and insurance. While the analysis shows that investors or corporations can hedge the pure risks, it also shows that only the hedging on corporate account changes incentives and increases value. The insurance contract provides one of the most efficient means of hedging pure risks because the contract is structured for just that purpose. The corporation can purchase one insurance contract and achieve the alteration of incentives that it would take a whole portfolio of options to accomplish.
This analysis is also an attempt at clarifying the role of risk management. A risk management theory should provide not only a menu of the tools that management has available for hedging risk but also some indication when tools should be used. The analysis here takes a few steps in that direction. Theorem two shows that, other things being equal, the firm should insure whenever there is a positive probability of distress; the analysis also shows that the risk of distress can be controlled by unlevering the firm. Both results are known but had to be verified in this generalization of the financial market model. Theorem three provides a tax result that is new in the literature. Tax results in the literature are comparisons of the insured versus the uninsured firm value and rely on the convex form of the tax liability function. Using a method equivalent to first order stochastic dominance, theorem three introduces a corporate tax and shows that it is optimal to insure and lever the firm because that maximizes the value of the tax shelter and so the current shareholder value; hence, theorem three also establishes an optimal capital structure result. Finally, the furtures section shows that hedging with futures cannot always generate the same results that can be achieved with insurance; if the probability of distress exceeds the probability of a gain on a futures contract then futures and insurance are not substitutes; in this case, the value of the insured firm exceeds that of the firm hedged with futures[RDM6]. Theorem four concludes by showing that the firm can insure to manage the pure risk and then if a positive probability of bankruptcy remains that is less than the probability of a gain on the futures then it is optimal to hedge with futures contracts in addition to the insurance. Hence, theorem four provides a case in which it is optimal to jointly use insurance and futures to manage the pure and speculative risks.
Consider the production decision of the levered firm. Let q^{l } denote the production decision that is implicitly defined by the following first order condition
(1)
Let q^{u} denote the optimal production decision of the firm with no distress risk. The production decision q^{u} is implicitly defined by the condition
(2)
Observe that the production level is greater with distress risk, i.e., q^{l} > q^{u}. This is a variant of the riskshifting problem. This result is summarized in the following theorem.
Theorem 1. If the principle of increasing uncertainty holds and there is a positive probability of insolvency then, ceteris paribus, the production decision of the levered firm is greater than that of the unlevered firm, i.e., q^{l} > q^{u}.
Next, consider how the production decision is affected by the financing and insurance decisions. The production decision will be denoted as q(b, d) if such a function exists. The next lemma establishes the existence of the function and its derivative properties.
Lemma 1: Let the second partial of the stock value with respect to the output be negative and let the marginal corporate payoff be negative at the boundary of the distress event.^{[16]} Then a function q(b, d) exists and is increasing in each of its arguments for all b and d such that the probability of distress is positive.
Proof. By the Implicit Function Theorem, the derivatives are
The inequalities in (3) and (4) follow because d is increasing in both b and d. QED
The lemma shows how the production decision is affected by the financing and insurance decisions in the previous stage.
The firm will make financing and insurance decisions in the first stage of the decision sequence knowing what impact those decisions have on the subsequent production decision. Hence, the firm will make those decisions to maximize G º V  I  i, where i is the insurance premium
V is corporate value; V º B + S, where B represents the bond value. Note that the bond value is
(5)
Hence, the corporate value is
(6)
The next lemma establishes G = V  I  i as the objective function.
Lemma 2. The objective function is G = V  I  i.
Proof: The manager makes the finance and insurance decisions to maximize current shareholder value subject to the constraint, i.e.,
maximize S^{o}
subject to S^{n} + D = I + i
Let the LaGrange function be L = S^{o} + l (S^{n} + D  I  i).
The lemma follows because this first order condition yields a LaGrange multiplier equal to one. QED
The objective function is expressed with the leverage b as an argument rather than the number of new shares and the leverage; once the leverage is determined the number of new shares is determined to raise the remaining amount required.
The following theorem characterizes the optimal deductible when all else is equal.
Theorem 2. Ceteris paribus, insuring is optimal if there is a positive probability of insolvency.
Proof. The capital structure and insurance decisions maximize F(b, d). The optimal deductible is implicitly defined by
If the derivative is evaluated at q^{l} then
QED
This theorem shows that the firm has an incentive to insure in order to reduce distress risk. The second equality in (7) follows because, with no change in production, the insurance decision yields a zero risk adjusted net present value. The change in production allows a change in the risk adjusted net present value. Ceteris paribus, the corporation has an incentive to reduce its deductible, equivalently, increase its insurance coverage, as long as the riskshifting problem exists. Current shareholders receive the additional value created by reducing what would otherwise be an agency cost. If the firm eliminates distress risk then it follows that the production choice is socially efficient.
While the theorem demonstrates an incentive to insure, it should be noted that other instruments could be used to achieve the same results. As in theorem two, differentiating the corporate objective with respect to b and evaluating the derivative at the optimal production level yields
Hence, ceteris paribus, it is optimal to reduce leverage if there is a positive probability of distress.
By inspecting the derivatives in (7) and (9), it is apparent that the firm will be indifferent between increasing its insurance coverage and decreasing it leverage in this version of the model. The indifference is in terms of firm value. If the firm increases its leverage by a dollar and decreases its deductible by a dollar then it is moving in a direction v = (v_{1}, v_{2}) = (1,  1) and the change in the value of the objective function in the direction v is D_{v}G where
There is, however, a linkage between the bond and insurance contracts. If the firm increases its leverage enough to make the probability of distress positive then it must hedge through instruments such as insurance or lose value!
EN.REFLIST
^{[1]}The economics literature does show that risk averse agents have an incentive to purchase insurance from risk neutral agents, i.e., insurance companies, in a competitive insurance market but the analysis has not been replicated in a financial market setting. In fact, if it were and the usual definition of complete financial markets was employed then we could obtain another irrelevance result in which risk averse individuals would be indifferent between purchasing insurance and purchasing a portfolio of securities.
^{[2]}The current paradigm noted here is the complete financial market model rather than the capital asset pricing model.
^{[3]} The literature in finance also provides a number of motivations for hedging but does not typically distinguish between hedging instruments or provide guidance on the use of the many hedging instruments available, e.g. see Grossman, S. (1975). "The Existence of Futures Markets, Noisy Rational Expectations and Informational Externalities." Journal of Econometrics 3: 25572, Holthausen, D. (1979). "Hedging and the Competitive Firm Under Price Uncertainty." American Economic Review 69: 98995, Feder, G., R. E. Just, et al. (1980). "Futures Market and the Theory of the Firm Under Price Uncertainty." Quarterly Journal of Economics XCIV: 31728, Stulz, R. M. (1984). "Optimal Hedging Policies." Journal of Financial and Quantitative Analysis 19(2): 12740, Smith, C. W. and R. M. Stulz (1985). "The Determinants of Firms' Hedging Policies." Journal of Financial and Quantitative Analysis 20(4): 391405, Fabozzi, F. J., Ed. (1988). Advances in futures and options research, A Research Annual; Greenwich, Conn and London; JAI Press, Leuthold, R. M., J. C. Junkus, et al. (1989). "The theory and practice of futures markets." xviii, 410, Campbell, T. S. and W. A. Kracaw (1990). "Corporate Risk Management and the Incentive Effects of Debt." Journal of Finance: 167386, Ladd, G. W. and S. D. Hanson (1991). "PriceRisk Management with Options: Optimal Market Positions and." Journal of Futures Markets: 73750, Priovolos, T. and R. C. e. Duncan (1991). "Commodity risk management and finance." xii, 173.; also see Froot, K. A., D. S. Scharfstein, et al. (1993). "Risk Management: Coordinating Corporate Investment and Financing Policies." Journal of Finance 48(5): 162958. and Stulz, R. M. (1996). "Rethinking Risk Management." Journal of Applied Corporate Finance 9(3): 824..
^{[4]}The 1958 ModiglianiMiller theorem at least implicitly assumes that capital structure is the only variable. The theorem does not allow for changes in either investment or operating decisions. Hence, although changing the capital structure may have implications for the probability of bankruptcy or financial distress which, in turn, can impact the investment or operating decisions, those implications are not explored. The other things being equal assumption here refers to the fixed investment and operating decision implicit in the 1958 ModiglianiMiller theorem.
^{[5]} Pure risks are introduced here because insurance contracts are being investigated but the same method could be used to introduce other risks.
^{[6]}This assumption is not necessary but it is made here for expediency. One could also assume that the corporation is managed by a real agent and then derive the objective function that the agent uses in making decisions for the corporation, e.g., see MacMinn, R. D. (1990). "Uncertainty, Financial Markets, and the Fisher Model." University of Texas, MacMinn, R. D. and F. H. Page (1991). "Stock Options and the Corporate Objective Function." University of Texas..
^{[7]} The portfolios would not be necessary if a call and put can be written with a random exercise price. In such a case it would be possible to write calls and puts on the corporate stock to achieve the same result.
^{[8]} I am indebted to Henri Loubrege for making this observation.
^{[9]}This RiemannStieltjes form for the integral allows us to consider either a continuum of states or a finite number of states in X.
^{[10]} If the corporation does not fully insure the pure risk, or equivalently, select a zero deductible, then risk averse investors will hedge the pure risk. This hedging on personal account follows by Jensen’s Inequality.
^{[11]} The model could be constructed in a multiperiod format so that each set of decisions is made at a different date but that would required more complex notation without changing the results that are reported here.
^{[12]} The after tax earning of the corporation is P  t(P  D) where D represents the deductions, e.g., interest on debt, depreciation, etc.. If the tax is progressive and the function t(×) is convex then
The result is an application of Jensen’s Inequality. Since an insurance contract can replace the random earning with the expected earning it follows that the use of insurance is optimal. While the convex tax function is sufficient, it is not necessary for the result. Suppose that there is a positive probability that the earning will not exceed the deductions and suppose the tax rate is a constant; then the after tax earning becomes P  t max{0, P D} where max{0, P D} is convex. Hence
This is also an application of Jensen’s Inequality and shows that insurance is optimal.
^{[13]} The assumption is only made to simplify the analysis and make the models here approximately the same.
^{[14]}Of course, the 1958 ModiglianiMiller theorem also suggests that the firm will not be able to increase value by using either contract but theorem two in the previous section shows that once the ceteris paribus assumptions implicit in the MM 58 theorem are relaxed, it is possible to increase value by altering the firm's contract set.
^{[15]} The corporate payoff is the stockholder and bondholder payoff.
^{[16]} It was not necessary to assume the negative marginal payoff at d in models without the pure risk because that result was guaranteed by the PIU. Here it is not because stockholders do receive a payoff for states x < d if no accident occurs.
Page: 3
[RM1] I need to comment on the hedging literature and include some cites;
in particular, I need to include Stulz.
Page: 4
[RM2] This is where I should introduce the idea that the introduction of
the pure risk fundamentally changes the description of events and values. Explain why.
Page: 10
[RM3] I need to note that the introduction of a pure risk changes all the
expressions for stock value. I probably
need to say more about the structure of the basic model in the first paragraph
of this section because the introduction of the pure risk does change the
notion and description of events and values.
Page: 10
[RM4] I need to point out that the two theorems here are provided as
generalizations of results in the literature.
Because the model has been altered we do need to confirm that some known
results continue to hold. Cite the
literature that yields the results in the two theorems.
[RM5] Check this spelling and make sure this derivative is correct. I did check the derivative and the derivatives of the limits do vanish. It remains to check the spelling.
[RDM6] Add more explanation for the difference in values.