# Corporate insurance and the underinvestment problem: an extension.

Corporate Insurance and the Underinvestment Problem: An Extension

In an insightful paper in this journal, Mayers and Smith (MS)[1] posit corporate insurance as a device to control the underinvestment problem which occurs when a firm experiences a casualty loss. Such a loss produces a Myers-type[2] option feature in affected assets because the assets' value depends on further discretionary investment. MS show that in the presence of risky debt, shareholders can have incentives to forego the investment required to rebuild the affected assets even though the investment may have a positive net present value. No such incentive exists in the absence of risky debt. Thus, the stipulation of corporate insurance coverage as a bond covenant is a way of controlling this type of perverse incentive.

In their model, MS assume actuarially fair insurance policies. Given this assumption, they show that if the firm's debt is risky, it is always optimal to take out insurance coverage. In terms of their symbols, if F, the promised payment to the bondholders, exceeds the minimum possible value of [V.sup.*] - I(S) (where [V.sup.*] is the terminal value of the firm with no loss and I(S) is the investment required to restore the value of the firm conditional upon the state of nature, S) thus rendering the firm's debt risky, then it is optimal to take out coverage. On the other hand, if F is less than the minimum possible value of [V.sup.*] - I(S), the firm's debt is riskless and the firm is indifferent between obtaining and foregoing actuarially fair insurance coverage. Thus, the critical value of F equals the minimum of [V.sup.*] - I(S). This extension focuses on the implications of assuming insurance policies whose premiums incorporate a safety loading. It is shown that even if the firm's debt is risky, it may be optimal to forego insurance coverage. There is a new critical value of F, which exceeds the minimum value of [V.sup.*] - I(S), below which it is optimal to forego coverage and above which it is optimal to obtain coverage. In keeping with the development in MS, this extension assumes a binary choice situation where the firm decides to either insure or not to insure. Thus, the elements of the insurance contract, e.g. deductible, are not decision variables.

This extension does not change the basic result of MS regarding the use of insurance policies to control the underinvestment problem occasioned by a casualty loss. The aim here is merely to formalize and more fully articulate the summary comments in MS regarding the impact of a positive loading fee, to wit, "a loading fee for the insurance policy reduces the benefits of additional debt in the capital structure ... as long as the benefit is greater than the loading fee, the additional debt should be issued and an insurance policy purchased".

Following the development in MS, consider an insurance policy whose indemnity is I(S) with a deductible of I(Sa). Sa is the state of nature where [V.sup.*] - I(Sa) = F. Assuming a safety loading of [Lambda], the premium on such a policy would be given by (1) [Mathematical Expression Omitted] where g(S) is the state contingent price paid now for delivery of a dollar at the terminal date conditional on state S. Define [V.sub.u], [V.sub.o] and [V.sub.Lambda] as the present values of the unlevered firm, the levered firm with no insurance coverage and the levered firm with insurance coverage which incorporates a safety loading of [Lambda]. Define L(S) as the amount of the casualty loss or the reduction in the terminal value of the firm conditional on state S. Consistent with MS, assume that rebuilding is a positive net present value proposition, i.e. L(S) - I(S) [is greater than] O for all states S. Thus, (2) [Mathematical Expression Omitted] where Sc is the critical state of nature below which casualty losses are incurred and above which no such losses are incurred. (3) [Mathematical Expression Omitted] and (4) [Mathematical Expression Omitted] The first two terms of equation (4) constitute the payoff to the bondholders when S [is less than] Sa, i.e., [Mathematical Expression Omitted] a result which MS employ. Thus, equation (3) of MS is analogous to our equation (4). However, note parenthetically that Mayers and Smith's equation (3) contains a typographical error. The following term should be substituted for the last term of their equation (3). [Mathematical Expression Omitted]

Define [R.sub.o] as the difference between [V.sub.u] and [V.sub.o], i.e., the reduction in the value of the firm induced by the presence of financial leverage and no insurance coverage. Similarly, define [R.sub.[Lambda]] as the difference between [V.sub.u] and [V.sub.[Lambda]] i.e., the reduction in the value of the firm induced by the presence of financial leverage and insurance coverage with a safety loading of [Lambda]. Then, by invoking equations (2), (3) and (4), we obtain (5) [Mathematical Expression Omitted] and (6) [Mathematical Expression Omitted]

Clearly, it is optimal not to obtain insurance coverage if [R.sub.o] [is less than] [R.sub. [Lambda] whereas insurance coverage is called for if [R.sub.o] [is greater than] [R.sub.[Lambda]. In terms of the figure [R.sub.o] is the area ABCD whereas [R.sub.[Lambda] is [Lambda] times the area DCF. Since [Lambda] = 0 for actuarially fair insurance policies, the case examined by MS is the one where [R.sub.[Lambda] identically equals zero; thus, [R.sub.o] [is greater than] [R.sub. [Lambda] and insurance coverage is always optimal. A visual inspection of the figure makes clear that there is a critical value of F, call it [F.sup.*], where [R.sub.o] = [R.sub.[Lambda]. For F [is less than] [F.sup.*], [R.sub.o] [is less than] [R.sub.[Lambda]] and it is optimal for the firm not to obtain coverage whereas for F [is greater than] [F.sup.*], [R.sub.o] [is greater than] [R. sub.[Lambda] and it is optimal for the firm to obtain coverage.(1) Note that for the case of actuarially fair insurance policies, [F.sup.*] equals D or the minimum value of [V.sup.*] - I(S). In this case, if F [is less than or equal to] [F.sup.*], [R.sub.o] = [R.sub.[Lambda] = 0 and the firm is indifferent between obtaining and foregoing insurance coverage. It is clear from the graph that the [F.sup.*] assuming an insurance policy with a safety loading exceeds D. Thus the impact of a safety loading is to increase the critical amount of debt outstanding above which it is optimal to take out insurance coverage.

Summary

This extension to the Mayers and Smith paper has considered the case of insurance policies whose premiums incorporate a safety loading. In the absence of a safety loading, they have shown that if the firm's debt is risky, it is always optimal for the firm to take out insurance coverage whereas if the firm's debt is riskless, the firm is indifferent between obtaining and foregoing insurance coverage. However, in the presence of a safety loading, this extension has shown that even though a firm's debt is risky, it may not be optimal to insure if the amount of debt is sufficiently low. There is a critical amount of risky debt outstanding above which it is optimal to insure and below which it is optimal not to insure.

(1)The intuition invoked in the foregoing may be given a more rigorous basis by recasting the firm's decision process. Define [Rho] as the following (7) [Mathematical Expression Omitted] Then no insurance is optimal if [Rho] [is less than] [Lambda] whereas if [Rho] [is greater than] [Lambda] insurance is optimal. Consider the derivative of [Rho] with respect to F. Via Leibnitz' Rule, (8) [Mathematical Expression Omitted] (9) [Mathematical Expression Omitted] In equations (8) and (9) the functional dependence of Sa on F is made explicit. Clearly, increasing F raises the value of Sa, i.e., [Mathematical Expression Omitted] is positive. Thus, equation (8) is always positive while equation (9) is always negative. As [R.sub.o] and [R.sub.[Lambda]] are each positive, it follows that [Mathematical Expression Omitted] is positive. We conclude that, for F equal to a low value, [Rho] [is less than] [Lambda]. Raising F, raises [Rho]. When F = [F.sup.*], [Rho] = [Lambda] and for F [is greater than] [F.sup.*], [Rho] [is greater than] [Lambda].

Jacques A. Schnabel is Professor of Business and Head of the Finance Area at Wilfrid Laurier University, Waterloo, Ontario.

Ebrahim Roumi is Associate Professor of Management Information Systems at the University of New Brunswick, St. John, New Brunswick.

In an insightful paper in this journal, Mayers and Smith (MS)[1] posit corporate insurance as a device to control the underinvestment problem which occurs when a firm experiences a casualty loss. Such a loss produces a Myers-type[2] option feature in affected assets because the assets' value depends on further discretionary investment. MS show that in the presence of risky debt, shareholders can have incentives to forego the investment required to rebuild the affected assets even though the investment may have a positive net present value. No such incentive exists in the absence of risky debt. Thus, the stipulation of corporate insurance coverage as a bond covenant is a way of controlling this type of perverse incentive.

In their model, MS assume actuarially fair insurance policies. Given this assumption, they show that if the firm's debt is risky, it is always optimal to take out insurance coverage. In terms of their symbols, if F, the promised payment to the bondholders, exceeds the minimum possible value of [V.sup.*] - I(S) (where [V.sup.*] is the terminal value of the firm with no loss and I(S) is the investment required to restore the value of the firm conditional upon the state of nature, S) thus rendering the firm's debt risky, then it is optimal to take out coverage. On the other hand, if F is less than the minimum possible value of [V.sup.*] - I(S), the firm's debt is riskless and the firm is indifferent between obtaining and foregoing actuarially fair insurance coverage. Thus, the critical value of F equals the minimum of [V.sup.*] - I(S). This extension focuses on the implications of assuming insurance policies whose premiums incorporate a safety loading. It is shown that even if the firm's debt is risky, it may be optimal to forego insurance coverage. There is a new critical value of F, which exceeds the minimum value of [V.sup.*] - I(S), below which it is optimal to forego coverage and above which it is optimal to obtain coverage. In keeping with the development in MS, this extension assumes a binary choice situation where the firm decides to either insure or not to insure. Thus, the elements of the insurance contract, e.g. deductible, are not decision variables.

This extension does not change the basic result of MS regarding the use of insurance policies to control the underinvestment problem occasioned by a casualty loss. The aim here is merely to formalize and more fully articulate the summary comments in MS regarding the impact of a positive loading fee, to wit, "a loading fee for the insurance policy reduces the benefits of additional debt in the capital structure ... as long as the benefit is greater than the loading fee, the additional debt should be issued and an insurance policy purchased".

Following the development in MS, consider an insurance policy whose indemnity is I(S) with a deductible of I(Sa). Sa is the state of nature where [V.sup.*] - I(Sa) = F. Assuming a safety loading of [Lambda], the premium on such a policy would be given by (1) [Mathematical Expression Omitted] where g(S) is the state contingent price paid now for delivery of a dollar at the terminal date conditional on state S. Define [V.sub.u], [V.sub.o] and [V.sub.Lambda] as the present values of the unlevered firm, the levered firm with no insurance coverage and the levered firm with insurance coverage which incorporates a safety loading of [Lambda]. Define L(S) as the amount of the casualty loss or the reduction in the terminal value of the firm conditional on state S. Consistent with MS, assume that rebuilding is a positive net present value proposition, i.e. L(S) - I(S) [is greater than] O for all states S. Thus, (2) [Mathematical Expression Omitted] where Sc is the critical state of nature below which casualty losses are incurred and above which no such losses are incurred. (3) [Mathematical Expression Omitted] and (4) [Mathematical Expression Omitted] The first two terms of equation (4) constitute the payoff to the bondholders when S [is less than] Sa, i.e., [Mathematical Expression Omitted] a result which MS employ. Thus, equation (3) of MS is analogous to our equation (4). However, note parenthetically that Mayers and Smith's equation (3) contains a typographical error. The following term should be substituted for the last term of their equation (3). [Mathematical Expression Omitted]

Define [R.sub.o] as the difference between [V.sub.u] and [V.sub.o], i.e., the reduction in the value of the firm induced by the presence of financial leverage and no insurance coverage. Similarly, define [R.sub.[Lambda]] as the difference between [V.sub.u] and [V.sub.[Lambda]] i.e., the reduction in the value of the firm induced by the presence of financial leverage and insurance coverage with a safety loading of [Lambda]. Then, by invoking equations (2), (3) and (4), we obtain (5) [Mathematical Expression Omitted] and (6) [Mathematical Expression Omitted]

Clearly, it is optimal not to obtain insurance coverage if [R.sub.o] [is less than] [R.sub. [Lambda] whereas insurance coverage is called for if [R.sub.o] [is greater than] [R.sub.[Lambda]. In terms of the figure [R.sub.o] is the area ABCD whereas [R.sub.[Lambda] is [Lambda] times the area DCF. Since [Lambda] = 0 for actuarially fair insurance policies, the case examined by MS is the one where [R.sub.[Lambda] identically equals zero; thus, [R.sub.o] [is greater than] [R.sub. [Lambda] and insurance coverage is always optimal. A visual inspection of the figure makes clear that there is a critical value of F, call it [F.sup.*], where [R.sub.o] = [R.sub.[Lambda]. For F [is less than] [F.sup.*], [R.sub.o] [is less than] [R.sub.[Lambda]] and it is optimal for the firm not to obtain coverage whereas for F [is greater than] [F.sup.*], [R.sub.o] [is greater than] [R. sub.[Lambda] and it is optimal for the firm to obtain coverage.(1) Note that for the case of actuarially fair insurance policies, [F.sup.*] equals D or the minimum value of [V.sup.*] - I(S). In this case, if F [is less than or equal to] [F.sup.*], [R.sub.o] = [R.sub.[Lambda] = 0 and the firm is indifferent between obtaining and foregoing insurance coverage. It is clear from the graph that the [F.sup.*] assuming an insurance policy with a safety loading exceeds D. Thus the impact of a safety loading is to increase the critical amount of debt outstanding above which it is optimal to take out insurance coverage.

Summary

This extension to the Mayers and Smith paper has considered the case of insurance policies whose premiums incorporate a safety loading. In the absence of a safety loading, they have shown that if the firm's debt is risky, it is always optimal for the firm to take out insurance coverage whereas if the firm's debt is riskless, the firm is indifferent between obtaining and foregoing insurance coverage. However, in the presence of a safety loading, this extension has shown that even though a firm's debt is risky, it may not be optimal to insure if the amount of debt is sufficiently low. There is a critical amount of risky debt outstanding above which it is optimal to insure and below which it is optimal not to insure.

(1)The intuition invoked in the foregoing may be given a more rigorous basis by recasting the firm's decision process. Define [Rho] as the following (7) [Mathematical Expression Omitted] Then no insurance is optimal if [Rho] [is less than] [Lambda] whereas if [Rho] [is greater than] [Lambda] insurance is optimal. Consider the derivative of [Rho] with respect to F. Via Leibnitz' Rule, (8) [Mathematical Expression Omitted] (9) [Mathematical Expression Omitted] In equations (8) and (9) the functional dependence of Sa on F is made explicit. Clearly, increasing F raises the value of Sa, i.e., [Mathematical Expression Omitted] is positive. Thus, equation (8) is always positive while equation (9) is always negative. As [R.sub.o] and [R.sub.[Lambda]] are each positive, it follows that [Mathematical Expression Omitted] is positive. We conclude that, for F equal to a low value, [Rho] [is less than] [Lambda]. Raising F, raises [Rho]. When F = [F.sup.*], [Rho] = [Lambda] and for F [is greater than] [F.sup.*], [Rho] [is greater than] [Lambda].

Jacques A. Schnabel is Professor of Business and Head of the Finance Area at Wilfrid Laurier University, Waterloo, Ontario.

Ebrahim Roumi is Associate Professor of Management Information Systems at the University of New Brunswick, St. John, New Brunswick.

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Author: | Schnabel, Jacques A.; Roumi, Ebrahim |
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Publication: | Journal of Risk and Insurance |

Date: | Mar 1, 1989 |

Words: | 1459 |

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