Corporate Insurance With Safety Loadings: A Note

• DOI: http://doi.org/10.1111/j.1539-6975.2012.01486.x
• Date: 01 December 2013
• Published date: 01 December 2013
• Author: Lutz G. Arnold,Johannes Hartl

© The Journal of Risk and Insurance, 2013, Vol. 80, No. 4, 1087–1094

DOI: 10.1111/j.1539-6975.2012.01486.x

1087

CORPORATE INSURANCE WITH SAFETY LOADINGS:

AN

OTE

Lutz G. Arnold

Johannes Hartl

ABSTRACT

In a article in this journal, Schnabel and Roumi (1989) assert that if uninsured

debt is risky,a levered firm takes a casualty insurance with a positive safety

loading if, and only if, the amount of debt is sufficiently high. This note

shows that in marked contrast to this assertion, the correct conclusion from

their model is that the firm generally takes insurance for low levels of risky

debt, and it depends on the magnitude of the loading whether it also takes

insurance for high levels of debt.

Mayers and Smith (1987) show that corporate insurance resolves the problem that

the shareholders of a firm with risky debt Fmay not benefit from undertak-

ing a positive-NPV investment that mitigates the effects of a casualty loss if the

premium is actuarially fair. Elaborating on a remark in Mayers and Smith (1987,

p. 50), Schnabel and Roumi (1989) investigate the case of a positive safety load-

ing. They conclude: “There is a critical value of F, call it F∗.... For F<F∗,...it is

optimal for the firm not to obtain coverage, whereas for F>F∗,...it is optimal for

the firm to obtain coverage” (Schnabel and Roumi, 1989, p. 157). That is, curiously,

the firm takes insurance if, and only if, a sufficiently large portion of the indemnity

accrues to the debt holders. In this note, we show that the correct conclusion from

their model is in marked contrast to this assertion: the firm generally takes insurance

for low levels of risky debt; whether it also takes insurance for high levels of debt

depends on the magnitude of the loading.

There are two dates. States of nature at the latter date are indexed by S∈[0, ¯

S]

(¯

S>0). Payoffs are valued using state prices g(S). g(S) is positive and atomless for all

S∈[0, ¯

S]. Consider a levered firm. At the latter date, in states without a casualty loss,

viz., for S>Sc(0 <Sc<¯

S), the firm value is V∗(>0). For S≤Sc, a casualty loss L(S)

occurs at the second date. The firm’s assets can be reconstituted at cost I(S), where

0<I(S)<L(S)≤V∗for all 0 ≤S<Scand L(Sc)=I(Sc)=0. That is, repairing the

damage is a positive-NPV project. L(S)andI(S) are twice continuously differentiable

and strictly decreasing on the interval [0, Sc] (so states with a higher index Sare better)

with I(0) >−∞. At the first date, the firm generates no cash flow to its shareholders

Lutz G. Arnold and Johannes Hartl are with University of Regensburg, Germany. The authors

can be contacted via e-mail: lutz.arnold@wiwi.uni-regensburg.de