In a paper 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.

Johannes Hartl

Lutz G. Arnold

English

Research Papers in Economics

 
 
 
 
 
 
BGPE Discussion Paper 
 
No. 110 
 
 
Corporate Insurance with Safety Loadings:  
A Note 
 
 
Lutz G. Arnold 
Johannes Hartl 
 
 
 
 
 
 
 
 
 
 
November 2011 
 
 
 
ISSN 1863-5733 
 
Editor: Prof. Regina T. Riphahn, Ph.D.   
Friedrich-Alexander-University Erlangen-Nuremberg 
© Lutz G. Arnold, Johannes Hartl 
 Corporate Insurance with Safety Loadings: A Note
Lutz G. Arnold (corresponding author)
Johannes Hartl
University of Regensburg
Department of Economics
93 040 Regensburg, Germany
Phone: +49-941-943-2705
Fax: +49-941-943-1971
E-mail: lutz.arnold@wiwi.uni-regensburg.deCorporate Insurance with Safety Loadings: A Note
Abstract
In a paper in this journal, Schnabel and Roumi (1989) assert that if uninsured debt is risky, a levered rm takes a
casualty insurance with a positive safety loading if, and only if, the amount of debt is suciently high. This note
shows that, in marked contrast to this assertion, the correct conclusion from their model is that the rm 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 rm
with risky debt F may not benet from undertaking a positive-NPV investment which mitigates the eects
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 loading. They conclude: \there is a
critical value of F, call it F ... For F < F , ... it is optimal for the rm not to obtain coverage, whereas
for F > F , ... it is optimal for the rm to obtain coverage" (Schnabel and Roumi, 1989, p. 157). That is,
curiously, the rm takes insurance if, and only if, a suciently 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 rm 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 2 [0;  S] ( S > 0). Payos are
valued using state prices g(S). g(S) is positive and atomless for all S 2 [0;  S]. Consider a levered rm. At
the latter date, in states without a casualty loss, viz., for S > Sc (0 < Sc <  S), the rm value is V  (> 0).
For S  Sc, a casualty loss L(S) occurs at the second date. The rm's assets can be reconstituted at cost
I(S), where 0 < I(S) < L(S)  V  for all 0  S < Sc and L(Sc) = I(Sc) = 0. That is, repairing the damage
is a positive-NPV project. L(S) and I(S) are twice continuously dierentiable and strictly decreasing on
the interval [0;Sc] (so states with a higher index S are better) with I0(0) >  1. At the rst date, the rm
generates no cash ow to its shareholders in the absence of insurance. Shareholder value is the sum of the
certain cash ows to shareholders at date one (i.e., zero in the absence of insurance) and the value of the
residual claim on second-date cash ow, given state prices g(S). The rm repairs the damage if, and only if,
this does not reduce shareholder value.
For F < V   I(0), shareholders have an incentive to reconstitute the rm's assets in each state of nature in
which a damage occurs; there is no underinvestment. This is because the ensuing residual claim V  I(S) F
is positive. Shareholder value is
R Sc
0 [V    I(S)   F]g(S)dS +
R  S
Sc(V    F)g(S)dS. As the debt is safe, there
is no need to insure it, and the proceeds of the debt issue are
R  S
0 Fg(S)dS. The (rst-best) value of the rm
(i.e., the sum of shareholder value and the value of the debt) as of the rst date is
Vu =
Z Sc
0
[V    I(S)]g(S)dS +
Z  S
Sc
V g(S)dS:
It is proportional to the area of the pentagon O SHJD in Figure 1 (linearity of the curves is assumed for ease
of exposition).
1Suppose to the contrary that the rm's debt F to be paid at the second date is large, in that F  V  I(0).
Then for any F in the interval [V    I(0); V ], there is Sa determined by
V    I(Sa) = F (1)
(see Figure 1). For S < Sa, the rm value falls short of the face value of debt F, even if the rm decides
to reconstitute the assets. Ignore corporate insurance to begin with. Then the shareholders prefer not to
repair the damage in states S < Sa, even though this is a positive-NPV project. This is the underinvestment
problem identied by Myers (1977, p. 153) and Mayers and Smith (1987, p. 48).1 Because of default in states
S < Sa, the debt is risky. Its value as of the rst date is
D0 =
Z Sa
0
[V    L(S)]g(S)dS +
Z  S
Sa
Fg(S)dS: (2)
Shareholder value is
S0 =
Z Sc
Sa
[V    I(S)   F]g(S)dS +
Z  S
Sc
(V    F)g(S)dS: (3)
The value of the rm V0 = D0 + S0 is
V0 =
Z Sa
0
[V    L(S)]g(S)dS +
Z Sc
Sa
[V    I(S)]g(S)dS +
Z  S
Sc
V g(S)dS:
The reduction in the value of the rm compared to the no-underinvestment case F < V  I(0), i.e., Vu V0,
is
R0 =
Z Sa
0
[L(S)   I(S)]g(S)dS: (4)
In terms of Figure 1, the value of the risky debt issue D0 is proportional to the area of the hexagon 0 SGCBA,
and shareholder value S0 is proportional to the area of CGHJ, so V0 is proportional to the area of 0 SHJCBA,
and the deadweight loss R0 is represented by the sum of the lightly and heavily shaded areas (the tetragon
ABCD). In what follows, we focus on risky debt, i.e., on the case F  V    I(0).
Now consider corporate insurance. Following Mayers and Smith (1987, pp. 49-50) and Schnabel and Roumi
(1989, p. 156), consider an insurance policy that pays I(S)   I(Sa) conditional on S  Sa. I(Sa) is the
maximum deductible that is consistent with providing the rm with an incentive to reconstitute its assets in
all states of nature in which a damage occurs: the second-date payo of the insured rm, if it reconstitutes
its assets, becomes V    I(S) + [I(S)   I(Sa)]   F = 0 for S  Sa, so it repairs the damage and the debt is
safe. The fair insurance premium is
R Sa
0 [I(S) I(Sa)]g(S)dS. Following Schnabel and Roumi (1989, p. 157),
we assume that the actual insurance premium is
P = (1 + )
Z Sa
0
[I(S)   I(Sa)]g(S)dS; (5)
1Hau (2007) shows that overinvestment can occur when asset reconstitution is risky. Consider a state S in which the rm
cannot repay its debt without asset reconstitution (i.e., V    L(S) < F), and the expected gain to the risky process of asset
reconstitution is negative. If, however, V    I(S) > F for some favorable outcome of the reconstitution process, the rm
\gambles for resurrection".
2Figure 1: Deadweight losses
where  ( 0) is the safety loading. With fair insurance (i.e., if  = 0), decreasing the deductible below
I(Sa) and raising the premium accordingly leaves the rm's choice between taking and not taking insurance
unaected (see Mayers and Smith, 1987, p. 51). In the presence of a safety loading (i.e., if  > 0), the use
of the maximum deductible consistent with removing the underinvestment problem is ecient, because this
minimizes the fair premium and, therefore, also the loading cost.2
Following Schnabel and Roumi (1989), we assume that the face value of debt F is the same with or without
insurance. Garven and MacMinn (1993, p. 636) call this the \cum dividend interpretation" of the model and
point out that it requires careful interpretation, since it implies that the date-one cash ow to shareholders
is no longer zero. For one thing, the rm has to pay the insurance premium P at date one. For another,
since insurance makes the debt safe, the value of the debt becomes
D =
Z  S
0
Fg(S)dS; (6)
so the proceeds of the debt issue rise by D  D0 ( 0). The rm now pays a dividend D  D0  P at the
rst date. From (1), (2), (5), and (6),
D   D0   P =
Z Sa
0
[L(S)   I(S)]g(S)dS   
Z Sa
0
[I(S)   I(Sa)]g(S)dS: (7)
The rst integral in the dierence on the right-hand side is the deadweight loss without insurance R0 (cf.
(4)). The second term
R = 
Z Sa
0
[I(S)   I(Sa)]g(S)dS (8)
is the deadweight loss caused by the safety loading (the fair part of the premium is not a deadweight loss).
Let S denote shareholder value with insurance. The rm takes insurance exactly if S  S0. Since the
2Leaving the connes of the model, there are other reasons for focusing on this insurance policy. If the probability of a
casualty loss depends on hidden actions taken by the rm, a high deductible serves to reduce moral hazard. If there is a xed
cost of processing claims, insurance policies which also pay o in higher states cause higher expected cost. A counterargument
is that in the worst states (i.e., for S close to zero) the rm is most cash-strained and, therefore, dependent on a low deductible.
3shareholders' residual claim at the second date is the same as in the uninsured case (cf. (3)), shareholder
value with insurance is S = S0+D D0 P = S0+R0 R. So S  S0 is equivalent to D D0 P  0
and to R0  R. That is, the rm takes insurance if, and only if, this yields a non-negative dividend at the
rst date or, equivalently, if the deadweight loss is no greater with insurance than without. In Figure 1, the
curve EC depicts V  I(S) [I(S) I(Sa)]. As in the uninsured case, the value of the shareholders' residual
claim on second-date cash ow is represented by CGHJ. The value of the insured debt D is proportional to
the rectangle 0 SGF; it exceeds the value without insurance by ABCF. This comes at the cost of P, which
is the sum of DCF (the fair premium) and the lightly shaded triangle DCE (the loading cost). The fair part
of the insurance premium DCF cancels out, so on net the shareholders gain ABCD (i.e., the removal of the
deadweight loss R0) at the expense of DCE (i.e., the deadweight loss R).
We have also analyzed the model under the alternative assumption proposed by Garven and MacMinn (1993)
that the insured rm adjusts F such that the proceeds of the debt issue are just sucient to raise what the
uninsured debt issue raised and pay the insurance premium, so that the date-one cash ow to shareholders
remains zero (see Remark 2 to the theorem below).
The question raised by Schnabel and Roumi (1989) is: how does the level of debt F aect the decision to
take insurance or not in the presence of a safety loading? Figure 1 provides a neat graphical intuition. For
F = V   I(0), ABCD and DCE are both zero, so R0 = R = 0. When debt F rises marginally, the increase
in the deadweight loss without insurance R0 is proportional to BC, while the increase in R is proportional
to EC. For F close to V    I(0), the length of EC is itself close to zero, while the length of BC is not. So
the impact of a marginal change in F on R0 is an order of magnitude greater than the impact on R for F
close to V    I(0). As a consequence, the condition for taking insurance R0  R is satised for F slightly
above V    I(0). That is, the rm generally takes insurance for low levels of risky debt. Conversely, as F
rises towards V , BC goes to zero, while EC is bounded away from zero. So for F large enough, R0   R
falls as F rises. Whether or not it becomes negative, so that the rm does not take insurance, depends on
the magnitude of the loading .
To put this formally, let  (F) be the composite function that relates R0  R to F, where R0  R is given
by the right-hand side of (7) and Sa is given by (1).   maps [V    I(0); V ] on the reals (see Figure 2). It
is twice continuously dierentiable. The rm takes out insurance if, and only if,  (F)  0. Let
  =
R Sc
0 L(S)g(S)dS
R Sc
0 I(S)g(S)dS
  1 (9)
(> 0) denote the value of the loading such that  (V ) = 0. The rm's decision to ensure is easy to characterize
when   is strictly concave (the general case is treated in Remark 3 below):
Theorem: Suppose  00(F) < 0 for all F 2 [V    I(0); V ]. (a) For    , the rm takes out insurance for
all F 2 [V   I(0); V ]. (b) For  >  , there is F 2 (V   I(0); V ) such that the rm takes insurance for
F 2 [V    I(0); F ] and does not take insurance for F 2 (F; V ].
Before proving the theorem, let us relate its implications to Schnabel and Roumi's (1989) assertion quoted
in the rst paragraph that there is a critical level of debt at which the rm switches from not taking
4Figure 2: Insurance versus no insurance
insurance to taking insurance. The theorem states, to the contrary, that the rm generally takes insurance
for suciently low levels of risky debt. According to part (a) of the theorem, for suciently low levels of the
safety loading, there is no critical value of F at which the rm's insurance decision changes. Part (b) states
that for suciently high loading, the rm switches from taking to not taking out insurance as F rises above
some critical level. From (9), the latter case arises (i.e.,  >  ) if the loading  and the investment outlays
I(S) are large enough relative to the damages L(S). The proof of the theorem essentially elaborates on the
graphical argument given above:
Proof: For F = V   I(0), we have Sa = 0, so from (4) and (8), R0 = R = 0, i.e.,  (V   I(0)) = 0. From
(1), dSa=dF =  1=I0(Sa) (> 0). So from (4) and (8),
 0(F) = [L(Sa)   I(Sa)]g(Sa)
 1
I0(Sa)
  
Z Sa
0
g(S)dS; (10)
where Sa is given by (1), and in particular
 0(V    I(0)) = [L(0)   I(0)]g(0)
 1
I0(0)
> 0:
From (1), Sa = Sc for F = V . It follows from (4) and (8) that
 (V ) =
Z Sc
0
L(S)g(S)dS   (1 + )
Z Sc
0
I(S)g(S)dS:
(a) From the denition of   and the fact that  (F) falls when  rises,  (V )  0 if    . Together
with  (V    I(0)) = 0 and  00(F) < 0, it follows that  (F)  0, and the rm takes out insurance, for all
F 2 [V    I(0); V ]. (b) For  >  , we have  (V ) < 0, so there is a unique F such that  (F)  0 for
F 2 [V    I(0); F ] and  (F) < 0 for F 2 (F; V ]. This completes the proof of the theorem.
Remark 1: Schnabel and Roumi (1989, fn. 1, p. 157) state mistakenly that
dR
dF
=  
Z Sa
0
dSa
dF
g(S)dS < 0:
5They argue that since dR0=dF > 0 and, allegedly, dR=dF < 0, R0   R rises as F rises (i.e., insurance
becomes relatively more attractive) and conclude that there exists a critical value F at which the rm
switches from not taking to taking out insurance. Actually, dR=dF is given by the term 
R Sa
0 g(S)dS (> 0)
in (10). Both R0 and R rise, as F rises, and R0   R is not monotonic. R0   R becomes positive at rst,
but it turns negative for F large if the loading is suciently high.
Remark 2: A set of simple conditions which imply that the concavity condition is satised is uniformity of
g(S) = g on [0;  S] and linearity of L(S) = (Sc   S) and I(S) = (Sc   S) (where 0 <  <   V =Sc). In
this case, (1) and (10) become Sa = Sc   (V    F)= and
 0(F) =
g


(    + )
V    F

  Sc

;
respectively. The critical value   in (9) is   = = 1. It can be shown that in this linear-uniform special case,
the assertion of the theorem also holds true under Garven and MacMinn's (1993) alternative assumption
that the insured rm adjusts the face value of debt such that the date-one cash ow remains zero, and that
the critical value F is the same as in the cum dividend case. Generally, from (10),
 00(F) =
1
I0(Sa)

d
dSa

L(Sa)   I(Sa)
I0(Sa)

g(Sa) +
L(Sa)   I(Sa)
I0(Sa)
g0(Sa) + g(Sa)

;
where Sa is given by (1). A simple set of sucient conditions (generalizing the linear-uniform example) for
 00(F) < 0 is I00(S)  0, L0(S) I0(S)  0, and g0(S)  0 for all S 2 [0;Sc]. The rst two inequalities imply
that the rst term in the sum in braces is non-negative, the third inequality ensures that the second term is
also non-negative.
Remark 3: The fact that  (V    I(0)) = 0 and  0(V    I(0)) > 0 does not depend on concavity of
 . So the rm generally takes insurance for suciently small levels of risky debt. However, when  (F) is
non-concave, there can be multiple values of F 2 (V    I(0); V ) at which  (F) = 0 crosses the F-axis.
In that case, the rm switches back and forth between taking and not taking insurance as F rises. To
see that this is a possible outcome of the model, consider the following example. Let Sc = 1, V  = 1,
I(S) = 0:9004498875   0:9(S + 0:001)0:5, L(S) = I(S) + ( S3 + S2) + 0:001(1   S), and
g(S) =
e
 0:5(
S 0:2
0:1 )
2
0:1
p
2
for S in [0; 1] (i.e., the distribution of state prices is truncated normal). The example is constructed such
that for S small, L(S)   I(S) is small and I0(S) is large, so that from (10),  0(F) < 0 for F and, hence,
Sa small enough. (a) For  = 0:05,  (F) turns negative at F = 0:1769684113 and positive again at F =
0:3433426436 and terminates at 0:01207642332 for F = 1. That is, there is an intermediate range of debt
levels (F 2 (0:1769684113; 0:3433426436)) for which the rm does not insure. (b) For  = 0:1,  (F) turns
negative at F = 0:1502631825, becomes positive again at F = 0:4160717224, nally turns negative at
F = 0:8714716708, and terminates at  0:01256042295. That is, the rm does not insure at intermediate
levels of debt (F 2 (0:1502631825; 0:4160717224)) or with high indebtedness (F 2 (0:8714716708; 1]).
6In sum, we have shown that what keeps a rm from insuring a casualty is the combination of a high
safety loading and high indebtedness. Against the background of Mayers and Smith's (1987) analysis of
fair insurance, this means in essence that the model outcome is \continuous" in model parameters: a rm
generally takes out fair insurance if it can. A rm takes out insurance with a safety loading if the fair premium
and/or the loading and, therefore, the deadweight loss of insurance are small enough. For a suciently small
safety loading, this condition is satised for all levels of risky debt. Taking insurance becomes unattractive
if, and only if, the loading is suciently high and debt is suciently large.
References
Garven, James R., and Richard D. MacMinn (1993), \The Underinvestment Problem, Bond Covenants,
and Insurance", Journal of Risk and Insurance 60, 635-46.
Hau, Arthur (2007), \Insurance, Bond Covenants, and Under- or Over-investment With Risky Asset Re-
constitution", Journal of Risk and Insurance 74, 3-22.
Mayers, David, and Cliord W. Smith (1987), \Corporate Insurance and the Underinvestment Problem",
Journal of Risk and Insurance 54, 45-54.
Myers, Stewart C. (1977), \Determinants of Corporate Borrowing", Journal of Financial Economics 5,
147-75.
Schnabel, Jacques A., and Ebrahim Roumi (1989), \Corporate Insurance and the Underinvestment Problem:
An Extension", Journal of Risk and Insurance 56, 155-9.
7

Corporate Insurance and the Underinvestment Problem: An Extension&quot;,

Corporate Insurance and the Underinvestment Problem&quot;,

Determinants of Corporate Borrowing&quot;,

Insurance, Bond Covenants, and Under- or Over-investment With Risky Asset Reconstitution&quot;,

The Underinvestment Problem, Bond Covenants, and Insurance&quot;,

Corporate Insurance with Safety Loadings: A Note

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Corporate Insurance with Safety Loadings: A Note

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