A.1 Proof of Proposition 1
For a stock insurer, the actuarially fair premium Pf air(C) as a function of capital C is implicitly de…ned by Pf air(C) = E Xi 1f iXi nPf air(C)+Cg+ Ii(X1; :::; Xn) 1f iXi>nPf air(C)+Cg . where Fn is the n-fold convolution of F1 and thus the distribution function of the aggregate lossPn
i=1Xi. Pf air(0) = 0 is therefore the unique solution to (10).
For C = +1, the company is never insolvent and the actuarially fair premium is given by Pf air(1) = E [X1].
As f is a continuous function in P the intermediate value theorem implies that there exists a unique solution Pf air(C) > 0 for f (Pf air(C)) = 0.
Implicitly di¤erentiating (10) with respect to C yields
nPf air0 (C) = nPf air0 (C) + 1 (1 Fn(nPf air(C) + C))
for all C > 0. The actuarially fair premium is thus strictly increasing in the amount of risk capital.
For a mutual insurer, suppose policyholders provide a premium Pm. This premium is actuarially fair if and only if
Pm= E Xi+ Pm 1 n
Xn
i=1Xi 1f iXi nPmg+ Ii(X1; :::; Xn) 1f iXi>nPmg
withPn
i=1Ii(X1; :::; Xn) = nPm and E [Ii(X1; :::; Xn)] = Pmfor all i = 1; :::; n. Summing over all policies yields that total premiums provided, nPm, are actuarially fair. Since all policies have the same expected value of payout, each single premium provided is actuarially fair.
A.2 Proof of Proposition 2
Suppose that the insolvency rule speci…es a pro-rata rule, i.e.
Ii(X1; :::; Xn) = Xi
Pn
i=1Xi T C,
and let denote the loading in excess of the actuarially fair premium.6 The …nal level of wealth of a policyholder is then given by
where wS(C; ) and wIS(C; ) are the levels of …nal wealth in solvency and insolvency states, i.e. if Pn
i=1Xi n (Pf air(C) + ) + C andPn
i=1Xi > n (Pf air(C) + ) + C, respectively. The actuarially fair premium Pf air(C) is implicitly de…ned by (10). The policyholder’s expected utility of …nal wealth is given by
i=2xi. Di¤erentiating expected utility with respect to yields
@E [u (W1s(C; ))]
6For expositional purposes, we focus on the pro-rata rule as bankruptcy rule. The results, however, are robust to any “reasonable” bankruptcy rule that allow to create the form of coinsurance decsribed above. More precisely, the bankruptcy rule must be such that the marginal bene…t of an extra dollar under bankruptcy is increasing in the realized size of the loss.
The second derivative is given by
Expected utility is globally concave in and any inner solution (C) to the FOC
@E [u (W1s(C; ))]
@ j = (C)= 0 (11)
is the unique global maximum. For C = 1, we have Pf air(1) = E [X1] and the …rst derivative is given by
@E [u (W1s(C; ))]
@ jC=1= u0(w0 E [X1] ) < 0.
As expected utility is decreasing in , we get the corner solution (1) = 0.7 As (C) is continuous in C, (C) = 0 for large values of C. For C = 0, we have Pf air(0) = 0 (see Proposition A.1) and the …rst Evaluating this derivative at = 0 yields
@E [u (W1s(C; ))]
7The participation constraint for risk-neutral shareholders providing capital imposes 0.
We have
Total di¤erentiation of the FOC (10) with respect to C and implies d (C)
As expected utility is globally concave in we derive sign d (C)
dC = sign @2E [u (W1s(C; ))]
@ @C j = (C) . (12)
The cross-derivative is then given by
@2E [u (W1s(C; ))]
In Proposition A.1, we have shown that Pf air0 (C) > 0 which implies
@2E [u (W1s(C; ))]
Introducing the constant coe¢ cient of absolute risk aversion Ra= uu000(w)(w) yields The FOC (11) for (C) implies
Z 1
is a one-to-one, increasing mapping between C and P . Let P be given and de…ne the function f by
This shows that for all P there exists a unique C = C (P ) such that (13) is satis…ed.
Implicitly di¤erentiating (13) with respect to P yields
C0(P ) = (1 ) nFn(C (P ) + nP ) 1 (1 ) Fn(C (P ) + nP ). Thus C0(P ) > 0 for 0 < 1 and C0(P ) = 0 for = 1.
The solution P can be decomposed in the actuarially fair premium Pf air= E Xi 1f iXi C +nP g+ Xi
Pn i=1Xi
(C + nP ) 1f iXi>C +nP g and a premium loading Pload= P Pf air. Summing over all policies yields
nPf air= Eh
min Xn
i=1Xi; C + nP i . Combining this equation with (13) implies
nPload=
First we show that there exists a strictly positive solution (C; P ) of the optimization problem. From Lemma 1 we know that for each level of capital C there exists a unique P = P (C) such that the ex-ante zero expected pro…t condition
C = (1 ) E C + nP (C) Xn i=1Xi
+
is satis…ed. As argued in the text, the premium P (C) that results from the maximization of policyholders’
expected utility given a level of capital C must include a strictly positive, optimal loading, (C), that compensates shareholders for the frictional cost of capital, i.e.
n (C) = E C + n (Pf air(C) + (C)) Xn
i=1Xi + . De…ne the function f by
f (C) = n (C) E C + n (Pf air(C) + (C)) Xn
i=1Xi + .
We note that
f (0) = n (0) E n (0) Xn
i=1Xi +
> 0
as (0) > 0 (see Proposition 2). We have also shown in Proposition 2 that there exists a critical threshold level of capital ^C > 0 above which the optimal loading is strictly zero, i.e. (C) = 0 for all C C.^ Evaluating the function f at C = ^C yields
f C^ = E C + nP^ f air C^ Xn
i=1Xi +
< 0.
The intermediate value theorem implies that there exists a solution C with 0 < C < ^C such that f (C) = 0.
The optimal premium P is then given by P = P (C) and satis…es P > 0.
To show that C < C and P < P , we …rst show that C C and P P and then C < C and P < P . (C ; P ) is de…ned by
(C ; P ) = arg max
C;P E [u (W1s(C; P ))]
subject to the participation constraint. The incentive-compatibility constraint for the stock insurer without commitment
P (C) = arg max
P E [u (W1s(C; P ))] (14)
implies that P (C ) P . Otherwise, if P (C ) > P then (C (P (C )) ; P (C )), where C (P (C )) > C is de…ned by the participation constraint, dominates (C ; P ). This contradicts the optimality of (C ; P ).
Now suppose C > C . Then P (C) > P P (C ). Since P (C) is strictly decreasing in C (this can be shown by implicitly di¤erentiating the FOC of (14)) P (C) > P (C ) implies C < C which is a contradiction to the assumption. Therefore, C C . This, in turn, implies P P since the participation constraint provides an increasing mapping (see Lemma 1).
Last, we show that C < C and P < P . For a stock insurer facing the commitment problem, the optimal loading (C) provided by policyholders satis…es the following FOC (see Proof A.2 of Proposition 2)
For a stock insurer not facing the commitment problem, policyholders’ expected utility of …nal wealth is given by and C (P ) is the unique level of capital that satis…es the zero expected pro…t condition (see Lemma 1).
Di¤erentiating expected utility with respect to P yields
Evaluating this derivative at the optimal premium level of the stock insurer facing the commitment problem, i.e. at P = P (C) = Pf air(C) + (C), yields P .8As the participation constraint provides an increasing mapping between capital and premium (see Lemma 1) we conclude C > C.
A.5 Proof of Proposition 4
Since nP + C = nPm the premium P under the stock form can be expressed as P = Pm 1
nC (P ) = Pm 1
n(1 ) E nPm Xn
i=1Xi +
where C (P ) is de…ned by the shareholders’ participation constraint (1). The policyholder’s …nal wealth distribution (3) under the stock form can then be written as
W1s(C; P ) = W1s(Pm) Subtracting the policyholder’s …nal wealth distribution (8) under the mutual form yields
W1s(C; P ) W1m(Pm) = 1 distri-bution under the stock form over the wealth distridistri-bution under the mutual form, i.e. E [u (W1s(C; P ))] >
E [u (W1m(Pm))].
A.6 Proof of Corollary 1
Let Pm denote the solution to the optimization problem (7) of the mutual insurer. De…ne the premium P0 of an stock insurer such that it is equally capitalized to the mutual insurer, i.e. P0 satis…es
nP0+ C (P0) = nPm
8Note that, for this conclusion, we do not need the concavity of expected utility in P , since we have shown above that P P.
where C (P0) is uniquely de…ned by the shareholders’ participation constraint (1).9 Proposition 4 implies that the stock insurer dominates the mutual insurer, i.e. E [u (W1s(C (P0) ; P0))] > E [u (W1m(Pm ))]. For the optimal level of total capital, nP + C (P ), of the stock insurer we have E [u (W1s(C (P ) ; P ))] >
E [u (W1s(C (P0) ; P0))] and therefore E [u (W1s(C (P ) ; P ))] > E [u (W1m(Pm ))].
A.7 Proof of Proposition 5
Stock insurer with and without commitment. The optimization problem (2) with k subgroups is By de…ning c = n1C as shareholders’capital per policyholder we can rewrite this problem as
(c ; P ) = arg max
This optimization problem is equivalent to the one with k policyholders facing independent and identically distributed losses. The same comparison holds for the optimization problem of the stock insurer without commitment.
Mutual insurer. For the mutual, the optimization problem with k subgroups is Pm = arg max
Again, the optimization problem of the mutual insurer is exactly identical to the one with k policyholders facing independent and identically distributed losses.
9Note that P0 exists and is unique since nP + C (P ) is increasing in P from 0 to +1 accoridng to Lemma (1).
A.8 Proof of Proposition 6
Incentives stemming from the e¤ect on the utility are higher under the mutual form if
Um(n) Um(q) > Us(n) Us(q) . (15)
We prove (15) by showing that the managerial incentives to expand a mutual insurer are increasing in the capital of the company. We thus have to show that
Um(njPm ) Um(qjPm ) > Um(nj Pm ) Um(qj Pm )
We will prove the inequality by using the Central Limit Theorem (CLT) to show that Eh
(nPm Pn
i=1Xi)+i Eh
(n Pm Pn
i=1Xi)+i
is increasing in n. The CLT implies
E nPm Xn
n . The …rst term of the product is increasing in n.
Di¤erentiating the second term with respect to n yields
@
De…ne the function f by
f ( ) = (Pm E [X1]) Z z2(n)
0
e 12z2dz ( Pm E [X1]) Z z1(n)
0
e 12z2dz.
If Pm E [X1] 0 then f ( ) > 0 (as Pm > E [X1]) and (16) is increasing in n. Suppose Pm E [X1] >
0. Then f (1) = 0 and
f0( ) = Pm
Z z1(n) 0
e 12z2dz ( Pm E [X1]) Pm (X1) =p
ne 12z1(n)2 < 0.
Since f is strictly decreasing and f (1) = 0, we derive that f ( ) > 0 for all 0 < 1. (16) is therefore increasing in n.
References
[1] Cagle, J.A.B. and S.E. Harrington (1995) “Insurance Supply with Capacity Constraints and Endogenous Insolvency Risk”, Journal of Risk and Uncertainty 11, p. 219-232
[2] Cummins, J.D. and P.M. Danzon (1997) “Price, Financial Quality, and Capital Flows in Insurance Markets”, Journal of Financial Intermediation 6, p. 3-38
[3] Doherty, N.A. and G. Dionne (1993) “Insurance with Undiversi…able Risk: Contract Structure and Organizational Form of Insurance Firms”, Journal of Risk and Uncertainty 6, p. 187-203
[4] Doherty, N.A. and J.R. Garven (1986) “Price Regulation in Property-Liability Insurance: A Contingent-Claims Approach”, Journal of Finance 41, p. 1031-1050
[5] Doherty, N.A. and H. Schlesinger (1990) “Rational Insurance Purchasing: Consideration of Contract Nonperformance”, Quarterly Journal of Economics 55, p. 243-253
[6] Fletcher, L.P. (1966) “Motivations Underlying the Mutualization of Stock Life Insurance Com-panies”, Journal of Risk and Insurance 33, p. 19-32
[7] Gron, A. (1994) “Capacity Constraints and Cycles in Property-Casualty Insurance Markets”, RAND Journal of Economics 25, p. 110-127
[8] Gruendl, H. and H. Schmeiser (2002) “Pricing Double-Trigger Reinsurance Contracts: Finan-cial Versus Actuarial Approach”, Journal of Risk and Insurance 69, p. 449-468
[9] Hellwig, M. (2001) “Corporate Governance and the Financing of Investment for Structural Change”, in: Deutsche Bundesbank (ed.), Investing Today for the World of Tomorrow:
Studies on the Investment Process in Europe, Springer Verlag, Berlin, Heidelberg, New York, p. 203-223
[10] Hellwig, M. (2000) “On the Economics and Politics of Corporate Finance and Corporate Con-trol”, in: X. Vives (ed.), Corporate Governance, Cambridge University Press, Cambridge, p. 95-134
[11] Lamm-Tennant, J. and L. Starks (1993) “Stock versus Mutual Ownership Structures: The Risk Implications”, Journal of Business 66, p. 29-46
[12] Mahul, O. and B.D. Wright (2004a) “Implications of Incomplete Performance for Optimal Insurance”, Economica 71, p. 661-670
[13] Mahul, O. and B.D. Wright (2004b) “E¢ cient Risk Sharing within a Catastrophe Insurance Pool”, working paper
[14] Mayers, D. and C.W. Smith (2005) “Agency Problems and the Corporate Charter”, Journal of Law, Economics, & Organization 21, p. 417-440
[15] Mayers, D. and C.W. Smith (1981) “Contractual Provisions, Organizational Structure, and Con‡ict Control in Insurance Markets”, Journal of Business 54, p. 407-434
[16] Myers, S.C. and N.S. Majluf (1984) “Corporate Financing and Investment Decisions when Firms Have Information that Investors do not have”, Journal of Financial Economics 13, p. 187-221
[17] Smith, B.D. and M. Stutzer (1995) “A Theory of Mutual Formation and Moral Hazard with Evidence from the History of the Insurance Industry”, Review of Financial Studies 8, p.
545-577
[18] Smith, B.D. and M. Stutzer (1990) “Adverse Selection, Aggregate Uncertainty, and the Role for Mutual Insurance Contracts”, Journal of Business 63, p. 493-510
[19] Stein, J.C. (1997) “Internal Capital Markets and the Competition for Corporate Resources”, Journal of Finance 52, p. 111-133
[20] Viswanathan, K.S. and J.D. Cummins (2003) “Ownership Structure Changes in the Insurance Industry: An Analysis of Demutualization”, Journal of Risk and Insurance 70, p. 401-437 [21] Winter, R. (1988) “The Liability Crisis and the Dynamics of Competitive Insurance Markets”,
Yale Journal of Regulation 5, p. 455-499
[22] Zanjani, G. (2007) “Regulation, Capital, and the Evolution of Organizational Form in U.S.
Life Insurance”, American Economic Review 97, p. 973-983
[23] Zanjani, G. (2004) “Mutuality and the Underwriting Cycle: Theory with Evidence from the Pennsylvania Fire Insurance Market, 1873-1909”, working paper