Participating insurance contracts and the
Rothschild-Stiglitz equilibrium puzzle
Pierre Picard
Ecole Polytechnique
July 2009
Pierre Picard (Ecole Polytechnique)
July 2009
1 / 25
Overview
1
Motivation and model
2
Market equilibrium with participating contracts.
3
Deferred premium variations.
4
Concluding comments
Pierre Picard (Ecole Polytechnique)
July 2009
2 / 25
Motivation
The RS (1976) model : one of the most in‡uential contributions in
the insurance economics litterature.
An enigma : no equilibrium exists when second-best e¢ ciency
requires cross-subdization between contracts...
... which stimulated a lot of research: Wilson (1977), Miyazaki
(1977), Spence (1977), Riley (1979), Hellwig (1987), Engers and
Fernandez (1987), in a dynamic setting.
Problem with these models: the timing is very arbitrary... which limits
the relevance of the conclusions.
Pierre Picard (Ecole Polytechnique)
July 2009
3 / 25
This paper heads in a di¤erent direction: focus on the nature of
insurance contracts. RS restrict attention to non-participating
insurance contracts. There is no ground (neither theoretical nor
empirical) for such a restriction.
Insurers may also o¤er participating contracts (i.e. contracts with
policy dividends or supplementary call). In the real world mutuals
(and sometimes stock insurers) o¤er participating contracts. The
mutual market share at the end of 2006 was 28% for non-life business
worldwide (40% in Germany, 40% in France, 36% in Japan, 30% in
USA...).
Transferring underwriting pro…ts to reserves and increasing or
decreasing premiums according to the level of accumulated surplus
may act as a substitute to policy dividend or supplementary call
Pierre Picard (Ecole Polytechnique)
July 2009
4 / 25
A few papers have adressed the role of mutuals o¤ering participating
contracts in the RS environnement: Boyd, Presscott and Smith
(1988), Smith and Stutzer (1990), Ligon and Thistle (2005)... but
without focusing attention on the equilibrium existence issue and on
the nature of equilibrium (with or without cross-subsidization between
contracts).
Our starting point is not an ex ante institutional distinction between
corporate forms (stock insurers and mutuals). We consider an
insurance market where insurers (entrepreneurs) trade with risk
adverse insurance seekers and possibly with risk neutral capitalists.
The nature of contracts, and thus the corporate form, are
endogenous. If an insurer only trades with insurance seekers by
o¤ering them participating contracts, we may call it a mutual. If an
insurer o¤ers non-participating contracts to insurance seekers and
transfers pro…ts to capitalists, it is a stock insurer.
Pierre Picard (Ecole Polytechnique)
July 2009
5 / 25
Main intuition of the paper : when second-best e¢ ciency requires
cross-subsidization between risk types, participating contracts act as
an implicit threat againts deviant insurers who would like to attract
low risks only.
The paper predicts that individuals should be pooled in subgroups
with cross-subsidization within the subgroups, and no
cross-subsidization between subgroups (as in Spence, 1978).
Participating contracts allow subgroups that include more than one
type to be robust to competitive attacks.
Prediction: the variance of the loss ratio between contracts should be
larager for mutuals than for stock insurers.
Pierre Picard (Ecole Polytechnique)
July 2009
6 / 25
Notations
Eu = (1
π ) u ( WN
k + D ) + πu (WA + x + D )
where
u 0 > 0, u 00 < 0,
π
WN , W A
A
k
x
D
2 (0, 1) = probability of an accident,
= wealth without (with) accident,
= WN WA = loss in case of an accident,
= insurance premium
= net indemnity,
= policy dividend.
D = 0 for a non-participating policy (stock insurer); D 6= 0 depends on
the insurer’s pro…t for a participating policy (mutual). D is deterministic
because of the law of large numbers.
Pierre Picard (Ecole Polytechnique)
July 2009
7 / 25
Notations
W 1 = WN
k + D = …nal wealth if no accident,
2
= WA + x + D = …nal wealth in case of an accident,
π 2 fπ 1 , π 2 , ..., π n g with 0 < π n < π n 1 ... < π 1 < 1,
W
n
λi
=
proportion of type i individuals, with
∑ λi = 1.
i =1
Pierre Picard (Ecole Polytechnique)
July 2009
8 / 25
The Rothschild-Stiglitz equilibrium
Pierre Picard (Ecole Polytechnique)
July 2009
9 / 25
Nonexistence of equilibrium in the RS model
Equilibrium doesn’t exist when λ1
Pierre Picard (Ecole Polytechnique)
λ
July 2009
10 / 25
Extended RS-model
We characterize a subgame perfect equilibrium of a two stage game, with
m insurers and a continuum of individuals with types i = 1, ..., n:
Stage 1: each insurer j = 1, ..., m o¤ers a menu of contracts,
Stage 2: individuals respond by choosing the contract they prefer among
the o¤ers of the insurers.
This is called the extended RS-model, because we allow insurers to o¤er
either participating or non-participating contracts, while RS restrict
attention to non-participating contracts. For a given individual, the
attractiveness of a participating contract depends on the distribution of
types of the other purchasers of the same contract, hence a participating
contract may act as an implicit threat to deter competitive attacks.
Pierre Picard (Ecole Polytechnique)
July 2009
11 / 25
Illustration in a simple case
Insurer j o¤ers contract C j = (k j , x j ) with policy dividend D j = γj P j ,
j
where P j = pro…t per policyholder and γj 2 [0, 1]. E F = average
fair-odds line corresponding to the distribution of types among individuals
who purchase C j . C j generates the lottery C1j if γj = 1 or C2j if
γj 2 (0, 1).
Pierre Picard (Ecole Polytechnique)
July 2009
12 / 25
The two type case
Pierre Picard (Ecole Polytechnique)
July 2009
13 / 25
Proposition 1
An equilibrium always exists in the extended RS model with n = 2; it is
generically unique and it coincides with the MSW allocation. When
λ1 λ , the separating contracts of the RS model C1 ,C2 are o¤ered at
equilibrium without cross-subsidization and they may be participating or
e1 , C
e2 are
non-participating. When λ1 λ , the separating contracts C
e1 which is
o¤ered at equilibrium with cross-subsidization. Contract C
e
chosen by type 1 individuals is participating, while C2 which is chosen by
type 2 individuals may be participating or non-participating. The menu of
contracts o¤ered at an equilibrium with cross-subsidization maximizes the
type 2 expected utility under the zero-pro…t constraint and incentive
compatibility conditions.
Pierre Picard (Ecole Polytechnique)
July 2009
14 / 25
The n type problem
Strategy of insurer j:
Menu of n contracts = C j = (C1j , C2j , ..., Cnj , D j (.)) where Chj = (khj , xhj ),
Policy dividend strategy = D j (.) = (D1j (.), ..., Dnj (.)), with
Dhj (N1j , P1j , ..., Nnj , Pnj ) for contract Chj
where
Nhj = number of individuals who choose Chj ,
Phj = pro…t per policyholder for Chj .
C j is fully participating if
n
∑
Nhj Dhj (N1j , P1j , ..., Nnj , Pnj )
h =1
while Dhj (N1j , P1j , ..., Nnj , Pnj )
Pierre Picard (Ecole Polytechnique)
n
∑ Nhj Phj
h =1
0 for all h when C j is non-participating
July 2009
15 / 25
Candidate equilibrium allocation
A sequence of reservation expected utility levels u i from Spence (1978):
problem P1
u 1 = max(1
π 1 )u (W 1 ) + π 1 u (W 2 )
with respect to W 1 , W 2 , subject to
(1
and for 2
i
π 1 ) W 1 + π 1 ( W 2 + A ) = WN
n, problem Pi
u i = max(1
π i )u (Wi1 ) + π i u (Wi2 )
with respect to Wh1 , Wh2 , h = 1, ..., i , subject to
(1 π h )u (Wh1 ) + π h u (Wh2 )
(1 π h )u (Wh1 ) + π h u (Wh2 )
for h < i,
i
∑ [(1
u h for h < i,
(1
π h )u (Wh1+1 ) + π h u (Wh2+1 )
π h )Wh1 + π h (Wh2 + A)] = WN .
h =1
Pierre Picard (Ecole Polytechnique)
July 2009
16 / 25
When n = 2, the optimal solution to P2 is the Miyazaki-Wilson
equilibrium allocation.
c1, W
c 2 ), i = 1, ..., ng = the optimal solution to Pn .
f(W
i
i
Trade o¤: in Pn , it is costly to increase the h - type expected utility EUh
above u h for all h < n, but it relaxes the h - type incentive compatibility
constraint.
When n = 2, the trade-o¤ tips in favor of increasing EU1 above u 1 if
λ1 < λ . In that case there is cross-subsidization between types in P2 .
More generally, at an optimal solution to Pn , risk types are pooled in
subgroups, with cross-subsidization within the subgroups and no
cross-subsidization between subgroups.
Pierre Picard (Ecole Polytechnique)
July 2009
17 / 25
Lemma 1
There exists `θ 2 f0, ..., ng, θ = 0, ..., θ + 1 with `0 = 0 < `1
... `θ < `θ +1 = n such that for all θ = 0, ..., θ
h
∑
λ i [ WN
(1
λ i [ WN
(1
i =`θ +1
` θ +1
∑
i =`θ +1
ci1
π i )W
ci1
π i )W
Furthermore, we have
(1
(1
`2
ci2 + A)] < 0 for all h = `θ + 1, ..., `θ
π i (W
ci2 + A)] = 0.
π i (W
ci1 ) + π i u (W
ci2 ) = u i if i 2 f`1 , `2 , ..., ng,
π i )u (W
ci1 ) + π i u (W
ci2 ) > u i otherwise.
π i )u (W
Pierre Picard (Ecole Polytechnique)
July 2009
18 / 25
Pierre Picard (Ecole Polytechnique)
July 2009
19 / 25
Lemma 2
There does not exist any incentive compatible allocation
f(Wh1 , Wh2 ), h = 1, ..., ng such that
(1
and
π `θ )u (W`1θ ) + π `θ u (W`2θ )
n
∑ λ h [ WN
(1
π h )Wh1
u `θ for all θ = 1, ..., θ + 1
π h (Wh2 + A)] > 0.
h =1
Pierre Picard (Ecole Polytechnique)
July 2009
20 / 25
Proposition 2
c1, W
c 2 ), i = 1, ..., ng is an equilibrium allocation sustained by a
f(W
i
i
symmetric equilibrium of the market game where insurers o¤er
bi
participating contracts. Type i individuals choose Ci = C
(kbi , b
xi ) with
1
2
c ,b
c
kbi = WN W
x
=
W
W
and
D
(
.
)
is
such
that
i
A
i
i
n
∑ Di (N1 , P1 , ..., Nn , Pn )
i =1
n
∑ Ni Pi
i =1
λ1
b1 ), ..., λn , Π(C
bn )) = 0 for all i = 1, ..., n
, Π (C
m
m
D`θ (N1 , P1 , ..., Nn , Pn ) 0 for all θ = 1, ..., θ + 1.
Di (
Pierre Picard (Ecole Polytechnique)
July 2009
21 / 25
Intuition of Proposition 2
Consider the allocation induced by C j0 6= C = (C1 , ..., Cn ) o¤ered by a
deviant insurer j0 . It corresponds to a compound lottery that mixes C j0
and C but with the same pro…t as C j0 alone, because insurers j 6= j0 o¤er
full participating contracts (they do not provide positive residual pro…ts
whatever the risk types of policyholders). Furthermore, all types `θ get at
least the same expected utility as in the equilibrium allocation. Lemma 2
shows that this allocation cannot be pro…table, hence deviant insurer j0
does not make positive pro…t.
Pierre Picard (Ecole Polytechnique)
July 2009
22 / 25
Deferred premium variations
Reserves as a shock absorber : mutuals may shift the payment of
underwriting pro…t to their members by transferring current pro…t to
reserves and by later increasing or decresing premiums according to
the level of accumulated surplus.
Deferred premium variations are substitutes to policy dividends or
supplementary calls.
Paper also includes an extension to an overlapping generation setting,
where individuals live for two periods.
If transaction costs prevent individuals from changing their insurers
between periods 1 and 2: no qualitative change in the results.
Otherwise, more complex because moving to another insurer may
signal risk type.
Pierre Picard (Ecole Polytechnique)
July 2009
23 / 25
Concluding comments
Initial motivation of the paper: an inquiry on the nonexistence of
equilibrium in the RS model, starting with the observation that R&S
restrict the set of insurance contracts to non-participating policies.
Result is striking: Removing this restriction guarantees the existence
of equilibrium. The equilibrium allocation coincides with the MSW
allocation, but the underlying game form coincides with the RS model.
Participating policies (or deferred premium variations in a dynamic
setting) act as an implicit threat which prevents deviant insurers to
attract low risk individuals only.
Pierre Picard (Ecole Polytechnique)
July 2009
24 / 25
A new explanation about why mutuals are so widespread in insurance
markets and why they coexist with stock insurers, besides other
explanations (reduction in agency costs, risk screening device or
ability to cover undiversi…able risks).
Our main conclusion is that mutuals are robust to competitive attacks
in insurance markets with adverse selection, which may not be the
case for stock insurance companies.
Pierre Picard (Ecole Polytechnique)
July 2009
25 / 25