Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Optimal insurance for catastrophic risks:
theory and application to nuclear corporate
liability
Alexis Louaas & Pierre Picard
December 17, 2015
EM Lyon
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Research question
What level of coverage should societies enforce against nuclear
and other large scale accidents?
• Loss correlation generates aggregate loss volatility
• Exposure differences leads to different coverage choices and
non-unanimity issues
• Uncertainty about the probability of occurrence makes it
difficult to determine optimal coverage
• Low probability of occurrence and high severity of the loss
To various degrees climate change, global economic crises,... also
possess these characteristics
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Main Contributions
We highlight the role of risk aversion in the individual’s worst
case
We find that optimal coverage tends to a limit when the loss
probability tends to zero
• The uncertainty concerning the loss probability is of little
relevance for low probability events
• The exposure differences do not yield significant
disagreements between agents
We provide estimates of the optimal coverage for a nuclear
accident in France
Introduction
Individual risk
Collective risk
Nuclear coverage
Outline
Introduction
Conclusion
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Outline
Introduction
Risk premium and insurance demand for catastrophic risks
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Outline
Introduction
Risk premium and insurance demand for catastrophic risks
Optimal catastrophic coverage for a population
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Outline
Introduction
Risk premium and insurance demand for catastrophic risks
Optimal catastrophic coverage for a population
Nuclear coverage
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Outline
Introduction
Risk premium and insurance demand for catastrophic risks
Optimal catastrophic coverage for a population
Nuclear coverage
Conclusion
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Outline
Introduction
Risk premium and insurance demand for catastrophic risks
Optimal catastrophic coverage for a population
Nuclear coverage
Conclusion
Appendix
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Model presentation
An agent with vNM utility function u(x) twice continuously
differentiable
Has initial wealth w and faces a loss L with probability p
m(p, L) = pL and σ 2 (p, L) = p(1 − p)L2 are the expected value
and variance of the r.v. ˜l
00
(x)
A(x) = − uu0 (x)
is her index of absolute risk aversion
T (x) =
1
A(x)
is her absolute risk tolerance
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Normalized risk premium
The certainty equivalent C, defined as
u(w − C) = pu(w − L) + (1 − p)u(w)
Is a function of p and L
Let θ(p, L) be the normalized risk premium
θ(p, L) =
C(p, L) − m(p, L)
σ 2 (p, L)
We call θ(0, L) = limp→0 θ(p, L)
(1)
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Proposition 1
Proposition
For all L > 0, we have
θ(0, L) =
1
2
Z w
Z w
A(t)dt}]dx
[k(x)A(x) exp{
w−L
where k(x) = 2[x − (w − L)]/L2 and
x
Rw
w−L k(x)dx
= 1.
θ(0, L) is a weighted sum of all levels of risk aversion between
w − L and w
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Corollary 1
Corollary
For all L > 0, we have
1
θ(0, L) >
2
Z w
k(x)A(x)dx.
w−L
θ(p, L) has a lower bound
Under DARA, this lower bound can be very large if A(w − L) is
very large
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Proposition 2
Proposition
Assume R(x) ≡ xA(x) < γ̄, for all x ∈ [w − L, w].
Then under non-increasing absolute risk aversion, we have
θ(0, L) <
(γ̄ + 1)A(w − L)
2
θ(0, L) can be non-negligible only if A(w − L) is large enough
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Proposition 3
Proposition
Assume
• T (x) ≡ t(x, ε), with > 0,
• t(w − L, 0) = t0x (w − L, 0) = t00xx (w − L, 0) = 0,
• t0x (x) > 0 for x > w − L.
Then for all M > 0, θ(0, L) > M if ε is small enough.
The normalized risk premium can be made arbitrarily large if
• Risk tolerance does not increase too fast in the
neighborhood of the loss state
• Risk tolerance in the loss state is sufficiently small
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Insurance choice
Agent can buy a loaded contract (P , I), with P = (1 + σ)pI
The optimal contract is a choice of I among the following
lotteries
w1 = w − P
(1 − p)
w2 = w − P − L + I
(p)
In the (w1 , w2 ) plan, feasible lotteries are below the line
w1 [1 − (1 + σ)p] + w2 (1 + σ)p = w − (1 + σ)pL
Graph
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Proposition 4
Proposition
When p goes to 0, the optimal insurance coverage I goes to a
limit I ∗ defined as
u0 (w − L + I ∗ ) = (1 + σ)u0 (w)
proof
Policy implication : for events with sufficiently low probability of
occurrence, ambiguity and exposure differences are not the main
issues.
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Effect of a change in p
−3
w = 10000, L = 5000, u(x) = − x 3 (CRRA), p = 0.5
Model description
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Effect of a change in p
−3
w = 10000, L = 5000, u(x) = − x 3 (CRRA), p = 0.2
Model description
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Effect of a change in p
−3
w = 10000, L = 5000, u(x) = − x 3 (CRRA), p = 0.05
Model description
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Effect of a change in p
−3
w = 10000, L = 5000, u(x) = − x 3 (CRRA), p ∈ ]0, p̄]
Model description
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
A model of collective risk with correlated losses
Population of unit mass composed of n groups
A fraction αi of the population belongs to group i
An accident occurs with probability π
Conditionally on the occurrence of the accident, agents in group i
have probability qi of being affected
Conditionally on being affected agent in group i faces a loss
x̃i ∈ [0, x̄i ]
Distributed with p.d.f. f (xi ) = F 0 (xi )
The total cost
of an accident is therefore
R x¯i
Pn
α
q
[
x
i f (xi )dxi ]
i=1 i i 0
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
An insurance contract
We assume that a firm buys an insurance contract, and passes
the prices on to the consumers
A contract specifies an indemnity schedule Ii (xi )∀xi ∈ [0, x̄i ]
For a loaded price
P = c(π, K)
Where K = (1 + σ) ni=1 αi qi [
required to pay all claims
P
R x¯i
0
Ii (xi )f (xi )dxi ] is the amount
And c(π, K) is the cost of contingent capital
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Optimal insurance
We call Ci the certainty equivalent for an agent of group i
We consider a benevolent social planner, who chooses the
contracts (Ii (xi )i=1,...,n , P )
In order to minimize the social cost
n
X
αi Ci
i=0
Under the constraint that the chosen contracts belong to the
feasible set
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Proposition 5
Proposition
Assume c0K (π, K) ≥ π and limπ→0 c0K (π, K) = 0. Then, when π
goes to zero, the optimal indemnity schedules Ii (xi ) converge
toward a unique straight deductible
I ∗ (xi ) = max{xi − d∗ , 0}
The deductible d∗ and the contingent capital K ∗ are jointly
defined by
c0K (π, K ∗ )
,
π→0
π
u0 (w − d∗ ) = (1 + σ)u0 (w) lim
and
∗
K =
n
X
i=1
"Z
xi
αi qi
d∗
#
∗
(xi − d )f (xi )dxi .
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Policy implication
• In a frictionless world with risk neutral investors, we would
have
c(π, K) = πK =>
c0K (π,K)
π
=1
• Ambiguity and exposure differences can be neglected
• We can decide a global indemnity envelope
Rank the agents with respect to loss severity
Indemnify agent with loss xi < x0i only if agent with loss x0i
has received at least x0i − xi
Introduction
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Collective risk
Nuclear coverage
Illustration
w − x1
w − x2
Conclusion
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Illustration
x2 − x1
w − x1
w − x2
Conclusion
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Illustration
K ∗ −(x2 −x1 )
2
K ∗ −(x2 −x1 )
2
x2 − x1
w − x1
w − x2
Conclusion
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Illustration
d∗
K ∗ −(x2 −x1 )
2
d∗
K ∗ −(x2 −x1 )
2
x2 − x1
w − x1
w − x2
Conclusion
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Methodology
Our objective is to characterize the optimal level of coverage for
a major nuclear accident
We rely on studies written by safety specialists (PSA)
And on a paper by Eeckhoudt et al. (2000)
The PSA provides information about the number and the type of
victims
From this aggregate information we can reconstruct the
individual lotteries Details
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Pricing cat-bonds
Between 2011 and 2014, information is available for 107 traded
tranches of cat bonds
We estimate
si = β0 E`i + β1 E`2i + γXi + εi
Where si is the spread, E`i the expected capital loss (as a
fraction of par value)
We consider a simple cat bond with E` = 1
Hence ŝ = βˆ0 Eπ + βˆ1 Eπ 2 + γ̂Xi
And ĉ(π, K) = K[ŝ(π) − ŝ(0)]
OLS estimation
(2)
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
The utility function
We use HARA functions
Except for the CARA and CRRA cases, HARA functions are
DARA and IRRA
We perform simulations letting R and R vary between 1 and 5
R is the index of relative risk aversion in the worst individual
state
R is the index of relative risk aversion in the best individual state
Introduction
Individual risk
Collective risk
Nuclear coverage
Social cost
K ∗ minimizes SC(K)
Figure: π = 10−6 , R = 1, R = 2
Details
Conclusion
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Social cost
K ∗ minimizes SC(K)
Figure: π = 10−5 , R = 1, R = 2
Details
Conclusion
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Social cost
K ∗ minimizes SC(K)
Figure: π = 10−4 , R = 1, R = 2
Details
Conclusion
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Result 1
Result
The optimal choice of coverage K ∗ does not change with the
accident probability π
This is a direct application of proposition 5
Policy implication: lack of knowledge of the precise probability π
should not affect coverage choice
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Simulations
Result
The central scenarios (π = 10−5 ) are characterized by optimal
levels of coverages higher than the 700 million euros provided for
by the 2004 revision of the Paris convention.
R
VSL
S1
S2
S3
S4
S5
1
3.39e+06
1.245e+09
2.475e+09
3.720e+09
4.965e+09
6.195e+09
2
4.93e+06
1.305e+09
2.595e+09
3.900e+09
5.190e+09
6.495e+09
Table: Coverage, R = 2
R
VSL
S1
S2
S3
S4
S5
1
3.39e+06
5.231e-02
9.643e-02
1.341e-01
1.667e-01
1.951e-01
2
4.93e+06
9.595e-02
1.702e-01
2.293e-01
2.775e-01
3.176e-01
Table: Welfare gain, R = 2
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Wrap-up
In this article we
• Characterized what is a catastrophic risk for an individual
• Analyzed the optimal insurance scheme for a risk of
catastrophic event generated by a productive activity
• Applied our approach to the case of a nuclear accident
Our results suggest that
• A(w − L) is critical to qualify a risk as disaster risk
• Infinitesimally small probability risk should be covered
• Ambiguity and exposure differences are not the main issues
• The french legislation concerning nuclear liability law could
be more ambitious than it currently is
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Conclusion
More research effort could be devoted to
• Try to elicit individual’s risk aversion around the loss state
• Understand better the complementarities between health
and wealth
• Understand the mechanisms underlying cat bond pricing
Introduction
Individual risk
Collective risk
Nuclear coverage
Thank you!
Conclusion
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Coverage in the limit
presentation
Our first order condition was
(1 − p)(1 + σ)u0 (w1 ) = [1 − (1 + σ)p]u0 (w2 )
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Coverage in the limit
presentation
Our first order condition was
(1 − p)(1 + σ)u0 (w1 ) = [1 − (1 + σ)p]u0 (w2 )
Recall
w1 = w − (1 + σ)pI
and
w2 = w − L + (1 − (1 + σ)p)I
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Coverage in the limit
presentation
Our first order condition was
(1 − p)(1 + σ)u0 (w1 ) = [1 − (1 + σ)p]u0 (w2 )
Hence
limp→0 w1 (p, L) = w
and
limp→0 w2 (p, L) = w − L + I
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
Coverage in the limit
presentation
Our first order condition was
(1 − p)(1 + σ)u0 (w1 ) = [1 − (1 + σ)p]u0 (w2 )
Hence
limp→0 w1 (p, L) = w
and
limp→0 w2 (p, L) = w − L + I
When p → 0, the FOC implies
(1 + σ)u0 (w) = u0 (w − L + I ∗ )
The optimal contract therefore converges to a limit (I ∗ , P ∗ ) with
I ∗ = u0−1 ((1 + σ)u0 (w)) − w + L and
P∗ = 0
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Appendix
The model
Two groups i = 1, 2. Type i = 1 agents are located close (100km)
to a power plant, type i = 2 are further away
n1 = 2 millions, n2 = 56 millions
In case of accident, agent of type i can face S situations
Each characterized by a probability fis
An insurance contract entitles to an indemnity Ii (s, K) in state s
For a price P (K) with
K = (1 + σ)
presentation
n
n
1
2
1 X
29 X
f1s I1 (s, K) +
f2s I2 (s, K)
30 i=1
30 i=1
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Individual lotteries
State
s=1
s=2
s=3
s=4
s=5
s=6
Direct loss
848 250
848 250
400 000
260 000
100 000
0
Total loss L1s
848 250
848 250
401 624
261 624
101 624
1 624.2
f1s (conditional)
1.2500e-06
2.4875e-04
2.5000e-06
4.9750e-04
4.9963e-03
9.9425e-01
Table: lotteries for type h = 1
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
Conclusion
Individual lotteries
State
s=1
s=2
s=3
Direct loss
848 250
260 000
0
Total loss L2s
848 250
261 624
1 624.2
f2s (conditional)
5.3571e-05
1.0714e-04
9.9984e-01
Table: lotteries for type h = 2
presentation
Appendix
Introduction
Individual risk
Collective risk
Nuclear coverage
OLS results
E`i
E`2i
R2
ŝ(10−5 )
Estimate
2.3156∗∗∗
(8.8725)
−7.6155∗∗∗
(−3.3070)
0.8204
2.3156*10−5
presentation
Conclusion
Appendix