Agenda
Intro
Setup
MV problem
Examples
Optimal Retention Levels
in
Dynamic (Re)insurance Markets
Enrico Biffis
Faculty of Actuarial Science and Insurance
Cass Business School, London
Montreal, 10th August 2006
1/12
Conclusion
Agenda
Intro
Setup
Outline
2/12
1. Introduction
2. Setup
3. Dynamic constrained MV problem
4. Examples
5. Conclusion
MV problem
Examples
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Intro & Motivation
3/12
General problem statement:
optimal dynamic insurance strategy for given risk exposure,
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Intro & Motivation
3/12
General problem statement:
optimal dynamic insurance strategy for given risk exposure,
where:
- optimality: maximize expected utility from terminal wealth/dividend
payouts, minimize ruin probability...
my setup:
◦ (re)insurance decision in a MV setting (De Finetti, 1940)
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Intro & Motivation
3/12
General problem statement:
optimal dynamic insurance strategy for given risk exposure,
where:
- optimality: maximize expected utility from terminal wealth/dividend
payouts, minimize ruin probability...
my setup:
◦ (re)insurance decision in a MV setting (De Finetti, 1940)
- setting: sources of randomness, financial market...
my setup:
◦ Brownian filtration
◦ financial market
◦ randomness in claims/market parameters
Conclusion
Agenda
Intro
Setup
MV problem
Examples
The model
4/12
. B
(Ω, F , F, P), with F = F (B Brownian motion)
Wealth and shocks:
◦ Wealth/exposure process: X (t) $
◦ Proportional wealth shocks: dX (t) = −X (t) [δ(t)dt + σ(t) · dB(t)]
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Conclusion
The model
. B
(Ω, F , F, P), with F = F (B Brownian motion)
Wealth and shocks:
◦ Wealth/exposure process: X (t) $
◦ Proportional wealth shocks: dX (t) = −X (t) [δ(t)dt + σ(t) · dB(t)]
Insurance market:
◦ Exposure: w (t) $ covered by insurance, v (t) = X (t) − w (t) $ retained.
◦ Premium rate per unit of insured capital/wealth: π(t)
w (t)π(t)dt = premium paid to instantaneously insure exposure w (t)
4/12
Agenda
Intro
Setup
MV problem
Examples
Conclusion
The model
4/12
. B
(Ω, F , F, P), with F = F (B Brownian motion)
Wealth and shocks:
◦ Wealth/exposure process: X (t) $
◦ Proportional wealth shocks: dX (t) = −X (t) [δ(t)dt + σ(t) · dB(t)]
Insurance market:
◦ Exposure: w (t) $ covered by insurance, v (t) = X (t) − w (t) $ retained.
◦ Premium rate per unit of insured capital/wealth: π(t)
w (t)π(t)dt = premium paid to instantaneously insure exposure w (t)
Financial market:
◦ money market account: r (t) risk-free rate
◦ risky stocks: dSi (t) = Si (t) [µi (t)dt + σ i (t) · dB(t)]
◦ δ, σ, µ, σ are F-adapted
(i = 1, . . . , m)
Agenda
Intro
Setup
MV problem
Examples
Insurance-investment strategies
Insurance/investment strategy: u(t) = (v (t), v 1 (t), . . . , v m (t))T .
5/12
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Insurance-investment strategies
Insurance/investment strategy: u(t) = (v (t), v 1 (t), . . . , v m (t))T .
Wealth dynamics:
dX (t) =X (t)(r (t) − π(t))dt + v (t) [(π(t) − δ(t))dt − σ(t) · dB(t)]
+
m
X
i=1
5/12
v i (t) [(µi (t) − r (t))dt + σ i (t) · dB(t)]
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Insurance-investment strategies
Insurance/investment strategy: u(t) = (v (t), v 1 (t), . . . , v m (t))T .
Wealth dynamics:
dX (t) =X (t)(r (t) − π(t))dt + v (t) [(π(t) − δ(t))dt − σ(t) · dB(t)]
+
m
X
v i (t) [(µi (t) − r (t))dt + σ i (t) · dB(t)]
i=1
Constraints: v (t) ≥ 0 and. . .
5/12
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Insurance-investment strategies
5/12
Insurance/investment strategy: u(t) = (v (t), v 1 (t), . . . , v m (t))T .
Wealth dynamics:
dX (t) =X (t)(r (t) − π(t))dt + v (t) [(π(t) − δ(t))dt − σ(t) · dB(t)]
+
m
X
v i (t) [(µi (t) − r (t))dt + σ i (t) · dB(t)]
i=1
Constraints: v (t) ≥ 0 and. . .
◦ no shorting: 0 ≤ v k+1 (t), . . . , v m (t) (k = 0, . . . , m + 1);
Pk
◦ ‘safer’ assets S1 , . . . , Sk , ‘riskier’ assets Sk+1 , . . . , Sm :
i=1 v i (t) ≥ v (t);
◦ combinations of the above.
Conclusion
Agenda
Intro
Setup
MV problem
MV problem and solution
The Optimization Problem
6/12
• Constrained dynamic MV problem:

min V [X (T )]



sub



E [X (T )] = z ∗
(X , u) admissible
.
u ∈ C = constraints set
Examples
Conclusion
Agenda
Intro
Setup
MV problem
Examples
MV problem and solution
The Optimization Problem
• Constrained dynamic MV problem:

min V [X (T )]



sub



E [X (T )] = z ∗
(X , u) admissible
.
u ∈ C = constraints set
• Stochastic LQ control and BSDE theory: Hu-Zhou (2005).
• Efficient strategy:
∗
u (t) =

[X (t) − A(t)] ξ1∗ (t)

[A(t) − X (t)] ξ2∗ (t)
X (t) > A(t)
X (t) ≤ A(t)
...and mean-variance frontier:
E [X (T )] = f
6/12
³p
´
V [X (T )]; A(0), X (0)
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Key BSDEs
A(·) depends on the solutions to the following BSDEs:

∗ )dt + Λ
 dP1,2 (t) = f (t, P1,2 , Λ1,2 , H1,2
1,2 (t) · dB(t)

P1,2 (T ) = 1
P1,2 (t) > 0 a.s.t ∈ [0, T ]
.
∗
H1,2
(t, ω) = min H1,2 (t, ω, u, P1,2 , Λ1,2 )
u∈C
7/12
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Key BSDEs
A(·) depends on the solutions to the following BSDEs:

∗ )dt + Λ
 dP1,2 (t) = f (t, P1,2 , Λ1,2 , H1,2
1,2 (t) · dB(t)

P1,2 (T ) = 1
P1,2 (t) > 0 a.s.t ∈ [0, T ]
.
∗
H1,2
(t, ω) = min H1,2 (t, ω, u, P1,2 , Λ1,2 )
u∈C
Look to H1 when X (t) > A(t), to H2 when X (t) ≤ A(t).
7/12
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Key BSDEs
A(·) depends on the solutions to the following BSDEs:

∗ )dt + Λ
 dP1,2 (t) = f (t, P1,2 , Λ1,2 , H1,2
1,2 (t) · dB(t)

P1,2 (T ) = 1
P1,2 (t) > 0 a.s.t ∈ [0, T ]
.
∗
H1,2
(t, ω) = min H1,2 (t, ω, u, P1,2 , Λ1,2 )
u∈C
Look to H1 when X (t) > A(t), to H2 when X (t) ≤ A(t).
ξ1,2 (·) are given by:
.
ξ1,2 (t, ω, P1,2 , Λ1,2 ) = arg min H1,2 (t, ω, u, P1,2 , Λ1,2 )
u∈C
7/12
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Conclusion
Insurance and financial market example
Example: u(t) = (v (t), v (t))T
8/12
3
retention
2
1
0
stock
−1
H2
−2
H1
−2
−1
0
1
2
3
Agenda
Intro
Setup
MV problem
Examples
Conclusion
Insurance and financial market example
Example: v (t) ≥ 0, v (t) unrestricted
9/12
3
retention
2
1
ξ
2
ξ
1
0
stock
−1
H2
−2
H1
−2
−1
0
1
2
3
Agenda
Intro
Setup
MV problem
Examples
Conclusion
Insurance and financial market example
Example: v (t), v (t) ≥ 0
10/12
3
retention
2
1
ξ
2
0
ξ
stock
1
−1
H2
−2
H1
−2
−1
0
1
2
3
Agenda
Intro
Setup
MV problem
Conclusion
11/12
• De Finetti’s result in continuous-time:
- Insurance market only:
v ∗ (t) =
π(t) − δ(t)
(A(t) − X ∗ (t))
σ 2 (t)
Examples
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Conclusion
• De Finetti’s result in continuous-time:
- Insurance market only:
v ∗ (t) =
π(t) − δ(t)
(A(t) − X ∗ (t))
σ 2 (t)
• De Finetti & Markowitz:
- efficient frontier
→ for fixed ǫ ∈ (0, 1), x ∗ ≥ 0 choose z ∗ such that P (X (T ) < x ∗ ) ≤ ǫ
- constant retention levels are inefficient
→ market portfolio inefficient
11/12
Conclusion
Agenda
Intro
Setup
MV problem
Examples
Conclusion
Some references
Some references
12/12
◦ Bäuerle N. (2005), Benchmark and mean-variance problems for insurers, MMOR.
◦ Briys E. (1986), Insurance and consumption: the continuous-time case, JRI.
• De Finetti B. (1940), Il problema dei pieni, GIIA.
◦ Gollier C. (1994), Insurance and precautionary capital accumulation in a continuous-time
model, JRI.
◦ Hipp C. (2003), Stochastic control with application in insurance, LNM, Springer.
◦ Hojgaard B. and M. Taksar (1998), Optimal proportional reinsurance policies for diffusion
models with transaction costs, IME.
• Hu Y. and X.Y. Zhou (2005), Constrained LQ control with random coefficients, and
application to portfolio selection, SIAM JCO.
◦ Mnif M. (2002), Optimal risk control under proportional reinsurance contract: a dynamic
programming duality approach, WP.
◦ Schmidli, H. (2001), Optimal Proportional Reinsurance Policies in a Dynamic Setting, SAJ.
◦ Touzi N. (2000), Optimal Insurance Demand Under Marked Point Process Shocks, AAP.