Transcript Pricing and capital allocation for unit
Pricing and capital allocation for unit-linked life insurance contracts with minimum death guarantee C. Frantz, X. Chenut and J.F. Walhin Secura Belgian Re
The problem
1,2
Capital sous risque dans une garantie plancher
Sum at risk 1 Insurer’s liability for a death at time t: max(
K
,
S t
)
S t
max(
K
S t
, 0 ) 0,8 0 1 2 3 4 5 6 7 8 9 10 • How to price it ?
• Capital allocation ?
Two approaches …
The financer: it is a contingent claim Solution: hedging on the financial market Black-Scholes put pricing formula
The actuary: it is an insurance contract Solution: equivalence principle Expected value of future losses
… and two risk managements
Financial approach : hedging on financial markets
Actuarial approach : reserving and raising capital
Agenda
Actuarial vs financial pricing Monte Carlo simulations Cash flow model Open questions
First question: actuarial or financial pricing?
Hypotheses :
– Complete and arbitrage-free financial market – Constant risk-free interest rate – Financial index follows a GBM:
dS t
S t dt
S t dW t
Simple expressions for the single pure premium in both approaches
Single pure premiums
Actuarial pricing :
SPP Act
k T
1
Ke
rk
(
d
2
Act
( 0 ,
k
))
k p x q x
k
S
0
k T
1
e
(
r
)
k
(
d
1
Act
( 0 ,
k
))
k p x q x
k
Financial pricing :
SPP Fi
k T
1
Ke
rk
(
d
2
Fi
( 0 ,
k
))
k p x q x
k
S
0
k T
1 (
d
1
Fi
( 0 ,
k
))
k p x q x
k
with
d
2
Act
(
t
,
T
)
d
1
Act
(
t
,
T
) log(
S t
/
K
) (
r
d
2
Act
(
t
,
T
)
T T
t
t
2 / 2 )(
T
t
)
d
2
Fi
(
t
,
T
)
d
1
Fi
(
t
,
T
) log(
S t
/
K
)
d
2
Fi
(
t
,
T
) (
T T
2
t
t
/ 2 )(
T
t
)
Monte Carlo simulations
Goal : distribution of the future costs 3 processes to simulate :
– Financial index – Death process – Hedging strategy (financial approach only)
1 0,8 0,6 0,4 0,2 0 0
Probability distribution functions
10 20 30
Discounted future costs
40 50 Actuarial Financial 60
Sensitivity analysis
1,00 0,80 0,60 0,40 0,20 0,00 0
Distribution of DFC - variation of
-
10 20 30 40
DFC Act
50 60 70 80 20% 15% 10% 8,5% 5% 0% -5% -10% -15% -20% No Stock
1 0,8
FI
0,6 0,4 0,2 0 6
Sensitivity analysis Distribution of DFC Fi - variation of
-
7 8 9 10
DFC Fi
11 12 13 -10% -5% 0% 5% 8,50% 10% 15% 20% 14
Conclusion
Difficult to put into practice (especially for the reinsurer)
Financial approach is better BUT only makes sense if the hedging strategy is applied !
Conclusion : actuarial approach has to be used
Second question : How to fix the price ?
Base : single pure premium + Loading for « risk »
Answer : cash flow model
Cash flow model
Insurance contract = investment by the shareholders
Investment decision: cash flow model t 1 2
5 … P C t
R t
K t r t (R) r t (K) Taxes
Price P fixed according to the NPV criterion
Open questions
How much capital to allocate?
How to release it through time?
What is the cost of capital?
Risk measures and capital allocation
Coherent risk measures (Artzner et al.) Conditional tail expectation (CTE):
CTE
(
X
) [
X X
V
(
X
) ]
where
V α
(
X
) inf
V
:
X
V
Capital to be allocated at time t:
k t
CTE
(
DFC t
)
p t
One-period vs multiperiodic risk measures
Problem: intermediate actions during development of risk
Addressed recently in by Artzner et al.
Capital at time t :
– to cover all the discounted future losses?
– to pay the losses for x years and set up provisions at the end of the period?
We applied the one-period risk measure to the distribution of future losses at each time t
Simulation of provisions and capital
Two possibilities:
– Independent trajectories
P
(
t
)
K
(
t
)
E
DFC E
DFC
(
t
(
t
) )
DFC
(
t
)
V
(
DFC
(
t
)) – Tree simulations
P
(
t
)
K
(
t
)
E E E
E E
DFC DFC
DFC
(
t
) (
t
(
t
) )
S t DFC DFC
,
N
(
t t
)
E
DFC
(
t
) , (
t
)
V
(
V
(
DFC DFC
(
t
)) (
t
.
)),
S t
,
N t
Independent trajectories
P(t) K(t) t = 1
Tree simulations
P 1 (t) K 1 (t) P N (t) K N (t) t = 1
P
(
t
)
i N
1
P i
(
t
)
N K
(
t
)
i N
1
K i
(
t
)
N
Comparison with non-life reinsurance business
Number of claims : Poisson(
l
) Severity of claim : Pareto(A,
) Let
Fix
l
vary so that we obtain the same pure premium
Compare premium with both models For usual values of
(1,52,5)
, results not significantly different
Cost of capital
CAPM :
COC
r
b (
r m
r
)
What is the
b
for this contract?
– Same b for the whole company?
– Specific b for this line of business?
How to estimate it?
Conclusions
Actuarial approach Pricing and capital allocation using simulations
Other questions:
– Asset model: GBM, regime switching models, (G)ARCH, …?
– Risk measure? Threshold ?
– Capital allocation and release through time?