pricing and capital allocation for unit-linked life insurance contracts with minimum death guarantee...
TRANSCRIPT
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
Capital sous risque dans une garantie plancher
0,8
1
1,2
0 1 2 3 4 5 6 7 8 9 10
Années
Val
eur
de l'
UC
Sum at risk
Fi n
an
cia
l in
de
x S
t
Time t
)0,max(),max( ttt SKSSK
Insurer’s liability for a death at time t:
• 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:
Simple expressions for the single pure premium in both approaches
tttt dWSdtSdS
Single pure premiums
T
kkxxk
Actkr
T
kkxxk
ActrkAct
qpkdeS
qpkdKeSPP
11
)(0
12
)),0((
)),0((
T
kkxxk
Fi
T
kkxxk
FirkFi
qpkdS
qpkdKeSPP
110
12
)),0((
)),0((
Actuarial pricing :
Financial pricing :
tTTtdTtd
tT
tTrKSTtd
ActAct
tAct
),(),(
))(2/()/log(),(
21
2
2
tTTtdTtd
tT
tTKSTtd
FiFi
tFi
),(),(
))(2/()/log(),(
21
2
2
with
Monte Carlo simulations
Goal : distribution of the future costs 3 processes to simulate :
– Financial index – Death process– Hedging strategy (financial approach only)
Probability distribution functions
0
0,2
0,4
0,6
0,8
1
0 10 20 30 40 50 60
Discounted future costs
Actuarial
Financial
Sensitivity analysis
Distribution of DFCAct - variation of -
0,00
0,20
0,40
0,60
0,80
1,00
0 10 20 30 40 50 60 70 80 DFC Act
20% 15% 10% 8,5% 5% 0% -5% -10% -15% -20% No Stock
Sensitivity analysis
Distribution of DFC Fi - variation of -
0
0,2
0,4
0,6
0,8
1
6 7 8 9 10 11 12 13 14 DFC Fi
FI
-10% -5% 0% 5% 8,50% 10% 15% 20%
Conclusion
Financial approach is better BUT only makes sense if the hedging
strategy is applied ! Difficult to put into practice (especially
for the reinsurer) 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 modelt 1 2 5 …
P
Ct Rt Ktrt(R)
rt(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):
where
Capital to be allocated at time t:
])([)( XVXXXCTE
VXVXVα :inf)(
ttt pDFCCTEk )(
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
.))(()()(
,)),(()()()(
,)(,)()(
tDFCVtDFCtDFCE
NStDFCVtDFCtDFCEEtK
tDFCENStDFCEEtP
tt
tt
– Tree simulations
))(()()()(
)()(
tDFCVtDFCtDFCEtK
tDFCEtP
Two possibilities:– Independent trajectories
Independent trajectories
P(t)
K(t)
t = 1
Tree simulations
P1(t)
K1(t)
PN(t)
KN(t)
t = 1
N
tPtP
N
ii
1
)()(
N
tKtK
N
ii
1
)()(
Comparison with non-life reinsurance business
Number of claims : Poisson() Severity of claim : Pareto(A,) Let vary Fix so that we obtain the same pure
premium Compare premium with both models For usual values of , results
not significantly different
Cost of capital
CAPM :
What is the for this contract?– Same for the whole company?– Specific for this line of business?
How to estimate it?
)( rrrCOC m
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?