c. frantz, x. chenut and j.f. walhin secura belgian re
DESCRIPTION
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. Sum at risk. Insurer’s liability for a death at time t:. Financial index S t. Time t. The problem. How to price it ? - PowerPoint PPT PresentationTRANSCRIPT
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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
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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 ?
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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
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… and two risk managements
Financial approach : hedging on financial markets
Actuarial approach : reserving and raising capital
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Agenda
Actuarial vs financial pricing Monte Carlo simulations Cash flow model Open questions
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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
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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
tTKSTtd
ActAct
tAct
),(),(
))(2/()/log(),(
21
2
2
tTTtdTtd
tT
tTrKSTtd
FiFi
tFi
),(),(
))(2/()/log(),(
21
2
2
with
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Monte Carlo simulations
Goal : distribution of the future costs 3 processes to simulate :
– Financial index – Death process– Hedging strategy (financial approach only)
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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
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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
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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%
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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
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Second question :How to fix the price ?
Base : single pure premium + Loading for « risk »
Answer : cash flow model
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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
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Open questions
How much capital to allocate? How to release it through time? What is the cost of capital?
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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 )(
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One-period vs multiperiodic risk measures
Problem: intermediate actions during development of risk
Addressed recently 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
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Simulation of provisions and capital
Two possibilities:– Independent trajectories
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Independent trajectories
P(t)
K(t)
t = 1
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Simulation of provisions and capital
– Tree simulations
))(()()()()(
)()(
tDFCVtDFCtDFCEtPtK
tDFCEtP
Two possibilities:– Independent trajectories
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Tree simulations
P1(t)
K1(t)
PN(t)
KN(t)
t = 1
N
tPtP
N
ii
1
)()(
N
tKtK
N
ii
1
)()(
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Simulation of provisions and capital
.))(()()(
,)),(()()()()(
,)(,)()(
tDFCVtDFCtDFCE
NStDFCVtDFCtDFCEEtPtK
tDFCENStDFCEEtP
tt
tt
– Tree simulations
))(()()()()(
)()(
tDFCVtDFCtDFCEtPtK
tDFCEtP
Two possibilities:– Independent trajectories
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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
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Conclusions
Actuarial approach Pricing and capital allocation using
simulations Other questions:
– Asset model: GBM, regime switching models, (G)ARCH, Jump diffusion, …?
– Risk measure? Threshold ?– Capital allocation and release through time?