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?