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
Sum at risk
Insurer’s liability for a
death at time t:
Valeur de l'UC
1,2
max( K , St )  St  max( K  St ,0)
1
0,8
0
1
2
3
4
5
Années
Time
t
• How to price it ?
• Capital allocation ?
6
7
8
9
10
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 dWt
Simple expressions for the single
pure premium in both approaches
Single pure premiums
T
Actuarial pricing :
SPP
Act
  Ke  rk  ( d 2Act (0, k )) k p x q x  k
k 1
T
 S 0  e (   r ) k  ( d1Act (0, k )) k p x q x  k
k 1
T
Financial pricing : SPP   Ke  rk  (d 2Fi (0, k )) k p x q x  k
Fi
k 1
T
 S 0   (d1Fi (0, k )) k p x q x  k
k 1
with
log( St / K )  (    2 / 2)(T  t )
Act
d 2 (t , T ) 
 T t
Act
1
d
(t , T )  d
Act
2
(t , T )   T  t
log( St / K )  (r   2 / 2)(T  t )
d (t , T ) 
 T t
Fi
2
d1Fi (t , T )  d 2Fi (t , T )   T  t
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
1
0,8
0,6
0,4
0,2
Actuarial
Financial
0
0
10
20
30
Discounted future costs
40
50
60
Sensitivity analysis
Distribution of DFCAct - variation of  1,00
20%
15%
0,80
10%
8,5%
5%
0,60
0%
-5%
-10%
0,40
-15%
-20%
No Stock
0,20
0,00
0
10
20
30
40
Act
DFC
50
60
70
80
Sensitivity analysis
Distribution of DFC
Fi
- variation of  -
1
0,8
0,6
FI
-10%
0,4
-5%
0%
5%
8,50%
0,2
10%
15%
20%
0
6
7
8
9
10
DFC Fi
11
12
13
14
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 model
t
1
2


5
…
P
Ct
 Rt
 Kt
rt(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):
CTE ( X )  [ X X  V ( X )]
where
Vα ( X )  inf V : X  V    
 Capital to be allocated at time t:
k t  CTE ( DFCt )  pt
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
Simulation of provisions and
capital
 Two possibilities:
– Independent trajectories
Independent trajectories
P(t)
K(t)
t=1
Simulation of provisions and
capital
 Two possibilities:
– Independent trajectories
P(t )  EDFC (t )
K (t )  P(t )  EDFC (t ) DFC (t )  V ( DFC (t ))
– Tree simulations
Tree simulations
N
P1(t)
K1(t)
P(t ) 
 P (t )
i 1
i
N
N
PN(t)
KN(t)
t=1
K (t ) 
 K (t )
i
i 1
N
Simulation of provisions and
capital
 Two possibilities:
– Independent trajectories
P(t )  EDFC (t )
K (t )  P(t )  EDFC (t ) DFC (t )  V ( DFC (t ))
– Tree simulations
P(t )  E E DFC (t ) St , N t    EDFC (t ),
K (t )  P(t )  E E DFC (t ) DFC (t )  V ( DFC (t )), St , N t  
 E DFC (t ) DFC (t )  V ( DFC (t )) .
Cost of capital
 CAPM :
COC  r  b (rm  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, Jump diffusion, …?
– Risk measure? Threshold ?
– Capital allocation and release through time?