1 mon 17.00_life_christiansen_products_presentation

Sensitivity Analysis and Worst-Case Analysis
– Making use of netting effects when
designing insurance contracts
Marcus C. Christiansen
September 6, 2009
IAA LIFE Colloquium 2009 in Munich, Germany

Risks in life insurance
present value of future benefits and premiums depends on ...
(a) random pattern of states (Xt )
-
r
r
r
r
r Xt
t0
d
i
a
→ biometrical risk:
unsystematic biometrical risk (diversifiable)
systematic biometrical risk (non-diversifiable)
(b) investment return / interest rate ϕ(t)
→ financial risk (non-diversifiable)
(c) ...

-
r
r
r
r
r Xt
t0
d
i
a

-
r
r
r
r
r Xt
t0
d
i
a
• unsystematic biometrical risk (diversiﬁable)

-
r
r
r
r
r Xt
t0
d
i
a
• systematic biometrical risk (non-diversiﬁable)
(c) ...

-
r
r
r
r
r Xt
t0
d
i
a
• systematic biometrical risk (non-diversiﬁable)
1970 1980 1990 2000
0
2
4
6
8
10
(c) ...

Getting rid of the non-diversiﬁable risks?
(i) beneﬁts guaranteed
Policyholder Third PartyInsurer

(ii) beneﬁts according to surplus
e.g. with proﬁts policies such as DC pension plans

• contrary to customer demand (in Germany)!

(iii) beneﬁts guaranteed by third party
e.g. reinsurance, securitisation, ...

(iii) beneﬁts guaranteed by third party
e.g. reinsurance, securitisation, ...
• enough capacity for risk transfer?

(iv) reality (in Germany): compositions of (i) – (iii)

• still a signiﬁcant load of systematic risks for the insurer !

Question A: effect of policy design on risk load of the insurer ?
Policyholder Third Party
⇐ §§ ⇒
Insurer

⇐ §§ ⇒
Insurer
(v) netting effects
e.g. survival beneﬁts vs. death beneﬁts
±§§
§§
§§

⇐ §§ ⇒
Insurer
(v) netting effects
e.g. survival beneﬁts vs. death beneﬁts
±§§
§§
§§
Question B: which combinations give strong netting effects ?

Effect of changes of the valuation basis on premiums
and reserves
qualitative results:
Lidstone (1905), Norberg (1985), Hoem (1988), Ramlau-Hansen (1988),
Linnemann (1993), ...

and reserves
quantitative results:
Bowers et al. (1987, 1997, constant interest), Dienst (1995, yearly invalidity),
Kalashnikov and Norberg (2003, general single parameter), C. and Helwich
(2008, yearly interest & yearly mortality),

and reserves
f(x + ∆x) f(x) + x f, ∆x
interpret x f = f (x) as sensitivity of f at x

and reserves
f(x + ∆x) f(x) + x f, ∆x
interpret x f = f (x) as sensitivity of f at x
• general approach for all valuation basis parameters and
with continuous time ?

General sensitivity analysis (C., 2008)
LET
Vt,a(ϕ, µad , µai , ...) := prospective reserve at time ’t’ in state ’a’
with a valuation basis (ϕ, µad , µai , ...) whose entries are either
(a) vectors of yearly interest rates, mortality probabilities, etc.
(b) or intensity functions for interest, mortality, etc.
(c) or cumulative intensity functions for interest, mortality, etc.

LET
THEN there exists a generalized ﬁrst-order Taylor expansion
Vt,a(ϕ + ∆ϕ, µad + ∆µad , ...) Vt,a(ϕ, µad , ...)
+ ϕVt,a , ∆ϕ + µad
Vt,a , ∆µad + ...

LET
THEN there exists a generalized ﬁrst-order Taylor expansion
Vt,a(ϕ + ∆ϕ, µad + ∆µad , ...) Vt,a(ϕ, µad , ...)
+ ϕVt,a , ∆ϕ + µad
Vt,a , ∆µad + ...
with generalized gradients / sensitivities
ϕVt,a (s) = −1s>t · v(t, s) ·
k
P(Xs = k|Xt = a) · Vs−,k
µad Vt,a (s) = 1s>t · v(t, s) · P(Xs− = a |Xt = a) · S@Rad (s)

Example
endowment insurance with disability waiver
30
1
252015
0,6
0,2
0,4
5
-0,2
0 35
0,8
0
10
(a,d)
(i,d)
(i,a)
ϕ
(a,i)
time
sensitivities of
the prospective reserve
at time t = 0 in state a =’active’

Example: mortality sensitivities
policyholder at time t = 0 is a 30 year old male
10 20 30 40 50 60 70
K2
K1
0
1
pure endowment ins.
(i,d)
annuity ins.
Φ
temporary life ins.
time
mortality sensitivities of
the prospective reserve
at time t = 0 in state a =’active’

Example
combinations of insurance contracts:
sensitivity of i αi · policyi = i αi · sensitivity of policyi

Example
pure endowment ins. + temporary life ins. = combined policy
2
+
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
1,0
+
+
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
1,0
=
+
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
1,0

Example
pure endowment ins. + temporary life ins. = combined policy
2
+
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
1,0
+
+
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
1,0
=
+
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
1,0
2×
+
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
1,0
+
+
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
1,0
=
+
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
1,0

Conclusion for Tool 1: Sensitivity Analysis
Pros
• yields graphic images of risk structures
Cons

Pros
• sensitivity is linear with respect to combinations of policies
Cons

Pros
very helpful tool to ﬁnd netting effects
Cons

Pros
Cons
• disregards the variability of the arguments

Pros
Cons
• no real-valued risk measure

Pros
Cons
• no real-valued risk measure
how to compare policies (with respect to their systematic
mortality risk) ?

Comparing policies
temporary life ins.
0 10 20 30
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
clear
yearly premium
lump sum premium

Comparing policies
temporary life ins. pure endowment ins.
0 10 20 30
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
clear
10 20 30
K0,20
K0,15
K0,10
K0,05
0
clear
yearly premium
lump sum premium
yearly premium
lump sum premium

Comparing policies
0 10 20 30
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
clear
10 20 30
K0,20
K0,15
K0,10
K0,05
0
clear
disability vs. pure endowm. ins.
10 20 30
K0,4
K0,3
K0,2
K0,1
0
unclear
yearly premium
lump sum premium
yearly premium
lump sum premium
pure endowm. ins.
disability ins.

Comparing policies
0 10 20 30
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
clear
10 20 30
K0,20
K0,15
K0,10
K0,05
0
clear
disability vs. pure endowm. ins. pure endowm. & temp. life ins.
10 20 30
K0,4
K0,3
K0,2
K0,1
0
unclear
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
unclear
yearly premium
lump sum premium
yearly premium
lump sum premium
pure endowm. ins.
disability ins.
’1+1’
’2+1’

Comparing / Measuring policies
IDEA (risk measure for systematic mortality risk):
use the SCR according to the standard formula in Solvency II
systematic mortality risk = Lifelong & Lifemort

• reﬂects the practical consequences for the insurer

• approximation of the risk measure V@R

• cost-of-capital method: proportional to some (hypothetical)
market price

• cost-of-capital method: proportional to some (hypothetical)
market price
BUT risk approximation by the standard formula is too rough
here !

Calculation of Lifemort and Lifelong
∆NAV = changes in the net value of assets and liabilities due to ...
40 45 50 55 60 65
0,000
0,005
0,010
0,015
0,020
+10% mortality shock
−25% longevity shock
mortality rate
± mixed shock

Calculation of Lifemort and Lifelong
∆NAV = changes in the net value of assets and liabilities due to ...
40 45 50 55 60 65
0,000
0,005
0,010
0,015
0,020
+10% mortality shock
−25% longevity shock
mortality rate
± mixed shock
• Do we really study the crucial scenarios?

MARKT/2505/08, TS.XI.B.3 & TS.XI.C.3
For those contracts that provide benefits both in case of death
and survival, one of the following two options should be chosen
[...]:
1 Contracts [...] should not be unbundled. [...] the mortality
scenario should be applied fully allowing for the netting
effect provided by the ’natural’ hedge between the death
benefits component and the survival benefits component.
[...]

[...]:
[...]
2 All contracts are unbundled into 2 separate components:
one contingent on the death and other contingent on the
survival of the insured person(s). [...]

[...]:
[...] → not the crucial scenarios
2 All contracts are unbundled into 2 separate components:
one contingent on the death and other contingent on the
survival of the insured person(s). [...] → no netting effect

Finding the crucial scenario(s)
40 45 50 55 60 65
0,000
0,005
0,010
0,015
0,020
+10% shock UPPER BOUND
−25% shock LOWER BOUND
worst/best scenario ?

Finding the crucial scenario(s)
40 45 50 55 60 65
0,000
0,005
0,010
0,015
0,020
+10% shock UPPER BOUND
−25% shock LOWER BOUND
worst/best scenario ?
• Which scenarios lead to the highest ∆NAV ?

Optimization problem
If we focus on the liabilities only then ∆NAV = ∆V0,a .

Problem Find scenario µad with
V0,a(µad ) = max V0,a(µad ) LowBound ≤ µad ≤ UppBound

ﬁrst-order Taylor approximation
V0,a(µad + ∆µad ) = V0,a(µad ) + µad
V0,a , ∆µad + Remainder
where µad V0,a (s) = 1s>0 · v(0, s) · P(Xs− = a |X0 = a) · S@Rad (s)

Conclusion 1 For all s
sign µad
V0,a(s) = sign S@Rad (s)

Conclusion 1 For all s
sign µad
V0,a(s) = sign S@Rad (s)
Conclusion 2 Because of the maximality property of µad :
µad
V0,a , ∆µad ≤ 0 for all LowBound ≤ µad + ∆µad ≤ UppBound

Solution of the optimization problem
Hence
µad =
UppBoundad : sign S@Rad > 0
LowBoundad : sign S@Rad < 0
& Thiele’s rekursion/differential/integral equation
THEOREM (C., 2008)
The Worst-Case rekursion/differential/integral equation has a
unique solution Vt,a that is maximal.
Conclusion
solution Vt,a =⇒ S@Rad =⇒ µad

Hence
µad =
= Worst-Case rekursion/differential/integral equation

Hence
µad =
THEOREM (C., 2008)

Hence
µad =
THEOREM (C., 2008)
Conclusion
solution Vt,a =⇒ S@Rad =⇒ µad

Example 1
Task Design a combination of
1×pure endowment ins. + β× temporary life ins.

Example 1
Question Which ratio
death beneﬁt : survival beneﬁt = β : 1
leads to the lowest systematic mortality risk ?

Example 1
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
sensitivity: temp. life ins.
sensitivity: pure endowm. ins.

Example 1
10 20 30
K0,2
0
0,2
0,4
0,6
0,8
0 1 2 3
0
0,002
0,004
0,006
0,008
0,010
0,012
0,014
0,016
0,018
sensitivity: temp. life ins.
sensitivity: pure endowm. ins.
systematic mortality risk
β : 1

Example 2
1×disability ins. + β× temporary life ins.
death beneﬁt : yearly disability beneﬁt = β : 1

Example 2
0 1 2 3 4
0
0,01
0,02
0,03
synchronised changes of (a, d) and (i, d)
β : 1 β : 1

Example 2
0 1 2 3 4
0
0,01
0,02
0,03
0 1 2 3 4
0
0,01
0,02
0,03
synchronised changes of (a, d) and (i, d)
independent changes of (a, d) and (i, d)
β : 1 β : 1

Example 3
1×annuity ins. + β× temporary/whole life ins.
death beneﬁt : yearly annuity beneﬁt = β : 1

Example 3
0 10 20 30 40
0
0,1
0,2
0,3
0,4
temporary life ins.
β : 1 β : 1

Example 3
0 10 20 30 40
0
0,1
0,2
0,3
0,4
0 10 20 30 40
0
0,1
0,2
0,3
0,4
temporary life ins.
whole life ins.
β : 1 β : 1

literature
• Christiansen, M.C. (2008): A sensitivity analysis concept
for life insurance with respect to a valuation basis of inﬁnite
dimension. Insurance: Mathematics and Economics 42,
680-690.
• Christiansen, M.C. (2008): A sensitivity analysis of typical
life insurance contracts with respect to the technical basis.
Insurance: Mathematics and Economics 42, 787-796.
• Christiansen, M.C., Helwich, M. (2008): Some further
ideas concerning the interaction between insurance and
investment risks. Blätter der DGVFM 29 (2), 253-266.
• Christiansen, M.C. (2008): Biometrical worst-case and
best-case scenarios in life insurance. Preprint. Available at
http://www.uni-rostock.de/∼christiansen/

contact information
http://www.uni-rostock.de/∼christiansen/

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