Sensitivity Analysis and Worst-Case Analysis Making use of netting effects when designing insurance contracts

Transcription

1 Sensitivity Analysis and Worst-Case Analysis Making use of netting effects when designing insurance contracts Marcus C. Christiansen September 6, 29 IAA LIFE Colloquium 29 in Munich, Germany

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

3 Risks in life insurance present value of future benefits and premiums depends on... (a) random pattern of states (X t ) a i d X t t

4 Risks in life insurance present value of future benefits and premiums depends on... (a) random pattern of states (X t ) a i d biometrical risk: unsystematic biometrical risk (diversifiable) X t t

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

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

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

8 Getting rid of the non-diversifiable risks? (i) benefits guaranteed Policyholder Insurer Third Party (ii) benefits according to surplus e.g. with profits policies such as DC pension plans Policyholder Insurer Third Party

9 Getting rid of the non-diversifiable risks? (i) benefits guaranteed Policyholder Insurer Third Party (ii) benefits according to surplus e.g. with profits policies such as DC pension plans Policyholder Insurer Third Party contrary to customer demand (in Germany)!

10 Getting rid of the non-diversifiable risks? (i) benefits guaranteed Policyholder Insurer Third Party (ii) benefits according to surplus e.g. with profits policies such as DC pension plans Policyholder Insurer Third Party contrary to customer demand (in Germany)! (iii) benefits guaranteed by third party e.g. reinsurance, securitisation,... Policyholder Insurer Third Party

11 Getting rid of the non-diversifiable risks? (i) benefits guaranteed Policyholder Insurer Third Party (ii) benefits according to surplus e.g. with profits policies such as DC pension plans Policyholder Insurer Third Party contrary to customer demand (in Germany)! (iii) benefits guaranteed by third party e.g. reinsurance, securitisation,... Policyholder Insurer Third Party enough capacity for risk transfer?

12 Getting rid of the non-diversifiable risks? (iv) reality (in Germany): compositions of (i) (iii) Policyholder Insurer Third Party

13 Getting rid of the non-diversifiable risks? (iv) reality (in Germany): compositions of (i) (iii) Policyholder Insurer Third Party still a significant load of systematic risks for the insurer!

14 Getting rid of the non-diversifiable risks? (iv) reality (in Germany): compositions of (i) (iii) Policyholder Insurer Third Party still a significant load of systematic risks for the insurer! Question A: effect of policy design on risk load of the insurer? Policyholder Insurer ( Third Party )

15 Getting rid of the non-diversifiable risks? (iv) reality (in Germany): compositions of (i) (iii) Policyholder Insurer Third Party still a significant load of systematic risks for the insurer! Question A: effect of policy design on risk load of the insurer? Policyholder Insurer ( Third Party ) (v) netting effects e.g. survival benefits vs. death benefits Policyholder } ± Insurer ( Third Party )

16 Getting rid of the non-diversifiable risks? (iv) reality (in Germany): compositions of (i) (iii) Policyholder Insurer Third Party still a significant load of systematic risks for the insurer! Question A: effect of policy design on risk load of the insurer? Policyholder Insurer ( Third Party ) (v) netting effects e.g. survival benefits vs. death benefits Policyholder } ± Insurer ( Third Party ) Question B: which combinations give strong netting effects?

17 TOOL 1: Sensitivity Analysis

18 Effect of changes of the valuation basis on premiums and reserves qualitative results: Lidstone (195), Norberg (1985), Hoem (1988), Ramlau-Hansen (1988), Linnemann (1993),...

19 Effect of changes of the valuation basis on premiums and reserves qualitative results: Lidstone (195), Norberg (1985), Hoem (1988), Ramlau-Hansen (1988), Linnemann (1993),... quantitative results: Bowers et al. (1987, 1997, constant interest), Dienst (1995, yearly invalidity), Kalashnikov and Norberg (23, general single parameter), C. and Helwich (28, yearly interest & yearly mortality),

20 Effect of changes of the valuation basis on premiums and reserves qualitative results: Lidstone (195), Norberg (1985), Hoem (1988), Ramlau-Hansen (1988), Linnemann (1993),... quantitative results: Bowers et al. (1987, 1997, constant interest), Dienst (1995, yearly invalidity), Kalashnikov and Norberg (23, general single parameter), C. and Helwich (28, yearly interest & yearly mortality), f (x + x) f (x) + xf, x interpret xf = f (x) as sensitivity of f at x

21 Effect of changes of the valuation basis on premiums and reserves qualitative results: Lidstone (195), Norberg (1985), Hoem (1988), Ramlau-Hansen (1988), Linnemann (1993),... quantitative results: Bowers et al. (1987, 1997, constant interest), Dienst (1995, yearly invalidity), Kalashnikov and Norberg (23, general single parameter), C. and Helwich (28, yearly interest & yearly mortality), f (x + x) f (x) + xf, x interpret xf = f (x) as sensitivity of f at x general approach for all valuation basis parameters and with continuous time?

22 General sensitivity analysis (C., 28) LET V t,a (ϕ, µ ad, µ ai,...) := prospective reserve at time t in state a with a valuation basis (ϕ, µ ad, µ ai,...) 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.

23 General sensitivity analysis (C., 28) LET V t,a (ϕ, µ ad, µ ai,...) := prospective reserve at time t in state a with a valuation basis (ϕ, µ ad, µ ai,...) 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. THEN there exists a generalized first-order Taylor expansion V t,a (ϕ + ϕ, µ ad + µ ad,...) V t,a (ϕ, µ ad,...) + ϕ V t,a, ϕ + µad V t,a, µ ad +...

24 General sensitivity analysis (C., 28) LET V t,a (ϕ, µ ad, µ ai,...) := prospective reserve at time t in state a with a valuation basis (ϕ, µ ad, µ ai,...) 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. THEN there exists a generalized first-order Taylor expansion V t,a (ϕ + ϕ, µ ad + µ ad,...) V t,a (ϕ, µ ad,...) + ϕ V t,a, ϕ + µad V t,a, µ ad +... with generalized gradients / sensitivities ϕv t,a (s) = 1 s>t v(t, s) P(X s = k X t = a) V s,k k µad V t,a (s) = 1 s>t v(t, s) P(X s = a X t = a) S@R ad (s)

25 Example endowment insurance with disability waiver 1,8 sensitivities of the prospective reserve at time t = in state a = active,6,4 (a,d),2 (a,i) (i,a) ,2 (i,d) ϕ time

26 Example: mortality sensitivities policyholder at time t = is a 3 year old male 1 temporary life ins pure endowment ins. mortality sensitivities of the prospective reserve at time t = in state a = active K1 K2 (i,d) Φ annuity ins. time

27 Example combinations( of insurance contracts: ) sensitivity of i α i policy i = ) i α i (sensitivity of policy i

28 Example combinations( of insurance contracts: ) sensitivity of i α i policy i = ) i α i (sensitivity of policy i pure endowment ins. + temporary life ins. = combined policy 1, 1, 1,,8,8,8,6,6,6,4,4,4 2 +,2 K, ,2 K, = +,2 K,

29 Example combinations( of insurance contracts: ) sensitivity of i α i policy i = ) i α i (sensitivity of policy i pure endowment ins. + temporary life ins. = combined policy 1, 1, 1,,8,8,8,6,6,6,4,4,4 2 +,2 K, ,2 K, = +,2 K, , 1, 1,,8,8,8,6,6,6,4,4,4 2 +,2 K, ,2 K, = +,2 K,

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

31 Conclusion for Tool 1: Sensitivity Analysis Pros yields graphic images of risk structures sensitivity is linear with respect to combinations of policies Cons

32 Conclusion for Tool 1: Sensitivity Analysis Pros yields graphic images of risk structures sensitivity is linear with respect to combinations of policies very helpful tool to find netting effects Cons

33 Conclusion for Tool 1: Sensitivity Analysis Pros yields graphic images of risk structures sensitivity is linear with respect to combinations of policies very helpful tool to find netting effects Cons disregards the variability of the arguments

34 Conclusion for Tool 1: Sensitivity Analysis Pros yields graphic images of risk structures sensitivity is linear with respect to combinations of policies very helpful tool to find netting effects Cons disregards the variability of the arguments no real-valued risk measure

35 Conclusion for Tool 1: Sensitivity Analysis Pros yields graphic images of risk structures sensitivity is linear with respect to combinations of policies very helpful tool to find netting effects Cons disregards the variability of the arguments no real-valued risk measure how to compare policies (with respect to their systematic mortality risk)?

36 Comparing policies temporary life ins.,9,8,7,6 yearly premium,5,4 lump sum premium,3,2, clear

37 Comparing policies temporary life ins.,9,8,7,6 yearly premium,5,4 lump sum premium,3,2, clear K,5 K,1 K,15 K,2 pure endowment ins yearly premium lump sum premium clear

38 Comparing policies temporary life ins.,9,8,7,6 yearly premium,5,4 lump sum premium,3,2, clear disability vs. pure endowm. ins K,5 K,1 K,15 K,2 pure endowment ins yearly premium lump sum premium clear K,1 pure endowm. ins. K,2 K,3 K,4 disability ins. unclear

39 Comparing policies temporary life ins.,9,8,7,6 yearly premium,5,4 lump sum premium,3,2, K,1 K,2 K,3 K,4 clear disability vs. pure endowm. ins pure endowm. ins. disability ins. unclear pure endowment ins K,5 K,1 yearly premium K,15 K,2 lump sum premium clear pure endowm. & temp. life ins.,8,6 1+1,4 2+1, K,2 unclear

40 TOOL 2: Worst-Case Analysis

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

42 Comparing / Measuring policies IDEA (risk measure for systematic mortality risk): use the SCR according to the standard formula in Solvency II systematic mortality risk = Life long & Life mort reflects the practical consequences for the insurer

43 Comparing / Measuring policies IDEA (risk measure for systematic mortality risk): use the SCR according to the standard formula in Solvency II systematic mortality risk = Life long & Life mort reflects the practical consequences for the insurer approximation of the risk measure V@R

44 Comparing / Measuring policies IDEA (risk measure for systematic mortality risk): use the SCR according to the standard formula in Solvency II systematic mortality risk = Life long & Life mort reflects the practical consequences for the insurer approximation of the risk measure V@R cost-of-capital method: proportional to some (hypothetical) market price

45 Comparing / Measuring policies IDEA (risk measure for systematic mortality risk): use the SCR according to the standard formula in Solvency II systematic mortality risk = Life long & Life mort reflects the practical consequences for the insurer approximation of the risk measure V@R cost-of-capital method: proportional to some (hypothetical) market price BUT risk approximation by the standard formula is too rough here!

46 Calculation of Life mort and Life long NAV = changes in the net value of assets and liabilities due to...,2 +1% mortality shock mortality rate,15 25% longevity shock ± mixed shock,1,5,

47 Calculation of Life mort and Life long NAV = changes in the net value of assets and liabilities due to...,2 +1% mortality shock mortality rate,15 25% longevity shock ± mixed shock,1,5, Do we really study the crucial scenarios?

48 MARKT/255/8, 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. [...]

49 MARKT/255/8, 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). [...]

50 MARKT/255/8, 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). [...]

51 MARKT/255/8, 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. [...] 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

52 Finding the crucial scenario(s),2 +1% shock UPPER BOUND,15 worst/best scenario? 25% shock LOWER BOUND,1,5,

53 Finding the crucial scenario(s),2 +1% shock UPPER BOUND,15 worst/best scenario? 25% shock LOWER BOUND,1,5, Which scenarios lead to the highest NAV?

54 Optimization problem If we focus on the liabilities only then NAV = V,a.

55 Optimization problem If we focus on the liabilities only then NAV = V,a. Problem Find scenario µ ad with V,a (µ ad ) = max { V,a (µ ad ) LowBound µ ad UppBound }

56 Optimization problem If we focus on the liabilities only then NAV = V,a. Problem Find scenario µ ad with V,a (µ ad ) = max { V,a (µ ad ) LowBound µ ad UppBound } first-order Taylor approximation V,a (µ ad + µ ad ) = V,a (µ ad ) + µad V,a, µ ad + Remainder where µad V,a (s) = 1 s> v(, s) P(X s = a X = a) S@R ad (s)

57 Optimization problem If we focus on the liabilities only then NAV = V,a. Problem Find scenario µ ad with V,a (µ ad ) = max { V,a (µ ad ) LowBound µ ad UppBound } first-order Taylor approximation V,a (µ ad + µ ad ) = V,a (µ ad ) + µad V,a, µ ad + Remainder where µad V,a (s) = 1 s> v(, s) P(X s = a X = a) S@R ad (s) Conclusion 1 For( all s ) sign µad V,a (s) ( ) = sign S@R ad (s)

58 Optimization problem If we focus on the liabilities only then NAV = V,a. Problem Find scenario µ ad with V,a (µ ad ) = max { V,a (µ ad ) LowBound µ ad UppBound } first-order Taylor approximation V,a (µ ad + µ ad ) = V,a (µ ad ) + µad V,a, µ ad + Remainder where µad V,a (s) = 1 s> v(, s) P(X s = a X = a) S@R ad (s) Conclusion 1 For( all s ) sign µad V,a (s) ( ) = sign S@R ad (s) Conclusion 2 Because of the maximality property of µ ad : µad V,a, µ ad for all LowBound µ ad + µ ad UppBound

59 Solution of the optimization problem Hence & µ ad = { UppBoundad : sign S@R ad > LowBound ad : sign S@R ad < Thiele s rekursion/differential/integral equation THEOREM (C., 28) The Worst-Case rekursion/differential/integral equation has a unique solution V t,a that is maximal. Conclusion solution V t,a = S@R ad = µ ad

60 Solution of the optimization problem Hence & µ ad = { UppBoundad : sign S@R ad > LowBound ad : sign S@R ad < Thiele s rekursion/differential/integral equation = Worst-Case rekursion/differential/integral equation

61 Solution of the optimization problem Hence & µ ad = { UppBoundad : sign S@R ad > LowBound ad : sign S@R ad < Thiele s rekursion/differential/integral equation = Worst-Case rekursion/differential/integral equation THEOREM (C., 28) The Worst-Case rekursion/differential/integral equation has a unique solution V t,a that is maximal.

62 Solution of the optimization problem Hence & µ ad = { UppBoundad : sign S@R ad > LowBound ad : sign S@R ad < Thiele s rekursion/differential/integral equation = Worst-Case rekursion/differential/integral equation THEOREM (C., 28) The Worst-Case rekursion/differential/integral equation has a unique solution V t,a that is maximal. Conclusion solution V t,a = S@R ad = µ ad

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

64 Example 1 Task Design a combination of 1 pure endowment ins. + β temporary life ins. Question Which ratio death benefit : survival benefit = β : 1 leads to the lowest systematic mortality risk?

65 Example 1 Task Design a combination of 1 pure endowment ins. + β temporary life ins. Question Which ratio death benefit : survival benefit = β : 1 leads to the lowest systematic mortality risk?,8,6 sensitivity: temp. life ins.,4,2 K, sensitivity: pure endowm. ins.

66 Example 1 Task Design a combination of 1 pure endowment ins. + β temporary life ins. Question Which ratio death benefit : survival benefit = β : 1 leads to the lowest systematic mortality risk?,18,16,8,6,4,2 K,2 sensitivity: temp. life ins sensitivity: pure endowm. ins.,14,12,1,8,6,4,2 systematic mortality risk 1 2 β : 1 3

67 Example 1 Task Design a combination of 1 pure endowment ins. + β temporary life ins. Question Which ratio death benefit : survival benefit = β : 1 leads to the lowest systematic mortality risk?,18,16,8,6,4,2 K,2 sensitivity: temp. life ins sensitivity: pure endowm. ins.,14,12,1,8,6,4,2 systematic mortality risk 1 2 β : 1 3

68 Example 2 Task Design a combination of 1 disability ins. + β temporary life ins. Question Which ratio death benefit : yearly disability benefit = β : 1 leads to the lowest systematic mortality risk?

69 Example 2 Task Design a combination of 1 disability ins. + β temporary life ins. Question Which ratio death benefit : yearly disability benefit = β : 1 leads to the lowest systematic mortality risk?,3,2 synchronised changes of (a, d) and (i, d), β : 1 4 β : 1

70 Example 2 Task Design a combination of 1 disability ins. + β temporary life ins. Question Which ratio death benefit : yearly disability benefit = β : 1 leads to the lowest systematic mortality risk?,3,2 synchronised changes of (a, d) and (i, d), β : 1 4 β : 1

71 Example 2 Task Design a combination of 1 disability ins. + β temporary life ins. Question Which ratio death benefit : yearly disability benefit = β : 1 leads to the lowest systematic mortality risk?,3,3,2 synchronised changes of (a, d) and (i, d),2 independent changes of (a, d) and (i, d),1, β : β : 1 4

72 Example 3 Task Design a combination of 1 annuity ins. + β temporary/whole life ins. Question Which ratio death benefit : yearly annuity benefit = β : 1 leads to the lowest systematic mortality risk?

73 Example 3 Task Design a combination of 1 annuity ins. + β temporary/whole life ins. Question Which ratio death benefit : yearly annuity benefit = β : 1 leads to the lowest systematic mortality risk?,4,3,2 temporary life ins., β : 1 4 β : 1

74 Example 3 Task Design a combination of 1 annuity ins. + β temporary/whole life ins. Question Which ratio death benefit : yearly annuity benefit = β : 1 leads to the lowest systematic mortality risk?,4,3,2 temporary life ins., β : 1 4 β : 1

75 Example 3 Task Design a combination of 1 annuity ins. + β temporary/whole life ins. Question Which ratio death benefit : yearly annuity benefit = β : 1 leads to the lowest systematic mortality risk?,4,4,3,2 temporary life ins.,3,2,1,1 whole life ins β : β : 1 4

76 Example 3 Task Design a combination of 1 annuity ins. + β temporary/whole life ins. Question Which ratio death benefit : yearly annuity benefit = β : 1 leads to the lowest systematic mortality risk?,4,4,3,2 temporary life ins.,3,2,1,1 whole life ins β : β : 1 4

77 literature Christiansen, M.C. (28): A sensitivity analysis concept for life insurance with respect to a valuation basis of infinite dimension. Insurance: Mathematics and Economics 42, Christiansen, M.C. (28): A sensitivity analysis of typical life insurance contracts with respect to the technical basis. Insurance: Mathematics and Economics 42, Christiansen, M.C., Helwich, M. (28): Some further ideas concerning the interaction between insurance and investment risks. Blätter der DGVFM 29 (2), Christiansen, M.C. (28): Biometrical worst-case and best-case scenarios in life insurance. Preprint. Available at christiansen/

78 contact information christiansen/