Biometrical worst-case and best-case scenarios in life insurance

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1 Biometrical worst-case and best-case scenarios in life insurance Marcus C. Christiansen 8th Scientific Conference of the DGVFM April 30, 2009

2 Solvency Capital Requirement: The Standard Formula

3 Calculation of Life mort and Life long NAV = changes in the net value of assets and liabilities due to... 0, % mortality shock mortality rate 0,015 25% longevity shock ± mixed shock 0,010 0,005 0,

4 Do we really study the crucial scenarios? NAV = changes in the net value of assets and liabilities due to... 0, % mortality shock mortality rate 0,015 25% longevity shock ± mixed shock 0,010 0,005 0,

5 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). [...]

6 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). [...]

7 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). [...]

8 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. [...] eventually, 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

9 Finding the crucial scenario(s) 0, % shock UPPER BOUND 0,015 worst/best scenario? 25% shock LOWER BOUND 0,010 0,005 0, Which scenarios lead to the highest/lowest NAV?

10 Finding the crucial scenario(s) 0, % shock UPPER BOUND 0,015 worst/best scenario? 25% shock LOWER BOUND 0,010 0,005 0, Which scenarios lead to the highest/lowest NAV?

11 Optimization problem If we focus on the liabilities only then NAV = V a,0. Definition Prospective Reserve given state a at time t V a,t = ( ) MeanPresentValue FutureBenefits FuturePremiums state a at time t Problem Find scenario µ with V a,0 (µ) = max { V a,0 (µ) LowerBound µ UpperBound } Remark discrete modeling: µ = vector of yearly transition probabilities absolutely continuous modeling: µ = transition intensity general continuous modeling: µ = cumulative transition intensity

12 Optimization problem If we focus on the liabilities only then NAV = V a,0. Definition Prospective Reserve given state a at time t V a,t = ( ) MeanPresentValue FutureBenefits FuturePremiums state a at time t Problem Find scenario µ with V a,0 (µ) = max { V a,0 (µ) LowerBound µ UpperBound } Remark discrete modeling: µ = vector of yearly transition probabilities absolutely continuous modeling: µ = transition intensity general continuous modeling: µ = cumulative transition intensity

13 Optimization problem If we focus on the liabilities only then NAV = V a,0. Definition Prospective Reserve given state a at time t V a,t = ( ) MeanPresentValue FutureBenefits FuturePremiums state a at time t Problem Find scenario µ with V a,0 (µ) = max { V a,0 (µ) LowerBound µ UpperBound } Remark discrete modeling: µ = vector of yearly transition probabilities absolutely continuous modeling: µ = transition intensity general continuous modeling: µ = cumulative transition intensity

14 Optimization problem If we focus on the liabilities only then NAV = V a,0. Definition Prospective Reserve given state a at time t V a,t = ( ) MeanPresentValue FutureBenefits FuturePremiums state a at time t Problem Find scenario µ with V a,0 (µ) = max { V a,0 (µ) LowerBound µ UpperBound } Remark discrete modeling: µ = vector of yearly transition probabilities absolutely continuous modeling: µ = transition intensity general continuous modeling: µ = cumulative transition intensity

15 Characterization of maximizing scenarios first-order Taylor approximation V a,0 (µ + µ) = V a,0 (µ) + µ V a,0, µ + Remainder Because of the maximality: µ V a,0, µ 0 for all LowerBound µ + µ UpperBound Proposition (C., 2008) µ i = UpperBound i sign( µ V a,0 ) i = signr i > 0 µ i = LowerBound i sign( µ V a,0 ) i = signr i < 0 where R i is the sum-at-risk at time i

16 Characterization of maximizing scenarios first-order Taylor approximation V a,0 (µ + µ) = V a,0 (µ) + µ V a,0, µ + Remainder Because of the maximality: µ V a,0, µ 0 for all LowerBound µ + µ UpperBound Proposition (C., 2008) µ i = UpperBound i sign( µ V a,0 ) i = signr i > 0 µ i = LowerBound i sign( µ V a,0 ) i = signr i < 0 where R i is the sum-at-risk at time i

17 Characterization of maximizing scenarios first-order Taylor approximation V a,0 (µ + µ) = V a,0 (µ) + µ V a,0, µ + Remainder Because of the maximality: µ V a,0, µ 0 for all LowerBound µ + µ UpperBound Proposition (C., 2008) µ i = UpperBound i sign( µ V a,0 ) i = signr i > 0 µ i = LowerBound i sign( µ V a,0 ) i = signr i < 0 where R i is the sum-at-risk at time i

18 Solution Dynamic Programming Thiele s differential equation with initial condition V a,t = 0 d dt V a,t = rate benefits/premium (t) + V a,t rate interest (t) R t µ t Theorem (C., 2008) Given that V a,t = 0, there exists a unique solution for d dt V a,t = rate benefits/premium (t) + V a,t rate interest (t) R t (1 Rt >0 UpperBound t + 1 Rt <0 LowerBound t ). The solution is maximal.

19 Solution Dynamic Programming Thiele s differential equation with initial condition V a,t = 0 d dt V a,t = rate benefits/premium (t) + V a,t rate interest (t) R t µ t Theorem (C., 2008) Given that V a,t = 0, there exists a unique solution for d dt V a,t = rate benefits/premium (t) + V a,t rate interest (t) R t (1 Rt >0 UpperBound t + 1 Rt <0 LowerBound t ). The solution is maximal.

20 Solution Dynamic Programming Thiele s differential equation with initial condition V a,t = 0 d dt V a,t = rate benefits/premium (t) + V a,t rate interest (t) R t µ t Theorem (C., 2008) Given that V a,t = 0, there exists a unique solution for d dt V a,t = rate benefits/premium (t) + V a,t rate interest (t) R t (1 Rt >0 UpperBound t + 1 Rt <0 LowerBound t ). The solution is maximal.

21 Example Excel Sheet

22 Example Excel Sheet

23 Example Excel Sheet

24 Example Excel Sheet

25 Example Excel Sheet

26 Example Excel Sheet

27 Concluding remarks wort-case calculation is possible for any state transition: lapse, choice of cash option,... best-case calculation is completely analogous to worst-case calculation GOAL: including the risk mitigating effect of future profit sharing GOAL: dependence of assets on biometrical scenarios

28 Concluding remarks wort-case calculation is possible for any state transition: lapse, choice of cash option,... best-case calculation is completely analogous to worst-case calculation GOAL: including the risk mitigating effect of future profit sharing GOAL: dependence of assets on biometrical scenarios

29 Concluding remarks wort-case calculation is possible for any state transition: lapse, choice of cash option,... best-case calculation is completely analogous to worst-case calculation GOAL: including the risk mitigating effect of future profit sharing GOAL: dependence of assets on biometrical scenarios

30 Concluding remarks wort-case calculation is possible for any state transition: lapse, choice of cash option,... best-case calculation is completely analogous to worst-case calculation GOAL: including the risk mitigating effect of future profit sharing GOAL: dependence of assets on biometrical scenarios

31 References 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, Christiansen, M.C. (2008): Biometrical worst-case and best-case scenarios in life insurance. Preprint. Available at christiansen/ Example Excel Sheet Available at christiansen/

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