Decomposition of life insurance liabilities into risk factors theory and application to annuity conversion options

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1 Decomposition of life insurance liabilities into risk factors theory and application to annuity conversion options Joint work with Daniel Bauer, Marcus C. Christiansen, Alexander Kling Katja Schilling University of Ulm March 2014

2 Page 2 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Introduction Risk decomposition methods from literature Life insurance modeling framework MRT approach Application to annuity conversion options Outlook

3 Page 3 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Introduction Risk decomposition methods from literature Life insurance modeling framework MRT approach Application to annuity conversion options Outlook

4 Page 4 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Motivation British insurance companies during the 1980s vs. 1990s: GAOs Equity Interest Mortality Question: Which are the most relevant risk drivers? Why is that important? To be able to take adequate risk management strategies such as Product modifications Hedging

5 Page 5 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Research objectives Situation: It is common to measure the total risk by advanced stochastic models. The question of how to determine the most relevant risk driver is not very well understood. Our paper (1) Theory: How to allocate the randomness of liabilities to different risk sources? (2) Application: What is the dominating risk in annuity conversion options? Note: we focus on the distribution under the real-world measure P.

6 Page 6 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Introduction Risk decomposition methods from literature Life insurance modeling framework MRT approach Application to annuity conversion options Outlook

7 Page 7 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Setting Insurance product with maturity T Insurer s liability as from time 0: L 0 Two risk drivers: X 1 := (X 1 (t)) 0 t T and X 2 := (X 2 (t)) 0 t T Liability L 0 Risk X 1 Risk X 2 Question: How to decompose L 0 with respect to X 1 and X 2?

8 Page 8 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Variance decomposition approach Step 1: L 0 = E (L 0 X 1 ) + [L 0 E (L 0 X 1 )] }{{}}{{} =:R 1 =:R 2 R1 represents the randomness of L 0 caused by X 1 R2 represents the randomness of L 0 caused by X 2 Step 2: Var (L 0 ) = Var (R 1 ) + Var (R 2 )

9 Page 9 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Variance decomposition approach Step 1: L 0 = E (L 0 X 1 ) + [L 0 E (L 0 X 1 )] }{{}}{{} =:R 1 =:R 2 R1 represents the randomness of L 0 caused by X 1 R2 represents the randomness of L 0 caused by X 2 Step 2: Var (L 0 ) = Var (R 1 ) + Var (R 2 ) Desirable property: full distribution of R 1 and R 2 Bühlmann (1995): annual loss = financial loss + technical loss Example: L 0 = X 1 (T )X 2 (T ), X 1, X 2 independent Brownian motions L0 = E(L 0 X 1) + [L 0 E(L 0 X 1)] = }{{} 0 + X 1(T )X 2(T ) }{{} =R 1 =R 2 L0 = E(L 0 X 2) + [L 0 E(L 0 X 2)] = 0 }{{} =R 2 + X 1(T )X 2(T ) }{{} =R 1

10 Page 10 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Variance decomposition approach Step 1: L 0 = E (L 0 X 1 ) + [L 0 E (L 0 X 1 )] }{{}}{{} =:R 1 =:R 2 R1 represents the randomness of L 0 caused by X 1 R2 represents the randomness of L 0 caused by X 2 Step 2: Var (L 0 ) = Var (R 1 ) + Var (R 2 ) Desirable property: full distribution of R 1 and R 2 Bühlmann (1995): annual loss = financial loss + technical loss Example: L 0 = X 1 (T )X 2 (T ), X 1, X 2 independent Brownian motions L0 = E(L 0 X 1) + [L 0 E(L 0 X 1)] = 0 + X 1(T )X 2(T ) L0 = E(L 0 X 2) + [L 0 E(L 0 X 2)] = 0 + X 1(T )X 2(T ) Desirable property: symmetric definition (uniqueness)

11 Page 11 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Further approaches Sensitivity analysis Analyzing the effect of changes in the input parameters/variables on the insurer s liability Usually based on derivatives Desirable property: comparability of the risk contributions Taylor expansion approach Function of random variables first-order Taylor expansion Desirable property: L 0 E(L 0 ) = R R n Local method: expansion point is relevant Desirable property: no problem-specific choices (uniqueness)

12 Page 12 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Introduction Risk decomposition methods from literature Life insurance modeling framework MRT approach Application to annuity conversion options Outlook

13 Page 13 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Risk driving processes (1) 1.) State process X (t): financial and demographic factors Risky assets Short rate Mortality intensity Assumption X = (X 1 (t),..., X n (t)) 0 t T is an n-dimensional diffusion process satisfying d dx i (t) = θ i (t, X (t))dt + σ ij (t, X (t))dw j (t), i = 1,..., n, j=1 with deterministic initial value X (0) = x 0 R n. W = (W1(t),..., W d (t)) 0 t T d-dimensional standard Brownian motion G = (Gt) 0 t T augmented natural filtration generated by W

14 Page 14 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Risk driving processes (2) 2.) Counting process N(t): actual occurrence of death Portfolio of m homogeneous policyholders of age x at time 0 τ i x : remaining lifetime of the i-the policyholder as from time 0 first jump time of a doubly stochastic process with intensity µ = (µ(t)) 0 t T µ is assumed to be continuous, G-adapted, and non-negative N(t) = m i=1 1 {τ i x t} : number of policyholders who died until time t I i = (I i t) 0 t T augmented natural filtration generated by (1 {τ i x >t}) 0 t T We assume: (Ω, F, F, P) with F = G m i=1 Ii

15 Page 15 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Insurer s net liability The life insurance contract implies: Cash flows C(t k ), independent of the policyholder s survival Cash flows C a (t k ), in case the policyholder survives until time t k Cash flows C ad (t), in case the policyholder dies at time t The insurer s time-t net liability is given by the sum of the (possibly discounted) future cash flows as from time t: L t = C(t k ) + T (m N(t k ))C a (t k ) + C ad (v)dn(v). k: t k t k: t k t t In what follows: we focus on the insurer s net liability L 0 at time 0.

16 Page 16 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Introduction Risk decomposition methods from literature Life insurance modeling framework MRT approach Application to annuity conversion options Outlook

17 Page 17 Decomposition of life insurance liabilities ICA 2014 Katja Schilling MRT decomposition Idea Decompose L 0 E P (L 0 ) into Itô integrals with respect to the compensated risk driving processes, i.e. n T T L 0 E P (L 0 ) = ψi W (t)dmi W (t) + ψ N (t)dm N (t) (1) i=1 0 0 }{{}}{{} =:R i =:R n+1 for some F-predictable processes ψ W i dm W i (t) = d j=1 σ ij (t, X (t))dw j (t) dm N (t) = dn(t) (m N(t ))µ(t)dt. Existence and uniqueness (t) and ψ N (t), where Assume that n = d, det σ(t, x) 0 for all (t, x) [0, T ] R n, and L 0 is F T -measurable. Then the MRT decomposition in eq. (1) exists and is unique.

18 Page 18 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Properties of the MRT decomposition MRT decomposition n L 0 E P (L 0 ) = i=1 T ψi W (t)dmi W (t) 0 }{{} =:R i T + ψ N (t)dm N (t). 0 } {{ } =:R n+1 List of desirable properties: Full distribution of each risk contribution R i Symmetric definition No problem-specific choices It holds: L 0 E(L 0 ) = R R n Comparability of the risk contributions Unsystematic mortality risk is diversifiable Appropriate dealing with correlations

19 Page 19 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Properties of the MRT decomposition MRT decomposition n L 0 E P (L 0 ) = i=1 T ψi W (t)dmi W (t) 0 }{{} =:R i T + ψ N (t)dm N (t). 0 } {{ } =:R n+1 List of desirable properties: Full distribution of each risk contribution R i Symmetric definition No problem-specific choices It holds: L 0 E(L 0 ) = R R n Comparability of the risk contributions Unsystematic mortality risk is diversifiable Appropriate dealing with correlations

20 Page 20 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Specification of the MRT decomposition Exemplarily, we decompose L 0 = (m N(T ))C a (T ). Special case Let the assumptions for existence and uniqueness hold. If E P (e T t µ(s)ds C a (T ) G t ) = f (t, X (t)), 0 t T, for some sufficiently smooth function f, then Itô s lemma yields n T L 0 E P (L 0 ) = (m N(t )) f (t, X (t)) dmi W (t) x i Existence of f : i=1 0 T 0+ f (t, X (t)) dm N (t). C a (T ) = h(x (T )) for some Borel-measurable function h : R n R Smoothness of f : Conditions from Theorem 1 in Heath and Schweizer (2000)

21 Page 21 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Introduction Risk decomposition methods from literature Life insurance modeling framework MRT approach Application to annuity conversion options Outlook

22 Page 22 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Guaranteed annuity option (GAO) Special type of annuity conversion option (cf. UK) Fund Annuity conversion option 0 T T T+1 T+2 Guaranteed annual annuity = g (conversion rate) A T (account value) Insurer s liability (= option payoff at time T): L GAO 0 = 1 {τx >T } max {ga T a T A T, 0} = 1 {τx >T }ga T max {a T 1g }, 0 τx: remaining lifetime of a policyholder aged x at time 0 at : time-t value of an immediate annuity of unit amount per year

23 Page 23 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Stochastic model Insurer s total liability (= option payoff at time T for a portfolio) m L GAO 0 = 1 {τ i x >T } ga T max {a T 1g }, 0 i=1 }{{}}{{} =m N(T ) =C a(t ) Risk Process Model Fund risk S(t) GBM Interest risk r(t) CIR model Systematic mortality risk µ(t) time-inhomogeneous CIR model Unsystematic mortality risk N(t) Binomial distribution Assumption: Processes S, r and µ are independent

24 Page 24 Decomposition of life insurance liabilities ICA 2014 Katja Schilling MRT decomposition of GAO (1) It can be shown: C a (T ) = h(s(t ), ( r(t ), µ(t )) for some measurable function h f (t, X (t)) := E P e ) T t µ(s)ds C a (T ) G t is sufficiently smooth This yields: L GAO 0 E P ( L GAO ) 0 = T (m N(t )) f (t, X (t))σ S S(t)dW S (t) x 1 0 T T 0 T 0+ (m N(t )) f x 2 (t, X (t))σ r r(t)dwr (t) (m N(t )) f x 3 (t, X (t))σ µ (t) µ(t)dw µ (t) f (t, X (t))dm N (t). } =: R } =: R 1 } =: R 2 } =: R 3 } =: R 4

25 Page 25 Decomposition of life insurance liabilities ICA 2014 Katja Schilling MRT decomposition of GAO (2) 1 Guaranteed annuity option (GAO) Empirical cdf of standardized risk R R1 (fund) R2 (interest) R3 (syst. mortality) R4 (unsyst. mortality) Unsystematic mortality plays a minor role (m = 100) Distributions of fund, interest and systematic mortality risk are comparable

26 Page 26 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Introduction Risk decomposition methods from literature Life insurance modeling framework MRT approach Application to annuity conversion options Outlook

27 Page 27 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Future research Model Extension to Lévy processes (instead of Brownian motions) Application Further annuity conversion options, e.g. modified GAOs Taking into account hedging

28 Page 28 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Contact Institute of Insurance Science University of Ulm Katja Schilling Thank you very much for your attention!

29 Page 29 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Literature Bühlmann, H. (1995). Life insurance with stochastic interest rates. In: Ottaviani, G. Financial Risk in Insurance. Springer. Christiansen, M.C. (2007). A joint analysis of financial and biometrical risks in life insurance. Doctoral thesis, University of Rostock. Christiansen, M.C., Helwich, M. (2008). Some further ideas concerning the interaction between insurance and investment risk. Blätter der DGVFM, 29: Dahl, M., Møller, T. (2006). Valuation and hedging of life insurance liabilities with systematic mortality risk. Insurance: Mathematics and Economics, 39(2): Heath, D., Schweizer, M. (2000). Martingales versus PDEs in finance: an equivalence result with examples. Journal of Applied Probability, 37(4): Kling, A., Ruß, J., Schilling, K. (2012). Risk analysis of annuity conversion options in a stochastic mortality environment. Working Paper, University of Ulm.

30 Page 30 Decomposition of life insurance liabilities ICA 2014 Katja Schilling Model parameters Description Parameter Value Age x 50 Term to maturity T 15 Single premium P 0 1 Conversion rate g 0.07 Limiting age ω 121 Number of realizations (outer) N 10,000 number of realizations (inner) M 100 Number of discretization steps per year n 100 Number of contracts m 100 GBM drift µ S 0.06 GBM volatility σ S 0.22 CIR initial value r(0) CIR speed of reversion κ ( κ) 0.2 (0.2) CIR mean level θ ( θ) (0.025) CIR volatility σ r ( σ r ) (0.075) Correlation ρ 0