Safety margins for unsystematic biometric risk in life and health insurance

Transcription

1 Safety margins for unsystematic biometric risk in life and health insurance Marcus C. Christiansen June 1, th Conference in Actuarial Science & Finance on Samos

2 Seite 2 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Introduction What is the problem?

3 Seite 3 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 biometric risk / demographic risk calculated number actual number of deaths, disabilities, lapses, etc. risk parts unsystematic risk: deviations from expected values estimation risk: estimations deviate because of finite sample sizes systematic risk: demographic changes not anticipated today

4 Seite 4 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 dealing with unsystematic biometric risk define L = future liabilities of a portfolio of m (independent) policies then because of the law of large numbers / diversifiability E(L) = equivalence premium yet the insurer charges the premium E(L) + safety margin = conservative premium E (L) = conservative premium

5 Seite 5 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 EXAMPLE pure endowments homogeneous portfolio of m pure endowment policies survival benefit of 1 after n years policies start at age x remaining lifetimes Tx 1,..., Tx m discounting factor v with P(T i x > n) = n p x actual number of survival benefits m 1 T i x >n Binomial(m, np x ) i=1 equivalence premium E(L) = E (v m n i=1 1 T i x >n ) = v n m n p x

6 Seite 6 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 EXAMPLE pure endowments explicit safety margin factor: choose s expl such that ( P s expl }{{} safety loading v n m n p x equivalence premium conservative premium v n m 1 T i x >n i=1 liabilities ) implicit safety margin factor: choose s impl such that ( P s }{{} impl np x 1 m ) 1 }{{} m T i x >n safety loading survival probability i=1 conservative survival probability both approaches are equivalent here empirical survival probability α }{{} confidence level α }{{} confidence level

7 Seite 7 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 EXAMPLE long term care homogeneous portfolio of m long term care policies state space {a, c 1, c 2, c 3, l, d} yearly care annuities of R 1 < R 2 < R 3 Markovian process Xt i gives state of policyholder i at time t discounted liabilities L = m ω i=1 s {1,2,3} t=x v t x R s 1 X i t =c s? ( Normal)

8 Seite 8 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 EXAMPLE long term care explicit safety margin factor: choose s expl such that ( ) P s }{{} expl α safety loading E(L) }{{} equivalence premium conservative premium L }{{} liabilities implicit safety margin factors: choose s impl jk,t ( P s impl jk,t }{{} safety loading P(Xt+1 i = k X t i = j) transition probability conservative transition probability such that {X i t+1 = k, X i t = j} } {Xt i = j} {{ } empirical transition probability (1) relation is unclear (2) confidence levels have different meanings (3) solution of implicit approach not unique ) j, k, t α

9 Seite 9 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 (1) relation is unclear sensitivity analysis (Linnemann (1993), Dienst (1995), Christiansen (2008),...) (2) confidence levels have different meanings top-down approach: choose P (Xt+1 i = k X t i = j) = s impl jk,t P(Xt+1 i = k X t i = j) such that ( ) P E (L) L α (Bühlman (1985), Pannenberg (1997),...) (3) solution of implicit concept not unique twostate case: allocation principles (Pannenberg (1997),... ) multistate case:?

10 Seite 10 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Basic Modeling

11 Seite 11 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 cumulative transition intensity matrix Markovian jump processes Xt 1,..., X t m deterministic initial states Xx 1 =... = Xx m transition probability matrix ( ) p(s, t) := P(X t = k X s = j) (j,k) S 2, = a x s t ω transition intensity matrix µ(t) := d dτ τ=t p(t, τ) cumulative transition intensity matrix q(t) = (x,t] µ(s)ds Nelson-Aalen estimator for q jk (t) Q (m) 1 jk (t) := jk (u) (x,t] I (m) j (u ) dn(m) I (m) j (u )= number of policies in state j at time u N (m) jk (u)= number of jumps from j to k till time u

12 Seite 12 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 portfolio liability (m policies): L = L (m) prospective reserve (at starting age x in initial state a) E(L (1) ) =: V (q) = v(x, t) p aj (x, t) db j (t) j S (x,ω] + v(x, t) b jk (t) p aj (x, t ) dq jk (t) (j,k) J LEMMA: L (m) = m V (Q (m) ) (x,ω] economic implication of random fluctuations L (m) = m V (Q (m) ) true loss = m V (q) mean loss + m(v (Q (m) ) V (q)) unsystematic risk

13 Seite 13 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 PROBLEM explicit confidence estimate: find s such that P ( s m V (q) m V (Q (m) ) ) α implicit confidence estimate: depending on the sign of the sum-at-risk find lower bound: choose s such that P ( Q (m) q s ) α upper bound: choose s such that P ( Q (m) q + s ) α top-down confidence estimate: find s such that (a) functional ( confidence condition: ) P m V (Q (m) ) m V (q ± s) α (b) risk proportional: ds jk (t) proportional to uncertainty in dq (m) jk (t)

14 Seite 14 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 asymptotic distribution

15 Seite 15 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Theorem (cf. Andersen et al. (1991)) Let q be absolutely continuous with density function µ, and let P(X t = j) > 0 for all j S and t (x, ω). Then m ( Q (m) jk q jk )(j,k) J d (U jk ) (j,k) J, m, on [x, ω], where (U jk ) (j,k) J is a vector of stochastically independent Gaussian processes with zero mean and covariance functions Cov(U jk (s), U jk (t)) = (x,s t] 1 P(X u = j) dq jk(u). Interpretation: uncertainty in the number of transitions jk at t dq (m) jk (t) dq jk (t) d 1 m du jk (t)

16 Seite 16 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Theorem (C., 2011) Let q n = q n + ε n h n with q n q 0, h n h 0, ε n 0, and the functions q, q n, and q n of uniformly bounded variation. Then we have for all 0 s T 1 ( V (qn ) V (q n ) ) D q V (h) ε n 0, n, where D q V (h) is a supremum norm continuous linear mapping in h that equals D q V (h) = v(x, t) p aj (x, t ) R jk (t) dh(t), (j,k) J (x,ω] interpreted for h not of bounded variation by formal integration by parts.

17 Seite 17 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 limit theorem for the Nelson-Aalen estimator + Hadamard differentiability of V + functional delta method =... Theorem (C., 2011) Under the assumptions of the previous theorems we have m ( V (Q (m) ) V (q) ) d D q V (U), m on [x, ω], where U = (U jk ) (j,k) J is a vector of independent Gaussian processes. and D q V (U) = v(x, t) p aj (x, t) R jk (t) du jk (t) (j,k) J (x,ω] (defined by integration by parts).

18 Seite 18 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Interpretation L (m) E(L (m) ) = mv (Q (m) ) mv (q) d m (j,k) (x,ω] total unsystematic risk 1 v(x, u) p aj (x, u) R jk (u) du jk (u) m weighting factor independent risk sources additive decomposition to independent risk contributions

19 Seite 19 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 risk proportional allocation

20 Seite 20 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 according to the Hadamard differentiability of q V (q) mv (q + s) mv (q) m (j,k) total margin (x,ω] v(x, u) p aj (x, u) R jk (u) ds jk (u) weighting factor partial margin Definition: γ-risk-proportional given that risk measure γ is additive for independent risks, let asymptotically in m ( 1 ) γ v(x, u) p aj (x, u) R jk (u) du jk (u) m = const v(x, u) p aj (x, u) R jk (u) ds jk (u)

21 Seite 21 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Theorem (C., 2011) Safety margin s meets ( the functional confidence ) condition P m V (Q (m) ) m V (q ± s) α is γ-risk-proportional asymptotically in m if (a) γ( ) = Var( ) and ds jk (u) = u α v(x, u) R m Var(L jk(u) dq jk (u) (1) ) (b) γ( ) = E Q ( ) and ds jk (u) = const m E Q (du jk (u)) Remark Ramlau-Hansen (1988), Linnemann (1993), etc. sign ds jk (u) = sign R jk (u)

22 Seite 22 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 numerical examples

23 Seite 23 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Corollary: For γ( ) = Var( ) we have ( µ jk (t) = µ u ) α jk(t) 1 + m Var(L (1) ) v(x, t) R jk(t) In the following we calculate the safety margin factor functions ( u ) α t 1 + m Var(L (1) ) v(x, t) R jk(t)

24 Seite 24 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Example: pure endowment policies homogeneous portfolio of m = 1000 policies starting age x = 35 and contract period of n = 30 years yearly constant premium, yearly interest rate of 2.25% top-down method with confidence level of 0.95 DAV 2004 R: constant line below 1

25 Seite 25 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Example: temporary life policies homogeneous portfolio of m = 1000 policies yearly constant premium... DAV 2004 T: constant line above 1

26 Seite 26 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Example: endowment policies survival benefit two times the death benefit...

27 Seite 27 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Example: disability policies homogeneous portfolio of m = 1000 policies constant yearly disability annuities and premiums state space {active, disabled, dead} active to dead active to disabled DAV 1997: constant line at 1.076

28 Seite 28 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 disabled to active DAV 1997: constant line at 0.79 disabled to dead DAV 1997: constant line at 0.78

29 Seite 29 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Conclusion decomposition of unsystematic biometric risk: independent addends with respect to transitions (j, k) and times t risk proportional allocation of margins variance principle: extends the classical theory

30 Seite 30 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 safety margins for transition estimation risk

31 Seite 31 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 transition estimation risk GIVEN: data of m policyholders TASK: estimate q q }{{} true distribution economic implication = } Q {{ (m) } + q Q (m) estimated distribution estimation risk V (q) = V (Q (m) ) + V (Q (m) ) V (q) mean loss estimated mean loss estimation risk

32 Seite 32 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 estimation risk Problem: find s such that (a) functional ( confidence condition: ) P V (Q (m) ± s) V (q) α (b) risk proportional: ds jk (t) proportional to uncertainty in dq (m) jk (t)

33 Seite 33 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Theorem (C., 2011) Safety margin s meets the functional confidence condition is γ-risk-proportional asymptotically in m if (a) γ( ) = Var( ) and u ds jk (t) = α v(x, t) R m Var(L jk(t; q) dq jk (t) (1) ;q) (b) γ( ) = E Q ( ) and ds jk (t) = const m E Q (du jk (t)) Corollary We can substitute (a) by ds jk (t) = 1 m Var(L (1) ; Q (m) ) v(x, t) R jk(t; Q (m) ) dq (m) (t) jk