Biomogorov Kolmogorov S Forward Integro-diff

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1 D E P A R T M E N T O F M A T H E M A T I C A L S C I E N C E S UNIVERSITY OF COPENHAGEN Cash Flows and Policyholder Behaviour in the semi-markov life insurance setup Kristian Buchardt Ph.D.-student Jointly with: T. Møller & K. B. Schmidt PFA Pension University of Copenhagen Department of Mathematical Sciences May 30, 2014 Slide 1/14

2 Main Results 1 (Rediscovery of) Kolmogorov s forward integro-diff. eq. 2 Free policy modelling with extra duration eliminated by a modified version of Kolmogorov s forward integro-diff. eq. 3 Significant impact from the modelling of policyholder behaviour: yearly payment Basic Surrender Sur+free pol Slide 2/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

3 Semi-Markov life insurance setup Stochastic process Z(t) J = {0,1,...,J}. Duration in current state U(t). Assume Z(t) is semi-markov (Z(t), U(t)) is Markov. Transition rates allowed to be duration dependent, µ ij (t,u). The transition probabilities are duration dependent p ij (t,s,u,v) = P(Z(s) = j,u(s) v Z(t) = i,u(t) = u) 0, active 1, disabled 2, dead Slide 3/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

4 Life insurance payments Time t, duration u: While in state j: b j (t,u), Transition from i to j: b ij (t,u). Conditional on (Z(t) = i,u(t) = u): The (expected) cash flow at time s, da i,u (t,s) ( ) = j J p ij (t,s,u, dv) b j (s,v) + µ jk (s,v)b jk (s,v) k:k j ds. The prospective reserve, V i,u (t) = e s t r(τ)dτ da i,u (t,s). t Slide 4/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

5 Life insurance payments Time t, duration u: While in state j: b j (t,u), Transition from i to j: b ij (t,u). Conditional on (Z(t) = i,u(t) = u): The (expected) cash flow at time s, da i,u (t,s) ( ) = j J p ij (t,s,u, dv) b j (s,v) + µ jk (s,v)b jk (s,v) k:k j ds. The prospective reserve, V i,u (t) = e s t r(τ)dτ da i,u (t,s). t Primary question: How to calculate the distr. p ij (t,s,u, dv)? Slide 4/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

6 Kolmogorov s forward integro-diff. eq. Theorem For D(t) = d + t 0, d ds p ij(t,s,u,d(s)) = D(s) 0 + k J k j p ij (t,t,u,d) = 1 {i=j} 1 {u d}, p ij (t,s,u,0) = 0, s > t. Similar results in Hoem (1972) and Helwich (2008). p ij (t,s,u, dz)µ j. (s,z) p ik (t,s,u, dz)µ kj (s,z), Slide 5/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

7 Kolmogorov s forward integro-diff. eq. Theorem For D(t) = d + t 0, d ds p ij(t,s,u,d(s)) = D(s) 0 + k J k j p ij (t,t,u,d) = 1 {i=j} 1 {u d}, p ij (t,s,u,0) = 0, s > t. Similar results in Hoem (1972) and Helwich (2008). p ij (t,s,u, dz)µ j. (s,z) p ik (t,s,u, dz)µ kj (s,z), One solve yields p ij (t,s,u,d(s)) for all s (t,t ) and j J. solve only once! Slide 5/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

8 The (Danish) setup: With profit products The Policy 1 policyholder agrees to pay a premium 2 life/pension insurance company guarantees certain benefits The policy is valued with 2 valuation bases Technical basis: Safe-side Determines relation between premium and guarantee. Conservative (low) interest rate r Safe-side mortality rate, disability rate, etc. Market basis: Best estimate Determines balance-sheet value of the liabilities (guaranteed payments). Market inferred forward interest rate f r. Best estimate mortality rate, disability rate, etc. Slide 6/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

9 Policyholder behaviour: 2 options Surrender cancel all future payments, and receives policy value, according to the technical basis Free policy (eqv. paid-up policy) cancel all future premiums, and the benefits are reduced, according to the technical basis Payments are calculated on the technical basis: Introduces risk on the market basis. Market based valuation should include policyholder behaviour. Slide 7/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

10 Surrender modelling 3, surrender 0, active 1, disabled µ sur 2, dead J Surrender transition: payment V 0,0 (t) Cash flow da s i,u(t,s) ( ) = p ij (t,s,u, dv) b j (s,v) + µ jk (s,v)b jk (s,v) ds j J k:k j + p i0 (t,s,u, dv)µ sur (s,v)v0,0(s)ds. Slide 8/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

11 State space: Surrender & free policy 3, surrender µ sur 0, active 1, disabled µ free 2, dead J 7, surrender free policy µ sur 4, active free policy 6, dead free policy 5, disabled free policy J f Free policy at time t: Premiums cancelled Future payments reduced by factor ρ(t) New duration W(t): Time since free policy conversion. Slide 9/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

12 Free policy cash flow Define ρ-modified transition probabilities p ρ ij (t,s,u,z) = s u+τ t t 0 p i0 (t,τ,u, dv)µ free (τ,v)ρ(τ)p ActFree,j (τ,s,0,z)dτ Slide 10/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

13 Free policy cash flow Define ρ-modified transition probabilities p ρ ij (t,s,u,z) = s u+τ t t 0 Proposition The cash flow is, p i0 (t,τ,u, dv)µ free (τ,v)ρ(τ)p ActFree,j (τ,s,0,z)dτ da fs i,u(t,s) = da s i,u(t,s) + j J f b + j + p ρ ij (t,s,u, dz) ( (s,z) + µ jk (s,z)b jk (s,z) )ds + k J f k j p ρ i,actfree (t,s,u, dz)µ sur(s,z)v,+ 0,0 (s)ds Slide 10/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

14 p ρ forward integro-differential equation Theorem p ρ ij (t,s,u,v) satisfy, with D(s) = d + s t 0, d R, d ds pρ ij (t,s,u,d(s)) = 1 {j=actfree} p i0 (t,s,u, dz)µ free (s,z)ρ(s) D(s) p ρ ij (t,s,u, dz)µ j.(s,z) 0 + p ρ ik (t,s,u, dz)µ kj(s,z) k J f k j p ρ ij (t,t,u,d) = 0, pρ ij (t,s,u,0) = 0, s > t. Slide 11/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

15 p ρ forward integro-differential equation Theorem p ρ ij (t,s,u,v) satisfy, with D(s) = d + s t 0, d R, d ds pρ ij (t,s,u,d(s)) = 1 {j=actfree} p i0 (t,s,u, dz)µ free (s,z)ρ(s) D(s) p ρ ij (t,s,u, dz)µ j.(s,z) 0 + p ρ ik (t,s,u, dz)µ kj(s,z) k J f k j p ρ ij (t,t,u,d) = 0, pρ ij (t,s,u,0) = 0, s > t. Compare to Kolmogorov forward integro-diff. eq. d D(s) ds p ij(t,s,u,d(s)) = p ij (t,s,u, dz)µ j. (s,z) 0 + p ik (t,s,u, dz)µ kj (s,z) k J k j Slide 11/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

16 Total cash flow yearly payment Basic Surrender Sur+free pol age Slide 12/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

17 Conclusion Reviewed Kolmogorov s forward integro-diff. eq. for efficient life insurance cash flows, in the semi-markov setup. Cash flows efficiently calculated with policyholder behaviour with a modified Kolmogorov forward integro-diff.-eq. Policyholder behaviour has a huge effect on cash flows. Essential for interest rate sensitivity analysis. With policyholder modelling, significantly less interest rate hedging is needed. Slide 13/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014

18 References K. Buchardt, K. B. Schmidt and T. Møller (2014), Cash flows and policyholder behaviour in the semi-markov life insurance setup. to appear in Scandinavian Actuarial Journal. K. Buchardt and T. Møller (2013), Life insurance cash flows with policyholder behaviour. Preprint, Department of Mathematical Sciences, University of Copenhagen and PFA Pension. Thank you! Slide 14/14 Kristian Buchardt Cash Flows and Policyholder Behaviour May 30, 2014