Stochastic Analysis of Life Insurance Surplus Natalia Lysenko Department of Statistics & Actuarial Science Simon Fraser University Actuarial Research Conference, 2006 Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 1 / 24
Outline 1 Introduction 2 Model Assumptions 3 Methodology 4 Results 5 Future Work Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 2 / 24
Motivations How risky is the portfolio of life policies? How likely is the insurance company to become insolvent in any given year? Are premiums and level of initial surplus adequate to ensure high probability of solvency? Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 3 / 24
Framework Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 4 / 24
Risks Facing Insurance Industry Mortality Investment Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 5 / 24
Risks Facing Insurance Industry Mortality Investment Expenses Persistency Other Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 5 / 24
Decrements due to Mortality: Model K x : curtate-future-lifetime of a person aged x number of complete years remaining until death Notation: P(K x = k) = k q x, k = 0, 1, 2,... P(K x > n) = n p x Nonparametric life table Canada 1991, Age Nearest Birthday, Male, Aggregate, Population Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 6 / 24
Stochastic Rates of Return: Notation δ(k): force of interest in period (k 1, k], k = 1, 2,..., n δ k : realization of δ(k) Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 7 / 24
Stochastic Rates of Return: Notation δ(k): force of interest in period (k 1, k], k = 1, 2,..., n δ k : realization of δ(k) I(s, r): force of interest accumulation function I(s, r) = { r j=s+1 δ(j) if s < r, 0 if s = r. Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 7 / 24
Stochastic Rates of Return: Notation I(s, r): force of interest accumulation function I(s, r) = { r j=s+1 δ(j) if s < r, 0 if s = r. Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 8 / 24
Stochastic Rates of Return: Model AR(1) model δ(k) δ = φ [δ(k 1) δ] + ε(k), where ε(k) N(0, σ 2 ) δ: long-term mean of the process φ < 1 (stationarity) conditional on starting value δ(0) = δ 0 Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 9 / 24
More Assumptions... Assumptions Future lifetimes are i.i.d. Lifetimes are independent of rates of return Identical contracts (i.e., homogeneous portfolio) Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 10 / 24
Notation: Life Insurance Policy n: term of contract x: age at issue b: death benefit payable at the end of the year of death c: pure endowment benefit payable upon survival to time n π: premium payable at the beginning of each year Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 11 / 24
Notation: Homogeneous Portfolio { 1 if policyholder i aged x survives for j years, L i,j (x) = 0 otherwise L j (x)= m i=1 L i,j(x) BIN(m, j p x ) number of policies in force at time j Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 12 / 24
Notation: Homogeneous Portfolio { 1 if policyholder i aged x survives for j years, L i,j (x) = 0 otherwise L j (x)= m i=1 L i,j(x) BIN(m, j p x ) number of policies in force at time j { 1 if policyholder i aged x dies in year j, D i,j (x) = 0 otherwise D j (x)= m i=1 D i,j(x) BIN(m, j 1 q x ) number of deaths in year j, j 1 Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 12 / 24
Retrospective Gain Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 13 / 24
Retrospective Gain Definition where RG r = r RCj r e I(j,r) j=0 RC r j : net cash flow at time j prior to time r, 0 j r RC r j = π L j (x) 1 {j<r} b D j (x) 1 {j>0} Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 14 / 24
Prospective Loss Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 15 / 24
Prospective Loss Definition where n r PL r = PCj r e I(r,r+j) PC r j : net cash flow j time units after time r, 0 j n r j=0 PCj r = b D r+j (x) 1 {j>0} + c L n (x) 1 {j=n r} π L r+j (x) 1 {j<n r} Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 16 / 24
Surplus Stochastic Surplus S stoch r = RG r PL r Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 17 / 24
Surplus Stochastic Surplus S stoch r = RG r PL r Accounting Surplus S acct r = RG r r V where r V r V (L r, δ(r)) is the actuarial reserve at time r different ways to calculate r V r V = E[PL r L r, δ(r)] Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 17 / 24
Towards Distribution Function... Recall: S acct r = RG r r V ( L r, δ(r) ) Observation Given values of L r and δ(r), r V ( L r, δ(r) ) is constant cdf of S acct r can be obtained from cdf RG r via P[S acct r ξ L r = m r, δ(r) = δ r ] = = P[RG r ξ + r V (m r, δ r ) L r = m r, δ(r) = δ r ] Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 18 / 24
Distribution Function: Recursive Approach Let G t = t j=0 RCr j e I(j,t), 0 t r Note: G r = RG r It can be shown: G t = G t 1 e δ(t) + RC r t Consider a function g t (λ, m t, δ t ) given by g t (λ, m t, δ t ) = P[G t λ L t = m t, δ(t) = δ t ] P[L t = m t ] f δ(t) (δ t ) Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 19 / 24
Recursive Formula for g t (λ, m t, δ t ) Result For 1 < t r, g t (λ, m t, δ t ) = = Z X m λ ηt «P[L t = m t L t 1 = m t 1 ] g t 1 e m t 1 =m δ, m t 1, δ t 1 t t f δ(t) (δ t δ(t 1) = δ t 1 ) dδ t 1 where η t is the realization of RC r t for given values of m t 1 and m t, j π mt b (m η t = t 1 m t ), 1 t r 1, b (m t 1 m t ), t = r with the starting value given by j P[L1 (x) = m g 1 (λ, m 1, δ 1 ) = 1 ] f δ(1) (δ 1 ) if G 1 λ 0 otherwise. Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 20 / 24
Distribution Function of Accounting Surplus It is easy to show: Result P[Sr acct ξ] = m r =0 m g r (ξ + r V (m r, δ r ), m r, δ r ) dδ r Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 21 / 24
Figure: CDF of Accounting Surplus (1000 5-yr Temporary Policies) r = 2 r = 3 0.0 0.2 0.4 0.6 0.8 1.0 0% 20% 0.0 0.2 0.4 0.6 0.8 1.0 0% 20% 15 10 5 0 5 10 15 10 5 0 5 10 surplus per policy surplus per policy r = 4 r = 5 0.0 0.2 0.4 0.6 0.8 1.0 0% 20% 0.0 0.2 0.4 0.6 0.8 1.0 0% 20% 15 10 5 0 5 10 15 10 5 0 5 10 surplus per policy surplus per policy Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 22 / 24
Figure: CDF of Accounting Surplus (Limiting Portfolio) r = 2 r = 3 0.0 0.2 0.4 0.6 0.8 1.0 0% 2% 3% I.S. 0.0 0.2 0.4 0.6 0.8 1.0 0% 2% 3% I.S. 0.4 0.2 0.0 0.2 0.4 0.6 0.4 0.2 0.0 0.2 0.4 0.6 surplus per policy surplus per policy r = 4 r = 5 0.0 0.2 0.4 0.6 0.8 1.0 0% 2% 3% I.S. 0.0 0.2 0.4 0.6 0.8 1.0 0% 2% 3% I.S. 0.4 0.2 0.0 0.2 0.4 0.6 0.4 0.2 0.0 0.2 0.4 0.6 surplus per policy surplus per policy Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 23 / 24
Future Work Under the same assumptions... Probability of solvency over all years Distribution of stochastic surplus Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 24 / 24
Future Work Under the same assumptions... Probability of solvency over all years Distribution of stochastic surplus Extend model to... general portfolio include expenses, lapses Natalia Lysenko (SFU) Analysis of Life Insurance Surplus ARC 2006 24 / 24