The Impact of Natural Hedging on a Life Insurer s Risk Situation

Transcription

1 The Impact of Natural Hedging on a Life Insurer s Risk Situation Longevity 7 September 2011 Nadine Gatzert and Hannah Wesker Friedrich-Alexander-University of Erlangen-Nürnberg

2 2 Introduction Motivation Demographic risk can significantly impact a life insurer s solvency level Increase in life expectancy poses serious problems to life insurers selling annuities However, risk of unexpected high mortality (e.g. due to pandemics) has increased as well; problem for term life But: Hedging instruments are still scarce Natural Hedge between term life insurance (death benefit) and annuities (lifelong survival benefits) is effective alternative Use opposed reaction of term life insurance and annuities towards shocks to mortality Hedge shocks to mortality internally through portfolio composition

3 3 Introduction Aim of paper Previous literature: Cox/Lin (2007), Bayraktar/Young (2007), Gründl/Post/Schulze (2006), Wang et al. (2010), Wetzel/Zwiesler (2008) Aim of this paper: 1. Quantify impact of natural hedging on a life insurance company s insolvency risk Holistic model, take into account dynamic interaction between assets and liabilities for a two-product life insurer 2. Simultaneously immunize an insurer s solvency situation against changes in mortality and fix the absolute level of risk Use investment strategy

4 4 Model framework Modeling and forecasting mortality Extension of the Lee-Carter (1992) model by Brouhns/Denuit/ Vermunt (2002): D ~ Poisson E t x, t x, t x ( µ ( )) µ x ( t) = exp( ax + bx kt ) qx ( t) = µ x ( t) 1 exp( ) D x,t Poisson-distributed number of deaths, E x,t exposure at risk a x and b x indicating the general shape of mortality over age k t indicating the general level of mortality in the population (with negative drift) Forecasting of k t (and µ x (t)) by ARIMA process for estimated time series of k t

5 5 Model framework Modeling systematic mortality risk Analyze systematic mortality risk in two ways: 1. Shock to (decreasing) mortality time trend: e*k t Leads to an unexpected change in the level and future development of mortality Shocks e > 1: mortality rates decrease (longevity scenario) Shocks e < 1: mortality rates increase (pandemic scenario) How to compose a portfolio of term life and annuities in order to immunize the portfolio against shocks to mortality? 2. Use empirically observed changes in mortality Analyze usefulness of natural hedging under realized changes in mortality Similar results

6 6 Model framework Model of a life insurance company Simplified balance sheet: Assets A(t) Liabilities E(t) B A (t) B L (t) L(t) o A(t) : market value of assets at time t o B A (t) : book value of liabilities for annuities at time t o B L (t) : book value of liabilities for term life insurance at time t o E(t) : equity at time t Default of the insurance company, if L(t) = B L (t) + B A (t) > A(t)

7 7 Model framework Liabilities Premium and benefit calculation Premiums and benefits: use actuarial equivalence principle Term life insurance T 1 T 1 t ( ) ( ) ( t ) P p + r = DB p q + r t x t x x+ t t= 0 t= 0 Life-long immediate annuity T 1 t= 0 t x ( ) ( t+ 1 1 ) SP = a p + r Improve comparability and isolate effect of natural hedging: Calibrate input parameters such that volume of both contract types is identical at inception Fix the number of contracts sold

8 8 Model framework Liabilities Book value of liabilities Use actuarial reserve to determine book value of liabilities Value of one term life insurance contract: T t 1 ( ) ( ) ( ) ( ) ( s+ 1 1 ) s L = s x+ t s+ x+ t + s x+ t ( ) ( 1+ ) s= 0 B t DB p e q e i P p e i Value of one annuity: T t 1 ( ) ( ) ( ) ( s+ 1 = 1+ ) B t a p e i A s x+ t s= 0 Mortality rates are subject to shock e Value of liabilities L(t): ( ) = ( ) ( ) + ( ) ( ) L t n t B t n t B t A A L L

9 9 Model framework Assets Assets follow a geometric Brownian motion: P da( t) = µ A( t) dt + σ A( t) dw ( t) Development of asset base depends on cash-flows of insurance portfolio t = 0 + t = 1 - t = 1 + t = E 0 - n A (1) a + n L (1) P - n A (2) a + n A (0) SP + n L (0) P - d L (0) DB - div Number of life insurance contracts active in t = 1 - d L (1) DB - div Number of annuity contracts active in t = 1 Constant dividend to shareholders Number of of life insurance annuity contracts policyholders Constant dividend who active died in during t = 2 t = 0to shareholders Number of life insurance policyholders who died during t = 1

10 10 Model framework Risk measurement Probability of default (PD): with { } ( ) ( ) ( ) ( d ) PD = P T T T = T + 1 inf t : A t < L t, t = 1,..., T. d Mean Loss (ML): ( max (( ( ) ( )) ( 1 ) T d,0 ) 1 { }) d d d ML = E L T A T + r T T Expected Shortfall (ES) ES = ML PD Contractual Payment Obligations (CP) T 1 ( ) ( ) ( ) ( t+ 1 + n 0 1 ) A a t px e + r ( ) ( ) ( ) ( ) ( t ) CP = n DB p e q e + r L t x x+ t t= 0 Only liability side Linear in portfolio composition T 1 t= 0

11 11 Numerical results Input parameters Liabilities Age at inception of term life 30 Max. duration of term life 35 Age at inception of annuity 65 Premium for life insurance (P) 417 Single premium for annuity (SP) 10,000 Yearly annuity (a) 725 Death benefit (DB) 88,724 Total number of contracts sold 10,000 Assets Drift of assets (µ) 6% Volatility of assets (σ) 10% Risk-free interest rate (r) 3%

12 12 Numerical results Risk under different shocks to mortality Change: 2.2-7% risk immunizing Expected Shortfall (ES) risk minimizing in Mio Change: +21% Portfolio of annuities fraction of life insurance d Portfolio of term life e = 1.1 mortality rates decrease (longevity scenario) initial death rates e = 0.9 mortality rates increase (pandemic scenario)

13 13 Numerical results Varying the investment strategy optimal fraction of life insurance d* ) Find immunizing portfolio: 17.0% life insurance CP PD ML ES Optimal hedge ratio for different investment strategies µ =4%, µ =5%, µ =6%, µ =7%, µ =8%, µ =9%, µ =10%, σ =5% σ =7.5% σ =10% σ =12.5% σ =15% σ =17.5% σ =20% investment strategy in % PD ML ES 1) Fixing absolute level of risk (e.g. PD=0,38%) 6,000 5,000 4,000 3,000 2,000 1,000 in T Corresponding level of insurer s default risk for optimal hedge ratio 0.3 µ =4%, µ =5%, µ =6%, µ =7%, µ =8%, µ =9%, µ =10%, σ =5% σ =7.5% σ =10% σ =12.5% σ =15% σ =17.5% σ =20% 0 Here: for a shock to mortality of e = 1.1 (longevity scenario) investment strategy

14 14 Summary Results show: Natural hedging can considerably reduce absolute risk level of an insurer and immunize it against shocks to mortality Optimal portfolio composition depends on risk measure Holistic consideration of mortality risk with respect to insurer s overall risk level is vital (focus on liability side only underestimates risk) Investment strategy can have substantial impact on the effectiveness of natural hedging Use investment strategy to simultaneously fix a risk level and immunize the portfolio against shocks to mortality Changing the investment strategy requires adjustment of portfolio mix to immunize portfolio against changes in mortality

15 The Impact of Natural Hedging on a Life Insurer s Risk Situation Thank you very much for your attention! Longevity 7 September 2011 Nadine Gatzert and Hannah Wesker University of Erlangen-Nürnberg