Dynamic Hybrid Products in Life Insurance: Assessing the Policyholders Viewpoint

Transcription

1 Dynamic Hybrid Products in Life Insurance: Assessing the Policyholders Viewpoint Samos, May 2014 Alexander Bohnert *, Patricia Born, Nadine Gatzert * * Friedrich-Alexander-University of Erlangen-Nürnberg, Florida State University

2 Introduction: Motivation Currently: Low interest rates, volatile capital markets, regulation, cost pressure, demographic development, changing consumer needs, Innovations in life & pension of high relevance Dynamic hybrid products: Ensure guarantee by a periodical rebalancing process (CPPI) of account value between 1) conventional reserves ( risk-free investment), 2) guarantee fund (at most 20% loss per period, 3) equity fund (risky) Combine stability of traditional life insurance (conventional policy reserves) with upside potential of unit-linked policies (through investing in a guarantee and / or equity fund) 2

3 Introduction: Aim of paper But: To what extent do interaction effects arise between traditional policies and dynamic hybrids? What is the impact of these interactions on fair value, risk, and policyholders willingness-to-pay? Provide a model for a life insurer selling Traditional participating life insurance contracts (PLI) and Dynamic hybrid products (DHP) Examine the impact of dynamic hybrid products on Fair value and insurer s risk situation Policyholders willingness to pay using mean-variance preferences Study interaction effects between PLIs and DHPs 3

4 Model framework: Insurance company Consider life insurer with a product portfolio comprising PLIs: Invest entirely in the actuarial policy reserves (PR) DHPs: Contracts funds are dynamically allocated between reserves, a guarantee fund (GF) and an equity fund (EF) Simplified balance sheet at time t: Assets A long-term A short-term GF A EF A Liabilities PR PLI PR DHP GF L EF L Buffer PR Policy reserves annually compounded at least with guaranteed interest rate (and surplus generated by insurer s investment portfolio) Total account value (AV) of hybrids Dynamic (short-term, procyclical) reallocation! Policy reserves (PR DHP ) Guarantee fund (GF) Equity fund (EF) 4

5 Model framework: Development of assets and PLI Development of policy reserves with r B + = max r, α γ PR + PR + + t t P G t t PLI DHP PR PR r PLI / DHP PLI / DHP P = + ( ) ( 1+ ) t + 1 t t B t+ : buffer = company s total assets (A t ) minus policyholders accounts B = A PR PR GF EF PLI DHP L L ( t+ t ) ( t+ t ) ( t+ t ) ( t+ t ) ( t + t ) ( t+ t ) Investments all evolve according to a geometric Brownian motion i i i P di = µ I dt + σ I dw, 1,2,3, i = t i t i t t i Guarantee fund equivalent to equity fund with a hedge that ensures a maximum loss of λ percent within one period ( t) 5

6 Model framework: Dynamic hybrid products DHP guarantee to policyholders (single premium P): Distribution of account value (Kochanski and Karnarski, 2011): PR DHP t + GF t L + ( 1 λ ) DHP DHP DHP Gt + t AV + t Gt + t, if > 1 t DHP G = ( 1+ r ) 1+ λ ( 1 λ ) AV + t 0, otherwise DHP DHP DHP Gt + t AV + PR +, if > 1 t t DHP ( 1 λ ) AV + t = DHP Gt + t, otherwise 1 λ EF = AV PR GF L DHP DHP L t t t t = x P Concrete shifting mechanism depends on account value t t needed to ensure that guarantee can be met at maturity (=> product design!) G DHP T DHP G + DHP 6

7 Model framework: Dynamic hybrid products (cont.) Two types of shifting mechanisms: Less risky (constant guarantee with higher portion of riskless assets in the portfolio, i.e. policy reserves and guarantee fund): DHP DHP DHP t t T { 0,,, } G + = G = x P t t T More risky (discounted maturity guarantee with higher portion in the risky asset, i.e. fund investments): ( ) ( T t 1 t ) DHP DHP G DHP G + = G + r with GT = x P t t T DHP Different product designs in the market; strong impact on risk-return profiles 7

8 Model framework: Development of liabilities Buffer at the end of a the period t given by B = A PR PR GF EF PLI DHP L L ( t + t ) ( t + t ) ( t + t ) ( t + t ) ( t + t ) ( t + t ) At maturity T, buffer is distributed to equityholders and policyholders Equityholders receive a buffer payback ( ( ( ) T 0 ) ) BP = max min B, B 1 + b,0 T + Policyholders receive remainder as an optional terminal bonus ( ) TB = max 0, B BP T T T Terminal bonus is distributed between PLIs and DHPs (i = PLI, DHP) TB = TB PR PR + PR T t T t i i PLI DHP ( ) ( ) ( ) T T k t k t k t k = 1 k = 1 8

9 Model framework: Fair valuation and risk measurement Buffer interest rate b is calibrated to ensure a fair situation for equityholders! + T r ( f ) Q B = E BP e 0 Shortfall probability T long term short term ( ) { } SP = P T T, T = inf t : A + A < PR, t = 1,..., T s s t t t Total payout to PLI policyholders ( ) 1{ } 1{ } V = PR + TB T > T + RF T = t PLI PLI PLI PLI T T T s t s Total payout to DHP policyholders ( ) 1{ } 1{ } V = AV + TB T > T + RF T = t DHP DHP DHP DHP T T T s t s 9

10 Model framework: The policyholder s perspective Present values from the policyholders perspective (( ) T rf t rf { }) ( { } T T s t s ) PLI Q PLI PLI Q PLI PV0 = E PR + TB e 1 T > T + E RF e 1 T = t (( ) T rf t rf { }) ( { } T T s t s ) DHP Q DHP DHP Q DHP PV0 = E AV + TB e 1 T > T + E RF e 1 T = t Maximum willingness to pay (mean-variance preferences) Φ, j r f T j a 2 j WTP0 = e E ( VT ) σ ( VT ), j = PLI, DHP 2 10

11 Numerical results: Input parameters Input parameters: Single premiums Participating life insurance 100 Dynamic hybrid products 100 Guaranteed interest rate 1.75% Length of a period t one month Contract duration in years 10 Maximal loss of the guarantee fund per period

12 Numerical results: Case without dynamic hybrids Value of traditional contracts = premium of 100 (=> calibrated to get fair contracts) present value P =0 participating life insurance shortfall probability (right axis) shortfall probability Portion of dynamic hybrid products in the portfolio is zero, single premium of trad. contracts = 100 Insurer s shortfall probability guaranteed interest rate r G Varying guaranteed interest rates (for funds in policy reserves) 12

13 Numerical results: Case with dynamic hybrids Now introduce dynamic hybrids, equal portions, i.e. single premium = 100 for both contracts Value of traditional contracts Value of dynamic hybrids Policies are no longer fair Value and risk level strongly depend on guarantees (dynamic hybrid guarantee level and guaranteed interest rate for conventional policy reserves) present value P =100, G =1.0 P participating life insurance dynamic hybrid product shortfall probability (right axis) guaranteed interest rate r G shortfall probability Dynamic hybrid guarantee level: money-back guarantee 13

14 Numerical results: Impact of guarantees Now introduce dynamic hybrids, equal portions, i.e. single premium = 100 for both contracts Value of traditional contracts Value of dynamic hybrids Policies are no longer fair Value and risk level strongly depend on guarantees (dynamic hybrid guarantee level and guaranteed interest rate for conventional policy reserves) present value P =100, G =0.5 P guaranteed interest rate r G shortfall probability Dynamic hybrid guarantee level: 50% of upfront premium 14

15 Numerical results: Portfolio composition But: concrete effects depend on the portfolio composition! Value of trad. contracts => fair Shortfall probability given fair contracts: Minimum for approx. 125:75-portion in the portfolio (for considered example) present value G =1.0 P participating life insurance dynamic hybrid product shortfall probability (right axis) single premiums dynamic hybrid product P shortfall probability Dynamic hybrid: money-back guarantee Value of dynamic hybrid contracts => fair Keep total premium volume fix: P PLI = P DHP 15

16 Numerical results: Willingness to pay for PLIs and DHPs Less risk averse policyholder (full money-back guarantee) a=0.005, G t =1.0 P (less risky shifting) a=0.005, G T =1.0 P (more risky shifting) willingness to pay W T P P LI W T P P P L I = P willingness to pay guaranteed interest rate r G guaranteed interest rate r G 16

17 Numerical results: Willingness to pay for PLIs and DHPs More risk averse policyholder (full money-back guarantee) a=0.01, G t =1.0 P (less risky shifting) a=0.01, G T =1.0 P (more risky shifting) willingness to pay W T P P LI W T P P P L I = P willingness to pay guaranteed interest rate r G guaranteed interest rate r G 17

18 Numerical results: Willingness to pay for PLIs and DHPs The impact of portfolio composition on the policyholders WTP a =0.01 (more risk averse) a =0.001 (less risk averse) willingness to pay P P L I WTP P L I P W TP willingness to pay P P LI =200-P single premiums dynamic hybrid product P single premiums dynamic hybrid product P 18

19 Numerical results: Willingness to pay for PLIs and DHPs The impact of portfolio composition on the policyholders WTP P LI payoff quartiles payoff quartiles single premiums dynamic hybrid product P single premiums dynamic hybrid product P 19

20 Summary Model and assess portfolio effects for innovative dynamic hybrids Results emphasize strong interaction effects with traditional policies Consideration of the portfolio as a whole is vital when introducing new products Higher guarantee (i.e. riskiness of shifting mechanism, DHP guarantee level, guaranteed interest) does not necessarily imply an increase in consumers willingness to pay or present value DHPs: Minimum guarantee must be offered for WTP DHP > P DHP, otherwise contracts are not purchased Contracts attractiveness strongly depends on contract design PLIs: WTP remained almost unchanged; traditional contract payoffs low volatile and stable 20

21 Thank you very much for your attention!

22 Dynamic Hybrid Products in Life Insurance: Assessing the Policyholders Viewpoint Samos, May 2014 Alexander Bohnert *, Patricia Born, Nadine Gatzert * * Friedrich-Alexander-University of Erlangen-Nürnberg, Florida State University

23 Numerical results: Impact of surplus participation rate P =100, G =1.0 P present value participating life insurance dynamic hybrid product shortfall probability (right axis) shortfall probability surplus participation rate α A. Bohnert and N. Gatzert Dynamic Hybrid Products in Life Insurance 23

24 Numerical results: Average partition of the account value G =1.0 P, r G =0.01 G =1.0 P, r G =0.025 average value EF t L G F t L PR t PR t P LI average value time t time t + A. Bohnert and N. Gatzert Dynamic Hybrid Products in Life Insurance 24

25 Numerical results: Average partition of the account value G =0.75 P, r G =0.01 G =0.5 P, r G =0.01 average value EF t L G F t L PR t PR t P LI average value time t time t + A. Bohnert and N. Gatzert Dynamic Hybrid Products in Life Insurance 25

26 Numerical results: Average partition of the account value G =1.0 P, α=0.1 G =1.0 P, α=0.7 average value EF t L G F t L PR t PR t P LI average value time t time t + A. Bohnert and N. Gatzert Dynamic Hybrid Products in Life Insurance 26

27 Numerical results: Willingness to pay for PLIs and DHPs More risk averse policyholder for a partial money-back guarantee a=0.01, G t =0.5 P (less risky shifting) a=0.01, G T =0.5 P (more risky shifting) willingness to pay W T P P LI W T P P P L I = P willingness to pay guaranteed interest rate r G guaranteed interest rate r G 27