Creating Customer Value in Participating

Transcription

1 in Participating Life Insurance Nadine Gatzert, Ines Holzmüller, and Hato Schmeiser

2 Page 2 1. Introduction Participating life insurance contracts contain numerous guarantees and options: - E.g., minimum interest rate guarantee, guaranteed annual surplus participation, terminal bonus payment, Appropriate pricing is crucial for insurer's stability Two perspectives for valuation: policyholder and insurer Derived prices on contract level (including embedded options must meet customer demand

3 Page 3 Insurer perspective: - Present value calculation (risk-neutral or fair valuation: based on insurer's ability to duplicate cash flows Policyholder perspective: - May not be able to duplicate claims via capital market instruments - Contract valuation is generally based on individual preferences - Willingness to pay (customer value will be different from fair premium calculated by insurance company

4 Page 4 Literature Aim of this paper Combine perspective of insurer and policyholder ld Identify contract parameters guaranteed interest rate, annual and terminal surplus participation leading to a 1 fixed fair contract value (from the insurer's perspec- tive and 2 will maximize i customer value Distinguish between deterministic and stochastic basis wealth of policyholder (different degree of diversification

5 Page 5 2. Life Insurance Contract and Valuation Standard d Black-Scholes l setting - Assets A(t follow geometric Brownian motion - Policyholders pay up-front premium - Equityholders make initialiti contribution tib ti P ( = β A Eq ( = ( 1 β A - Total initial payments invested in assets A = P + Eq - P(t policyholder account ( ( ( ( ( α γ ( ( Pt = Pt 1 1+ g + max At At 1 gpt 1, Guaranteed interest rate Annual surplus participation rate Relation of book to market values: build up hidden reserves

6 Page 6 Terminal bonus participation: ( ( = δ β ( ( BT max AT PT, Default put option (insurer may get insolvent: ( ( = ( ( D T max P T A T, Final payoffs - Policyholders' payoff: ( = ( + δ ( ( LT PT BT DT - Equityholders' payoff: ( δ ( ( ( ( ( ( Eq T = AT LT = maxat PT, BT

7 Page 7 Insurer perspective - Risk-neutral valuation (under pricing measure Q ( ( ( δ ( ( ( ( E e LT E e PT BT E e DT Π = Q rt = Q rt + Q rt DPO =Π Π - Fair contracts: calibrate ( g, α, δ to satisfy * Π = P - Provides lower end of premium agreement range - Necessary premium to conduct adequate risk manage- ment measurers (e.g., equity capital - Infinite number of contract specifications with same fair value, but different customer value

8 Page 8 Customer perspective - In general, there a very different ways to derive the customer value of a life insurance contract - E.g., mean-variance preferences (under real-world measure P Determine upper end of premium agreement area, i.e., customer's willingness to pay P Φ using the following preference function: a Φ= E ZT σ 2 2 ( ( Z T Z is policyholder's ld wealth, ais the degree of risk aversion (a >

9 Page 9 Derivation of customer's willingness to pay P Φ - Compare preference functions for case without insurance (WI a 2 Φ = E( ZT σ ( Z T, Z = Z 2 NI NI NI NI and with insurance (NI WI WI a 2 WI WI Φ = E( ZT + L( T σ ( ZT + L( T, = 2 Z Z P Φ - Maximum willingness to pay is price at which customer becomes indifferent between the two cases: WI NI Φ =Φ

10 Page 1 Deterministic wealth (policyholder cannot diversify - Invest in risk-free asset or purchase life insurance ( = Z t Z e rt Φ rt a 2 P = e E( L( T σ ( L( T 2 Stochastic wealth (policyholder can diversify: - Invest in stochastic assets or purchase life insurance ( 2 ( = ( μ σ + σ ( ( Z t Z t 1 exp Z Z /2 Z WZ t WZ t 1, dwadwz = ρdt NI 2 NI NI ( ( ( ( ( ( 2 ( ( ( ( ( ( ( LT ( Φ E ZT Cov Z T,ZT Cov Z T,L T σ LT + Cov Z T,L T E LT EZT Cov Z T,ZT Cov Z T,L P = a a a σ ( Z ( ( T σ ZT σ ZT

11 Page Customer value differs, even though contracts have the same fair value (based on risk-neutral valuation Idea: keep contracts fair from insurer perspective: Φ max g, αδ, * ( Q rt ( αδ ( P max such that P =Π g,, = E e L T. Customer value under the real world measure P Fair contract under the risk neutral measure Q For fixed nominal premium, choose fair combination ( g, α, δ that leads to highest customer value, while pro- viding at least a risk-adequate returns for shareholders

12 Page Numerical Examples Input parameters - Risk-free rate r = 4.46% - Asset drift μ Α =7%, - Volatility of assets σ Α = 6% - Fair premium P = 1 - Contribution of equityholders Eq =3 - Relation of book to market values γ = 5% - TimetomaturityT=1

13 Page 13 Table 1: Fair contracts and corresponding customer value for deterministic and stochastic wealth. Fair contract parameters (insurer perspective Customer value P Φ (policyholder perspective Terminal Annual Part A: Part B: Guaranteed * Shortfall participation p participation p Π deterministic stochastic ρ =.9 σ bbili Ζ = 8% Z = 2 a =.685 interest rate (g probability rate (δ rate (α (a =.685 (a =.15 Panel A: Contract with regulatory restrictions: 2.25% 68% 9% 1.2% Panel B: Simple contracts with one parameter only: 4.56% % % 1.69% % 99.89% % 1.2% % % 13% 1.9% Panel C: Maximizing customer value: 2.% % 113% 1.3% 3% % 15% 1.2% % 85% 1.1% % % 77% 1.26% % 58% 1.22% % 47% 1.21% % % 67% 1.39% % 62% 1.39% % 45% 1.37% % % 62% 1.42% % 37% 1.38% % 9% 1.38% % % 56% 1.45% % 39% 1.42% % 26% 1.42%

14 Page Summary Examine how insurers can generate customer value for participating life insurance contracts Insurer: preference-free f approach of risk-neutral valuation Customer value defined d as policyholder ld willingness to pay Calculated l based on mean-variance preferences First calibrate contract parameters to the same fair value, then derivee corresponding customer value of these contracts

15 Page 15 Find that customer value varies substantially, even if contracts have same value Customer segmentation is a viable tool for increasing insurer profit and achieving shareholder return above risk- adequate rate Design contracts to specifically increase customer value compared to standard contracts Preferred contracts may be simple ones (e.g., only one instead of three embedded options Further research: Analyzing the impact of different valuation techniques for policyholders (e.g., regarding the shortfall risk of a contract