ON COMPETING NON-LIFE INSURERS

Transcription

1 ON COMPETING NON-LIFE INSURERS JOINT WORK WITH HANSJOERG ALBRECHER (LAUSANNE) AND CHRISTOPHE DUTANG (STRASBOURG) Stéphane Loisel ISFA, Université Lyon 1 2 octobre 2012

2 INTRODUCTION Lapse rates Price elasticity Competition : different markets, different features Customer behavior : a complex topic Classical explanatory variables : age of the policyholder / of the policy, cross-selling, sales channel, declared claims last year,... Focus on some particular competition mechanism Impact of Solvency constraints Disclaimer : I m not a game theory specialist. For questions about game theory, please ask Prof. Jean Lemaire! 3/21 Stéphane Loisel 2 octobre 2012

3 FRAMEWORK Usually, competition involves 1 market cycles models (time series) 2 lapse rates and new business (regression techniques) 3 position w.r.t. estimated market price (optimal control) 5/21 Stéphane Loisel 2 octobre 2012

4 FRAMEWORK Usually, competition involves 1 market cycles models (time series) 2 lapse rates and new business (regression techniques) 3 position w.r.t. estimated market price (optimal control) Game theory, prisoner s dilemma : Two suspects if one testifies against his partner who remains silent, he is free and the other one gets a 10-year sentence. If both of them testify against each other, they get a 5-year sentence. If both of them remain silent, they get a 6-month sentence. P1 P2 silent (S) betrays (B) silent (S) (-1/2, -1/2) (-10, 0) betrays (B) (0, -10) (-5, -5) Each player tries to minimize its potential sentence. (B,B) is a Nash equilibrium, i.e. a couple of strategies such that no player can decrease its sentence unilaterally. 5/21 Stéphane Loisel 2 octobre 2012

5 SIMPLE LAPSE MODEL Consider I insurers on a market with n policyholders. Let x [x, x] I correspond to the price vector. The choice C i of policyholder number i is given by a multinomial r.v. M I (1, p j (x)) where p j (x) = (p j 1 (x),..., p j I (x)) and j is the previous insurer of customer i. Probability P(C i = k; j) = p j k (x) is given by 1 1+ if j = k, e f j (x j,x l ) l j p j k (x) = e f j (x j,x k ) (1) if j k, e f j (x j,x l ) 1+ l j where function f j (x j, x l ) represents price sensitivity fj (x j, x l ) = µ j + α j x j x l et f j (x j, x l ) = µ j + α j (x j x l ). Portfolio size of insurer j is N j (x) = B jj (x) + I k=1,k j B kj (x). where B kj B(n k, p k j (x)) and n k is the initial portfolio size. 6/21 Stéphane Loisel 2 octobre 2012

6 SIMPLE CLAIM MODEL Assume that for policyholder i, the aggregate claim amount is M i Y i = Z i,l, where M i is the number of claims, (Z i,l ) l are the i.i.d. amounts and M i (Z i,l ). l=1 Assume that the (Y i ) i are independent. The total loss for insurer j is N j (x) S j (x) = Y i. 2 particular cases are considered for illustrations : Poisson Lognormal (PLN) and Negative Binomial Lognormal (NBLN). i=1 7/21 Stéphane Loisel 2 octobre 2012

7 SIMPLE OBJECTIVE FUNCTION Choice of objective function x O j (x) is guided by economic criteria : for given x j, x j O j (x) must be decreasing in x j and depend on a threshold π j, mathematical constraints : x j O j (x) must be strictly concave. One chooses O j (x) = n ( ( )) j xj 1 β j n m j (x) 1 (x j π j ) (2) where threshold premium π j and average market premium m j (x) are given by π j = ω j a j,0 + (1 ω j )m 0 and m j (x) = 1 x k. I 1 a j,0, m 0, ω j respectively represent the average actuarial premium, the average market premium and the credibility factor. If one did not take competition into account, one would have O j (x) = O j (x j ). k j 8/21 Stéphane Loisel 2 octobre 2012

8 SIMPLE SOLVENCY CONSTRAINT One needs x j gj 1 (x j ) to be closed-form and concave. We choose the following solvency constraint : K j + n j (x j π j )(1 e j ) k 99.5%σ(Y ) n j, where e j is the commission rate and k 99.5% is the coefficient such that ( E(Y )n j + k 99.5%σ(Y ) nj ) n j VaR 99.5% Y i. In practice we take k 99.5% = 3. Constraint g j is defined by i=1 g 1 j (x j ) = K j + n j (x j π j )(1 e j ) k 99.5%σ(Y ) n j 1 g 2 j (x j ) = x j x (3) g 3 j (x j ) = x x j 9/21 Stéphane Loisel 2 octobre 2012

9 SIMPLE GAME SEQUENCE During one period, the game is as follows. 1 Insurers determine their offered price after a Nash equilibrium, by solving for all j {1,..., I} x j arg max O j (x j, x j ). x j,g j (x j ) 0 2 Policyholders randomly choose their new insurer according to p k j (x ) : on obtient N j (x ). 3 During the year, the aggregate claim amounts S j (x ) are simulated according to portfolio sizes. 4 The underwriting result is determined as UW j (x ) = N j (x )x j (1 e j ) S j (x ). 10/21 Stéphane Loisel 2 octobre 2012

10 PROPERTIES OF THE SIMPLE MODEL PROPOSITION (ADL(2012A)) The game considered with objective and constraints described by (2) and (3) admits a unique Nash equilibrium. ELEMENTS OF PROOF. O j continuous + x j O j (x) quasiconcave existence, x j O j (x) strictly concave uniqueness. PROPOSITION (ADL(2012A)) Let x be the equilibrium premium of the I-insurer game. For each player j, if xj ]x, x[, the equilibrium premium xj depends on parameters in the following manner : it increases in threshold premium π j, in Solvency constraint k 99.5%, in claims volatility σ(y ), in commission fee e j ; and decreases in sensitivity premium β j and in capital K j. ELEMENTS OF PROOF. KKT Conditions and implicit function theorem. 11/21 Stéphane Loisel 2 octobre 2012

11 SIMPLE MODEL : NUMERICAL RESULTS Consider 3 players with customers, i.e. n = 10000, I = 3, and (n 1, n 2, n 3 ) = (4500, 3200, 2300) as well as K i such that the coverage ratio is 133%. P1 P2 P3 market E(X) σ(x) PLN NBLN TABLE: Actuarial premium ā j,0 and market premium m 0 Equilibrium premia are given in the table below. x1 x2 x3 ˆN 1 ˆN 2 ˆN 3 PLN-ratio PLN-diff NBLN-ratio NBLN-diff TABLE: Equilibrium premium 12/21 Stéphane Loisel 2 octobre 2012

12 LESS SIMPLE MODEL MAIN CHANGES Better taking N j into account : its mean N j (x) is given by N j (x) = n j p j j (x) + l j n l p l j (x). We choose the following functions : Objective function Solvency constraint Õ j (x) = n jp j j (x) (x j π j ), (4) n g j 1 (x) = K j + n j (x j π j )(1 e j ) 1. (5) k 99.5%σ(Y ) Nj (x) 13/21 Stéphane Loisel 2 octobre 2012

13 LESS SIMPLE MODEL : PROPERTIES PROPOSITION (ADL(2012A)) The game with objective functions and constraints defined by (4) and (5) admits a generalized Nash equilibrium if for each player j = 1,..., I, g 1 j (x) > 0. 14/21 Stéphane Loisel 2 octobre 2012

14 DYNAMIC MODEL Consider premium volume GWP j,t, portfolio size n j,t, and capital K j,t for player j at time t. At the beginning of each period, one determines m t 1 = 1 d d u=1 N j=1 GWP j,t u xj,t u 1 1 et ā j,t = GWP.,t u 1 e j,t d d u=1 s j,t u n j,t u. So, one has π j,t = ω j ā j,t + (1 ω j ) m t 1. Objective functions and constraints become O j,t (x) = n j,t n ( ( )) xj 1 β j,t m j (x) 1 (x j π j,t ), g 1 j,t(x j ) = K j,t + n j,t (x j π j,t )(1 e j,t ) k 995 σ(y ) n j,t 1. 15/21 Stéphane Loisel 2 octobre 2012

15 REPEATED SIMPLE MODEL ; GAME SEQUENCE For period t, the game is described as follows : 1 Insurers maximize their objective function sup x j,t O j,t (x j,t, x j,t ) tel que g j,t (x j,t ) 0. 2 Once equilibrium premium xt is determined, policyholders lapse or do not lapse. We get new values nj,t of N j,t (x ). 3 The aggregate claim amount S j,t is simulated according to PLN or NBLN. One obtains s j,t. 4 The underwriting result of insurer j is then determined : 5 Eventually, the capital is given by UW j,t = n j,t x j,t (1 e j ) s j,t. K j,t = K j,t 1 + UW j,t. Player j is said to be ruined or insolvent if K j,t < 0 or n j,t = 0 16/21 Stéphane Loisel 2 octobre 2012

16 REPEATED SIMPLE MODEL : PROPERTIES PROPOSITION (ADL(2012A)) For the 1-period game, if k j, x j,t x k,t and x j,t (1 e j,t ) x k,t (1 e k,t ), then underwriting results by policy uw j,t are stochastically ordered : uw j,t icx uw k,t. ELEMENTS OF PROOF. Majorization order techniques and convex order properties. PROPOSITION (ADL(2012A)) For the game repeated infinitely many times, the probability that at least two insurers remain solvent up to time t geometrically decreases in t. ELEMENTS OF PROOF. Bounding probability P(Card(I t) > 1). 17/21 Stéphane Loisel 2 octobre 2012

17 REPEATED SIMPLE MODEL : ONE SAMPLE PATH FIGURE: NBLN Claim model and sensitivity f j 18/21 Stéphane Loisel 2 octobre 2012

18 EMPIRICAL PROBABILITY TO BE LEADER Simulation of sample paths over T = 20 periods. Ruin before Ruin before Leader Leader Leader t = 10 t = 20 at t = 5 at t = 10 at t = 20 Insurer 1 6.1e e Insurer Insurer TABLE: Ruin probability and probability to be leader Min. 1st Qu. Median Average 3rd Qu. Max. Insurer Insurer Insurer TABLE: Underwriting result by policy at t = 20 19/21 Stéphane Loisel 2 octobre 2012

19 CYCLES Calibrate an AR(2) process X t = a 1 X t 1 + a 2 X t 2 + E t. If ( a 2 < 0) and a a 2 < 0, then (X t) is p-periodic with p = 2π arccos a1 2. a 2 prime marché Histogramme des périodes prime marché Q5% Q50% Q95% traj. 1 traj. 2 effectif temps périodes FIGURE: Market premium 20/21 Stéphane Loisel 2 octobre 2012

20 SOME PERSPECTIVES Work in progress. Leader - follower models. Direct sales channel / UK market data. 21/21 Stéphane Loisel 2 octobre 2012