Bonus Malus and No Claim Discount Systems: Effects on the Solvency of a Non Life Insurance Company Rocco Roberto Cerchiara Università della Calabria Dipartimento di Scienze Aziendali Email: rocco.cerchiara@unical.it 21 aprile 2005

Rocco Roberto Cerchiara Introduction The goal of this paper is a comparative analysis between BM and NCD systems with special reference to the: Insureds Distribution in the classes of BM Effects on solvency of a new Non Life Insurance Company, with run off and going concern approaches on a long time horizon (Risk Reserve U and solvency ratio U/B). Approach: Risk Theory Montecarlo Simulation (MATLAB) 14-Nov-18 Rocco Roberto Cerchiara

Rocco Roberto Cerchiara The Model For a portfolio of policies with homogeneous a priori characteristics, it’s sufficient to know the transition rules of the BMS, the claim number and the severity distributions - k t e Z i,t – to calculate, for every time t: the premium the aggregate loss the insureds distribution in the classes of BM 14-Nov-18 Rocco Roberto Cerchiara

Transition Rules of Italian and Switzerland BMS and Sweden NCD 14-Nov-18 Rocco Roberto Cerchiara

Premium Coefficients j of Italian and Switzerland BMS and Sweden NCD ITALY SWITZERLAND SWEDEN 14-Nov-18 Rocco Roberto Cerchiara

Rocco Roberto Cerchiara The Model Collective approach of Risk Theory Independence between claim number and claim size Average claim size is constant and equal to Euro 3.250, derived from database of the Italian Association of Insurers (ANIA) - 2001 Run-off and Going on approach (only new policies) No hunger for bonus (cfr. Lemaire 1995): all claims are reported Time horizon: 60 years 14-Nov-18 Rocco Roberto Cerchiara

Rocco Roberto Cerchiara Number of Claims: Poisson Distribution The unique parameter is  : - no short-term fluctuations - no long-term cycles 14-Nov-18 Rocco Roberto Cerchiara

Weighted Claim Frequency of 2001 =  = 9,22% Statistics of ANIA referred to the 67% of the Italian TPML Market “First Sector” – only “regular” passenger cars Claim Frequency 2001 Weighted Claim Frequency of 2001 =  = 9,22% 14-Nov-18 Rocco Roberto Cerchiara

Variable Claim Frequency for each class of BM We have built three frequency curves, for the two BMS and NCD considered, by an interpolation and extrapolation of the claim frequency observed in italian market in the 2001 (figure in the precedent slide). These curves are built by introducing some assumptions and constraints; in particular we have supposed the Swiss BMS has a better discrimination power between “good” and “bad” drivers, so we should have a lower claim frequency in the first classes (of bonus), respect to the other two systems, and higher in the next classes (of malus). The calibration of parameters is based on these constraints: 14-Nov-18 Rocco Roberto Cerchiara

Variable Claim Frequency for each class of BM The Weighted Claim Frequency of population for the first 18 classes of Swiss BM and for the 7 classes of Sweden NCDS has to be equal to 9,22% (Italian Market Frequency). In the Going On approach, the new insureds of each year, who enter in the initial class of each system, must have the same claim frequency (about 13,8%). Initial Class of each system 14-Nov-18 Rocco Roberto Cerchiara

Insureds Distribution in the classes of BM 1000 simulations T = 60 years Run-Off Approach: in t = 0 we have 100 insureds in the initial class of each system (Yit = 14 , Ych = 10, Yncd = 7) Going-On Approach: 3% and 6% of insureds at the beginning of each year (589 and 3.299 insureds in T = 60) 14-Nov-18 Rocco Roberto Cerchiara

AVERAGE CLAIM FREQUENCY OF INSUREDS IN ALL CLASSES The behaviour of the AVERAGE CLAIM FREQUENCY of insured population along the time horizon (60 years)   AVERAGE CLAIM FREQUENCY OF INSUREDS IN ALL CLASSES TYPE OF BMS T=0 T = 60 RUN OFF GOING ON (3%) GOING ON (6%) BM ITALY (YINITIAL = 14) 13,8% 7.5% 9.1% 10.1% BM SWITZERLAND (YINITIAL = 10) 10.8% 12.0% 12.5% NCD SWEDEN (YINITIAL = 7) 7.4% 8.4% 9.2% 14-Nov-18 Rocco Roberto Cerchiara

Insureds Distribution in the first class Insured Distribution in the first class (T=30, 60 years and 1.000 simulations) with Run Off and Going On Approach (3% and 6%)   Insureds Distribution in the first class T = 30 RUN OFF GOING ON (3%) GOING ON (6%) BM Italy 78% 47% 29% BM Switzerland 63% 44% 32% NCD Sweden 84% 67% 54% T = 60 83% 49% 30% 66% 45% 32% of SWISS BMS has due to the slowing down of convergence process to the stationary distribution. In fact new insureds has the same claim frequency (13,8%), but the convergence to the first class is much slower for italian BMS (Y=14) and new insureds become much more significant. 14-Nov-18 Rocco Roberto Cerchiara

“Equilibrium Premiums” g = 0 14-Nov-18 Rocco Roberto Cerchiara

Risk Reserve and Solvency Ratio 14-Nov-18 Rocco Roberto Cerchiara

The effect of BMS on the Claim Number and the Aggregate Loss 14-Nov-18 Rocco Roberto Cerchiara

Standard deviation of solvency ratio u=U/B In Beard et al. (1984) and Savelli (2003) standard deviation formula of solvency ratio is inverse function of square root of the expected value of claim number and the growth rate of portfolio. In this model the expected value of claim number decreases by the effect of BMS; this reduction can be expressed by a percentage BM , depending on the type of BMS used and on the time (for a certain t is increasing and then constant). The decreasing of claim frequency produces an increasing of process variability. In fact the SDV of U(t) decreases, but the grow with t. 14-Nov-18 Rocco Roberto Cerchiara

The Parameters of the model 1000 simulations; N0 =100; T = 60; U0 =100% B0; Going on approach: real growth year rate g = 6% in the initial class; Annual Rate of Investment Return j = 4%; Claim Inflation Year Rate i = 5% ; Coefficient of safety loadings λ = 5%. 14-Nov-18 Rocco Roberto Cerchiara

Standard Deviation of Solvency Ratio u=U/B With Run Off approach (g=0) St. DEV. has divergence values for the three systems as shown in Cerchiara (2003). With Going On Approach (g=6%) St. DEV., as shown in Beard and Savelli: grows to a maximum value, depending on different time t when the population claim frequency become constant and so dependent from the type of BMS, then decrease to the same value for the three systems (t = 50). 14-Nov-18 Rocco Roberto Cerchiara

RUIN PROBABILITY FOR EVERY T 14-Nov-18 Rocco Roberto Cerchiara

CONCLUSIONS AND FUTURE IMPROVEMENTS With the assumptions and the database used in this work, Swiss BMS results to be better then Italian BMS and Sweden NCDS. This preference is based only on solvency profile. 14-Nov-18 Rocco Roberto Cerchiara

CONCLUSIONS AND FUTURE IMPROVEMENTS Other elements would be introduced to evaluate BMS: Hunger for Bonus (not all the claims are reported) efficiency, moral hazard, etc. a higher number of simulations Another distribution for a better fit of the claim number distribution (Negative Binomial, Poisson-Inverse Gaussiana, GPD, etc.) However this approach is an useful method to analyse the underwriting risk and solvency (SOLVENCY II). 14-Nov-18 Rocco Roberto Cerchiara

Rocco Roberto Cerchiara BIBLIOGRAPHY Beard et al. (1984) Cerchiara Rocco Roberto (2003) Daykin C. D., Pentikainen T., Pesonen M. (1994) Gigante P., Pichech L., Sigalotti L., (2001) Lemaire Jean, (1995) Savelli N. (2003) 14-Nov-18 Rocco Roberto Cerchiara

THANK YOU VERY MUCH FOR YOUR ATTENTION 14-Nov-18 Rocco Roberto Cerchiara