1. 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
21 aprile 2005

2. 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

3. 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

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

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

6. 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)
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

7. 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

8. 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

9. 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

10. 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

11. 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 insureds in T = 60)
14-Nov-18
Rocco Roberto Cerchiara

12. 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

13. Insureds Distribution in the first class
Insured Distribution in the first class (T=30, 60 years and 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

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

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

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

17. 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

18. 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

19. 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

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

21. 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

22. 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

23. 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

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