Solvency Testing Model SD-CNSF

México

October, 2002

Contents

1. Background

2. Solvency Testing Model

3. Dynamic Solvency System

4. Perspectives

Background

Background

In Mexico, the insurance operations are carry out under a dynamic behavior of financial and risk variables. This situation occurs mainly because a business line high rotation, competition and inflation effects. At the same time, traditional insurance companies are not specialized and they can manage both, life insurance and non-life insurance, increasing the administration complexity.

Background

For this reason it is important to have efficient risk analysis tools, for identifying business line risk factors and analyzing capital sufficiency in the medium and short term.

Background The National Insurance and Surety Commission of

Mexico (CNSF) has been developing a Dynamic Solvency Testing Model to encourage self-regulatory practices by insurance enterprises, and at the same time strengthen preventive supervision.

Based upon the mathematical analysis solvency model, a dynamic solvency testing computing system (SD-CNSF) was developed, which allows a prospective analysis of both solvency and risk exposure factors.

Solvency Testing Model

Solvency Testing Model

The dynamic solvency model of the CNSF incorporates aspects of the Mexican regulation, as well as the laws of the behavior of the risk variables of each insurance line of business in Mexico.

Probability density functions have been fitted using the last five years statistical information from the Mexican market.

The statistical data correspond to each company’s claim amount for a specific business line i, MRi(t). The claims amount is expressed as a percentage of the written premium PEi(t), loss ratio. The loss ratio t for business line i at year t is expressed as:

Solvency Testing Model

)(

)()(

tPE

tMRtX

i

ii

For each insurance business line, it was proved through statistical analysis, that the loss ratio random variable, has the typical characteristics of a gamma probability function, whose mathematical expression is:

Solvency Testing Model

otherwise,0

0,,0)(

1),;(

1

xex

xf

x

X

αβXE )(

2)( αβXVar

Consequently, gamma density functions were fitted, for each business line, using Mexican insurance companies statistical data.

Solvency Testing Model

Seguros de Vida

0.0000

0.5000

1.0000

1.5000

2.0000

2.5000

3.0000

3.5000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

Empírica Teórica Gama

Parámetros

9.0000

0.0450

A Kolmogorov-Smirnov goodness-of-fit test was performed to prove the probability distribution functions’ adequacy on each business line. The Kolmogorov-Smirnov goodness-of-fit test is based on the absolute value of the maximum difference between the sample cumulative distribution values and the hypothetical cumulative distribution:

Solvency Testing Model

)()( 0 xFxSD nn

The probability density functions for each business line are as shown:

Solvency Testing Model

The company’s capital position at time t (CAPt), can be expressed as the company’s capital position at time t-1 (CAPt-1) plus the capital contributions at time t (ACt) plus the operation flow (profit or loss) at time t (Rt) :

Solvency Testing Model

tttt RACCAPCAP 1

Where:

tCAP = company’s capital position at time t,

tAC = capital contributions at time t,

tR = operation flow at time t (profit or loss).

The insurance company’s solvency margin at time t, is calculated as a portion of company’s assets at time t (), minus the solvency requirement at time t :

Solvency Testing Model

))()(( tDtRSCAPMS tt

)(tMS : solvency margin at time t. : portion of assets allowed by Mexican regulation, for capital matching. )(tD: Other types of assets (corresponding assets to special reserves). )(tRS: solvency requirement at time t.

The company’s solvency requirement (RS(t)) is obtained by adding the solvency requirements of every business line:

Solvency Testing Model

k

i

k

ii tStPRftRStRS

11

))(),(()()(

The operational flow of the insurance company at time t, is calculated as the difference between inflow (premium, investment earnings) and outflow (expenses, premium reserves, ceded premium, claims) at time :

Solvency Testing Model

)()()()()()()( tREStRENDtCOtCAtStPCtPERt

Written premium (PE), is simulated based on the historic premium growth rate of the company in the last five years, for each business line, making the growth rate fluctuate, within a small interval of values around the historicby using a uniform distribution function:

Solvency Testing Model

)(1)1()( ttPEtPE kkk

))(ε1(δ)(δ 0 ttk

-

5,000.00

10,000.00

15,000.00

20,000.00

25,000.00

30,000.00

35,000.00

40,000.00

45,000.00

50,000.00

H5 H6 H7 H8 H9 B P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

esenario 1

esenario 2

esenario 3

esenario 4

esenario 5

esenario 6

esenario 7

esenario 8

esenario 9

esenario 10

Retention Premium (PR) is calculated by multiplying a historic rate of retention premium, by the written premium, making the rate of retention premium fluctuate, within a small interval of values around the historic value by using a uniform distribution function.

Solvency Testing Model

)()()( ttPEtPR

))(1()( 0 ttk

60%

65%

70%

75%

80%

85%

90%

95%

100%

H5 H6 H7 H8 H9 B P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

esenario 1

esenario 2

esenario 3

esenario 4

esenario 5

esenario 6

esenario 7

esenario 8

esenario 9

esenario 10

Acquisition Cost (CA) is calculated as a percentaje of the retention premium (historical percentage), making the percentage fluctuate, within a small interval of values around the historical value by using a uniform distribution function :

Solvency Testing Model

)()()( tPRttCA kkk

))(1()( 0 ttk

Costos de Adquisisión

300.00

500.00

700.00

900.00

1,100.00

1,300.00

1,500.00

1,700.00

1,900.00

2,100.00

H5 H6 H7 H8 H9 B P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

esenario 1

esenario 2

esenario 3

esenario 4

esenario 5

esenario 6

esenario 7

esenario 8

esenario 9

esenario 10

Administrative Cost (CO) is calculated with a formula that involves a part as fixed cost, and another part as variable cost that depends on the premium:

Solvency Testing Model

)1(

)())1()( 11 tPE

tPECOINFCOtCO ttt

Investement Earnings (PF), is calculated as the amount of assets multyplied by their asset yield rates, making each rate fluctuate, within a small interval of random values in accordance with the expected trend:

Solvency Testing Model

)()()(1

trtItREND j

m

ji

))(1()( 0 trtrj

-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

H5 H6 H7 H8 H9 B P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

esenario 1

esenario 2

esenario 3

esenario 4

esenario 5

esenario 6

esenario 7

esenario 8

esenario 9

esenario 10

Premium reserve is calculated, in the case of short term insurance, as a percentage of the retention premium. The portion of unearned premium reserve is calculated by a formula which involves the average retention premium of last two years:

Solvency Testing Model

)(

2

1)1(

2

1(t))( ω tPRtPRtRRC

))(1()( 0 ttk

The simulation process, is based on the so-called “inversion method”, which consists in generating random numbers with a continuous uniform distribution on (0,1), and then applying the inverse of the cumulative distribution function of the loss ratio random variable:

Solvency Testing Model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 5 9

13

17

21

25

29

33

37

41

45

49

53

57

61

65

Número simulado (dist uniforme)

Siniestralidad simulada

Función de distribución de la siniestralidad

Finally, with the model it is possible to generate several scenarios and calculate a ruin probability as well as the expected value of future capital necessities.

Solvency Testing Model

Dynamic Solvency System

Dynamic Solvency System Based on the dynamic solvency model, the CNSF

has developed a dynamic solvency computing system (SD-CNSF), which carries out simulations of stochastic processes based on each of the insurance business lines’ probability functions, as well as on the company’s business plan scenarios.

The input information of the system is a data base, which contains all financial information of the company in the last five year.

Financial Statementes

Business Plan

Macroeconomical

Expectations

Invested Assets

Estimated Risk Factors

Projected Financial

Statements

Future Capital

Necessities

Sensitivity Analysis Results

Processes •Stochastic Proceeses•Simulations•Scenarios•Index Calculatios•Grafhs

SD-CNSF

The system functioning is based on the information, processes and the next results:

Risk Probability Functions

Dynamic Solvency System

Next, we are going to show the Dynamic Solvency Computing System (SD-CNSF).

SD-CNSF

Dynamic Solvency System

Perspectives

Perspectives

The development of the Dynamic Solvency Computing System (SD-CNSF), is in an initial phase. We hope to improve and to incorporate new routines in order to increase its efficiency.

The Dynamic Solvency Computing System (SD-CNSF), will allow to implement preventive regulation.

The Dynamic Solvency Computing System (SD-CNSF), is a quite flexible tool that could be updated to the normative changes.

October, 2002

Solvency Testing Model SD-CNSF