On RBC-type Solvency Monitoring Systems for Non-life Insurance

Petr Mandl and Monika Šťástková

The Czech Society of Actuaries Prague, 2nd April 2004

Solvency II project

Solvency II project - initiated by the European Commission in 2001 to review the European framework for prudential supervision of insurance companies.

Phase 1 – general design of the new supervisory regime.

Phase 2 – drafting of the proposal of a new framework directive (beginning 2005), preparation of implementing measures after the adoption of the directive.

A risk based capital system

A risk based capital system is a system in which the minimum capital requirement is based on the risk – or risks – facing an insurance company. (MARKT/2085/01-EN).

Broad definition which includes the European minimum margin rule.

In narrow sense the US system employing the risk based capital formula is understood.

Two capital levels

Two capital levels will be established by Solvency II (MARKT/2539/03-EN):

A minimum capital level (safety margin) – trigger level for ultimate supervisory action.

The target capital – which will reflect the economic capital that a company would need to operate with a quantified low probability (x) of failure within a give time period (y). It is conjectured that x should be less than 1 % and y not less than one year.

Target capital

The probability assumptions for the target capital level will be the main policy issue in the Solvency II project. (MARKT/2085/01-EN Par. 95)

To determine the target capital (not the safety margin) it will be allowed to use internal risk models. The supervisory authority will be empowered to require the use of internal models for part or all the business written by the company if there are indications that result obtained by the standard formula would not be accurate enough.

Target capital II. Consider the definition of the target capital (1)

u - the target capital level, U - the increment of free capital during period

Assuming that the distribution of can be approximated by the normal distribution

(2)

where (z1-) = 1- .

)uU(P

))U(

UEu(1)uU(P

UVar)U(

1u

)U(z

u

UE1

The capital adequacy line

Using the NP2 approximation for (1)

(3)

(2) is a relation between the expected return from

capital and its standard deviation. It fits into the framework of the Markowitz mean – variance theory of investment as a shortfall constraint.

(2) says that the point with coordinates ((U)/u, EU/u) lies on the straight-line

r = z1- - 1

in the (, r) plane (the capital adequacy line).

1u

)U())1z(

6z(

u

UE 21

11

3

3

1 )(

)(

U

UEUE

Mathematical model Mathematical model of U An insurance company running only one line of business – to avoid the

explanation of the aggregation of the lines (see [2]) is considered. The risk reserve (amount of free capital) of the company at the end of

year (period) t is expressed as (4) Ut = u +Bt + Jt –St – Et ,where u – risk reserve at the end of i ,

Bt - the earned gross premium income during t,

Jt - the investment income,

St - the loss regulation expenses,

Et - the operating expenses.

All quantities are assumed to be net from reinsurance.

Mathematical model II.

Let Yt denote the amount paid for insured losses from year t until the end of t.

According to the assumptions of the chain ladder method the estimated total amount paid for losses from year t is expressed as the sum of portions paid in subsequent years

Yt (1+d1 +…+ dr) = Yt (1 + D1).

Consequently,St = Yt (1+ D1) + Zt,

where Zt is the deviation from the amount reserved to be spent in t for losses from previous years.

Mathematical model III.

Bt´- the premiums written in year t The earned premiums are Bt = B´t–1+(1–) B´t. The operating expenses are modeled to be

equal to the expense loading of the premium Et = c Bt.

The premiums are dealt with as non-random quantities.

Mathematical model IV.

The investment income Jt in equation (4) is modeled as resulting from a random net rate of return It applied to the cash flows.

Further - the weighted average of the times of the

premium payments during the year, - the average time of the loss payments. U in equation (1) is defined as the increment of the

risk reserve net of investment earnings on allocated capital U = Ut – u(1+It).

Mathematical model V. Provided that It is uncorrelated to Yt and Zt,

(5) U - EU = It – EIt A-

–Yt–EYt (1+D1+(1-)EIt )–(1-)Y–EYtI–EI – –Zt–EZt (1+(1-)EIt)–(1-)Zt–EZtIt–EIt,

where EU = (1-c)(Bt-1´+(1-) Bt ´) + EIt A- EYt(1+D1)-EZt,

A = (1-c)(Bt-1´+(1-)(1-) Bt´) + (1-)Bt´+

+ (yt-1D1 + yt-2D2+ … +yt-rDr) -

- (1-)(yt-1d1 + yt-2d2+ … +yt-rdr)-(1-)EYt-(1-)EZt.

Mathematical model VI.

Assuming the random variables It, Yt, Zt mutually independent,

(U)2 = V1+V2+V3 ,

where V1 = A2 Var Yt

V2=(1+D1+(1-)EIt)2Var Yt+ (1-)2VarIt VarYt

V3=(1+(1-)EIt)2VarZt+(1-)2VarIt VarZt

EU = E1+E2+E3

E1=EIt A

E2=(1-c)(Bt-1´+(1-)Bt´) - EYt(1+D1)

E3=-EZt.

Mathematical model VII.

Decomposition into components corresponding to investment risk, underwriting risk and reserve risk.

Inserting into (2) the target capital formula is obtained. It is a model based formula. US non-life RBC formula

3213211 EEEVVVzu

25

24

23

22

210 RRRRRRR

Dynamic model

Dynamic model of the risk reserve Essentially a repetitive application of the one

period model with additional use of third central moments of random variables involved.

This enables the use of NP2 approximation to calculate the quantiles.

In the basic version used in the course on Non-life insurance It=i is non-random.

The probability assumption

Internal models Safety margin The probability assumption Even for much simpler tasks than is the

modeling of free capital of an insurance company it is impossible to define tail events of probability 1 % with reasonable accuracy.

should be regarded as a parameter of the model the value of which should be determined aposteriori.

The probability assumption II.

Introduce a capital adequacy ratio analogous to Standard and Poor’s.

To do this rewrite the equation of the capital adequacy line u = z1-(U) - EU

as and set .

4C3C1C)EV)U(

z(u ii

3

1i

1

4C3C

1Cu

The probability assumption III.

Standard & Poor’s states that its ratio for companies with rating A stays in the range 125 % - 150 %.

This contains the hint, how to estimate . Select a class of insurance companies whose

rating guaranties, by statistical evidence, the desired smallness of their failure probability, and find z1-, which makes the average (the median) of capital adequacy ratios equal to 100 %.

Example of application Example of application The transition from the ex lege motor third-party

liability insurance to the compulsory contractual form in Czech Republic took place on 1 January 2000 after several years of preparation. The insurers applying for licence had to submit the business plan for first three financial years.

Examination of the business plans using the dynamic model

The dependence of the moments of Yt (payments during t for claims incurred in t in millions of Czech crowns) on the number of insured vehicles (in thousands) was assumed to have the form

EYt=c1M, VarYt=c2M, E(Y-EYt)3=c3M.

Example of application II.

The values of the constants employed for 2000 – 02 were

c1=0,773; c2=3,669; c3=143,68 Constants of the claim development d1=0,8; d2=0,3; d3=0,15; d4=0,1; d5=0,05;

d6=d7=…=0 The error Zt in the estimate of the reserve

(t=1 for year 2000) EZt=0, E(Zt-EZt)3=0

Var Z1=0; Var Z2=2,56VarY1;

Var Z3=2,56VarY2+0,36VarY1.

Example of application III.

Values of other model parametres i=5%; =0,25; =0,5. Numerical illustration the business plan of an

insurer who intends to insure about 350 thousands of vehicles.

50 % of his production are ceded to reinsurers.

Business plan

Year 2000 2001 2002

1 Gross premiums written 1131,0 1285,4 1420,0

2 Gross premiums written ceded to reinsurers

565,5 642,7 710,0

3 Change in provisions for unearned premiums

78,300 10,692 9,316

4 Change in provisions for unearned premiums –reinsurers’ share

39,150 5,346 4,658

5 Income from financial placement 11,310 12,854 28,400

6 Claims paid 250,12 484,38 616,49

7 Claims paid – reinsurers’ share 125,06 242,19 308,25

8 Change in the provision for claims 350,17 197,892 137,19

9 Change in the provision for claims reinsurers’ share

175,08 98,946 68,595

Business plan II.

Year 2000 2001 200210 Acquisition costs for insurance

contracts 79,170 89,98 99,40

11 Reinsurance commissions and profit participation

141,37 160,6 177,5

12 Administrative expenses 226,20 154,2 170,4

13 Contribution to Czech Insurers’ Bureau 56,550 64,27 71,00

14 Other technical charges, net of reinsurance

11,310 77,12 42,60

15 Change in equalization provision 16,965 19,28 21,30

16 Result of technical account -11,30 64,85 129,7

Example of application IV.

The numbers Mt of insured vehicles reinsurance, the earned premiums Bt and the operating expenses Et (net from reinsurance) presented in Table 1 were obtained from the business plan.

The other parameters were set at the general values presented above.

Table 1: t Mt Bt Et

1 345,000 526,350 231,855

2 355,350 637,354 224,952

3 358,904 705,342 205,902

Example of application V.

Software was produced for the evaluation of the business plans. Its application on the shown plan yielded the data in Table 2, where U0=0 and u25, u75 denote the quartiles of the distribution of Ut.

Table 2:

t EUt Var Ut E(Ut-EUt)3 u25 u75

1 -7,883 1860,7 -88 329 -32,666 25,525

2 86,944 4883,5 -196 563 48,539 134,205

3 58,706 8559,7 -329 511 208,641 317,897

Example of application VI.

The solvency margin 90,48 equals to16% of the premiums for 2000.

The desirable level of allocated capital was calculated as (90,48-u25)/(1+i)=117,29.

Survey of models and recent publications Survey of models and recent publications Single period models [1] M. Šťástková: A comparison between investments

into non-life insurance and investments on the capital market. Pojistné rozpravy 12 (2002)

pp. 85 – 100, in Czech. [2] D. Pilcová: The free capital development of a non-

life insurance company. Pojistné rozpravy 12 (2002) pp. 75 – 84, in Czech. A model of multi-line business taking into account the correlations between the lines.

[3] M. Čížek: The risk in health insurance. Pojistné rozpravy 12 (2002) pp. 25-35, in Czech. A special model for permanent health insurance.

Survey of models and recent publications Multi-period models

[4] P. Finfrle: Simulation analysis of the risk reserve model. Pojistné rozpravy 12 (2002) pp.36-51, in Czech.

[5] D. Chládková: A contribution to probability modelling of the non-life insurance reserve. Pojistné rozpravy 12 (2002) pp. 52-62, in Czech. Extension of the basic model to stochastic rates of return from investment.

[6] V. Unzeitigová: A model of accident insurance cash flows. Pojistné rozpravy 12 (2002)pp. 120-132, in Czech. A special model for accident insurance.

[7] P. Mandl: Mathematische Modelle für die Reform of Kfz-Haftpflicht in Tschechien. Münchener Blätter zur Versicherungsmathematik 34 (2001), 26 – 33.

References References: [8] European Commission – Internal Market DG: Risk-

based capital systems.MARKT/2085/01-EN. [9] European Commission – Internal Market DG:

Solvency II – Reflections on general outline of a framework directive and mandates for further technical work. MARKT/2539/03-EN.

[10] J. Balling, A. M. Levin: Standard and Poor’s property / casualty capital adequacy model. www.insure.com,1997.

[11] R. E. Beard, T. Pentikäinen, E. Pesonen: Risk Theory. Chapman & Hall, London 1984.

[12] P. Mandl, L. Mazurová: Mathematical Foundations of Non-Life Insurance. Matfyzpress, Prague 1999. In Czech.

References [13] R. Schnieper: Capital allocation and solvency

testing. SCOR, Paris 1996. [14] R. Schnieper: Solvency testing. Mitteilungen der

SAV (1999) 11-45. [15] M. Šťástková, E. Všetulová: Capital adequacy

and the business plans of non-life insurance companies. Seminář z aktuárských věd 1999/2000, 118–125, in Czech.

[16] E.Všetulová:Mathematical modeling of non-life insurance–applications to motor third-party liability insurance.Dissertation,Charles University 1998,in Czech.

[17] E. Všetulová: MTP - Liability program in Excel. Ministry of Finance, Prague 1999.