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)
P( U  u )  
 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

u  EU
( U )  VarU
P ( U   u )  1   (
)
( U )
(2)
EU
(  U )
 z 1 
1
u
u
where (z1-) = 1- .
The capital adequacy line
 Using the NP2 approximation for (1)
EU
1 2
(  U )
 (3)
 ( z 1   ( z 1   1))
1
u
6
u
E (U  EU )3
1 
 (U )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, EU/u) lies on the straight-line
r = z1-  - 1
in the (, r) plane (the capital adequacy line).
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 - EU =  It – EIt  A–Yt–EYt (1+D1+(1-)EIt )–(1-)Y–EYtI–EI –
–Zt–EZt (1+(1-)EIt)–(1-)Zt–EZtIt–EIt,
where
EU = (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
EU = 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
u  z1 V1  V2  V3  E1  E2  E3
is obtained. It is a model based formula.
 US non-life RBC formula
R  R 0  R12  R 22  R 23  R 42  R 52
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) - EU
3
as
z 1 
u  (
Vi  Ei )  C1  C3  C4
i  1 (  U )
and set
u  C1 .
C 3  C4
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 2002
10 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.