CEIOPS’ Advice for Level 2
Implementing Measures on Solvency II:
Article 130 - Calibration
of the MCR (8 April 2010)
1. Introduction
1.1. In its letter of 19 July 2007, the European
Commission requested CEIOPS to provide final, fully consulted advice
on Level 2 implementing measures by October 2009 and recommended
CEIOPS to develop Level 3 guidance on certain areas to foster
supervisory convergence.
On 12 June 2009 the European Commission sent a letter
with further guidance regarding the Solvency II project, including
the list of implementing measures and timetable until
implementation.
1.2. This Paper aims at providing
advice with regard to the calculation of the Minimum Capital
Requirement (MCR) as requested in Article 130 of the Solvency II
Level 1 text.
1.3. This paper follows CP 55 which was published in
June 2009, and for which the consultation period closed on 11
September 2009.
1.4. The objective of this paper is to give draft
advice on the calibration of the MCR, in
particular on the calibration of the linear function referred to in
Article 127(1b) of the Level 1 text.
1.5. The calibration proposals in this paper reflect
CEIOPS’ best knowledge on the basis of QIS4 data, also taking into
account CEIOPS’ revised proposals for the calibration of the SCR
standard formula.
CEIOPS suggests that the calibration of the MCR
should be further revised after the results of
the new calibration of the SCR standard formula become available
following QIS5.
2. Extract from Level 1 Text
Legal basis for the implementing
measure
2.1. Article 130 – Implementing
measures:
The Commission shall adopt implementing measures
specifying the calculation of the Minimum Capital Requirement,
referred to in Articles 128 and 129.
Other relevant Level 1 text for providing background
to the advice
2.2. Recitals:
(42) When the amount of eligible basic own funds falls
below the Minimum Capital Requirement, the authorisation of
insurance and reinsurance undertakings should be withdrawn, if those
undertakings are unable to re-establish the amount of eligible basic
own funds at the level of the Minimum Capital Requirement within a
short period of time.
(43) The Minimum Capital
Requirement should ensure a minimum level below which the amount of
financial resources should not fall.
It is necessary that it is calculated in accordance
with a simple formula, which is subject to a defined floor and cap
based on the risk-based Solvency Capital Requirement in order to
allow for an escalating ladder of supervisory intervention and that
it is based on the data which can be audited.
2.3. Articles:
Article 129 –
Calculation of the Minimum Capital Requirement
(1) The Minimum Capital Requirement shall be
calculated in accordance with the following principles:
[...]
(c) the linear function referred to in paragraph 2 used to calculate
the Minimum Capital Requirement shall be
calibrated to the Value-at-Risk of the basic own funds of an
insurance or reinsurance undertaking subject to a confidence level
of 85% over a one-year period;
[...]
(2) Subject to paragraph 3 the Minimum
Capital Requirement shall be calculated as a linear function of a
set or sub-set of the following variables: the undertaking’s
technical provisions, written premiums, capital-at-risk, deferred
tax and administrative expenses. The variables used shall be
measured net of reinsurance.
(3) Without prejudice to point (d) of paragraph 1, the
Minimum Capital Requirement shall not fall
below 25% nor exceed 45%, of the undertaking’s Solvency Capital
Requirement, calculated in accordance with Chapter VI,
Section 4, Sub-sections 2 or 3, and including any capital add-on
imposed in accordance with Article 37. [...]
3. Advice
3.1. Background
3.1. The MCR approach tested in QIS4 combined a linear
formula with a cap of 50% and a floor of 20% of the SCR. Overall,
this approach was found workable in QIS4. The Level 1 text sets out
an MCR calculation method similar to QIS4, yet
with a narrower corridor (25% to 45% of the SCR).
3.2. The calibration of the linear component of the
MCR in QIS4 was regarded as satisfactory for non-life business,
whereas it was also concluded that the calibration of the linear
formula for life business would need improvement.
The subject of this paper is the refinement of
the QIS4 calibration, as well as its adjustment to post-QIS4 changes
of the MCR and the SCR.
3.3. This paper builds on the the definitions and
notations used in CEIOPS’ advice on Article 128: Calculation of the
MCR (CEIOPS-DOC-47/09).
3.4. The Level 1 text requires that the MCR linear
formula is calibrated to a 85% Value-at-Risk
confidence level over a one-year time horizon.
It is not expected, however, that a simple
linear formula will accurately reflect a prescribed level of
confidence. Therefore, instead of an independent modelling of
the 85% VaR confidence level, CEIOPS calibrated the MCR linear
formula relative to the SCR standard formula.
The life linear formula was fitted to a benchmark
percentage (35%) of the SCR standard formula; whereas the non-life
calibration was built on the standard deviation parameters used in
the premium and reserve risk submodule of the SCR standard formula.
3.5. Admittedly, the relationship
between the 85% and 99.5% confidence levels can not be described by
a fixed percentage across all probability distributions.
CEIOPS however considers that the
35% ratio – which corresponds to the middle of the 25%–45% corridor
– is broadly consistent with the range of distribution assumptions
used in the SCR standard formula.
3.6. From this approach it follows that the
calibration of the MCR linear formula is closely linked to the
calibration of the SCR standard formula. This also means that
when there is a significant change in the
calibration of the SCR standard formula, the MCR linear formula
should also be recalibrated.
3.7. Accordingly, the change of the level of the
linear formula in this advice relative to QIS4 largely mirror the
impact of CEIOPS’ revised proposals for SCR standard formula
calibrations.
3.8. The calibration exercise described in this paper
has been carried out on the basis of QIS4 data, taking into account
the proposed changes in the calibration of the SCR standard formula.
CEIOPS suggests that the
calibration of the MCR should be further revised after the
results of the new calibration of the SCR standard formula become
available following QIS5
3.9. It is also noted that, from the 20%–50% corridor
used in QIS4, the Level 1 text narrowed down the corridor to between
25% and 45% of the SCR.
Therefore it is expected that, despite calibration
refinements, a larger percentage of linear formula results will fall
outside the corridor than was observed in QIS4.
3.1.1 Estimating the impact of SCR
standard formula changes
3.10. For the purpose of adjusting the calibration of
the MCR to the revised SCR standard formula, a single-factor
adjustment technique was used.
That is, CEIOPS estimated a single factor reflecting
the average change in the overall SCR standard formula.
3.11. The SCR adjustment factors were informed by an
impact assessment study, carried out by CEIOPS with the aim of
delivering an estimate of the overall impact of proposed calibration
changes relative to QIS4.
A detailed description of this impact assessment study
is provided in a separate document published by CEIOPS.
3.12. Separate SCR adjustment factors were estimated
in respect of the life and for non-life components of the MCR linear
formula. The non-life adjustment factor was set at 1.45, reflecting
the estimated SCR impact for the undertakings affected (non-life,
composite, reinsurance and captive).
The life adjustment factor, reflecting the estimated
SCR impact for life and composite undertakings, was set at 1.5.
3.13. Both of the above factors are rounded and are of
an approximate nature. Given the simplified treatments used in the
calculation, the results are regarded as a preliminary indication.
QIS5 will allow a far more accurate assessment of the
impact, on the basis of undertaking-by-undertaking data.
3.2 Non-life linear formula
3.14. Following CEIOPS’ advice in CEIOPS-DOC-47/09 on
the calculation of the MCR, similarly to the QIS4 approach, the
non-life linear formula is expressed as a function of net technical
provisions and net written premiums according to the segmentation
defined below.
The linear formula charge for each line of business is
the higher of a fixed percentage of technical provisions and a fixed
percentage of written premiums. The non-life linear formula is the
sum of charges over all lines of business.
3.15. Following the results of QIS4, CEIOPS concluded that the QIS4
calibration approach was broadly satisfactory for non-life
undertakings.
CEIOPS therefore retains its general approach to the
calibration of the non-life linear formula, whereby the factors were
derived from the SCR standard formula premium and reserve risk
parameters.
3.16. The suggested linear formula factors are derived
from the SCR premium and reserve risk standard deviations as
follows:
where the steps of the process, and the meaning of the
ρ(σ) function and the adjustment factor K are explained below:
3.17. Step 1 – Determine the 85%
VaR factor corresponding to the premium and reserve risk standard
deviations:
Following the lognormal assumptions of the SCR premium
and reserve risk module, this is done by applying the ρ(σ) function
similar to that used in the SCR standard formula (see
CEIOPS-DOC-41/09 on the non-life underwriting risk), but reflecting
a 85% quantile instead of 99.5%:
where N0.85 is the 85% quantile of the
standard normal distribution. (An indicative value of the ρ85%(σ) to
ρ99.5%(σ) ratio is 0.35, varying slightly according to line of
business.)
3.18. For reserve risk, the net
standard deviations by line of business are directly available from
CEIOPS’ advice on the calibration of the non-life and health
underwritng risk modules.
For premium risk, the net parameters are derived from
the gross parameters by using undertaking-specific adjustment
factors (the NCR/GCR ratio in each line of business).
As in the SCR impact assessment prepared by CEIOPS, it
is assumed that, for the purpose of this paper, the overall effect
of the gross-to-net adjustments can be reflected by adjustment
factors equal to 100%.
3.19. Step 2 – Apply an
adjustment factor to reflect risks other than premium and reserve
risk:
In the first step, only premium and reserve risk has
been explicitly reflected. To implicitly reflect all other risks in
the SCR (non-life CAT risk, market risk, operational risk,
counterparty default risk etc.) an adjustment factor K is applied.
3.20. On the basis of the QIS4 calibrations of the SCR
standard formula, the correct choice of the adjustment factor would
have been 1.18. This means that a 1.18 factor would scale up the ρ85%
factors such that the weighted average of the linear formula to SCR
ratio for non-life undertakings is equal to 35%, where the SCR is
calculated by the QIS4 standard formula.
3.21. Referring to the assessment of the impact of SCR
calibration changes it is estimated that, after the calibration
changes suggested by CEIOPS, the overall increase in the SCR premium
and reserve risk sub-modules (both under life and health
underwriting risk) could be reflected by a factor of 1.454, whereas
the overall SCR increase for the undertakings affected is estimated
by a factor of 1.45.
This leads to an adjustment
factor of K = 1.18 · 1.45/1.45 = 1.18.
3.22. The results of the above steps are the following
(in step 2, the factors are rounded):
3.23. The results of Step 2 reflect the factors
suggested by CEIOPS. The MCR factors have been derived based on the
factors calibrated for the SCR standard formula. Therefore in case a
different calibration is adopted in the SCR standard formula, the
calibration of the MCR linear formula factors should be adjusted
accordingly, following the procedure described above.
3.3 Life linear formula
3.3.1
Linear fitting techniques
3.24. Following CEIOPS’ advice
in advice in CEIOPS-DOC-47/09 on the calculation of the MCR, the
life linear formula is expressed as a function of the volume
measures listed below.
The formula specified in CEIOPS’ advice is a linear
combination of the variables, with the exception of the application
of the with-profit floor, which sets a minimum value for the capital
charge for participating contracts.
3.25.
It is noted that, in its draft advice in CP55 CEIOPS
had suggested more granular capital-at-risk factors (with three
segments depending on the outstanding term of contract, factors
C.4.1 to C.4.3). Following stakeholder feedback on CP55,
capital-at-risk is now treated as a single segment. However, a large
part of the calibration work had been completed before this change;
therefore some of the following explanations refer to multiple
capital-at-risk factors.
3.26.
To derive the calibration
of the life linear formula factors, CEIOPS applied least-squares
linear regression techniques to the data collected in QIS4, using
35% of the SCR standard formula as a proxy for the target confidence
level (85% VaR).
3.27.
The linear properties of these
techniques allowed to carry out a linear fitting exercise without
collecting individual undertaking data in a central database. This
was possible because the coefficient matrices of the resulting
linear equation systems are additive across populations of
undertakings. Therefore it was sufficient to collect the relevant
coefficient matrices for each country market instead of centralising
individual undertaking data.
3.28.
In light of the QIS4
results, it was expected that applying linear fitting techniques to
the problem of life MCR calibration would face significant
difficulties, including the following ones:
possible significant
non-linearity in the target function,
possible material effect of
hidden variables, e.g. market risk of assets and deferred taxes,
lack of consistent interpretation or comparability of part of the
data, especially with regard to future discretionary benefits.
3.29.
Aware of these difficulties, CEIOPS tested several
variants of least-squares linear regression techniques on QIS4 data
in two iterations, and compared their results against each other and
against expert judgement. The factors resulting from linear fitting
tests were treated with extreme caution. It was recognised that
linear regression alone, without expert judgement, was unlikely to
lead to a satisfactory calibration.
3.30.
Linear fitting
was attempted both on an absolute distance and on a relative
distance basis. The absolute vs. relative distance approaches seek
to minimize, respectively, the following square distance functions:
3.31. Regarding the choice of the target function, net
and gross fitting approaches were both tested. By net and gross we
refer to the adjustment of the SCR standard formula for the risk
absorbing effect of future profit sharing.
3.32. In the net approach, the target function was
35% of the SCR of each undertaking, that is,
Note that in this approach αC.1.2 (the factor for
technical provisions for discretionary benefits) is finally derived
as the sum of a positive and a negative fitted factor (resulting
from the Zgross target and the ZFDB target, respectively).
3.34. For the target function of the life side of
composite undertakings, a proxy “life SCR” was disaggregated from
the overall SCR result.
This was calculated by decomposing the market risk and
counterparty default risk modules, as well as the adjustment for
deferred taxes according to the ratio of the life technical
provisions to the total technical provisions, and by recalculating
the operational risk charge from the life side volume measures.
3.35. The more unknown parameters are included in the
fitting test or are calibrated at the same time, the less reliable
the result becomes. Therefore the number of fitted factors was
reduced to five in the first iteration, and to just two in the
second one:
• In the 5-factor fitting, the αC.2.2/αC.2.1,
αC.4.2/αC.4.1 and αC.4.3/αC.4.1 ratios were fixed identically to
QIS4, i.e. only one independent factor was left for unit-linked
technical provisions and capital-at-risk each.
• In the 2-factor fitting, all factors except αC.1.1
and αC.1.2 (the factors for technical provisions for guaranteed and
discretionary benefits in respect of participating contracts) were
fixed.
The setting of the fixed factors was identical to the
respective QIS4 parameters; however a set of increased
capital-at-risk factors (1.5 times higher than in QIS4) was also
tested in parallel to inform expert judgement.
• Furthermore, only when fitting for the gross target
in the gross approach, no distinction was made between the
guaranteed and discretionary part of technical provisions (a single
factor was fitted for both).
3.3.2. Linear fitting results
3.36. The first iteration of the exercise took into
account QIS4 data of 334 life and 225 composite undertakings in 29
countries. The second iteration included QIS4 data of 340 life and
225 composite undertakings in 29 countries. (Some undertakings whose
data were thought to be grossly unreliable were excluded by national
QIS analysts.)
3.37. Generally, the relative distance approaches
failed to yield meaningful factors (most of the fitted factors were
very close to zero). In the relative distance approach, small and
large undertakings influence the outcome by an equal weight,
however, this approach apparently introduced such a level of noise
to the target function that masked any possible linear trend.
3.38. In the absolute distance approaches, in the
5-factor fitting exercise the raw fitted factors were the following:
3.39. These results illustrate the limitations of the
linear fitting technique. Apart from the differences between the
results of the net and gross approaches, country-by-country results
also showed significant variations. There were examples of the
fitted factors falling outside the acceptable range (e.g. a result
falling below zero where a positive factor was expected).
3.40. Furthermore, it appeared that the outcome was
heavily driven by the first two factors (relating to participating
contracts). CEIOPS therefore focused its efforts to find the most
appropriate factors αC.1.1 and αC.1.2, hoping that reducing the
number of factors would lead to more reliable results. Even for
these two factors, the net and the gross approaches yielded markedly
different overall results:
• Net fitting results (absolute distance basis):
3.41. Combined with further analysis of data, these
results indicated that the gross SCR had a stronger linear
relationship with the volume measures than the net SCR.
The analysis also indicated that in the net approach,
both factors are more sensitive to the non-linear effects and random
distortions in the QIS4 data regarding discretionary benefits and
SCR adjustments.
For these reasons, the results of the gross fitting
approach were selected as the starting point for the choice of the
linear formula factors.
3.42. It is noted that, despite the effort put
into finding the correct linear factors, a major improvement in the
overall quantitative effect relative to QIS4 cannot be expected.
In QIS4, the linear formula
result was inside the 25%–45% SCR band for 154 out of 558 life and
composite undertakings (28% of the results).
Since QIS4, CEIOPS back-tested a range of alternative
calibration proposals on the QIS4 datasets.
In addition to testing the results of the above linear
fitting approaches, independent expert adjustments of the QIS4
factors were also tested.
None of the tested alternatives did materially
increase the proportion of the results falling within the 25%–45%
corridor.
Restricting the majority of the life linear
formula results to the corridor between 25% and 45% of the SCR would
require a strong linear relationship between the volume measures and
the (net) SCR, while the analysis of the data indicates that such a
strong linear relationship is not present.
3.3.3. Choice of factors
3.43. Following the above analysis, the choice of the
life linear formula factors was derived in the following
three steps:
3.44. Step 1 – Set initial
calibration to reflect the results of the gross fitting approach:
Following the gross fitting results, αC.1.1 and αC.1.2 were set to
4.8% and –8.5% respectively.
For the remaining factors, the QIS4 calibration was
retained, adjusted for changes in the segmentation (these QIS4
factors had been informed by expert judgement during the preparation
for QIS4, reflecting a ranking of the risks of the respective
segments):
• for the αC.2.1 and αC.2.2 factor in respect of
unit-linked contracts, the QIS4 factors were retained;
• for
the αC.3 factor in respect of non-participating contracts, a 2.8%
value was chosen, which was between the gross and net fitted
factors, and also fell in between the QIS4 factors of the former
sub-segments;
• for the αC.4 factor in respect of capital-at-risk, a
0.095% factor was chosen, leading to the same aggregate risk charge
as the former more granular factors in QIS4.
• The with-profit floor parameter was also left
unchanged at 1.5%. This parameter resulted too from expert
judgement, however the net fitting results indicate that the choice
of this parameter was in the correct range (we note that the
with-profit floor parameter has been included to keep the
with-profit charge in a reasonable range for those countries where,
due to the specificities of the profit sharing regime, the gross
approach does not work well).
3.45. Step 2 – Remove bias
from the weighted average: Next, the weighted averages of the linear
formula to SCR ratio (weighted by the SCR) were calculated for each
country, and the weighted average of the country weighted averages
was calculated (where countries were weighted according to the
number of relevant undertakings in the QIS4 sample).
The initial calibration was then adjusted by a factor
of 0.85 to adjust the weighted average of country averages to the
35% target.
3.46. Step 3 – Adjust for
changes in SCR calibration: A single-factor adjustment was applied
to the calibration in order to take into account to overall change
in the level of the SCR standard formula following the proposed new
calibrations (the resulting factors were also rounded).
The setting of the adjustment factor (1.5) took into
account the assessment of the impact of SCR calibration changes
described in section 3.1.1.
3.48. The results of Step 3 reflect the factors
suggested by CEIOPS.
The MCR factors have been derived based on the factors
calibrated for the SCR standard formula. Therefore in case a
different calibration is adopted in the SCR standard formula, the
calibration of the MCR linear formula factors should be adjusted
accordingly, following the procedure described above.
3.4 CEIOPS’ Advice
3.49. The advice below
supplements CEIOPS’ advice in advice in CEIOPS-DOC-47/09 on the calculation of
the MCR.
The present advice covers the calibration of the
parameters of the MCR linear formula, using the definitions and
notations in the paper referred to above.
3.50. The calibration of the MCR
shall be standardised: all undertakings falling under the scope of
Solvency II should use the same linear formula factors.
3.51. The non-life technical provision and written
premium factors by line of business are defined below (the same
factors apply to linear formula components A – Non-life activities
practised on a non-life technical basis and D – Life activities:
supplementary obligations practised on a non-life technical basis):
3.52. The life technical provision and capital-at-risk
factors are defined below (the same factors apply to linear formula
components C – Life activities practised on a life technical basis
and B – Non-life activities technically similar to life):
3.53. The above factors have been calibrated so that,
on the average, the linear formula match the
centre of a 25%–45% corridor based on the SCR standard formula.
The calibration proposals in this paper reflect
CEIOPS’ best knowledge on the basis of QIS4 data, also taking into
account CEIOPS’ 16/17 revised proposals for the calibration of the
SCR standard formula.
CEIOPS suggests that the calibration of the MCR should
be further revised after the results of the new calibration of the
SCR standard formula become available following QIS5.
3.54. CEIOPS suggests that in case there is a
significant change in the calibration of the SCR standard formula,
the MCR linear formula should also be recalibrated.
