ï»¿ WPS6306
Policy Research Working Paper 6306
Reinsurance as Capital Optimization Tool
under Solvency II
Eugene N. Gurenko
Alexander Itigin
The World Bank
Finance and Private Sector Development
Non-Banking Financial Institutions
January 2013
Policy Research Working Paper 6306
Abstract
This paper compares solvency capital requirements relief attained by the insurer by applying different
under Solvency I and Solvency II for a sample mid-size possible adjustments in the reinsurance structure.
insurance portfolio. According to the results of a study, To evaluate the efficiency of reinsurance as the solvency
changing the solvency capital regime from Solvency I to capital relief instrument, the authors introduce a cost-
Solvency II will lead to a substantial additional solvency of-capital based approach, which puts the achieved
capital requirement that might represent a heavy burden capital relief in relation to the costs of extending the
for the companyâ€™s shareholders. reinsurance protection. This approach allows a direct
One way to reduce the capital requirement under comparison of reinsurance as a capital relief instrument
Solvency II is to increase reinsurance protection, which with debt instruments available in the capital market.
will reduce the net retained risk exposure and hence also With the help of the introduced approach, the authors
the solvency capital requirement. Therefore, this paper show that the best capital relief efficiency under all
proposes an extended reinsurance structure that, under examined reinsurance alternatives is achieved when a
Solvency II, brings the capital requirement back to the financial quota share contract is chosen for proportional
level of that required under Solvency I. In a step-by-step reinsurance.
approach, the paper demonstrates the extent of solvency
This paper is a product of the Non-Banking Financial Institutions, Finance and Private Sector Development. It is part of
a larger effort by the World Bank to provide open access to its research and make a contribution to development policy
discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org.
The author may be contacted at egurenko@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Reinsurance as Capital Optimization Tool
under Solvency II
Eugene N. Gurenko 1
Alexander Itigin2
Keywords: Solvency I, Solvency II, Reinsurance, Insurance Portfolio, Non-Life, Capital
Requirement, Risk Management
JEL code: G22
Sector Board: Financial Sector
1
Eugene N. Gurenko is a Lead Insurance specialist at the World Bank Capital Markets and Non-Bank Finance Practice.
Email: egurenko@worldbank.org
2
Alexander Itigin is a World Bank Consultant and a senior actuary at a leading global and a member of the German Society
of Certified Actuaries.
We are very grateful for useful comments provided by our colleagues Serap Gonulal, Senior Insurance Specialist, World
Bank and Craig Thorburn, Lead Insurance Specialist, World Bank.
1. Introduction
For a sample insurance portfolio, this paper compares the solvency capital requirements
under Solvency I and Solvency II (QIS5 Standard Formula), with the latter taking into account
only two risk categories: non-life underwriting risk and counterparty default risk due to
reinsurance 3. The results of this comparison are presented in Section 2.
In Section 3, the paper shows that under Solvency II substantially more extensive
reinsurance protection is required to achieve the same minimum level of solvency capital
requirements as under Solvency I which results in considerably higher costs of reinsurance.
We show that these costs can be reduced only partially by changing the structure of
reinsurance coverage. In Section 3 we also introduce the concept of financial quota share
reinsurance as a powerful tool for reducing reinsurance costs when reinsurance is used as a
solvency relief instrument. Using the capital cost of different instruments as a benchmark,
we develop an approach for evaluating the efficiency of reinsurance as a substitute for
solvency capital.
Section 4 presents the conclusions and summarizes the main results of the paper.
2. Solvency capital requirements under Solvency I and Solvency II for a sample insurance
portfolio
2.1 Sample insurance portfolio and its reinsurance protection
We assume that the sample portfolio contains Motor Liability, Motor Hull, Marine, Fire and
General Liability insurance business, cf. Table 2.1. Besides presenting gross premium per line
of business, the table contains the estimates of the ultimate loss ratio 4 and ultimate loss for
last business year.
Table 2.1.1: Sample portfolio gross premium composition and ultimate losses 5
3
For the purposes of this comparison, we ignored such Solvency II risk categories as market risk, operational
risk and default risk due to counterparties other than reinsurance as these risk categories have no equivalents
in the â€œold worldâ€? of Solvency I.
4
In this paper, the term â€œultimate loss ratioâ€? is used to describe the ratio between the ultimate underwriting
loss under an annual contract period after its full development and the gross insurance premium written under
the contract during the same annual term.
5
Throughout this paper, all figures presented in tables are denominated in thousands (â‚¬â€™000).
2
Further, we assume that the sample portfolio is protected by proportional reinsurance for
such lines of business as Marine and Fire and by the non-proportional excess of loss
reinsurance for all lines of business. The following table summarizes the parameters of
proportional reinsurance for the sample portfolio.
Table 2.1.2: Proportional reinsurance for the sample portfolio
The non-proportional XL (excess of loss) reinsurance is chosen in the way that the companyâ€™s
net retention due to proportional reinsurance is protected up to the level of retained
Probable Maximum Loss (PML) for a selected return period with the self-retention of
â‚¬500,000 for each line of business. Such self-retention represents a common retention
choice for a portfolio of given size in emerging markets. The following table illustrates the
approach and provides a summary of parameters for the companyâ€™s XL reinsurance.
Table 2.1.3: Non-proportional reinsurance for the sample portfolio
Please note that under the line of business Fire along with the â€œpureâ€? fire exposure there is
also the natural catastrophe exposure due to earthquake that requires reinsurance cover.
This necessitates two XL reinsurance contracts for this line of business â€“ one for the non-
natural-catastrophe per risk exposure and one for the earthquake per event exposure.
According to Solvency I, cf. Section 2.2 below, for calculating solvency capital requirements
the total gross and net retained losses for the last three years are required. The following
table calculates these parameters assuming (a) that the loss burdens per line of business for
the years 2009 and 2010 were equal to the ones of the year 2011 and (b) that within last
three years the non-proportional reinsurance contracts paid out â‚¬500,000 each.
3
Table 2.1.4: Total gross and net underwriting loss for the last three business years
2.2 Solvency capital requirements for the sample insurance portfolio according to Solvency I 6
First, the Retention Ratio needs to be calculated. The following table provides the result for
the sample portfolio whereas the figures in columns A and B were taken from the Table
2.1.4, columns K and N.
Table 2.2.1: Solvency I, Retention Ratio
Then we can calculate the Premium Index (Table 2.2.2) and the Claims Index (Table 2.2.3)
Table 2.2.2: Solvency I, Premium Index (EUR 000)
Adjusted gross premium in column D of Table 2.2.2 is computed as the total gross premium
plus 50% of the premium for general liability, cf. Table 2.1.1 and Annex A1.
Table 2.2.3: Solvency I, Claims Index (EUR 000)
The adjusted gross loss in column H of Table 2.2.3 is the product of the total gross loss plus
50% of the loss for general liability, cf. Table 2.1.4 and Annex A1.
6
For more details on calculating solvency capital requirements under Solvency I, see Annex A1.
4
The solvency capital requirement according to Solvency I is calculated as the maximum
between the Premium Index and Claims Index. For the sample portfolio, we obtain
Solvency Capital Requirement = max (Premium Index; Claims Index)
= max (7,765; 7,691)= â‚¬7,765
2.3 Solvency capital requirements for the sample insurance portfolio under Solvency II
This section provides the results of capital requirement calculation under Solvency II for such
risk categories as non-life underwriting risk and counterparty default risk due to reinsurance.
More details of the capital requirement calculation are provided in Annexes A2 and A3.
In our calculation, for simplification purposes we ignored such Solvency II risk categories as
market risk, operational risk and default risk of counterparties, other than that of reinsurers,
as these Solvency II risk categories have no equivalent under the â€œold worldâ€? of Solvency I.
The capital requirement for non-life underwriting risk is the total of such sub-modules as
non-life premium and reserve risk, lapse risk sub-module and catastrophe risk. For the
purposes of our calculation, we assumed that the lapse risk can be neglected. Furthermore,
we assumed that the sub-module man-made catastrophe risk can be neglected as well.
Therefore, under the risk category for non-life underwriting risk, only the sub-modules for
premium and reserve risk and natural catastrophe risk need to be considered.
2.3.1 Capital requirement for premium and reserve risk
For calculating the capital requirement for the sub-module premium and reserve risks, we
first calculate the net retained premium and reserves. The net retained premium is
calculated as the gross premium income net of reinsurance premium. The gross premium
and the premium retained after proportional reinsurance for the previous year (2011) were
already provided in Section 2.1, Table 2.1.2. We assume that for the next year (2012) these
premiums are equal to the ones of 2011. Table 2.3.1 provides the 2012 reinsurance
premiums for non-proportional reinsurance and finally the net premium retained under both
proportional and non-proportional reinsurance. For non-proportional reinsurance, we
assumed that in 2012 the sample portfolio is protected by the same XL contracts as in 2011.
Table 2.3.1: XL reinsurance premium and net retained premium
Reinsurance rates on premium in Column C of Table 2.3.1 were calculated based on market
benchmarks for corresponding loss frequencies and loss severities (see Annex A4 for more
5
details). These are gross rates, i.e. they cover both the expected layer loss as well as the
reinsurerâ€™s costs of capital and administrative expenses allocated to the XL layer.
Along with the net retained premium, one also needs to estimate the provisions for
outstanding claims for calculating capital requirements in the premium and reserve risk sub-
module (see Annex A2). Table 2.3.2 summarizes our assumptions regarding the provisions
for outstanding claims.
Table 2.3.2: Provisions for outstanding claims
Assumptions for gross provisions in Column F were made based on the gross premium (Table
2.1.1, Column A) and the characteristic payout patterns for each line of business. Net
retained provisions in Column G were assumed based on the gross provisions in column F
and the risk mitigating effect of both proportional and non-proportional reinsurance.
The first step in calculating the capital requirement for premium and reserve risks is to
calculate the volume measures for premiums and reserves, cf. Table 2.3.3.
Table 2.3.3: Solvency II, premium and reserve risk, volume measures
The next step is to calculate standard deviations per line of business, cf. Table 2.3.4. This
calculation takes into account the NP factors (column L) which represent the volatility
reduction effect of non-proportional reinsurance.
Table 2.3.4: Solvency II, premium and reserve risk, standard deviations
6
Table 2.3.5 completes the calculation. It provides the results for the total volume measure
(column Q), total standard deviation (column R), the function í µí¼Œ of the total standard
deviation, which approximates the 99.5% VaR (column S), and finally the capital requirement
for non-life premium and the reserve risk (column T).
Table 2.3.5: Solvency II, capital requirement for premium and reserve risk
For a detailed explanation of the calculation steps presented in Tables 2.3.3, 2.3.4 and 2.3.5,
please refer to Annex A2, section A2.2.
2.3.2 Capital requirement for natural catastrophe risk
The only natural catastrophe risk covered by the sample portfolio is the earthquake risk.
According to Table 2.1.3, the PML for the earthquake risk is â‚¬ 100m. We assume that this
PML is the 200 years loss. The capital requirement for earthquake risk according to Solvency
II is equal to the 200 years loss net of the risk mitigation due to reinsurance. The following
Table 2.3.6 summarizes the calculation of risk mitigation and provides the resulting capital
requirement.
Table 2.3.6 Capital requirement for natural catastrophe risk
For more details on the calculation of capital requirement for natural catastrophe risk,
please refer to Annex A2, section A2.4.
2.3.3 Total capital requirement for premium and reserve risk and natural catastrophe risk
In this section, we calculate the total capital requirement for premium and reserve risks and
the natural catastrophe risk. Capital requirements for these sub-modules taken individually
were calculated in sections 2.3.1 and 2.3.2. The total capital requirement results from both
individual capital requirements and some diversification effects, cf. Annex A2, section A2.1.
Table 2.3.7 provides the resulting capital requirement.
7
Table 2.3.7 Total capital requirement for premium, reserve risk and natcat risk
2.3.4 Capital requirement for counterparty credit default risk from reinsurance
According to Annex A3, LGD (loss given default) for each reinsurance contract is calculated as
í µí°¿í µí°ºí µí°·í µí±– = maxï¿½(1 âˆ’ í µí±…í µí±… ) âˆ™ (í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±– + í µí±…í µí±€í µí±– ); 0ï¿½
where
í µí±…í µí±… = Recovery rate for reinsurance arrangements. According to SCR6.52, for
reinsurance arrangements, í µí±…í µí±… = 50% can be assumed.
í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±– = Best estimate recoverables from the reinsurance contract i. We
assume that í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±– = expected loss ceded to the reinsurance
contract i plus reinsurance share on provisions for outstanding claims.
According to SCR6.17 and SCR6.32, Risk Mitigation í µí±…í µí±€í µí±– can be calculated as
í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±–
í µí±…í µí±€í µí±– = ï¿½í µí±†í µí°¶í µí±…í µí±”í µí±Ÿí µí±œí µí± í µí± âˆ’ í µí±†í µí°¶í µí±…í µí±›í µí±’í µí±¡ ï¿½ âˆ™
í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±¡í µí±œí µí±¡í µí±Ží µí±™
where
í µí±†í µí°¶í µí±…í µí±”í µí±Ÿí µí±œí µí± í µí± = solvency capital requirements for underwriting risk calculated without
reinsurance
í µí±†í µí°¶í µí±…í µí±›í µí±’í µí±¡ = solvency capital requirements for underwriting risk taking reinsurance
into account
Therefore, for calculating LGD for each counterparty and then the capital requirement for
counterparty credit default risk, we first need to calculate the recoverables and risk
mitigation.
The following Table 2.3.8 provides the results for the expected loss ceded to proportional
reinsurance.
8
Table 2.3.8: Expected loss ceded to proportional reinsurance
The next step is to calculate the expected loss ceded to non-proportional reinsurance. This
calculation is carried out in Table 2.3.9. Further, this table (column I) provides the result for
the total expected loss ceded to both proportional and non-proportional reinsurance.
Table 2.3.9: Expected loss ceded to XL reinsurance and to both proportional and XL
reinsurance
As explained above, the reinsurance recoverable consists of the expected ceded loss and
reinsurance share on provisions for outstanding claims. The expected ceded loss has been
calculated in the Tables 2.3.8 and 2.3.9. Table 2.3.10 provides the results for claims
provisions and the results for reinsurance recoverables (column M).
Table 2.3.10: Reinsurance share on provisions for outst. claims and reinsurance recoverables
Apart from reinsurance recoverables, Loss Given Default also includes a Risk Mitigation
component, which is calculated in Table 2.3.11.
9
Table 2.3.11: Risk Mitigation
We further assume that the reinsurance panel is made up of four reinsurance companies,
each taking one fourth of the total reinsurance program and having an A rating. Table 2.3.12
provides the calculation results for the Loss Given Default for each of four reinsurance
companies, please refer to Annex A3 for more details.
Table 2.3.12: Reinsurance recoverables, Risk Mitigation and Loss Given Default per company
Applying the capital requirement calculation methodology explained in Annex A3 to the
results for Loss Given Default provided in Table 2.3.12 results in the capital requirement
provided in Table 2.3.13.
Table 2.3.13: Capital requirement for counterparty credit default risk from reinsurance
2.3.5 Total capital requirement for non-life underwriting risk and credit default risk
In this section, we calculate the total capital requirement for non-life underwriting risk and
counterparty credit default risk from reinsurance. Taken individually, capital requirements
for these sub-modules were calculated in sections 2.3.3 and 2.3.4. The total capital
requirement takes into account both individual capital requirements and partial
diversification effects (see Annex A2, section A2.5). Table 2.3.13 provides the resulting
capital requirement.
Table 2.3.13: Total capital requirement for non-life underwriting risk and credit default risk
2.3.6 Comparison of capital requirements according to Solvency I and Solvency II
In sections 2.2 and 2.3, capital requirements for the chosen sample portfolio under Solvency
I (â‚¬ 7.8m) and Solvency II (â‚¬ 11.0m) were calculated. As can be seen, the Solvency II capital
10
requirement is more than 40% higher than that of Solvency I which might represent a heavy
burden for the companyâ€™s shareholders. One of the ways to deal with the increased capital
requirement is to increase the cession to reinsurance which will reduce the net retained risk
exposure and hence the capital requirements. However, this will also increase the costs of
reinsurance if no restructuring of reinsurance portfolio is undertaken. In the next section 3,
we derive the reinsurance structure which will bring the Solvency II capital requirement to
the level of that for Solvency I, discuss different alternatives for the reinsurance structure
and compare the resulting costs of reinsurance.
3. Reinsurance portfolio for reducing Solvency II capital requirement to Solvency I level
As demonstrated in Section 2, the capital requirement for premium and reserve risk
accounts for the largest â€œportionâ€? of the overall capital requirement. For this reason, our
first solution for reducing the solvency capital requirement is to buy more of proportional
reinsurance. While in the original sample portfolio there was proportional reinsurance for
some lines of business (e.g. Motor, Aviation and Transport (50% cession) and Fire (30%
cession)), now we introduce a proportional 50% cession for all lines of business, cf. Table 3.1.
Table 3.1: Proportional cessions
As far as non-proportional reinsurance is concerned, we first decided not to introduce any
material changes. We still have to adjust the required non-proportional capacities to the
new values of PMLs retained under proportional reinsurance. Table 3.2 provides the
summary of the non-proportional reinsurance.
Table 3.2: Non-proportional reinsurance (Scenario 1)
The reinsurance premium rates in column H were calculated in the same way as for the
original contracts, cf. Table 2.3.1, column C and Annex A4. The rate for the Marine XL is the
same as in the original portfolio because the protected exposure has not changed. For all
other XL contracts, the rates have dropped because the exposure to the layers has
decreased due to the higher proportional cession of 50%.
11
Table 3.3 provides the solvency capital requirement which results from the above
reinsurance structure defined by Table 3.1 and Table 3.2.
Table 3.3: Total capital requirement for non-life underwriting risk and credit default risk
As we see, we have achieved a substantial reduction of solvency capital requirement
compared to the original portfolio (Table 2.3.13), however the level of Solvency I (â‚¬ 7.8m)
has not yet been achieved.
Now, we will adjust the non-proportional reinsurance and see what effect on the capital
requirement such an adjustment will have. We will reduce the retentions of non-
proportional contract to a â‚¬ 250k. The following Table 3.4 summarizes the new non-
proportional reinsurance portfolio.
Table 3.4: Non-proportional reinsurance (Variant 2)
Pricing rates in column H were calculated in the same way as for the original contracts, cf.
Table 2.3.1, column C and Table 3.2, column H and Annex A4.
With these adjusted non-proportional reinsurance contracts, the resulting capital
requirement is as per Table 3.5.
Table 3.5: Total capital requirement for non-life underwriting risk and credit default risk
(Scenario 2)
One can see that although the resulting capital requirement is getting closer to the one of
Solvency I, it is still above it by about â‚¬ 1.1m. The comparison of Table 3.3 and Table 3.5
shows that the capital requirement for the underwriting risk has dropped while the capital
requirement for the counterparty default risk has increased. Both movements can be
explained by the more extensive XL reinsurance covers in Scenario 2 - the net retained
12
underwriting risk exposure has dropped and the exposure to the credit default risk has
increased due to the higher expected loss transferred to reinsurance.
As the next instrument for reducing the capital requirement, we now will examine the effect
of improving the reinsurersâ€™ credit quality i.e. their credit ratings. In the previous
calculations, we assumed that all reinsurance companies in the panel have an A rating. Now
we change this assumption and assume that all reinsurance companies have the rating of an
AA. Table 3.6 provides the parameters of non-proportional portfolio with the AA rating.
Table 3.6: Non-proportional reinsurance (Scenario 2, rating AA)
While the reinsurance premium for proportional reinsurance does not depend on the rating,
the premium for non-proportional reinsurance does. We assumed that a one notch upgrade
in rating, i.e. a change from A to AA, costs 10% more in additional reinsurance premium.
Pricing rates in Table 3.6, column H we obtained by applying a 10% loading on reinsurance
pricing rates listed in Table 3.4, column H.
The resulting capital requirement is provided in Table 3.7.
Table 3.7: Total capital requirement for non-life underwriting risk and credit default risk
(Option 2, rating AA)
The resulting capital requirement is now nearly equal to the one of Solvency I, i.e. the
desired effect has been achieved. The capital requirement relief is equal to â‚¬ 11.0m â€“ 7.8m =
3.2m.
A comparison of Table 3.5 and Table 3.7 shows that the capital requirement for the
underwriting risk has not changed much whereas the capital requirement for the
counterparty credit default risk has dropped substantially. Such a substantial positive effect
on the capital requirement is a result of improved credit quality of the involved reinsurance
companies.
Now we compare the costs of reinsurance for the original reinsurance portfolio (cf. Table
2.1.2 and Table 2.3.1) and the updated reinsurance portfolio (Scenario 2, rating AA) which
13
under Solvency II leads to the same capital requirement as that of the original portfolio
under Solvency I (cf. Table 3.1 and 3.4). Table 3.8 summarizes this comparison.
Table 3.8: Reinsurance premiums, original and updated reinsurance portfolio
As can be seen from the last row of columns C and F in Table 3.8, the total reinsurance
premium of the updated portfolio is about â‚¬ 18 million higher than that of the original
portfolio. The achieved relief of â‚¬ 3.2 million in the capital requirement has been achieved at
the cost of â‚¬ â‚¬28.2m â€“ â‚¬10.4m = â‚¬17.8 million of additional reinsurance premium! At first
glance, it may appear that seeking capital relief through reinsurance, which costs over 500%
more than the capital relief itself, makes no economic sense! However, one can easily
observe that in the case of proportional reinsurance, the reinsurance premium does not
reflect well the actual capital cost of reinsurance protection. It can be shown that if the
expected loss ratio materializes, the real cost (we called it P & L statement costs) of
proportional reinsurance can be calculated as shown below.
â€œP & L Statement Costs of prop RIâ€? =
â€œReinsurance Premiumâ€? * (1 â€“ â€œExpected LRâ€? â€“ â€œRI Commission [%]â€?).
Table 3.9 summarizes the P & L statement costs of proportional reinsurance in the original
and updated portfolio.
Table 3.9: P & L statement costs of proportional reinsurance, original and updated portfolio
In columns H and I of Table 3.9, we present reinsurance commissions (flat) for different lines
of business which were calculated based on the assumed 10% profit margin for the
reinsurance company7 when it has an A rating (original reinsurance, column H) and a 10%
lower commission when it has an AA rating (updated reinsurance structure, column I).
7
Reinsurance company Profit Margin as percentage of the premium = 100% reinsurance premium minus ceded
loss as [%] of reinsurance premium minus reinsurance commission as [%] of reinsurance premium.
14
Now we can compare the overall costs of the original and updated reinsurance portfolio
made of reinsurance premium for non-proportional XL reinsurance and P & L statement
costs of proportional reinsurance, cf. Table 3.10.
Table 3.10: total P & L statement costs of reinsurance, original and updated portfolio
Based on the P & L statement costs of reinsurance, the additional costs for the updated
reinsurance structure calculates to â‚¬ 6.2m â€“ 4.1m = 2.1m â€“ the amount which no longer
exceeds the achieved â‚¬ 3.2 million relief in capital requirement.
To evaluate the efficiency of reinsurance as the solvency capital relief instrument, we now
introduce the relative costs of capital ratio which is calculated as the cost of reinsurance
divided by the achieved capital relief. Obviously, the lower this ratio, the better the
efficiency of the given reinsurance structure as the capital relief instrument. Furthermore,
the straight forward cost of capital approach afforded by this ratio also allows a direct
comparison of reinsurance as a capital relief instrument with capital borrowed from the
capital markets. That is if the rates offered by the capital markets are higher than the
relative cost of capital ratio resulting for a prospected reinsurance structure, then a given
reinsurance structure has a cost advantage over debt instruments as a solvency capital relief
instrument.
In our case, we calculate the relative cost of capital 2.1m / 3.2m = 65.6% (cost of capital
relief divided by the capital relief) which clearly is very high. Most likely, the insurance
company would be able to borrow from the capital markets at a much lower rate.
To increase the attractiveness of reinsurance as a solvency capital relief instrument
compared to borrowing, the cost of reinsurance protection should be reduced substantially.
One conventional way to reduce the costs of reinsurance protection is to replace
conventional proportional contracts with flat reinsurance commissions with the so called
financial quota share (FQS) contracts â€“ a type of proportional contracts which usually has a
broad sliding scale reinsurance commission that depends on the underwriting performance
of the contract. Such contracts are called financial because along with the transfer of
insurance risk they are usually motivated also by financing objectives such as achieving a
solvency capital relief.
The diagram below provides an example of a sliding scale reinsurance commission. In the
loss ratio range of 45%-85% the reinsurance commission is chosen in the way that the total
of the loss ratio and commission is kept constant at 95%. Obviously, the resulting reinsurerâ€™s
margin in this broad loss ratio range is 5%.
15
The cost of reinsurance for a financial quota share contract is equal to the reinsurerâ€™s
margin, in the case of the above diagram 5% of the reinsurance premium. We would like to
comment that typically a 5% margin is rather high, but we chose it at 5% to be conservative;
in the international reinsurance markets the margins of 2-3% are quite common.
We now calculate the costs of reinsurance for the updated portfolio when the conventional
quota share contracts have been replaced with a financial quota share contract for the full
amount of proportional reinsurance, with the reinsurerâ€™s margin of 5%. The financial quota
share contracts do not necessarily need to be taken out for the whole account, as the per-
line-of-business contracts are possible and can be often seen in practice.
Table 3.11 summarizes the P & L statement costs of proportional reinsurance in the original
and updated portfolio.
Table 3.11: P & L statement costs of proportional reinsurance, original and updated portfolio
(with FQS)
We can now compare the overall costs of the original and updated reinsurance portfolio
made of reinsurance premium for non-proportional XL reinsurance and P & L statement
costs of proportional reinsurance, cf. Table 3.12.
Table 3.12: total P & L statement costs of reinsurance, original and updated portfolio (with
FQS)
16
Now, the additional reinsurance costs for the updated reinsurance structure amount to only
â‚¬ 4.4m â€“ 4.1m = 0.3m. With the achieved capital requirement relief of â‚¬ 7.8m we calculate
the relative cost of capital as 0.3/3.2 = 9.4%. Hence, by introducing the financial quota share
reinsurance we have achieved a substantial reduction in the cost of capital. Most likely, the
resulting cost of capital will compare favorably with the borrowing costs for an emerging
market mid-size insurance company. Reinsurance has proven to be a competitive capital
relief instrument!
We must point out however that to make good use of financial quota share contracts for
achieving cost efficient solvency relief, one needs to ensure that such contracts must fulfill
certain requirements to qualify as reinsurance for regulatory purposes and hence achieve
the desired solvency relief effect. In many cases, the ceding company will be required by law
to prove to the market regulator that the contract transfers a substantial amount of risk and
hence can be considered a reinsurance contract. If this cannot be proved, the contract will
be considered not as reinsurance but as a purely financial instrument and therefore no
solvency capital relief effect will be granted. Please refer to the paper â€œInsurance Risk
Transfer and Categorization of Reinsurance Contractsâ€? by the same authors.
4. Conclusions
In this paper, we have compared solvency capital requirements under Solvency I and
Solvency II for a sample mid-size insurance portfolio. We have shown that the capital
requirement under the Solvency II regime is substantially higher than under the Solvency I
regime. Hence, according to our result, changing the solvency capital regime from Solvency I
to Solvency II will lead to a substantial additional solvency capital requirement which might
represent a heavy burden for the companyâ€™s shareholders.
One way to reduce the capital requirement under Solvency II is to increase reinsurance
protection, which will reduce the net retained risk exposure and hence also the solvency
capital requirement. Therefore, we proposed an extended reinsurance structure which
under Solvency II brings the capital requirement back to the level of that required under
Solvency I. In a step-by-step approach we demonstrate the extent of solvency relief attained
by the insurer by applying different possible adjustments in the reinsurance structure:
increasing the cession to proportional reinsurance, reducing the retention of non-
proportional reinsurance, and selecting better rated reinsurers for its reinsurance program.
To evaluate the efficiency of reinsurance as the solvency capital relief instrument, we have
introduced a cost-of-capital based approach which puts the achieved capital relief in relation
to the costs of extending the reinsurance protection. This approach allows a direct
comparison of reinsurance as capital relief instrument with debt instruments available in the
capital market. With the help of the introduced approach, we have shown that the best
capital relief efficiency under all examined reinsurance alternatives is achieved when a
financial quota share contract is chosen for proportional reinsurance. For the sample
insurance portfolio and the proposed reinsurance structure, we achieved the cost of capital
required for the capital relief of less than 10% - a highly competitive rate in comparison with
the cost of corporate borrowing in the capital market.
17
Annex A1
Solvency Capital Requirements according to Solvency I
In order to calculate Solvency Capital Requirements, first one needs to calculate the
Retention Ratio
Retention Ratio = ( + retained loss report yr.
+ retained loss 1st yr. before report yr.
+ retained loss 2nd yr. before report yr. ) /
( + gross loss report yr.
+ gross loss 1st yr. before report yr.
+ gross loss 2nd yr. before report yr. ).
Then the Premium Index and the Claims Index can be calculated. The Premium Index is
calculated as follows
Premium Index =
Retention Ratio * ( + 0.18 * min (adjusted gross premium for report yr., â‚¬ 57.5 M)
+ 0.16 * max (adjusted gross premium for report yr. â€“ â‚¬ 57.5 M, 0) ),
The expression in brackets in the above formula for the Premium Index is equal to the
adjusted gross premium for the report year up to the limit of â‚¬ 57.4 M multiplied by 0.18
plus the premium in excess of â‚¬ 57.5 M, if any, multiplied by 0.16. Adjusted gross premium
is equal to the gross premium plus 50% of the premium from the lines Aviation Liability,
Water Transport Liability and General Liability (if any).
Claims Index is calculated as follows
Claims Index =
Retention Ratio * ( + 0.26 * min (mean gross loss of last three yrs., â‚¬ 40.3 M)
+ 0.23 * max (mean gross loss of last three yrs. â€“ â‚¬ 40.3 M, 0) ),
where mean gross loss of last three years is calculated as
mean gross loss of last three yrs =
( + adjusted gross loss report yr.
+ adjusted gross loss 1st yr. before report yr.
+ adjusted gross loss 2nd yr. before report yr. ) / 3
The expression in brackets in the above formula for the Claims Index is equal to the mean
gross loss of last three years up to the limit of â‚¬ 40.3 M multiplied by 0.26 plus the mean
gross loss of last three years in excess of â‚¬ 40.3 M if any multiplied by 0.23. Adjusted gross
loss is equal to the gross loss plus 50% of the gross loss from the lines Aviation Liability,
Water Transport Liability and General Liability (if any).
18
Finally, the Solvency Capital Requirement is calculated as maximum of both Premium Index
and Claims Index
Solvency Capital Requirement = max (Premium Index, Claims Index)
If the resulting Solvency Capital Requirement for the report year is lower than for the
previous year, the following further calculation needs to be carried out.
Loss reserve ratio =
max (1, loss reserves at the end of report yr. / loss reserves at the beginning of report yr.)
Solvency Capital Requirement = max (max (Premium Index, Claims Index),
Loss reserve ratio * Solvency Capital Requirement previous yr.)
19
Annex A2
Solvency Capital Requirements for non-life underwriting risk under Solvency
II (QIS5) standard formula
This Annex describes the calculation of solvency capital requirements for non-life
underwriting risk according to Solvency II (QIS5) standard formula, as defined by EU
document â€œQIS5 Technical Specificationâ€? (Brussels, July 2010).
The Annex is organized as follows. Section A2.1 describes the overall approach for
calculating the solvency capital requirement for non-life underwriting risk based on the
results of three sub-modules: non-life premium and reserve risk, non-life lapse risk and non-
life catastrophe risk sub-module. Sections A2.2 and A2.4 then provide a detailed explanation
of the capital requirement calculation for the sub-modules non-life premium and reserve risk
(Section A2.3) and non-life catastrophe risk (Section A2.4). Section A2.3 provides an
explanation on the objective and motivation behind the non-life lapse risk sub-module
However, the section provides no calculation details as we assume this risk category does
not materialize for the sample portfolio under consideration. Finally Section A2.4 explains
the aggregation of the capital requirements for the non-life underwriting risk module and
the counterparty default risk module (cf. Annex 3). This aggregation takes into account some
correlation assumptions between these two risk categories.
A2.1 Calculation of the overall capital requirement for non-life underwriting risk
The non-life underwriting risk module consists of the following sub-modules:
â€¢ The non-life premium and reserve risk sub-module
â€¢ The non-life lapse risk sub-module
â€¢ The non-life catastrophe risk sub-module
Capital requirements for each module are denoted by
í µí±?í µí°¿í µí±?í µí±Ÿ = Capital requirement for non-life premium and reserve risk
í µí±?í µí°¿í µí±™í µí±Ží µí±?í µí± í µí±’ = Capital requirement for non-life lapse risk
í µí±?í µí°¿í µí°¶í µí°´í µí±‡ = Capital requirement for non-life catastrophe risk
The resulting solvency Capital requirement for non-life underwriting risk is denoted by
í µí±†í µí°¶í µí±…í µí±›í µí±™ = Capital requirement for non-life underwriting risk
It calculates by combining the capital requirements for the non-life sub-risks using a
correlation matrix as follows
í µí±†í µí°¶í µí±…í µí±›í µí±™ = ï¿½ï¿½ í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí±?í µí°¿í µí±Ÿ,í µí±? âˆ™ í µí±?í µí°¿í µí±Ÿ âˆ™ í µí±?í µí°¿í µí±?
where
í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí±?í µí°¿í µí±Ÿ,í µí±? = The entries of the correlation matrix í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí±?í µí°¿
20
í µí±?í µí°¿í µí±Ÿ , í µí±?í µí°¿í µí±? = Capital requirement for individual non-life underwriting sub-risks
according to the rows and columns of correlation matrix í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí±?í µí°¿
and where the correlation matrix í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí±?í µí°¿ is defined by
With this correlation matrix, the overall solvency capital requirement for non-life
underwriting risk calculates to
2 2
í µí±†í µí°¶í µí±…í µí±›í µí±™ = ï¿½í µí±?í µí°¿2
í µí±?í µí±Ÿ + í µí±?í µí°¿í µí±™í µí±Ží µí±?í µí± í µí±’ + í µí±?í µí°¿í µí°¶í µí°´í µí±‡ + 0.25í µí±?í µí°¿í µí±?í µí±Ÿ í µí±?í µí°¿í µí°¶í µí°´í µí±‡
A2.2 Calculation of the capital requirement for non-life premium and reserve risk
This module considers two main sources of underwriting risk, premium risk and reserve risk.
Premium risk results from fluctuations in the timing, frequency and severity of insured
events. Premium risk entails the risk that premium provisions turn out to be insufficient to
cover claims in full or need to be increased. Reserve risk results from fluctuations in the
timing and amount of claim settlements.
In order to carry out the non-life premium and reserve risk calculation, the following
parameters need to be determined
í µí±ƒí µí°¶í µí±‚í µí±™í µí±œí µí±? = Provisions for claims outstanding (best estimate) for each LoB net of
recoverables from reinsurance and special purpose vehicles
í µí±¡ ,í µí±¤í µí±Ÿí µí±–í µí±¡í µí±¡í µí±’í µí±›
í µí±ƒí µí±™í µí±œí µí±? = Estimate for net written premium for each LoB during the forthcoming
year
í µí±¡ ,í µí±’í µí±Ží µí±Ÿí µí±›í µí±’í µí±‘
í µí±ƒí µí±™í µí±œí µí±? = Estimate for net earned premium for each LoB during the forthcoming
year
í µí±¡âˆ’1,í µí±¤í µí±Ÿí µí±–í µí±¡í µí±¡í µí±’í µí±›
í µí±ƒí µí±™í µí±œí µí±? = Net written premium for each LoB during the previous year
í µí±ƒí µí±ƒ
í µí±ƒí µí±™í µí±œí µí±? = Present value of net premiums of existing contracts which are
expected to be earned after the following year for each LoB
í µí±ƒí µí±ƒ
The term í µí±ƒí µí±™í µí±œí µí±? is only relevant for contracts with a coverage period that exceeds the
í µí±ƒí µí±ƒ
following year. For annual contracts without renewal options í µí±ƒí µí±™í µí±œí µí±? is zero. Undertakings may
í µí±ƒí µí±ƒ í µí±¡ ,í µí±’í µí±Ží µí±Ÿí µí±›í µí±’í µí±‘
not calculate í µí±ƒí µí±™í µí±œí µí±? where it is likely not to be material compared to í µí±ƒ í µí±™í µí±œí µí±? .
The capital requirement for premium and reserve risk is determined as follows:
í µí±?í µí°¿í µí±?í µí±Ÿ = í µí¼Œ(í µí¼Ž) âˆ™ í µí±‰
where
21
í µí±‰ = Volume measure
í µí¼Ž = Combined standard deviation
í µí¼Œ(í µí¼Ž) = A function of the combined standard deviation
The function í µí¼Œ(í µí¼Ž) is specified as follows:
exp(í µí±?0.995 âˆ™ ï¿½log(í µí¼Ž 2 + 1))
í µí¼Œ(í µí¼Ž) = âˆ’1
âˆší µí¼Ž 2 + 1
where
í µí±?0.995 = 99.5% quantile of the standard normal distribution (â‰ˆ2.576)
The function í µí¼Œ(í µí¼Ž) is set in such a way that, assuming a lognormal distribution of the
underlying risk, a risk capital requirement consistent with the VaR 99.5% calibration
objective is produced. Roughly, í µí¼Œ(í µí¼Ž) â‰ˆ 3 âˆ™ í µí¼Ž.
The volume measure V and the combined standard deviation Ïƒ for the overall non-life
insurance portfolio are determined in two steps as follows:
â€¢ For each individual LoB, the standard deviations and volume measures for both
premium risk and reserve risk are determined;
â€¢ The standard deviations and volume measures for the premium risk and the reserve
risk in the individual LoBs are aggregated to derive an overall volume measure V and
a combined standard deviation Ïƒ.
The calculations needed to perform these two steps are set out below.
Step 1: Volume measures and standard deviations per LoB
The following numbering of LoBs applies for the calculation:
For each LoB, the volume measures and standard deviations for premium and reserve risk
are denoted as follows:
22
í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š,í µí±™í µí±œí µí±?) = The volume measure for premium risk
í µí±‰(í µí±Ÿí µí±’í µí± ,í µí±™í µí±œí µí±?) = The volume measure for reserve risk
í µí¼Ž(í µí±?í µí±Ÿí µí±’í µí±š,í µí±™í µí±œí µí±?) = Standard deviation for premium risk
í µí¼Ž(í µí±Ÿí µí±’í µí± ,í µí±™í µí±œí µí±?) = Standard deviation for reserve risk
The volume measure for premium risk in the individual LoB is determined as follows:
í µí±¡ ,í µí±¤í µí±Ÿí µí±–í µí±¡í µí±¡í µí±’í µí±› í µí±¡,í µí±’í µí±Ží µí±Ÿí µí±›í µí±’í µí±‘ í µí±¡âˆ’1,í µí±¤í µí±Ÿí µí±–í µí±¡í µí±¡í µí±’í µí±› í µí±ƒí µí±ƒ
í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š,í µí±™í µí±œí µí±?) = maxï¿½í µí±ƒ
í µí±™í µí±œí µí±? ; í µí±ƒ
í µí±™í µí±œí µí±? ; í µí±ƒ
í µí±™í µí±œí µí±? ï¿½ + í µí±ƒí µí±™í µí±œí µí±?
If the undertaking has committed to its regulator that it will restrict premiums written over
the period so that the actual premiums written (or earned) over the period will not exceed
its estimated volumes, the volume measure is determined only with respect to estimated
premium volumes, so that in this case:
í µí±¡ ,í µí±¤í µí±Ÿí µí±–í µí±¡í µí±¡í µí±’í µí±› í µí±¡ ,í µí±’í µí±Ží µí±Ÿí µí±›í µí±’í µí±‘ í µí±ƒí µí±ƒ
í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š,í µí±™í µí±œí µí±?) = maxï¿½í µí±ƒ
í µí±™í µí±œí µí±? ; í µí±ƒ
í µí±™í µí±œí µí±? ï¿½ + í µí±ƒí µí±™í µí±œí µí±?
The market-wide estimates of the net standard deviation for premium risk for each line of
business are:
The adjustment factor for non-proportional reinsurance í µí±?í µí±ƒí µí±™í µí±œí µí±? of a line of business allows
undertakings to take into account the risk-mitigating effect of particular per risk excess of
loss reinsurance. Undertakings may choose for each line of business to set the adjustment
factor to 1 or to calculate it according to the following algorithm.
Box: Calculation of í µí±?í µí±ƒí µí±™í µí±œí µí±? factors
The adjustment factor for non-proportional reinsurance should only be calculated in relation to per risk excess
of loss reinsurance which complies with the following conditions:
â€¢ it covers all insurance claims that the insurance or reinsurance undertaking may incur in the segment
during the following year;
â€¢ it allows for reinstatement;
23
â€¢ it meets the requirements for risk mitigation techniques, set out in subsection SCR.13 of the defined
by EU document â€œQIS5 Technical Specificationâ€? (Brussels, July 2010).
1 + (Î©í µí±›í µí±’í µí±¡ í µí±›í µí±’í µí±¡ 2
í µí±™í µí±œí µí±? âˆ• í µí±€í µí±™í µí±œí µí±? )
í µí±?í µí±ƒí µí±™í µí±œí µí±? = ï¿½ í µí±”í µí±Ÿí µí±œí µí± í µí± í µí±”í µí±Ÿí µí±œí µí± í µí± 2
1 + (Î©í µí±™í µí±œí µí±? âˆ• í µí±€í µí±™í µí±œí µí±? )
where
í µí±›í µí±’í µí±¡ í µí±”í µí±Ÿí µí±œí µí± í µí±
í µí±€í µí±™í µí±œí µí±? = í µí±€í µí±™í µí±œí µí±? â‹… ï¿½1 âˆ’ í µí°¹í µí±š+í µí¼Ž2,í µí¼Ž (í µí±Ž + í µí±?) + í µí°¹í µí±š+í µí¼Ž2,í µí¼Ž (í µí±Ž)ï¿½ + í µí±Ž âˆ™ ï¿½í µí°¹í µí±š,í µí¼Ž (í µí±Ž + í µí±?) âˆ’ í µí°¹í µí±š,í µí¼Ž (í µí±Ž)ï¿½ âˆ’ í µí±? âˆ™ ï¿½1 âˆ’ í µí°¹í µí±š,í µí¼Ž (í µí±Ž + í µí±?)ï¿½
í µí±”í µí±Ÿí µí±œí µí± í µí± 2 í µí±”í µí±Ÿí µí±œí µí± í µí± 2
Î©í µí±›í µí±’í µí±¡
í µí±™í µí±œí µí±? = ï¿½ï¿½ï¿½Î©í µí±™í µí±œí µí±? ï¿½ + ï¿½í µí±€í µí±™í µí±œí µí±? ï¿½ ï¿½ âˆ™ ï¿½1 âˆ’ í µí°¹í µí±š+2í µí¼Ž2,í µí¼Ž (í µí±Ž + í µí±?) + í µí°¹í µí±š+2í µí¼Ž2,í µí¼Ž (í µí±Ž)ï¿½ + í µí±Ž2 âˆ™ ï¿½í µí°¹í µí±š,í µí¼Ž (í µí±Ž + í µí±?) âˆ’ í µí°¹í µí±š,í µí¼Ž (í µí±Ž )ï¿½
1/ 2
í µí±”í µí±Ÿí µí±œí µí± í µí± í µí±›í µí±’í µí±¡ 2
âˆ’ 2í µí±? âˆ™ í µí±€í µí±™í µí±œí µí±? âˆ™ ï¿½1 âˆ’ í µí°¹í µí±š+í µí¼Ž2,í µí¼Ž (í µí±Ž + í µí±?)ï¿½ + í µí±? 2 âˆ™ ï¿½1 âˆ’ í µí°¹í µí±š,í µí¼Ž (í µí±Ž + í µí±?)ï¿½ âˆ’ (í µí±€í µí±™í µí±œí µí±? ) ï¿½
í µí±”í µí±Ÿí µí±œí µí± í µí± 2
Î©í µí±™í µí±œí µí±?
í µí¼Ž = ï¿½ln ï¿½1 + ï¿½ í µí±”í µí±Ÿí µí±œí µí± í µí± ï¿½ ï¿½
í µí±€í µí±™í µí±œí µí±?
í µí±”í µí±Ÿí µí±œí µí± í µí± í µí¼Ž 2
í µí±š = ln í µí±€í µí±™í µí±œí µí±? âˆ’
2
í µí±”í µí±Ÿí µí±œí µí± í µí±
ï¿½í µí±™í µí±œí µí±?
í µí±”í µí±Ÿí µí±œí µí± í µí± í µí±€ if í µí±† â‰¥ 1
í µí±€í µí±™í µí±œí µí±? =ï¿½ í µí±”í µí±Ÿí µí±œí µí± í µí±
í µí±† âˆ™ í µí±€ï¿½í µí±™í µí±œí µí±? otherwise
í µí±”í µí±Ÿí µí±œí µí± í µí±
ï¿½ í µí±™í µí±œí µí±?
í µí±”í µí±Ÿí µí±œí µí± í µí± Î© if í µí±† â‰¥ 1
Î©í µí±™í µí±œí µí±? =ï¿½ í µí±”í µí±Ÿí µí±œí µí± í µí±
í µí±† âˆ™ Î©ï¿½ í µí±™í µí±œí µí±? otherwise
í µí±› âˆ™ í µí¼Ž(2 2
í µí±?í µí±Ÿí µí±’í µí±š ,í µí±”í µí±Ÿí µí±œí µí± í µí± ,í µí±™í µí±œí µí±? ) âˆ™ í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š ,í µí±”í µí±Ÿí µí±œí µí± í µí± ,í µí±™í µí±œí µí±? )
í µí±† = ï¿½ í µí±”í µí±Ÿí µí±œí µí± í µí± 2 í µí±”í µí±Ÿí µí±œí µí± í µí± 2
ï¿½
í µí±? âˆ™ ï¿½ï¿½Î© ï¿½
í µí±™í µí±œí µí±? ï¿½ + ï¿½í µí±€í µí±™í µí±œí µí±? ï¿½ ï¿½
The terms used in these formulas are defined as follows
í µí±”í µí±Ÿí µí±œí µí± í µí±
ï¿½í µí±™í µí±œí µí±?
í µí±€ = Average cost per claim gross of reinsurance per LoB estimated from the claims of the
last í µí±› years where í µí±› â‰¥ 1
í µí±”í µí±Ÿí µí±œí µí± í µí±
ï¿½ í µí±™í µí±œí µí±?
Î© = Standard deviation of the cost per claim gross of reinsurance per LoB estimated with
the standard estimator from the claims of the last í µí±› years where í µí±› â‰¥ 1
í µí±Ž = Retention of non-proportional reinsurance contract
í µí±? = Limit of non-proportional reinsurance contract
í µí°¹í µí±š,í µí¼Ž = Distribution function of a Lognormal random variable with parameters (í µí±š, í µí¼Ž)
í µí°¹í µí±š+í µí¼Ž2,í µí¼Ž = Distribution function of a Lognormal random variable with parameters (í µí±š + í µí¼Ž 2 , í µí¼Ž)
í µí°¹í µí±š+2í µí¼Ž2,í µí¼Ž = Distribution function of a Lognormal random variable with parameters (í µí±š + 2í µí¼Ž 2 , í µí¼Ž)
ï¿½í µí±™í µí±œí µí±? í µí±”í µí±Ÿí µí±œí µí± í µí± í µí±”í µí±Ÿí µí±œí µí± í µí±
ï¿½ í µí±™í µí±œí µí±?
í µí±› = Number of years used in the estimation of í µí±€ and Î©
í µí±? = Number of claims during the last í µí±› years
24
í µí¼Ž(í µí±?í µí±Ÿí µí±’í µí±š,í µí±”í µí±Ÿí µí±œí µí± í µí± ,í µí±™í µí±œí µí±? ) = Standard deviation for premium risk gross of reinsurance, calculated by
putting the adjustment factor í µí±?í µí±ƒí µí±™í µí±œí µí±? to 1.
í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š,í µí±”í µí±Ÿí µí±œí µí± í µí± ,í µí±™í µí±œí µí±? ) = Volume measure for premium risk gross of reinsurance, calculated in the
same way as the usual volume measure but based on gross premiums
instead of net premiums
Where the excess of loss reinsurance contract has no limit the adjustment factor for non-proportional
reinsurance of a line of business shall be calculated in the same way as set out above, but with the following
changes:
í µí±›í µí±’í µí±¡ í µí±”í µí±Ÿí µí±œí µí± í µí±
í µí±€í µí±™í µí±œí µí±? = í µí±€í µí±™í µí±œí µí±? â‹… í µí°¹í µí±š+í µí¼Ž2,í µí¼Ž (í µí±Ž) + í µí±Ž âˆ™ ï¿½1 âˆ’ í µí°¹í µí±š,í µí¼Ž (í µí±Ž)ï¿½
1/ 2
í µí±”í µí±Ÿí µí±œí µí± í µí± 2 í µí±”í µí±Ÿí µí±œí µí± í µí± 2
Î©í µí±›í µí±’í µí±¡
í µí±™í µí±œí µí±? = ï¿½ï¿½ï¿½Î©í µí±™í µí±œí µí±? ï¿½ + ï¿½í µí±€í µí±™í µí±œí µí±?
í µí±›í µí±’í µí±¡ 2
ï¿½ ï¿½ âˆ™ í µí°¹í µí±š+2í µí¼Ž2,í µí¼Ž + í µí±Ž2 âˆ™ ï¿½1 âˆ’ í µí°¹í µí±š,í µí¼Ž (í µí±Ž)ï¿½ âˆ’ (í µí±€í µí±™í µí±œí µí±? ) ï¿½
End Box
The volume measure for reserve risk for each individual LoB is determined as follows:
í µí±‰(í µí±Ÿí µí±’í µí± ,í µí±™í µí±œí µí±?) = í µí±ƒí µí°¶í µí±‚í µí±™í µí±œí µí±?
The market-wide estimates of the net of reinsurance standard deviation for reserve risk for
each line of business are:
The standard deviation for premium and reserve risk in the individual LoB is defined by
aggregating the standard deviations for both sub-risks under the assumption of a correlation
coefficient of í µí»¼ = 0.5:
2 2
ï¿½ï¿½í µí¼Ž(í µí±?í µí±Ÿí µí±’í µí±š,í µí±™í µí±œí µí±?) í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š,í µí±™í µí±œí µí±?) ï¿½ + 2í µí»¼í µí¼Ž(í µí±?í µí±Ÿí µí±’í µí±š,í µí±™í µí±œí µí±? ) í µí¼Ž(í µí±Ÿí µí±’í µí± ,í µí±™í µí±œí µí±?) í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š,í µí±™í µí±œí µí±?) í µí±‰(í µí±Ÿí µí±’í µí± ,í µí±™í µí±œí µí±?) + ï¿½í µí¼Ž(í µí±Ÿí µí±’í µí± ,í µí±™í µí±œí µí±?) í µí±‰(í µí±Ÿí µí±’í µí± ,í µí±™í µí±œí µí±? ) ï¿½
í µí¼Ží µí±™í µí±œí µí±? =
í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š,í µí±™í µí±œí µí±?) + í µí±‰(í µí±Ÿí µí±’í µí± ,í µí±™í µí±œí µí±?)
25
The overall volume measure for each LoB, í µí±‰í µí±™í µí±œí µí±? is obtained as follows:
í µí±‰í µí±™í µí±œí µí±? = (í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š,í µí±™í µí±œí µí±?) + í µí±‰(í µí±Ÿí µí±’í µí± ,í µí±™í µí±œí µí±?) ) âˆ™ (0.75 + 0.25 âˆ™ í µí°·í µí°¼í µí±‰í µí±™í µí±œí µí±? )
where
2
âˆ‘í µí±—ï¿½í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š,í µí±—,í µí±™í µí±œí µí±?) + í µí±‰(í µí±Ÿí µí±’í µí± ,í µí±—,í µí±™í µí±œí µí±?) ï¿½
í µí°·í µí°¼í µí±‰í µí±™í µí±œí µí±? = 2
ï¿½âˆ‘í µí±—ï¿½í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š,í µí±—,í µí±™í µí±œí µí±?) + í µí±‰(í µí±Ÿí µí±’í µí± ,í µí±—,í µí±™í µí±œí µí±?) ï¿½ï¿½
and where the index j denotes the geographical segments as set out in Annex M to the
Technical Specification and í µí±‰(í µí±?í µí±Ÿí µí±’í µí±š,í µí±—,í µí±™í µí±œí µí±?) and í µí±‰(í µí±Ÿí µí±’í µí± ,í µí±—,í µí±™í µí±œí µí±?) denote the volume measures as
defined above but restricted to the geographical segment j.
However, the factor í µí°·í µí°¼í µí±‰í µí±™í µí±œí µí±? should be set to 1 for the line of business credit and suretyship
and where the standard deviation for premium or reserve risk of the line of business is an
undertaking-specific parameter.
For an undertaking offering insurance in one geographic segment only, the factor is equal to
1.
Undertakings active in more than one geographic segment may choose to allocate all of their
business in a line of business to the main geographical segment in order to simplify the
calculation.
Step 2: Overall volume measures and standard deviations
The overall volume measure í µí±‰ for all LoBs is determined as the total of the volume
measures for each LoB:
í µí±‰ = ï¿½ í µí±‰í µí±™í µí±œí µí±?
í µí±Ží µí±™í µí±™ í µí°¿í µí±œí µí°µí µí±
The overall standard deviation í µí¼Ž for all LoBs is determined as follows:
1
í µí±Ÿ âˆ™ í µí±‰
í µí¼Ž = ï¿½ 2 âˆ™ ï¿½ í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí°¿í µí±œí µí°µí µí±Ÿ,í µí±? âˆ™ í µí¼Ží µí±Ÿ âˆ™ í µí¼Ží µí°¶ âˆ™ í µí±‰ í µí±?
í µí±‰
í µí±Ÿ ,í µí±?
where
í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí°¿í µí±œí µí°µí µí±Ÿ,í µí±? = The entries of the correlation matrix í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí°¿í µí±œí µí°µ
í µí¼Ží µí±Ÿ , í µí¼Ží µí±? = Standard deviations for individual line of business according to the
rows and columns of correlation matrix í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí°¿í µí±œí µí°µ
í µí±? =
í µí±Ÿ , í µí±‰
í µí±‰ Volume measures for individual line of business according to the rows
and columns of correlation matrix í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí°¿í µí±œí µí°µ
26
The correlation matrix í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí°¿í µí±œí µí°µ is defined as follows
A2.3 Non-life lapse risk
Non-life insurance contracts can include policyholder options which significantly influence
the obligations arising from them. Examples for such options are options to terminate a
contract before the end of the previously agreed insurance period and options to renew
contracts according to previously agreed conditions. Where such policyholder options are
included in a non-life insurance contract, the calculation of premium provisions is based on
assumptions about the exercise rates of these options. Lapse risk is the risk that these
assumptions turn out to be wrong or need to be changed.
Where non-life insurance contracts do not include policyholder options or where the
assumptions about the exercise rate of such options have no material influence on premium
provisions, the contracts do not need to be included in the calculations of the lapse risk sub-
module. Where this is the case for the whole portfolio of an undertaking (except for a non-
material part) the three components of the sub-module can be set to zero. In this Annex, we
assume that this is the case and do not provide calculation details for the non-life lapse risk.
A2.4 Calculation of the capital requirement for non-life catastrophe risk
Under the non-life underwriting risk module, catastrophe risk is defined in the Solvency II
Framework Directive (Directive 2009/138/EC) as: â€œthe risk of loss, or of adverse change in
the value of insurance liabilities, resulting from significant uncertainty of pricing and
provisioning assumptions related to extreme or exceptional events.â€?
CAT risks stem from extreme or irregular events that are not sufficiently captured by the
capital requirements for premium and reserve risk. The catastrophe risk capital requirement
has to be calibrated at the 99.5% VaR (annual view).
27
The CAT risk sub-module under the standard formula should be calculated using one of the
following alternative methods (or as a combination of both): Standardized Scenarios Method
(method 1) or Factor Based Method (method 2) whereas the method 1 should be chosen for
all exposures where possible. As usually this will be the case, in this Annex we do not provide
any details on the method 2.
The CAT risk sub-module differentiates between natural catastrophes and man-made
catastrophes. Perils considered under natural catastrophes are: Windstorm, Flood,
Earthquake, Hail, Subsidence. Man-made catastrophes are extreme or exceptional events
arising from Motor, Fire, Marine, Aviation, Liability, Credit & Suretyship, Terrorism.
The capital requirement for non-life catastrophe risk is calculated as follows:
2 2
í µí±?í µí°¿_í µí°¶í µí°´í µí±‡ = ï¿½ï¿½í µí±?í µí°¿_í µí°¶í µí°´í µí±‡í µí±?í µí±Ží µí±¡_í µí±?í µí±Ží µí±¡ ï¿½ + ï¿½í µí±?í µí°¿_í µí°¶í µí°´í µí±‡í µí±€í µí±Ží µí±›_í µí±ší µí±Ží µí±‘í µí±’ ï¿½
where
í µí±?í µí°¿_í µí°¶í µí°´í µí±‡í µí±?í µí±Ží µí±¡_í µí±?í µí±Ží µí±¡ = Catastrophe capital requirement for natural catastrophes net of
risk mitigation
í µí±?í µí°¿_í µí°¶í µí°´í µí±‡í µí±€í µí±Ží µí±›_í µí±ší µí±Ží µí±‘í µí±’ = Catastrophe capital requirement for man-made net of risk
mitigation
In the following, we assume that the man-made catastrophe exposure is remote and do not
provide any further details on this sub-module.
For calculating í µí±?í µí°¿_í µí°¶í µí°´í µí±‡í µí±?í µí±Ží µí±¡_í µí±?í µí±Ží µí±¡ , firstly natural catastrophe capital requirements at country
level should be aggregated to estimate the capital requirement at peril level:
í µí°¶í µí°´í µí±‡í µí±?í µí±’í µí±Ÿí µí±–í µí±™ = ï¿½ï¿½ í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí±?í µí±¡í µí±Ÿí µí±¦,í µí±–,í µí±— âˆ™ í µí°¶í µí°´í µí±‡í µí±?í µí±’í µí±Ÿí µí±–í µí±™_í µí±?í µí±¡í µí±Ÿí µí±¦,í µí±›í µí±’í µí±¡,í µí±– âˆ™ í µí°¶í µí°´í µí±‡í µí±?í µí±’í µí±Ÿí µí±–í µí±™_í µí±?í µí±¡í µí±Ÿí µí±¦,í µí±›í µí±’í µí±¡,í µí±—
í µí±–,í µí±—
where
í µí°¶í µí°´í µí±‡í µí±?í µí±’í µí±Ÿí µí±–í µí±™ = Catastrophe capital requirement for each peril type = Windstorm,
Earthquake, Flood, Hail and Subsidence
í µí°¶í µí°´í µí±‡í µí±?í µí±’í µí±Ÿí µí±–í µí±™_í µí±?í µí±¡í µí±Ÿí µí±¦,í µí±›í µí±’í µí±¡,í µí±–,í µí±— = Catastrophe capital requirement for each peril type by country. Where
there are separate reinsurance programs for each country the
aggregations (across countries) are done net of reinsurance.
í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí±?í µí±¡í µí±Ÿí µí±¦,í µí±–,í µí±— = The entries of the correlation matrix between countries i, j.
Parameters of the correlation matrix are provided in the â€œTechnical Specificationâ€?. As
usually, insurance undertaking will be operating in one country only, we do not provide this
lengthy correlation matrix in this Annex.
Then, capital requirements at peril level should be aggregated to estimate the catastrophe
capital requirements at total level:
28
í µí±?í µí°¿_í µí°¶í µí°´í µí±‡í µí±?í µí±Ží µí±¡_í µí±?í µí±Ží µí±¡ = ï¿½ï¿½ í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí±?í µí±’í µí±Ÿí µí±–í µí±™,í µí±–,í µí±— âˆ™ í µí°¶í µí°´í µí±‡í µí±?í µí±’í µí±Ÿí µí±–í µí±™,í µí±›í µí±’í µí±¡,í µí±– âˆ™ í µí°¶í µí°´í µí±‡í µí±?í µí±’í µí±Ÿí µí±–í µí±™,í µí±›í µí±’í µí±¡,í µí±—
í µí±–,í µí±—
where
í µí±?í µí°¿_í µí°¶í µí°´í µí±‡í µí±?í µí±Ží µí±¡_í µí±?í µí±Ží µí±¡ = Catastrophe capital requirement for non-life net of risk mitigation
í µí°¶í µí°´í µí±‡í µí±?í µí±’í µí±Ÿí µí±–í µí±™,í µí±›í µí±’í µí±¡,í µí±–,í µí±— = Catastrophe capital requirement for each peril. Where there are
separate reinsurance programs for each country the aggregations
(across countries) are done net of reinsurance.
í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí±?í µí±’í µí±Ÿí µí±–í µí±™,í µí±–,í µí±— = The entries of the correlation matrix between perils i, j.
The peril correlation matrix í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí±?í µí±’í µí±Ÿí µí±–í µí±™ is as follows
Notice that among all perils, some correlation is assumed for the pairs (windstorm; flood)
and (windstorm; hail) only.
The capital requirements per peril, country í µí°¶í µí°´í µí±‡í µí±?í µí±’í µí±Ÿí µí±–í µí±™_í µí±?í µí±¡í µí±Ÿí µí±¦ gross of reinsurance are obtained
from the sum insured aggregates per cresta zone, relativity factor for each zone and the
zone aggregation matrix per country. Then the recoverables of reinsurance contract are
taken into account. This provides the capital requirements per peril and country net of
reinsurance í µí°¶í µí°´í µí±‡í µí±?í µí±’í µí±Ÿí µí±–í µí±™_í µí±?í µí±¡í µí±Ÿí µí±¦,í µí±›í µí±’í µí±¡ .
For example, for earthquake, the calculation is as follows:
í µí°¶í µí°´í µí±‡í µí°¸í µí±„_í µí±?í µí±¡í µí±Ÿí µí±¦ = í µí±„í µí°¶í µí±‡í µí±…í µí±Œ ï¿½ï¿½ í µí°´í µí°ºí µí°ºí µí±Ÿ .í µí±? âˆ™ í µí±Ší µí±‡í µí°¼í µí±‰í µí±?í µí±‚í µí±?í µí°¸,í µí±Ÿ âˆ™ í µí±Ší µí±‡í µí°¼í µí±‰í µí±?í µí±‚í µí±?í µí°¸,í µí±?
í µí±Ÿ,í µí±?
where
í µí±Ší µí±‡í µí°¼í µí±‰í µí±?í µí±‚í µí±?í µí°¸ = í µí°¹í µí±?í µí±‚í µí±?í µí°¸ âˆ™ í µí±‡í µí°¼í µí±‰í µí±?í µí±‚í µí±?í µí°¸
í µí±‡í µí°¼í µí±‰í µí±?í µí±‚í µí±?í µí°¸ = í µí±‡í µí°¼í µí±‰í µí±?í µí±‚í µí±?í µí°¸_í µí°¹í µí±–í µí±Ÿí µí±’ + í µí±‡í µí°¼í µí±‰í µí±?í µí±‚í µí±?í µí°¸_í µí±€í µí°´í µí±‡
í µí±‡í µí°¼í µí±‰í µí±?í µí±‚í µí±?í µí°¸_í µí°¹í µí±–í µí±Ÿí µí±’ = Total insured value for Fire and other damage by zone
í µí±‡í µí°¼í µí±‰í µí±?í µí±‚í µí±?í µí°¸_í µí±€í µí°´í µí±‡ = Total insured value for Marine by zone
í µí±„í µí°¶í µí±‡í µí±…í µí±Œ = 1 in 200 year factor for each country. These factors for different perils,
countries are provided in Annex L5 to the â€œTechnical Specificationâ€?
29
í µí°´í µí°ºí µí°ºí µí±Ÿ,í µí±? = Rows and columns of aggregation matrix by country. These values are
provided in the excel spreadsheet â€œparameters for non-life
catastropheâ€? available from EIOPA website (European Insurance and
Occupational Pensions Authority) under the QIS5 helper tabs
í µí°¹í µí±?í µí±‚í µí±?í µí°¸ = Relativity factors for each zone by country. These values are provided
in the excel spreadsheet â€œparameters for non-life catastropheâ€?
available from EIOPA website (European Insurance and Occupational
Pensions Authority) under the QIS5 helper tabs
í µí±Ší µí±‡í µí°¼í µí±‰í µí±?í µí±‚í µí±?í µí°¸ = Geographically weighted total insured value by zone
Finally through taking into account risk mitigating effect of reinsurance, the capital
requirement for earthquake net of reinsurance
í µí°¶í µí°´í µí±‡í µí°¸í µí±„_í µí±?í µí±¡í µí±Ÿí µí±¦,í µí±›í µí±’í µí±¡
can be calculated.
Capital requirements for other perils can be calculated analogously, for more details consult
â€œTechnical Specificationâ€?.
Undertakings should note that netting down would not always be possible at a country level.
For example you may have a European windstorm program in which case this would still be
gross and not adjusted for risk mitigation until aggregating at country level. When netting
down, undertakings should take care to adjust and interpret formulae accordingly.
A2.5 Aggregation of capital requirements for non-life underwriting risk and default risk
Aggregation of capital requirements for non-life underwriting risk and counterparty default
risk takes into account some correlation assumptions. These assumptions are summarized by
the following matrix (Technical Specification, SCR1.32)
The aggregation calculates with the following
í µí±†í µí°¶í µí±…í µí±›í µí±™+í µí±‘í µí±’í µí±“ = ï¿½ï¿½ í µí°¶í µí±œí µí±Ÿí µí±Ÿí µí±–,í µí±— âˆ™ í µí±†í µí°¶í µí±…í µí±›í µí±™ âˆ™ í µí±†í µí°¶í µí±…í µí±‘í µí±’í µí±“
í µí±–,í µí±—
30
Calculation of the capital requirement for the counterparty default í µí±†í µí°¶í µí±…í µí±‘í µí±’í µí±“ risk is described
in Annex A3.
31
Annex A3
Solvency capital requirements for reinsurance counterparty default risk under
Solvency II (QIS5) standard formula
This specification follows â€œTechnical Specification QIS5â€?, Section SCR.6 (SCR Counterparty
risk module), pp. 193-203.
QIS5 differentiates between two kinds of counterparty risk exposure, denoted by type1 and
type2 exposures. The class of type1 exposures covers the exposures which may not be
diversified and where the counterparty is likely to be rated. Reinsurance arrangements
belong to this type 1 class of exposures. The class of type2 exposures covers the exposures
which are usually diversified and where the counterparty is likely to be unrated. Example of
this class are receivables from intermediaries, policyholders debtors etc. Please refer to QIS5
Sections SCR6.2-6.4 for more details. We will consider only credit default risk exposure from
reinsurance arrangements (type1) and assume thereâ€™s no exposure of type 2.
Solvency capital requirements for the type 1 of exposure can be calculated as (SCR6.6):
í µí±†í µí°¶í µí±… = min(í µí°¿í µí°ºí µí°·1 + í µí°¿í µí°ºí µí°·2 + â‹¯ + í µí°¿í µí°ºí µí°·í µí±? ; í µí±ž âˆ™ âˆší µí±‰ )
where
í µí°¿í µí°ºí µí°·í µí±– = Loss-given-default for counterparty i
í µí±ž = Quantile factor
í µí±‰ = Variance of the loss distribution for the credit risk exposure
According to SCR6.16, í µí°¿í µí°ºí µí°·í µí±– can be calculated as
í µí°¿í µí°ºí µí°·í µí±– = maxï¿½(1 âˆ’ í µí±…í µí±… ) âˆ™ (í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±– + í µí±…í µí±€í µí±– âˆ’ í µí°¶í µí±œí µí±™í µí±™í µí±Ží µí±¡í µí±’í µí±Ÿí µí±Ží µí±™í µí±– ); 0ï¿½
where
í µí±…í µí±… = Recovery rate for reinsurance arrangements. According to SCR6.52, for
reinsurance arrangements, í µí±…í µí±… = 50% can be assumed.
í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±– = Best estimate recoverables from the reinsurance contract i. We
assume that í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±– = expected loss ceded to the reinsurance
contract i plus reinsurance share on provisions for claims provisions.
í µí°¶í µí±œí µí±™í µí±™í µí±Ží µí±¡í µí±’í µí±Ÿí µí±Ží µí±™í µí±– = Risk-adjusted value of collateral in relation to the reinsurance
arrangement i. We assume í µí°¶í µí±œí µí±™í µí±™í µí±Ží µí±¡í µí±’í µí±Ÿí µí±Ží µí±™í µí±– = 0.
According to SCR6.17 and SCR6.32, Risk Mitigation í µí±…í µí±€í µí±– can be calculated as
í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±–
í µí±…í µí±€í µí±– = ï¿½í µí±†í µí°¶í µí±…í µí±”í µí±Ÿí µí±œí µí± í µí± âˆ’ í µí±†í µí°¶í µí±…í µí±›í µí±’í µí±¡ ï¿½ âˆ™
í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±¡í µí±œí µí±¡í µí±Ží µí±™
where
32
í µí±†í µí°¶í µí±…í µí±”í µí±Ÿí µí±œí µí± í µí± = solvency capital requirements for underwriting risk calculated without
reinsurance
í µí±†í µí°¶í µí±…í µí±›í µí±’í µí±¡ = solvency capital requirements for underwriting risk taking reinsurance
into account
í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±– = Best estimate recoverables from the reinsurance contract i.
í µí±…í µí±’í µí±?í µí±œí µí±£í µí±’í µí±Ÿí µí±Ží µí±?í µí±™í µí±’í µí± í µí±¡í µí±œí µí±¡í µí±Ží µí±™ = Best estimate recoverables from all reinsurance contracts together.
According to SCR6.55, Quantile factor í µí±ž can be chosen as
í µí±ž = ï¿½ 3 if âˆší µí±‰ â‰¤ 5% â‹… (í µí°¿í µí°ºí µí°·1 + í µí°¿í µí°ºí µí°·2 + â‹¯ + í µí°¿í µí°ºí µí°·í µí±? )
5 else
Calculation of the variance í µí±‰ is described in SCR6.7. We differentiate between
counterparties belonging to different rating classes from AAA to CCC. We assume that in our
case there are M rating classes for reinsurance counterparties. For each rating class j we first
calculate í µí±¦í µí±— and í µí±§í µí±— according to
í µí±¦í µí±— = í µí°¿í µí°ºí µí°·1 + í µí°¿í µí°ºí µí°·2 + â‹¯ + í µí°¿í µí°ºí µí°·í µí±?í µí±—
2 2 2
í µí±§í µí±— = í µí°¿í µí°ºí µí°·1 + í µí°¿í µí°ºí µí°·2 + â‹¯ + í µí°¿í µí°ºí µí°·í µí±? í µí±—
where the sums are over all counterparties in the rating class j and í µí±?í µí±— denotes the total
number of counterparties in the rating class j.
Then, we calculate
í µí±‰ = (í µí±¢11 âˆ™ í µí±¦1 âˆ™ í µí±¦1 + í µí±¢12 âˆ™ í µí±¦1 âˆ™ í µí±¦2 + â‹¯ + í µí±¢1í µí±€ âˆ™ í µí±¦1 âˆ™ í µí±¦í µí±€
+ í µí±¢21 âˆ™ í µí±¦2 âˆ™ í µí±¦1 + í µí±¢22 âˆ™ í µí±¦2 âˆ™ í µí±¦2 + â‹¯ + í µí±¢2í µí±€ âˆ™ í µí±¦2 âˆ™ í µí±¦í µí±€
+ â‹¯ + í µí±¢í µí±€1 âˆ™ í µí±¦í µí±€ âˆ™ í µí±¦1 + í µí±¢í µí±€2 âˆ™ í µí±¦í µí±€ âˆ™ í µí±¦2 + â‹¯ + í µí±¢í µí±€í µí±€ âˆ™ í µí±¦í µí±€ âˆ™ í µí±¦í µí±€ )
+ (í µí¼ˆ1 âˆ™ í µí±§1 + í µí¼ˆ2 âˆ™ í µí±§2 + â‹¯ + í µí¼ˆí µí±€ âˆ™ í µí±§í µí±€ )
Finally, the coefficients í µí±¢í µí±–í µí±— and í µí¼ˆí µí±– are calculated from the default probabilities valid for each
different rating class
í µí±?í µí±– âˆ™ (1 âˆ’ í µí±?í µí±– ) âˆ™ í µí±?í µí±— âˆ™ (1 âˆ’ í µí±?í µí±— )
í µí±¢í µí±–í µí±— =
(1 + í µí»¾ ) â‹… ï¿½í µí±?í µí±– + í µí±?í µí±— ï¿½ âˆ’ í µí±?í µí±– â‹… í µí±?í µí±—
(1 + 2í µí»¾ ) â‹… í µí±?í µí±– â‹… (1 âˆ’ í µí±?í µí±– )
í µí¼ˆí µí±– =
(2 + 2í µí»¾ âˆ’ í µí±?í µí±– )
where
í µí»¾ = 0.25
and probabilities í µí±?í µí±– are according to the following table
33
Rating í µí±?í µí±–
AAA 0,002%
AA 0,01%
A 0,05%
BBB 0,24%
BB 1,20%
B 6,04%
CCC or lower 30,41%
34
Annex A4
Indicative pricing of non-proportional reinsurance for the sample portfolio
The indicative pricing of non-proportional reinsurance in this paper is based on the following
formalism.
For an arbitrary non-proportional reinsurance layer, we first introduce the parameter called
layer centroid which has the following property: Expected loss covered by a layer A xs B with
the liability limit A and retention B is equal to the liability level of the layer multiplied by the
expected number of losses higher than the centroid of this layer.
í µí°¸í µí±¥í µí±?. í µí°¿í µí±œí µí± í µí± í µí°´ í µí±¥í µí± í µí°µ = í µí°´ âˆ™ í µí°¸í µí±¥í µí±?. í µí±?í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¶ (A4.e1)
where í µí°¸í µí±¥í µí±?. í µí°¿í µí±œí µí± í µí± í µí°´ í µí±¥í µí± í µí°µ denotes the expected loss covered by the layer A xs B and
í µí°¸í µí±¥í µí±?. í µí±?í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¶ denotes the expected number of losses higher than the layer centroid.
From the above relation, we then derive for the net rate on line of the layer A xs B:
í µí°¸í µí±¥í µí±?.í µí°¿í µí±œí µí± í µí± í µí°´ í µí±¥í µí± í µí°µ
í µí±?í µí±’í µí±¡í µí±…í µí±œí µí°¿í µí°´ í µí±¥í µí± í µí°µ â‰œ = í µí°¸í µí±¥í µí±?. í µí±?í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¶ (A4.e2)
í µí°´
Therefore, the layer price parameter NetRoL which is per definition equal to the expected
loss covered by the layer divided by the layer liability limit can be calculated as the expected
number of losses higher than the layer centroid.
For calculating í µí°¸í µí±¥í µí±?. í µí±?í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¶ we use the following relation
í µí°¸í µí±¥í µí±?. í µí±?í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¶ = í µí°¸í µí±¥í µí±?. í µí±?í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¿í µí°¿ â‹… í µí±ƒ(í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¶ ) (A4.e3)
= í µí°¸í µí±¥í µí±?. í µí±?í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¿í µí°¿ â‹… ï¿½1 âˆ’ í µí±ƒ(í µí±™í µí±œí µí± í µí± â‰¤ í µí°¿í µí°¶ )ï¿½
= í µí°¸í µí±¥í µí±?. í µí±?í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¿í µí°¿ â‹… ï¿½1 âˆ’ í µí°¶í µí°·í µí°¹ (í µí°¿í µí°¶ )ï¿½
where í µí°¸í µí±¥í µí±?. í µí±?í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¿í µí°¿ denotes the expected number of losses higher than lower loss limit
(LLL) â€“ the lower threshold of the loss severity distribution; í µí±ƒ(í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¶ ) denotes the
probability that the loss severity is higher than layer centroid LC according to the chosen loss
severity distribution and í µí°¶í µí°·í µí°¹ (í µí°¿í µí°¶ ) denotes the cumulative distribution function of the
chosen loss severity distribution. We further assume that í µí°¿í µí°¶ > í µí°¿í µí°¿í µí°¿ and that due to the fact
that í µí°¿í µí°¿í µí°¿ is the lower threshold of loss severity distribution í µí±ƒ(í µí±™í µí±œí µí± í µí± â‰¤ í µí°¿í µí°¿í µí°¿) = í µí°¶í µí°·í µí°¹(í µí°¿í µí°¿í µí°¿) =
0.
For calculating NetRoL according to (A4.e2) and (A4.e3), the value of the layer centroid is
required. It can be shown that the following expression provides a good proxy:
Layer centroid of the layer A xs B â‰… ï¿½í µí°µ âˆ™ (í µí°´ + í µí°µ ) (A4.e4)
With the proxy (A4.e4), we can calculate NetRoL of an arbitrary layer A xs B which expresses
the expected loss covered by the layer divided by the liability limit A. The market price of
layer can be expressed with help of the parameter called GrossRoL which expresses the total
of the expected loss covered by the layer and the costs allocated to the layer (costs of capital
and management expenses) divided by the liability limit A.
35
í µí°¶í µí±œí µí± í µí±¡í µí±
í µí°ºí µí±Ÿí µí±œí µí± í µí± í µí±…í µí±œí µí°¿í µí°´ í µí±¥í µí± í µí°µ = í µí±?í µí±’í µí±¡í µí±…í µí±œí µí°¿í µí°´ í µí±¥í µí± í µí°µ + í µí°´
(A4.e5)
Instead of estimating the costs in (A4.e5), we apply an approach as adopted by capital
markets when pricing catastrophe bonds. It estimates the GrossRoL as a product of NetRoL
and the factor called Multiple which expresses the relation of the gross layer price (which
includes the expected loss and the allocated costs) to the pure expected loss.
í µí°ºí µí±Ÿí µí±œí µí± í µí± í µí±…í µí±œí µí°¿í µí°´ í µí±¥í µí± í µí°µ = í µí±€í µí±¢í µí±™í µí±¡í µí±–í µí±?í µí±™í µí±’ â‹… í µí±?í µí±’í µí±¡í µí±…í µí±œí µí°¿í µí°´ í µí±¥í µí± í µí°µ (A4.e6)
Expected loss covered by the layer A xs B and the reinsurance premium can be calculated
from the net and gross rate on line according to the following:
í µí°¸í µí±¥í µí±?. í µí°¿í µí±œí µí± í µí± í µí°´ í µí±¥í µí± í µí°µ = í µí°´ â‹… í µí±?í µí±’í µí±¡í µí±…í µí±œí µí°¿í µí°´ í µí±¥í µí± í µí°µ (A4.e7)
í µí±…í µí°¼ í µí±ƒí µí±Ÿí µí±’í µí±ší µí±–í µí±¢í µí±ší µí°´ í µí±¥í µí± í µí°µ = í µí°´ â‹… í µí°ºí µí±Ÿí µí±œí µí± í µí± í µí±…í µí±œí µí°¿í µí°´ í µí±¥í µí± í µí°µ (A4.e8)
Often, the expected loss and the price of an XL layer are expressed as rate on GNPI (gross
net premium income) â€“ premium retained under preceding proportional reinsurance (if any):
í µí±?í µí±’í µí±¡í µí±…í µí±Ží µí±¡í µí±’í µí°´ í µí±¥í µí± í µí°µ = í µí°¸í µí±¥í µí±?. í µí°¿í µí±œí µí± í µí± í µí°´ í µí±¥í µí± í µí°µ /í µí°ºí µí±?í µí±ƒí µí°¼ (A4.e9)
í µí°ºí µí±Ÿí µí±œí µí± í µí± í µí±…í µí±Ží µí±¡í µí±’í µí°´ í µí±¥í µí± í µí°µ = í µí±…í µí°¼ í µí±ƒí µí±Ÿí µí±’í µí±ší µí±–í µí±¢í µí±ší µí°´ í µí±¥í µí± í µí°µ /í µí°ºí µí±?í µí±ƒí µí°¼ (A4.e10)
For the loss severity we chose censored Pareto distribution with the following cumulative
distribution function
í µí°¿í µí°¿í µí°¿âˆ’í µí»¼ âˆ’í µí±¥ âˆ’í µí»¼
í µí°¶í µí°·í µí°¹í µí±ƒí µí±Ží µí±Ÿí µí±’í µí±¡í µí±œ (í µí±¥) = í µí°¿í µí°¿í µí°¿âˆ’í µí»¼âˆ’í µí±ˆí µí°¿í µí°¿âˆ’í µí»¼ (A4.e11)
where í µí»¼ denotes the shape parameter of the Pareto distribution, LLL and ULL denote the
lower and the upper loss limit respectively. The following Table A4.1 summarizes the chosen
parameter of Pareto distribution per line of business. The values of Pareto Alpha in column C
were chosen according to market benchmarks.
Table A4.1: Parameters of the loss severity distribution (Pareto).
The following Table A4.2 provides the estimates for í µí°¸í µí±¥í µí±?. í µí±?í µí±™í µí±œí µí± í µí± > í µí°¿í µí°¿í µí°¿ â€“ the expected number
of losses higher than LLL, cf. (A4.e3). The values in column D are the assumed frequencies
p.a. per â‚¬ 1 m premium. For simplicity reasons, we assumed the same frequency for each
36
line of business, in-line with market benchmarks. The values in column F (except for fire
earthquake) were obtained by multiplying the premiums in column E by the frequencies per
â‚¬ 1m premium (Table A4.1, column D). Obviously, the frequency of the earthquake loss does
not depend on premium. It was estimated directly assuming the return period of 10 years
for the loss higher than the lower loss threshold which translates into the frequency of 0.1.
Table A4.2: Loss frequencies
Based on the formalism presented above (A4.e1 â€“ A4.e11) and the parameters of loss
severity and loss frequency distributions summarized in Tables A4.1 and A4.2, we estimated
the prices of all XL layers throughout the paper. Below we provide calculation details for the
layers of the original reinsurance structure, cf. Section 2.1. All other layers were priced
analogously.
Table A4.3 summarizes the parameters of the original reinsurance portfolio.
Table A4.3: Parameters of the original reinsurance portfolio.
Based on the layer parameters provided in Table A4.3, columns I and J, we then calculated
layer centroids and the expected number of losses higher than layer centroid, cf. Table A4.4.
37
Table A4.4: Layer centroids and expected number of losses higher than layer centroid
For calculating column M, we used (A4.e4) and for calculating column N (A4.e3) and
(A4.e11). Please notice that in (A4.e11) the lower and upper loss limits from Table A4.4
columns K and L were used. These loss limits were obtained from the original loss limits
(Table A4.1, columns A and B) through applying the retention ratio under the proportional
reinsurance.
Table A4.5 provides the results for NetRoL and GrossRoL per line of business.
Table A4.5: Net and gross RoL
For calculating column O, we used (A4.e2). The Multiples in column P were assumed based
on market benchmarks. For calculating column Q, we used (A4.e6).
Finally, Table A4.6 calculates the expected loss covered by the XL layers and the reinsurance
premium as â‚¬ figures as well as the net and gross rates on GNPI.
Table A4.6: Expected covered loss, reinsurance premium, net and gross rate on GNPI
For calculating column R, we used (A4.e7), for column S (A4.e7), for column T (A4.e9) and for
column U (A4.e10). In the paper, these results were used in Tables 2.3.1 and 2.3.9, cf.
Sections 2.3.1 and 2.3.4.
38