Geographical diversification and Solvency II
A proposal by Lloyd’s
The proposal
This is a proposal for the recognition of geographical diversification in the non-life
underwriting risk module of the SCR standard formula.
Why should the standard formula recognise geographical diversification?
Recognition of geographical diversification reflects Solvency II’s intentions because:

Solvency II is based on an economic approach to the setting of capital, an approach
that requires recognition of diversification.

A key form of diversification for insurance undertakings is diversification of their
underwriting portfolios by location – geographical diversification. This recognises
reduced correlation between losses in different places – e.g. losses arising from bad
weather in the UK have no connection to claims experiences in Greece.

The Solvency II Directive therefore makes explicit provision for diversification.

Insurance undertakings carrying on business in different regions across the world
lessen the concentration of risk that occurs if they are restricted to a single country.
This has a material impact on the amount of capital they require.
A detailed rationale for recognition of geographical diversification is set out in Appendix 1.
Recognition of geographical diversification by Solvency II
The non-life module in QIS4 included geographical diversification. CEA’s January 2009
QIS4 feedback to the European Commission said:
“The introduction of geographical diversification into QIS4 was a valuable
improvement in the SCR standard formula compared to QIS3 - it enabled a much
more realistic recognition of insurers' risk profiles.”
However, subsequently, CEIOPS decided not to include geographical diversification. Its
advice on non-life underwriting risk1 said:
“While this change is crucial for reinsurers and cross-border groups, it was seen
as introducing unnecessary complexity at solo level, in view of the materiality of
the reduction in capital requirement they could obtain from the calculation.”
This decision elicited much comment when it was first outlined in CP48 (see Appendix 2).
CEIOPS’ report on QIS4 recognised that geographical diversification is a material factor
1
DOC-41-09, October 2009
1
for many solo entities such as reinsurers and specialists in transport insurance, but has
little or no effect on others. CEIOPS recognises that undertakings with geographically
diversified portfolios are seriously affected by the non-recognition of geographical
diversification. It means that the standard formula is not an appropriate method of
calculating their SCR and requires them to use undertaking-specific parameters (if
appropriate) or to develop internal models. On the other hand, the issue is less relevant
to undertakings carrying on business on a regionally restricted basis, who want to avoid
additional complexity in their SCR calculations.
The EU’s insurance sector includes many undertakings which carry on insurance
business across the world: important and successful components of the EU’s financial
services industry. The standard formula is intended “…to reflect the risk profile of most
insurance and reinsurance undertakings” (Directive preamble 26) and “…to enable all
insurance and reinsurance undertakings to assess their economic capital” (Directive
preamble 65). The removal of geographical diversification means that the standard
formula does not reflect the risk profile of globally active undertakings and cannot be
used by them to assess their economic capital.
The aim is to incorporate geographical diversification in the standard formula so that it
can reflect the risk profiles of globally active undertakings, whilst avoiding the imposition
of complexity on undertakings whose business is not geographically diversified. This
proposal does this in two ways:

It includes a simplification whereby undertakings finding allocating risks to
geographical regions too complex can report all risks within a single region.

It makes the approach to geographical diversification less complex than that
adopted by QIS4. The number of regions into which premiums and reserves must
be broken down is reduced from 54 to 18.
Geographical diversification: specification
The approach set out here is similar to, but not exactly the same as, that set out in the
QIS4 specification.
Incorporation of geographical diversification into the non-life premium and reserve risk
module would be standard with an acceptable simplification of allocating all risk to one
region, at the option of the undertaking.
If an undertaking wishes to claim geographical diversification benefits it will calculate a
Herfindahl index based on the geographical location of the risks underlying the
premiums and reserves for each line of business (LoB). The Herfindahl index (also
known as the Herfindahl-Hirschman Index) is commonly used to calculate the size of
firms in relation to an industry and so to indicate the extent of competition within the
industry. Here it is used to calculate the geographical concentration of a portfolio of
premiums and reserves. The maximum value of the Herfindahl index is 1, implying no
2
geographical diversification. The closer the Herfindahl index is to 0, the less
concentrated the portfolio is and the greater the diversification effect.
The index which represents a measure of the implicit geographical diversification is
calculated for each line of business as follows:
 (V
( prem , j ,lob )
DIVpr ,lob 
 V( res , j ,lob ) ) 2
j


  (V( prem , j ,lob )  V( res , j ,lob ) ) 


 j

2
In this formula, for each individual line of business, V(prem, j, lob) and V(res, j, lob) are the
standard non-life volume measures for premium and reserve risk, respectively split
further by the underlying geographical regions which are represented by j.
The overall volume measure V to be used in the standard formula calculation would then
be determined as follows:
V 
V
Lob
lob
Vlob is the line of business volume measure for premium and reserve risk which can be
defined as:
Vlob = ( V(prem,lob) + V(res, lob)) * (0.75+0.25 * DIVpr,lob)
For undertakings choosing not to claim geographical diversification, a simplification
would be applied whereby all data was entered within one geographical region and
DIVpr,lob would implicitly be set to 1 for all lines of business.
Maximum diversification benefit
The maximum diversification benefit available would be 25%. Appendices 3 – 5 discuss
the factors underlying the selection of this figure. This does implicitly assume moderate
(50%) correlation between non-identical regions.
Materiality threshold
If an undertaking has more than 95% of its non-life activities (premium and reserves) in
the same geographical area, it cannot benefit from geographical diversification.
Lines of business
Geographical diversification can be applied to any line of business, other than Credit and
suretyship.
3
Non-proportional reinsurance classes will have less need for explicit recognition of
geographical diversification as this is better reflected in the calibration of the standard
factors. This leaves two options for non-proportional reinsurance:

To reduce the maximum allowance from 25%; or

To remove geographical diversification entirely.
Lloyd’s believes some additional diversification would remain and proposes a maximum
of 10% to recognise this.
Regional breakdown
We consider that the 54 regions contained in QIS4 can be streamlined to make
calculation easier. Our approach is based on the United Nations geo-scheme, developed
by the UN Statistics Division for statistical purposes, which divides the world into “macrogeographical regions”. The UN confirms that these geographical divisions do not imply
any assumptions regarding political or other affiliations of countries or territories.
The geo-scheme consists of 22 sub-regions. From an insurance point of view it is
possible to aggregate these further, to take account of the size of national markets. In
addition, we believe that it is necessary to take special account of the US, in view of the
size of the US non-life market and of South America, in view of its importance to some
geographically diversified EU insurers. The US can be divided into regions using the
NAIC’s regional split.
The regional split that Lloyd’s recommends is as follows:
1. Central & Western Asia (UN geo-scheme Central Asia and Western Asia, less
Cyprus)
Armenia
Azerbaijan
Iraq
Israel
Kuwait
Kyrgyzstan
Palestinian Territories Qatar
Tajikistan
Turkey
Uzbekistan
Yemen
Bahrain
Jordan
Lebanon
Saudi Arabia
Turkmenistan
Georgia
Kazakhstan
Oman
Syrian Arab Republic
United Arab Emirates
2. Eastern Asia (UN geo-scheme Eastern Asia)
China
Mongolia
Hong Kong
North Korea
Japan
South Korea
4
Macao
Taiwan
3. South and South-Eastern Asia (UN geo-scheme Southern Asia and SouthEastern Asia)
Afghanistan
Cambodia
Lao PDR
Nepal
Sri Lanka
Bangladesh
India
Malaysia
Pakistan
Thailand
Bhutan
Indonesia
Maldives
Philippines
Timor-Leste
Brunei Darussalam
Iran
Myanmar
Singapore
Vietnam
4. Oceania (UN geo-scheme Oceania region)
American Samoa
Australia
French Polynesia
Guam
Micronesia
Nauru
Niue
Norfolk Island
Papua New Guinea Pitcairn
Tokelau
Tonga
Wallis & Futuna Islands
Cook Islands
Kiribati
New Caledonia
N. Mariana Islands
Samoa
Tuvalu
Fiji
Marshall Islands
New Zealand
Palau
Solomon Islands
Vanuatu
5. Northern Africa (UN geo-scheme Northern Africa and Western Africa plus
Cameroon, Central African Republic and Chad)
Algeria
Cape Verde
Egypt
Guinea-Bissau
Mauritania
Saint Helena
Togo
Benin
Central African Rep.
Gambia
Liberia
Morocco
Senegal
Tunisia
Burkina Faso
Chad
Ghana
Libya
Niger
Sierra Leone
Western Sahara
Cameroon
Cote d’Ivoire
Guinea
Mali
Nigeria
Sudan
6. Southern Africa (UN geo-scheme Southern Africa, Eastern Africa and Middle
Africa other than countries specified under Northern Africa)
Angola
Dem Rep of Congo
Ethiopia
Madagascar
Mozambique
Rwanda
South Africa
Tanzania
Botswana
Djibouti
Gabon
Malawi
Namibia
Sao Tome & Principe
Swaziland
Zambia
Burundi
Equatorial Guinea
Kenya
Mauritius
Rep of the Congo
Seychelles
Uganda
Zimbabwe
5
Comoros
Eritrea
Lesotho
Mayotte
Reunion
Somalia
United Rep. of
7. Eastern Europe (UN geo-scheme Eastern Europe)
Belarus
Moldova
Slovakia
Bulgaria
Poland
Ukraine
Czech Republic
Romania
Hungary
Russian Federation
8. Northern Europe (UN geo-scheme Northern Europe)
Aland Islands
Faeroe Islands
Republic of Ireland
Lithuania
United Kingdom
Channel Islands
Finland
Isle of Man
Norway
Denmark
Guernsey
Jersey
Svalbard, Jan Mayen
Estonia
Iceland
Latvia
Sweden
9. Southern Europe (UN geo-scheme Southern Europe, plus Cyprus)
Albania
Cyprus
Macedonia
San Marino
Vatican City
Andorra
Gibraltar
Malta
Serbia
Bosnia
Greece
Montenegro
Slovenia
Croatia
Italy
Portugal
Spain
10. Western Europe (UN geo-scheme Western Europe)
Austria
Liechtenstein
Switzerland
Belgium
Luxembourg
France
Monaco
Germany
Netherlands
11. Northern America excluding the USA (UN geo-scheme Northern America, less
the USA)
Bermuda
Canada
Greenland
St Pierre & Miquelon
12. Caribbean & Central America (UN geo-scheme Caribbean and Central America)
Anguilla
Barbados
Costa Rica
El Salvador
Haiti
Mexico
Panama
St Lucia
Turks & Caicos Is’ds
Antigua & Barbuda
Belize
Cuba
Grenada
Honduras
Montserrat
Puerto Rico
St Martin
US Virgin Islands
Aruba
British Virgin Islands
Dominica
Guadeloupe
Jamaica
Netherlands Antilles
St-Barthelemy
St Vincent
6
Bahamas
Cayman Islands
Dominican Republic
Guatemala
Martinique
Nicaragua
St Kitts & Nevis
Trinidad & Tobago
13. Eastern South America (UN geo-scheme South America divided)
Brazil
Paraguay
Falkland Islands
Suriname
French Guiana
Uruguay
Guyana
14 Northern, southern and western South America (UN geo-scheme South
America divided)
Argentina
Ecuador
Bolivia
Peru
Chile
Venezuela
Colombia
15. North-east US (NAIC North-eastern zone)
Connecticut
Maryland
New York
Delaware
Massachusetts
Pennsylvania
District of Columbia
New Hampshire
Rhode Island
Maine
New Jersey
Vermont
16. South-east US (NAIC South-eastern zone, less US Virgin Islands)
Alabama
Kentucky
Puerto Rico
W. Virginia
Arkansas
Louisiana
South Carolina
Florida
Mississippi
Tennessee
Georgia
North Carolina
Virginia
Iowa
Missouri
Oklahoma
Kansas
Nebraska
South Dakota
17. Mid-west US (NAIC Midwestern zone)
Illinois
Michigan
North Dakota
Wisconsin
Indiana
Minnesota
Ohio
18. Western US (NAIC Western zone, less American Samoa and Guam)
Alaska
Hawaii
New Mexico
Washington
Arizona
Idaho
Oregon
Wyoming
California
Montana
Texas
Colorado
Nevada
Utah
The suggested regional split is indicated on the attached maps (Appendix 6). Regional
non-life insurance markets’ relative sizes are indicated below. This shows the regions
more likely to be significant sources of business to EU insurers and reinsurers
transacting business globally. Regions are not intended to be of broadly equal size, but
to be sufficiently large to justify their separate treatment.
The proposal that countries be grouped into regions is an integral part of this proposal.
Nevertheless, we are receptive to suggestions, from supervisors, insurers and other
7
interested parties, for change to the precise number of regions and their make up, if
alternatives are considered to be more satisfactory.
2008 Non-life premium income (Source: Sigma
3/2009)
USD bn
Central & West Asia
Eastern Asia
South & South –eastern Asia
Oceania
Northern Africa
Southern Africa
Eastern Europe
Northern Europe
Southern Europe
Western Europe
Northern America (excl. USA)
Caribbean & Central America
E South America
NSW South America
North-east US
South-east US
Mid-west US
Western US
25.3
192.5
28.0
34.6
6.2
10.5
69.4
157.4
115.5
355.7
57.7
14.1
27.4
24.4
165.3
165.3
165.3
165.3
Note: figures for US are approximations only, derived by dividing total US non-life
premiums of USD 661.2bn between the four regions.
Allocation of transactions to particular countries
We suggest that the method of defining risk location should be:

Simple and easy to apply.

Objective and not open to subjective interpretation.

At the overall level, give a good approximation of the spread of risk in the
undertaking’s account.
We therefore suggest the following methods of allocation:
Motor – location of insured vehicle registration.
Fire & other property damage – physical location of the insured risk.
General liability – insured’s location, unless the coverage provided is:
8
(i)
Products liability or professional indemnity; and
(ii)
The insured provides goods or services to regions other than its home
region and
(iii)
The main risk of loss arises in a region other than its home region.
The risk should then be allocated to the region where the greatest risk of
insured loss arises.
All other classes (including MAT) – determined by the undertaking to reflect the
location of loss. Depending on the precise way the contract is written, this may be
the insured’s location or the location at which an insured asset has its greatest
exposure to insured perils.
It is important to bear in mind that geographical diversification operates at an aggregate
portfolio level, rather than at the level of individual risks. Some of the factors that give
rise to geographical diversification are a consequence of diversity in the physical
locations at which losses arise; others, such as diversity in risk and loss cultures and
legal environments reflect the countries in which insurance undertakings are carrying on
business. Any rule will produce anomalies, but these will not give rise to serious
distortions at the aggregate portfolio level.
The premium and reserves for a single contract insuring risks located in several regions
(“global contracts”) should be split and allocated to regions on the basis of the underlying
locations of risks. A contract insuring £100m of risk spread across locations in 10
countries worldwide evidences reduced risk correlation, in comparison with a contract
insuring £100m of risk in a single or small number of locations in one country/region.
Lloyd’s
April 2010
9
Appendix 1
Rationale for recognising geographical diversification
Diversification and insurance
Solvency II is based on an economic approach to the setting of capital, an approach that
requires recognition of diversification. The European Commission’s Impact Assessment
Report2 says (page 104):
“An economic risk based approach also takes account of the specific risk profile
of the insurance undertaking and the impact of risk mitigation techniques, as well
as size and diversification effects.”
This reflects international thinking on insurer capital requirements. The IAIS Common
Structure for the Assessment of Insurer Solvency (February 2007) says (para 79):
“The IAIS would thus suggest that an overall capital requirement should take into
account diversification between risk factors where this can be substantiated with
sufficient robustness.”
The Chief Risk Officer Forum (CRO) report “A framework for incorporating diversification
in the solvency assessment of insurers” (June 2005) points out that concentration of risk
is bad for the insurance industry and consumers and that diversifying strategies are the
basis of sound risk management. It further states that existing regulatory solvency
approaches do not adequately take diversification into account, an issue which it
considered regulators should address.
Diversification and the Solvency II Directive
The Solvency II Directive makes explicit provision for diversification. Directive Article
13(39) defines “diversification effects” as:
“…the reduction in the risk exposure of insurance and reinsurance undertakings
and groups related to the diversification of their business, resulting from the fact
that the adverse outcome from one risk can be offset by a more favourable
outcome from another risk, where those risks are not fully correlated”
Preamble 64 says:
“…economic capital should be calculated on the basis of the true risk profile of those
undertakings, taking account of the impact of possible risk-mitigation techniques, as well
as diversification effects.”
Article 104(4) includes the following:
2
[SEC 2007] 871
10
“Where appropriate, diversification effects shall be taken into account in the
design of each risk module.”
Geographical diversification
An important form of diversification for insurers is diversification across locations, i.e.
geographical diversification. A geographically diversified portfolio shows lower
correlation of loss than one which is limited to a single country. Insured losses arise in
particular locations at particular times and for the majority of classes the underlying
sources are not dependent. Loss frequency and severity reflect factors that vary
geographically, such as catastrophe exposure, physical geography, climate, culture,
legal systems and levels of prosperity. The KPMG study of insurance prudential
supervision, prepared for the European Commission in May 20023, commented:
“International companies experience risk reduction due to geographical
diversification of risks.”
This study discussed at some length the diversification of a portfolio of insurance risks,
including geographical diversification.
The CRO June 2005 report classified diversification benefits into four distinct categories,
including “Level 4 – Across geographies or regulatory jurisdictions” (section 4.4). Its
survey of reports analysing the historical impairments of insurance companies concluded:
“…these spikes in insolvencies were largely driven by insurers with highly
concentrated risk profiles (either geographically or in terms of asset mix), which
were particularly adversely impacted by the large loss events of these periods.
Geographical concentration is particularly important. For example, Hurricane
Andrew led to higher impairment rates among P&C insurers, including 11
insolvencies. Yet no large multi-state insurer (e.g. more geographically diverse)
was impaired, even though many such insurers did suffer losses from the same
natural catastrophe event.”
The European Central Bank published a report entitled “Potential impact of Solvency II
on financial stability” in July 2007. This assumed that Solvency II would recognise risk
diversification across locations and saw this as contributing to a positive expected
outcome of improved efficiency and competitiveness of EU insurers4. It contrasted this
with existing insurance regulatory systems in EU countries which:
“…do not adequately account for risk diversification in the insurance business, so
that the risk of insurance groups being engaged in many different…geographical
areas could potentially be overestimated, and capital requirements may appear
artificially high.”
3
4
Contract no: ETD/2000/BS-3001/C/45. See para 3.4.16
Section 3.1
11
Appendix 2
Reactions to the exclusion of geographical diversification from the Standard Formula NonLife underwriting module
The following are responses received by CEIOPS to its decision in CP 48 “Standard
Formula – Non-life underwriting risk” not to apply geographical diversification. They are
taken from CEIOPS website. Lloyd’s own comments are not included.
Organisation
Comment
Association of British
Insurers (ABI)
We strongly disagree with CEIOPS. There is a strong case for
recognising geographical diversification. Omitting recognition would
be a serious departure from the Directive and lead to substantial
additional prudence.
AMICE
We suggest recognizing geographical diversification as was done
during QIS 4. Geographical diversification should be recognised
using a blending formula for business underwritten or commitments
existing in different geographical areas.
Belgian Coordination
Group Solvency II
(Assuralia/RABA)
No allowance for geographical diversification for non-life business
will be applied. This will decrease as well the incentive to spread
risks over different geographies. CEIOPS should review the decision
to exclude geographical diversification.
CEA
All diversification effects need to be considered appropriately in
the standard formula. The introduction of geographical
diversification in QIS4 was a valuable improvement to the SCR
formula. Failing to recognise it would decrease the incentive
undertakings have to spread risks across different geographies.
Ceiops should review the decision to exclude geographical
diversification and should reward sound risk management.
CRO Forum
Danish Insurance
Association
Not allowing for geographic diversification is not recognising one of
the key principles of insurance, which is giving credits for well
diversified portfolios.
The decision to not apply geographical diversification for non-life
business across the globe as proposed in the QIS 4 exercise should
be reconsidered. We agree that some geographical diversification
benefits will be reflected in the estimated volatilities when calibrated
on European historical data, however, excluding the possibility for
company specific geographical diversification is not in line the level 1
directive and the economic risk-based approach to solvency.
FFSA
FFSA thinks that geographical diversification should be taken into
account.
GDV
Diversification effects should be considered appropriately in the
standard formula. There is a strong case for recognising
geographical diversification.
12
Groupama
We suggest recognizing geographical diversification as it was done
during QIS 4. Being geographically well-diversified is an important
element which reduces risk exposure
Groupe Consultatif
…although removal of the diversification benefit might well simplify
the application of the SCR but, at the same time, it could increase
the SCR for some insurers by amounts that might be material.
Do not agree that this should be left to internal models. Small
companies and captives may write significant amounts of business
across borders (e.g. in Ireland many companies write north & south
of the border). Not allowing for geographical diversification will push
up their capital requirement and put them at a disadvantage.
International
Underwriters’ Association
(IUA)
We are disappointed that geographical diversification is not
adequately allowed for in this module…We believe that it is important
for the economic realities of risks written are recognised in the
standard formula, especially if some companies rely on the standard
formula whilst gaining internal model approval. ..We do not believe
the complexity issue is a relevant one;
KPMG
We disagree with the proposal to exclude geographical
diversification as it:
a. goes against theory
b. goes against the principles of Solvency II (an economic
assessment)
c. goes against the views of various respected international
associations such as the IAIS and IAA
d. actively discriminates against a significant portion of the EU
insurance/reinsurance market. That is the large, cross border
or reinsurance undertakings
e. implies that certain firms will get internal (or partial internal)
model approval or will use undertaking specific parameters.
This is an inappropriate assumption when forming the
standard formula parameters and approaches
f. incorrectly states the alternatives are complex or impractical.
There are alternative that are completely aligned with the
principle of proportionality (in that only those who will benefit
have to do any significant extra work)
g. proposes implicitly allowances that will be inadequate if not
calibrated correctly…
h. proposes implicit allowances that will knowingly (and
avoidably) understate the capital requirements for a large
number of (re)insurance undertakings
i. ignores realistic improvements to the QIS4 approach (rather
than the alternative suggested). The introduction of
geographical diversification in QIS4 was widely welcomed
Legal & General
We strongly disagree. We believe that geographical diversification
should be allowed, and is under the level 1 directive
Munich Re
Geographical diversification should be recognised also in the
standard formula as this is one of the main principles of insurance.
Companies should not be forced to use partial internal model in
13
order to recognise this positive feature of a balanced portfolio.
RBS Insurance
Whilst not currently material for us, we believe that allowing
geographical diversion for catastrophe risk is important, and in
keeping with solvency II principles. We believe policyholders will be
better protected when their insurers (and in particular the
catastrophe reinsurers of the insurers) have a geographically
diversified portfolio.
ROAM
We suggest recognizing geographical diversification as it was done
during QIS 4. Geographical diversification should be recognised
using a blending formula for business underwritten or commitments
existing in different geographical areas. An important argument is an
argument of level playing field. Recognizing geographical
diversification on solo level is necessary to allow companies with
foreign branches to be treated on diversification level equivalent to
companies with subsidiaries who file group SCRs etc...
RSA Insurance Group
We do not agree with the conclusion to drop geographical
diversification.
UNESPA- Association of
Spanish Insurers and
Reinsurers
Diversification effects should be considered appropriately in the
standard formula
There is a strong case for recognising geographical diversification.
The introduction of geographical diversification into QIS4 was a
valuable improvement to the SCR formula. Omitting recognition
would be a serious departure from the Directive and lead to
substantial additional prudence.
XL Capital Group
The removal of geographical diversification is very disappointing and
goes against the interests of firms writing large volumes of global
risks and global programs. We do not believe that “complexity” is a
valid ground for its exclusion, particularly given the level of
complexity noted elsewhere in the standards formula.
14
Appendix 3
Allowance for geographical diversification
A derivation of optimal geographical diversification allowances to apply under Solvency II
is covered in an ASTIN paper by Hürlimann (2009)5. We recommend interested parties
refer to this paper.
Lloyd's has deduced two key findings from the papers that are relevant to the current
discussion:

CEIOPS’ 25% maximum diversification used in QIS4 is equivalent to assuming a
50% correlation between regions. A rework of this derivation is provided in
Appendix 4

25% is not optimal and using a range of plausible underlying distribution
assumptions the maximum credit should be between 30-33%.
Lloyd's has supplemented the theoretical results with two illustrative examples to show
the extent that diversification does exist within lines of business. An outline of the
example is:
-
conducted for two major lines of business (Property & Casualty)
-
based on largest 5 currencies to give approximate geographical diversification
without making any assumptions on exact geographical splits
-
only designed to support that the proposed levels are appropriate
-
based on 1:200 year reserving risk to estimate effects of diversification within a
given line of business
This analysis shows similar results for both property and casualty lines which are
consistent with theory. Specifically that, even using a simplified approach, the implicit
diversification is 33% for casualty and 32% for property. This is consistent with the
theoretical findings of Hürlimann. Further details of the analysis are shown in Appendix 5.
Lloyd's proposes the maximum allowance of 25% for geographical diversification (as
used in QIS4) remains. The rationale for this is:
5
-
It is a prudent approach.
-
The underlying assumption of 50% correlations between regions appears
reasonable.
-
A theoretical derivation suggests a more appropriate maximum of 33%.
-
Lloyd’s illustrative example supports the theory.
http://www.actuaries.org/ASTIN/Colloquia/Helsinki/Papers/S3_3_Hürlimann.pdf
15
Appendix 4
Derivation of maximum allowance for geographical diversification assuming 50% correlation
between selected regions
Let V 
n
V
j 1
j
be the geographical decomposition of the volume measure of a line of
business into n geographical regions. Let us assume that the correlation coefficient (rho)
between any two non-identical regions is 50%. This can be represented by
 ij 
1, i  j
1 1
  ij ,  ij  
,
2 2
0, i  j
(X.1)
and that diversification can be measured by the intra-portfolio correlation coefficient
n
n
Q    ij wi w j  [1,1], wi 
i 1 j 1
Vi
,
V
(X.2)
where wi represent the portfolio weights of the non-life risks in the geographical regions.
If we use H 
n
w
i 1
2
i
to denote the Herfindahl-Hirschman index, and based on assumption
(X.1), Q can be expressed as
1
Q  (1  H ) .
2
(X.3)
The economic capital of the insurance risk portfolio to the confidence level  is supposed to
depend only on the random loss L and is denoted by EC [L] . In the standard Solvency II
approach, the economic capital is defined to be the value-at-risk ( VaR ) of the random loss
taken at the confidence level   99.5% , so we have EC [ L]  VaR[ L] . Adjusting for
geographical diversification the QIS4 non-life risk capital can be represented as
1
(1  Q)  EC [ L] .
2
(X.4)
It can bee seen that if Q  1 (perfect positive dependence between the regions), no
reduction for diversification occurs while if Q  1 (perfect negative dependence), the nonlife risk capital charge vanishes. And we assume a linear dependence structure between
these two perfect dependences.
In this simple model, based on (X.3) and (X.4), the diversified non-life risk capital charge
reads
(0.75  0.25  H )  EC [ L] .
16
(X.5)
The maximum diversification benefit of 25% can be achieved when H reaches the lower
limit of 0. This is consistent with the QIS4 technical specification on geographical
diversification cap.
17
Appendix 5
Lloyd’s examples
We give two numerical examples, which compare the proposed 25% diversification benefit
cap with Lloyd’s own diversification modelling results. These analyses are gross of
reinsurance and are included for illustrative purposes – they are designed to show that the
theoretical and practical results are similar and support a maximum credit for diversification
of at least 25%
Lloyd’s market aggregate data covering 1993 to 2009 years of account is used in our
analysis. The data is split into five major currencies. This gives approximate geographical
diversification without making any assumptions on exact geographical splits. The classes of
business being modelled are Casualty and Property (D&F). A standard Bootstrap approach
is used to produce the reserve distribution with the mean of the distribution scaled to our
best estimate reserves. Required capital is taken as the difference between the 99.5th
percentile reserve and the best estimate reserve. The analysis is carried out on an
underwriting year basis, which will include an element of unearned and unincepted
exposures at the valuation date.
The effects of diversification are estimated by comparing the sum of the capital requirements
of the individual currencies with the capital requirement when modelling the class as a whole.
Table Y.1 and Table Y.2 summarise the diversification benefit results from our modelling
exercises. The five major currencies were: AUD, CAD, EUR, GBP, and USD. Please note
that the total best estimate reserves were scaled to 100 for consistency.
Table Y.1: Diversification Results on Casualty
Best Estimate Reserve
AUD
7.9
CAD
3.7
EUR
16.6
GBP
21.4
USD
50.3
Pre-diversified Total
100.0
Post-diversified Total
100.0
Diversifciation Benefit as % of Pre-diversified Total
Table Y.2: Diversification Results on Property (D&F)
Best Estimate Reserve
AUD
3.2
CAD
10.0
EUR
8.0
GBP
24.6
USD
54.2
Pre-diversified Total
100.0
Post-diversified Total
100.0
Diversifciation Benefit as % of Pre-diversified Total
99.5th Reserve
11.2
5.3
22.5
28.9
66.1
134.0
122.7
99.5th Reserve
5.5
14.0
14.4
34.5
75.2
143.7
129.9
Capital
3.3
1.6
5.9
7.4
15.8
34.0
22.7
33%
Capital
2.4
4.0
6.4
9.9
21.0
43.7
29.9
32%
From the tables above, we can see that the diversification benefit is consistent between the
two classes modelled. The results are 33% for Casualty and 32% for Property (D&F). Both
are also consistent with the theoretical derivation and are higher than the QIS4 cap of 25%.
The analysis is on a gross of reinsurance underwriting year basis and has not had
catastrophic losses removed. It should not therefore be compared to the proposed standard
formula factors which are net of reinsurance and exclude the impact of catastrophes.
18
Appendix 6
Geographical diversification: proposed regional split
N. Europe
E. Europe
N. America
C & W Asia
W. Europe
S. Europe
US (see separate
map)
E. Asia
N. Africa
Caribbean & C.
America
S & SE. Asia
S. Africa
Eastern S.
America
Oceania
NSW S. America
19
Geographical diversification: regional split (US)
Mid- west US
North- east US
Western US
South - east US
20