January 2013
Insights
Economic capital for life insurers
The ‘state of the art’ – an overview
Welcome...
...to the first in a planned series of papers
examining the various facets of economic
capital – with a focus on its role and
importance in the life insurance industry.
We start in this issue with how approaches
to modelling economic capital have evolved
and the strong influence that Solvency II has
exerted, and continues to exert, on what is
considered ‘state of the art’.
Conscious that ‘state of the art’ can only
ever be a temporary label, we conclude by
looking at some of the emerging methods
and technologies, and Towers Watson’s role
in developing them, that will shape the future
business and financial benefits available from
the accurate measure of risk exposure.
I trust you will find this and future issues in
the series useful and thought provoking. Of
course, we would value your insights and the
opportunity to inform our own thinking, so
please feel free to contact me or one of my
colleagues listed at the end of the article.
John Rowland
Global Leader – Life Capital Modelling
Towers Watson
Introduction
Economic capital, representing the economic
resources required by firms to protect themselves
against unforeseen events up to a given risk
tolerance, has been used by many insurers as part of
a risk management framework – their enterprise risk
management (ERM) framework – for over 10 years.
ERM typically starts with a formulation of a firm’s risk
appetite: that is, an assessment of the risks a firm
is willing and unwilling to accept. This leads naturally
to a measurement question: how do you measure
‘how much’ risk you are willing to accept? Economic
capital, focusing on extreme outcomes, is part of
the answer; with an assessment of less extreme
outcomes, such as earnings volatility, forming another
part of the assessment.
The development of economic capital models
in Europe has been given significant impetus by
Solvency II. The Directive sets into EU law an
approach to regulation based on, amongst other
things, management of ‘own funds’ that include
a measure of regulatory capital (Solvency Capital
Requirement or SCR) that has a standard economic
capital paradigm at its heart. Furthermore, Solvency II
provides insurers with the option of defining their
regulatory capital requirement using an internal
(economic capital) model reflecting a firm’s internal
assessment of its capital requirements. Solvency II is
also fairly prescriptive regarding how it expects firms
to develop and validate an internal model and this
is influencing internal thinking regarding economic
capital too. As a result, for European life firms a
standard economic capital paradigm is emerging: the
1-year value at risk (VaR).
Insights | January 2013
•• An approach to ‘valuation’ of the balance sheet:
Solvency II bases its valuation on a marketconsistent approach, with notable adjustments.
But the 1-year VaR method could equally be
defined in terms of IFRS or GAAP earnings as well.
•• Risk measure: the Solvency II definition adopted
VaR as opposed to say, tail value at risk, as
adopted under the Swiss Solvency Test.
The concept is simple: it defines economic capital
by following the process below:
•• To what risks is the firm exposed? And how do
those risks evolve over a one-year time horizon –
allowing for the dependent nature of the evolution
of risk?
•• Given all possible evolutions of risks over a
one-year time frame, what are the possible
net asset (or own fund) values?
•• Given all possible net asset values, the SCR
(or economic capital) is then measured by
considering the 99.5th percentile value.
The merits of these choices and other core issues
are not the focus of this article. We shall focus on
how firms have built their models and the ‘state of
the art’ from a modelling perspective.
This approach is illustrated in Figure 01 below.
When implementing this approach, a number
of crucial choices are necessary or will have
been made:
•• Time horizon: risks are measured by their
evolution over one year, not 10 days (as was
standard in banking when VaR was first commonly
used) or five years. This choice is somewhat
arbitrary but prescribed in Solvency II, and
firms should consider risk emergence over
all time horizons. Under Solvency II, this
requirement is captured under the Own Risk
Solvency Assessment (ORSA) requirements.
Figure 01. The SCR is defined using a 1-year VaR approach
(2) For each possible state,
(1) How do the risks we
what is the value of the
are exposed to evolve
balance sheet?
over a one-year time frame?
Available economic capital
Market
value
assets
(3) Given all possible balance
sheets, the SCR is determined
by the 99.5th percentile result.
OF
Probability of
outcome
RM
BEL
Scen 1
Expected
Market
value
assets
OF
RM
BEL
OF – Own fund
RM – Risk margin
BEL – Best estimate liability
Market
value
assets
Scen 1,000,000
T=0
2 towerswatson.com
T=1
OF
RM
BEL
0.5% Probability
Insights | January 2013
Implementation – the challenges
Clearly economic capital pre-dates Solvency II, and
before Solvency II firms had used the 1-year VaR
approach to measure their economic capital. Few,
however, had attempted to implement the nested
stochastic approach displayed in Figure 01.
Some quick mathematics shows that, if assumptions
are made about the distribution of risks, the
dependency structure between risks and the
behaviour of the balance sheet, a simple ‘stress
test/correlation matrix’ calculation can be adopted.
For many firms, economic capital started with such a
calculation and the Solvency II Standard Formula is
a formula that uses the simplicity of this approach.
Similar to Basel II, Solvency II permits simple
methods – the Standard Formula – implementable by
any insurance entity alongside an option for complex
methods – an internal model – suitable for firms with
more complex risk exposures.
As noted above, simplifying assumptions are
necessary to set up a stress test/correlation
matrix-style economic capital calculation. As a
consequence, the Standard Formula is appropriate
for simpler organisations, where it can provide a
reasonable assessment of capital requirements.
However, it does not provide much detail about
the risk profile of an organisation. Stress test
approaches have been in use for a number of
years and the limitations of such an approach are
well known. Solvency II, with an explicit focus on
management of risk as well as measurement of
the SCR, seeks to address these by permitting
the use of internal economic capital models. This
leads to a wish list for an internal economic capital
model – at least if the model is to prove useful. This
wish list consists of both technical and business
considerations.
Business considerations
•• Allow quick/daily estimates of economic capital
requirements. Effectively this implies that the
model needs to be able to estimate (accurately)
the economic capital requirement without
requiring underlying asset and liability matching
(ALM) models to be run. For most firms, this is
equivalent to separating a calibration process
from the risk reporting process.
•• A methodology and system that is simple to
understand by management and to implement.
•• Be flexible enough to enable ‘what if’ analysis –
for example, ‘what if’ a restructure happened;
what if a new product was launched; what if a
new re-insurance arrangement was adopted.
•• Multiple controlled user access with auditability,
repeatability and workflow capability.
The technical challenges led firms to go back to
the original 1-year VaR definition. Adopting this –
implementing a full Monte Carlo simulation – enables
an economic capital model to explicitly address
the technical challenges. However, for all bar the
simplest liabilities, full Monte Carlo simulation fails
to meet the business requirements due to run-time
issues associated with the production of ‘all possible
balance sheets’. Thus, firms have found that to
implement the full Monte Carlo simulation, methods
that efficiently estimate the balance sheet are
required. These methods or approximate models are
known as proxy models.
Technical considerations
•• Address the shortcuts implicit in the Standard
Formula/stress test correlation matrix approach –
allow for complex risk distributions and more
general dependency structures. Allow for non-linear
and non-separable behaviour of the balance sheet.
•• Allocate capital across the legal and management
hierarchy by risk and product.
•• Explicitly model capital fungibility and transferability
restrictions, re-insurance and tax.
“Stress
“
test approaches have been in use for a number of years
and the limitations of such an approach are well known.
Solvency II...seeks to address these by permitting the use of
internal economic capital models.”
towerswatson.com 3
Insights | January 2013
Proxy models – replicating portfolios,
loss functions and Least Squares
Monte Carlo
In the present context, proxy models started in the
mid-2000s with firms seeking to construct replicating
portfolios of their liabilities. A replicating portfolio
is a portfolio of assets with the same risk return
characteristics as the liabilities being replicated.
They are typically constructed using optimisation
techniques with either present values of liabilities
or individual simulated liabilities (cash flows) being
targeted. Towers Watson was at the forefront of
developments and implemented a number of
economic capital models using replicating portfolios,
and replicating portfolio-based models form the basis
of a number of internal models across Europe.
Replicating portfolios, by construction, focus on
market and credit risk. When using a replicating
portfolio, capital for insurance and operational risk
is usually modelled separately and aggregated
to generate the SCR. The separation, for certain
types of liabilities, can be a material shortcoming.
For example, liabilities such as pay-out annuities
have material non-linear interaction between
longevity/mortality and market/credit risk, and with
participating business there is often significant
interaction between lapse risk and market/credit
risk. To address this issue, replicating portfolios
were generalised, first to include non-traded ‘asset’
functions of insurance risk and then polynomials in
all risk factors directly. In a sense the polynomial
can be thought of as a Taylor series expansion of
the replicating portfolio. In time, the connection
between replicating portfolios and the polynomial
has dropped and many firms just construct the
polynomial without reference to its replicating
portfolio origin.
The polynomials have become known as loss
functions. Two methods, illustrated in Figure 02,
are used to generate loss functions: curve fitting or
Least Squares Monte Carlo (LSMC).
Figure 02. Loss function generation – curve fitting and LSMC
(1) Curve fitting
A small number of
realisations of risk
factors − for example,
50−100.
For each realisation, value
the balance sheet
accurately (for example,
1,000−5,000 simulations).
(2) LSMC
A large number of
realisations of risk
factors − for example,
1,000−10,000.
Portfolio value
Portfolio value
100
100
80
80
60
60
40
40
20
20
0
0
20 40 60 80 100 120 140 160 180 200
-20
20 40 60 80 100 120 140 160 180 200
-20
Portfolio value in base case and sensitivities
Loss function fitted to base + sensitivities
4 towerswatson.com
For each realisation,
value the balance sheet
approximately (for example,
2−10 simulations).
Portfolio value from simulations
Loss function based on LSMC
Insights | January 2013
Curve fitting refers to the process of constructing
a loss function by interpolation or regression from
a relatively small number of full balance sheet
valuations. Curve fitting can be seen as a natural
generalisation of the stress test approach, as stress
tests can be thought of as linear loss functions.
Curve fitting has strong benefits, not least its
simplicity – you run the ALM model and fit the curve.
There are challenges, however, if the loss function
is truly to be a generalisation of the stress test
approach. Liabilities are multidimensional, so that
the best-fit polynomial (now thought of as the Taylor
series expansion of the liabilities) may be a function
of 10–15 risk factors with 100 or so non-zero terms,
and the process of curve fitting does not in itself
prescribe the structure of the polynomial. However,
many firms have adopted a curve fitting approach as
a valuable extension of the stress test/correlation
matrix approach and have utilised grids to ensure
efficient production of the 100 or 200+ balance
sheets required.
Whereas curve fitting adopts a brute force approach
to the construction of loss functions, LSMC offers a
more efficient alternative by exploiting mathematical
methods. LSMC originated in work undertaken to
estimate American option prices using Monte Carlo
simulation. In insurance terms, LSMC builds on the
early replicating portfolio work that fitted replicating
portfolios to Monte Carlo simulations of liabilities.
Essentially with LSMC, the polynomial is constructed
as a regression of very approximate values of the
liabilities against values of risk factors. Done well,
using what we now call ‘advanced LSMC’, from a
single projection of 20,000–100,000 simulations,
the method constructs and fits an optimal
polynomial without requiring foreknowledge of
the optimal structure of the polynomial.
“Done
“
well...‘advanced
LSMC’...constructs and fits an
optimal polynomial without
requiring foreknowledge
of the optimal structure of
the polynomial.”
RiskAgility EC
From a business perspective, a workable
methodology is only part of the solution.
Life firms need software to deploy the methodology.
Towers Watson developed RiskAgility EC specifically
to deliver the Monte Carlo simulation of the 1-year
VaR definition of economic capital (using proxy
models) required by Solvency II. The system is
constructed around the risk factor loss function
paradigm outlined above and was designed to
address both the technical and business challenges
described earlier.
The software, now in version 2.2, has matured
so that firms selecting this software are able to
implement an internal model from inception to
deployment in a business-as-usual (BAU) context in
three to four months.
A specific goal with RiskAgility EC was to create a
system that broke the link between the underlying
ALM systems and management reporting. ALM
models are used as calibration tools, with most
firms choosing to move towards a ‘hard close’
calibration cycle of either two or four calibrations
per year. Once calibrated, management reporting
of risk and capital metrics is available daily as
RiskAgility EC allows users to monitor risk exposures
and update values for variations in financial markets
and volumes of business in force. With RiskAgility
EC monthly, weekly or even daily monitoring of
economic capital positions are possible.
Quick solvency monitoring is not new, but
RiskAgility EC industrialises the process and
provides a tool that enables much more robust
monitoring of solvency across the business.
Another key facet of RiskAgility EC is its ‘enterprise
nature’. Installation models vary, but it is not
desktop software; rather, it is a solution designed
to allow multiple users in multiple locations
access in a controlled way to the same model.
Consequently, the solution was developed with
embedded audit and governance features – users
check out elements of the model they wish to review
or change; runs are managed via job scheduling
tools; and versions of models and assumptions are
strictly controlled – last year’s run is repeatable as
the system captures it automatically. Experience
shows that the governance features initially make
some actuaries nervous – but the IT infrastructure
meets firms’ IT departments’ needs, as it fits in with
the typical emerging strategy for systemisation and
automation of core actuarial models.
towerswatson.com 5
Insights | January 2013
Validation of proxy models – advanced
LSMC and the direct method
Having developed an internal model, either with
RiskAgility EC or other aggregation software, the
final challenge under Solvency II is validation of
the model – either for internal use or as part of an
application for use as an approved internal model.
Solvency II has an exhaustive set of validation
requirements, and firms undertaking a formal
Solvency II validation need to map validation actions
against specific Solvency II requirements. More
broadly, validation falls into two categories:
•• Technical – does the model generate materially
accurate estimates of economic capital? In
addition to the general methodology, this would
be expected to cover: modelling of technical
provisions; modelling of risks including
risk dependency; use of proxy models; and
convergence of the Monte Carlo simulation itself
(given that a 99.5th percentile risk metric is used).
•• Use – is the model being used to assist with
the management of the business; is use of the
model feeding back into future improvements of
the model.
A key challenge in the technical validation is
validation of the proxy model. As noted above,
these can be constructed using either replicating
portfolios or loss functions. Until recently, proxy
model validation relied on ad hoc methods
because, despite widespread use of grids, direct
validation – against a full nested stochastic
simulation of the balance sheet is still not
possible – and mathematical methods to prove
accuracy appeared not to exist.
Recent work by Towers Watson has changed this, by
explicitly addressing two issues:
1. E
fficient construction of loss functions that can
be demonstrated to be robust, in the sense that
they fit the underlying balance sheet and produce
an error in the SCR that can be measured.
2. D
erivation of an estimate of the error in the SCR
when using a proxy model.
The first we have described before in this article:
‘advanced LSMC’. The second is called the ‘direct
method’ of estimating the SCR. The advances
rely on:
•• Specific (nested) scenario sets that are used to
create a very approximate value of the liabilities
across several thousand realisations of risk factors.
•• Analysis of the sources of breakdown and error
arising in the SCR from methods used to date to
construct replicating portfolios or loss functions
(either using curve fitting or LSMC).
6 towerswatson.com
The advances have been tested with clients in
Europe and we expect them to become standard
as economic capital modelling continues to evolve.
In test cases we have seen replicating portfolios
giving rise to material errors in the SCR, which the
new techniques can measure and correct. Critically,
these techniques have a mathematical foundation
and hence can be used to prove accuracy of
the SCR. Given that some European regulators
are challenging the use of proxy models without
‘proper formal validation’, these are significant
developments. Both will be subject to further
articles in this series.
Conclusion – the ‘state of the art’
So, after 10 or more years of development of economic capital models
in the life insurance industry, what is the ‘state of the art’? In Europe, it
is fair to say that Solvency II dictates much of what would be considered
best practice, but what of ‘state of the art’. The answer here has both
business and technical facets. The following, by highlighting ‘use’,
captures the business requirements:
“All
“ models are wrong; some are useful.”
George Box
Following George Box’s lead, we argue that the best economic capital
models are those that are useful…and used…to manage the business.
A well-understood (approximate) model used in day-to-day management
of a life insurer, is of more use to management, and would give more
comfort to a Board of Directors and also regulators, than a slow to
update, highly accurate and overly complex model. The best economic
capital models are models such as RiskAgility EC that are able to
provide firms with daily solvency reports, aggregating and allocating
risk and capital across a legal and management hierarchy and allowing
flexible analysis quickly addressing the questions asked by management
as they manage the business.
The ‘state of the art’ will push boundaries and go further than this.
Technology and methodology does not stand still and armed with new
tools, actuaries and management can ask new questions and seek
new levers in the battle for understanding and competitive advantage.
Approximations inherent in the use of proxy models, even if these
approximations are understood, create demand for new improved
techniques. The new methods developed by Towers Watson addressing
the accuracy of proxy models will enable actuaries to give management
more confidence in the accuracy of these models, thus enabling more
complex analysis and decisions to be taken with greater confidence.
We expect to see use of the new techniques described above becoming
standard and a model such as RiskAgility EC incorporating the new
analysis will remain, temporarily at least, the ‘state of the art’.
Insights | January 2013
Financial life modelling software
Global contacts
Clients in more than 30 countries – leading P&C and life insurance
companies, multinationals, pension funds, mutual funds and asset
managers – use our systems for enhanced risk and capital management.
Towers Watson
RiskAgility
Towers Watson
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RiskAgility SF
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John Rowland
+44 20 7170 3853
[email protected]
Bernhard Gose
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Peter Murphy
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Insights | January 2013
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