English Appendix
OPTIMUM CAPITAL DETERMINATION BY SIMULATION TECHNIQUES
«It is only after a long course of uniform experiments in any kind, that we
attain a firm reliance and security with regard to a particular event»
David Hume /1711 - 1776) Enquiry Concerning Human Understanding
In these troubled times the financial solvency of companies
is coming under much closer scrutiny, and their equity now needs
to be properly determined to ensure they can meet their
commitments with the highest guarantees.This is one of the main
aims of the project known as Solvency II. One way of
estimating the capital requirement is by using simulation
techniques.This paper examines the possibilities of two of them,
Monte Carlo and Bootstrapping, both of which have shown
themselves to be particularly useful in determining the capital
to match market risk and technical risks.
IRENE ALBARRÁN LOZANO
Universidad Carlos III de Madrid
PABLO ALONSO GONZÁLEZ
Universidad de Alcalá
Change has become one of the
bywords of our times,
sometimes gradual, often
sudden. Everything is in flux:
consumer tastes, their needs.The
business response needs to be correspondingly fleet-footed, without
any hidebound clinging to formulae or
procedures that have worked in the
past.There is now no choice but to
adapt to changing circumstances even
when this involves a costly learning
process. In fact this outlay should not
be perceived as a cost but rather as an
essential investment to keep afloat in
the market.As the Spanish saying goes:
Camarón que no se mueve, se lo lleva la
corriente (any shrimp that stays still is
swept away by the current).
These switchback changes in our surrounding circumstances mean that what was once sure ground has now become much less dependable underfoot.The only clear conclusion we can draw from this morass of uncertainty is that a good insurance policy is now essential. For this protection to fulfil its ends, however, two things need to be in place. Firstly, a clear-sighted perception of the circumstances and events that might trigger the insurance-related compensation mechanism. Secondly, the organisations offering this service to society need to be able to respond properly when called upon, without letting down all the people that have pinned their hopes on them.What do these two prerequisites mean? Simply that insurance companies should be capable of correctly weighing up the likelihood and intensity of certain events covered by their insurance and also should be backed up by sufficient resources to guarantee the compensation agreed in the policies.
The first axiom involves factors related to the design of products, such as prices, application conditions and compensation to be paid to policyholders.The second involves the procedure for evaluating their financial capacity for coping with any coverable events.
Risk and capital: BASEL II AND SOLVENCY II
In recent years, under the influence of these ever-changing circumstances, special heed has been paid to the analysis of this latter factor. This concern about sufficient financial capacity is not limited to insurance companies or to the European sphere. Nor is it anything new. Indeed the legislation on this subject goes back a long way, not only in the insurance world but also in the banking world. Nonetheless, this legislation took a general, across-the-board approach without considering the idiosyncrasies of each company. For some years now banks have shown a certain interest in ensuring that their equity matches the activities they take on. In a nutshell, companies that take on risks in the greatest number or intensity need to have more available resources than other companies that opt for more conservative activities or a lower contracting level.The Bank for International Settlements (BIS) undertook some years ago what has come to be known as the Basel II process, which is really nothing more than the practical implementation of the abovementioned principles.
The equivalent in the insurance world has taken many guises.The one impinging most on the EU countries, enshrined in a Directive, is colloquially known as Solvency II. Its ultimate declared aim is to ensure a better defence of European policyholders. This is its purpose.The means for achieving this end is a proper evaluation of the risk.This in turn implies not only identifying the various causes that might entail equity losses for insurers but also quantifying them correctly.To do so it is necessary to employ the technical arsenal that may prove necessary.
The whole process is hence based on two premises: firstly, a proper knowledge of all potential lossproducing situations and secondly, an evaluation of how much might be lost in each one of them.This depends on an estimation of the occurrence probability of the phenomena under consideration. For the first task the EU has drawn up a series of field studies through the Committee of European Insurance and Occupational Pensions Supervisors (CEIOPS), striving to identify each and every possible equity-threatening situation.These are known as Quantitative Impact Studies (QIS). In brief, they aim to classify risks into modules, pooling those that are akin to each other.At the date of writing –autumn 2008–, four series have been brought out, with possibilities of a fifth or even sixth in the future. But the aim in view here is not only to identify risks individually but also, and more ambitiously, to quantify them and evaluate the relationships they bear to each other. Long debates might be held about these two questions, and the soundness of the methods used might be called into question. It might be thought, for example, that risk calibration would be based on the figures of a given insurance class or situation corresponding to all EU countries that will be affected by the reform of Solvency II.This is not in fact the case, different data having been used according to the situation under analysis. In the market risk module, for example, the data used were the following:
For the interest rate risk, the German zero coupon rates for securities with a residual life of one year and euro IRS rates.
For the equity risk, the yields obtained from the MSCI Developed Markets Index, pooling listed prices from 23 countries around the world.
For property risk, the IPD indices corresponding to Holland, France, Germany, Sweden and Great Britain.
For currency risk, the euro exchange rates of a set of currencies (US dollar, pound sterling,Argentine peso, Japanese yen, Swedish krona, Swiss franc and Australian dollar), forming a basket whose weights are those corresponding to the positions held by the Dutch financial institutions.
For spread risk, the Moody´s series based on US treasury securities.
The next step is then to evaluate
possible losses, leading to an estimate
of the capital exposed to risk.To do so
it is first necessary to define the desired
level of security for this purpose. In
other words, the capital sum needed to
preserve the firm’s financial strength
will vary directly with the range of
situations to be protected and the
required level of confidence.This takes
us into the field of probability and the
available tools for evaluating the
necessary capital level.As regards this
latter aspect, the chosen measuring rod
is VaR -Value at Risk-, which has a
long track record in the banking
world. In most cases this concept
necessarily goes hand in hand with the
assumption of a given probability
distribution function.The normal or
Gaussian distribution function has
been chosen for the QIS tests, its main
advantage being its ease of use.
Nonetheless, it is somewhat rash to
assume this behaviour without having
performed the corresponding statistical
crosschecks to ascertain the actual
distribution.Witness the following example, based on the daily
EUR/USD quotation from 2/1/2004
to 10/10/8 (n = 1229), obtaining the
following values for their basic
statistics:
| Average daily yield | (%) | 0,007% |
| Daily standard deviation | (%) | 0,547% |
Thus, the daily VaR1 at different probability levels, obtained from historical figures and based on a normality hypothesis, is the following:
| Empirical | Normal | Difference | |
|---|---|---|---|
| al 95,0% | -0,76% | -1,221% | 0,345% |
| al 97,5% | -1,015% | -1,396% | 0,381% |
| al 99,0% | -1,420% | -1,727% | 0,307% |
| al 99,5% | -1,765% | -1,921% | 0,148% |
This use of stochastic models is not a completely new feature of Solvency II. On the contrary there are historical precedents dating right back to 1953 when Finland began to use specific capital models for each company.The stochastic character of the insurance business was also taken on board in the Special Equalization Reserves.The most recent precedent is to be found in the Swiss Solvency Test or SST.This technique pursues a twofold aim, seeking not only to ascertain the risk-related capital figure but also its statistical distribution. Its calculation is based on the combined use of stochastic models and scenarios. The main thrusts of SST, very similar to those of Solvency II, are the following:
- Market price valuation of both assets and liabilities.
- The market value of the payment commitments is equal to the best estimate plus a risk margin.
- The key risks in any company are market, credit and technical risks.
- The risk measurement used is the expected loss in a single year or Tail VaR.
- The key risks have a standard model distribution.
- The target capital is obtained from a standard stochastic model together with a set of scenarios.
- In the event of financial difficulties, the policyholders are protected by the risk margin.
- Internal models may be used; their hypotheses have to be duly recorded in a report and sent to the regulator.
- Reinsurance is considered.
Monte Carlo ASSET PRICING
Both in the Swiss Test and in Solvency II the simulation-related situations we might consider would be those associated with the market pricing not only of assets but also of liabilities, specifically known as the Best Estimate.These values then serve as the basis for estimating the distribution of values in the particular phenomenon under consideration.
As regards asset pricing, the Monte Carlo method gives us the final value of a financial asset, or a portfolio thereof, after replicating thousands of times the trajectory it might follow over time.This exercise works from the simplest case, valuing a single basic asset such as a bond or share, up the most complex case for valuing a portfolio comprising several assets and/or derivatives. In all cases the basic premise is the assumption of a given hypothesis over the stochastic behaviour of the prices.Traditionally, the financial theory assumes that the prices follow a lognormal distribution or that the yields follow a normal distribution, in which case the value of the asset at any point in time can be expressed as follows:
where ì is the average yield and ó the volatility. Take the example of a share with the current price of ?25, with an average annual yield of 15% and volatility of 25%. If we want to know the probable loss it might suffer, we have only to generate thousands of random numbers following a standard normal distribution, feed each one into the above expression and obtain the year-end price for each one of them.
After performing 10,000 replicas, the frequency distribution obtained for the final price is shown in the following graph:
The VaR and TVaR values for different levels of confidence are shown in the following table:
| Associated loss (%) | Associated price | |||
|---|---|---|---|---|
| Confidence | VaR | Tail VaR | VaR | Tail VaR |
| 90 | 11,99% | 19,53% | 22,003 | 20,12 |
| 95 | 17,94% | 24,29% | 20,515 | 18,93 |
| 96 | 19,91% | 25,63% | 20,023 | 18,59 |
| 97 | 21,71% | 27,22% | 19,573 | 18,20 |
| 99 | 27,90% | 32,80% | 18,026 | 16,80 |
| 99,5 | 31,19% | 31,19% | 17,204 | 15,99 |
At the required Solvency II level, the VaR is 31.19% and the capital amount is the difference between the initial price (€ 25) and the price associated with this loss (€ 17.204), i.e., € 7.796.
BOOTSTRAPPING DETERMINATION OF RESERVE LEVEL
As regards estimating the most probable value of technical provisions, the technique used is resampling or bootstrapping. Put simply, the simulation here involves repeating a sample-generating process enough times, for example 10,000, to be able to draw inferences therefrom. By repeating and generating data samples, the aim is to study the accuracy associated with the various statistics we wish to use, such as the mean or median.The number of possible different samples that might be drawn is determined from the following expression2:
As already pointed out, one of the most widespread applications of resampling in the actuarial world is the estimation of the reserve figure. In general, the idea is to combine the use of this methodology with such widespread reserve calculation systems as the Chain Ladder.To do so,working from a model that explains the sums paid out in claims, the aim is to make a resampling of the residuals of this model to ascertain the provisions that should be made against future claims. The first step is therefore the choice of the model to be used. But this subject goes beyond our remit here; readers interested in pursuing it further will find a plethora of scientific articles on the matter, particularly works by Mack, England,Verrall and Renshaw.
The estimation process is based
on the performance of a series of
phases for obtaining the desired figure.
This process has to be repeated a very high number of times, building up the
overall result distribution from the
results obtained in each iteration. In
brief, the procedure starts by
performing a traditional Chain Ladder,
which involves obtaining the
development factors, estimation of the
cumulative values on the basis of these
factors and calculation of associated
annual increments. From these annual
values we obtain what are known as
unscaled Pearson residuals, on the basis
of which the Bootstrapping is carried
out by random drawing. From then on
we follow the opposite path to the one
followed hitherto, i.e.,working
backwards from this new set of
residuals we obtain the values
associated with the annual figures,
building up the cumulative figures
therefrom.These then serve as the basis
for calculating the new development
factors; finally, the reserves are
calculated from the results of the
regenerated sample.The process ends
with the estimation of the current
value of the reserves, since this is one of the Solvency II requirements.To do
so, it is necessary to set up a path of
future rates for performing a cash-flow
discount and obtain the value sought.
This whole process has to be repeated a number of times, for example 10,000, to build up an estimated payment distribution.This then serves as the basis for calculating, at the desired confidence level, the VaR or selected risk measurement.
FINAL CONSIDERATIONS
This paper offers a brief overview of simulation techniques for insurance companies, mainly Monte Carlo and Bootstrapping, describing them and explaining how they might be applied. Despite their enormous flexibility, all of them work from starting models or premises. It is vital to check the validity of these premises and also be ready to modify and update them, especially in view of the unprecedented rate of breakneck change we have to deal with today.3
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REFERENCES
(1)Although Solvency II lays it down that capital figures are to be calculated over the timeframe of one year, here, for merely demonstrative effects, they have been calculated on a daily basis.
(2)Hall, P. (1992): The Bootstrap and the Edgeworth Expansion, Springer-Verlag, Appendix I.
(3)This paper stems from the book «Análisis del riesgo en seguros en el marco de Solvencia II: técnicas estadísticas avanzadas. Monte Carlo y Bootstrapping», by the same authors. This book has been published by FUNDACIÓN MAPFRE as number 119 of the collection Cuadernos de la Fundación and it was funded by a Risk and Insurance Grant conceded to the authors by the Fundación in 2005.