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Life conference and exhibition 2010
Steven Morrison and Alex McNeil

Application of Extreme
Value Theory to Risk
Capital Estimation
7-9 November 2010
© 2010 The Actuarial Profession  www.actuaries.org.uk

Application of EVT to risk capital estimation:
Agenda

• Motivation
• Background theory
• VaR case study
• Summary
• Questions or comments?

1

Application of extreme value theory to risk capital estimation

Motivation


Motivation

• Measures of risk capital are based on the (extreme) tail of a distribution
– Value at Risk (VaR)
– Conditional Value at Risk (CVaR) / Expected Shortfall

• In particular, Solvency II SCR is defined as a 99.5% VaR over a one year
horizon

• Generally needs to be estimated using simulation
1. Generate real-world economic scenarios for all risk drivers affecting the
balance sheet over one year
2. Revalue the balance sheet under each real-world scenario
– e.g. Monte Carlo („nested stochastic), Replicating Formula, Replicating Portfolio

3. Estimate the statistics of interest
3

Motivation

• An insurer who has gone through such a simulation exercise states
– “Our Solvency Capital Requirement is £77.5m”

• How confident can we be in this number?

• Many sources of uncertainty
– Choice of economic scenario generator (ESG) models and their
calibration
– Liability model assumptions e.g. dynamic lapse rules
– Choice of scenarios sampled i.e. choice of real-world ESG random
number seed

4

Motivation

• The same insurer re-runs their internal model using a different random
number seed (but all other assumptions are unchanged)
– “Our Solvency Capital Requirement is now £82.8m”

• So, simulation-based capital estimates are subject to statistical uncertainty
– Can we estimate this statistical uncertainty?
– How can we reduce the amount of statistical uncertainty?

• In this presentation, we will address these questions using a statistical
technique known as Extreme Value Theory (EVT)

5


Background theory



VaR case study


VaR Case study

Liability book
• UK-style with profits
– Management actions, dynamic EBR, dynamic bonus rates,
regular premiums

Valuation methodology
• Nested stochastic
– 1,000 real-world outer scenarios
– 1,000 risk-neutral inner scenarios per outer scenario

8

Estimated distribution of liability value at end of year
(empirical quantile method)

• Estimate „empirical
quantiles‟ by ranking
1,000 scenarios

• Estimated 99.5% VaR =
£77.5m
(995th worst-case
scenario)

• Note that estimated
distribution is „lumpy‟,
particularly as we go
further out in the tail

9

Estimated distributions using different scenario sets
(empirical quantile method)

Initial set of 1,000 real-
world scenarios
• 99.5% VaR = £77.5m

Second set of 1,000
real-world scenarios
• 99.5% VaR = £82.8m

10

What is the ‘true’ VaR?

• Two different estimates for VaR
– £77.5m
– £82.8m
– Which is „correct‟?

• Both use same (subjective) modelling assumptions
– Same economic scenario generator and calibration
– Same liability model assumptions e.g. dynamic lapse rules

• Difference is purely due to different random number streams
used to generate the economic scenarios
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Two important questions

1. Can we reduce the sensitivity of the estimate to the choice of
random numbers?
– Run more scenarios
– May not be feasible because of model run-time

– Find a „better‟ estimator than the empirical quantile

2. Given a particular estimate of the 99.5% VaR, can we estimate
the uncertainty around this?

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Application of Extreme Value Theory

• Recall that Extreme Value Theory tells us something about the
shape of the distribution in the tail
– Distribution of liability value beyond some threshold is
(approximately) Generalised Pareto
– Parameterised by 2 parameters

• Estimate the tail of the distribution by:
1. Picking a threshold
2. Fitting the 2 parameters of the Generalised Pareto
Distribution to values in excess of the threshold

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Choice of threshold

20
• Choice of threshold is subjective
– But examination of „mean excess

Mean Excess (£m)
15
function‟ helps identify a suitable
choice 10

5
• We have judged that a threshold of 20 40 60 80 100
Threshold (£m)
£40m is suitable for this particular 20
case study

Mean Excess (£m)
– Approximately 26% of scenarios 15

exceed the threshold
10

5
20 40 60 80 100
Threshold (£m)

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Estimated distributions using different scenario sets
(Extreme Value Theory method)
Initial set of 1,000 real-world
scenarios
• 99.5% VaR = £75.9m
• 95% confidence interval =
[71.1m, 84.3m]

Second set of 1,000 real-
world scenarios
• 99.5% VaR = £77.8m
• 95% confidence interval =
[72.2m, 87.7m]

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Summary


Summary

• Simulation-based measures of risk capital, e.g. VaR, are subject
to statistical uncertainty

• Extreme Value Theory provides a robust method for estimating
VaR
– Allows statistical uncertainty to be estimated
– Statistical uncertainty lower than „naïve‟ quantile estimation
– Provides an estimate of entire tail of distribution, allowing estimate of
more extreme VaR, CVaR etc.

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Questions or comments?

Expressions of individual views by
members of The Actuarial Profession
and its staff are encouraged.
The views expressed in this presentation
are those of the presenter.

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Application of Extreme Value Theory to Risk Capital Estimation

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Application of Extreme Value Theory to Risk Capital Estimation