© 2009 the actuarial profession estimating var in the presence of model uncertainty parit jakhria...

© 2009 The Actuarial Profession www.actuaries.org.uk

Estimating VaR in the presence of model uncertainty

Parit Jakhria & Stuart Jarvis

11 September 2012

Acknowledgements

• Joint work with Andrew Smith as part of the extreme events working party

2© 2009 The Actuarial Profession www.actuaries.org.uk

Agenda

• Background– Modelling– Value at Risk– Types of uncertainty

• Model Uncertainty – Known model and parameters– Unknown model and parameters!

• Parameter Uncertainty– Example– Adjusting for parameter uncertainty

• Other Tools– Confidence and Predictive intervals– Frequentist vs Epistemic probability, Bayesian techniques– Considering families of models

3

Modelling

• Modelling seems to be an inescapable part of Actuarial Life

• In this context, we think of models as tools / processes that:– Use information from the past (history)– Together with knowledge about a particular problem (judgement)– To model future (uncertain) outcomes– (and hence help make decisions about the future)

• A model is necessarily a simplified representation of the real world!

4

Value at Risk

• Definition!

5

Types of Uncertainties

• A model is necessarily a simplified representation of the real world!

• The process of stripping down to the bare useful components and ‘calibrating’ the resultant model has (necessarily) got a large amount of judgement associated with it.

• This judgement can manifest itself in various different ways. Broadly speaking, some of the ways we encounter decisions over the process of actuarial modelling

– Choice of overall framework for the model

– Choosing individual parts of the model (e.g. distribution)

– Choice of calibration methodology

– Choice of parameters, overriding certain parameters if necessary

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For professional clients / qualified investors only

Model certainty

If we know true distribution, can just read off quantile

e.g. normal distribution with known

• 99.5th percentile is

e.g. t distribution with 10 degrees of freedom

• 99.5th percentile is 3.17

etc.

7

Model Uncertainty!

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What does ‘1 in 200’ even mean?

• If underlying model for observations X is known, then– 99.5% quantile q is calculable as F-1(0.995)– Prob(X<=q)=99.5%

• If model is unknown then q is unknown.

• Consider two estimators VaR(1) and VaR(2) defined by:– E(VaR(1))=q i.e. VaR(1) is an unbiased estimate of q– P(X<=VaR(2))=0.995

• Often this will mean VaR(2)>VaR(1)!



Example: normal distributionUnknown parameters, n observations

• Standard unbiased estimates of mean and standard deviation

• i.e. ,

• Then is an unbiased estimate of , the quantile we are after

BUT

• It is not true that the next observation has a 0.5% chance of exceeding this estimate (or that 0.5% of the next m observations will exceed this level)

•

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Adjusting the estimate

• Even though , we have

• This is to be expected: the tail cdf function is convex in the region of interest and the estimator is uncertain, so this behaviour is familiar from Jensen’s inequality.

𝑭 (𝑥 )

• An adjustment can be determined analytically

VarEstimator =

Such that

•

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Confidence intervals v prediction intervals

• Confidence interval

• Statement about parameters

• Can be Monte Carlo tested

• Parameter unobservable so untestable in practice


• Prediction interval

• Statement about observations

• Readily tested in practice

• This is exactly the VaR backtest: count ‘exceptions’ in a time series

Approaches to dealing with uncertainty

Situation Responses

Known parameters Exact formula available

Unknown parameters Create estimators of known exact formula.

Typically hit desired probability level exactly

Calculate prediction interval for next observation.

Typically hit desired probability level exactly

Unknown model Bayesian approach

Bayesian prior over family of models.

Calculate desired quantile of posterior distribution.

Probability of observation exceeding VaR may be >0.5% for some models.

Robust approach

Family of models form an ambiguity set.

Determine VaR so that quantile exceeded for all models.

Probability of exceeding VaR may be <0.5% for some models.


Bayesian approach

• Prior / posterior probabilities associated to parameters quantify ‘degree of belief / knowledge’: epistemic probabilities

• Choice of prior required– Classroom cases: conjugate family; non-informative prior– Building knowledge on foundation of ignorance?– Jeffreys prior responds to parameterisation problem

• Posterior distribution of parameters -> posterior predictive distribution of observations, from which quantile can be read off


Robust approach

• Seek VaR such that for all

• This is not as tough as it sounds: don’t need since VaR is a random variable (it depends on the data)– Often results in quite reasonable answers

• In classroom cases, where there are sufficient statistics, pivot quantities, the choice of VaR is quite straightforward– Eg mean + k. standard deviation in the normal case


Robust approach in practice(Very simple) example

• Models = uniform distributions on for all

• Observations x[1],…,x[n]. Maximum m is sufficient statistic.

• Easy to show that – Provided i.e.

• So the random variable is a VaR statistic


Bayesian approach to previous example

• Non-informative prior:

• Posterior:

• Posterior predictive: for

• p’th quantile of this is

• Identical answer to the robust approach!


Normal exampleUnknown mean & variance – still classroom • Bayesian: uninformative prior , )

– Posterior distribution for parameters is Normal inverse gaussian

– Posterior predictive )– Desired percentile is

• Robust:, so– is a VaR(p) for all parameters

• Again identical! Beware classroom examples.


Practical issues

• In general robust & Bayesian approaches won’t agree

• They don’t even provide a recipe – generic guidance at best– In absence of conjugate family, [non-informative] prior difficult

to determine. Eg what prior over PIV family?– In robust case, VaR might be any functional of the data. In

absence of a pivot quantity, difficult to narrow this down

• Shimmer of objectivity is probably illusory anyway– Already making judgements on class of models



Inherent model uncertainty

• In practice never know that we’re sampling from (say) a normal distribution

• Normal distribution may even provide a good fit to true distribution overall but be too thin in the tails

• Clever mathematics to determine exact interval may be irrelevant

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Other sources of error

• Time-variation in returns. ‘True’ parameters will evolve over time; distribution may be conditional on current state of the world, which may itself be difficult to identify at the time

• This may lead to pro-cyclical behaviour

• Variation of factor exposures. Even if data are available on well-understood risk factors, the business exposure to these factors may vary over time and not be readily identifiable

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Next steps

• Move away from classroom distributions– gave nice relationships; learned something but need more

• Direction 1: bigger families of more practical interest– Eg Pearson IV, hyperbolic– Explore VaR choices & trade offs

• Direction 2: time-variation in returns– Eg Markov-modulated regime-switching model with

conditionally normal returns– Explore procyclicality & model fitting


Contact

• Stuart Jarvis – [email protected]

• Parit Jakhria – [email protected]

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mailto:[email protected]

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.


© 2009 the actuarial profession estimating var in the presence of model uncertainty parit jakhria...

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