1. © 2009 The Actuarial Profession  www.actuaries.org.uk Estimating VaR in the presence of model uncertainty Parit Jakhria & Stuart Jarvis 11 September 2012

2. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Acknowledgements Joint work with Andrew Smith as part of the extreme events working party 1 © 2009 The Actuarial Profession  www.actuaries.org.uk

3. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 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 2

4. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 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! 3

5. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Value at Risk Definition! 4

6. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 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 5

7. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Model certainty For professional clients / qualified investors only 6

8. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Model Uncertainty! 7

9. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 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)! 8 © 2009 The Actuarial Profession  www.actuaries.org.uk

10. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Example: normal distribution Unknown parameters, n observations For professional clients / qualified investors only 9

11. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Adjusting the estimate For professional clients / qualified investors only 10

12. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Confidence intervals v prediction intervals 11 © 2009 The Actuarial Profession  www.actuaries.org.uk

13. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Approaches to dealing with uncertainty SituationResponses Known parametersExact formula available Unknown parametersCreate estimators of known exact formula. Typically hit desired probability level exactly Calculate prediction interval for next observation. Typically hit desired probability level exactly Unknown modelBayesian 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. 12 © 2009 The Actuarial Profession  www.actuaries.org.uk

14. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Bayesian approach 13 © 2009 The Actuarial Profession  www.actuaries.org.uk

15. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Robust approach 14 © 2009 The Actuarial Profession  www.actuaries.org.uk

16. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Robust approach in practice (Very simple) example 15 © 2009 The Actuarial Profession  www.actuaries.org.uk

17. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Bayesian approach to previous example 16 © 2009 The Actuarial Profession  www.actuaries.org.uk

18. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Normal example Unknown mean & variance – still classroom 17 © 2009 The Actuarial Profession  www.actuaries.org.uk

19. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 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 18 © 2009 The Actuarial Profession  www.actuaries.org.uk

20. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 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 For professional clients / qualified investors only 19

21. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 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 For professional clients / qualified investors only 20

22. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 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 21 © 2009 The Actuarial Profession  www.actuaries.org.uk

23. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 Contact Stuart Jarvis – stuart.jarvis@blackrock.com Parit Jakhria – parit.jakhria@prudential.co.ukparit.jakhria@prudential.co.uk 22

24. Colour palette for PowerPoint presentations Actuarial Bright Green R193 G216 B47 Actuarial Slate R65 G89 B104 Secondary Olive Green R120 G162 B47 Secondary colour palette Primary colour palette Secondary Bottle Green R0 G147 B127 Secondary Turquoise R0 G138 B176 Secondary Aqua Blue R31 G190 B202 Secondary Pastel Green R126 G205 B195 Secondary Light Purple R185 G199 B212 Secondary Purple R117 G129 B191 Secondary Ecru R186 G163 B171 Secondary Yellow R255 G196 B37 Secondary Orange R249 G157 B49 Secondary Red R238 G52 B36 Secondary Rubine Red R226 G1 B119 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. 23 © 2009 The Actuarial Profession  www.actuaries.org.uk