Time Based Model CFE-ERCIM'11_light2

8/2/2019 Time Based Model CFE-ERCIM'11_light2

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A new time based approach

for operational risk

Duc PHAMHI

Dept. of Financial Engineering

CFE-ERCIM 2011

CS 16: Risk management

Dec 17, 20111 Duc Pham-Hi


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The current model in AMA Loss distribution approach

Not a market model

Not a time based model good reasons for this : rare events

Based on Monte Carlo Convolution between frequency and severity distributions

Practical unsatisfactory in agregation and scenarios



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Most frequent practice of operational risk For a single type of loss:

But in practice, yearly losses are given by

=

=1

)()(k

kXjL

)(p

with p ~ Poisson and Xk~ lognormal, GPD, etc.

Product of 2 distributions

Computed with FFT

Computed with Panjers formula

But in most cases with Monte Carlo

=

=1k



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LDA schematics

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No time dimension AMA models are currently time insensitive

Switching time axis produce sameValue at risk !

They are even almost risk insensitive

Add a last loss worth a VAR

The new VAR can be moved only by a few percentdepending on the size of time series of losses

No real time risk monitoring possible !



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Value-at-Risk level is insensitive to risk

evolution

Illustration :

Value At Risk at 99,9 % in these 2 situations is the same !

For the bank with more risks and the bank with diminishing risks

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Even worse : Regulatory time space

Regulatory capital asValue at risk computed quarterly

New losses as new data But recalculation of parameters only a yearly exercise

Qualitative data updated less frequently.

Parameters of model to evolve with enormous inertia, due tothis batch process.

Updates are slow and not welcome Many theoretical difficulties still untreated



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Difficulties at each step bottom-up

Local loss distribution

Total loss distribution

Risk Mitigation (1)Capital requirement

Aggregationmechanism

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Data collection

Data refinement and EDA

Statistically homogeneous risk class ?

Quantification (Severity-Frequency)both LDA and SBA

Correlations /dependencies

s m ga on

Fittingmethodology

Data quality

Dec 17, 2011


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Why Supervisors should pay attention to

modeling process

As a consequence :

Top management cares only aboutValue at Risk

Managers are not familiar with probability density

functions (Pareto, Weibull, Lognormal ) enough to careabout parameters.

Modelers are the only ones who care, but they have nomandate

Parameters do not evolve , or only as long term stairs

function.



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More frequent Value-at-risk watch

When in crisis, value-at-risk should be watched daily e.g. Kerviel case : top management needs ways of tracking how the

value-at-risk of not only portfolios, but also of operational risks,evolve as a time function of the unwinding of those massive illicittransactions.

, A statistical approach demand too much (scarce) data for

calibration Price of sophistication can outweight benefits (g-and-h)

Alternatives Use Lvy process Use exploratory modeling Maybe both



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Modelin time based o erational

risk with dynamic control

Dec 17, 2011Duc Pham-Hi11


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Modelling Loss using Lvy process Small losses as Lognormal

Large losses as jumps

(heterogeneous Poisson) Evolution of aggregate loss is sum

If are added other business characteristics for the bank :

)exp(0, tAA BXX =

tBB LXX exp0,=

tBtAt XXZ ,, +=

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a drift which is the rate of earning a subordinator representing a provisioning system that

compensates for the

cumulated loss

put together, they yield the wealth evolution equation (much like

Itos formula applied to deriveZt )

dtdR =

)(),(),(~

dzdtdzdtNdzdtN =

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Incorporating time into operational risk

models Differential form for loss equation, derived from Itos formula

The banks evolution equation is ++=

< **R 1

),(~

)1()()1(R

z

z

zdzdtNedtdzzebZdZ 1

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where trepresents the part of losses unabsorbed by reserves

This modelization with time flow enables risk management policy to

be formulated as an optimization problem

tt

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Ex erimentation :

human fraud modeling

(not Levy based)



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On Fraud modeling Op risk modeling in banks needs to take new tracks

Supervisors and regulators should encourage moreexploration of human risks

Fraud

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Litigations and lawsuits which are hidden and cumulative,then reach unexpected punitive levels

Time is of the essence !

But is until now absent in models

Risk policy making should be modeled too.

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Exploratory forward direct modelling

Assume an organization which produce errors, controls errors,

make mistakes or voluntary fraud, some of which accumulateuntil discovered

Experiments with computational models of 2 interacting

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departments Lognormal x Poisson as usually encountered in banks model,

but add 2 parameters :

Generation of fraud rate/threshold

Detection of fraud frequency/threshold

Small, single risk models

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Fat tails are

generated

naturally

200

400

600

800

1000

1200

1400

1600

140

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risk prevention cancut tails even shorter

0 2 4 6 8 10 12

x 10

5

0 1 2 3 4 5 6 7

x 105

0

20

40

60

80

100

120

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The rate of bursting

the bubble can becontrolled

3.5x 10

5

1

2

3

4

5

6

7x 10

5

Duc Pham-Hi181.3 1.35 1.4 1.45 1.5 1.55 1.6

x 105

0

0.5

1

1.5

2

2.5

31.3 1.35 1.4 1.45 1.5 1.55 1.6

x 105

0

Dec 17, 2011


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Risk states: what about effects of fighting

fraud, environment of fraud?

Wealth WWealth

W+dWTransition Probability

Environt var.

)()( ttF =


Control variables ,

=)(

0

)()(tN

jt xKLG

amountfixedHwherexHxK jj == )(),(inf)(

1)(0).()(


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Risk Management dynamic control factors

Reducing small losses through better process management, e.g.

where t is the loss reduction factor whose cost is expense

))(()( ttF =

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Reducing impact of catastrophic events

through insurance, or recovery plans, at cost where losses xj maybe capped or reduced ,

=)(

0

)()(tN

jt xKLG

1)(0).()(


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Valuing risk management policy Wealth evolution is :

We get state of wealth at time

( ) =

+++=)(

1

0, )()()()(tN

j

jtAt xKdBXdtRdW

=

),,),(,,( ttttdWW

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which leads us to formulate policy whose economic value underutility function Uis :

where t is the given of a pair ((t) , (t))

0

[ ]dttWUrtV =

0

)()exp(

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Risk management

as an optimal policy search through HJB Objective is to maximize a time based value

Resolution strategies

Solve as pure Hamilton-Jacobi-Bellman through Galerkin

[ ]

= dttWUrtEV

0

)()exp(max

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reminds of Basel II matrix)

Neural techniques/ supervised learning, if experience base available

and sufficient. Adaptive, Reinforcement Learning , TD learning, Q online sampling

MCMC exploratory techniques instead of backward fitting from data

= kkrJ

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About solving HJB Known cases

Linear Quadratic

in Merton's problem and Black Scholes context New : with Levy processes, but only for HARA and

CARA utility function : explicit solutions


But still, classic obstacles

Unknown P(x,y) POMDP in Markovian case

Curse of dimensionality

Too large sets { y } for eachx

Too large sets {} for eachx

Too strong nonlinearities


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Ex lorator solution strate ies for

dynamic op risk equation



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Learning as quick sampling of state space

wrt. rewards To use Q-learning, (Watson) philosophy is Action-Reward :

using time based action at:

at=

(x

t)then reward the (stochastic) consequence

{ }),(),( axrEaxR =


Classically , define value function is the total of what can be expectedin the future (here zero terminal value)

Introducing a discount rate and taking the expected value(stochastic case) :

==

0)),(,()),(,(

ttt dttxxrtxxV

==

=0

0))(,()(t

tt

txxxxrExV


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Dual problem : solving for strategies or for

economic value of risk Rationality of the risk center is to seek maximum of value, starting from state t x0 ,

to "learn" policy maximisingV over set A of admissible actions satisfying

where V(x)is the the consequence of following policy from initial situation x[ ]

)(min)(00

*

xVxVA

=

t


Now introduce transition mechanism between states xand y:

If we want to solve for optimal policy , it requires dealing with nonlinear equation

+=

=0

* ))(,())(,(minarg)(t

tt

txxrExxRx

=0

,,

t

tt

+= ++y

tyxtt yVPxRxV )(..),()( ,11


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Solving for risk with Temporal Differences We turn to solving for value ( if G is terminal value)

by reasoning in terms of discrete time. Alternately, in terms of discrete states y,as possible outcomes of state x, and introducing action at :

[ ]),,()1),((min),( tuxGtxfVtxVUu

++=

+= a

yVyxPxarV )(),(),(min*


We can iterate on V since the problem is linear.

let t be the proxy for V at time t ; we iterate thus :

As a special case, Q-learning is particularly easy to set up and is model-free

=y yxPxyxP 0),(,1),(y

+ + y tt

yyxPxrxV )(),(.),(min)(1

),().,(),(),(1 ayQyspasgasQ tt +=+


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Forward solutions : filterin and

online learning



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Operational risks and expert opinions Role of scenarios

Mandatory internal surveys

Quantification ?

How LDA and scenarios mix

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Often labeled bayesian approach Its rather belief networks

Another variant, Bayesian networks Essentially a factor network

Quantified through exposure, occurrences, severity parameters

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Positioning bayesian and non-bayesian

inferences

Exploration (previously seen) requires online states sampling

Q learning (Watson)

Reinforced learning (Barto & Sutton)

Temporal Differences (Tsitsiklis & Van Roy)

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Too little data ? Learn from simulation

Interacting Particle Systems (N. Shephard & Flury)

but if and only if model has no bias

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Transpose real time filtering techniques. One of the easier ways in filtering is to try some moving average

with or without weighting systems that let one attribute a decaying role to

older data as one goes back in time.

One of the more sophisticated filters is the Kalman filter.

considers the whole distribution of probability,

if it can be assumed that it is Gaussian, and that the distortion is linear.

But in risk situations that are extreme, neither of these assumptions holds.

Sequential Monte Carlo (SMC), or Interacting Particle Filters (IPS).

has been applied to Probabilistic Robotics [Thrun][De Freitas] or signal

processing [Doucet]



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An example of IPS tracking a noisy evolution



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Getting proactivity into equations of

operational Losses

just feed the projection of new losses into the usual batch process for theValue-at-risk

However, as already discussed, this process suffers from too muchinertia.

A more proactive way to take into account new data : Bayesian blending of

expert based opinions and hard, collected loss data. This method has long been discussed, as various implementations of

credibility theory were proposed. One of the more rigoroustreatments of the topic is by [Lambrigger ShevchenkoWuthrich].

Even if expert opinions are off target, they bring about a quickerconvergence of the inferred parameters to the real hidden values.[Peters]



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Lambrigger-Shevchenko-Wuthrich approach Give weights to sources of information

)()1()()()(exp212int1

xFwwxFwxFwxFertext

++=

)()( =PoissonNd

Given a set of observations at timeJ,N = (N1 ,, NJ )

If furthermore we have a set of expert opinions = ( 1 ,, )

)(~

=

J

j

jNPN1

)()()(



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Theorem For a Poisson-Gamma-Gamma model,

( ))exp(

)2(2

)/(,

1

2/)1(

=

+

+

KN

/1 +=dd

Nj

0

1

+= J

=m

m

+

+ =0

2/)/1(

12

1)( dueuzK uuz



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Computing loss by L-S-W

Posterior of , given N and , has a Generalized Inverse

Gaussian (GIG) distribution Algorithm

1. Simulate from GIG accordin to revious

2. Simulate N from Poisson ()

And obtain the empirical distribution for the number of losses

Or use MCMC method



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e.g. Poisson-Gamma

Poisson ( = 0.6) ( 0 = 3.4 ; 0 = 0.15) = 4Vs. Maximum likelihood



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To conclude

AMA op risk models as of today are not

Forward looking Time Risk sensitive


There are ways to model time into them And there are ways to solve these models

The Regulator should encourage these efforts

T h t t t ti li k b t IPS d


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Tech note: tentative links between IPS and

expert opinion

Linking Resampling IPS, classic Importance Sampling (IS), Sequential MCafter a spectrum of variants has been introduced to minimize the effects ofdegeneration in particles.

They can both be viewed as changes of measure in Markov chains but, inthe case of IPS, implicit Feynman-Kac change of measure is lessdemandin on data than the im licit Girsanov chan e of measure in IS.

In the field of Finance, there has been a first application into Credit Risk[Carmona]. [Crepey] investigated, as a consequence, the advantages of IPSover IS in Contagion cases in Credit portfolio risks.

[Shephard] showed that IPS can be very useful for learning out of simulated

samples, thus reducing the need for real data. This is a great remedy tothe scarcity of data in extreme losses in operational risks.



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Thank ou !

[email protected]


Time Based Model CFE-ERCIM'11_light2

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