Operational Risk: EVT Models

(1)

Operational Risk:

EVT

Models

Jack L. King, Ph.D. Genoa (UK) Limited

(2)

Presentation Outline

• Introduction (The Problem)

• Methodology Survey

• Extreme Value Theory

• Example

• Assumptions and Issues

• Summary

(3)

The Problem

• What is the maximum amount of loss

due to operational risk that can be

expected in a specific business unit over

a period of one year at a very high

(4)

The Environment

• Business unit with no market or credit risk.

• Processes composed of people, systems, and

infrastructure.

• Business model is based on earning fees for

services.

(5)

Approach to Methodology

• Factor approach

– Identify key indicators and relationships to losses. – Use a factor model to calculate the capital

requirement directly, (e.g. 12% of income).

• Delta approach

– Calculate volatility of profits using error propagation technique that relates volatility of profits to volatility of key indicators using sensitivities.

– Choose a factor on profit volatility based on

distributional assumption (normal or lognormal) and required confidence interval, plus a ‘fudge’ factor.

(6)

Approach to Methodology

• Loss model

– Record losses made and fit a parametric

distribution to their severity and one to their frequency.

– Use simulation to create cumulative loss distribution for the period based on

frequency and severity.

– Choose quantile of resulting distribution, plus a ‘fudge’ factor

(7)

Delta Method

• Factor model that is predictive.

– Uses measure at hand and linear relationship to predicts measure not at hand.

• Theory

– Partial differential equations (PDE’s).

• Example

– Change in interest rate used to predict change in bond price.

f

p

∆

∂

=

∆

(8)

Delta Method

• Assumption

– Causal relationship between measured indicator and predicted measure.

– Linear relationship (approximated)

• Advantages

– Predictive – Understandable

• Disadvantages

– Factor interactions – Non-linear relationships – Unknown factors

(9)

Loss Models

• Use historical results to predict future values

of a measurement based on an intrinsic

regularity and symmetry of underlying system.

• Theory

– Statistical distributions

• Example

– Yearly claims loss of automobile insurance

business for defined set of policy characteristics.

}

,...

{

1 t n t t

x

₊

=

Φ

₋

(10)

Loss Models

• Assumptions

– Underlying regularity and symmetry are known and stable.

• Advantages

– Allows prediction without understanding causes. – Large samples tend toward a regularity and

symmetry.

• Disadvantages

– Irregularity and shocks to the system.

– Not easily influenced because no known causes. – ‘Post-predictor’ that is insensitive to outliers.

(11)

Advanced Models -- Causal Models

• Causal Models

– Advantages of factor model – Non-linear relationships

– Mitigates interrelationships of factors

D om ic il e G 10 F a r Ea s t E a ste rn Eu r. .. S o uth A me ri. .. M ex ic o 65 .2 10 .3 5.1 7 5.1 7 14 .1 P rod uc t B ra d ie B o nd G ov er nm ent. .. S u praBo n d C orp or ate B o... I nd ex D eriv ative F ID e rivativ e E q uityD eriv a... 1 3.4 5 9.0 1 5.2 1 2.2 . 05 2 . 05 2 . 05 2 V o lu me Lo w Me d H ig h 2 5 .0 5 0 .0 2 5 .0 A ve ra ge V alu e Lo w Me d H ig h V ery H ig h 0 .31 2 7 .1 6 1 .9 1 0 .7 3 .1 e +0 0 6 ± 3.8 e +0 0 6 S e ttl em e nt L o s s -1 e 5 to 1 1 to 5 0 50 t o 5 00 50 0 t o 1e 6 7 9.0 1 .25 1 1.6 8 .07 0 ± 1 .7e +00 5 D el a y d a y0 d a y1 d a y2 d a y3 d a y4 d a y5 d a y6 orm o re 7 9 .0 1 0 .8 5 .07 2 .04 2 .02 0 .51 0 .49 0.4 1 ± 0.9 9 E xpo su re Lo w Me dium H ig h V ery H ig h 1 0 .6 4 9 .7 2 6 .4 1 3 .3 1 .7e +00 3 ± 2 .7 e +0 0 3 B uy _ S ell B u y S e ll 58 .9 41 .1 E x ch a n g e O TC C ED EL L IFF E D AX E u ro clea r 70 .0 10 .0 10 .0 5.0 0 5.0 0 C ou n terp a rt y T o p50 B an k C orp ora te B a nk F u nd A filia te G ov ernm ent S m allIn s t itu t. .. 37 .7 16 .9 31 .3 1.8 8 3.7 7 3.7 7 4.7 1 D ays D elay 0 1 2 3 4 5 6 1 0 0 7 9.0 1 0.8 5 .0 7 2 .0 4 2 .0 2 0 .5 1 0 .4 4 . 04 9 0 .5 ± 2 .4

(12)

Extreme Value Theory Approach

• Loss Model

– Uses one of the extreme value distributions for severity distribution.

– Fits distribution based on extreme events only.

• Generalized Pareto Distribution (GPD)

– Fits the excess losses above a threshold. ξ β ξ 1 ) 1 ( ) ( −     ₊ − = x threshold x GPD

(13)

Advantages of EVT Loss Models

Data:

Only need large losses

(not all losses).

Analysis:

Not influenced by volume of

small losses.

Theory:

Well-founded theory beginning

with Fisher-Tippet.

Application:

Widely applied in physical sciences

(14)

Advanced Model -- Delta-EVT

™

• Uses the Delta factor model for high frequency,

low severity losses.

• Uses EVT loss model for low frequency, high

severity losses.

• Combines the two approaches using the threshold

value for large losses.

(15)

Delta-EVT

™

Combination Methodology

• Combines Delta and EVT to link performance of the

business to the losses and provide a built-in validation of capital charges. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 Size of Loss (X) P( X) Threshold u

Operating Loss Excess Loss

(16)

Example (Genoa Bank)

• Business Unit Description

– Profit center in financial services – No market or credit risk

• Problem

– What is the maximum amount of loss due to operational risk that can be expected in a this business unit over a period of one year at a very high confidence level (e.g. 99%) ?

(17)

Example Business Process

(1) 260 days/year

Transactions/day 1,000

Transactions/year (1) 260,000

Average transaction value 200,000 €

Transaction margin 100 bp

Transaction net value 200 €

Net per year 52,000,000 €

Proposed capital ratio 12%

Capital Required 6,240,000 €

(18)

Example Delta Loss

Loss rate 3%

Loss transactions/day 30

Loss transactions/year 7,800

Average loss/loss transaction 200 €

Average total loss/year 1,560,000 €

But: € (~lognormal)

And actual year's loss is: 5,529,860 €

Delta Loss

500

=

(19)

Poisson-Lognormal Simulation

Poisson-Lognormal Simulation 1400000 1450000 1500000 1550000 1600000 1650000 1700000 1750000 1800000 1850000 1900000 1 ₄₀ ₇₉ 118 157 196 235 274 313 352 391 430 469 508 547 586 625 664 703 742 781 820 859 898 937 976 Run Y ea rl y Lo ss

quantile events loss

90% 7,911 1,757,224 95% 7,940 1,771,004 99% 8,020 1,816,982 99.9% 8,068 1,834,619

(20)

Example Loss History

All Transaction Losses for Year

0 500 1000 1500 2000 2500 3000 3500 4000 4500 100 200 300 400 500 600 700 800 900 ₁₀00 ₁₁00 ₁₂00 ₁₃00 ₁₄00 ₁₅00 Amount Fr eq ue n cy

(21)

Large Losses

ID Date Amount ('000) Loss 1 01/01/98 -35.1 35 2 01/19/98 -40.0 40 3 01/30/98 -55.0 55 4 02/11/98 -80.9 81 5 02/12/98 -508.0 508 6 02/17/98 -3.5 4 7 03/18/98 -48.8 49 8 03/20/98 -12.0 12 9 03/27/98 -168.9 169 10 03/27/98 -98.0 98 11 04/07/98 -128.0 128 12 06/01/98 -21.6 22 13 06/11/98 -100.0 100 14 07/01/98 -770.0 770 15 07/24/98 -142.0 142 16 08/18/98 -61.5 62 17 08/30/98 -129.4 129 18 10/05/98 -1450.0 1450 19 10/12/98 -30.0 30 20 10/15/98 -17.0 17 21 10/22/98 -8.0 8 22 11/30/98 -12.0 12 23 12/28/98 -50.0 50

Internal Large Loss History

0 200 400 600 800 1000 1200 1400 1600 1/ 1/ 9 8 2/ 1/ 9 8 3/ 1/ 9 8 4/ 1/ 9 8 5/ 1/ 9 8 6/ 1/ 9 8 7/ 1/ 9 8 8/ 1/ 9 8 9/ 1/ 9 8 1 0 /1/9 8 1 1 /1/9 8 1 2 /1/9 8 Date A m ount (' 000 E u ro s )

(22)

Poisson-GPD Simulation Results

Poisson-GPD Simulation 0 2,000,000 4,000,000 6,000,000 8,000,000 10,000,000 12,000,000 14,000,000 16,000,000 1 62 12 3 18 4 24 5 30 6 36 7 42 8 48 9 55 0 61 1 67 2 73 3 79 4 85 5 91 6 97 7 Run Y e ar ly L o ss

quantile events loss

90% 30 2,557,100 95% 32 3,262,250 99% 35 7,846,340 100% 39 14,381,519

(23)

Summary Results

(Factor model using 12% of income was € 6,240,000)

events loss events loss events loss

90% 7,911 1,757,224 30 2,557,100 7,941 4,314,324 95% 7,940 1,771,004 32 3,262,250 7,972 5,033,254 99% 8,020 1,816,982 35 7,846,340 8,055 9,663,322 99.9% 8,068 1,834,619 39 14,381,519 8,107 16,216,138 In Control (Poisson-Lognormal) Out of Control (Poisson-GPD) Total

Summary of Yearly Losses in Business Unit at High Confidence Levels

(24)

Assumptions and Issues

• Delta Model

– There is little affect from the mean loss figure. Most of the information is found in the volatility (sigma). The calculation of the volatility of the losses does not need large losses (or any losses), but needs to be validated using a sample of losses. – Is not affected by large losses (or any previous

(25)

Assumptions and Issues

• Lognormal Loss Model

– Contributes little to the delta methodology when there is a large sample of losses available, since most of the information is present in the volatility of the losses.

– Could be used for validation of delta method, or reduced to a small sample size.

– Is not affected by the inclusion of large losses when the sample size is large.

(26)

Assumptions and Issues

• EVT Loss Model

– Captures the effect of large losses and shows an order of magnitude difference in the use of a delta or lognormal loss model when large losses are