Operational Value at Risk

Metoda Pengukuran Risiko Operasional dengan Advanced Measurement

Approach (AMA – Loss Distribution)

2

Pengukuran Risiko Operasional

Tahapan Measuring OR: Statistical Approach: Severity, Frequency Severity Distributions Frequency Distributions Aggregation

3

Pengukuran Risiko Pasar

Building the Operational VaR

1) Estimating Severity

2) Estimating Frequency

3) Aggregating Severity and FrequencyMonte Carlo SimulationValidation and Backtesting

Choosing the distribution Estimating Parameters Testing the Parameters

PDFs and CDFsQuantiles

4

Pemodelan Severity of Losses

Procedure:1) Choose a few distributions (severity and frequency) and estimate parameters (we will try here lognormal and exponential for severity)2) Check which distribution has the best fit3) Find confidence intervals for the parameters

Berikut Ini adalah loss harian rata-rata dalam satu bulan untuk kasus penyalahgunaan kartu kredit

  1992 1993 1994 1995 1996

1 45,354 55,000 330,000 30,000 91,000

2 42,250 32,500 197,500 19,734 37,500

3 36,745 27,800 65,000 13,000 21,300

4 27,500 10,732 20,503 12,417 21,166

5 20,300 10,000 17,500 11,955 16,600

6 18,000 8,000 10,000 8,250 14,742

7 18,000 7,854 8,800 6,000 11,500

8 17,500 6,000 6,488 5,800 11,468

9 11,018 3,919 5,477 4,344 10,527

10 9,122 2,602 5,352 4,181 6,421

11 3,400 2,595 5,350 3,759 6,133

12 2,500 2,375 3,230 2,635 4,477

Rp 000

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(1) (2) (3)

No. of Events Observed Freq. (1) x (2)

0 221 0

1 188 188

2 525 1050

3 112 336

4 73 292

5 72 360

6 44 264

7 40 280

8 14 112

9 7 63

10 2 20

11 2 22

12 4 48

13 3 39

14 2 28

15 1 15

lamda (λ) 2.3794  

POISSON PDF

-

0.05

0.10

0.15

0.20

0.25

0.30

0 5 10 15 20

POISSON CDF

0.00

0.20

0.40

0.60

0.80

1.00

0 5 10 15 20

Pemodelan Frecquency of Losses

0

0

kk

kk

n

kn

6

Distribusi Frekuensi Kerugian

# Events/Day Observed Frequency0 2211 1882 5253 1124 735 726 447 408 149 7

10 211 212 413 314 215 1

3338

Another example, comparing Poisson and Negative Binomial DistributionsFrauds Database

Distribution Parameter(s)

Poisson = 2.379

Negative Binomial r = 3.51 = 0.67737

2

00k

2

0

k)(1r

and

n

kn

n

n

n

knr

kkk

kk

Parameter estimation of thenegative binomial is a bit more complexand it is based on solving this systemof equations

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Distribusi Frekuensi Kerugian

Number of Frauds = 102

January February March April May June July August

95 82 114 74 79 160 110 115 91%118 95%126 99%

Poisson

Poisson PDF

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

0 50 100 150 200

Poisson CDF

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

0 20 40 60 80 100 120 140 160

!)(

0 k

exf

kx

k

Poisson Distribution:

Other popular distributions toestimate frequencyare the geometric,negative binomial,binomial, Weibull, etc

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Aggregation: Estimate the Operational VaR

No analyticalsolution!

Need to be solvedby simulation

Prob

Number of Losses

Prob

Frequency

Losses sizes

Prob

Aggregated losses

Aggregated Loss Distribution

)(0

* xFpn

nXn

Alternatives:1) Fast Fourier Transform2) Panjer Algorithm3) Recursion

Severity

9

SeverityEksponential/Lognormal/weibull/pareto

FrequencyPoisson/neg.binomial

Operational VaR

)(0

* xFpn

nXn

Aggregation: Estimate the Operational VaR

10

Lakukanlah agregasi dengan @Risk dengan prosedur berikut 1. Data severity dan frequency dicari distribusinya untuk mendapatkan

parameter dalam simulasi Monte Carlo 2. Pertama kali yang disimulasi adalah parameter distribusi frequency,

buatlah 1.000 iterasi3. Identifikasikan numbers of #event dengan fungsi Excel

COUNTIF(range,criteria). Ex. COUNTIF(a1:a1000;1)=220. Artinya dalam 1000 simulasi, ada 220 kejadian dimana fraud terjadi sekali

4. Akumulasikan #event (tentunya terkecuali untuk 0 event), untuk menentukan berapa iterasi yang diperlukan untuk simulasi kedua yakni simulasi atas distribusi severity. Misalnya kita harus memperoleh 2.370 data severity data untuk membangun (aggregate) operational loss distribution

5. Lakukanlah agregasi (lihat slide berikut) dan sortirlah untuk memperoleh the worst 1% (data ke 11 dari hasil sortiran), itulah nilai VaR

6. VaR = unexpected loss, sedangkan Capital at Risk adalah VaR – expected loss. Bagaimana cara menghitung Expected loss ?

Agregasi Operational VaR Dengan Simulasi MC

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How to prepare frequency distribution for aggregation…

Aggregation: Estimate the Operational VaR

Result of Monte Carlo Simulation for Frequency Distribution

0 926 01 2204 90742 2621 68703 2079 42494 1237 21705 589 9336 233 3447 79 1118 24 329 6 8

10 1 211 1 112 0 013 0 014 0 015 0 0

10000 23794#iteration for Monte Carlo

Simulation of Severity Distribution

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How to operate the aggregation in @Risk and Excel from 10.000 iteration

Aggregation: Estimate the Operational VaR

#iteration 1 2 3 4 5 6 7 8 9 10 11 TOTAL SORTED TOTAL1 139.403 25.355 3.028 37.287 2.413 62.683 3.077 106.145 17.996 29.145 28.931 455.462 2.794.407 2 5.906 15.094 60.111 6.412 5.717 2.888 6.190 4.368 13.120 12.693 132.498 2.302.650 3 41.016 34.273 17.829 10.913 121.993 31.014 3.013 4.311 2.867 267.227 1.838.147 4 3.010 16.466 71.539 3.668 24.766 64.436 4.789 2.848 1.036.967 1.228.489 1.589.285 5 5.372 16.507 12.280 229.791 396.221 3.133 3.356 2.820 14.135 683.615 1.442.909 6 19.552 5.364 2.544 5.840 15.704 11.879 10.091 3.044 9.696 83.713 1.372.917 7 2.817 22.719 9.117 12.405 26.192 3.262 6.648 2.848 17.606 103.614 1.371.524 8 4.491 29.917 4.240 4.270 11.240 41.110 2.943 8.490 5.850 112.552 1.262.464 9 17.105 10.360 6.097 7.844 21.148 69.398 42.012 6.649 4.104 184.717 1.228.489

10 2.311 11.268 33.293 58.336 7.880 4.487 71.697 9.249 198.521 1.208.609 11 10.619 28.960 39.238 7.351 2.273 397.857 17.500 22.171 525.969 1.194.787 98 4.044 20.590 12.202 8.690 5.730 236.285 36.835 324.377 455.850 99 40.555 32.769 52.625 107.587 2.755 4.314 3.747 244.353 455.462

100 15.605 25.863 89.315 3.224 62.638 19.859 12.503 229.006 448.611 101 5.584 4.667 19.408 6.858 4.147 2.814 3.533 47.010 447.740 102 21.416 8.238 5.680 8.168 13.596 3.667 15.001 75.766 442.711 103 7.437 13.141 66.185 13.844 5.912 10.419 32.618 149.557 442.142 104 7.152 5.699 11.974 5.746 2.813 2.551 15.965 51.900 441.322 105 6.927 5.045 10.536 4.530 26.766 2.612 5.228 61.644 439.977

Unexpected Loss Rp 447.740.000

Expected Loss Rp 25.265.000

Capital at Risk Rp 422.475.000

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Size of losses

Fre

qu

en

cy o

f lo

sses

Expect

ed

Loss

es

Capital at Risk (Rp 422.475.000)=

Unexpected losses – Expected Losses

Income Capital Insurance

Sustaining losses in Operational Risk

1%

447.74025.265