Operational Value at Risk is estimated using an Advanced Measurement Approach involving statistical modeling of operational risk loss data. Loss severity and frequency are each modeled separately using distributions like lognormal or Poisson. Monte Carlo simulation is then used to aggregate the severity and frequency distributions and estimate unexpected losses, expected losses, and the capital at risk needed to sustain operational losses exceeding expectations.

1. 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
2

3. Pengukuran Risiko Pasar
Building the Operational VaR
Choosing the distribution
1) Estimating Severity Estimating Parameters
Testing the Parameters
2) Estimating Frequency PDFs and CDFs
Quantiles
3) Aggregating Severity and Frequency
Monte Carlo Simulation
Validation and Backtesting
3

4. Pemodelan Severity of Losses
Rp 000
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
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 fit
3) Find confidence intervals for the parameters 4

5. Pemodelan Frecquency of Losses
POISSON PDF
(1) (2) (3)
0.30
No. of Events Observed Freq. (1) x (2)
0.25
0 221 0 0.20
0.15
1 188 188 0.10
2 525 1050 0.05
-
3 112 336 0 5 10 15 20
4 73 292
5 72 360
POISSON CDF
6 44 264
7 40 280
1.00
8 14 112 0.80
9 7 63 0.60
0.40
10 2 20
0.20
11 2 22 0.00
0 5 10 15 20
12 4 48
13 3 39
∞
14 2 28
∑ kn k
15 1 15 λ= k =0
∞
lamda (λ) 2.3794
∑n
k =0
k
5

6. Distribusi Frekuensi Kerugian
Another example, comparing Poisson and Negative Binomial Distributions
Frauds Database
# Events/Day Observed Frequency
0 221
1 188 Parameter estimation of the
2 525 negative binomial is a bit more complex
3 112 and it is based on solving this system
4 73
5 72 of equations
6 44
7 40
∞
8 14
9 7
∑ kn k
rβ = k =0
10 2 n
11 2 and
12 4 2
∞
 ∞ 
13 3 ∑ k nk  ∑ knk
2

14 2 rβ (1 + β ) = k =0 −  k =0 
15 1 n  n 
 
 
3338
Distribution Parameter(s)
Poisson λ = 2.379
Negative Binomial r = 3.51
β = 0.67737
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7. Distribusi Frekuensi Kerugian
Poisson Distribution:
Number of Frauds λ= 102
x
e− λ λ k
f ( x) = ∑
January February March April May June July August
95 82 114 74 79 160 110 115 91% k=0 k!
118 95%
126 99%
Poisson
Poisson PDF Poisson CDF
4.50% 100.00% Other popular
4.00% 90.00%
distributions to
80.00%
3.50%
70.00%
estimate frequency
3.00%
2.50%
60.00% are the geometric,
2.00%
50.00%
40.00%
negative binomial,
1.50%
30.00%
binomial, Weibull, etc
1.00%
20.00%
0.50%
10.00%
0.00%
0.00%
0 50 100 150 200 0 20 40 60 80 100 120 140 160
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8. Aggregation: Estimate the Operational VaR
Severity Frequency
Prob Prob
Number of Losses
Losses sizes
Aggregated Loss Distribution
Prob
Need to be solved
by simulation
∞
Aggregated losses ∑p F n
*n
X ( x ) No analytical
n =0 solution!
Alternatives:
1) Fast Fourier Transform
2) Panjer Algorithm
3) Recursion 8

9. Aggregation: Estimate the Operational VaR
Severity Frequency
Eksponential/Lognormal/weibull/pareto Poisson/neg.binomial
∞
∑pF
n =0
*n
n X ( x)
Operational VaR
9

10. Agregasi Operational VaR Dengan Simulasi MC
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 iterasi
3. 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 ?
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11. Aggregation: Estimate the Operational VaR
How to prepare frequency distribution for aggregation…
Result of Monte Carlo Simulation for Frequency Distribution
0 926 0
1 2204 9074
2 2621 6870
3 2079 4249
4 1237 2170
5 589 933
6 233 344
7 79 111
8 24 32
9 6 8
10 1 2
11 1 1
12 0 0
13 0 0
14 0 0
15 0 0
#iteration for Monte Carlo
10000 23794 Simulation of Severity Distribution
11

12. Aggregation: Estimate the Operational VaR
How to operate the aggregation in @Risk and Excel from 10.000 iteration
#iteration 1 2 3 4 5 6 7 8 9 10 11 TOTAL SORTED TOTAL
1 139.403 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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13. Sustaining losses in Operational Risk
Capital at Risk (Rp 422.475.000)
Frequency of losses
=
Unexpected losses – Expected Losses
ses d
e
ect
1%
Exp
Los
25.265 447.740
Size of losses
Income Capital Insurance
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