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![Page 1: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/1.jpg)
Random Number GenerationUsing Low Discrepancy
PointsDonald Mango, FCAS, MAAA
Centre Solutions
June 7, 1999
1999 CAS/CARe Reinsurance Seminar
Baltimore, Maryland
![Page 2: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/2.jpg)
What is Discrepancy?
• Large # of points inside a unit hypercube :n-dimensional hypercube of length 1 on each side
• For any “sub-volume” of the hypercube,
Discrepancy = the difference between
the proportion of points inside the volumeand
the volume itself
![Page 3: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/3.jpg)
Low Discrepancy Point Generator:
• Method to generate a set of points which fills out a given n-dimensional unit hypercube, with as little discrepancy as possible
• Attempt to be systematic and efficient in filling a space, given the number of points
• My paper discusses “Faure” Points, just one of many alternatives
• Faure method relies on prime numbers
![Page 4: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/4.jpg)
Other Low Discrepancy Point Generators:
• Named after number theorists: Sobol’, Neiderreiter, Halton, Hammersley, ...
• More advanced methods use “irreducible polynomials” -- polynomial equivalents of prime numbers (cannot be factored)
• More complex algorithms
• Less flexible than Faure
![Page 5: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/5.jpg)
Linear Congruential Generator:• Xn+1 = (aXn + c) mod m
• Used in spreadsheets -- RAND() in Excel, @RAND in Lotus
• Sequential
• Cyclical, with a long cycle length or “period”
• “Randomized” in spreadsheets by using a random seed value ( X0 ) = the system clock
![Page 6: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/6.jpg)
LDPMAKER Excel 97 Workbook:
• Available in the 1999 Spring Forum section of the CAS Website:
www.casact.org/pubs/forum/99spforum/99spftoc.htm
• Includes both:
• A spreadsheet-only calculation (recalc-driven), and
• A Visual Basic for Applications (VBA) macro-driven generator (run with a button)
![Page 7: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/7.jpg)
LDPMAKER Excel 97 Workbook:
• “Example” sheet is spreadsheet-only calculation
• Demonstrates formulas
• Not very flexible
![Page 8: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/8.jpg)
Example: 4 Dimensions, 24 Iterations• Dimension #1:
• First, convert each iteration number N to base Prime (= 5)
• Iteration 1 = 01base5
Iteration 10 = 20base5
• F(N, 1) = Faure point (Iteration N, Dimension 1)F(1,1) = 0/52 + 1/5 = 0.20F(10,1) = 2/52 + 0/5 = 0.08
![Page 9: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/9.jpg)
Example: 4 Dimensions, 24 Iterations• Dimension #2:
• Start with the base Prime digits from Dimension #1 and “shuffle” them
• Using combinations, sum of digits and MOD operator
• First digit in Dimension #2 = [ Sum (first digit, second digit) from Dimension #1 ] MOD Prime
•Dimension #1, Iteration 10 = 20base5
Dimension #2, Iteration 10 = 22base5
• Formula for F(N,2) is the same
![Page 10: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/10.jpg)
Example: 4 Dimensions, 24 Iterations• Dimensions #3 and higher:
• Start with the base Prime digits from the previous dimension and “shuffle” them
• Formula for F(N,3) ... is the same
![Page 11: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/11.jpg)
Loops in the Faure Algorithm:
• Fills out the space in ever-larger loops of ever-smaller spacing
• Fills out the space sequentially
• There MAY be an issue with ending the iterations in the middle of one of these loops
• Examples later in the test results...
![Page 12: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/12.jpg)
Visual Basic for Applications (VBA) Version:
• VBA = real programming language
• Recursive algorithm using “dynamic arrays” - arrays which are dimensioned (sized) at run-time
• Generalization of spreadsheet-only calculations
• FAST
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Performance Test #1:Sum of Limited Paretos
Test # /Pareto #
B Q Policy Limit Limited Expected Value
1 / 1 10,000 1.10 100,000 21,321
1 / 2 15,000 1.30 250,000 28,874
Test # 1 Theoretical Result 50,194
2 / 1 10,000 1.10 50,000 16,4042 / 2 15,000 1.30 25,000 12,7452 / 3 25,000 1.20 40,000 21,7442 / 4 12,500 1.40 50,000 14,8342 / 5 30,000 2.00 25,000 13,636
Test # 2 Theoretical Result 79,364
Table 2 (from Paper) - Pareto Parameters
![Page 14: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/14.jpg)
Performance Test #1:Sum of Limited Paretos
# of Iterations LDP Value LDP % Error RAND() Value RAND() % Error
250 49,170 -2.04% 47,573 -5.22%
728 50,022 -0.34% 50,267 0.15%
1,000 49,769 -0.85% 49,640 -1.10%
1,500 49,903 -0.58% 51,307 2.22%
2,186 50,137 -0.11% 50,737 1.08%
Table 3: Sum of 2 Limited Paretos
![Page 15: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/15.jpg)
Performance Test #1:Sum of Limited Paretos
Table 4: Sum of 5 Limited Paretos
# of Iterations LDP Value LDP % Error RAND() Value RAND() % Error
342 79,319 -0.06% 80,179 1.03%
1,000 79,201 -0.21% 78,837 -0.66%
1,500 79,206 -0.20% 79,088 -0.35%
2,000 79,280 -0.11% 79,049 -0.40%
2,400 79,358 -0.01% 79,154 -0.27%
![Page 16: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/16.jpg)
Performance Test #2:Sum of Poissons
Table 5: Sum of 2 Poissons ( = 8)
# of Iterations LDP % Error RAND() % Error
250 -0.42% 1.30%
728 -0.03% 0.64%1,000 -0.22% 0.23%2,000 -0.09% -0.08%2,186 -0.01% 0.17%
![Page 17: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/17.jpg)
Performance Test #2:Sum of Poissons
Table 6: Sum of 5 Poissons ( = 8)
# of Iterations LDP % Error RAND() % Error
342 -0.24% 0.78%
1,000 -0.20% 0.59%2,000 -0.11% -0.22%2,400 -0.04% -0.23%
![Page 18: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/18.jpg)
Performance Test #3:Low Frequency Events
Pareto # B Q Policy Limit Limited Expected Value
1 10,000 1.30 50,000 13,860
Test #1 Theoretical Result 693
2 25,000 1.60 50,000 20,113
3 5,000 1.10 50,000 10,660
Test #2 Theoretical Result 2,232
Table 7 - Pareto Parameters used for Severity
![Page 19: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/19.jpg)
Performance Test #3:Low Frequency Events
Table 8: One Event, 5% Prob of Occurrence
# of Iterations LDP Value LDP % Error RAND() Value RAND() % Error
250 563 -18.82% 1,009 45.60%
728 615 -11.19% 657 -5.23%1,000 670 -3.27% 569 -17.93%1,500 667 -3.81% 613 -11.58%2,186 690 -0.50% 662 -4.45%
![Page 20: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/20.jpg)
Performance Test #3:Low Frequency Events
Table 9: Two Events, each with 5% Prob of Occurrence
# of Iterations LDP Value LDP % Error RAND() Value RAND() % Error
342 2,199 -1.46% 3,175 42.26%
1,000 2,251 0.86% 2,456 10.04%1,500 2,221 -0.49% 2,295 2.83%2,400 2,204 -1.22% 2,348 5.20%
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Performance Test #4:99th Percentile of Sum of NormalsTable 10 - Normal Parameters
Test # /Normal #
Mean StdDev
99th
Percentile
1 / 1 2,000 750 -
1 / 2 1,000 500 -
1 Combined 3,000 901.4 5,097
2 / 1 1,000 300 -2 / 2 1,000 800 -2 / 3 500 300 -2 / 4 750 600 -2 / 5 2,000 100 -
2 Combined 5,250 1090.9 7,788
![Page 22: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/22.jpg)
Performance Test #4:99th Percentile of Sum of NormalsTable 11 - 99th Pctle of Sum of 2 Normals
# of Iterations LDP Value LDP % Error RAND() Value RAND() % Error
250 5,084 -0.25% 4,800 -5.82%
728 5,036 -1.19% 4,898 -3.91%1,000 4,995 -2.00% 4,934 -3.19%1,500 5,047 -0.98% 4,989 -2.12%2,186 5,070 -0.52% 4,967 -2.55%
![Page 23: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/23.jpg)
Performance Test #4:99th Percentile of Sum of NormalsTable 12 - 99th Pctle of Sum of 5 Normals
# of Iterations LDP Value LDP % Error RAND() Value RAND() % Error
342 7,661 -1.63% 7,524 -3.38%
1,000 7,808 0.26% 7,653 -1.73%1,500 7,808 0.26% 7,650 -1.76%2,400 7,804 0.21% 7,703 -1.09%
![Page 24: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/24.jpg)
Performance Test #5:Mixed Bag
• Sum of 5 each from:
• LogNormal
• Pareto
• Uniform
• Normal
• Testing variability of estimates over 10 runs
![Page 25: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/25.jpg)
Performance Test #5:Mixed Bag
# of Iterations LDP Average %
Error
LDP Std Dev of
% Error
Rand Average
% Error
Rand Std Dev
of % Error
250 -10.39% 0.33% -0.36% 5.51%
500 -2.28% 0.71% -3.03% 7.79%
1,000 -0.47% 1.36% -0.76% 4.39%
1,500 -0.41% 0.69% -0.67% 4.62%
2,000 -0.41% 0.62% -1.40% 4.01%
3,000 -0.72% 0.47% -1.17% 2.79%
Table 14 - Avg % Error and Std Dev of % Error over 10 runs
![Page 26: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/26.jpg)
Possible Concerns in Using LDPs
• Unused Dimensions:
• Example: modeling Excess Claims
• # of Excess claims between 0 and 30
•requires 30 dimensions
• If # claims < 30, are the “used” dimensions still filled out with low discrepancy?
• Dr. Tom?
![Page 27: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/27.jpg)
Possible Concerns in Using LDPs
• Time Series:
• Example: Probability of 2 consecutive years of loss ratio exceeding 75%
• How many dimensions is this problem?
• Can’t use a single dimension of LDPs, because they are sequentially dependent
• Need to know “over how many years”, then set dimensions
![Page 28: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/28.jpg)
Possible Concerns in Using LDPs
• Correlation:
• If two variables are
•100% correlated ==> 1 dimension
• 0% correlated ==> 2 dimensions
• x% correlated ==> ? dimensions
• Is promise of “low discrepancy” still fulfilled?
• How to implement?
![Page 29: Random Number Generation Using Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore,](https://reader035.fdocuments.in/reader035/viewer/2022062715/56649d835503460f94a69bd7/html5/thumbnails/29.jpg)
Possible Concerns in Using LDPs
• Loop Boundaries:
• Faure algorithm fills out space sequentially in ever-expanding loops of ever-finer granularity
• If iteration count does not finish on a loop boundary (depends on Prime), there may be potential bias...
• See Appendix B of paper