Operation Risk - Joint Probability Distribution For Two Department Losses

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Operation Risk - Joint Probability Distribution For Two Department Losses

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This exercise has as goal to show how correlations affect the distributions of the total losses that a firm may incur. Consider a company with two divisions. In each one of the two divisions, the probability of one operational risk event happening in one year is 0.5 and the probability of no event happening is 0.5. If an event happens in division i (i =1,2), then the random damage has a Normal distribution with mean 30 and standard deviation of 3. If both divisions are hit by an event in the same year, then the two random damages are Normally distributed random variables that are independent from one another and the total damage to the firm is the sum of these two independent random variables that are Normally distributed. Let Xi denote the binary random variable of an event happening in division i, i =1,2. The binary random variable Xi may assume either the value 0 or 1. Consider the following three scenarios. (A) Consider the case where the joint probabilities of events happening in the divisions are: P(X1 = 0, X2 = 0) = 0.25 P(X1 = 0, X2 = 1) = 0.25 P(X1 = 1, X2 = 0) = 0.25 P(X1 = 1, X2 = 1) = 0.25 The above joint probability distribution basically corresponds to a scenario where the probability of an event happening in one division is completely independent of the probability of an event happening in the other division. In this scenario the firm with probability 0.25 does not incur any losses; with probability 0.5 the firm incurs a loss of one Normally distributed random variable. With probability 0.25 the firm incurs a loss that is the sum of two independently Normally distributed random variables. (The correlation coefficient of the two random variables in this case is 0, since the outcomes of the two random variables are independent of one another.) (B) Consider the case where the joint probabilities of events happening in the two divisions are: P(X1 = 0, X2 = 0) = 0.5 P(X1 = 0, X2 = 1) = 0.0 P(X1 = 1, X2 = 0) = 0.0 P(X1 = 1, X2 = 1) = 0.5 The above joint probability distribution corresponds to the scenario where the probability of an event happening in one division is highly positively correlated with an event happening in the other division as well. In this scenario the firm with probability 0.5 does not incur any losses at all and with probability 0.5 incurs a loss that is the sum of two independent Normally distributed random variables. (The correlation coefficient of the two random variables is now +1.) (C) Consider the case where the joint probability of events in the two divisions is: P(X1 = 0, X2 = 0) = 0.0 P(X1 = 0, X2 = 1) = 0.5 P(X1 = 1, X2 = 0) = 0.5 P(X1 = 1, X2 = 1) = 0.0 The above joint probability distribution corresponds to the scenario where the probability of an event happening in one division is highly negatively correlated with an event happening in the other division. In this scenario the firm with probability 1 incurs a loss that is a single Normally distributed random variable. (The correlation coefficient of the two random variables is -1.) I) Compute for all three scenarios A, B, and C the 95% VaR value for the annual total losses incurred by the firm. II) Consider scenarios A and B again. When in a given year an event occurs in both divisions, then the amount of loss incurred in the two divisions are not independent but actually the same random variable that is Normally distributed. So the amounts of the losses are not any more independent. The damage to the firm is in such a case twice the value of a random variable that is Normally distributed with mean 30 and standard deviation of 3. Compute again the 95% VaR for cases A and B.

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Listed Derivatives Day Loss Errors Trades Downtime Trade /head Error / Head ln loss/ headcnt ln loss HeadCnt Trades Downtime 1 109,158 65 6469 8 1078.166667 10.833333 0 0 6 6469 8 2 14,387 54 5581 2 697.625000 6.750000 -0.2533089191 -2.0264713529 8 5581 2 3 16,417 25 5768 4 721.000000 3.125000 0.0164990453 0.1319923623 8 5768 4 4 8,564 30 6532 6 1088.666667 5.000000 -0.1084583354 -0.6507500126 6 6532 6 5 29,517 42 5698 5 814.000000 6.000000 0.1767712854 1.2373989978 7 5698 5 6 9,292 65 6260 4 569.090909 5.909091 -0.1050738685 -1.1558125536 11 6260 4 7 24,241 68 5804 8 829.142857 9.714286 0.1369845142 0.9588915996 7 5804 8 8 42,938 143 5547 3 924.500000 23.833333 0.0952853 0.5717118003 6 5547 3 9 120,400 15 4973 10 994.600000 3.000000 0.2062124636 1.0310623181 5 4973 10 10 144,569 35 5227 11 1306.750000 8.750000 0.0457343423 0.1829373694 4 5227 11 11 15,818 31 5447 4 907.833333 5.166667 -0.3687680617 -2.2126083701 6 5447 4 12 4,300 27 5242 3 873.666667 4.500000 -0.2170889182 -1.3025335094 6 5242 3 13 41,658 24 4993 6 998.600000 4.800000 0.4541756809 2.2708784043 5 4993 6 14 5,150 19 4781 4 531.222222 2.111111 -0.2322774125 -2.0904967123 9 4781 4 15 38,503 55 5448 5 605.333333 6.111111 0.2235266051 2.0117394457 9 5448 5 16 48,554 53 5533 3 553.300000 5.300000 0.023194042 0.2319404203 10 5533 3 17 33,612 55 5287 4 528.700000 5.500000 -0.0367793435 -0.3677934346 10 5287 4 18 5,614 95 5275 5 439.583333 7.916667 -0.1491349723 -1.7896196681 12 5275 5 19 104,004 27 5676 9 946.000000 4.500000 0.486527647 2.919165882 6 5676 9 20 35,393 17 4733 5 788.833333 2.833333 -0.1796525499 -1.0779152995 6 4733 5 21 306,670 11 6315 15 1578.750000 2.750000 0.5398145476 2.1592581904 4 6315 15 22 21,420 33 6513 3 1085.500000 5.500000 -0.443574531 -2.6614471859 6 6513 3 23 26,239 71 6964 2 1160.666667 11.833333 0.0338202981 0.2029217886 6 6964 2 24 24,850 31 4902 3 817.000000 5.166667 -0.0090648502 -0.0543891011 6 4902 3 25 18,780 30 1037 4 172.833333 5.000000 -0.0466775465 -0.2800652788 6 1037 4 26 176,490 41 5872 8 978.666667 6.833333 0.3734119573 2.2404717438 6 5872 8 27 9,649 43 5736 2 956.000000 7.166667 -0.4844016558 -2.9064099345 6 5736 2 28 21,094 82 5531 2 921.833333 13.666667 0.1303557261 0.7821343568 6 5531 2 29 42,015 31 4846 3 807.666667 5.166667 0.1148396763 0.6890380575 6 4846 3 30 255,438 30 4568 12 1142.000000 7.500000 0.4512381477 1.8049525907 4 4568 12 31 35,241 29 5281 4 880.166667 4.833333 -0.3301281851 -1.9807691106 6 5281 4 32 346,771 41 5191 15 1730.333333 13.666667 0.7621518076 2.2864554227 3 5191 15 33 45,566 35 5201 4 866.833333 5.833333 -0.3382506293 -2.0295037758 6 5201 4 34 79,086 33 5444 6 907.333333 5.500000 0.0918956739 0.5513740436 6 5444 6 35 4,550 30 4669 3 778.166667 5.000000 -0.4759014392 -2.855408635 6 4669 3 36 35,810 53 5322 3 887.000000 8.833333 0.3438499918 2.063099951 6 5322 3 37 26,413 68 5566 3 927.666667 11.333333 -0.0507284785 -0.3043708708 6 5566 3 38 38,141 18 5881 2 980.166667 3.000000 0.0612389176 0.3674335058 6 5881 2 39 20,770 19 5972 4 995.333333 3.166667 -0.1012966969 -0.6077801811 6 5972 4 40 19,940 30 5425 3 904.166667 5.000000 -0.0067969789 -0.0407818734 6 5425 3 41 2,716 37 6228 4 1038.000000 6.166667 -0.3322612591 -1.9935675548 6 6228 4 42 66,364 27 5529 7 921.500000 4.500000 0.5326657552 3.1959945311 6 5529 7 43 38,225 33 5914 5 985.666667 5.500000 -0.0919441648 -0.551664989 6 5914 5 44 1,297 47 5913 4 985.500000 7.833333 -0.5639059744 -3.3834358465 6 5913 4 45 221,633 29 5664 9 1416.000000 7.250000 1.2852422389 5.1409689554 4 5664 9 46 11,525 46 5847 4 974.500000 7.666667 -0.492750712 -2.9565042719 6 5847 4 47 13,775 32 5373 3 895.500000 5.333333 0.0297227944 0.1783367662 6 5373 3 48 13,484 34 5272 4 878.666667 5.666667 -0.0035585929 -0.0213515577 6 5272 4 49 3,569 28 5272 3 878.666667 4.666667 -0.2215363921 -1.3292183529 6 5272 3 50 9,407 24 4430 3 738.333333 4.000000 0.1615281081 0.9691686485 6 4430 3 51 4,735 18 5123 4 853.833333 3.000000 -0.1144120611 -0.6864723664 6 5123 4 52 1,640 55 5540 5 923.333333 9.166667 -0.1767142475 -1.0602854848 6 5540 5 53 320,025 27 5666 12 1416.500000 6.750000 1.3184258049 5.2737032196 4 5666 12 54 16,040 32 4641 3 773.500000 5.333333 -0.4988856432 -2.993313859 6 4641 3 55 784,200 35 4335 16 1445.000000 11.666667 1.2965261625 3.8895784874 3 4335 16 56 -5,631 22 4744 4 237.200000 1.100000 0 0 20 4744 4 57 14,514 13 4649 4 774.833333 2.166667 0 0 6 4649 4 58 30,000 17 5240 6 873.333333 2.833333 0.1210139468 0.7260836808 6 5240 6 59 6,413 23 4533 3 755.500000 3.833333 -0.2571450336 -1.5428702015 6 4533 3 60 1,772 19 4358 4 726.333333 3.166667 -0.2143697213 -1.286218328 6 4358 4 61 68,000 16 5317 5 886.166667 2.666667 0.6078998088 3.647398853 6 5317 5 62 1,865 41 5191 3 865.166667 6.833333 -0.599374442 -3.5962466521 6 5191 3 63 6,387 29 5071 4 845.166667 4.833333 0.2051672702 1.2310036215 6 5071 4 64 467,665 43 5312 14 1770.666667 14.333333 1.4311625177 4.2934875531 3 5312 14 65 6,621 15 3626 4 604.333333 2.500000 -0.7095843019 -4.2575058117 6 3626 4 66 4,750 15 5448 3 908.000000 2.500000 -0.0553502996 -0.3321017979 6 5448 3 67 4,721 26 5963 4 993.833333 4.333333 -0.0010206627 -0.0061239765 6 5963 4 68 345,000 11 5000 16 1666.666667 3.666667 1.4305079252 4.2915237755 3 5000 16 69 3,250 19 4620 5 770.000000 3.166667 -0.7774815701 -4.6648894207 6 4620 5 70 0 24 1285 3 214.166667 4.000000 0 0 6 1285 3 71 269,951 26 4402 11 1467.333333 8.666667 0 0 3 4402 11 72 4,169 31 4630 3 771.666667 5.166667 -0.6950940437 -4.1705642621 6 4630 3 73 -7,990 25 4936 3 246.800000 1.250000 0 0 20 4936 3 74 0 12 4113 4 685.500000 2.000000 0 0 6 4113 4 75 87,125 9 5102 7 850.333333 1.500000 0 0 6 5102 7 76 57,598 30 5175 5 862.500000 5.000000 -0.068976004 -0.4138560242 6 5175 5 77 18,267 72 5115 3 852.500000 12.000000 -0.1913986153 -1.1483916916 6 5115 3 78 11,441 12 4760 3 793.333333 2.000000 -0.0779821264 -0.4678927586 6 4760 3 79 19,449 49 5588 4 931.333333 8.166667 0.0884320434 0.5305922601 6 5588 4 80 6,141 28 5049 3 841.500000 4.666667 -0.1921346766 -1.1528080595 6 5049 3 81 14,880 42 5689 4 948.166667 7.000000 0.1475050723 0.8850304341 6 5689 4 82 22,781 86 5229 3 871.500000 14.333333 0.0709848043 0.4259088258 6 5229 3 83 6,783 388 4720 5 786.666667 64.666667 -0.2019178955 -1.2115073732 6 4720 5 84 3,185,749 65 5402 18 1800.666667 21.666667 2.0506744077 6.1520232232 3 5402 18 85 15,507 28 4838 3 806.333333 4.666667 -0.8875251951 -5.3251511703 6 4838 3 86 3,906 107 5376 4 896.000000 17.833333 -0.2297962837 -1.3787777024 6 5376 4 87 3,380 28 5435 4 905.833333 4.666667 -0.0241063538 -0.144638123 6 5435 4 88 86,784 109 5860 6 976.666667 18.166667 0.5409244272 3.2455465634 6 5860 6 89 2,000 1039 3456 2 576.000000 173.166667 -0.6283791821 -3.7702750923 6 3456 2 90 -5,587 33 5388 3 898.000000 5.500000 0 0 6 5388 3 91 12,038 31 5565 3 927.500000 5.166667 0 0 6 5565 3 92 24,818 21 5349 4 891.500000 3.500000 0.1205834805 0.7235008832 6 5349 4 93 58,266 21 5407 7 901.166667 3.500000 0.1422415894 0.8534495365 6 5407 7 94 14,900 28 5228 3 871.333333 4.666667 -0.2272762533 -1.3636575199 6 5228 3 95 6,493 6 4200 5 700.000000 1.000000 -0.1384394232 -0.8306365394 6 4200 5 96 9,038 36 5618 4 936.333333 6.000000 0.0551188729 0.3307132375 6 5618 4 97 3,345 26 6231 4 1038.500000 4.333333 -0.1656618696 -0.9939712174 6 6231 4 98 27,385 58 6416 3 1069.333333 9.666667 0.3504214541 2.1025287247 6 6416 3 99 51,866 44 5789 5 964.833333 7.333333 0.1064446752 0.6386680513 6 5789 5 100 155,838 22 5284 10 880.666667 3.666667 0.1833589228 1.1001535366 6 5284 10 101 16,666 64 6357 4 1059.500000 10.666667 -0.3725743817 -2.2354462902 6 6357 4 102 2,660 57 5564 2 927.333333 9.500000 -0.3058407655 -1.8350445932 6 5564 2 103 83,285 28 6155 6 1025.833333 4.666667 0.5739903897 3.4439423381 6 6155 6 104 0 171 5110 3 851.666667 28.500000 0 0 6 5110 3 105 21,900 33 5388 3 898.000000 5.500000 0 0 6 5388 3 106 21,681 24 6285 3 1047.500000 4.000000 -0.001675056 -0.0100503359 6 6285 3 Assignment 1: a) Using Excel (or some other data analysis software), come up with a statistical model to predict Losses as a function of System Downtime, Number of Employees (Headcount), Cancels/Corrects, and Number of Transactions. b) Which composite factors would you think of using? c) What type of nonlinearity would you build in your function? d) Give an estimate for the probability that you will lose on any given day more than $20,000. 1 2 3 4 5 6 A B C Day Loss Errors 1 109,158 65 2 14,387 54 3 16,417 25 4 8,564 30 5 29,517 42

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{Question: This exercise has as goal to show how correlations affect the distributions of the total losses that a firm may incur. Consider a company with two divisions. In each one of the two divisions, the probability of one operational risk event happening in one year is 0.5 and the probability of no event happening is 0.5. If an event happens in division i (i =1,2), then the random damage has a Normal distribution with mean 30 and standard deviation of 3. If both divisions are hit by an event in the same year, then the two random damages are Normally distributed random variables that are independent from one another and the total damage to the firm is the sum of these two independent random variables that are Normally distributed. Answer: Please find attached all the solutions. I used R for the second part. Best Regards.}

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