1 CHAPTER 1: CLAIM TRIANGLES In many types of insurance, there are IBNR (Incurred but not reported) claims. It could be that claims take months or even years to be reported and then paid. Insurance companies need a good way to determine the “ultimate” number of claims given the information they have at any given moment. For instance, imagine the following table represents claims from a particular set of policies by “policy year” and “development year”. I’ve simulated these values so I know the true number of ultimate claims in each year. Development Year Year 0 1 2 3 4 5 6 7 8 9 10 2008 3031 1108 409 153 47 17 6 5 2 1 0 2009 2871 1117 401 159 65 22 6 3 1 1 2010 2962 1080 407 169 57 26 6 3 0 2011 3411 1262 450 160 65 12 9 4 2012 3239 1191 452 152 65 26 9 2013 3166 1185 425 150 64 16 2014 3295 1265 453 191 70 2015 3122 1119 443 160 2016 3148 1212 403 2017 3139 1121 2018 3301 The basic method works like this: The first step is to create the cumulative claims table: Development Year Year 0 1 2 3 4 5 6 7 8 9 10 2008 3031 4139 4548 4701 4748 4765 4771 4776 4778 4779 4779 2009 2871 3988 4389 4548 4613 4635 4641 4644 4645 4646 2010 2962 4042 4449 4618 4675 4701 4707 4710 4710 2011 3411 4673 5123 5283 5348 5360 5369 5373 2012 3239 4430 4882 5034 5099 5125 5134 2013 3166 4351 4776 4926 4990 5006 2014 3295 4560 5013 5204 5274 2015 3122 4241 4684 4844 2016 3148 4360 4763 2017 3139 4260 2018 3301 2 Every development year has a development factor computed for that year. To compute that development factor, you take all calendar years in which the ending cumulative claims are known. You take the ending cumulative claims and divide by the beginning cumulative claims to get that year’s development factor. Year-by-year, these factors are: 0|1 4139 3988 4042 4673 4430 4351 4560 4241 4360 4260 3031 2871 2962 3411 3239 3166 3295 3122 314 1.3715268 3 8 3139 9d 1|2 4548 4389 4449 5123 4882 4776 5013 4684 4763 4139 3988 4042 4673 4430 4351 4560 4241 1.09908725 4 2 360 d 2|3 4701 4548 4618 5283 5034 4926 5204 4844 4548 4389 4449 5123 4882 4776 5013 1.03417494 4684 2d 3|4 4748 4613 4675 5348 5099 4990 5274 4701 4548 4618 5283 5034 4926 520 1.0126 5 4 187 6d 4|5 4765 4635 4701 5360 5125 5006 4748 4613 4675 5348 1.004037594 5099 4990 d 5|6 4771 4641 4707 5369 5134 4765 4635 4701 5360 512 1.00146424 5 8d 6|7 4776 4644 4710 5373 4771 4641 4707 5369 1.000769704d 7|8 4778 4645 4710 4776 4644 4710 1.000212314d 8|9 4779 4646 4778 1.000212247 4645 d 9|10 4779 4779 1.000000000d We can then use these factors to determine the ultimate claims for each year. For instance the cumulative claims for 2018 would be filled in as follows: Year 1 3301 1.371526893 4527.410273 Year 2 4527.410273 1.099087252 4976.018918 And so on giving ultimate claims of 5245.97165 Similarly, for 2017, the Year 2 claims are 4260 1.099087252 4682.111696 and ultimate claims are 4936.119742 . 3 The fully developed table is (values rounded to two decimal places): Development Year Year 0 1 2 3 4 5 6 7 8 9 10 2008 3031 4139 4548 4701 4748 4765 4771 4776 4778 4779 4779 2009 2871 3988 4389 4548 4613 4635 4641 4644 4645 4646 4646.00 2010 2962 4042 4449 4618 4675 4701 4707 4710 4710 4711.00 4711.00 2011 3411 4673 5123 5283 5348 5360 5369 5373 5374.14 5375.28 5375.28 2012 3239 4430 4882 5034 5099 5125 5134 5137.95 5139.04 5140.13 5140.13 2013 3166 4351 4776 4926 4990 5006 5013.33 5017.19 5018.25 5019.32 5019.32 2014 3295 4560 5013 5204 5274 5295.29 5303.05 5307.13 5308.26 5309.38 5309.38 2015 3122 4241 4684 4844 4905.13 4924.93 4932.14 4935.94 4936.99 4938.03 4938.03 2016 3148 4360 4763 4925.78 4987.93 5008.07 5015.40 5019.27 5020.33 5021.40 5021.40 2017 3139 4260 4682.11 4842.12 4903.22 4923.02 4930.23 4934.02 4935.07 4936.12 4936.12 2018 3301 4527.41 4976.02 5146.07 5211.01 5232.05 5239.71 5243.75 5244.86 5245.97 5245.97 The final ultimate claims is the sum of the last column: 4779 4646.00 4711.00 5375.28 5140.13 5019.32 5309.38 4938.03 5021.40 4936.12 5245.97 55,121.64 The current claims is the sum of the diagonal: 4779 4646 4710 5373 5134 5006 5274 4844 4763 4260 3301 52,090 Therefore, the company needs to hold an IBNR reserve for the difference of $3,031.64. It turns out in my simulation that the actual future claims will be $3,022 and the company will be slightly over-reserved. How do we “weight” experience in different years In the method above we weighted by number of claims, but we could weight equally by year, so that we just take the arithmetic average. That is: 0|1 1 4139 3988 4042 4673 4430 4351 4560 4241 4360 4260 10 3031 2871 2962 3411 3239 3166 3295 3122 3 1.371567792 148 3139 d 1|2 1 4548 4389 4449 5123 4882 4776 5013 4684 4763 9 4139 3988 4042 4673 4430 4351 4560 4241 436 1.09914427 0 7d 2|3 1 4701 4548 4618 5283 5034 4926 5204 4844 8 4548 4389 4449 5123 4882 4776 5013 46 1.0342 84 35931d 4 3|4 1 4748 4613 4675 5348 5099 4990 5274 7 4701 4548 4618 5283 5034 49 1. 26 520 0126 16 4 13 6d 4|5 1 4765 4635 4701 5360 5125 5006 6 4748 4613 4675 53 1.004076 48 5099 4990 727d 5|6 1 4771 4641 4707 5369 5134 5 4765 4635 1.0014530 4701 5360 5125 41d 6|7 1 4776 4644 4710 5373 4 4771 4641 4707 536 1.0007691 4 9 9d 7|8 1 4778 4645 4710 3 4776 4644 4710 1.000211364d 8|9 1 4779 4646 2 4778 46 1.0002122 5 8 4 9d 9|10 4779 4779 1.000000000d And the resulting table looks like this: Development Year Year 0 1 2 3 4 5 6 7 8 9 10 2008 3031 4139 4548 4701 4748 4765 4771 4776 4778 4779 4779 2009 2871 3988 4389 4548 4613 4635 4641 4644 4645 4646 4646.00 2010 2962 4042 4449 4618 4675 4701 4707 4710 4710 4711.00 4711.00 2011 3411 4673 5123 5283 5348 5360 5369 5373 5374.14 5375.28 5375.28 2012 3239 4430 4882 5034 5099 5125 5134 5137.95 5139.04 5140.13 5140.13 2013 3166 4351 4776 4926 4990 5006 5013.27 5017.13 5018.19 5019.26 5019.26 2014 3295 4560 5013 5204 5274 5295.50 5303.20 5307.27 5308.40 5309.52 5309.52 2015 3122 4241 4684 4844 4905.10 4925.09 4932.25 4936.05 4937.09 4938.14 4938.14 2016 3148 4360 4763 4926.07 4988.20 5008.53 5015.81 5019.67 5020.73 5021.80 5021.80 2017 3139 4260 4682.35 4842.66 4903.74 4923.73 4930.89 4934.68 4935.72 4936.77 4936.77 2018 3301 4527.55 4976.43 5146.80 5211.72 5232.96 5240.57 5244.60 5245.71 5246.82 5246.82 The final ultimate claims is the sum of the last column: 4779 4646.00 4711.00 5375.28 5140.13 5019.26 5309.52 4938.14 5021.80 4936.77 5246.82 55,123.70 The current claims is the sum of the diagonal: 4779 4646 4710 5373 5134 5006 5274 4844 5 4763 4260 3301 52,090 Therefore, the company needs to hold an IBNR reserve for the difference of $3,033.70. In practice, the reserves will not usually be this close together. Taking Inflation into account The inflation rates for the years 2008 to 2018 in the US were: 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 3.84% -0.36% 1.64% 3.16% 2.07% 1.46% 1.62% 0.12% 1.26% 2.13% 2.53% These rates can be used to adjust the claims in the initial table (claims per year, not cumulative claims per year). Three quick examples: The 2008 inflation adjusted claims in development year 0 would be: 3031 1 0.0384 1 0.0036 1 0.0164 1 0.0316 1 0.0207 1 0.0146 1 0.0162 1 0.0012 1 0.0126 1 0.0213 3582.98 The 2008 inflation adjusted claims in development year 1 would be: 1108 1 0.0036 1 0.0164 1 0.0316 1 0.0207 1 0.0146 1 0.0162 1 0.0012 1 0.0126 1 0.0213 1261.34 The 2009 inflation adjusted claims in development year 0 would be: 2871 1 0.0036 1 0.0164 1 0.0316 1 0.0207 1 0.0146 1 0.0162 1 0.0012 1 0.0126 1 0.0213 3268.34 And so on… The claim table then looks like this… 6 Development Year Year 0 1 2 3 4 5 6 7 8 9 10 2008 3582.98 1261.34 467.29 171.98 51.21 18.15 6.31 5.18 2.07 1.02 0.00 2009 3268.34 1276.18 450.75 173.25 69.39 23.15 6.21 3.10 1.02 1.00 2010 3384.12 1214.00 443.48 180.42 59.97 26.92 6.21 3.06 0.00 2011 3834.22 1375.13 480.40 168.35 67.30 12.41 9.19 4.00 2012 3529.35 1271.45 475.59 157.38 67.22 26.55 9.00 2013 3379.85 1246.84 440.05 155.13 65.36 16.00 2014 3466.94 1309.79 468.48 195.07 70.00 2015 3232.55 1157.23 452.44 160.00 2016 3255.56 1237.82 403.00 2017 3205.86 1121.00 2018 3301.00 The cumulative claims table looks like this …. Development Year Year 0 1 2 3 4 5 6 7 8 9 10 2008 3582.98 4844.33 5311.61 5483.60 5534.81 5552.96 5559.27 5564.45 5566.52 5567.54 5567.54 2009 3268.34 4544.52 4995.28 5168.53 5237.92 5261.07 5267.28 5270.38 5271.41 5272.41 2010 3384.12 4598.12 5041.60 5222.02 5281.99 5308.91 5315.12 5318.18 5318.18 2011 3834.22 5209.35 5689.75 5858.10 5925.40 5937.81 5947.00 5951.00 2012 3529.35 4800.80 5276.39 5433.77 5500.99 5527.54 5536.54 2013 3379.85 4626.68 5066.73 5221.86 5287.22 5303.22 2014 3466.94 4776.74 5245.21 5440.28 5510.28 2015 3232.55 4389.78 4842.22 5002.22 2016 3255.56 4493.38 4896.38 2017 3205.86 4326.86 2018 3301.00 The development factors are then computed by one of the two methods above. The resulting completed table looks like this: 7 Development Year Year 0 1 2 3 4 5 6 7 8 9 10 2008 3582.98 4844.33 5311.61 5483.60 5534.81 5552.96 5559.27 5564.45 5566.52 5567.54 5567.54 2009 3268.34 4544.52 4995.28 5168.53 5237.92 5261.07 5267.28 5270.38 5271.41 5272.41 5272.41 2010 3384.12 4598.12 5041.60 5222.02 5281.99 5308.91 5315.12 5318.18 5318.18 5319.17 5319.17 2011 3834.22 5209.35 5689.75 5858.10 5925.40 5937.81 5947.00 5951.00 5952.14 5953.25 5953.25 2012 3529.35 4800.80 5276.39 5433.77 5500.99 5527.54 5536.54 5540.39 5541.45 5542.48 5542.48 2013 3379.85 4626.68 5066.73 5221.86 5287.22 5303.22 5310.32 5314.01 5315.02 5316.02 5316.02 2014 3466.94 4776.74 5245.21 5440.28 5510.28 5531.00 5538.40 5542.25 5543.31 5544.34 5544.34 2015 3232.55 4389.78 4842.22 5002.22 5061.79 5080.81 5087.61 5091.15 5092.12 5093.07 5093.07 2016 3255.56 4493.38 4896.38 5057.14 5117.37 5136.60 5143.48 5147.05 5148.03 5148.99 5148.99 2017 3205.86 4326.86 4744.51 4900.30 4958.65 4977.29 4983.95 4987.41 4988.37 4989.30 4989.30 2018 3301.00 4506.81 4941.83 5104.09 5164.87 5184.29 5191.23 5194.83 5195.83 5196.80 5196.80 And the sum of the last column minus the sum of the diagonal elements gives: 58,943.36 55,985.63 2957.73 . At this point, the book suggests that since this reserve is lower than the reserve previously computed, neglecting inflation might cause a company to underreserve. On the other hand, this analysis neglects future inflation. If we assume that future inflation will be 2% per year, it will increase all of the red values in the table above. We first need to deaccumulate the table to get the claims in specific years: Year 0 1 2 3 4 5 6 7 8 9 10 2008 3582.98 1261.34 467.29 171.98 51.21 18.15 6.31 5.18 2.07 1.02 0.00 2009 3268.34 1276.18 450.75 173.25 69.39 23.15 6.21 3.10 1.02 1.00 0.00 2010 3384.12 1214.00 443.48 180.42 59.97 26.92 6.21 3.06 0.00 0.99 0.00 2011 3834.22 1375.13 480.40 168.35 67.30 12.41 9.19 4.00 1.14 1.11 0.00 2012 3529.35 1271.45 475.59 157.38 67.22 26.55 9.00 3.85 1.06 1.03 0.00 2013 3379.85 1246.84 440.05 155.13 65.36 16.00 7.10 3.69 1.02 0.99 0.00 2014 3466.94 1309.79 468.48 195.07 70.00 20.71 7.40 3.85 1.06 1.03 0.00 2015 3232.55 1157.23 452.44 160.00 59.57 19.03 6.80 3.53 0.97 0.95 0.00 2016 3255.56 1237.82 403.00 160.77 60.22 19.24 6.87 3.57 0.98 0.96 0.00 2017 3205.86 1121.00 417.65 155.78 58.35 18.64 6.66 3.46 0.95 0.93 0.00 2018 3301.00 1205.81 435.02 162.26 60.78 19.42 6.94 3.61 0.99 0.97 0.00 Then we need to inflate the yearly projections: Year 0 1 2 3 4 5 6 7 8 9 10 2008 3582.98 1261.34 467.29 171.98 51.21 18.15 6.31 5.18 2.07 1.02 0.00 8 2009 3268.34 1276.18 450.75 173.25 69.39 23.15 6.21 3.10 1.02 1.00 0.00 2010 3384.12 1214.00 443.48 180.42 59.97 26.92 6.21 3.06 0.00 1.01 0.00 2011 3834.22 1375.13 480.40 168.35 67.30 12.41 9.19 4.00 1.16 1.15 0.00 2012 3529.35 1271.45 475.59 157.38 67.22 26.55 9.00 3.92 1.10 1.10 0.00 2013 3379.85 1246.84 440.05 155.13 65.36 16.00 7.24 3.84 1.08 1.07 0.00 2014 3466.94 1309.79 468.48 195.07 70.00 21.13 7.70 4.08 1.15 1.14 0.00 2015 3232.55 1157.23 452.44 160.00 60.76 19.80 7.22 3.83 1.08 1.07 0.00 2016 3255.56 1237.82 403.00 163.98 62.65 20.41 7.44 3.94 1.11 1.10 0.00 2017 3205.86 1121.00 426.01 162.07 61.93 20.18 7.35 3.90 1.10 1.09 0.00 2018 3301.00 1229.93 452.60 172.19 65.79 21.44 7.81 4.14 1.16 1.16 0.00 And then reaccumulate: Year 0 1 2 3 4 5 6 7 8 9 10 2008 3582.98 4844.33 5311.61 5483.60 5534.81 5552.96 5559.27 5564.45 5566.52 5567.54 5567.54 2009 3268.34 4544.52 4995.28 5168.53 5237.92 5261.07 5267.28 5270.38 5271.41 5272.41 5272.41 2010 3384.12 4598.12 5041.60 5222.02 5281.99 5308.91 5315.12 5318.18 5318.18 5319.19 5319.19 2011 3834.22 5209.35 5689.75 5858.10 5925.40 5937.81 5947.00 5951.00 5952.16 5953.32 5953.32 2012 3529.35 4800.80 5276.39 5433.77 5500.99 5527.54 5536.54 5540.47 5541.57 5542.67 5542.67 2013 3379.85 4626.68 5066.73 5221.86 5287.22 5303.22 5310.46 5314.30 5315.38 5316.45 5316.45 2014 3466.94 4776.74 5245.21 5440.28 5510.28 5531.41 5539.11 5543.19 5544.34 5545.48 5545.48 2015 3232.55 4389.78 4842.22 5002.22 5062.98 5082.77 5089.99 5093.82 5094.89 5095.96 5095.96 2016 3255.56 4493.38 4896.38 5060.36 5123.01 5143.43 5150.87 5154.81 5155.92 5157.03 5157.03 2017 3205.86 4326.86 4752.87 4914.94 4976.87 4997.04 5004.40 5008.30 5009.39 5010.48 5010.48 2018 3301.00 4530.93 4983.52 5155.71 5221.51 5242.94 5250.76 5254.90 5256.06 5257.22 5257.22 And the sum of the last column minus the sum of the diagonal elements gives: 59,037.75 55,985.63 3052.11 , which is back to the normal range. Average Cost per Claim Method To implement this method, we need to know both the number and the average size of claims. Assume the table we’ve been using is for “Number of Claims”. We complete it as we have been doing. Assume the total dollar amount of claims are as in this table: Claims Development Year Year 0 1 2 3 4 5 6 7 8 9 10 9 2008 $4,407,650 $1,658,410 $579,745 $187,091 $55,899 $17,053 $5,469 $3,271 $1,428 $719 $0 2009 $4,506,494 $1,789,747 $564,700 $205,554 $82,425 $22,025 $6,936 $2,718 $867 $601 2010 $4,738,519 $1,634,353 $641,033 $216,606 $65,814 $30,819 $6,718 $2,617 $0 2011 $5,613,005 $1,888,078 $635,756 $179,344 $88,885 $13,504 $11,011 $2,839 2012 $4,910,583 $1,836,558 $669,448 $185,873 $89,627 $29,472 $9,903 2013 $5,102,072 $1,814,010 $598,353 $214,152 $85,185 $15,528 2014 $4,786,811 $1,840,727 $626,993 $253,640 $78,915 2015 $4,808,567 $1,813,809 $582,155 $192,541 2016 $5,418,338 $1,945,527 $505,725 2017 $5,274,933 $1,722,629 2018 $5,385,714 10 The Cumulative dollar amounts are then: Cumulative Development Year Year 0 1 2 3 4 5 2008 $4,407,650 $6,066,060 $6,645,805 $6,832,896 $6,888,795 $6,905,848 2009 $4,506,494 $6,296,241 $6,860,941 $7,066,495 $7,148,920 $7,170,945 2010 $4,738,519 $6,372,872 $7,013,905 $7,230,511 $7,296,325 $7,327,144 2011 $5,613,005 $7,501,083 $8,136,839 $8,316,183 $8,405,068 $8,418,572 2012 $4,910,583 $6,747,141 $7,416,589 $7,602,462 $7,692,089 $7,721,561 2013 $5,102,072 $6,916,082 $7,514,435 $7,728,587 $7,813,772 $7,829,300 2014 $4,786,811 $6,627,538 $7,254,531 $7,508,171 $7,587,086 2015 $4,808,567 $6,622,376 $7,204,531 $7,397,072 2016 $5,418,338 $7,363,865 $7,869,590 2017 $5,274,933 $6,997,562 2018 $5,385,714 6 7 8 9 10 $6,911,317 $6,914,588 $6,916,016 $6,916,735 $6,916,735 $7,177,881 $7,180,599 $7,181,466 $7,182,067 $7,333,862 $7,336,479 $7,336,479 $8,429,583 $8,432,422 $7,731,464 11 And the “Average Claim Size” can be found by dividing the two tables: Avg Claim Size Development Year Year 0 1 2 3 4 5 2008 $1,454.19 $1,465.59 $1,461.26 $1,453.50 $1,450.88 $1,449.29 2009 $1,569.66 $1,578.80 $1,563.21 $1,553.76 $1,549.73 $1,547.13 2010 $1,599.77 $1,576.66 $1,576.51 $1,565.72 $1,560.71 $1,558.64 2011 $1,645.56 $1,605.20 $1,588.30 $1,574.14 $1,571.63 $1,570.63 2012 $1,516.08 $1,523.06 $1,519.17 $1,510.22 $1,508.55 $1,506.65 2013 $1,611.52 $1,589.54 $1,573.37 $1,568.94 $1,565.89 $1,563.98
2014 $1,452.75 $1,453.41 $1,447.14 $1,442.77 $1,438.58 2015 $1,540.22 $1,561.51 $1,538.12 $1,527.06 2016 $1,721.20 $1,688.96 $1,652.23 2017 $1,680.45 $1,642.62 2018 $1,631.54 0.9932835 0.9912125 0.9942143 0.997837 0.9987964 12 6 7 8 9 10 $1,448.61 $1,447.78 $1,447.47 $1,447.32 $1,447.32 $1,546.62 $1,546.21 $1,546.06 $1,545.86 $1,558.08 $1,557.64 $1,557.64 $1,570.05 $1,569.41 $1,505.93 0.9996022 0.9996207 0.9999004 0.9998811 1 The development factors are listed below the tables. They are less than one, implying that average claim size *decreases* with time. The average claim size triangle would be completed like this: Development Year Year 0 1 2 3 4 5 2008 1454.19 1465.5859 1461.2588 1453.4984 1450.8835 1449.286 2009 1569.66 1578.7966 1563.2128 1553.7588 1549.7334 1547.1294 2010 1599.7701 1576.663 1576.5127 1565.7235 1560.7112 1558.6352 2011 1645.56 1605.1964 1588.2957 1574.1403 1571.6283 1570.6291 2012 1516.08 1523.0567 1519.1702 1510.2229 1508.5485 1506.646 2013 1611.5199 1589.5385 1573.3742 1568.9377 1565.8862 1563.9832 2014 1452.7499 1453.4075 1447.1436 1442.7692 1438.5829 1436.85 2015 1540.2201 1561.5129 1538.1151 1527.0586 1523.76 1521.92 2016 1721.2001 1688.9599 1652.2339 1642.67 1639.12 1637.15 2017 1680.4501 1642.6202 1628.19 1618.77 1615.26 1613.32 2018 1631.5401 1620.58 1606.34 1597.05 1593.59 1591.67 13 6 7 8 9 10 1448.6097 1447.7781 1447.4709 1447.3185 1447.318477 1546.6238 1546.2099 1546.0637 1545.8603 1545.86 1558.0756 1557.6389 1557.6389 1557.45 1557.45 1570.0471 1569.4067 1569.25 1569.06 1569.06 1505.9338 1505.36 1505.21 1505.03 1505.03 1563.36 1562.77 1562.61 1562.43 1562.43 1436.28 1435.74 1435.59 1435.42 1435.42 1521.32 1520.74 1520.59 1520.41 1520.41 1636.50 1635.88 1635.71 1635.52 1635.52 1612.68 1612.07 1611.91 1611.71 1611.71 1591.04 1590.44 1590.28 1590.09 1590.09 And multiplied by the completed “number of claims” table on page 3 to get: Completed Triangle Development Year Year 0 1 2 3 4 5 2008 $4,407,650 $6,066,060 $6,645,805 $6,832,896 $6,888,795 $6,905,848 2009 $4,506,494 $6,296,241 $6,860,941 $7,066,495 $7,148,920 $7,170,945 2010 $4,738,519 $6,372,872 $7,013,905 $7,230,511 $7,296,325 $7,327,144 2011 $5,613,005 $7,501,083 $8,136,839 $8,316,183 $8,405,068 $8,418,572 2012 $4,910,583 $6,747,141 $7,416,589 $7,602,462 $7,692,089 $7,721,561 2013 $5,102,072 $6,916,082 $7,514,435 $7,728,587 $7,813,772 $7,829,300 2014 $4,786,811 $6,627,538 $7,254,531 $7,508,171 $7,587,086 $7,608,551 2015 $4,808,567 $6,622,376 $7,204,531 $7,397,072 $7,474,212 $7,495,358 2016 $5,418,338 $7,363,865 $7,869,590 $8,091,446 $8,175,827 $8,198,958 2017 $5,274,933 $6,997,562 $7,623,347 $7,838,261 $7,920,002 $7,942,409 2018 $5,385,714 $7,337,039 $7,993,184 $8,218,523 $8,304,230 $8,327,724 14 6 7 8 9 10 $6,911,317 $6,914,588 $6,916,016 $6,916,735 $6,916,735 $7,177,881 $7,180,599 $7,181,466 $7,182,067 $7,182,067 $7,333,862 $7,336,479 $7,336,479 $7,337,164 $7,337,164 $8,429,583 $8,432,422 $8,433,372 $8,434,159 $8,434,159 $7,731,464 $7,734,480 $7,735,352 $7,736,074 $7,736,074 $7,837,645 $7,840,703 $7,841,586 $7,842,318 $7,842,318 $7,616,661 $7,619,632 $7,620,491 $7,621,202 $7,621,202 $7,503,347 $7,506,274 $7,507,120 $7,507,821 $7,507,821 $8,207,697 $8,210,899 $8,211,824 $8,212,591 $8,212,591 $7,950,875 $7,953,976 $7,954,873 $7,955,615 $7,955,615 $8,336,601 $8,339,853 $8,340,792 $8,341,571 $8,341,571 The sum of the diagonal is $80,665,491 and the sum of the final column is $85,087,317 leading to a reserve of $4,421,826. If, instead, we had taken the overall average claim size of existing claims as a stable $1,548.58 and multiplied that by the 3031.64 expected ultimate claims, we would have a reserve of $4,694,731. Bornhuetter-Ferguson The “Bornhuetter-Ferguson” method combines two relatively independent measures of anticipated claims. The first is based the claim ladder result we’ve already discussed. So, as a first step, compute the claim ladder result including development factors. The second measure is based on the “loss ratio”. In general, claims should be a stable proportion of premiums. Let’s assume claims are expected to be 90% of premiums. The second step is then to find the “emerging liability” which is the expected total claims computed from the loss ratio minus the expected claims at the moment Let’s assume the claim table on Page 1. Assume premiums are $5,000 in 2008 and increase by $100 every year to reach $6,000 in 2018. Finally, assume a loss ratio of 90% expected in all years. As a reminder, the development factors on page 1 are: Development Year 0 1 2 3 4 5 6 7 8 9 10 1.371527 1.099087 1.034175 1.012619 1.004038 1.001464 1.00077 1.000212 1.000212 1 You can get the percentage completed at each duration by assuming 100% at time 10 and then dividing by the development factors at each time. For instance, at time 9: 15 9 1/1 1PctComplete At time 8: 8 1/1.000212 0.999788PctComplete At time 7: 7 0.999788 /1.000212 0.999576PctComplete And so on, giving: Development Year 0 1 2 3 4 5 Factor 1.371527 1.099087 1.034175 1.012619 1.004038 PctComp 0.629245 0.863026 0.948541 0.980957 0.993336 0.997346 PctIncomp 0.370755 0.136974 0.051459 0.019043 0.006664 0.002654 Development Year 6 7 8 9 10 Factor 1.001464 1.00077 1.000212 1.000212 1 PctComp 0.998807 0.999576 0.999788 1 1 PctIncomp 0.001193 0.000424 0.000212 0 0 The Percentage Incomplete is 1-PctComp. The reserves can be found by multiplying premium by loss ratio by PctIncomp and then adding up, like this: Times Emerging Premium Loss Ratio Liability 5000 4500 5100 4590 0.00 5200 4680 0.99 5300 4770 2.02 5400 4860 5.80 5500 4950 13.14 5600 5040 33.59 5700 5130 97.69 5800 5220 268.62 5900 5310 727.33 6000 5400 2002.08 3151.26 The reserve is $3,151.26 compared to an original reserve of $3,031.64. 16 For CT6 exam purposes, the loss ratio is often calculated from the most recent completed row in the claim lag table. In this case, the loss ratio would be 4779 0.9558 5000 . Using this instead of 0.90 would give: Times Emerging Premium Loss Ratio Liability 5000 4779 5100 4874.58 0.00 5200 4970.16 1.05 5300 5065.74 2.15 5400 5161.32 6.16 5500 5256.9 13.95 5600 5352.48 35.67 5700 5448.06 103.75 5800 5543.64 285.27 5900 5639.22 772.43 6000 5734.8 2126.21 3346.63 17 CHAPTER 2: CLASSICAL LOSS DISTRIBUTIONS The distributions we might use to fit claims are discussed below. We’ll give the CDF, PDF, mean, variance, standard deviation and (often) the moment generating function in terms of the parameters, as well as interesting features of the distributions. The Exponential Distribution ( ) 1 xF x e for 0x ( ) xf x e for 0x 1E X 2 1( )Var X 1( )x ( )XM t t for t The exponential distribution has a “memoryless” property, meaning Pr | PrX M x X M X x The Pareto Distribution ( ) 1F x x for 0x 1 ( )f x x for 0x 1 E X 2 2( ) 1 2 Var X ( ) 1 2 x The Pareto distribution remains Pareto when subject to an “excess” or “deductible”: Pr | MX M x X M M x 18 The Gamma Distribution , ( ) x F x for 0x 1( ) xf x x e for 0x E X 2( )Var X 1( )x ( )XM t t for t If events are occurring at a uniform (Poisson) rate per unit time, the time to the rth event follows a gamma distribution with parameters r and . The Weibull Distribution ( ) 1 cxF x e for 0x 1( ) cxf x c x e for 0x 11 11E X c 2 2 1 2 1( ) 1 1Var X c 2 1 1 2 1( ) 1 1x c It has heavier tails than the exponential distribution, but not so heavy as Pareto distribution 19 The Lognormal Distribution ln( ) xF x N for 0x 2 2 (ln ) 21( ) 2 x f x e x for 0x 2 2E X e 2 22( ) 1Var X e e 2 2 2( ) 1x e e This is used for stock values, for instance Mixture Distributions Discrete: 1 2( ) ( ) (1 ) ( )F x pF x p F x Continuous: 1 1 1( ) ( | ,… ) ( ,… ) , …n n nf x f x g d d Example 1: x is Poisson, with parameter distributed as a gamma distribution… 1 0 ( ) ! ( ) xe ef x d x 11 0( ) ! x e d x 1 0 1 ( ) ! 1 x uu e du x 1 0( ) ! 1 x u x u e dux ( ) 1 ( ) ! 1 1 x x x which is a negative binomial distribution with parameters and 1 p . Example 2: x is exponential, with parameter distributed as a gamma distribution… 1 0 ( ) ( ) x ef x e d ( ) 0( ) xe d 0( ) uu due x x 1 0( ) uu e du x 1 ( 1) ( ) x 1x Which is a Pareto distribution with parameters and . Fitting distributions using Method of Moments… If there is a single parameter, set expected value equal. For instance, let M be the mean of the actual data. 20 For the exponential distribution, 1E X M so 1 M . If there are two parameters, set expected value and variance (or standard deviation or 2E X ) equal to each other. For instance, let M be the mean of the actual data and S be the standard deviation of the actual data. For the Pareto Distribution, 1 E X M and 2 2 2 2( ) 21 2 Var X S M . This implies 2 22 S M 2 2 2 S M 2 2 2 22 1 S S M M 2 2 2 2S S M . We can then find 1 M For the gamma distribution, E X M and 2 2( )Var X S . This implies 2 M S and 2 2 MM S For the Weibull Distribution the method is too difficult to apply by hand, although a computer could do it. For the Lognormal Distribution, 2 2E X e M and 2 2 22 2 2( ) 1 1Var X e e M e S This implies 2 2ln 1 M S and 2 ln 2 M The Maximum Likelihood Method involves maximizing 1 ( ) | n i i L f x or 1 ( ) ln | n i i l f x . Take the derivatives with respect to the parameters and set equal to “0”. For the exponential distribution, 1 ln i n x i l e 1 ln n i i x 1 ln n i i n x Now: 1 0 n i i l n x implies 1 1 n i i n Mx , the same as the method of moments. 21 For the Pareto Distribution, 11 , ln n i l x 1 ln ln 1 ln n i i x 1 ln ln 1 ln n i i n n x . Therefore 1 ln ln 0 n i i l n n x implying 1 ln 1 n i i n x Similarly, 1 11 0 n i i l n x implying 1 1 1n i i n i i i x x x . Setting them equal gives a non-linear equation for that can only be solved numerically. 1 1 1 0 ln 1 n i i n i i i i x n x x x . The value of and then be used to find . For the gamma distribution, numerical methods are again necessary. For the Weibull distribution, numerical methods are again necessary. For the Lognormal distribution, you would find the MLE parameters to be: 1 ln n i i x n and 2 1 ln n i i x n The book discusses various test, such as Kolmogorov-Smirnov, Anderson-Darling, Chi-Square and AIC. These are not part of the CT6 syllabus and are beyond the scope of ACTS 336. Types of reinsurance 1. Excess of Loss Insurer pays the first M and the reinsurer pays 0 if X M and X M otherwise. The mean amount paid by the insurer is 0 ( ) ( ) M M xf x dx M f x dx The mean amount paid by the reinsurer is ( ) M x M f x dx 22 The pdf of the reinsurers payout *conditional* on payout occurring is ( )( ) 1 ( ) f w Mg w F M MLE can be done by either the insurer or the reinsurer. For the reinsurer, just use ( | )g w for the likelihood of a payout w and for the insurer use 1 ( )F M for the likelihood of each claim that exceeds the deductible (assuming you don’t know the actual amount). For example: Exponential Distribution Assume there are n values of ix as well as m values of M . In this case 1 1 ln lni n m x M i i l e e 1 ln n i i n x mM . Now 1 0 n i i l n mM x implies 1 n i i n mM x 2. Proportional In this case, the company pays X and the reinsurer pays 1 X . The means, variances, and maximum likelihood estimators are straightforward to determine. While they are not discussed specifically in this chapter, we will use the following discrete distributions later: Binomial: Pr 1 n iiN i p p for 0,1,..,i n ; E X np ; ( ) (1 )Var X np p ; ( ) 1 ntSM t pe p Negative Binomial: 1 !Pr (1 ) !( 1)! k nk nN n p p n k for 0,1,2…n ; 1E X pk p ; 2( ) 1r X p V ka p ; ( ) 1 (1 ) kk tSM t p p e . The Geometric Distribution is the special case 1k . And the following continuous distribution will be used later: Beta: 11( ) 1f x x x for 0 1x ; E X ; 2 ( ) 1 Var X ; the moment generating function is too complicated to include. 23 Chapter 3: RISK THEORY Let S be a random loss from a business line, reflected in two random variables: the number of claims (or “Frequency”) N , and N values of the loss per claim (or “Severity”) iX so that 1 N i i S X . In general: E S E N E X 2( ) ( ) ( )Var S E N Var X E X Var N And ( ) ln ( )S N XM t M M t Some Specific Cases: Compound Poisson Distribution: Pr ! neN n n for 0,1,2…n E S E X 2( )Var S E X ln ( ) 1 ( ) 1( ) M tX X e M t SM t e e A sum of independent compound poisson random variables is also compound poisson. The new parameters 1 n i i and 1 1( ) ( ) i n X i X i F x F x . For example: Let 1S be compound poisson with 1 and a severity distribution of 100% chance of $20,000. Let 2S be compound poisson with 2 and a severity distribution of 100% chance of $10,000. 1 2S S S is also compound poisson. 1 2 3 and 1 2 1 2( ) ( ) ( ) 3 3 F x F x F x . In this case: 1( ) 0F x if 20,000x and 1( ) 1F x if 20,000x 2 ( ) 0F x if 10,000x and 2 ( ) 1F x if 10,000x Leading to: ( ) 0F x if 10,000x 2( ) 3 F x if 10,000 20,000x ( ) 1F x if 20,000 x . In order to check this, we can add 1S and 2S directly via convolution: 24 1S 2S 1 2S S S Value Prob Prob Prob $0 1e 2e 1 2 0.049787068e e $10,000 0 22e 1 22 0.099574137e e $20,000 1e 22e 1 2 1 22 0.149361205e e e e $30,000 0 2 4 3 e 1 2 1 24 2 0.1659568953e e e e $40,000 1 2 e 22 3 e 1 1 2 1 2 22 2 0.15765905 3 2 ee e e e e $50,000 0 2 4 15 e 1 1 2 1 2 24 4 2 0.129446378 15 3 2 ee e e e e $60,000 1 6 e 24 45 e 1 1 1 2 1 2 2 24 2 2 45 3 2 6 e ee e e e e e 0.095701809 Or, we can determine the probabilities from the compound distribution: 0N 1N 2N 3N 4N 5N 6N Prob 3e 33e 3 9 2 e 39 2 e 327 8 e 381 40 e 381 80 e Value ( | )f s N ( | )f s N ( | )f s N ( | )f s N ( | )f s N ( | )f s N ( | )f s N $0 1 0 0 0 0 0 0 $10,000 0 2/3 0 0 0 0 0 $20,000 0 1/3 4/9 0 0 0 0 $30,000 0 0 4/9 8/27 0 0 0 $40,000 0 0 1/9 12/27 16/81 0 0 $50,000 0 0 0 6/27 32/81 32/243 0 $60,000 0 0 0 1/27 24/81 80/243 64/729 25 Value Total $0 3 0.049787068e $10,000 32 3 0.0995741373 e $20,000 3 31 4 93 0.1493612053 9 2e e $30,000 3 3 4 9 8 9 0.165956895 9 2 27 2 e e $40,000 3 3 3 1 9 12 9 16 27 0.15765905 9 2 27 2 81 8 e e e $50,000 3 3 3 6 9 32 27 32 81 0.129446378 27 2 81 8 243 40 e e e $60,000 3 3 3 3 1 9 24 27 80 81 64 81 0.095701809 27