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8 Pages

### Assignment 3 Solutions

Course: ACTSC 463, Fall 2011
School: Waterloo
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Word Count: 600

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Solution Calculate 1. ultimate AY and RY loss development factors: Cumulative LDFs Development Interval 1-2 Years 2-3 Years 3-4 Years 4-5 Years 5-Ultimate Accident Cumulative Year AY 1.500 2.382 1.250 1.588 1.100 1.271 1.100 1.155 1.050 1.050 Report Year 1.100 1.100 1.000 1.000 1.000 Cumulative RY 1.210 1.100 1.000 1.000 1.000 Calculate ultimate Accident Year and Report Year losses. AY Incurred to AY RY...

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Solution Calculate 1. ultimate AY and RY loss development factors: Cumulative LDFs Development Interval 1-2 Years 2-3 Years 3-4 Years 4-5 Years 5-Ultimate Accident Cumulative Year AY 1.500 2.382 1.250 1.588 1.100 1.271 1.100 1.155 1.050 1.050 Report Year 1.100 1.100 1.000 1.000 1.000 Cumulative RY 1.210 1.100 1.000 1.000 1.000 Calculate ultimate Accident Year and Report Year losses. AY Incurred to AY RY Incurred AY/RY Date AY LDF Ultimate to Date 2005 17,000 1.050 17,850 10,000 2006 21,000 1.155 24,255 16,000 2007 26,000 1.271 33,046 23,000 2008 25,000 1.588 39,700 28,000 2009 20,000 2.382 47,640 32,000 Total 109,000 162,491 109,000 Estimated unreported losses = AY Ultimate - RY Ultimate 43,971 2. Solution: i) Limited data ii) leveraging effect RY LDF 1.000 1.000 1.000 1.100 1.210 RY Ultimate 10,000 16,000 23,000 30,800 38,720 118,520 3. Solution Since loss development factors are for paid losses we can calculate unpaid losses directly: Expected Accident Earned Ult Claim Year Premium Ratio % Unpaid Unpaid 2005 1,000,000 0.65 0.0% 2006 1,000,000 0.65 9.5% 61,765 2007 1,000,000 0.70 20.0% 140,000 2008 1,000,000 0.70 37.5% 262,500 2009 1,000,000 0.80 50.0% 400,000 Total 5,000,000 864,265 4. Solution a) Incurred Expected Accident Earned Incurred LDF to % Loss Year Premium Loss Ultimate Unreported Ratio 2006 15,000 4,500 1.500 33.3% 65% 2007 16,000 4,800 1.750 42.9% 65% 2008 17,000 3,200 2.000 50.0% 65% 2009 18,000 2,500 4.500 77.8% 65% B-F Method 3,250 4,457 5,525 9,100 Solution b) Incurred Accident Earned Incurred LDF to % % Used Up Cape Cod Cape Cod Year Premium Loss Ultimate Unreported Reported Premium LR IBNR 2006 15,000 4,500 1.500 33.3% 66.7% 10,000 45.0% 2,370 2007 16,000 4,800 1.750 42.9% 57.1% 9,143 52.5% 3,251 2008 17,000 3,200 2.000 50.0% 50.0% 8,500 37.6% 4,029 2009 18,000 2,500 4.500 77.8% 22.2% 4,000 62.5% 6,637 Total 66,000 15,000 31,643 47.4% 16,287 Solution c) Depends on how much information you have about the expected loss ratio of 65%. The Cape Cod uses actual experience to select the loss ratio so is more appropriate in some situations. 5. Solution Select disposal Ratios. Report Year 0-12 2006 0.50 2007 0.45 2008 0.55 2009 0.55 Select 0.50 12-24 0.25 0.15 0.20 24-36 0.15 0.24 37-Ult 0.10 0.20 0.20 Calculate future claim closings Report Year 0-12 12-24 2006 60 30 2007 63 21 2008 88 32 2009 110 36 0.10 Calculated Trended Paid Severities Report Year 0-12 12-24 24-36 2006 926 1,597 2,248 2007 926 1,573 2,258 2008 945 1,650 2009 950 Select 937 1,607 2,253 37-Ult 3,042 3,042 Actual Paid Severities Report Year 0-12 12-24 2006 800 1,200 2007 840 1,300 2008 900 1,500 2009 950 1,607 Select 937 1,607 24-36 18 34 27 36 24-36 1,600 1,800 2,253 2,253 2,253 37-Ult Ultimate 12 120 22 140 13 160 18 200 37-Ult 2,000 2,300 2,645 3,042 3,042 Report Ultimate RY Year Losses 2006 136,800 2007 191,300 2008 222,413 2009 298,198 Total 848,712 6. Solution a) IBNR would be overstated. The incurred development technique reacts appropriately to l oss ratio deterioration, but will overstate when there is case strengthening. b) The B-F method understates when there is loss ratio deterioration and overstates when there is case reserve strengthening. c) The expected loss technique will appropriately reflect case reserve strengthening, but doesn't reflect loss ratio deterioration so will understate unless you adjust the LR. d) Paid development is independent of case reserves so is unaffected by strengthening. It also reflects loss ratio deterioration appropriately. 7. Solution a) Calculate case outstanding triangle. AY 12 24 2007 11,700 20,000 2008 24,600 26,250 2009 18,880 Calculate average case o/s triangle AY 12 24 2007 390 500 2008 410 525 2009 472 Trend average case backwards at 5% AY 12 24 2007 428.12 500.00 2008 449.52 525.00 2009 472.00 Calculate case adjusted triangle AY 12 24 2007 12,844 20,000 2008 26,971 26,250 2009 18,880 36 17,100 36 855 36 855.00 36 17,100 b) Calculate disposal ratios and select latest diagonal AY 12 24 36 2007 44.44% 55.56% 77.78% 2008 38.46% 61.54% 2009 28.57% Select 28.57% 61.54% 77.78% Calculate adjusted paid losses by interpolation of factors AY 12 24 36 2007 18,669 48,335 65,800 2008 28,303 70,400 2009 32,000 c) Sum the adjusted case and adjusted paid triangles. AY 12 24 36 2007 31,512 68,335 82,900 2008 55,274 96,650 2009 50,880 d) Calculate development factors and use geometric average to select AY 12-24 24-36 36-Ult 2007 2.169 1.213 2008 1.749 Select 1.947 1.213 1.05 Cumulative 2.480 1.274 1.05 IBNR = (2.137-1)(50,880) 75,323.44 8. Solution Determine weighted number of open claims. =110+75 = 185 Expense per open claim is =100,000/185 540.54 9. Solution Calculate LDFs and make selections: AY 12 24 36 2006 2.00 1.25 1.10 2007 2.00 1.25 2008 2.00 Select 2.00 1.25 1.10 Cumulative 2.75 1.375 1.1 Use B-F formula to calculate the ratio Expected Unreported = Loss Ratio((1-1/1.1)(1400)+(1-1/1.375)(1500)+(1-1/2.75)(2000)) 1300=Loss Ratio(1809.09) Loss Ratio = 71.9% 10. Solution a) Sum of the credit spread, timing risk and mismatch risk =25bp + .1(5%) + 0 0.75 =75 basis points b) Calculate payment pattern Accident Year 0-12 12-24 68.1% 61.4% 84.3% 74.0% 72.1% 72.0% 72.0% 100.0% 2006 2007 2008 2009 2010 Select Incremental Cumulative 36-48 48-60 94.7% 92.0% 87.8% 90.5% 24-36 97.8% 99.0% 93.9% 100.0% 100.0% 100.0% 91.3% 19.3% 28.0% 96.9% 5.6% 8.7% 100.0% 3.1% 3.1% 100.0% 0.0% 0.0% 2013 2014 Ultimate 9,300 9,900 8,646 11,078 10,706 Calculate discount factors Accident Year 2010 2011 0.0% 0.0% 100.0% 64.5% 68.8% 2006 2007 2008 2009 2010 Discounted @ 5% Accident Year 2006 2007 2008 2009 2010 Accident Year 2007 2008 2009 2010 Total 0.0% 0.0% 0.0% 35.5% 20.1% 2012 0.0% 0.0% 0.0% 0.0% 11.1% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 5% Discounted @ PFAD rate = 4% 0.5 1.5 2.5 3.5 4.5 Accident Year 0.0% 0.0% 97.6% 63.0% 67.2% 0.0% 0.0% 0.0% 33.0% 18.7% 0.0% 0.0% 0.0% 0.0% 9.8% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% Outstandin g 523 1,054 2,991 4,568 Disc @ 5% 511 1,011 2,861 4,382 Disc @ 4.0% 513 1,019 2,886 4,418 Claim Int PFAD PFAD % 2 8 25 35 10% 10% 10% 10% 10% Total 0.0% 0.0% 97.6% 95.9% 95.7% Claim PFAD 51 101 286 438 2005 2006 2007 2008 2009 Total PFAD 54 109 311 474 4% 0.5 1.5 2.5 3.5 4.5 0.0% 0.0% 98.1% 63.3% 67.5% 0.0% 0.0% 0.0% 33.5% 19.0% 0.0% 0.0% 0.0% 0.0% 10.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% APV 564 1,120 3,172 4,856 Total 0.0% 0.0% 98.1% 96.7% 96.5% 11. Solution (note: the ideal solution would use loss ratios instead of loss dollars, but the derivation is the same. AY 2005 Ultimate = 7100 x 1.1 7,810 AY 2006 Ultimate = 9500 x 1.1 10,450 AY 2007 Ultimate = as below 6,490 AY 2008 Ultimate = as below 5,457 AY 2009 Ultimate = as below 7,655 AY 2010 Ultimate = as below 4,763 Start with AY 2007: Use data from AY 2005 and 2006 Average Y: 9130 Average XY: 36,789,500 b: 2.4 AY 2007 Ultimate: 1973.333 4214133 10618.99 Average X: Average X^2: a: 1487.5 3038825 6966.686 Average X: Average X^2: a: 1540 3804450 4083.936 7,655 AY 2010: Use data from AY 2005 - 2009 Average Y: 7,572 Average XY: 14,907,134 b: 2.265208309 AY 2010 Ultimate: Average X: Average X^2: a: 5,457 AY 2009: Use data from AY 2005 - 2008 Average Y: 7,552 Average XY: 11,558,094 b: 0.393292738 AY 2009 Ultimate: 3950 15905000 -350 6,490 AY 2008: Use data from AY 2005 - 2007 Average Y: 8250 Average XY: 15,895,733 b: -1.200499861 AY 2008 Ultimate: Average X: Average X^2: a: 4,763
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