Exercise 1. In a PC insurance company, the estimated average Loss Cost per exposure unit for 2009 is 382.78 and the Expense Ratio for 2009 is determined to be 24%. Find the premium to be charged for a unit exposure during 2009. The premium is simply the Loss Cost grossed up for expenses: $382.78 1 0.24 $503.66.
Exercise 2. At ABC insurance company, for accident year 2007, the expected loss ratio was 64% and earned premium was $20,890,000. Through the first year of development, to 12/31/2008, cumulative paid losses were $11,987,654. Historical loss development factors are given as: 1/0 = 1.35; 2/1=1.11; 3/2=1.05; 4/3=1.02. All claims are fully developed after 5 years. Determine the estimated loss reserve for 12/31/2008 using the expected loss ratio method. Solution: The Estimated Ultimate Losses (EUL), after claims are fully developed, equal premiums times expected loss ratio, and in this case this equals EUL = $20,890,000 0.64 = $13,369,600. This is the amount we expect to pay in total over 5 years. But in 2008, cumulative amount of payments due to claims are $11,987,654, so the reserve is equal to the Estimated Ultimate Losses minus the amount already paid: EUL $11,987,654 = $13,369,600 $11,987,654 = $1,381,946.
Exercise 3. At ABC insurance company, for accident year 2007, the expected loss ratio was 64% and earned premium was $20,890,000. Through the first year of development, to 12/31/2008, cumulative paid losses were $11,987,654. Historical loss development factors are given as: 1/0 = 1.35; 2/1=1.11; 3/2=1.05; 4/3=1.02. All claims are fully developed after 4 years. Determine the estimated loss reserve for 12/31/2008 using the chain ladder method. For chain ladder method, the Estimated Ultimate Losses (EUL) are equal to the Cumulative Paid Losses (CPL) given in some year, transformed to the end of the claim development period by using historical loss development factors. Since we are given CPL in 2008, the first development factor from 2007 to 2008 is already built in into the value 11,987,654. We have EUL = $11,987,654 1.11 1.05 1.02 $14,251,043. The loss reserve equals the difference of EUL and CPL: EUL $11,987,654 = $14,251,043 $11,987,654 = $2,263,389.
Exercise 4. At ABC insurance company, for accident year 2007, the expected loss ratio was 64% and earned premium was $20,890,000. Through the first year of development, to 12/31/2008, cumulative paid losses were $11,987,654. Historical loss development factors are given as: 1/0 = 1.35; 2/1=1.11; 3/2=1.05; 4/3=1.02. All claims are fully developed after 5 years. Determine the estimated loss reserve using the Bornhuetter-Ferguson Method. Using the Bornhuetter-Ferguson Method, Estimated Loss Reserve = EUL (1-(1/ f ult )), where EUL is computed as in Loss Ratio Method: EUL = $20,890,000 0.64 = $13,369,600. Note that fult is the factor we would use to compute the EUL in chain ladder method, so f ult = 1.11 1.05 1.02 = 1.18881. Substituting these values, we obtain: ELR = 13,369,600 (1 (1/1.18881) = 2123396.
Exercise 5. Incurred claims at ABC insurance company for past months are shown in the following, all claims are assumed to be fully paid after 3 months of development: Development Month Claim Month 0 1 2 3 12/2008 3067137 393726 215260 19569 01/2009 3375641 373608 274829 02/2009 4441270 490330 03/2009 4763283 Using the chain ladder method,a reasonable set of loss-development factors are 1.116 for 1/0, 1.068 for 2/1, and 1.005 for 3/2. Using these loss-development factors, determine an estimated loss reserve at 03/2009. Cumulative Claim Payments Claim Month 0 1 2 3 12/2008 3067137 3460863 3676123 3695692 01/2009 3375641 3749249 4024078 02/2009 4441270 4931600 03/2009 4763283 Based on this, we calculate the reserve as of 03/2009, first separately for each claim month: 01/2009: 4024078 1.005 4024078 = 20120, 02/3009: 4931600 1.068 1.005 4931600 = 361684, 03/2009: 4763283 1.116 1.068 1.005 4763283 = 942403. Total reserve is: 20120 + 362684 + 942403 = 1324207.
Exercise 6. You are given the following tables of cumulative amounts paid Development Year Accident year 0 1 2 3 Paid as of 12/31/2009 2006 260 330 360 390 390 2007 300 360 400 380 2008 330 430 420 2009 350 150 You are also given these incurred development factors 0/1: 1.2 1/2: 1.1 2/3: 1.1 Calculate the IBNR reserve as of 12/31/2009. Using the development factors we complete the table Development Year Accident year 0 1 2 3 2006 260 330 360 390 2007 300 360 400 1.1 400 = 440 2008 330 430 1.1 430 = 473 1.1 473 = 520.3 2009 350 1.2 350 1.1 420 = 462 1.1 462 = 508.2 = 420 The total ultimate amount paid is 390 + 440 + 520.3 + 508.2 = 1858.5. The amount already paid is 390 + 380 + 420 + 150 = 1340. The IBNR reserve is 1858.5 1340 = 518.5.
Exercise 7. You have established for your company the following: Expected Losses on group of autos in 2003 = $195,490 (trended and developed). Earned exposure units on group of autos in 2003 = 168.52. Permissible Loss Ratio = 65%. What is New Average Gross Rate? New Average Loss Cost = Expected Losses in Effective Period (Trended and Developed) / Number of Earned Exposure Units So New Average Loss Cost = $195,490/168.52 = $1160.04 Furthermore New Average Gross Rate = New Average Loss Cost / Permissible Loss Ratio Thus New Average Gross Rate = $1160.04/0.65 = $1784.68.
Exercise 8. You Bet Your Car Insurance Co. paid $275,000 of losses in 2001, its unpaid loss reserves were $1,200,000 on December 31, 2000, and they increased by 10% by the end of 2001. Find the total amount of losses incurred by this company in 2001. We have Incurred losses (2001) = Paid Losses (2001) + Δ Reserves (2001) so that Incurred losses (2001) = $275,000 + $120,000 = $395,000.
Exercise 9. Calculate the new average gross rate given the following information: Expected effective incurred losses (trended and developed) $30,000,000 Earned exposure units 1,000,000 Earned premium at current rates $45,000,000 Present average manual rate 45 Permissible loss ratio (equal to 1 Expense ratio) 0.70 This can be calculated using the lost cost method and the loss ratio method. Either solution will lead to the same result, as you have learned. First, we consider the loss cost method, as follows: Expected Effective Incurred Loss Expected Effective Loss Cost = = Earned Exposure Units = 30,000,000 1,000,000 = 30, Expected Effective Loss Cost Indicated Average Gross Rate = Permissible Loss Ratio Alternatively, using the loss ratio method: Expected Effective Incurred Loss Expected Effective Loss Ratio = Earned Premiums at Current Rates = = 30,000,000 45,000,000 = 0.6666... Expected Effective Loss Ratio Indicated Rate Change Factor = 1 = Permissible Loss Ratio = 0.6666... 1 = 0.04762. 0.70 Indicated Average Gross Rate = 45 1 0.04762 ( ) = 42.86. = 30 0.70 = 42.86.
Exercise 10. A homeowner has a house valued at 400,000, but has insured it for 250,000 with an insurer that requires 80% of full coverage before it reimburses losses in full. If coverage is less than 80% of full coverage, then any loss is reimbursed of a pro-rata basis of what would have been paid had the 80% requirement been met. The homeowner has a kitchen fire estimated at 40,000 on a replacement cost basis. How much will the insurer pay toward reimbursing the homeowner for this loss? For full coverage, the homeowner needed insurance equal to at least 80% of 400,000, i.e., 320,000, as measured at the time of the loss. But there was only 250,000 worth of coverage, i.e., 78.125% of what was needed. For this 40,000 kitchen fire loss, the insurer will pay 78.125% of 40,000, i.e., 31,250.