PART 6 Sample Essay Solutions Fall 2002 21. a) Link ratio estimate Æ y=cx where c=E(y)/E(x) = 1.346 Æ y = 1.346(60) = 80.77 Budgeted loss estimate Æ y=E(y)= 87.5 b) Least squares y = a+bx b=(E(xy) – E(x)E(y))/(E(x^2)-E(x)^2)= 0.5 E(xy) = (65(90)+50(80)+70(85)+75(95))/4 = 5731.25 E(x^2) = (65^2+50^2+70^2+75^2)/4 = 4312.5 a=E(y)-bE(x) = 87.5 - .5*65 = 55 Least Squares Estimates Æ y=55+.5(60) = 85 L(x) = Z(cx) + (1-Z)E(y) where Z = b/c = .5/1.346 = .3715 = .3715(80.77) + (1-.3715)(87.5) = 85 ties to least squares. PART 6 Sample Essay Solutions Fall 2002 22. (a) IBNR @ 12/31/01 using case incurred loss development. Case Incurred @ LDF to 12/31/01 (000’s) Ultimate IBNR Factor IBNR @ 12/01 AY 98 100 1.080 0.080 8.00 99 1,000 1.134 0.134 134.00 00 900 1.247 0.247 222.30 01 600 1.559 0.559 335.40 Total 2,600 $699.70 $699,700 = IBNR @ 12/01 (b) IBNR reserve using B/F method. AY 98 99 00 01 Total EP (000’s) 200 1000 1500 1500 ELR 0.80 0.80 0.80 0.80 IBNR Factor 0.0741 0.1182 0.1981 0.3586 IBNR @12/31/01 11.856 94.56 237.72 430.32 774.456 $774,456 (c) B/F method is preferred if there is little to no credible historical data. For example, on a new line of business. (d) When the loss development pattern in the data is consistent and if the loss ratio is deteriorating. (e) AY 98 99 00 01 Total EP (000’s) 200 1000 1500 1500 ELR 0.80 0.80 0.80 0.80 Factor for IBNR to emerge during CY 2002 0.0741 = 0.0741 0.1182 – 0.0741 = 0.0441 0.1981 – 0.1182 = 0.0799 0.3586 – 0.1981 = 0.1605 IBNR to emerge during CY 2002 11.856 35.280 95.880 192.60 335.616 $335,616 PART 6 Sample Essay Solutions Fall 2002 23. Reliance on Non-Actuarial Expertise: The loss event modules are non-actuarial in nature, and require understanding of complicated events. A reluctance for actuaries to rely on non-actuaries for such an important input. Black Box: These models are proprietary, and as such much is not disclosed because of the proprietary nature. Actuaries reluctant to accept without understanding the process of the model (complicated). Cost/Benefit Analysis: The models are expensive, so the benefit must outweigh the cost. 24. Cumulative Closed Claims Closed = (Cumulative Reported) - Open 0-12 12-24 24-36 36-48 1998 20 50 80 100 1999 21 53 86 2000 24 59 91 115 2001 25 Proportion Closed Ratios 0-12 12-24 24-36 36-48 25/120 = 0.208 (59-24)/(115-24) = 0.385 (86-53)/(110-53) = 0.579 1.000 0.579*(115-59)=32.4 32.4 + 59 = 91 Calculate Average Cost per Closed Claim for 2000 24-36: 4,725*1.05 = 4961.25 36-48: 7,500*(1.05^2) = 8,268.75 Accident Year 2000 Ultimate Loss = 1,653*24 + 3,308*(59-24) + 4,961.25*(91-59) + 8,268.75*(115-91) = 512,662 PART 6 Sample Essay Solutions Fall 2002 25. a) (i) I would expect the incurred loss method to produce a slightly overstated reserve. The method is not affected by the deteriorating loss ratio, but the slip in reserve adequacy and subsequent correction by claims would produce larger than necessary development factors. (ii) I would expect the expected loss method to understate reserves. Since the actuaries have made no adjustment, the expected loss ratio used is lower than what one would now expect. (iii) I would also expect the Percentage of Premium method to understate reserves, although I expect it to be a better estimate than the expected loss method. This method would be relying on previous years of lower losses for the factors so the reserves would be understated. b) Using paid losses to predict development. Once the correction is made by claims, I would expect the paid development to be represented by past development. PART 6 Sample Essay Solutions Fall 2002 26. Disposal Rate= Cumulative Closed Claims Projected Ultimate Claims AY 12 24 36 48 98 .325 .700 1.000 1.000 99 .350 .750 1.000 00 .375 .800 01 .375 Selected .400 .800 1.000 1.000 (Increasing Trend in disposal rates) Restated Closed Claims Disposal Rate .400 .800 1.000 1.000 AY 12 24 36 48 98 20000 40000 50000 50000 99 21000 42000 52500 52500 Restated Closed Claims for AY00 at 24 months = (55000)(.8)=44000 Restated Paid Loss (using linear interpolation w/ closed claims) AY @12 98 99 121875*(35000-20000)/(35000-16250) + 262500*(20000-16250)/(35000-16250) = 150000 137813*(39375-21000)/(39375-18375) + 295313*(21000-18375)/(39375-18375) = 157500.5 AY @24 98 99 262500*(50000-40000)/(50000-35000) + 375000*(20000-35000)/(50000-35000) = 150000 295313*(52500-42000)/(52500-39375) + 393750*(42000-39375)/(52500-39375) = 157500.5 Development Factors: AY 24-36 98 375000/300000=1.250 99 393750/315000.4=1.250 Selected 1.250 46-48 375000/375000=1.000 1.000 1.000 UltimateAY00 Loss = (330,000)(1.25)(1.00) =412,500 PART 6 Sample Essay Solutions Fall 2002 27. Adjustment to Total Known Losses a). Ultimate loss estimates for AY99,00,01,02 Expected Losses - average then adjust = 1/n(sum of Ka*) = 1/3(1400+1700+1200) = 2092,000 1/n(sum of 1/fa) 1/3(1/1.8+1/1.5+1/1.2) AY 99 00 01 02 Expected Ult Losses (000’s) 2,092 2,092 2,092 2,092 % unreported @ 12/31/01 16.7% 33.3% 44.4% 100% IBNR (000’s) 349 697 930 2,092 Known(000’s) @12/31/01 1,400 1,700 1,200 0 Ultimate Loss 1,748,667 2,397,333 2,129,778 2,092,000 % unreported = 1-1/age to ult factor b). The average of the estimates of ultimate losses should be equivalent to the expected ultimate losses used in the calculation ¼*(1,748,667+…+2,092,000) = 2,092,000 PART 6 Sample Essay Solutions Fall 2002 28. a) Ultimate LDF for unlimited loss @ 12 = 2.004 x 1.498 x 1.335 x 1.245 x 1.098 = 5.4785 Ultimate loss for AY 2001 = $220,000 x 5.4785 = $1,205,272 Ultimate LDF for deductible loss @ 12 = 1.894 x 1.419 x 1.251 x 1.201 x 1.049 = 4.2358 Ultimate deductible loss for AY 2001 = $220,000 x 4.2358 = $931,882 The reported excess loss as of Dec 31, 2001 for AY 2001 = $220,000 - $220,000 =0 IBNR for excess loss for AY 2001 = $1,205,272 - $931,882 = $273,390 b) Adv: 1) It estimates the IBNR for excess loss based on more stable loss development pattern from the limited loss triangle. 2) It provides the estimate of excess loss at maturities where they have not emerged. c) Service revenue asset = (Ultimate deductible loss – Ult losses excess of aggregate) x loss multiplier – known recoveries = ($931,882 - $256,944) (0.1) - $50,071 = $17,423 PART 6 Sample Essay Solutions Fall 2002 29. a) W2 is the weight given to the “experienced” adjustments for the policy year. Berry’s formula weights together expected (a priori) deviation, and deviation implied by experience to date. Since no substantial adjustments will have been paid by 21 months, Berry gives the “experience” portion of his formula no weight. Note that for a 12-month policy, adjusted 6 months after termination, with adjustment taking 2-3 months to appear in the data, we should not have any substantial adjustments before 21 months. b) W2 = 0 for 1 to 20 months, then increases linearly by 1/28 per month until reaching 1.00 at 48 months. Thus, at 33 months, W2 = 1/28 *(33 – 20) = 0.4643 c) Morrell believes that the informational value of actual paid deviations increases faster than linearly. He suggests something that looks like: W2 Months d) It is assumed that at 48 months, no reserve is required, as adjustments beyond 48 months are likely to be small & are equally likely to be positive or negative. e) % of 1st adjustments paid at 22 months = 1 / DPF at 22 mos. = 1 / 3.57 = 28% f) Add’l premium from 2nd & later adj. As % of incurred losses = LPF LPF at 46 months = 0.0005 = 0.05% PART 6 Sample Essay Solutions Fall 2002 30. A. Test 1: UEPR must provide for premium return if cancellation. Assume $800,000 of premium written at 1 July 2001, all policies are for 5 years, but only 6 months have been earned at 31 Dec 2001. Therefore unearned premium = (60 – 6) / 60 * 800,000 = $720,000 for Test 1. Test 2: UEPR = WP * a / b a = = = b = = = future losses and expenses 88,400 + 102,000 + 136,000 + 204,000 + 102,000 632,400 total loss and expense 632,400 + 47,600 + (.10)(800,000) 760,000 Therefore, UEPR = 800,000 * [632,400 / 760,000] = 665,684 for Test 2. Test 3: UEPR = present value of future loss and expense – present value of future guaranteed premiums (= 0) Discount rate = min [ .04, .05 - .015 ] = 3.5% Therefore, UEPR = (88,400)(1.035)-.5 + (102,000)(1.035)-1.5 + (136,000)(1.035)-2.5 + (204,000)(1.035)-3.5 + (102,000)(1.035)-4.5 = 308,555 + 268,230 = 576,785 B. Company UEPR (580,000) fails Test 1 and Test 2. It passes Test 3. 31. 1. Uses created year statistics - most companies don’t carry such data by created year - use accident year data instead – more readily available 2. F-ratio is an abstract quantity that is difficult to explain because the ratio of paid ALAE to losses disposed of has no real meaning. - use the ratio of paid ALAE to paid loss instead 3. The loss disposed of ratio is artificial and may be unstable. - the stability will depend on whether estimates of ultimate losses are revised since losses disposed of = losses paid + changes in O/S, thus it may be unsuitable for long tailed lines where ultimate loss estimates must be continually monitored. - use paid losses instead of losses disposed of. PART 6 Sample Essay Solutions Fall 2002 32. PDLD2 = [P2 – P1] / [L2 – L1] = [CL2 – CL1] / [L2 – L1] * TM * LCF CL / L = 1 - χ - LER CL1 / L1 = 1 – (.123 - .005) - .035 = .847 CL2 / L2 = 1 – (.134 - .004) - .035 = .835 PDLD2 = [ (CL2/L2 * ELR * % Loss2) – (CL1/L1 * ELR * % Loss1) ] / [ELR * % Loss2 – ELR * % Loss1] * TM * LCF = [.835*.9 - .847*.8] / [.9 - .8] * 1.03 * 1.20 = .09134 / .10 = .9134. 33. (a) (1) The price derived by the pricing algorithm. This incorporates company specific expense ratios, loss experience, investment performance and target market percentile to develop an indicated rate. (2) The market price. The price at which the company will be able to write its goal amount of business given its underwriting selectivity. This is limited by the price competitors are charging for coverage with a similar underwriting selectivity. (b) One must consider that a company’s price may decrease its allowable level of underwriting selectivity, which will bring in business with higher loss levels. This then increases the companies pricing formulas. PART 6 Sample Essay Solutions Fall 2002 34. Reinsurer A: 95,000 / (450,000 + 76,000) = 18% < 20% Therefore only reinsurance recoverable on paid losses & LAE > 90 days overdue are subject to penalty. Penalty = 95,000 x 0.20 = 19,000 Reinsurer B: 128,000 / (500,000 + 63,000) = 22.7% > 20% Therefore all reinsurance recoverables are subject to penalty. Penalty = (500,000 + 450,000 + 200,000 + 215,000 + 6,000 – 110,000) * 0.20 = 252,000 Total Penalty = 19,000 + 252,200 = 271,200 35. a) In SAP, policy acquisition costs are immediately expensed. In GAAP, they are matched with revenue (ie – earned evenly in the period). b) Since GAAP matches revenue with expenses, a deferred PAC is created as an asset on the expense not yet recognized. DPAC’s do not exist for SAP. PART 6 Sample Essay Solutions Fall 2002 36. (a) 1. It is probable that an asset has been impaired or a liability has been incurred at the date of the financial statement. 2. The amount of loss can be reasonably estimated. (b) Disclosure shall be noted in the financial statement. The estimated amount of loss shall be disclosed. If the amount can not be estimated, it should state that the amount can not be estimated. 37. a) 250,000 written premium. (250,000 x 0.75) = 187,500 = written premium ceded. (187,500 x 0.30) = 56,250 = ceding commission. (187,500 – 56,250) = 131,250 premium due reinsurer. 50,000 losses. 9,000 ALAE. (59,000 x 0.75) = 44,250 ceded and due from reinsurer. Premium – loss = 131,250 – 44,250 = 87,000 is due to the reinsurer. b) The purpose of the ceding commission is for the reinsurer to pay for the primary insurer’s policy acquisition costs that were incurred in obtaining the business ceded to the reinsurer. The ceding commission is adjusted as a way to price the reinsurance contract. PART 6 Sample Essay Solutions Fall 2002 38. Exposure Rate 50,000 XS 50,000 35,000 50,000 100,000 200,000 500,000 ILF 1.00 1.15 1.32 1.45 1.55 Premium 2,000,000 3,000,000 3,000,000 2,000,000 5,000,000 15,000,000 Exp Factor 0 0 17/132 17/145 17/155 Exp Prem 0 0 386,364 234,483 548,387 1,169,234 Prem = (1,169,234)(primary co. ELR)(ALAE loading)(Reinsurer exp & profit load) =(1,169,234)(1-0.35)(1.15)(1.20) =1,048,803 Exposure Rate = 1,048,803 / 15,000,000 = 0.0699 = 6.99% 39. State CA FL IL GA Total (1) Expected Loss = (Standard Premium)x(ELR) 2,360,000 (0.75) = 1,770,000 1,200,000 (0.80) = 960,000 800,000 (0.70) = 560,000 640,000 (0.70) = 448,000 (2) % of Loss in $700,000 XS of $300,000 0.08 – 0.03 = 0.05 0.125 – 0.055 = 0.07 0.025 – 0.005 = 0.02 0.040 – 0.010 = 0.03 (3) = (1) x (2) Excess Loss 88,500 67,200 11,200 13,440 180,340 Therefore, loss cost for $700,000 XS $300,000 = 180,340/Total Premium = $180,340 / 5,000,000 = 3.61% PART 6 Sample Essay Solutions Fall 2002 40. Before Loss Corridor Range of Loss Ratios 0-75% 75-85% 85%+ Avg. Loss Ratio in Range before Loss Corridor 65.0% 82.0% 93.5% Probability that Loss Ratio is in the Range 60% 25% 15% Reinsurer Loss Ratio = (.65)(.6) + (.82)(.25) + (.935)(.15) = 0.39 + 0.205 + 0.140 = 0.735 Reinsurer Loss Ratio without Loss Corridor = 73.5% After Loss Corridor Adjust Average Loss Ratio for Range 75-85% = 0.75+(0.82 - 0.75) * (1- 0.7) = 0.771 = 77.1% Adjust Average Loss Ratio for Range 85% + = 0.75 + (0.85 – 0.75) * (1-0.7) + (0.935 – 0.85) = 0.865 = 86.5% Reinsurer Loss Ratio After Loss Corridor = (0.65)(0.6) + (0.771)(0.25) + (0.865)(0.15) = 0.7125 Reinsurer Loss Ratio After Loss Corridor = 71.25% PART 6 Sample Essay Solutions Fall 2002 41. Ceding Co. Coverage Limit 100,000 200,000 300,000 400,000 500,000 Direct Premium 5,000,000 5,000,000 5,000,000 5,000,000 5,000,000 % of loss falling under ceding co. Retention 100% 84.6% 73.6% 66.3% 61.5% % of loss falling under retention & treaty limit 100% 100% 100% 100% 96.1% % of loss covered by reinsurer 0% 15.4% 26.4% 33.7% 34.6% Company expected loss and ALAE ratio = 0.70 * 1.15 = 0.805 Ceding Co. Coverage Lim. 100,000 200,000 300,000 400,000 500,000 Direct Premium 5,000,000 5,000,000 5,000,000 5,000,000 5,000,000 Expected Loss & ALAE ratio .805 .805 .805 .805 .805 Expected Ult. Loss & ALAE 4,025,000 4,025,00 0 4,025,000 4,025,000 4,025,000 % of loss covered by reinsurer 0% 15.4% 26.4% 33.7% 34.6% Loss & ALAE covered by reinsurer 0 619,850 1,062,600 1,356,425 1,392,650 4,431,525 Exposure Rate = {(4,431,525/25,000,000) * 1.1 * [1/(1-0.2)]} – 1 = 0.2437 42. A) 1) Both use an expected loss ratio to apply to premium. 2) Both use the unreported loss percentage to apply to the expected losses. B) 1) Bornhuetter-Ferguson use unadjusted earned premium, while StanardBuhlmann use earned premium that is adjusted (adjusted for on-level premium rates and trend). 2) Stanard-Buhlmann use data from all years to estimate an expected loss ratio, while Bornhuetter-Ferguson use separate estimates for each year without grouping all years together to estimate the loss ratios. PART 6 Sample Essay Solutions Fall 2002 43. Advantages 1) 2) can retain larger portion of premium on small profitable risks. Helps stabilize underwriting results. Disadvantages 1) no surplus benefit 2) increase in frequency of small losses can deteriorate the loss ratio. 44. a) 1. The assuming company must assume significant insurance risk (both timing and underwriting) 2. The assuming company must have a reasonable possibility of having a significant loss b) If no significant risk, it is fine if substantially all of the risk is transferred. This puts the reinsurer in essentially the same position as the ceding company. PART 6 Sample Essay Solutions Fall 2002 45. First, estimate future paid losses using loss development method. age-to-age (3-yr wtd avg) 12 – 24 3200/1600 = 2.000 24 – 36 3000/2000 = 1.500 36 – 48 2000/1500 = 1.333 age-to-ult LDF 4.000 2.000 1.333 Projected Paid Claims Pattern 98 99 12 500 500 24 1000 1000 36 1500 1500 48 2000 2000 00 600 1200 1800 2400 01 750 1500 2250 3000 Claims Projected Paid by Calendar Year 2002: 2003: 2004: 500 + 600 + 750 600 + 750 750 = 1,850 = 1,350 = 750 Discounted Future Payments: Commutation Price = [1850/(1.05)^.5] + [1350/(1.05)^1.5] + [750/(1.05)^2.5] = 3,724 b. Upon commutation, all reserves are released Current reserves = 0 + 400 + 600 + 750 = Since booked reserves < commutation price, Reinsurer will realize an underwriting loss 1,750
© Copyright 2024