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### AS463_Lec08

Course: ACTSC 463, Fall 2011
School: Waterloo
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Word Count: 1215

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463/863 Property ACTSC and Casualty Loss Reserving 1 Examples From Previous Calculate the ULAE reserve using both the traditional approach and the Kittel Refinement and the following information: CY 2010 Paid Losses Total Reserves at 12/31/2009 Total Reserves at 12/31/2010 IBNR at 12/31/2009 IBNR at 12/31/2010 CY 2010 Paid ULAE CY 2010 Incurred ULAE 900 9000 10000 3600 4000 90 210 2 Examples From Previous...

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463/863 Property ACTSC and Casualty Loss Reserving 1 Examples From Previous Calculate the ULAE reserve using both the traditional approach and the Kittel Refinement and the following information: CY 2010 Paid Losses Total Reserves at 12/31/2009 Total Reserves at 12/31/2010 IBNR at 12/31/2009 IBNR at 12/31/2010 CY 2010 Paid ULAE CY 2010 Incurred ULAE 900 9000 10000 3600 4000 90 210 2 Examples From Previous Using the Wendy Johnson technique for determining ULAE reserves, determine the weighted number of open claims for each future calendar year. Use the information below: AY 2007 2008 2009 End of Year 1 2 3 4 Cumulative Reported Claims at 12/31/2009: 45 41 40 % Reported % Closed 40% 30% 85% 60% 100% 85% 100% 3 Loss Development Using Credibility 4 Least Squares Development > Want to determine future development points using equations of the form = + > Use historical data to solve for and > = 2 2 and = > If is the loss at time then is the loss at time + 12 5 Example Use the given information to calculate the ultimate losses for AY 2008 using least-squares. Assume the tail factor from 48 months to ultimate is 1.0375. AY 2006 2007 2008 2009 2010 12 100 110 120 130 140 24 120 130 140 150 36 130 150 150 48 140 160 6 60 145 Example = 140 = 155.5 = 21,875 2 = 19,700 21,875 140 155.5 = = 1.05 19,700 19,600 = 155.5 1.05 140 = 8.5 2008 = 8.5 + 150 1.05 = 166 7 Least Squares Development > Special Cases and are totally uncorrelated: = 0, = (Expected loss technique) Loss development factors are all equal = 0, = (Loss development technique) Bornhuetter-Ferguson has similar equation = 1, = + > Recall that B-F is a credibility weighting of expected loss and loss development (remember Hugh Whites question) 8 Least Squares Development > Least squares is similar to B-F but has more flexibility by allowing the multiplier to known data to fluctuate (automatically adjusts weights in credibility formula) > Test hypothesis that least squares produces a better estimate of ultimate than traditional approaches using known distributions and Bayesian analysis 9 Bayesian Development > Define random variable Y for number of claims which can be 0 or 1 with equal probability > Define random variable X as the # reported by year end with 50% probability of claim being reported > We want to determine: = = , expected total claims = = = , Reserve 10 Bayesian Development > Remember Bayes Theorem! > = ( ) > In the example: > = 0 = 0 = > = 1 = 0 = 1 2 ( )( 1) 1 2 1 1 3 11 11 +( ) ( ) 22 = 2 3 Bayesian Development > 0 = 0 2 3 +1 1 3 = 1 3 > = 1 1 2 1 1 2 > = + , = , = 3 3 3 3 > Since both and are both not 0 the solution is different then what we would be achieved by the standard simple approaches, suggesting the least squares method would produce a better estimate > i.e. The least squares method could produce the exact solution given enough historical data 12 Bayesian Development > Whats the problem with Bayesian Development? We dont have the distributions! The real world is far more complex, then the illustrative example That leads to finding an approximation to the Bayesian estimate The best linear approximation is given by minimizing 2 Bayesian Credibility! 13 Bayesian Credibility > The best linear approximation to Q is given by: = , + () > This can be used to answer Hugh Whites Question (if actual losses are higher than expected what do you do?): If Cov(X,Y)<Var(X), decrease the reserve If Cov(X,Y)=Var(X), leave the reserve the same If Cov(X,Y)>Var(X), increase the reserve Easier to understand with an example 14 Hugh White Example , = + () > Suppose you have a line of business with expected ultimate losses of \$900 (E[Y]) and expected reported losses at time 1 of \$500 (E[X]) > If actual losses at time 1 are \$600 we can demonstrate the result on the previous slide > The expected reserve at time 1 is 900 500 = 400 > But 600 = 600 500 R 600 = 600 600 , 15 + 900 and the reserve is Hugh White Example > For Cov(X,Y)=Var(X), R(600)=400 > For Cov(X,Y)<Var(X), R(600)<400 > For Cov(X,Y)>Var(X), R(600)>400 16 Bayesian Credibility > , 2 , () 2 2 2 > , () > Substituting into L(X) gives the least squares estimate > It can be shown that the least squares estimate has less bias then the loss development technique > Least squares can also be used with Berquist-Sherman type adjustments 17 Bayesian Credibility > L(X) can be written in the form of a credibility weighting to give additional insight into the relative use of the loss development and expected loss techniques > Define: EVPV = VHM = > If there is a real number 0, = = , : > = > = () + + + 1 , = + 18 Bayesian Credibility > E(Y) is clearly the expected loss method > x/d is the loss development technique It can be shown that = and = 19 Final Exam > Saturday, December 10th 7pm > Will cover everything from lectures, slides, Friedland and Brosius Note: We didnt get to the Mack paper or the Clark paper so they will not be included on the exam > E-mail me with any questions 20
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