Making the Most of Catastrophe Modeling Output July 9 th, Presenter: Kirk Bitu, FCAS, MAAA, CERA, CCRA

Transcription

1 Making the Most of Catastrophe Modeling Output July 9 th, 2012 Presenter: Kirk Bitu, FCAS, MAAA, CERA, CCRA Kirk.bitu@bmsgroup.com 1

2 Agenda Database Tables Exposure Loss Standard Outputs Probability of Exceedence Curves Advance Analyses Pure Premium Calculators Allocating Loss / Costs Marginal Analyses Appendix Formulas 2

3 Database Tables Exposure Database Portfolio Information Loss Database Link to Exposure Database Policy Information Event Loss Tables Limits EP Curves Property Details Construction Occupancy Etc Stats Address detail 3

4 Database Tables Model A Event Loss Table Analysis Link To Exposure Loss Perspective Event Loss Amount Std Deviation Exposure Limit / Maximum Loss Rate Exceedence Probability Curves Analysis Link To Exposure Loss Perspective EP Curve Type (AEP / OEP) Loss Amount Probablity of Exceedence Stats Analysis Link To Exposure Loss Perspective EP Curve Type Loss Amount St Deviation 4

5 Database Tables Model B Event Loss Table Analysis (in table name) Link To Exposure Level of Analysis (Portfolio, Location, etc.) Peril Model Year Presentation assumes 10,000 Year Event Set Event Loss Amount Ground Up Gross Net 5

6 Database Tables Differences Differences in Model DBs Stats and EP tables are not in Model B. Calculated from ELT Each Analysis has its own table in Model B Model B has a Year associated with each Event, Model A does not If a given Event occurs Model A has uncertainty around the amount of loss from that event. Model B does not. (Secondary Uncertainty) 6

7 Standard Outputs: Probability of Exceedence Probability of exceedence curves Image - Calculate the PML of a portfolio Determine Retention and Reinsurance needed to mitigate risk to an acceptable level Source: RMS Methodology Documentation 7

8 Standard Outputs: Probability of Exceedence Probability of Exceedence: The probability of exceeding a loss threshold. Return Time: The inverse of the exceedence probability. Probability Return Earthquake of Time A EP OEP Non-Exceedence (Years) (000s) (000s) 99.99% 10,000 $1,107,388 $1,104, % 1,000 $483,791 $482, % 500 $312,960 $312, % 250 $144,878 $144, % 200 $104,109 $103, % 100 $28,352 $28, % 50 $4,388 $4, % 25 $290 $286 Limit (000s) Premium (000s) Risks $5,906,893 $12,778 15,500 Average Annual Loss 100 Yr PML:Premium 2.219:1 $2,123, :1 250 Yr PML:Premium : :1 Vulnerability (AAL per 1k TIV) AAL to Premium % % Loss Perspectives: Ground Up Client loss to prior to any insurance Gross Estimated insurer loss after the application of insurance policy terms Net Underlying Reinsurance Estimated insurer loss after the application of policy terms and any underlying reinsurance (FAC, XPR, QS). Net Cat Estimated loss amount retained by insurer after ceding loss to the applicable cat treaties. Loss amounts associated with the respective probability of exceedence. Aggregate Exceeding Probability (AEP) Aggregated losses associated with Return Time / Probability of Exceedence. Occurrence Exceeding Probability(OEP) Largest single event for one period associated with Return Time / Probability of Exceedence. Average Annual Loss (AAL): The long term average one period loss. Risk Management Metrics: PML to Premium Ratio The number of years of Premium required to cover the Probable Maximum Loss. Vulnerability (AAL per 1k TIV) The Average Annual Loss per $1,000 of Insured Value. AAL to Premium Ratio (Loss Ratio) The Average Annual Loss as a percentage of Premium. 8

9 Standard Outputs: Probability of Exceedence Model A (EP table) Use Table xxxep Create Table for specified Analysis Loss Perspective EP Type (AEP / OEP) Interpolate to find Loss based on EP EP based on Loss Model A (ELT table) Used for OEP only Find specified loss for each Event Appendix: Beta CDF Use Return Time Formula across entire event set Sum(1 (1-e^(- rate*(1-cdf)) To find Loss based on Return time use Solver or similar product. Model B Determine correct loss column (Ground Up, Gross, Net) For AEP sum losses within year to get AEP loss for year For OEP find Maximum Loss for each year Rank Losses in descending order Assume each year is 1/10000 probability Find Year / Loss associated with value. 9

10 Use EP Table Model A Standard Outputs: Probability of Exceedence Model A - EP Example Find the 1 in Model A From EP Table find value closest to but > then.01 From EP Table find value closest to but <.01 Loss Amount Probability of Exceeding the Loss Amount Return Time 1,100, ,000, Interpolate 1,000,000 + (1,100,000-1,000,000) * ( ) / ( ) 1,066, Find the Return Time for 1,066, Model A From EP Table find value closest to but > then 1,066, From EP Table find value closest to but < 1,066, Loss Amount Probability of Exceeding the Loss Amount Return Time 1,100, ,000, Interpolate ( ) * (1,100,000-1,066,666) / (1,100,000-1,000,000)

11 Standard Outputs: Probability of Exceedence Model A - ELT MODEL A Using ELT Table Find Loss Amount For Each Event Find 2,500,000 Exposed Limit (Max Loss Amount St Deviation Loss) Rate 2,000, ,000 10,000, Use Beta Function to Find CDF Mean Damage Ratio = Loss Amount / Exposed Limit 0.2 CV = St Dev / Loss Amount 0.25 Alpha (1 - MDR) / (CV^2) - MDR 12.6 Beta Alpha * (1-MDR) / MDR 50.4 Beta CDF To get Return Time use formula Sum(1 / (1-e^(-rate*(1-CDF))) 11

12 Standard Outputs: Probability of Exceedence Model B - ELT Model B Rank the 10,000 years and apply a probability or 1/10,000 to each year Rank Year Loss ,000, ,500, ,750, ,250,000 Note Model B also does not have AEP vs OEP so we need to calculate OEP Loss = Max(Loss) by Year AEP Loss = Sum(Loss) by Year Create EP table Rank EP Loss ,000, ,500, ,750, ,250,000 Interpolate - The 1/5,000 return time loss is 11,500,000 12

13 Advanced Analyses Pure Premium Calculator Calculate the desk price for a Reinsurance Layer Find St Deviation of loss within a layer Allocating Losses / Costs Allocate losses from portfolio level to more granular level Used to allocate Reinsurance Costs / Capital to States, Business Units, LOB s, location, etc. Marginal Analyses Find the impact of adding (removing) a location, new book of business etc. to an existing portofolio. Image - Image - esearch/structure/crust/nam. php Image -29/screens/006sr.jpeg Image Source: RMS Methodology Documentation 13

14 Advanced Analyses: Pure Premium Calculator The AAL within the retention and retention + limit Risk factor applied to Standard deviation to get risk transfer cost Layer % Placed Limit Retention Reinstatements Pure Premium Risk Load (b) (a) (c) 1 100% 25, ,111 50% 2 100% 50,000 25, ,940 50% 3 100% 250,000 50, ,184 50% 4 100% 1,000, , ,997 50% 5 100% 10,000,000 1,000, ,894 50% Standard Deviation Loaded Premium ROL Reinsurer Loss Ratio Entry Return Time Exhaustion Return Time Loss On Line 16,313 22, % 53.8% NA % 24,044 27, % 46.7% % 75,190 74, % 39.3% % 144, , % 25.8% % 143,689 94, % 13.7% % The Standard deviation of losses within the layer Estimated Deposit (or reinsurance) premium: Can be backed into from Reinsurer loss margins. = [(c) + (d) * (e)]/(1 expense fee) Reinsurance cost as a % of limit. i.e. Cost = $27,735 Limit = $50,000 RoL = $27,735/ $50,000 = 55.5% Pure Premium / Loaded Premium i.e. PP = $12,940 Cost = $27,735 RLR = $12,940 / $27,735 = 46.7% The Return times at the retention and retention + limit Pure Premiumas a % of limit. i.e. PP = 12,940 Limit = $50,000 RoL = $12,940 / $50,000 = 25.9% 14

15 Advanced Analyses: Pure Premium Calculator Model A (ELT table) Determine Layer size based on retention and limit Find Loss in Layer for Each Event Appendix: Beta Layer Loss Find Variance in Layer for each Event Beyond scope of presentation can be found using numerical analysis Var = E[X^2] E[X]^2 Use this relationship substituting E[X] and loss in Layer Model B Layer losses by Event Loss in Layer = Max(0,Min(Limit,Loss- Retention) Sum Losses within each year to get loss in layer by year Apply reinstatements Find Average loss and St Deviation of loss over 10,000 year Event Set 15

16 Advanced Analyses: Allocating Loss / Costs Pure Premium Allocation By Layer AAL Retention Below Layer 1 Layer 2 State $ % Tot $ % Tot $ % Tot $ % Tot MN $6,125, % $4,858, % $837, % $429, % WI $809, % $748, % $44, % $16, % Grand Total $6,935, % $5,607, % $882, % $445, % Deposit Premium* $1,380,000 $1,500,000 Risk Multipler State MN WI MN WI AAL $6,125,943 $809, % 11.7% RI Cost $2,755,111 $124, % 4.3% 16

17 Advanced Analyses: Allocating Loss / Costs Model A and B (ELT table) Run model (including ELT) at granularity (policy) which is being allocated to Aggregate up losses by Event to portfolio level Appendix: Model A Rolling up ELT Get percentage of loss from each event to layer From (policy) ELT multiply Loss Amount * % of loss in layer Roll up for each (policy) to get loss in layer. Apply further calculations as need e.g. RI Cost multiplier 17

18 Differences between Allocating Loss vs. Marginal Analysis Domino s Pizza example Assumptions 1 st Pizza Costs $10 2 nd Pizza Costs $8 3 rd Pizza Costs $6 Assume 3 people buy 3 pizzas and find Allocated cost Find Total cost = $24 Allocate to 3 people = $8 each Order does not matter Assume 2 people plan to buy pizza then 3 rd person elects to buy pizza as well The Marginal Cost for the 3 rd person is $24 (3 pizza cost) - $18 (2 person pizza cost) = $6 While allowing the 3 rd person in for $7 would benefit everyone they are not paying their allocated share of the entire portfolio. Order Matters!! 18

19 Advanced Analyses: Marginal Analysis Probability Return Portfolio New Portfolio Marginal Impact of Time Book + New Book Exceedence (Years) (000s) (000s) (000s) (000s) 0.01% 10,000 $342,210 $178,060 $342,210 $0 0.10% 1,000 $241,470 $91,863 $241,470 $0 0.20% 500 $176,740 $44,800 $177,513 $ % 250 $137,141 $25,397 $140,548 $3, % 100 $84,699 $11,835 $88,764 $4, % 50 $57,008 $6,699 $60,467 $3, % 20 $28,360 $3,183 $32,166 $3,806 Limit (000s) $84,345,235 $31,223,303 $115,568,538 $31,223,303 Premium (000s) $111,740 $37,713 $149,454 $37,713 Risks 105,837 38, ,065 38,228 Average Annual Loss $4,536,971 $1,250,326 $5,787,297 $1,250,326 19

20 Advanced Analyses: Marginal Analysis Model A (ELT table) Start with a Portfolio ELT Find Metrics same as in EP / Return time calculations Combine ELT s Appendix: Model A Combining ELTs Do this for each (policy) that is being analyzed Examine Key metrics to find Drivers of PML / Cost 20

21 Appendix Beta CDF From ELT table Loss Amount Standard Deviation = Independent Standard Deviation + Correlated Standard Deviation Exposed Limit (Max Loss) Find Alpha and Beta for Beta Function in Excel Betadist(, Alpha, Beta) = Beta X Mean Damage Ratio = Loss Amount / Exposed Limit CV = Standard Deviation / Loss Amount Alpha = Beta = CDF for Event Loss Table * Each Event has a specific rate I associated with it 21

22 Appendix Beta Layer Loss Loss in Layer = Limited Expected Retention + Limit Limited Expected Retention EV (Beta X) = (EV (Beta retention + limit) EV (Beta retention) ) / Loss Amount = % of Loss in Layer Loss in Layer for Event Set From ELT table 22

23 Appendix Model A Rolling Up / Combining ELT From ELT table Event Id Loss Amount Standard Deviation = Independent Standard Deviation + Correlated Standard Deviation Exposed Limit (Max Loss) Adding ELT s (use same procedure to roll up granulated ELT to portfolio) For Each Event ID Loss Amount = Standard Deviation = Exposure Limit = Rate = Rate for Event (Rate is consistent for each Event) Use same procedure to remove a Location from an ELT 23

24 Appendix Model B Rolling UP / Combining ELT Combined ELT s based on Event ID, Year and Model Sum Loss Amounts 24