making the most of catastrophe modeling output july 9th, 2012 the most of catastrophe... · of time...

© Copyright BMS Group 1

Making the Most of Catastrophe Modeling Output July 9th, 2012

Presenter: Kirk Bitu, FCAS, MAAA, CERA, [email protected]

© Copyright BMS Group

Agenda

2

Database Tables•Exposure• Loss

Standard Outputs•Probability of Exceedence Curves

Advance Analyses•Pure Premium Calculators•Allocating Loss / Costs•Marginal Analyses

Appendix•Formulas


Database Tables

3

Portfolio Information

Policy Information

Limits

Property Details•Construction•Occupancy•Etc

Address detail

Exposure Database

Link to Exposure Database

Event Loss Tables

EP Curves

Stats

Loss Database


Database TablesModel A

4

Event Loss Table

• Analysis• Link To

Exposure• Loss Perspective• Event• Loss Amount• Std Deviation• Exposure Limit /

Maximum Loss• Rate

ExceedenceProbability Curves


Exposure• Loss Perspective• EP Curve Type

(AEP / OEP)• Loss Amount• Probablity of

Exceedence

Stats


Exposure• Loss Perspective• EP Curve Type • Loss Amount• St Deviation


Database TablesModel B

5

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


Database TablesDifferences

6

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)


Prob

abili

ty o

f ex

ceed

ence

curv

es • Calculate the PML of a portfolio

• Determine Retention and Reinsurance needed to mitigate risk to an acceptable level

Image -http://sincedutch.wordpress.com

Source: RMS Methodology Documentation

Standard Outputs: Probability of Exceedence


Probability Returnof Time AEP OEP

Non-Exceedence (Years) (000s) (000s)99.99% 10,000 $1,107,388 $1,104,06599.90% 1,000 $483,791 $482,49299.80% 500 $312,960 $312,08099.60% 250 $144,878 $144,35599.50% 200 $104,109 $103,71199.00% 100 $28,352 $28,21498.00% 50 $4,388 $4,35996.00% 25 $290 $286

Limit (000s)Premium (000s)RisksAverage Annual Loss100 Yr PML:Premium 2.219:1 2.208:1250 Yr PML:Premium 11.338:1 11.297:1Vulnerability (AAL per 1k TIV) 0.359 0.359AAL to Premium 16.616% 16.616%

$2,123,217

$12,77815,500

Earthquake

$5,906,893

Probability of Exceedence: The probability of exceeding a loss threshold.

Return Time: The inverse of the exceedenceprobability.

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.

Loss Perspectives:Ground Up – Client loss to prior to any insuranceGross – Estimated insurer loss after the application of insurance policy termsNet 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.

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.




9

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 CDF @ 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.


Standard Outputs: Probability of ExceedenceModel A - EP

10

Model AUse EP Table

ExampleFind the 1 in 100 0.01Model AFrom EP Table find value closest to but > then .01From EP Table find value closest to but < .01

Loss AmountProbability of Exceeding

the Loss Amount Return Time1,100,000 0.009 111.111,000,000 0.012 83.33

Interpolate1,000,000 + (1,100,000 - 1,000,000) * (.01-.009) / (.011 - .009)

1,066,666.67

Find the Return Time for 1,066,666.66Model AFrom EP Table find value closest to but > then 1,066,666.66From EP Table find value closest to but < 1,066,666.66

Loss AmountProbability of Exceeding

the Loss Amount Return Time1,100,000 0.0090 111.111,000,000 0.0120 83.33

Interpolate.009 + (.012 - .009) * (1,100,000 - 1,066,666) / (1,100,000 - 1,000,000)

0.0100 100.00


Standard Outputs: Probability of ExceedenceModel A - ELT

11

MODEL AUsing ELT Table

Find CDF @ Loss Amount For Each EventFind CDF @ 2,500,000

Loss Amount St DeviationExposed Limit (Max

Loss) Rate2,000,000 500,000 10,000,000 0.02

Use Beta Function to Find CDF

Mean Damage Ratio = Loss Amount / Exposed Limit 0.2CV = St Dev / Loss Amount 0.25Alpha (1 - MDR) / (CV^2) - MDR 12.6Beta Alpha * (1-MDR) / MDR 50.4

Beta CDF 0.840813

To get Return Time use formulaSum(1 / (1-e^(-rate*(1-CDF)))


Standard Outputs: Probability of ExceedenceModel B - ELT

12

Model B

Rank the 10,000 years and apply a probability or 1/10,000 to each year

Rank Year Loss1 2345 12,000,0002 3876 11,500,0003 6797 10,750,0004 2597 10,250,000

Note Model B also does not have AEP vs OEP so we need to calculateOEP Loss = Max(Loss) by YearAEP Loss = Sum(Loss) by Year

Create EP table

Rank EP Loss1 0.0001 12,000,0002 0.0002 11,500,0003 0.0003 10,750,0004 0.0004 10,250,000

Interpolate - The 1/5,000 return time loss is 11,500,000


Pure

Pre

miu

m C

alcu

lato

r • Calculate the ‘desk price’ for a Reinsurance Layer

• Find St Deviation of loss within a layer

Allo

catin

g Lo

sses

/ C

osts • Allocate losses

from portfolio level to more granular level

• Used to allocate Reinsurance Costs / Capital to States, Business Units, LOB’s, location, etc.

Mar

gina

l Ana

lyse

s • Find the impact of adding (removing) a location, new book of business etc. to an existing portofolio.

Image -http://earthquake.usgs.gov/research/structure/crust/nam.php

Image –http://pubs.usgs.gov/dds/dds-29/screens/006sr.jpeg

Image –http://depositphotos.com

Image -http://sincedutch.wordpress.com

Source: RMS Methodology Documentation

Advanced Analyses


Advanced Analyses: Pure Premium Calculator

14

Layer % Placed Limit Retention ReinstatementsPure Premium Risk Load

(b) (a) (c) 1 100% 25,000 ‐ 99 12,111 50%2 100% 50,000 25,000 99 12,940 50%3 100% 250,000 50,000 99 29,184 50%4 100% 1,000,000 250,000 99 28,997 50%5 100% 10,000,000 1,000,000 99 12,894 50%

Standard Deviation

Loaded Premium ROL

Reinsurer Loss Ratio

Entry Return Time

Exhaustion Return Time

Loss On Line

16,313 22,519 90.1% 53.8% NA 3.3 48.4%24,044 27,735 55.5% 46.7% 3.3 5.6 25.9%75,190 74,199 29.7% 39.3% 4.5 17.3 11.7%144,067 112,256 11.2% 25.8% 14.3 84.2 2.9%143,689 94,154 0.9% 13.7% 58.1 0.1%

The AAL within the retention and retention + limit

Risk factor applied to Standard deviation to get risk transfer cost

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 = $27,735Limit = $50,000

RoL = $27,735/ $50,000 = 55.5%

Pure Premium / Loaded Premium

i.e. PP = $12,940

Cost = $27,735RLR = $12,940 / $27,735 =

46.7%

The Return times at the retention and retention +

limit

Pure Premiumas a % of limit.

i.e. PP = 12,940Limit = $50,000

RoL = $12,940 / $50,000 = 25.9%


Advanced Analyses: Pure Premium Calculator

15

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


Advanced Analyses: Allocating Loss / Costs

16

State $ % Tot $ % Tot $ % Tot $ % TotMN $6,125,943 88.3% $4,858,725 86.6% $837,722 94.9% $429,496 96.3%WI $809,567 11.7% $748,580 13.4% $44,598 5.1% $16,389 3.7%

Grand Total $6,935,510 100.0% $5,607,305 100.0% $882,320 100.0% $445,885 100.0%Deposit Premium*

Risk Multipler

State MN WI MN WIAAL $6,125,943 $809,567 88.3% 11.7%RI Cost $2,755,111 $124,889 95.7% 4.3%

1.564 3.364

$1,380,000 $1,500,000

Pure Premium Allocation By LayerAAL Retention Below Layer 1 Layer 2


Advanced Analyses: Allocating Loss / Costs

17

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


Differences between Allocating Loss vs. Marginal Analysis

18

Domino’s Pizza example

• Assumptions• 1st Pizza Costs $10• 2nd Pizza Costs $8• 3rd 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 3rd person elects to buy pizza as well• The Marginal Cost for the 3rd person is $24 (3 pizza cost) - $18 (2 person

pizza cost) = $6• While allowing the 3rd person in for $7 would benefit everyone they are not

paying their allocated share of the entire portfolio.• Order Matters!!


Advanced Analyses: Marginal Analysis

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Probability Return Portfolio New Portfolio Marginal Impactof Time Book + New Book

Exceedence (Years) (000s) (000s) (000s) (000s)0.01% 10,000 $342,210 $178,060 $342,210 $00.10% 1,000 $241,470 $91,863 $241,470 $00.20% 500 $176,740 $44,800 $177,513 $7730.40% 250 $137,141 $25,397 $140,548 $3,4071.00% 100 $84,699 $11,835 $88,764 $4,0652.00% 50 $57,008 $6,699 $60,467 $3,4595.00% 20 $28,360 $3,183 $32,166 $3,806

Limit (000s) $84,345,235 $31,223,303 $115,568,538 $31,223,303Premium (000s) $111,740 $37,713 $149,454 $37,713Risks 105,837 38,228 144,065 38,228Average Annual Loss $4,536,971 $1,250,326 $5,787,297 $1,250,326


Advanced Analyses: Marginal Analysis

20

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


Appendix – Beta CDF

21

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 CDF @ 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


Appendix – Beta Layer Loss

22

Loss in Layer = Limited Expected Value @ Retention + Limit – Limited Expected Value @ 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


Appendix – Model A Rolling Up / Combining ELT

23

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


Appendix – Model B Rolling UP / Combining ELT

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Combined ELT’s based on Event ID, Year and Model Sum Loss Amounts

making the most of catastrophe modeling output july 9th, 2012 the most of catastrophe... · of time...

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