Florida Public Hurricane Loss Model (FPHLM 3.0)

STATISTICAL STANDRADS

B. M. Golam KibriaDepartment of Statistics

Florida International UniversityMiami, FL 33199

S-1 Modeled Results and Goodness-of-Fit

A. The use of historical data in developing the model shall be supported by rigorous methods published in currently accepted scientific literature.

The historical data for the period 1900-2007 was modeled using scientifically accepted methods that had been published.

B. Modeled and historical results shall reflect agreement using currently accepted scientific and statistical methods.

Modeled and historical results are in agreement as indicated by appropriate statistical and scientific tests. Some of these tests are discussed below.

S1: Modeled Results and Goodness of Fit (GOF)

Comparison of modeled vs historical occurrences

0.0000

0.10000.2000

0.30000.4000

0.50000.6000

0.7000

1 2 3 4 5 6 7

Events per year

Occ

urre

nce

rate

Modeled ProbabilityHistorical Probability

Comparison of modeled vs historical occurrences

H0: Historical and modeled data follow the same distribution

Ha: They are from different distributions.

Chi-square test statistic

Chi square goodness of fit, p-value =0.82 (DF=2)

Since p-value is 0.82, we can not reject the null hypothesis at 5% level of significance.

Comparison between observed and modeled Holland B

0.5 1 1.5 2 0123456789

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Distribution of the B parameter

ObservedModel Scaled

B param eter

Occ

urre

nce

Model Holland B by Normal Distribution

Chi square goodness of fit Test: p-value = 0.57 (DF=8)

Kolmogorov-Smirnov goodness of fit Test: p-value= 0.85

Comparison between observed and modeled R Max

Gamma vs Observed

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5 10 15 20 25 30 35 40 45 50 55

ObservedGamma

Model Rmax by Gamma Distribution

Maximum likelihood estimators:

53547.5ˆ =k 67749.4ˆ =θ

Chi square goodness of fit test: p-value= 0.26 (DF= 5)

KS-

goodness of fit test: p-value = 0.80

Assessment of Uncertainty in Loss Costs

Coefficient of variation provides a relative measure of

data dispersion compared to the mean.

2.5 3.0 3.5 4.0 4.5 5.0 5.5CV

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Histogram of CVs

S-2 Sensitivity Analysis for Model Output

The modeler shall have assessed the sensitivity of temporal and spatial outputs with respect to the simultaneous variation of input variables using currently accepted scientific and statistical methods and have taken appropriate action.

We have assessed the sensitivity of temporal and spatial outputs with respect to the simultaneous variation of input variables using currently accepted scientific and statistical methods and submitted to the commission under 2007 standard.

S-2 Sensitivity Analysis (SA) for Model Output

Standardized Regression Coefficients:

A linear regression model that regress maximum sustained surface wind speed (MSSWS) on Rmax, VT, CP and Holland B (HB) could be fit for 966 grid points for each time point and for each hurricane category. The model is defined as

From a fitted model, we can to compare the regression coefficients in terms of the magnitude of their effects.

The magnitude of βs can not be compared directly due to their different unit of measurements.

Standardized regression coefficients (SRC) may be used to compare the importance of different regressors.

0 1 2 3Y= + Rmax + VT+ CP+ 4 Holland B+eβ β β β β

Standardized Regression Coefficients vs. Time at Grid Coordinates (30,0) for Cat 1

Grid 30,0

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (hr)

Stan

dard

ized

Reg

ress

ion

Coe

ffici

ent HollandB

RmaxVTCP

Standardized Regression Coefficients vs. Time at Grid Coordinates (30,0) for Cat 5

Grid 30,0

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

11.2

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (hr)

Stan

dard

ized

Reg

ress

ion

Coe

ffici

ent HollandB

RmaxVTCP

S-3 Uncertainty Analysis for Model Output

The modeler shall have performed an uncertainty analysis on the temporal and spatial outputs of the model using currently accepted scientific and statistical methods and have taken appropriate action. The analysis shall identify and quantify the extent that input variables impact the uncertainty in model output as the input variables are simultaneously varied.

We have performed an uncertainty analysis on the temporal and spatial outputs of the model using currently accepted scientific and statistical methods and submitted to the commission in 2007 submission.

S-3 Uncertainty Analysis (UA) for Model Output

Expected Percentage Reduction (EPR):

The goal of uncertainty analysis is to quantify the contributions of the input parameters to the uncertainty in maximum sustained surface wind speed. The simple model is

The variance of Y can be expressed as a conditional variance onX’s. Here X’s are Rmax, CP, VT and HB etc.

The expected percentage reduction in V(Y) is expressed as

1 2 3 4Y=X +X +X +X

Var(E(Y/X)EPR in Var(Y)= 100Var(Y)

×

Summary for UA

From the following figures, we notice that the major contribution to the uncertainty in the model is R-max followed by VT, Holland B and CP. At hour 0, Rmax produces the most uncertainty and at hour 12 VT contributed the highest uncertainty in the model. It is also noted that at hour 2 there is no uncertainty among the four parameters except VT.uncertainty of MSSWS depends on the time, grid points and the category of hurricanes.

Expected Percentage Reductions in the Var(MSSWS) for a Cat 1 Hurricane versus Time at Coordinate (30,0)

Grid 30,0

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Time (hr)

Expe

cted

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cent

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n HollandBRmaxVTCP

Expected Percentage Reductions in the Var(MSSWS) for a Cat 5 Hurricane versus Time at Coordinate (30,0)

Grid 30,0

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0 1 2 3 4 5 6 7 8 9 10 11 12Time (hr)

Expe

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cent

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HollandBRmaxVTCP

S-4 County Level Aggregation

At the county level of aggregation, the contribution to the error in loss cost estimates attributable to the sampling process shall be negligible

The error in the county level loss costs induced by the sampling process can be quantified by computing standard errors for the county level loss costs. These loss costs have been computed for all counties in the state of Florida using 53,500 years of simulation. The results indicate that the standard errors are less than 2.5% of the average loss cost estimates for all counties

S4: County Level Aggregation

Number of Simulation Years

For each county Y, define average loss cost by and standarddeviation by sy. The standard error of estimate (SE( )) will be 2.5% of the estimated mean, if the number of simulation years for county Y is:

Based on the initial 10,700 simulation runs, the highest minimumacceptable number is 35,080 for Lafayette County.

We use 53,500 years (500x107 years) of simulation for our final results. For the 53,500 simulation year run we find the standarderrors to be less than 2.5% of the loss cost for each county

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0.025YsN

x⎛ ⎞= ⎜ ⎟×⎝ ⎠

YX

YX

Scatter plot between number of years vs percentage of the loss cost estimate for Lafayette County

Scatter plot between number of simulation years vs percentage of the loss cost estimate for Lafayette

County

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0 10000 20000 30000 40000 50000 60000

Number of Years

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Scatter plot between number of years vs percentage of the loss cost estimate for Nassau County

Scatter plot between number of simulation years vs percentage of the loss cost estimate for Nassau

County

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Number oif Years

Per

cent

age

Erro

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Percent

S-5 Replication of Known Hurricane Losses

The model shall estimate incurred losses in an unbiased manner on a sufficient body of past hurricane events from more than one company, including the most current data available to the modeler. This Standard applies separately to personal residential and, to the extent data are available, to mobile homes. Personal residential experience may be used to replicate structure-only and contents-only losses. The replications shall be produced on an objective body of loss data by county or an appropriate level of geographic detail.

The following Table compares the modeled and actual total losses by storm and company for residential coverage. The following Figure, correlation and statistical results indicated a reasonable agreement between the observed and modeled losses

S5: Replication of Known Hurricane losses

In February 2008 submission (2.7), the validation set consisted of 25 events (company/hurricane).

We processed additional company data since the first pro-team visit.

In May 16 revised submission (v3.0) , the validation set consisted of 31 events.

We continue to process additional company data. In the response to the deficiency letter that we submitted on May 29, 2008, eight (8) new events are added.

Since the February submission we have processed and added 16 newevents and consolidated 4 events into two

We checked the validation of some prior company/hurricane portfolios and made some corrections to thesis portfolios

We now have a total of 39 data points

Total Actual Losses and Total Modeled losses

Name Event Total Actual Loss Total Modeled Loss

A Charley 110471361 135347764

A Frances 20201407 78044136

B Andrew 2984373067 2558464266

B Charley 1037108745 735800992

B Charley_Mob 23395988 25990713

B Frances 614006549 428145647

B Frances_Mob 18467176 8466678

B Erin 50519119 60458572

C Charley 63889029 32714722

C Frances 122776727 88435747

D Charley 274702333 239567424

D Frances 224656954 142289724

E Charley 62086256 53273107

E Frances 43799401 18568918

F Charley 111013524 269149904

F Frances 94272660 380701388

G Charley 952353 900161

G Frances 10007410 4176704

Scatter plot between Total Actual Losses vs. Total Modeled losses

Actual Losses

Mod

el L

osse

s

0 5*10^8 10^9 1.5*10^9 2*10^9 2.5*10^9 3*10^9

05*

10^8

10^9

1.5*

10^9

25*

10^9

Scatter plot between Total Actual Losses and Modeled Losses

Paired t Test

The paired t-test t = 0.9882, df = 38, p-value = 0.3293

We fail to reject the null hypothesis (the difference in paired mean values equals zero), and conclude that there is insufficient evidence to suggest a difference between actual and modeled losses.

Proportion of points above and below the straight line

About 51% of the actual losses are more than the corresponding model losses and 49% of the model losses are more than the corresponding actual losses.

Concordance correlation coefficient (CCC)

The CCC is defined as

CCC evaluates the degree (measuring agreement to the identity line) to which the pairs fall on the line (straight line)Where is a Bias correction factor (measure of accuracy) that measures how far the best-fit line deviates from the line (or how closely the fitted line agrees with the identity line).ρ evaluates how closely observations fall on the fitted line

*c bCρ ρ=

bC

Concordance correlation and Pearson correlation coefficients

Actual Loss

Mod

el L

oss

0 5*10^8 10^9 1.5*10^9 2*10^9 2.5*10^9 3*10^9

05*

10^8

10^9

1.5*

10^9

2.5*

10^9

Concordance correlation and Pearson correlation coefficients

*c bCρ ρ=

Concordance correlation coefficient (CCC)

Sample concordance correlation coefficient= 0.973 Sample correlation coefficient (precision)= 0.988Bias correction factor (accuracy)=0.985

Conclusion: Very good agreement between actual and model losses.

S-6 Comparison of Projected Hurricane Loss Costs

The difference, due to uncertainty, between historical and modeled annual average statewide loss costs shall be reasonable, given the body of data, by established statistical expectations and norms.

The difference, due to uncertainty, between historical and modeled annual average statewide loss costs is reasonable as shown in the following description.

Standard S6: Comparison of projected hurricane loss costs

(Standard S-4) The number of simulation years is 53500. The standard errors are within less than 2.5% of the means for all counties.

(From S4): The 95% CI on the difference between the mean of the historical and modeled losses contains 0 indicating that the modeled losses do not differ significantly from historical losses.

Form S-1: Probability of Florida Landfalling Hurricanes per Year

Number Historical ModeledOf Hurricanes Probability Probability

Per Year(Hurdat June

2006)0 0.5794 0.60261 0.2710 0.26682 0.1215 0.09393 0.028 0.02834 0 0.00835 0 0.00016 0 07 0 0

8 or more 0 0

Form S2: An Example of a Probable Maximum Loss Based on a Limited Hypothetical Data (Part A)

Return Time (Years)

Probability of Exceedance Estimated Loss

53500 0.00% $103,728,414

10000 0.01% $70,655,088

5000 0.02% $68,977,344

2000 0.05% $58,506,876

1000 0.10% $53,610,128

500 0.20% $47,154,509

250 0.40% $41,384,473

100 1.00% $33,235,114

50 2.00% $26,297,815

20 5.00% $16,867,502

10 10.00% $9,482,954

5 20.00% $3,245,185

Form S2, Part B

Mean $2,772,311

Median $0

Standard Deviation $6,709,689

Interquartile Range $1,690,706

Sample size53500 years of

Simulated storms

Form S-3: Five Validation and Comparisons

Comparison #1: Hurricane Charley and Company A by Coverage

Company Actual Modeled Difference

CoverageLoss/Expos

ure

Loss/Expos

ure

Building 0.01788 0.02297 -0.00508

Contents 0.00349 0.00310 0.00039

Appurtenant

s 0.02764 0.01191 0.01573

ALE 0.00227 0.00214 0.00013

Total 0.01298 0.01590 -0.00292

Comparison #2: Different Companies by Different Hurricanes

Company Actual Modeled Difference

Company EventLoss/Expos

ure Loss/Exposure

A Charley 0.01298 0.01590 -0.00292

B Andrew 0.10426 0.08938 0.01488

C Frances 0.01769 0.01275 0.00495

H Jeanne 0.01916 0.02093 -0.00177

M Jeanne 0.00916 0.01082 -0.00167

Comparison #3: Company B by Hurricane Andrew, Charley, Frances

Company Actual Modeled Difference

Company EventLoss/

Exposure Loss/Exposure

B Andrew 0.10426 0.08938 0.01488

B Charley 0.01573 0.01116 0.00457

B Frances 0.01288 0.00898 0.00390

Comparison #4: Construction Type for Hurricane Charley

Company Actual Modeled Difference

ConstructionLoss/Expo

sure

Loss/Exposure

Frame 0.01610 0.01678 -0.00068

Masonry 0.01445 0.01214 0.00231

Manufactured 0.04466 0.04962 -0.00495

Comparison #5: County-wide for Company A and Hurricane Charley

Company Actual Modeled Difference

County Loss/Exposure Loss/Exposure

Lake 0.00042 0.00029 0.00013

Osceola 0.01359 0.01428 -0.00069

Sarasota 0.00030 0.00083 -0.00053

Collier 0.00073 0.00024 0.00049

St. Johns 0.00013 0.00033 -0.00020

Form S-4: Average Annual Zero Deductible Statewide Loss Costs –Historical versus Modeled

Part A (2002 FHCF Exposure Data)

The 95% CI for (μ1- μ2) is [-1.24 and 0.91].

Testing of equality two means using parametric test t = -0.3043, df = 53605, p-value = 0.7609

Equality of CDFs using nonparametric test KS = 0.0629, p-value = 0.7887

Conclusion: Both parametric and non-parametric methods give high p-values. The average historical and modeled losses are not statistically different.

Part B (2007 FHCF Exposure Data) (Continue Form S4)

The 95% CI for (μ1- μ2) is [-2.18 and 1.53].

Testing of equality two means using parametric test t = -0.3472, df = 53605, p-value = 0.7284

Equality of CDFs using nonparametric test KS = 0.0636, p-value = 0.7766

Conclusion: Both parametric and non-parametric cases, we have high p-values. The average historical and modeled losses are not statistically different.