Introduction to Increased Limits Ratemaking Joseph M. Palmer, FCAS, - PowerPoint PPT Presentation

Introduction to Increased Limits Ratemaking Joseph M. Palmer, FCAS, MAAA, Joseph M. Palmer, FCAS, MAAA, CPCU CPCU Assistant Vice President Assistant Vice President Increased Limits & Rating Plans Division Increased Limits & Rating Plans Division Insurance Services Office, Inc. Insurance Services Office, Inc.

Increased Limits Ratemaking is the process of Increased Limits Ratemaking is the process of developing charges for expected losses at developing charges for expected losses at higher limits of liability. higher limits of liability.

Increased Limits Ratemaking is the process of Increased Limits Ratemaking is the process of developing charges for expected losses at developing charges for expected losses at higher limits of liability. higher limits of liability. Expressed as a factor --- --- an Increased Limit an Increased Limit Expressed as a factor Factor --- --- to be applied to basic limits loss to be applied to basic limits loss Factor costs costs

Calculation Method Expected Costs at the desired policy limit Expected Costs at the desired policy limit ________________________________________________________________ _____________________________________________________________________________________________________________ _____________________________________________ Expected Costs at the Basic Limit Expected Costs at the Basic Limit

KEY ASSUMPTION: KEY ASSUMPTION: Claim Frequency is independent independent of of Claim Frequency is Claim Severity Claim Severity

This allows for ILFs to be developed by This allows for ILFs to be developed by an examination of the relative an examination of the relative severities ONLY severities ONLY ( ) ( ) E Frequency E Severity × k ILF = k ( ) ( ) E Frequency E Severity × B ( ) E Severity k = ( ) E Severity B

Limited Average Severity (LAS) � Defined as the average size of loss, where Defined as the average size of loss, where � all losses are limited to a particular value. all losses are limited to a particular value. � Thus, the ILF can be defined as the ratio of Thus, the ILF can be defined as the ratio of � two limited average severities. two limited average severities. � ILF (k) = LAS (k) ILF (k) = LAS (k) ÷ ÷ LAS (B) LAS (B) �

Example Losses @100,000 Limit @1 Mill Limit @1 Mill Limit Losses @100,000 Limit 50,000 50,000 75,000 75,000 150,000 150,000 250,000 250,000 1,250,000 1,250,000

Example (cont’d) Losses @100,000 Limit @1 Mill Limit Losses @100,000 Limit @1 Mill Limit 50,000 50,000 50,000 50,000 75,000 75,000 75,000 75,000 150,000 100,000 150,000 100,000 250,000 100,000 250,000 100,000 1,250,000 100,000 1,250,000 100,000

Example (cont’d) Losses @100,000 Limit @1 Mill Limit Losses @100,000 Limit @1 Mill Limit 50,000 50,000 50,000 50,000 50,000 50,000 75,000 75,000 75,000 75,000 75,000 75,000 150,000 100,000 150,000 150,000 100,000 150,000 250,000 100,000 250,000 250,000 100,000 250,000 1,250,000 100,000 1,000,000 1,250,000 100,000 1,000,000

Example – Calculation of ILF Total Losses $1,775,000 Total Losses $1,775,000 Limited to $100,000 $425,000 Limited to $100,000 $425,000 (Basic Limit) (Basic Limit) Limited to $1,000,000 $1,525,000 Limited to $1,000,000 $1,525,000 Increased Limits Factor 3.588 Increased Limits Factor 3.588 (ILF) (ILF)

Insurance Loss Distributions � Loss Severity Distributions are Skewed Loss Severity Distributions are Skewed � � Many Small Losses/Fewer Larger Losses Many Small Losses/Fewer Larger Losses � � Yet Larger Losses, though fewer in number, Yet Larger Losses, though fewer in number, � are a significant amount of total dollars of are a significant amount of total dollars of loss. loss.

Basic Limits vs. Increased Limits � Use large volume of losses capped at basic Use large volume of losses capped at basic � limit for detailed, experience- -based based limit for detailed, experience analysis. analysis. � Use a broader experience base to develop Use a broader experience base to develop � ILFs to price higher limits ILFs to price higher limits

Loss Distribution - PDF ( x ) f 0 x Loss Size

Loss Distribution - CDF ( x ) F 1 Claims 0 x

Claims vs. Cumulative Paid $ $ 1 Liability 0 x ( x ) Claims F $ 0 x Property 0 x

A novel approach to understanding Increased A novel approach to understanding Increased Limits Factors was presented by Lee in the Limits Factors was presented by Lee in the paper --- --- “ “The Mathematics of Excess of The Mathematics of Excess of paper Loss Coverages and Retrospective Rating - - Loss Coverages and Retrospective Rating A Graphical Approach” ” A Graphical Approach

Lee (Figure 1) x n i x i x i 0 n

Limited Average Severity k ∫ ( ) [ 1 ( )] xdF x k F k + − 0 Size method; ‘vertical’ ∫ k [ 1 ( )] F x dx − 0 Layer method; ‘horizontal’

( ) 1 ( ) G x F x ∗ = − Size Method Loss Size ∫ k ( ) ( ) xdF x k G k + × 0 k x 0 1 ( x ) F

( ) 1 ( ) G x F x ∗ = − Layer Method Loss Size ∫ k ( ) G x dx 0 k x 0 1 ( x ) F

Empirical Data - ILFs Lower Upper Losses Occs. . LAS Lower Upper Losses Occs LAS 1 100,000 25,000,000 1,000 1 100,000 25,000,000 1,000 100,001 250,000 75,000,000 500 100,001 250,000 75,000,000 500 250,001 500,000 60,000,000 200 250,001 500,000 60,000,000 200 500,001 1 Million 30,000,000 30,000,000 50 500,001 1 Million 50 1 Million - 15,000,000 10 - 1 Million - 15,000,000 10 -

Empirical Data - ILFs LAS @ 100,000 LAS @ 100,000 (25,000,000 + 760 × × 100,000) 100,000) ÷ ÷ 1760 1760 (25,000,000 + 760 = 57,386 = 57,386 LAS @ 1,000,000 LAS @ 1,000,000 ( 190,000,000 + 10 × × 1,000,000 ) 1,000,000 ) ÷ ÷ 1760 1760 ( 190,000,000 + 10 = 113,636 = 113,636 Empirical ILF = 1.98 Empirical ILF = 1.98

“Consistency” of ILFs � As Policy Limit Increases As Policy Limit Increases � � ILFs should increase ILFs should increase � � But at a decreasing rate But at a decreasing rate � � Expected Costs per unit of coverage should Expected Costs per unit of coverage should � not increase in successively higher layers. not increase in successively higher layers.

Illustration: Consistency Loss Size k 3 x k 2 k 1 0 1 ( x ) F

“Consistency” of ILFs - Example Limit ILF Diff. Lim. Diff. ILF Diff. ILF Marginal Marginal Limit ILF Diff. Lim. 100,000 1.00 - - - 100,000 1.00 - - - 250,000 1.40 250,000 1.40 500,000 1.80 500,000 1.80 1 Million 2.75 1 Million 2.75 2 Million 4.30 2 Million 4.30 5 Million 5.50 5 Million 5.50

“Consistency” of ILFs - Example Limit ILF Diff. Lim. Diff. ILF Diff. ILF Marginal Marginal Limit ILF Diff. Lim. 100,000 1.00 - - - 100,000 1.00 - - - 250,000 1.40 150 0.40 250,000 1.40 150 0.40 500,000 1.80 250 0.40 500,000 1.80 250 0.40 1 Million 2.75 500 0.95 1 Million 2.75 500 0.95 2 Million 4.30 1,000 1.55 2 Million 4.30 1,000 1.55 5 Million 5.50 3,000 1.20 5 Million 5.50 3,000 1.20

“Consistency” of ILFs - Example Limit ILF Diff. Lim. Diff. ILF Diff. ILF Marginal Marginal Limit ILF Diff. Lim. 100,000 1.00 - - - 100,000 1.00 - - - 250,000 1.40 150 0.40 .0027 250,000 1.40 150 0.40 .0027 500,000 1.80 250 0.40 .0016 500,000 1.80 250 0.40 .0016 1 Million 2.75 500 0.95 .0019 1 Million 2.75 500 0.95 .0019 2 Million 4.30 1,000 1.55 .00155 2 Million 4.30 1,000 1.55 .00155 5 Million 5.50 3,000 1.20 .0004 5 Million 5.50 3,000 1.20 .0004

“Consistency” of ILFs - Example Limit ILF Diff. Lim. Diff. ILF Diff. ILF Marginal Marginal Limit ILF Diff. Lim. 100,000 1.00 - - - 100,000 1.00 - - - 250,000 1.40 150 0.40 .0027 250,000 1.40 150 0.40 .0027 500,000 1.80 250 0.40 .0016 500,000 1.80 250 0.40 .0016 1 Million 2.75 500 0.95 .0019* 1 Million 2.75 500 0.95 .0019* 2 Million 4.30 1,000 1.55 .00155 2 Million 4.30 1,000 1.55 .00155 5 Million 5.50 3,000 1.20 .0004 5 Million 5.50 3,000 1.20 .0004

Components of ILFs � Expected Loss Expected Loss � � Allocated Loss Adjustment Expense Allocated Loss Adjustment Expense � (ALAE) (ALAE) � Unallocated Loss Adjustment Expense Unallocated Loss Adjustment Expense � (ULAE) (ULAE) � Parameter Risk Load Parameter Risk Load � � Process Risk Load Process Risk Load �

ALAE � Claim Settlement Expense that can be Claim Settlement Expense that can be � assigned to a given claim --- --- primarily primarily assigned to a given claim Defense Costs Defense Costs � Loaded into Basic Limit Loaded into Basic Limit � � Consistent with Duty to Defend Insured Consistent with Duty to Defend Insured � � Consistent Provision in All Limits Consistent Provision in All Limits �