Modeling Retention & Effective Rate Impact

Robert J. Walling, FCAS, MAAAMHL/Paratus Ltd.

2002 CAS Risk & Capital Mgmt. Seminar

Objectives• Why do it?• What characteristics matter?• How do you model it?• What applications are there?

Why Do Retention Modeling?• Incomplete picture of your customers

and prospective customers• Incomplete picture of pricing impacts

on policy retention and premium• Underspecified financial models

Rate Impacts: The Current Problem

What’s the impact of a +25% rate change?Current Loss Ratio = Loss/PremiumProposed Loss Ratio = Loss/(Premium*1.25) = Loss/Premium*(1/1.25) = Loss/Premium*80% = 80% of Curr. Loss Ratio

The only answer is -20% on the Loss Ratio!

The Absurdity (If a little is good…)

What’s the impact of a 200% rate increase?Ignoring inflation momentarily.If Current Loss Ratio = Loss/PremiumProposed Loss Ratio = Loss/(Premium*3) = Loss/Premium*(1/3) = Loss/Premium*33.3% = 33% of Curr. Loss Ratio

Problem with the Current Pricing WorldNo response expected from policyholders:

• Likelihood of Renewal• Satisfaction of Policyholder• Book Churning/Adverse Selection• Mix of Business Shift• Consideration of Marketing/Underwriting• Satisfaction of Agent• Competition

Why Hasn’t Retention Modeling Done?

• Sensitive to many factors• Tough parameterization issues• New business penalty poorly understood• Not the “Coolest” area of research

What Characteristics Matter?

• Flexible Shape• Parameterization• Actuarially Intuitive Scenarios

• Decreasing Incremental Changes for larger rate actions

• Asymptotic Behaviors at Extremes• Different Retention Behavior for

Different Rating Characteristics

Renewal Rate (R)

Price (P)

100%

0%

Demand Curve

1P

1RR = f(P)

The Flexible Shape of the Retention Demand Curve

Renewal Behavior Characteristics

• Renewal Pricing Change (% or $)• Competitive Position• Market Conditions (Inflation, U/W

Cycle, etc.)• Customer Rating Characteristics

Renewal Behavior Rating Factors Characteristics

• Traditional Rating Factors• Age/Sex/Marital Status• Territory• Amount of Insurance• Premium Size (CML)• Industry Group

• Claims/MVR/Underwriting History• Age of Youngest Additional Driver• Satisfaction with Agent/Service• Number of Years Insured• Distribution Channel

How Do You Model Retention?• Premium Retention can be modeled as:

where: P1 = Proposed Rate LevelP0 = Current Rate LevelPM = Market Leveli = a segment of the rating plan

ii

m1

01i

i

PP

PP1

1r

Modeling Retention - Example• Premium Retention using:

where: P1 = 110

P0 = 100 r = 69.5%PM = 100

ii

m1

01i

i

PP

PP1

1r

Modeling Retention

0.00

0.20

0.40

0.60

0.80

1.00

1.20

-25%

-21%

-17%

-13% -9% -5% -1% 3% 7% 11

%15

%19

%23

%

% Rate Change

Ret

entio

n

Retention Factors 1 Retention Factors 2

a1 0.4 0.3p0 100 100pm 100 100b1 1 4g1 1 4

Retention Modeling DatabaseRisk#

Age Sex MS Terr Limit

Ren?

Comp Score

1 25 M S 1 2 Y 3 5002 64 F S 1 6 Y 2 5003 17 M S 2 1 Y 2 5254 36 F S 2 4 Y 1 5005 44 M S 1 4 N 5 5006 21 F M 1 2 N 2 6007 55 M M 2 5 N 2 6258 70 F M 2 6 Y 3 5009 29 M M 1 3 Y 1 50010 40 F M 2 4 Y 4 656

Multivariate Analysis Determines Renewal ProbabilityRisk#

Age Sex MS Terr Limit

Comp Score P(Ren)

1 25 M S 1 2 3 500 .852 64 F S 1 6 2 500 .863 17 M S 2 1 2 525 .874 36 F S 2 4 1 500 .805 44 M S 1 4 5 500 .706 21 F M 1 2 2 600 .927 55 M M 2 5 2 625 .948 70 F M 2 6 3 500 .809 29 M M 1 3 1 500 .85

10 40 F M 2 4 4 656 .91

Stochastic Approach to Retention Modeling• Model “Market Rate Level”• Model/Assume Company Rate

Actions• Model Retention

What Applications Are There?• Retention by class segment• Improved premium/policy/loss

ratio impacts of rate changes• Lifetime Customer Value • Optimal Rate Changes/

Effective Rate Impact

Reviewing Renewal Differences

LCV Definitions• Pr = Profit P = Premium• L = Losses E = Expenses• I = Investment Income t = time• P(Ren) = probability of renewal• P(Con) = probability of conversion• d = discount rate• E(Prt) = Pt + It – E(Lt) - Et

Lifetime Customer Value

Lifetime Customer Value (t): Expected profit at time t+1, t+2, etc. times the probability of realizing that profit in year t+1, t+2, etc. (renewal ratio) adjusted for the time value of money

E(Prt) E(Prt+1) x P(Rent+1) E(Prt+2) x P(Rent+2) ------- + ----------------------- + ----------------------- + …..(1+d) (1+d)2 (1+d)3

Effective Rate Impact

The effective rate impact is:the inverse of the percent change in expected

loss ratios created by the proposed rate change.

ERI = E[Loss Ratio without rate change] - 1.00 E[Loss Ratio reflecting rate change]

Effective Rate Impact - Example

Suppose current trended expected loss ratio is 60% and a proposed class plan is expected to result in a loss ratio of 54%

ERI = 0.60 - 1.00 = +10% 0.54

Effective Rate Impact - Example

Current Expected Loss Ratio 75.0%Current Average Rate 635

New/ Quotes/ Average Conversion/ Policies Expected LossSegment Renewal Curr. PIF Rate Retention Written Premium Losses Ratio ERIClass 1 New 400 495 20% 80 39,600 31,680 80.0%

Renewal 1000 550 80% 800 440,000 286,000 65.0%Class 2 New 250 900 23% 58 51,750 38,813 75.0%

Renewal 500 1050 70% 350 367,500 257,250 70.0%

Total 698 1,288 898,850 613,743 68.3% 10%10%

New/ Quotes/ Average Conversion/ Policies Expected LossSegment Renewal Curr. PIF Rate Retention Written Premium Losses Ratio ERIClass 1 New 400 405 26% 104 42,120 41,184 97.8%

Renewal 1000 450 88% 880 396,000 314,600 79.4%Class 2 New 250 1065 18% 45 47,925 30,375 63.4%

Renewal 500 1250 64% 320 400,000 235,200 58.8%

Total 705 1,349 886,045 621,359 70.1% 7%11%

Strategy 1

Strategy 2

Risk Premium

Model

Expenses

Renewal Model PRICE

Most LoyalMost Profitable

MOST VALUABLE

Optimisation Algorithm

Optimal Pricing Strategy