P/C RATEMAKING AND LOSS RESERVING by R. Brown and L. Gottlieb CHAPTER 5: INTERMEDIATE TOPICS 5.1 Introduction 5.2 Individual Rating Plans 5.3 Increased Limits Factor (ILF) 5.4 Deductible Pricing 5.5 Reinsurance

I. Individual Risk Rating Plans Chapter 4 showed how to get rates for a risk class that might contain many policyholder (i.e. by using “class” relativities). This is used widely for auto and homeowners. Underwriter tries to select above average Ph’s in each risk cell as customer base. For large (e.g. corporate) Ph, use risk rating plans to achieve fair premium. Called experience rating if Ph’s own historical experience is used (to some extent) to set premium. (Ph gets discount or surcharge on published, “manual”, rate). Adopted New Premium = Z*(Recent Ph Loss Cost) + (1-Z)*(Adopted Class Rate)

I. Individual Risk Rating Plans Under Schedule Rating, Ph gets discount or surcharge (within limits) for certain risk characteristics (e.g. a sprinkler system). Policyholder Dividends: From mutual P&C co. (may be affected by Ph’s loss experience) Retrospective Rating: Final price not determined until end of experience period Final Charge = Z*(Actual Ph Experience) + (1-Z)*(Industry Average Experience)

Class (Manual) Rating Automobile Liability Insurance Class Rates Premium = base premium * (primary rating factor + secondary rating factor) Base premium: policy limits; territory Primary rating factor (161 classes): use; age; sex; marriage status; driving training; good-students… Secondary rating factor: more than one car?; type of car; traffic points… E.g., base premium = $200 age 17, no driving training; low grades; four points $200(3.50+2.20)=$1,040 age 17, driving training; good grades; zero points $200(2.50+0.00)=$500

Case Discussion: GENCO GENCO is a small insurance company writing car insurance in a Canadian province. The management of the company decided in 1992 to implement an aggressive marketing strategy to increase its business. The rate-making department was using the classification system of drivers suggested by the Association of Automobile Insurers and based on 38 classes established from a multi criteria which included the following factors: Characteristics related to the driver (age, sex, occupation and driving experience) . Characteristics related to the vehicle (power, usage, mileage) . Territory in which the vehicle operates. The decision was taken to simplify the rating system and to reduce the number of classes to 20 by amalgamating some classes.

Case Discussion: GENCO For example class A and B formed a single new class. Observing that in a national sample of 100,000 policyholders, 80% belong from class A and only 20% belong from class B, the new average premium was fixed at $180.

Case Discussion: GENCO After only two years of operation, although the company had increased its business as expected, the overall financial results were disastrous as the company was losing money in several classes of business. For example, contrary to the national average, a sample of policyholders from the new class shown above revealed that about 67% were in fact policyholders belonging from original class B. The company should have charged an average premium estimated at $250.

Merit (Modification) Rating Further refinement of class rating Usually for large corporations Comparison between the insured exposure and standard exposure in the same class Issues being examined: past experience; size of exposure; detailed analysis of the quality of the exposure Schedule rating Experience rating Retrospective rating Premium-discount plans

Schedule Rating A schedule of debits and credits is applied to the insured exposure rate depends on a more detailed analysis of the exposure’s characteristics encourage loss-control efforts e.g., fire insurance on manufacturing plant Construction: physical characteristics Occupancy: use of the building Protection: fire dept. & protective devices Exposure: adjacent buildings Maintenance: housekeeping & overall upkeep

Experience Rating Insured’s past experience affects the premium charged Degree of influence from past loss experiences is tied to statistical credibility of the insured’s loss experiences Weight = C*A + (1-C)*E C: credibility factor of the insured A: insured’s past loss experiences E: loss experiences of exposures of the same class e.g., A=1.5; E=2.0; C=0.4 ==> Weight = 1.8 premium will be 1.8 / 2.0 = 90% of the standard rate Premium change = [(A - E) / E ] * C

Retrospective Rating Premium depends on the losses of the insured during the current policy period, subject to minimums and maximums. Actual premium is not known in advance. Premium = (basic premium + Claims * Claim-conversion factor) * Tax multiplier e.g., min=$51,000; max=$131,500; basic=$23,600 claim-conversion factor=1.125; tax multiplier=1.078 What’d be the premium if actual claims during the policy period = $40,000; $20,000; $200,000? Answer: $73,950.80; $51,000; $131,500. (Loading for claim-processing = 12.5% of claims; Loading for tax = 7.8% of premiums)

II. Increased Limits Factors Introduction: for liability insurance Data consideration Chap 3 methods are used to calculate rate for basic limits coverage If Ph buys a larger limit (e.g. auto liability), premium is determined as: (Increased Limits Factor (ILF)) x (Basic Limits Premium) Complicating factors Can’t use data written at lower limits to determine ILF Loss development factors (LDF) larger as limits increase Trend factors increase as limits increase Greater risk (variance) to I. C. at higher limits Some expenses are fixed; some vary with premium

II. Increased Limits Factors Data consideration Assume policy limit is $50,000; so all losses capped at $50,000. Data provide no info about loss distribution beyond $50,000. Also lawyers often settle at insured limits. Phs who buy increased limits may be different. So to study limit, L, look only at policies with limit L or +. Where appropriate, include defense costs in addition to insured loss. Then

Example 5.1: You are given data for an insurer that sells Personal Automobile Liability policies at the following combined single limits: $50,000, $100,000, $200,000 and $300,000. You are asked to calculate increased limit factors (ILF’s) to the $50,000 basic limits based on historic losses. For each policy limit the losses have been limited to several limits as shown below. Assume policy limits include ALAE. All amounts are in $000. Losses (including ALAE) Limited to Limit 50 100 200 300 50 1M 100 5M 7.5M 200 3M 5M 6M 300 2.5M 4M 5M 5.5M (For the $100,000 limit policies, losses limited to $50,000 total $5M, and are $7.5M at full policy limits of $100,000.)

II. Increased Limits Factors Loss Development Larger losses take longer to settle. Larger losses have greater range of possible outcomes. Thus, calculate loss development factors for limit, L, on historical experience on policies with limit, L. Trend General inflation hits higher limits more than lower limits. Increased limits get impact of inflation in both frequency and severity. Thus, best to use separate trend factors for each policy limit. (Example 5.2)

II. Increased Limits Factors Risk Higher limits are riskier (more variance). Higher risk should be compensated by higher premium (risk load). Expenses Some are a % of G.P. (commissions, tax). Some are fixed per policy (underwriting) Thus

Example 5.3: You are given the summarized loss data for a block of policies, all with limit of $1,000,000. The table shows the # of claims in each of 6 loss-size intervals. Assume the claims are all closed and have been trended to current cost level. Assume policy limits and the loss ranges include ALAE. The risk charge at any limit is the square of the limited average loss size divided by 100,000. The basic limit is $100,000. Find the increased limits factor for $500,000 and $1,000,000 limits. Bottom Top # of claims Average of Range of Range in Range in Range 1 1,000 1,000 350 1,001 10,000 1,250 2,500 10,001 50,000 1,500 27,000 50,001 100,000 1,000 62,000 100,001 500,000 200 190,000 500,001 1,000,000 50 800,000

Deductible Pricing Loss Elimination Ratios Example 5.5 Indicated deductible relativity = 1 – LER Example 5.5 Given the following loss data for car collision coverage. The insurer offers three deductible options: no deductible, $500, and $1,000.

LER & Indicated deductible relativity Size of loss # claims Ground-ip Total Losses incl. ALAE 1 - 500 1,240 396,800 501- 1,000 1,080 831,600 1,001 or + 2,180 9,352,200 Total $4,500 10,580,600 Calculate the LER for a deductible of $1,000 and the indicated $1,000 deductible relativity. Assume the base level deductible is $500.

Deductible Pricing Loss distributions x = b: x = l > b: Model the loss distribution statistically (e.g. Pareto). Then, total losses limited to basic limit x = b: 3. Total losses limited to increased limit x = l > b: ILF modified for risk, fixed expenses, etc.

Reinsurance Reinsurance is an arrangement by which the primary insurer that initially writes the insurance transfers to another insurer part or all of the potential losses associated with such insurance The primary insurer is the ceding company The insurer that accepts the insurance from the ceding company is the reinsurer The retention limit is the amount of insurance retained by the ceding company The amount of insurance ceded to the reinsurer is known as a cession 21

Reinsurance THE INSURED PRIMARY REINSURER INSURER CEDENT RETROCEDENT REINSURER REINSURER RETROCEDENT RETROCESSIONAIRE

Reinsurance & Retrocession LOMA, Figure 1-2

A Complex Reinsurance Relationship LOMA, Figure 1-3

Reasons for Reinsurance Reinsurance is used to: Increase underwriting capacity Stabilize profits Reduce the unearned premium reserve The unearned premium reserve represents the unearned portion of gross premiums on all outstanding policies at the time of valuation Provide protection against a catastrophic loss Retire from business or from a line of insurance or territory Obtain underwriting advice on a line for which the insurer has little experience 25

Example: Without reinsurance 8/31/2015 9/1/2015 Asset Cash $500,000 $4,000,000* Investment 1,500,000 1,500,000 Total assets $2,000,000 $5,500,000 Liabilities Unearned premium reserve $0 $5,000,000 Total Liabilities $0 $5,000,000 Policyholders’ surplus $2,000,000 $500,000 Total liability and surplus $2,000,000 $5,500,000 * Premiums ($5,000,000) and expenses ($1,500,000)

With Reinsurance W/O Reins. With Reins. Asset Cash $4,000,000 $2,250,000* Investment 1,500,000 1,500,000 Total assets $5,500,000 $3,750,000 Liabilities Unearned premium reserve $5,000,000 $2,500,000 Total Liabilities $5,000,000 $2,500,000 Policyholders’ surplus $500,000 $1,250,000 Total liability and surplus $5,500,000 $3,750,000 * 50% insurance ceded and received 30% ceding commission

Categories of Reinsurance TREATY FACULTATIVE PRO EXCESS PRO EXCESS RATA OF LOSS RATA OF LOSS QUOTA SURPLUS PER PER AGGREGATE SHARE SHARE RISK OCCRRENCE EXCESS

Types of Reinsurance Agreements There are two principal forms of reinsurance: Facultative reinsurance is an optional, case-by-case method that is used when the ceding company receives an application for insurance that exceeds its retention limit Facultative reinsurance is often used when the primary insurer has an application for a large amount of insurance Treaty reinsurance means the primary insurer has agreed to cede insurance to the reinsurer, and the reinsurer has agreed to accept the business All business that falls within the scope of the agreement is automatically reinsured according to the terms of the treaty 29

Methods for Sharing Losses There are two basic methods for sharing losses: Under the Pro rata method, where the ceding company and reinsurer agree to share losses and premiums based on some proportion Under the Excess method, where the reinsurer pays only when covered losses exceed a certain level Under a quota-share treaty, the ceding insurer and the reinsurer agree to share premiums and losses based on some proportion Under a surplus-share treaty, the reinsurer agrees to accept insurance in excess of the ceding insurer’s retention limit, up to some maximum amount An excess-of-loss treaty is designed for catastrophic protection A reinsurance pool is an organization of insurers that underwrites insurance on a joint basis 30

Surplus Share Example Five-line surplus share reinsurance treaty with a minimum line of $200,000 and max. capacity of the treaty of $1,000,000. Ex: Policy limit lines retained lines ceded % retained Risk 1 $200K 1 0 100% Risk 2 $500K 1 1.5 40% Risk 3 $1M 1 4 20% Risk 3 $1.2M 1 5 16.7%

Excess of Loss Treaty Example AICPCU, Exhibit 6-11

I.L.F. and Excess Reinsurance Pricing Example 5.4 The ABC Insurance Company writes $10 million in premiums of Automobile Liability Insurance. All of its policies are for $1 million limit. In order to stabilize its underwriting results, it wants to reinsure all losses in excess of $100,000. Using exposure rating, what are the expected ceded losses (i.e., paid by the reinsurer)? The expected loss ratio is 70%, and the I.L.F. based on industry statistics is: Limit ILF $25,000 1.00 $100,000 1.25 $1,000,000 1.40 $5,000,000 1.45 Assume the ILFs have not been loaded for risk and expenses.

Reinsurance Alternatives Some insurers use the capital markets as an alternative to traditional reinsurance Securitization of risk means that an insurable risk is transferred to the capital markets through the creation of a financial instrument, such as a futures contract Catastrophe bonds are corporate bonds that permit the issuer of the bond to skip or reduce the interest payments if a catastrophic loss occurs Catastrophe bonds are growing in importance and are now considered by many to be a standard supplement to traditional reinsurance. 34