Chapter 9 Experience rating

0 INTRODUCTION 1 Chapter 9 Experience rating 0 Introduction The rating process is the process of deciding on an appropriate level of premium for a particular class of insurance business. The contents of this chapter are The rating of general insurance business Experience-rating systems Definition of no claims discount systems Steady state analysis The effect of NCD systems on the propensity to claim

1 THE RATING OF GENERAL INSURANCE BUSINESS 2 1 The rating of general insurance business 1.1 Basic methodology The rating process may start with a calculation of the pure risk premium, before loadings are added for commission, expenses, profit and other contingencies to give the office premium. Alternatively, where there is an established rating structure, the process may be to identify changes that need to be made in the relative levels of premium for different categories within that structure, and then to determine the overall percentage adjustment that needs to be applied to the existing premiums to achieve the desired financial result.

1 THE RATING OF GENERAL INSURANCE BUSINESS 3 1.2 The risk premium The risk premium is derived from the base data and then projected, making allowance for any changes in cover, inflation and any expected experience trends.

1 THE RATING OF GENERAL INSURANCE BUSINESS 4 1.3 Data required In order to carry out an examination of the appropriateness of the premium structure an insurer needs to produce a specification of the data requirements, assuming that the insurer has maintained appropriate records for this purpose.

1 THE RATING OF GENERAL INSURANCE BUSINESS 5 1.4 Calculation of base values The premiums will be based on the past experience of either the insurer or the market. Premiums are usually quoted in relation to a unit of exposure. In practice it is more common to analyze the elements of claim frequency, cost per claim and exposure per policy separately. These elements are analyzed separately so that trends in experience can be spotted and projected into the future.

1 THE RATING OF GENERAL INSURANCE BUSINESS 6 Pure risk premium per unit of exposure = Expected claim amount per unit of exposure The basic elements of the pure risk premium can be derived by expanding the claim amount per unit of exposure as follows: = T otal claim amount Exposure No. of claims T otal claim amount Exposure No. of claims This gives the usual formula for the pure risk premium: Pure risk premium = Expected claim frequency Expected cost per claim

1 THE RATING OF GENERAL INSURANCE BUSINESS 7 1.5 Choice of base experience statistics Internal data An insurer that has been writing a class of business for some years should have a bank of past experience from which to derive the base values.

1 THE RATING OF GENERAL INSURANCE BUSINESS 8 External data Where an insurer has insufficient or unsuitable internal data, it will be necessary to make use of external data. These may take the form of aggregate market statistics, or competitors rates for a similar product.

1 THE RATING OF GENERAL INSURANCE BUSINESS 9 1.6 Adjusting the base values Many different situations may arise to cause the base experience to be different from that expected during the new rating period. Suitable adjustment will need to be made for: Unusually heavy/light experience Large or exceptional claims Trends in claim experience Changes in risk Changes in cover Changes in the cost of reinsurance

1 THE RATING OF GENERAL INSURANCE BUSINESS 10 1.7 Projecting the base values The total claim cost and exposure values produced from the initial analyses will be expressed in the money terms of the base period. Therefore, as well as allowing for future trends and any proposed risk or cover changes, the projections need to allow for the expected effect of inflation on claims between: the mean payment date of claims in the base period, and the mean payment date of claims arising during the exposure period of the new rating series

1 THE RATING OF GENERAL INSURANCE BUSINESS 11 When revaluing base values for future premium rates, there are two parts to the calculation: inflating base values to the present day using (broadly) known inflation rates projecting from the present day to the future using estimated future inflation rates

1 THE RATING OF GENERAL INSURANCE BUSINESS 12 1.8 Projecting exposure values In order to arrive at a risk premium rate, the projected claim cost must be divided by a corresponding projected value of the exposure. For example, with private motor insurance, the premium is quoted per vehicle-year. One vehicle-year is the unit of exposure.

1 THE RATING OF GENERAL INSURANCE BUSINESS 13 Where these exposure units are expressed in terms of monetary units, the base exposure values need to be projected at an appropriate rate of inflation. This may not be the same as that applied to claim cost. Here, the projection is only to the mid-point of the exposure period arising under the new rates. For example, in many forms of property insurance, the premium is quoted per 1,000 sum insured. In these cases, high inflation does not necessarily mean that the premium rate must be increased. If the exposure measure inflates as quickly as the average claim amounts, then premium rates might stay constant.

1 THE RATING OF GENERAL INSURANCE BUSINESS 14 For example, a premium rate of 2 per 1,000 sum insured set in 1956 might still be appropriate in 2005. But we very much doubt whether a, premium of 15 per vehicle year set in 1956 would still be acceptable in 2005!

1 THE RATING OF GENERAL INSURANCE BUSINESS 15 1.9 Allowing for investment income Insurers will be able to invest part of the premiums for a period of time. This can be particularly significant for the longertailed classes of business. For long-tail classes, premiums may be invested for many years before being needed to settle claims. The assumption regarding investment returns is then significant. Note also that the inflation assumptions are far more significant for long-tail classes.

1 THE RATING OF GENERAL INSURANCE BUSINESS 16 1.10 Adjustments for commission, expenses and other loadings Insurers adopt many different ways of loading the risk premium for commissions, expenses, the cost of reinsurance and other margins, and may allow for the investment income likely to be generated by holding the premium until claims are paid. Those who start by calculating pure risk premiums will load those premiums, either by applying a simple overall percentage addition or by allowing for expenses in a more detailed way, having regard to their fixed or variable nature. Those who estimate the overall percentage change required in the existing premium rates should make due allowance for expected changes in expense levels.

1 THE RATING OF GENERAL INSURANCE BUSINESS 17 1.11 Example of premium rating and premium rating formula An example of premium rating A sample premium rating formula

2 DEFINITION OF NO CLAIMS DISCOUNT SYSTEMS 18 2 Definition of no claims discount systems 2.0 Experience-rating systems An experience-rating system is one in which the premium for each individual risk depends, at least in part, on the actual claims experience of that risk. Concept underlying experience-rating system: HIGH RISK tends to remain HIGH RISK. Experience-rating system and SELECTION RISK Number-based / cost-based SYSTEM Prospective / retrospective BASIS

2 DEFINITION OF NO CLAIMS DISCOUNT SYSTEMS 19 2.0.1 Prospective vs retrospective basis With prospective rating, the premium at the renewal date depends on the experience of the risk prior to that renewal. The insurer takes on all underwriting risk in such and arrangement. NCD in private motor is a prospective system of experience-rating system. With retrospective rating, the premium for the current policy period is adjusted, based on the experience of that period of risk. A deposit premium, paid at the inception of the policy, will usually be followed by an adjustment premium, or refund, at the end of the period. The underwriting risk to the insurer is reduced with retrospective rating.

2 DEFINITION OF NO CLAIMS DISCOUNT SYSTEMS 20 2.0.2 Number-based systems With number-based system, the premium adjustments (whether prospective or retrospective) are based on the number of claims paid in respect of the policyholder, and the amounts of the claims are ignored. NCD system / bonus-malus system (BMS system)

2 DEFINITION OF NO CLAIMS DISCOUNT SYSTEMS 21 2.0.3 Cost-based systems With cost-based system, the premium adjustments (whether prospective or retrospective) are based on the total amounts of claims incurred in respect of the policyholder over a defined period. System based on the cost of claims tend to be used for larger risks or group of risks where the aggregate cost of claims experienced within a year may be a more suitable indicator of the relative level of the underlying risk. Motor fleet (for larger fleets) / employers liability

2 DEFINITION OF NO CLAIMS DISCOUNT SYSTEMS 22 Profit sharing, where the insurer charges a higher initial premium, and returns some profit to policyholders whose claims are lower than expected. This is a typical retrospective arrangement based on claim amount.

2 DEFINITION OF NO CLAIMS DISCOUNT SYSTEMS 23 The policyholder pays an end of year adjustment premium to reflect the amount of exposure during the year (e.g. as in employer s liability). This is not the example of experience-rating.

2 DEFINITION OF NO CLAIMS DISCOUNT SYSTEMS 24 2.1 Discount categories There are two parts to a NCD system: the discount categories which are often referred to as the number of claim free years. a set of rules for moving between these categories. In addition, in order to investigate the properties of a NCD system the chance that a policyholder makes a claim each year also needs to be known.

2 DEFINITION OF NO CLAIMS DISCOUNT SYSTEMS 25 Question 9.3 on page 12. A motor insurer operates an NCD system with discount levels of 0%, 30%, 40%, 50% and 60%. The rules are as follows: 1. At the end of a claim free year, a policyholder moves up one level (or remains on maximum discount). 2. At the end of a year in which exactly one claim was made, a policyholder drops back two levels (or moves to zero discount). 3. At the end of a year in which more than one claim was made, a policyholder drops back to zero discount. What premium does a motorist who first took out a policy on 1 January 1989 pay for insurance cover in the year 2000, if the policyholder made claims on 15 August 1990, 3 February 1994, 17 September 1994 and 14 November 1999, and the full premium for the year 2000 is 750pa?

2 DEFINITION OF NO CLAIMS DISCOUNT SYSTEMS 26 2.2 The transition matrix The probability that a policyholder in category i moves to category j from one year to the next can be written as a matrix of transition probabilities. p 00 p 01 p 02 p 10 p 11 p 12 p 20 p 21 p 22 where p ij is the probability that a policyholder moves from category i to category j. Question 9.4 on page 13. Given the transition matrix 0.2 0.8 0 0.2 0 0.8 0 0.2 0.8 for the system with States 0, 1 and 2, using the convention given above, what is the probability that a policyholder who starts in State 0 is in State 0 again 2 years later?

2 DEFINITION OF NO CLAIMS DISCOUNT SYSTEMS 27 2.3 Distribution of policyholders The transition matrix can be used to estimate how many policyholders are expected to be in each discount category each year. The expected proportion of policyholders in category i is denoted by π i. Note that π i = 1. Also, the proportions in the discount categories can be represented as a vector, π = (π 0, π 1, π 2,..., π n ). Then we can write Example on page 14. π (n+1) = π (n) P.

3 STEADY STATE ANALYSIS 28 3 Steady state analysis 3.1 The equilibrium distribution It is possible to continue finding π (n) for larger values of n. Under reasonable conditions, π (n) will tend to a limit as n. When this happens, the system has reached equilibrium or its steady state. This limit is denoted π. Letting n gives π = πp. This is a set of equations which can be solved to find π noting that π i = 1. An example on pages 16-17.

3 STEADY STATE ANALYSIS 29 3.2 Heterogeneity in the portfolio One of the reasons which is used to justify NCD systems is that they result in automatic premium rating. In other words, policyholders who make fewer claims pay less than those who make more claims. While this is obviously true, most do not work as well as is hoped and the premiums policyholders ultimately pay are not proportional to their likelihood of making a claim. This is partly because of the small number of categories of discount and the relatively low levels of discount that are offered. But it is also due to the relatively low probabilities of claims occurring and hence the high probabilities of all policyholders reaching the maximum discount level at some stage. Another reason why NCD systems do not work as well as might be hoped is because of noise in the system. People s actual claim rates differ from those that might be expected.

3 STEADY STATE ANALYSIS 30 Given the probabilities of claiming for all policyholders, it would in fact be possible, mathematically, to determine a NCD system that would result, over the long term, in all policyholders paying a pure premium that was directly proportional to their probability of claiming. However, this would be an extremely complex system to administer and understand. The following example takes the situation to the opposite extreme, by assuming that there are only two possible types of policyholder and there are only three categories of discount. However, even in this simple situation it is not easy to produce a system that matches premiums to the probabilities of claiming. An example on pages 18-19.

4 THE EFFECT OF NCD SYSTEMS ON THE PROPENSITY TO CLAIM 31 4 The effect of NCD systems on the propensity to claim 4.1 Reassessment of transition probabilities In what has been done so far, it has been assumed that the probability that a driver makes a claim is the same, no matter which discount category he or she is in. The policyholder may take into account the increases in future premiums when deciding whether to make a claim or not. This can be considered by comparing the change in premiums when a claim is made. The number of future years considered is called the policyholder s horizon, and the propensity to claim will also depend on this horizon. Example on page 21.

4 THE EFFECT OF NCD SYSTEMS ON THE PROPENSITY TO CLAIM 32 4.2 Calculating the transition probabilities It can be seen from this that the probability that a policyholder incurs a loss (eg has an accident) is not the same as the probability that a claim is made. If the distribution of the loss is known, the probability that a claim is made following an accident can be calculated. For example, considering the policyholder in the previous example, who has an infinite horizon, is at present in the 25% discount category and has just had an accident. A claim will only be made if the cost of the accident is greater than 275. If X is the random variable which represents the cost of an accident, then: P (Claim Accident) = P (X > 275) Since it is assumed that the distribution of X is known, this probability can be evaluated.