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
1
2
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
1.2
3
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
1.3
4
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
1.4
5
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
N o. of claims T otal claim amount
=
×
Exposure
N o. 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
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.
7
1 THE RATING OF GENERAL INSURANCE BUSINESS
• 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.
8
1 THE RATING OF GENERAL INSURANCE BUSINESS
1.6
9
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
1.7
10
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
1.8
12
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
1.9
15
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
1.10
16
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
1.11
17
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
2
18
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
2.0.1
19
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
2.0.2
20
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
2.0.3
21
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
2.1
24
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
• 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?
25
2 DEFINITION OF NO CLAIMS DISCOUNT SYSTEMS
2.2
26
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.


p00 p01 p02 · · ·
 p10 p11 p12 · · · 


 p20 p21 p22 · · · 


 · · · · · ·
· · · · · ·
where pij 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
2.3
27
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 πi.
P
• Note that
πi = 1.
• Also, the proportions in the discount categories
can be represented as a vector, π = (π0, π1, π2, . . . , πn).
• Then we can write
π (n+1) = π (n)P.
• Example on page 14.
3 STEADY STATE ANALYSIS
3
28
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
P
find π noting that
πi = 1.
• An example on pages 16-17.
3 STEADY STATE ANALYSIS
3.2
29
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
4
31
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
4.2
32
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.