1
Comparison of the Standard Rating Methods and the
New General Rating Formula
Muhamed Borogovac, Ph.D., ASA, MAAA
Boston Mutual Life Insurance Company
Abstract: A comparison between standard rating methodology in general insurance
and the new rating formula published in [1], which we call General Rating Formula
(GRF), is given in this paper. A proof and many advantages of the General Rating
Formula over the standard rating methods are explained. Equivalence of the General
Rating Formula and standard methods regarding credibility is proven.
1. Introduction
In property and casualty (P&C) insurance or general insurance, a rate change traditionally
involves the following Three Step Process:
Step 1. Determine the overall average rate change,
Step 2. Change all risk classification variables according to the past experience
projected to the future period of new rates,
Step 3. Balance back to overall rate change indicated, so that the overall change in
premium is actually the one determined in the Step 1.
Traditionally, the Three Step Process was accomplished by either a loss ratio approach
or a pure premium (or loss cost) approach. It is well known that those two methods are
equivalent (meaning the two methods produce the same rates) when losses are adjusted
for homogeneity. The problem with Three Step Process is that each step consists of
rather complicated and tedious calculations for either of the two methods. That requires a
lot of data in the process, which is not always available.
General Rating Formula, published first time by M. Borogovac in 2007, see [1],
eliminates the need for all three steps. The calculation of proposed rates will be
performed by a simple formula. The formula results in exactly the same indicated rates
as the traditional Three Step Process. The only input variables for the formula are:
number of earned exposure units for each risk cell, fully developed and trended loss
(FDTL) amounts for each cell, permissible loss ratio (PLR) and current values of rating
variables (class, territory,…, industry) Note that earned exposures, losses and PLR
characterize a specific product, while rating variables (risk factors) characterize external
forces.
We will first review Three Step Process and after that we will prove the General Rating
2
Formula.
The General Rating Formula can be used, not only for rerating of an existing product,
but also to calculate rates for a new product. It is understandable intuitively because
both, new rates and changed rates will be in effect for a future period. The only
difference is that for a new product we have to assume values of input variables,
exposures and losses for every risk cell, and risk factor while for rerating, the historic
experience of the input variables should be available to us. That is the reason that we can
use both names “General Rating” and “General Rerating” formula.
2. Notation and Introduction of the Rating Variables
In the calculations that follow, rate classification variables are assumed to be applied
multiplicatively. In order to simplify derivation, let us limit ourselves to only three risk
classification variables (or factors, relativities, differentials): class, territory and
industry. Class, territory and industry are generic risk classification variables introduced
only to ease communication and are not examples of any particular P&C insurance
product nor a group insurance product.
We could not use less than three classification variables for the preceding derivation of
the General Rerating formula because that would result in a loss of generality. In that
case, it would not be clear how formulae derived for two variables would extend to
three or more variables.
Let
denote the tensor (i.e. three‐dimensional array) of fully developed
and trended losses (FDTL). L is also called expected dollar losses (in effective period) or
projected losses.
We obtain element
in the cell
by multiplying the current loss of that cell by
the development factor and the trend factor. Less formally, we can say
is a
projected loss in the
,
, and
, where
, and
are numbers of risk factors for class, territory and
industry, respectively. Let us denote the Total Projected (or Expected) Loss by , i. e.
(2.1)
By keeping fixed one index at a time and by summing loss costs of the corresponding
slice, we get:
Vector of Loss Costs for Class,
,
Vector of Loss Costs for Territory,
is defined by:
,
is defined by
3
,
,
Vector of Loss Costs for Industry
,
is defined by:
.
Note on generalization: If we have N>3 classification parameters class, territory, …,
industry, i.e. if industry is Nth rather than third classification parameter, then the above
formula for Class i would be
,
,
meaning that only summation by index would be omitted. All other formulae would be
generalized accordingly.
Also, let us introduce:
Tensor
, where
is the Number of Earned Exposure Units in
C lass , Territory , and Industry . Let us denote the Total Number of Earned
Exposure Units by , i.e.
.
Tensor
, where
(2.2)
represents Current Manual Rate in the cells
.
As before, by summing Numbers of Earned Exposure units of a particular slice, we get
the following:
Vector of Earned Exposure Units for Class,
,
,
Vector of Earned Exposure Units for Territory,
,
is defined by
,
Vector of Earned Exposure Units for Industry
,
is defined by:
is defined by:
.
Earned Premium at Current rates is denoted by . Therefore:
4
.
(2.3)
Then Current Average Rate, denoted by , is:
.
(2.4)
By summing premium of a particular slice, we get the following:
Vector of Premiums for Class,
, is defined by:
,
,
Vector of Premiums for Territory,
,
is defined by
,
Vector of Premiums for Industry
,
is defined by:
.
The, Loss Ratios of the corresponding slices are:
,
.
,
.
,
.
In this setting we will deal with two sets of differentials. Let the vectors
represent current class, territory and industry differentials, respectively, and let vectors
represent new (indicated, proposed) class, territory and industry differentials,
respectively.
Let us call the cell
the base cell. That cell is usually the cell with the largest
amount of exposure for the line of business observed so that it has maximal statistical
credibility. Without loss of generality, we can denote
. This means,
5
we set
. We already mentioned that classification variables are applied
multiplicatively. Therefore, rate in the cell
is defined by:
(2.5)
From (2.5), it follows
.
,
.
3. Review of the Standard Three-Steps Rating Process
3.1 Overall rate change
If denotes new average rate and
change (RC) is defined as
denotes old average rate, then the average rate
, or equivalently,
.
(3.1)
In addition, let us introduce:
.
Symbolically,
.
(3.2)
Both, loss cost (or pure premium) method and loss ratio method historically were used
to calculate the overall rate change. In both, we will need Permissible Loss Ratio, PLR,
which is defined as
3.1.1 Loss Cost Method
In this method, we first calculate:
,
(3.3)
6
where
and
are given by formulae (2.1) and (2.2), respectively.
Then, according to this method, the new average rate, denoted by
will be calculated as:
.
Substituting form (3.3) gives:
(3.4)
3.1.2 Loss Ratio Method
In Loss Ratio method, we first calculate Expected Loss Ratio, defined by (3.2). Then rate
change ( ) is calculated as:
.
Symbolically,
.
Substituting
from here into (3.1) we get
(3.5).
It is well established that loss cost method and loss ratio method are equivalent for
overall rate change, Step 1. In our notation it means that formulae (3.4) and (3.5) are
equivalent.
3.2 Changing Risk Classification Differentials
In traditional Three-Step Process it is necessary to determine new risk classification
differentials.
In loss ratio method, the indicated differentials are calculated as:
,
,
(3.6C)
7
,
,
,
(3,6T)
,
(3,6I)
while, in the loss cost method new class differentials are defined by:
,
,
(3.7C)
,
,
(3,7T)
,
.
(3,7I)
We saw that loss ratio and loss cost methods are equivalent regarding the overall rate
change calculations. It would be convenient if that were the case here as well.
Unfortunately, the two methods do not calculate equal indicated differentials if the
process is not adjusted for heterogeneity, as the following example shows.
Example 3.1 Determine indicated differentials for Class B and Class C, using both
methods, given the following information and assuming full credibility of all classes, see
[2].
Class
Existing Differential ( )
Loss Cost (
A(
)
1.00
0.65
129
B(
)
0.85
0.71
120
C(
)
1.21
0.66
157
Solution: In our notation:
Loss Ratio method i.e. formulae
,
and
give:
)
8
,
,
Loss Cost method i.e. formulae
,
.
,
give:
and
,
. ///
Note that the results of the two methods are not equal.
4. Adjusting for Heterogeneity and Balancing Back
4.1 Understanding Heterogeneity
The uneven distribution of risks by cells is reason that loss cost and loss ratio methods
produce different indicated differentials, which means different indicated rates. The
uneven distribution of risks is called heterogeneity.
In this section, we will introduce variables
,
,
,
,
,
that will be used in
the derivation of the formula.
First, let us observe the simple example where we have only two classes and only two
territories. It will help us to intuitively understand the meaning of the above mentioned
variables.
Assume that Class 2 consists of risky drivers and that all risky drivers live in Territory 2.
In our notation, this means
and
. The formulae for indicated differentials
in Loss Cost method for Class 2 and Territory 2 are, respectively:
,
.
Assume now that everything is the same except for the distribution of risky drivers.
Assume that half of risky drivers’ losses belong to those risky drivers who live in the
Territory 1. This means that
. Then we have
,
.
Hence, the Class differentials did not change (as it should be the case) while Territory 2
differential changed considerably. This means that the risk that comes with territory is
affected by the risk that comes with the class, according to this calculation of , and
that should not be the case.
9
Let us now look at the Territory 2 differential,
in the same example, but by means of
the Loss Ratio method. By developing numerator and the denominator, and taking into
account
we get
In this formula, we deal with average loss
, which is practically a loss per unit
of exposure. Therefore, the distribution of losses by cells does not play a significant role
here. Note, that exposures in the denominator
corresponding risks
adjusted exposure.
and
are weighted by the
. Therefore, we can call the variable
total Class 2
4.2. Adjusting for Heterogeneity
Now we will apply the lesson that we learned in this example to general situation in order
to adjust for heterogeneity the formulae that follow.
As every unit of exposure in the cell
is expected to cost
,
we can introduce the following concept, the total Class i adjusted exposure
,
.
Similarly,
,
;
,
.
As before, C, T, and I stand for class, territory, and industry, respectfully.
The Loss Cost for Class i adjusted for heterogeneity, denoted by
,
.
, becomes:
(4.1C)
The Loss Cost for Territory j adjusted for heterogeneity, denoted by
, becomes:
10
,
.
(4.1T)
The Loss Cost for Industry k adjusted for heterogeneity, denoted by
,
.
, becomes:
(4.1I)
The base rate for class i (i.e. Class i, Territory 1, Industry 1 rate), denoted by
expressed as:
,
Obviously,
.
(4.2C)
Similarly,
(4.2T)
(4.2I)
Also, the base rates
represented by:
and
for Territory j and Industry k, respectively, can be
,
;
,
.
The Fully Developed and Trended Loss Ratio for Class i , denoted by
,
(Here we used formulae (4.1) and (4.2).) Similarly,
,
,
,
.
.
, is
, can be
11
4.3 Indicated Differentials Adjusted for Heterogeneity
In Loss Ratio method, after adjustment for heterogeneity we calculate class differentials
by
,
,
while, in the loss cost method new class differentials are defined by
,
.
We have
,
Here we used the previously proven equation
. (4.3C)
,
.
Similarly,
,
,
,
(4,3T)
.
(4,3I)
The superscripts , , and stand for class, territory, and industry, same as before.
The last three equations prove the following classical statement.
Corollary 4.1 Loss ratio method and loss cost method are equivalent if loss costs are
adjusted for heterogeneity.
The big advantage of the General Rating Formula over both standard methods is that we
will not have to actually calculate indicated risk classification differentials.
Remark 4.2 In the current texts, experience loss ratio is defined as current losses
divided by earned premiums. In this text, losses are fully developed and trended. One
might think that loss ratios used above are different from the experience loss ratios used
in the traditional texts. However, the values of , , are the same because the
12
development and trend factors are applied on both, numerators and denominators.
Therefore they cancel out in the above relations for , , .///
4.4 Balancing Back
Obviously, the indicated differentials will be different from the current ones except
.
However, the rates calculated by means of the indicated differentials in the standard
methods may give different rate increase than the desired increase calculated in Step 1. In
order to make a correction in the process we multiply those indicated rates by the Balance
Back Factor (BBF). As we know, BBF is defined by:
. In our notation it reduces to:
.
Example 4.3 Given:
Base rate:
Trend*Develop. Factor:
Permissible Loss Ratio:
Earned Exposures:
Class
Territory1
Territory2
Class1
12,000
3,000
Class2
4,500
2,000
Current Losses:
Class
Territory1
Territory2
Class1
840,000
300,000
Class2
500,000
250,000
Class Differentials:
Territory Differentials:
T
13
Calculate new rates.
Solution by Standard Method adjusted for heterogeneity
Step 1, Overall Rate Change:
Multiplying Current Losses by the Trend*Development Factor = 1.409 we get array
(matrix) of Fully Developed and Trended Losses.
Then, Total FDTL is the sum
Let us first calculate
.
. According to (2.3) we have:
According to (3.5) we have the average rate increase:
Step 2, Calculation of Indicated Differentials
In order to calculate indicated differentials adjusted for heterogeneity we first have to
calculate:
.
.
According to equations (4.1), simplified to two rating variables, we have:
14
Therefore:
Indicated class differentials:
Indicated territory differentials:
,
,
Step 3, Balance Back Factor
Finally we can calculate the indicated rates:
///
5. General Rating Formula
5.1 Derivation of the Formula
In the previous sections we introduced notation, terminology and equations that will
enable us to finish the derivation of the General Rating Formula. Recall, that the goal is
to find easier way to calculate same rates that are calculated in the standard Three-Step
Process.
.
Symbolically
15
.
The overall rate change
is equal to:
,
(5.1)
where PLR stands for Permissible Loss Ratio which is sometimes called Expected or
Target Loss Ratio.
Let us denote the indicated rates by . Then, form (5.1) it follows:
.
Let us substitute expressions for
and
.
According (2.3) and (2.5)
.
Hence, after canceling ,
Substituting
from formulae (4.3) we get
After canceling out constant
we get the final formula
(5.2)
16
Note that the number
(5.3)
in the denominator is a constant number for all cells
it only once and then we calculate rates by the formulae
.
. It means that we calculate
(5.4)
This is the General Rerating Formula that calculates exactly the same rates as standard
Three-Step Process. The advantages of the General Rerating Formula over the
traditional, Three Step Process is the following:
1. We only need to know losses and earned exposures for every cell, and current
differentials.
2. We do not even need to know current rates.
3. We do not need to calculate overall rate change.
4. We do not need to calculate indicated differentials.
5. We do not need to calculate balancing back factor.
6. The rates calculated by the General Rerating Formula are already adjusted for
heterogeneity.
7. If we want, we can calculate (uniquely determined) indicated differentials, by
simple formulae.
Remark About Generalization: In general case, when there are N>3 classification
parameters class, territory, …, industry, i.e. industry is Nth rather than third classification
parameter, then General Rating Formula is
.
(5.5)
As before, the formula can be written equivalently as:
;
,
to underline that the denominator is a constant for all cells
calculated only once.
, and therefore
17
General Rerating Formula for two rating variables: Assume that we have a product
with two classification parameters Class and Territory. Then the General Rerating
Formula simplifies to
,
.
where
Example 5.1 Assume given all information as in the Example 4.3, except base rate
which is not needed in for General Rating Formula. Hence,
Trend*Develop. Factor:
Permissible Loss Ratio:
Earned Exposures:
Class
Territory1
Territory2
Class1
12,000
3,000
Class2
4,500
2,000
Current Losses:
Class
Territory1
Territory2
Class1
840,000
300,000
Class2
500,000
250,000
Class Differentials:
Territory Differentials:
T
Calculate new rates.
Solution by General Rating Formula:
Multiplying Current Losses by the Trend*Development Factor = 1.409 we get
and
We need to find
.
, nd
.
and then
and
,
18
.
Now we have, according to equations (4.1), when simplified to two rating variables and
round:
Now we calculate the denominator constant
variables:
given by formula (5.3) adjusted to two
.
Now we have all that we need to substitute into General Rating Formula (5.4) for two
rating variables and calculate indicated rates:
;
;
;
.
The rates are the same as the rates obtained by standard method, except for small
rounding errors. (Corresponding excel calculations produce identical rates.)
19
Note that adjustment for heterogeneity is built in the GRF calculations. Also note that this
calculation has all the advantages listed earlier in the section General Rerating Formula,
including the fact that we did not need to calculate indicated differentials. However, if we
need to know indicated differentials, they are given by formulae (4.3), and they are
already adjusted for heterogeneity, therefore uniquely determined. Let us calculate them
here:
Indicated class differentials:
,
Indicated territory differentials:
,
///
5.2 Credibility of the New Rating Method
Credibility is not uniquely determined concept. There are different formulae, and even in the
same formula there are different definitions for the full credibility. For example, full credibility is
assigned to different numbers of exposures, sometimes depending on the judgment of the actuary.
The following section is the quote from [2] and describes how credibility was introduced
in the standard models.
“In practice, the actuary will usually do two separate average rate calculations. One is
based on data from the actuary’s own company; the other is based on industry wide data
which are normally available from one or more rating bureaus and/or statistical agents.
Unless the actuary’s company is very large, the indication based on its data will have
credibility Z<1. Then the final answer will be
Credibility-Weighted Average Indicated Rate=Z*(Company Indication)+(1-Z)*(Industry
Indication).”
In this section, we will describe how to adjust for credibility in the new method
characterized with the General Rerating Formula (5.3).
Credibility factor always depends only on company’s data and we will determine it the
same way as before by selecting one of the standard credibility formulas.
Using industry experience to make corrections on company’s rates makes sense only if
we deal with the same products. That means that our product and industry products
observed, theoretically have the same cells, or otherwise we would be mixing apples and
oranges. In practice, this means that sometimes we will have to adjust industry’s cells to
company’s usually by combining similar cells. For example, if industry data has three
rating variables, Class, Territory, and Industry, and company’s product have only two,
20
Class and Territory, then exposure in the cell
should be
same token we would adjust
and all other industry data.
. By the
Then, in order to be consistent with the above quoted standard approach to credibility, we
will do two separate rate calculations by means of the formula (5.3). Therefore, we will
get two rates for each cell, the company’s, and the industry’s rate. Let us prove that two
approaches remain equivalent after the credibility.
Theorem 5.2 Standard Three Step Process and General Rating Formula produce the
same future premiums if rates are adjusted for credibility by the same factor Z.
Proof: Let us denote the future earned exposures by
company rate by . Let us denote industry rates by
, company rates by
and average
and average industry rate by
.
Then, the rate adjusted for credibility, according to General Rerating Formula approach, in every
cell equals
(5.6)
Obviously
and
.
That means
(5.7)
(5.8)
Premium adjusted for credibility, calculated by means standard methods,
is
Premium adjusted for credibility, calculated by General rating Formula approach
Substituting into the last equations expressions from (5.7) and (5.8) we get
is
21
We have proven the statement. ///
References
[1] M. Borogovac, General Re-rating Formula, SOA Monographs, 2007 Enterprise Risk Management
Symposium.
[2] R. L. Brown & L. R. Gottlieb, Introduction to Ratemaking and Loss Reserving for Property and
Casualty Insurance, Second Edition, ACTEX Publications, Inc., Winsted, Connecticut
[3] R. L. Brown, On the Equivalence of the Loss Ratio and Pure Premium Methods of Determining
Property and Casualty Rating Relativities, Journal of Actuarial Practice, Vol. 1, NO 2, 1995