When to Use a Pure Loss Rating Matrix

When insureds are considering how to fund estimated liabilities for a future period, they will often use a loss projection and confidence interval to determine adequate funding. For company XYZ, Inc. projected workers compensation losses for the 2023 policy year can be seen below.

XYZ. Inc.
2023 Aggregate Loss Probability Distribution
(Losses are Limited to $100,000 per Occurrence)

                      Aggregate                         Loss Probability                      Aggregate                       Loss Distribution                         Risk                           Margin
Expected $340,000 100%
45% $320,000 95%
60% $350,000 101%
75% $380,000 107%
90% $440,000 113%


For insureds whose business activities are comprised of different entities, members, or locations, it can be useful to distinguish how each of these units contribute to the total projected losses. Questions the insured may be interested in considering could include:

  • Is the underlying exposure for each of these units equal?
  • Is the loss history consistent across each unit, or does it vary?
  • Can we pinpoint parts of the business with loss experience that deviates from the overall business average?

An insured may have enough historical loss data on each of the different units to complete a separate actuarial report and loss projection for each unit individually. However, often an insured does not have unique loss data for each unit, as some units may have never experienced a claim or may be newly acquired. Even if they do, it can be inefficient and costly to analyze and project losses for each unit separately. In cases like this, an insured may choose to use a pure loss rating matrix to determine if the individual units have enough unique loss experience (credibility) to be rated individually and compared to the overall business average and to allocate total projected losses from the overall business to each unit.


Considering Historical Loss Experience and Credibility

A pure loss rating matrix considers whether an individual unit has enough historical loss experience to be deemed credible for rating. Often, this is determined based on how long each unit has been included as part of the loss program and whether historical loss and exposures specific to this unit are available. In the following exhibit incurred losses and payroll for company XYZ, Inc. have been summarized by policy year (1/1/2020, 1/1/2021, and 1/1/2022) and unit (A, B, and C).

XYZ, Inc.
Workers Compensation
Loss and Exposure Summary by Unit

Incurred Losses by Policy Year:

Unit 1/1/2020 1/1/2021 1/1/2022 Total
A $15,000 $100,000 $30,000 $145,000
B $90,000 $100,000 $85,000 $275,000
C 0 0 $50,000 $50,000
Total $105,000 $200,000 $165,000 $470,000


Payroll by Policy Year (in 100’s):

Unit 1/1/2020 1/1/2021 1/1/2022 Total
A $24,500 $25,500 $26,500 $76,500
B $23,500 $24,500 $25,500 $73,500
C 0 0 $26,000 $26,000
Total $48,000 $50,000 $78,000 $176,000


If a specific unit is deemed credible enough for rating, the pure loss rating matrix considers whether that unit has significant deviation in loss experience compared to the overall business. Any units for which the loss or exposure data is not deemed to have enough credibility for rating will be assigned the base loss rate, which is the loss rate at the expected level for the overall business.


The Impact of the Frequency and Severity of Losses

When considering the historical loss experience of each unit, there are two factors to consider:  the frequency and severity of losses. For example, suppose unit A experienced one claim during the 1/1/2021-22 period worth $100,000. Meanwhile, unit B experienced ten claims over that same period, each worth $10,000. In total, losses over this period are equal for unit A and unit B, but unit A’s losses are due to adverse development on an individual claim (severity), while unit B’s losses are due to development on several claims (frequency). The severity of claims can often be more volatile than frequency, so it can be common practice to limit individual losses to a specific threshold, such as $50,000, so that specific units are not overly punished for one accident which may have been abnormally severe. For company XYZ, Inc. individual losses will be limited to $50,000, and a summary of limited incurred losses by policy year and unit is included below.

XYZ, Inc.
Workers Compensation
Limited Loss Summary by Unit

Limited Incurred Losses by Policy Year:

Unit 1/1/2020 1/1/2021 1/1/2022 Total
A $15,000 $50,000 $30,000 $95,000
B $90,000 $100,000 $85,000 $275,000
C 0 0 $50,000 $50,000
Total $105,000 $150,000 $165,000 $420,000


Calculating Pure Loss Rates by Unit

To measure the loss experience of each unit compared to the overall business, losses for each unit are divided by their correlated exposures to determine pure loss rates by accident year. These pure loss rates by unit are then compared to the loss rate at the expected level for the overall business to determine if any unit has significant deviation in loss experience. Pure loss rates for each policy year for units A, B, and C are calculated below.

XYZ, Inc.
Workers Compensation
Pure Loss Rate by Unit
(Losses are Limited to $100,000 per Occurrence)

Pure Loss Rate by Policy Year (Per $100 Payroll):

Unit 1/1/2020 1/1/2021 1/1/2022 Total
A $0.88 $3.11 $2.61 $2.21
B $5.48 $6.49 $7.70 $6.56
C N/A N/A $4.44 $4.44
Total $3.13 $4.78 $4.89 $4.36


Assigning More or Less Favorable Loss Rates

If an individual unit is determined to have significantly lower loss experience than the overall business, that unit may be assigned a more favorable loss rate.  Conversely, an individual unit determined to have significantly higher loss experience than the overall business may be assigned a less favorable loss rate. The more and less favorable loss rates are determined based on the loss rate at the expected level for the overall business, as well as the loss rates at various confidence levels around the expected level.

For company XYZ, the credibility and loss experience has been used to determine an appropriate loss rate for each unit. The assigned loss rates are then applied to projected payroll and projected losses are obtained below.

XYZ, Inc.
Workers Compensation
2023 Projected Losses by Unit 
(Losses are Limited to $100,000 per Occurrence)

Unit Credibility to Loss Experience Selected Pure Loss Rate Projected Payroll (in 100’s)  Projected  Losses         Risk           Margin
A Yes $3.75 $27,500 $103,000 88%
B Yes $4.75 $26,500 $126,000 112%
C No $4.20 $27,000 $113,000 100%
Total N/A $4.25 $80,000 $340,000 100%


The creation and review of a pure loss rating matrix allows an insured to determine how it may charge or fund each unit of its business relative to the others and whether there are any segments of its business that have more or less optimal loss experience. Once an insured has determined which units of the business deviate from the overall expectations, it may be interested in analyzing what practices implemented at each of these units causes them to be more or less optimal than others and whether expansion or reduction of activity in these units would be beneficial from a total cost of risk perspective.

Utilizing a pure loss rating matrix can provide an insured with an insightful look at how each individual component of the business impacts funding of the insured’s liability and what effect future activity in each component could have on the business in total.

If you would like more information on utilizing a Pure Loss Rating Matrix, please contact us at support@SIGMAactuary.com or schedule a call with one of our consulting actuaries.

© SIGMA Actuarial Consulting Group, Inc.