1. Introduction
For traditional rate indications, premium and loss trend selections are of vital importance to the overall accuracy of the analysis. For premium trends, the analysis typically includes written or earned premiums at current rate level. Similarly for loss trends, calendar period paid losses are used to calculate annual estimates based on various periods in time.[1] Trends can have a significant contribution to the final indications, so understanding what is driving these trends can lead to understanding how the mix of business is impacting trends. Accurate premium and loss trend estimation is crucial for setting rates that reflect future costs. Errors in trend projections can lead to underpricing, which threatens solvency, or overpricing, which risks loss of market share. Univariate analyses on specific rating characteristics can be done, but these may only account for one or a few pieces of the puzzle and do not consider the broader scope of what makes up the underlying trends.
In most property and casualty insurance lines, a significant portion of the premium and loss trends could be driven by changes in the mix of business. However, determining which rating characteristics are the main culprits of these shifts and to what extent they are impacting trends in the business is not easy. Combined with the complexity and length of modern rate algorithms, this adds even more components to the estimation of premium and loss trends.
1.1. Objective
Traditional actuarially sound trending methods continue to serve their purpose, but the goal of this paper is to show how modern on-leveling calculations, like the extension of exposures (EE), can be used to improve these methods. With granular details about the individual policy premiums re-rated to current rates, information about how each average rating variable has changed over time and the impact those changes are having on earned and written premiums is available for use. Historical premiums and losses can then be adjusted to account for how the book of business has changed over time to better align with the current market and most recent target business. This new methodology improves the premium and loss trend selections by coverage or form, etc., with more accurate, detailed, and transparent factors to account for how a shifting book of business impacts the average premium and loss per policy over time.
1.2. Outline
The remainder of the paper proceeds as follows. Section 2 is a reference for the notation used throughout the paper, and Section 3 highlights some of the key foundational papers upon which this paper builds. In Section 4, the fictional data created to help illustrate the concepts in this paper is described. The primary analysis and outcomes are described in detail in Section 5, which focuses on premium adjustments due to shifts in the mix of business. In Section 6, the application of adjustment factors is expanded to include loss exposures. In Section 7, the premium and loss mix adjustments are summarized to examine their combined impact on rate indications. In Section 8, various practical considerations are discussed. Sections 9 and 10 summarize some conclusions and thoughts on potential future research.
2. Notation
The paper uses the following notation in the formulas:
rating factor for individual policyholder
exposures for individual policyholder
average premium rating factor during experience period
average of all premium rating factors during experience period
premium mix adjustment factor for rating factor during experience period
premium mix adjustment factor for all rating factors during experience period
underwriting adjustment factor for rating factor
average loss rating factor during experience period
average of all loss rating factors during experience period
loss mix adjustment factor for rating factor during experience period
loss mix adjustment factor for all rating factors during experience period
the number of rating variables.
the experience period (when).
the total number of experience periods.
the current or most recent experience period.
3. Background
Premium trend refers to changes in premium levels over time due to factors such as inflation, coverage modifications, and shifts in policyholder behavior. Accurate premium trend projections are essential in property and casualty ratemaking to ensure that rates adequately cover future losses and expenses. Challenges in accurately estimating premium trends often intersect with similar issues in loss trend analysis. Both are essential components of the ratemaking process, and errors in either can result in inadequate or excessive rates. Several authors discuss these challenges and propose solutions to address them.
The Actuarial Standard of Practice (ASOP) No. 13 (Actuarial Standards Board 1991), which focuses on trending procedures, underscores the importance of selecting appropriate methods and data for projecting trends in both premium and loss. It advises actuaries to consider internal factors (e.g., changes in underwriting practices) and external influences (e.g., economic conditions) when developing trend factors. ASOP 9 (Actuarial Standards Board 2009), which deals with documentation and disclosure, emphasizes that actuaries must clearly communicate the rationale behind their premium trend selections, ensuring transparency in the assumptions and methodologies used. Together these standards provide a framework for actuaries to conduct thorough and transparent trend analyses, highlighting the critical role that premium and loss trends play in the ratemaking process.
McClenahan (1988) and Werner and Modlin (2010) outline foundational ratemaking principles, including methods for adjusting premiums to current rate levels. These adjustments, based on historical data, form the basis for projecting future premium trends. Similarly, Weinman (1990) provides a detailed overview of premium trend analysis in homeowners insurance, emphasizing adjustments for inflation and coverage increases. Weinman discusses how inflation guard endorsements automatically increase coverage limits, requiring premium trend projections to accurately reflect future policy periods.
Jones (2002) reinforces these fundamentals by outlining both direct adjustment and trending methodologies for premium estimation and highlights the necessity of aligning premium and loss trend projections on consistent exposure bases. He stresses that improperly matching the trend basis between numerator (losses) and denominator (premium) in loss ratio analysis can lead to flawed rate indications. His two-step trending method offers additional precision in tracking shifts in written premium and projecting them into the future.
Inflation significantly impacts premium trends. Butsic (1981) explores how inflation influences both premiums and losses, emphasizing its leveraged effect on excess loss coverages. He warns that failure to align premium trend adjustments with inflationary pressures can lead to inadequate rates. Similarly, Ball and Staudt (2011) discuss the broader effects of inflation on financial outcomes in insurance, including its impact on premium and loss. They discuss how economic inflation directly affects claim costs, particularly in property and liability insurance, where repair costs and legal settlements are sensitive to price levels. They emphasize that traditional models often fail to capture the compounding effects of inflation over time, leading to underestimation of future losses.
Commodore (1994) addresses the limitations of the parallelogram method, a common technique for trending premium and loss data. He points out that the method assumes uniform policy distribution throughout the year, which may not hold true in practice. This assumption can lead to significant distortions in premium trend projections, especially in rapidly growing or shrinking portfolios. Commodore’s alternative “rectangle method” aims to provide more accurate results by correcting these distortions.
Styrsky (2005) focuses on the impact of changing exposure levels on calendar year trends, arguing that ignoring these shifts can result in misleading trend analyses. For example, if an insurer significantly expands its market share, the increased exposure levels can artificially inflate calendar year premium and loss trends. Styrsky proposes a data reorganization method to account for these exposure level changes, ensuring that trends reflect true underlying shifts rather than artifacts of exposure growth.
Feldblum (1988) adds another dimension by discussing how social inflation—changing legal environments and jury award patterns—introduces variability into premium and loss trends that traditional methods may not adequately capture. This variability, particularly in long-tail lines like liability or workers’ compensation, where these effects are most pronounced, complicates the selection of appropriate trend factors.
Several authors highlight challenges in accurately capturing premium trends, pointing out the limitations of traditional methods and their potential to misrepresent actual trends. McCarthy (2000) critiques traditional methods that often oversimplify trend estimation by assuming straight-line projections for both premium and loss trends. These methods fail to account for variability in historical data, such as shifts in claim frequencies and severities, policyholder behavior shifts, or the distribution of policy coverage levels. McCarthy emphasizes that overlooking such factors can lead to biased rate indications.
Each of these authors identifies specific challenges tied to data assumptions, statistical methods, and external influences. Together, they advocate for refined methodologies that account for these complexities, ensuring more accurate premium and loss trend estimation. It is for these reasons that the idea of mix of business adjustments is being approached, because it can provide a way to capture exposure shifts over time and calculate factors to more accurately align historic exposures with more recent experience.
4. Fictional Data
While it would be ideal to use real company data, this is hard to obtain due to privacy issues and the proprietary nature of the data. Fortunately, the mechanics of analyzing the impact mix of business shifts have on trends can be explored using fictional data. The fictional data includes three main components, the rating plan, policyholder information, and claim information, all of which is in the companion Excel file.
4.1. The Rating Plan
Real rating plans typically contain dozens of rating variables, which could be multiplicative and/or additive. Including this level of complexity in fictional data could make it more realistic, but only a few variables are needed to illustrate the concepts in this paper. Thus, the fictional rating plan includes a base rate, an additive fixed fee, and only five multiplicative rating factors. The first two rating variables are labeled “Territory” and “Limit/Amount,” as these are typical for many types of insurance. The other three rating variables have generic names.
For the Territory rating variable, there are eight different territories or rate codes (and factors) by territory. For the Limit/Amount rating variable, there are seven different limits or rate codes (and factors) by limit/amount. For Rating Groups 1, 2, and 3, there are five, eight, and 10 rate codes (and factors), respectively. The premium for any fictional policyholder is calculated by multiplying the base rate times all applicable rate factors and adding the fixed fee.
While fictional, multiple rate changes over a period of five years are included to keep the data as realistic as possible. Starting with the initial base rate, fixed fee, and relativities for the five rating variables, the timing and changes to the rating plan are shown in the Rates tab in the Excel file and are summarized in Table 4.1.
4.2. Policyholders
Typically, new policyholders are added throughout the year, and most of the existing policyholders will renew at the end of their policy term. In addition, some policyholders will change their policy coverages over time, e.g., for auto they may increase their limits of liability or replace a vehicle. All these changes over time, and others, will cause the mix of business to shift over time.
To keep it simple, and to mimic real shifts over time, the policy term is six months and the fictional data includes one policyholder per day. For the first six months, each new policyholder is assigned one rate code (or rating factor) for each of the five rating variables, selected at random based on predetermined “target” weights by code within each of the five rating variables. After six months, each policyholder will either randomly renew or be replaced by a new policyholder, based on a predetermined retention rate. For example, if the retention rate is 90%, then approximately 90% of the policies will randomly renew every six months and the rest will be new business. To keep the preliminary new policyholders from distorting any shifting impacts, the first six months is prior to the start of the five-year experience period ending 12/31/2024.
Moving forward through time, for each policy that non-renews and is replaced with a new policy, a rate code (or factor) for each of the five rating variables is selected at random based on the predetermined “target” weights by code within each of the five rating variables. For the renewal business, all the rate codes will remain the same as they were six months prior.[2] The predetermined “target” weights by code within each of the five rating variables can be different by year to mimic shifts in addition to random changes in the mix of business.
4.2.1. Scenarios
The predetermined “target” weights by relativity within each of the five rating variables are organized into seven different “scenarios” to more effectively isolate how changes to the mix of business over time will impact the trend calculations. The scenario weights by year are shown in the Scenarios tab in the Excel file and are summarized in Table 4.2.
Note that scenario 6 from Table 4.2 was used to calculate the rate change impacts shown in Table 4.1. Since the impact of a rate change is calculated using the current in-force book of business at the time of the rate change, the rate change impacts (in Table 4.1) will be different for other scenarios.
4.3. Premiums and Exposures
In the Policies tab of the Excel file, the policyholder characteristics are randomly selected for each policy over time, starting six months prior to the five-year experience period. Next, the base rate, fixed fee, and rating factors are looked up for the rate level in effect at that time and then the total written premium is calculated. In addition, the base rate, fixed fee, and rate factors are looked up for the most recent rate level so that the extension of exposures method can be used to calculate the total written premium at the current rate level. Finally, the annual exposure for each policy term is 0.5, and the earned exposures and earned premium by quarter are calculated for use in the analysis.
To illustrate how a policyholder could evolve over time, the Policy X tab allows the selection of any date within the first 6 months to see how that policy date either renews or becomes new business over the next five years. For example, consider the policy dates illustrated in Table 4.3, which is based on day 1 for scenario 1 discussed in Section 5.1.[3]
For the example presented in Table 4.3, the policy renews in nine of the 10 periods, and the rate codes are new in the new business period. While the rating codes don’t change for the other four rating variables, the Limit and Group 2 factors do change because of changes in the rating plan. The key takeaway from Table 4.3 is in the last column, which shows the premiums recalculated at the most recent rate level, i.e., level 10. In the written premium at current rate level column, the “trend” from this one policy date is clearly being impacted as a result of the shift in the mix of business by Territory.
4.4. Growth
In the Growth tab of the Excel file, new policyholders are randomly added based on the annual growth rate scenario. For example, if a 10% growth rate is assumed, then approximately 36 new policies would be randomly added the first year.[4] Once a new policy is added for a given date, it will either renew or become new business every six months following the same random processes noted in Section 4.2.
Including policy growth in the fictional data adds a level of reality and allows for acceleration of the shift in the mix of business over time. In practice, significant exposure growth can lead to more significant impacts from the shift in the mix of business. More importantly, significant shifting can often be an underlying driver of growth, intentionally or unintentionally.
4.5. Claims
In both the Policies and Growth tabs of the Excel file, the policyholder characteristics are used to randomly simulate incurred claim values for every policy. Of course, it is not realistic to assume every policyholder will have a claim but given the limited size of the policyholders in the fictional data, using all policyholder data to generate claim data helps illustrate how a shift in the mix of business will also have an impact on the incurred claim costs over time. In practice there may be separate frequency and severity impacts, but this paper will only focus on pure premium impacts.[5]
To calculate each incurred claim, the starting point is the average Claim Severity, Coefficient of Variation (CoV), and Annual Inflation assumptions in the Scenarios sheet.[6] For each policy, a policy adjustment is calculated based on the product of all current rating factors.[7] The policy and inflation factors are used to adjust the expected mean of the claim value, then the lognormal distribution and CoV are used to randomly simulate a claim value. The impact of loss development is ignored, and all claim values are assumed to be at ultimate.
5. Premium Mix Adjustment
To illustrate the impact of using mix adjustment factors, scenarios ranging from a simple example where mix shifting occurs in one rating variable to more complex examples with shifting across all rating variables in the rating plan are used, as shown in Table 4.2. The goal for calculating a premium mix adjustment factor is to measure how the shift in exposures over time has impacted the earned premiums at current rate level. Thus, a comparison of the current mix to the mix during different parts of the overall experience period can be used to estimate premium trends related to the shift.[8]
5.1. Single Exposure Shift
Scenario 1 is a simple case where only the Territory is experiencing a shift in exposure and the remaining four rating variables are held constant. To simplify it even more, all codes will begin at the base factors (i.e., 1.00) for each rating code. Then, over time some policyholders will shift from territory 1 into other territories with higher or lower factors. In practice, this could represent a shift in the book of business from rural to more urban areas within a state or vice versa.
To calculate the average rating factor for a given experience period, it would be tempting to use the earned premiums as the exposure base, but even from the simple example in Table 4.3 both the earned premiums and the earned premiums at the current rate level are being distorted by the shift in rating factors. Thus, the thing that has been distorted cannot be used to measure how it is being distorted. A better exposure base would be the number of policies in force or the exposures earned during the experience period.[9] For each rating variable, the average rating factor in the experience period, can be calculated using formula (5.1).
\[F_{w}^{n} = \left\lbrack \sum_{All\ x\ in\ w}^{}f_{x}^{n} \times e_{x} \right\rbrack \div \sum_{All\ x\ in\ w}^{}e_{x} \tag{5.1}\]
In addition to, or as part of, the calculation of average rating variables by experience period, the averages for the most recent experience period would also be calculated. In the example data, since six-month policy terms are used the current average would consist of the most recent six months of policy in-force data. For annual experience periods, the current average would be different from the annual averages because of the six-month policy term. For all shorter experience periods, e.g., semi-annual, quarterly, or monthly, the current average should be the same. For experience periods shorter than six months, six months of in-force policies needs to be included because they would all be contributing to the earned premiums during those exposure periods.
Using the average rating factors by experience period, and the current average rating factor, the premium mix adjustment factors, for rating variable can be calculated using formula (5.2).[10]
\[P_{w}^{n} = F_{C}^{n} \div F_{w}^{n}\text{, for each experience period } w \tag{5.2}\]
For example, continuing with scenario 1, Table 5.1 illustrates the average rating factors by year, the current average rating factor, and the premium mix adjustment factors by year for the Territory rating variable. From Table 5.1, the average Territory factor increased from 1.0500 in 2020 to the current average of 1.1733, so to compensate for this shift adjustment factors ranging from 1.1174 to 1.0097 would need to be applied.
Even before these adjustment factors are used in the indications, the fact that the average has changed significantly is important diagnostic information. For example, did the company target specific territories in their marketing plan and is that what is driving this change? Would this shift be expected to impact the losses to the same extent (discussed in Section 6)? Answering these questions requires a deeper understanding and analysis, but it draws attention to the issue.
Table 5.2 shows, using the adjustment factors from Table 5.1, how this shift in business by Territory would impact the trend calculations. The details for Table 5.2 are from the Trends tab in the Excel file. From rows 2) and 3) in Table 5.2, the earned premiums grew significantly during the experience period, mostly due to the rate change history, and the earned premiums at current rate level do a good job of adjusting for the rate changes. For comparison, the earned premiums at current rate level (CRL) using the parallelogram factor (PF) method (rows 5 to 7) are also shown, but these calculations are ignored in the rest of the paper.[11]
While not generally used in practice unless there is no way to adjust for rate changes, the calculated trend for the earned premiums is 12.2%. Essentially, adjusting the premiums to the current rate level “explains” most of this 12.2% trend and reduces the “unexplained” trend to 2.3% in this example. By adjusting the earned premium at current rate level by the premium mix adjustment factors, the rest of the premium trend is “explained,” and the resulting trend is close to zero. In Table 5.3, this example is expanded to examine how these results would be used in a rate indication.
For comparison, the trend factors for the earned premiums are shown in rows 13) and 14), but these will be excluded in the rest of the paper. Typically, trend factors from the earned premiums at current rate level would be used, as shown in rows 15) and 16). Alternatively, separate factors for shifts in the mix of business could be used with the “net” trend after adjusting for the shift using the premium mix adjustment factors, as shown in rows 17) and 18) to produce row 19). The interesting takeaway from this comparison is that the combination of these two factors results in a nearly identical overall trend, except that the annualized trends by year now better represent the shift in mix that took place, as shown in rows 20) and 21). In other words, the combined factors in row 20) represent a consistent but more accurate trend by year in the earned premiums at current rate level.
5.2. Dual Exposure Shift
Scenario 2 continues with a slightly more complex case where Limit/Amount is experiencing a shift in exposure, in addition to the same shift experienced for Territory, and the remaining three rating variables are held constant. Over time some policyholders will shift from the basic limit into other limits with higher or lower factors. In practice, this could represent a shift in the book of business from lower to higher limits of liability, or amounts insured, within a state.
For example, consider the policy dates illustrated in Table 5.4, which is based on day 1 for scenario 2. Comparing Table 5.4 to Table 4.3, for this policy date both the territory and limit codes change for the one new business date. In the written premium at current rate level column, note that the “trend” from this one policy is impacted further as a result of the shift in the mix of business by Territory and Limit.
For scenario 2, Table 5.5 illustrates the average rating factors by year, the current average rating factor, and the premium mix adjustment factors by year for the Limit/Amount rating variable. From Table 5.5, the average Limit/Amount factor increased from 0.8800 in 2020 to the current average of 1.0296, so adjustment factors ranging from 1.1700 to 1.0124 need to be applied to compensate for this shift. Note that the average factors are below 1.00 in the earlier years because of the base limit change in rate level 4.[12]
Before examining the impact of this shift in Limit/Amount on the trends, note that the Limit/Amount impact is also affected by the shift in Territory, and statistically these two shifts are independent. Because the rating factors are multiplicative in the rating plan, the solution for combining the impact of multiple rating variables is to multiply all of the average factors by period, to get an average for all rating factors, as shown in formula (5.3). Like formula (5.2), the overall average rating factors by experience period, and the current overall average rating factor, can be used to calculate the premium adjustment factors for all rating variables combined, using formula (5.4).
\[F_{w} = \prod_{n}^{}F_{w}^{n} \tag{5.3}\]
\[P_{w} = F_{C} \div F_{w} \tag{5.4}\]
Table 5.6 summarizes combining the Territory and Limit/Amount averages factors using formulas (5.3) and (5.4).[13]
Table 5.7 examines, using the adjustment factors from Table 5.6, how this shift in business by Territory and Limit/Amount impacts the trend calculations. From rows 2) and 3) in Table 5.7, the earned premiums grew more significantly during the experience period than in Table 5.2, which is expected because the impact from both shifts were positive, and the earned premiums at current rate level continue doing a good job of adjusting for the rate changes. For comparison, the overall trend in the earned premium at current rate level increased from 2.3% in Table 5.2 to 6.2% in Table 5.7. As in scenario 1, the earned premium at current rate level times the premium mix adjustment factors results in a trend close to zero. In Table 5.8, this example continues to examine how these results would be used in a rate indication.
Typically, trend factors from the earned premiums at current rate level would be used, as shown in rows 15) and 16). Alternatively, separate factors for the shifts in the mix of business are used with the “net” trend after adjusting for the shift using the premium mix adjustment factors, as shown in rows 17) to 19). Even with the dual shifting in scenario 2, the combination of these two factors results in a similar overall trend, except that the annualized trends by year now better represent the shifting in mix that took place, as shown in rows 20) and 21).
5.3. All Exposures Shift
Continuing the process of adding more rating variables in scenarios 3, 4, and 5 is instructive, but a detailed analysis is left to the reader and the related exhibits for scenarios 3, 4, and 5 are shown in Appendices A, B, and C, respectively.[14] By reviewing the exhibits in the Appendices, notice that as additional rating variables are included, the trend in the earned premiums at current rate level continues to increase, yet the “net” trend after adjusting for the premium mix of business factors continues to be close to zero. In practice, not all rating variables will have increasing average factors, so there will be offsets from the rating variables that have decreasing average factors, and the overall earned premium at current rate level trend may not be as extreme as in these examples.
Scenarios 6 and 7 start with all rating variables in the fictional rating plan using equal “target” weights for all rating codes by rating variable. This means every rate code is equally likely for new business in the first six-month period. For scenario 6, the equal weights assumption continues throughout the five-year experience period, so changing the renewal rate assumption is the only way to allow the mix to randomly change over time. For scenario 7, the target weights change over time, so the mix will also shift based on the changing weights.[15]
Starting with scenario 6 and a 75% retention assumption, Table 5.9 shows the average premium factors and premium mix adjustment factors for each experience period. In practice, this represents an ideal situation where all rate factors are perfect and random changes in policyholders still cause some changes in the mix of business.
Using the adjustment factors from Table 5.9, how this change in the mix of business would impact the premium trend calculations can be examined in Table 5.10. In this scenario, the earned premiums at current rate level trend is close to zero, as might be expected from only random changes to a constant target mix of business. After including the premium mix adjustment factors, the calculated trend moves slightly closer to zero. In Table 5.11, this example continues to examine how these results would be used in a rate indication.
After combining the premium mix adjustment factors with the “net” premium trend, the implied combined trend moves further away from zero. However, this illustrates a nuanced benefit of using the mix adjustment factors, meaning that the mix factors provide a way to use variable trends instead of constant trends based on what is changing in the mix of business. In other words, without the mix of business factors the “real” trends by year can be partially “hidden” by looking at an overall five-year trend analysis. Alternatively, at the extreme the shift could be smooth, and the resulting trends would then still be constant, but without a detailed analysis there is no way to know if the shift is smooth or not.
Finally for scenario 7 and a 75% retention assumption, Table 5.12 shows the average premium factors and premium mix adjustment factors for each experience period. In practice, this moves away from the ideal situation where rate factors are not perfect and marketing, competition, adverse selection, etc., will cause the mix of business to shift.
Table 5.13 shows, using the adjustment factors from Table 5.12, how this change in the mix of business would impact the premium trend calculations.
In this scenario, the earned premiums at current rate level trend is positive. After including the premium mix adjustment factors, the calculated trend is closer to zero. In Table 5.14, this example continues to examine how these results would be used in a rate indication. After combining the premium mix adjustment factors with the “net” premium trend, the implied combined trend is very close to zero. Again, the benefit of using variable trends instead of constant trends based on what is changing in the mix of business is clearly illustrated.
5.4. Simulations
Since all the tables in the paper thus far are based on a single set of random values, it is possible that this could be a biased view of the impact of using premium mix of business factors. To test this idea, the Excel file includes the ability to test up to 1,000 iterations of different random values and to examine the results from any iteration. Table 5.15 summarizes the results for 1,000 iterations for scenario 7.[16]
Comparing Table 5.14 to the results in Table 5.15, it seems that earned premium at current rate level trend of 1.5% in Table 5.14 is below the average of 2.66% in Table 5.15. Similarly, the mix adjusted trend of –0.6% and implied annual trend of 0.3% are both below the averages of –0.19% and 1.13%, respectively. An interesting observation from Table 5.15 is that the standard deviation of the premium mix adjusted trends is less than half of the standard deviation of the unadjusted trends, whereas the standard deviation of the implied annual trends is nearly double. This supports the idea that the use of combined mix adjustment and “net” trend factors results in an improved trend that varies by year.
5.5. Growth
As noted in Section 4.4, adding policy growth to the data can be used to see how this will impact using mix adjustment factors. To maximize the impact of growth on shifting, an annual growth rate of 30% was assumed before rerunning the simulations. Table 5.16 summarizes the results for 1,000 iterations.[17]
Comparing Table 5.16 to Table 5.15, all the averages and standard deviations went down. This implies that the target mix percentages set for scenario 7 only marginally increased the average rating factor. In practice, it is possible to see double digit trends for average earned premiums at current rate level that drop to zero after adjusting for the premium mix of business factors.
6. Loss Mix Adjustment
If it can be assumed that the rating plan is 100% accurate, then there is a direct relationship between each rating factor and its corresponding exposure to loss. To better highlight this, coverage limit can be used as an example. If a policyholder chooses a higher loss limit, they will typically be charged a higher premium for this additional expected loss potential. Thus, for losses it could be assumed that all rating variables impact the exposure to loss in the same way the premium was impacted, i.e., all rating factors are 100% accurate in their predictive power. For example, if the coverage limit factors are accurate, then it would make sense to assume that a shift in the book of business from lower to higher limits is impacting the exposure to losses in the same way it is impacting the earned premiums.
In practice, however, there are various reasons why a measured shift in earned premiums may not be perfectly balanced to the loss exposures, some of which are noted below. Overall, since no rating plan is perfect, it seems logical to expect loss mix adjustments to differ from premium mix adjustments in aggregate. The degree to which loss and premium mix adjustments differ would depend on the imbalance in the rating plan.
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Some rating variables may not be intended to reflect exposures to loss, such as payment option type or advance purchase options.
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As part of a rating plan transition, the factor changes for some rating variables may be constrained to prevent large swings in policy premiums for the group(s) of policyholders most impacted by those rating variable changes.
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Some state regulators may prevent rating factors from matching the exposure to loss due to reasons such as affordability or other regulatory constraints.
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Competitive pressures may constrain rate factor changes or encourage factors negatively correlated to loss, such as new business discount rating variables.
To adjust the mix of business factors for use with loss exposures, an underwriting adjustment factor, can be added to the premium factor calculations for each rating variable. These underwriting adjustment factors would typically be between zero and one, but in theory they could be greater than one for a rating variable that is underpredicting exposure to losses. In practice, most underwriting adjustment factors would be selected judgmentally, but discussions with the factor modeling team could help refine the selections.[18] The formulas for the loss mix adjustment factors, (6.1) to (6.4), correspond to the premium mix adjustment factors, (5.1) to (5.4), shown above.
\[G_{w}^{n} = 1 + \left\lbrack (F_{w}^{n} - 1) \times U^{n} \right\rbrack\tag{6.1}\]
\[L_{w}^{n} = G_{C}^{n} \div G_{w}^{n}\text{, for each experience period } w \tag{6.2}\]
\[G_{w} = \prod_{n}^{}G_{w}^{n} \tag{6.3}\]
\[L_{w} = G_{C}\ \div \ G_{w} \tag{6.4}\]
Expanding scenario 5, shown in Appendix C, can be used to illustrate how underwriting factors impact the loss mix adjustment factors. For example, if the Group 1 premium factors were assumed to only have a 50% correlation to loss exposure, then the resulting loss mix adjustment factors are shown in Table 6.1. Choosing 50% is an arbitrary choice. This is simply to illustrate that actuarial judgment can be used to reduce the impact of the mix adjustment factors for specified rating variables if they are not expected to have the same impact on losses as they do premium. For example, when variables (like limit factors) are isolated, it could be assumed a change in purchased limit would have a similar effect on both the premiums and losses. However, for rating variables that are not intended to reflect exposures to loss, the underwriting factor could be set to 0%.[19] Selecting underwriting factors for other types of variables noted above would involve deeper discussions and analysis with the modeling team. The significance or sensitivity of each underwriting factor will depend on the premium mix adjustment factor for each variable compared to the other variables.
Comparing Table 6.1 to Table C.3, the Group 1 factors were reduced by 50%, which in turn reduced the overall factors. Using the simulated loss data and the adjustment factors from Table 6.1, how this shift in business would impact the loss trend calculations is shown in Table 6.2. The details for Table 6.2 are from the Trends tab in the Excel file.
From Table 6.2, the average pure premium grew significantly during the experience period. But how much of the 18.8% trend is being driven by the shift in the mix of business? After adjustment for the loss mix adjustment factors, the “net” loss trend drops to 5.4%, which is quite close to the 5.0% inflation parameter. Like the premium trends, trend factors from the average pure premium would normally be used, as shown in rows 6) and 7). Alternatively, separate factors for the shifts in the mix of business are used with the “net” trend after adjusting for the shift using the loss mix adjustment factors, as shown in rows 8) to 10). The combination of these two factors results in a similar overall loss trend, except that the annualized trends by year now better represent the shifting in mix that took place, as shown in rows 11) and 12).
In practice, the use of loss mix adjustment factors helps reduce, or explain, some of the loss trends, and combining the loss mix adjustment factors with the annual loss trend factors will have the impact of moving toward loss trends that vary by period. Like premiums, the combined loss mix adjustment and trend factors could result in a higher or lower implied annual trend. For loss trends, the “net” trend after mix adjustment is still subject to other considerations, such as economic or social inflation, benefit changes, law changes, etc., but the ability to analyze those issues should be improved.
7. Rate Indications
Now that the impact a shifting mix of business has on both premiums and losses has been reviewed, the parts can be combined and the total impact on rate indications can be examined. Starting with a sample rate indication using Tables C.4 and 6.2, the overall rate indication without any mix adjustment factors is shown in Table 7.1.
For comparison, Table 7.2 combines Tables C.5 and 6.2 so the overall rate indication with mix adjustment factors is shown. At first glance, it appears that a difference of only 1.0% (4.1% vs. 3.1%) may not be worth all the extra effort.[20] However, a deeper look shows that the projected premiums and losses are quite different and, to the extent these are part of a financial planning or similar process, the differences can be quite significant. It also ignores the potential value of early warning insights gained during the analysis.
Another reason that these sample indications are so close is the use of well-behaved data, meaning using a known distribution to simulate data, instead of real data that would include frequency in addition to severity changes. In practice, much more significant differences after including mix of business adjustments can be observed. Comparing Tables 7.1 and 7.2 also helps examine the idea of potential future mix shifting. Table 7.2 essentially assumes that there will either be no more shifting in the future or that all shifts in premium will be exactly matched by shifts in losses. By moving the projection assumptions toward the selections in Table 7.1, the results in Table 7.2 could then reflect some judgment about future mix changes and any imbalances in the rating factors.
8. Practical Considerations
Fictional data allowed the benefits of using mix adjustment factors to be illustrated, but in practice there are some limitations and potential constraints. First, the process of calculating the mix adjustment factors requires detailed information on all policies so that the extension of exposures method can be used to calculate the earned premium at current rate level. When the extension of exposures method cannot be used, mix adjustment factors cannot be used either.
Second, more complex rating algorithms may include additive rating variables in combination with a generally multiplicative rate plan. For some additive variables, such as the additive fixed fee in the fictional plan, the amount is constant across all policy premiums at current rate level, and it does not directly impact the trends. However, suppose there are additive features at multiple points in the plan. One way to approach this is to combine the additive factors into a multiplicative rating variable. For example, multiple driver factors could be combined into a household factor. Another option is to modify formula (5.3) to include additive elements.
A related question is how many variables to include in a mix of business analysis. In practice, combining variables as noted above may be necessary, but the goal should be to include all variables in the rating plan. This ensures that all explanatory variables are used, even if some have very little impact, and it will result in a robust early warning system.
Even when the rating algorithm is mostly multiplicative, the net trend after the premium mix adjustment factors may not get as close to zero as preferred. This could be caused by additional premium elements like endorsements and insured value adjustments to the extent that they are not part of the rating algorithm.[21] Similarly for auto physical damage coverages, if the rating plan uses age instead of model year, then as vehicles change from one age group to another, even without any change to the age factors, it could still impact the net trends. Throughout the examples, trends nearing zero were noted because the mix of business factors were used, but the underlying goal of this process is not necessarily to get to zero but to explain as much as possible about how the change in mix of business is impacting the analysis and profitability.
The examples shown in this paper only used annual data, but in practice trends are often estimated using semi-annual, quarterly, or monthly data. The semi-annual and quarterly examples for the fictional data are also included in the Excel file. While the examples are generic, in practice the mix adjustment factors would need to be applied at a granular level corresponding to the rate indication analysis, such as by coverage for auto or by form for home. Focus on a granular level is particularly important because some rating variables may not apply, such as limit of liability only applying to liability coverages, and each segment of data may require different rating variables.
Throughout this paper, the mix of business adjustments are calculated using the average rating factors of the most recent period. In essence this assumes that future business will be the same as the most recent mix of business, but what if the future mix is expected to change? In theory, if the rating factors are all correctly adjusted for the exposure changes by variable, then any shift in the future mix of business would impact premiums and losses consistently. For example, if the only future shift is a change in the average limit of liability, then the measured future shift in earned premiums at current rate level should also change the average loss by the same percentage. In other words, this is shifting the “final” mix from one point in time to another point in time but there should be no overall indication change since the trend selections are already “net” of the impact of shifting and the indications would be insulated from any future mix changes.
If instead some future mix of business is assumed, it could be allowing assumptions of this future mix to cloud the accuracy of the calculation. This may be adding more uncertainty to an already uncertain projection, and the extra calculations are likely not worth the effort. This is because a correct estimate of the future mix of business would have a minor impact compared with simply estimating a future trend, and an incorrect future mix estimate could lead to inaccurate rate indications.
Any difference between the most recent mix of business and a future assumed mix would be largely negligible as it most likely offsets itself in the ratio of premium mix adjustment factors to loss mix adjustment factors that were covered in Section 6. However, if there are known rating variable imbalances (meaning there is not a one-to-one relationship between the rating factors and their corresponding exposure to loss), then this could require estimating what the future mix of business would be for each rating variable in order to calculate the average rating factors of this future mix.[22]
Another nuance to this issue includes the selection of future projections compared to the historical trends. For example, if the premium trend prior to a mix adjustment is 3.0% and after the mix adjustment it is 0.5%, what future projection assumption should be used? Using a projection of 0.5% assumes that either all shifting is done or the future shifting in the premiums will be exactly offset by the shifting impact on the future losses. This returns the discussion to whether there is an imbalance in the rating factors. If the rating factors are in balance, then the 0.5% assumption can be used. But if they are not in balance, then either the premium or loss projection assumption could be adjusted (depending on the direction of the imbalance) in lieu of a more sophisticated analysis.
9. Conclusions
This paper was written to show that adjustment factors can be identified and calculated based on how a book of exposures changes over the experience period. Using the extension of exposures method, granular data can be used to calculate unique factors that more closely align past experience to current exposures. Instead of relying on a single trend factor across the experience period to adjust earned premium or incurred losses, each period more accurately accounts for trend based on the average exposure within the period. This not only provides additional insight into the change of the mix over time but also allows for more accurate pricing for future exposures.
A nuanced benefit of using the mix adjustment factors is that it provides a way to use variable trends instead of constant trends based on what is changing in the mix of business. Without the mix of business factors, the real trends by year can be partially hidden by looking at an overall trend analysis and applying constant trend factors.
In principle, this method is most effective when a book of business is experiencing significantly large premium trends and/or loss trends resulting from changes in mix of business rather than from external causes (e.g., inflation, changing legal environment, etc.). This adjustment can help explain a large portion of that trend and reduce the need for large trend selections. When traditional trend calculations result in small or minor trends in premium and loss, the mix of business will have a less significant impact, but it is nonetheless still useful.
A corollary benefit to analyzing all the rating variables is that a shift may alert the company to an unexpected adverse selection concern that could indicate the rating factor(s) in question are not completely aligned to exposure to loss. For example, if we convert Table 5.12 for scenario 7 to show the premium adjustment factors by rating variable in Table 9.1, diagnostically the shift in Group 1 had the most impact on the overall shift in 2020–21.[23]
10. Suggested Future Research
Research for this method and the description of its applications have been focused primarily on personal lines. Commercial lines present unique challenges due to the individualized risk focus of the business and reliance on underwriters to accurately price policies. Addressing these and other complexities within commercial lines would be a good topic for future research.
Modern rating plans are typically based on multiplicative rating variables, and while additive rating variables are used, this has not been the focus of this paper. Additional research can be done to identify the challenges associated with and need to modify this method to incorporate additive features.
The approach to adopting the premium mix adjustment factors to losses relies on pure premiums and actuarial judgment. Additional research on using rate factor modeling data to refine this process and using factors based on frequency and severity data could improve the loss mix adjustment factors.
Abbreviations and notations
Collected here in alphabetical order are all abbreviations and notations used in the paper
CoV, coefficient of variation
CRL, current rate level
EE, extension of exposures
PF, parallelogram factors
Acknowledgment
The authors gratefully acknowledge the many authors listed in the References and Selected Bibliography sections (and others not listed) who contributed to the foundation of actuarial pricing and trend models. The authors note that most of the Background section was written using ChatGPT to summarize the papers in the References section. We would like to thank all the peer reviewers, in particular April Truebe, Min Gu Lee, Valerie Albers, William Prucknic, and the members of the Ratemaking committee, for their many comments that greatly improved the quality of this paper.
Supplementary Material
Mix of Business Examples.xlsm – This file contains the fictional data used in this paper, as well as various scenarios used to explore how shifts in the mix of business impacts the trends. All tables in the paper can be replicated in this file.
Biographies of the Authors
Mark R. Shapland is an actuary at Applied Underwriters. He has a B.S. degree in integrated studies (actuarial science) from the University of Nebraska-Lincoln. He is a Fellow of the Casualty Actuarial Society, a Fellow of the Society of Actuaries, and a Member of the American Academy of Actuaries. He previously served as a member of the CAS Board of Directors and as the chair of the Audit Committee. He was the leader of Section 3 of the Reserve Variability Working Party, the chair of the CAS Committee on Reserves, co-chair of the Tail Factor Working Party, and co-chair of the Loss Simulation Model Working Party, as well as serving on numerous other CAS committees and task forces. He is also a co-developer and co-presenter of the CAS Reserve Variability Limited Attendance Seminar, and the European Actuarial Academy’s Stochastic Modelling Seminar and is a frequent speaker both within the CAS and internationally. He can be contacted at mrshapland@netzero.com.
Trevor M. Parish is a senior actuarial analyst at the Auto Club Group. He has a B.S. in statistics from Grand Valley State University and an M.A. degree in actuarial science from Ball State University. He can be contacted at tparish89@gmail.com.