Method to Include New Business When Measuring the Rate Change of an Excess Casualty Insurance Portfolio

1. Introduction

Measured rate changes are a significant factor in (re)insurance experience rating. Experience rating is based on restating historical losses and premiums. Historical losses are brought to prospective levels using trend and development. Historical premiums are brought to prospective levels using historical rate changes and on-level adjustments. This makes calculating an accurate rate change essential for reinsurance pricing.

2. Purpose

Traditional rate change calculations are often based on data from renewal policies, as discussed by Bodoff (2009). The rate calculations work on the assumption that the insurer has information about prior-year rates in addition to prior-year attachments and limits. This prior-year information is used to understand changes in loss potential and calculate rate changes policy by policy. However, renewal policies constitute only a portion of the total portfolio, with the remainder attributable to new business. New business may come in at significantly different rate levels than renewal business, so it is important to include new business in rate change calculations.

To calculate rate change inclusive of new business, the insurer must contemplate the total portfolio. Yet new policies do not provide the same prior-year information that renewal policies carry, meaning that new business rate change cannot be calculated by directly matching current policies to prior-year policies, as is done with renewal business. Thus, rate change on new business is difficult to quantify.

3. Motivation

The question of how to measure rate change inclusive of new business arises simply because there is no information about prior-year rates for new business. One possible solution is to measure the change in rate adequacy, where rate adequacy is defined in terms of actual premium and technical premium.

Let A(t) = actual premium for the total portfolio (new and renewal business) at time t.

Let T(t) = technical premium, obtained from an internal pricing or benchmarking tool for the total portfolio at time t.

Then

$$\text{Rate Adequacy}(t) = \frac{\text{A}(t)}{\text{T}(t)}\text{,} \tag{3.1}$$

$$\text{Rate Adequacy}(t + 1) = \frac{\text{A}(t + 1)}{\text{T}(t + 1)}\text{,} \tag{3.2}$$

and

$$\begin{align} \text{Rate Change}(t + 1) &= \left\lbrack \frac{\text{Rate Adequacy}(t + 1)}{\text{Rate Adequacy}(t)} \right\rbrack - 1 \\ &= \frac{\left\lbrack \frac{\text{A}(t + 1)}{\text{T}(t + 1)} \right\rbrack}{\left\lbrack \frac{\text{A}(t)}{\text{T}(t)} \right\rbrack} - 1\text{.} \tag{3.3} \end{align}$$

This approach works if a technical premium inclusive of new business is available from a pricing tool. However, such a premium is not commonly available in US reinsurance pricing scenarios. Often, only the actual premium is available, so we propose a method that will work in this scenario.

4. Proposed new method: Theory

4.1. Interconnectivity of rate level and rate change

We designate the rate level for year t as Rate Level(t).

We designate the rate level for year t + 1 as Rate Level(t + 1).

We then can derive the rate change for year t + 1 using the fundamental theorem of rate change:

$$\text{Rate Change}(t + 1) = \left\lbrack \frac{\text{Rate Level}(t + 1)}{\text{Rate Level}(t)} \right\rbrack - 1 \text{.} \tag{4.1a}$$

Provided the rate change for year t + 1 and the rate level for year t, we could derive the rate level for year t + 1 by rearranging Equation 4.1a as follows:

$$\small \text{Rate Level}(t + 1) = \text{Rate Level}(t)*\left\lbrack 1 + \text{Rate Change}(t + 1) \right\rbrack \text{.} \tag{4.1b}$$

Traditional renewal-to-renewal calculations use rate change to derive rate level, in line with Equation 4.1b. When measuring new business, we cannot start off with rate change because we do not have prior-year rates. However, using current information, we can measure the rate level of new business relative to that of renewal business. That is, we can measure the rate level of new business and then weight it with the rate level for renewal business to arrive at a total portfolio rate level. From there, we can calculate the total portfolio rate change. The idea of measuring the rate level for new business instead of rate change motivates the process described in the next section.

4.2. Applying the interconnectivity of rate level and rate change to our problem

Let the analysis begin with year t = 1.

Define the renewal rate level for year t = 1 as

$$\text{Renewal Rate Level}(1) = 1.0 \text{.} \tag{4.2a}$$

Designate the renewal rate change for year 2 (the rate change from year 1 to year 2) as Renewal Rate Change(2). Then the renewal rate level for year t = 2 is

$$\begin{gathered} \text{Renewal Rate Level}(2) = \\ \text{Renewal Rate Level}(1)*\left\lbrack 1 + \text{Renewal Rate Change}(2) \right\rbrack \text{.} \end{gathered} \tag{4.2b}$$

To derive a rate level for new business from this renewal rate level, we need to quantify the rate level of new business relative to the rate level of renewal business. This relativity is intended to measure the relationship of new business rate levels to renewal business rate levels. There may be multiple ways to measure this relationship. The approximation that we employ here defines the relative rate level as the ratio of new business rate per million (RPM) to renewal business RPM, where RPM is the total premium divided by the total policy limit in millions of dollars for a given year. Although the relativity of RPMs is not a perfect way to describe the relative rate levels of new business and renewal business, it is a reasonable approximation. In addition, in Section 6.1 we will discuss some specific quantitative adjustments to improve the RPM approximation to overcome known potential distortions, for example, when new business and renewal business have different limit and attachment profiles.

Let the RPM for renewal business for year 2 be designated as Renewal RPM(2) and the RPM for new business for year 2 be designated as New Business RPM(2). Then we can approximate the relativity for year 2 as follows:

$$\begin{gathered} \text{Relativity}(2) = \\ \frac{\text{New Business Rate Level}(2)}{\text{Renewal Rate Level}(2)} \approx \frac{\text{New Business RPM}(2)}{\text{Renewal RPM}(2)} \text{.} \end{gathered} \tag{4.2c}$$

We can derive the new business rate level for year 2 as

$$\scriptsize \text{New Business Rate Level}(2) = \text{Renewal Rate Level}(2)*\text{Relativity}(2)\text{.} \tag{4.2d}$$

Now that we have a rate level for both renewal and new business, we can weight them together to derive the total portfolio rate change.

Designate the weight for renewal business for year 2 as Renewal Weight(2) and the weight for new business for year 2 as New Business Weight(2). Then the total portfolio rate level for year 2 is

$$\begin{gathered} \text{Total Rate Level}(2) = \\ \text{Renewal Rate Level}(2) * \text{Renewal Weight}(2) + \\ \text{New Business Rate Level}(2) * \text{New Business Weight}(2) \text{.} \end{gathered} \tag{4.2e}$$

Note that for year 1, the total portfolio rate level is Total Rate Level(1) = 1.0.

The total portfolio rate change for year 2 is then

$$\text{Total Rate Change}(2) = \left\lbrack \frac{\text{Total Rate Level}(2)}{\text{Total Rate Level}(1)} \right\rbrack - 1 \text{.} \tag{4.2f}$$

For year 3, the renewal rate level applies the renewal rate change for year 3 to the total portfolio rate level for year 2:

$$\begin{gathered} \text{Renewal Rate Level}(3) = \\ \text{Total Rate Level}(2)*\left\lbrack 1 + \text{Renewal Rate Change}(3) \right\rbrack \text{.} \end{gathered} \tag{4.2g}$$

This is because every policy that was new in year 2 is now a renewal policy in year 3.

From here, the same steps can be followed to derive the total portfolio rate level and rate change for year 3:

$$\scriptsize \text{New Business Rate Level}(3) = \text{Renewal Rate Level}(3)*\text{Relativity}(3) \text{.} \tag{4.2h}$$

$$\begin{gathered}\text{Total Rate Level}(3) = \\ \text{Renewal Rate Level}(3) * \text{Renewal Weight}(3) + \\ \text{New Business Rate Level}(3) * \text{New Business Weight}(3) \text{.} \end{gathered} \tag{4.2i}$$

$$\text{Total Rate Change}(3) = \left\lbrack \frac{\text{Total Rate Level}(3)}{\text{Total Rate Level}(2)} \right\rbrack - 1 \text{.} \tag{4.2j}$$

These steps can be repeated for all subsequent years in the historical period of analysis.

We will proceed with a numerical example.

5. Proposed new method: Numerical example

This section uses a numerical example to show how to incorporate new business into rate change calculations. To overcome the lack of information for new business, we use a relativity approach. The process entails calculating the renewal business, new business, and (weighted) total portfolio rate levels iteratively each year. The process is iterative because a new policy in one year becomes a renewal policy in the following year.

We can use the following steps to derive the total portfolio rate change, starting with the earliest year in the historical period.

Step 1: Calculating renewal rate change has an established methodology and is thus a rational starting point. The renewal rate level is 1 + the renewal rate change, calculated using traditional methodology (Table 1).

Step 2: We need some way to quantify the difference between renewal and new business. The respective RPMs of renewal and new business offer a straightforward and practical way to measure the difference in the rate levels of renewal and new business. The relativity of new business RPM to renewal business RPM is applied to the renewal rate level to derive the new business rate level (Table 2). Section 6.1 shows how the relativity of 1.25 is derived.

Step 3: The total portfolio rate level is then derived using a weighting that comes from renewal business and new business premiums. We create volume-based premium weights to properly derive the total portfolio rate level (discussed further in Section 6.2 and Appendix A).

In Table 3, the 75% renewal business premium weight comes from $46,000,000 of renewal business premium as a percentage of the total premium ($61,600,000 = $46,000,000 + $15,600,000). The 25% new business premium weight comes from $15,600,000 of new business premium as a percentage of the total premium. As shown in Appendix A, volume-based premium weights are appropriate for deriving the total portfolio rate level. We adjust the renewal business premium weighting: 75% / 1.05. We adjust the new business premium weighting: 25% / 1.32. We then derive the volume-based renewal business premium weight as Renewal Weight(2015) = 79% = (75% / 1.05) / (75% / 1.05 + 25% / 1.32). The volume-based new business premium weight is the complement: New Business Weight(2015) = 21% = 100% – 79%.

The total portfolio rate level is Total Rate Level(2015) = 1.05 * 79% + 1.32 * 21% = 1.11. The 11% total portfolio implied rate change for 2015 is Total Rate Change(2015) = 1.11 – 1.

Step 4: Repeat this process iteratively. Each year, the renewal rate level is equal to the prior year’s total portfolio rate level multiplied by 1 + the current-year renewal rate change. Go through the same calculations of using the RPM relativity to calculate the new business rate level and using the weights to calculate the total portfolio rate level.

In Table 4, we get the renewal rate level for 2016 by applying the 2016 renewal rate change (10%) to the 2015 total portfolio rate level: Renewal Rate Level(2016) = 1.22 = 1.11 * (1 + 10%). We do so because what was a new policy in 2015 is now a renewing policy in 2016. We then proceed with the same steps to calculate the new business rate level and total portfolio rate level for 2016: Derive the new business rate level using the RPM relativity, calculate the volume-based premium weights, and arrive at the total portfolio rate level for 2016 of 1.26.

The total portfolio implied rate change for 2016 is 14%, calculated as Total Rate Change(2016) = 14% = (1.26 / 1.11) – 1.

Table 5 shows the calculations for the entire sample historical period for the example discussed in this section. The total portfolio implied rate change (column 11) exceeds the renewal rate change (column 2), which is calculated by the traditional method. This is because new business comes in at higher rate levels than renewal business. If, hypothetically, renewal business were coming in at higher rate levels, the total portfolio implied rate change would be less than that calculated by the traditional approach of using only renewal rate change. Whether this proposed method will result in higher rate changes than those derived using the traditional renewal method will depend on the relativity of new business rate levels to renewal business rate levels.

6. Proposed new method: Details of process

6.1. Overcoming biases in RPM

The goal of using the new business–to–renewal business relativity is to capture differences in rate level. We think a good estimator for this relativity is the ratio of new business RPM to renewal business RPM. As demonstrated in Section 4.2, the relativity for year t can be expressed as

$$\begin{gathered} \text{Relativity}(t) = \\ \frac{\text{New Business Rate Level}(t)}{\text{Renewal Rate Level}(t)} \approx \frac{\text{New Business RPM}(t)}{\text{Renewal RPM}(t)}\text{.} \end{gathered} \tag{4.2c}$$

However, an adjustment needs to be made. In practice, different parts of the insurance tower will have different loss dynamics and thus different rates. If an insurer writes both primary and excess policies, a differing portfolio composition of renewal and new business can distort this analysis. For example, suppose renewal business in a given year is more weighted to low-attaching excess business and new business coming in is more weighted to higher excess business. In this situation, it may appear that new business comes in at a lower rate level due to the low-frequency nature of excess business. This is not what the new business–to–renewal business rate level relativity is attempting to quantify. Instead, it is attempting to quantify the difference in rate level after adjusting for expected loss. Therefore, improved accuracy can be achieved by splitting up the analysis by attachment band to achieve apples-to-apples comparisons of new business to renewal business. We recommend not creating bands so granular that there is a small number of policies (and thus credibility) in each band.

Even when the analysis is split into attachment bands, the goal is to derive a total rate change for the entire portfolio across all attachment bands. We can split up premium, limit, RPM, and RPM relativity by attachment band and by renewal versus new business. We want the RPM relativity to be aggregated across all attachment bands, as the renewal rate change is commonly available only at the aggregated level.

Designate the renewal business premium, limit, and RPM for an attachment band as Renewal Premium(t, attachment), Renewal Limit(t, attachment), and Renewal RPM(t, attachment), respectively.

Designate the new business premium, limit, and RPM for an attachment band as New Business Premium(t, attachment), New Business Limit(t, attachment), and New Business RPM(t, attachment), respectively.

Designate the new business–to–renewal business RPM relativity for an attachment band as

$$\small \text{Relativity}(t \text{, attachment}) = \frac{\text{New Business RPM}(t \text{, attachment})}{\text{Renewal RPM}(t \text{, attachment})}\text{.} \tag{6.1a}$$

We can illustrate this process using a numerical example that ties to the analysis in Section 5. The data in Table 6 is for 2015.

The goal is to derive a total RPM relativity across all attachment bands. If no adjustments are made for limit and attachment, this would be calculated as the ratio of total new business RPM to total renewal business RPM, as shown in the Total row in Table 7.

The new business RPM is $7,429 across all attachment bands, and the renewal business RPM is $6,970 across all attachment bands. The relativity is $7,429 / $6,970 = 1.07. The issue with this approach is its assumption that the distribution of limit by attachment band is the same for both new business and renewal business. However, Table 8 demonstrates this is not the case in the example under discussion.

Renewal business has a higher concentration in lower attachments. Specifically, renewal business has 54% of limits concentrated in policies with attachments of $30 million and less. New business has only 26% of limits concentrated in policies with attachments of $30 million and less. The intention of the relativity is to measure the ratio of new business rate level to renewal business rate level. The 1.07 calculated above (= New Business RPM / Renewal RPM = $7,429 / $6,970) is biased due to differing distributions of limit by attachment band. The total renewal business RPM of $6,970 includes a greater portion of low-attaching policies than the total new business RPM of $7,429.

The approximation above, resulting in 1.07 for the relativity at time t, is

$$\text{Relativity}(t\text{, total}) \approx \frac{\text{New Business RPM}(t\text{, total})}{\text{Renewal RPM}(t\text{, total})}\text{,} \tag{6.1b}$$

where

$$\scriptsize \begin{gathered} \text{New Business RPM}(t\text{, total}) = \\ \frac{\sum_{}^{}{\ \left\lbrack \text{New Business Limit}(t\text{, attachment}) * \text{New Business RPM}(t\text{, attachment}) \right\rbrack}}{\sum_{}^{}{\text{New Business Limit}(t\text{, attachment})}} \end{gathered} \tag{6.1c}$$

and

$$\scriptsize \begin{gathered} \text{Renewal RPM}(t\text{, total}) = \\ \frac{\sum_{}^{}{\ \left\lbrack \text{Renewal Limit}(t\text{, attachment}) * \text{Renewal RPM}(t\text{, attachment}) \right\rbrack}}{\sum_{}^{}{\text{Renewal Limit}(t\text{, attachment})}}\text{.} \end{gathered} \tag{6.1d}$$

The total new business RPM is a weighted average of the new business RPMs for each attachment band, with the weights proportional to the new business limit. The same is true for the total renewal business RPM, using the renewal business limit as a weight. If the distribution of limit is different for new business and renewal business, then the ratio of total new business RPM to total renewal business RPM is biased. The relativity is meant to quantify the difference in rate level and should not be affected by differing limit and attachment profiles.

We present two equivalent ways to set new business and renewal business on the same level of limit and attachment.

Approach 1: The first approach uses a notional renewal business premium. For each attachment band, calculate the notional renewal business premium as the renewal business RPM multiplied by the new business limit. This is the renewal business premium as if it were written to the limits of new business.

$$\scriptsize \begin{gathered} \text{Notional Renewal Premium}(t\text{, attachment}) = \\ \text{Renewal RPM}(t\text{, attachment}) * \frac{\text{New Business Limit}(t\text{, attachment})}{1\text{M}}\text{.} \end{gathered} \tag{6.1e}$$

Next, calculate the percentage of notional renewal business premium in each attachment band. Using these percentages as weights, calculate a weighted average of the RPM relativities by band.

Define the total notional renewal business premium across all attachment bands as

$$\begin{gathered} \text{Notional Renewal Premium}(t\text{, total}) = \\ \sum_{}^{}{\text{Notional Renewal Premium}(t\text{, attachment})}\text{.} \end{gathered} \tag{6.1f}$$

Then the weight for the RPM relativity in a given attachment band is

$$\begin{gathered} \text{Weight Relativity}(t\text{, attachment}) = \\ \frac{\text{Notional Renewal Premium}(t\text{, attachment})}{\text{Notional Renewal Premium}(t\text{, total})}\text{.} \end{gathered} \tag{6.1g}$$

Taking a weighted average of the RPM relativities by attachment band provides an adjusted RPM relativity that accounts for varying levels of limit and attachment:

$$\scriptsize \begin{gathered} \text{Adjusted Relativity}(t\text{, total}) = \\ \sum_{}^{}\left\lbrack \text{Weight Relativity}(t\text{, attachment}) * \text{Relativity}(t\text{, attachment}) \right\rbrack \text{.} \end{gathered} \tag{6.1h}$$

The adjusted RPM relativity is 1.25, as shown in Table 9.

Column 11 in Table 9 calculates the notional renewal business premium. For the $0 million attachment band, we have Notional Renewal Premium(2015, 0M) = $1,538,462 = $12,308 * $125,000,000 / $1,000,000. We do this for all attachment bands and then calculate the percentage of the total for each band (column 12). These percentages are used to weight the unadjusted RPM relativities in column 8. Taking a weighted average of the RPM relativities across attachment bands provides Adjusted Relativity(2015, total) = 1.25. The 1.25 adjusted RPM relativity is for the total portfolio, and it normalizes for the varying limit and attachment profiles of renewal and new business.

Approach 2: The second approach is to directly calculate a notional total renewal business RPM that is normalized to the distribution of a new business limit. Using the percentages of the new business limits in each attachment band as weights, calculate a weighted average of the renewal RPMs by attachment band. This procedure results in the notional total renewal business RPM:

$$\scriptsize \begin{gathered} \text{Notional Renewal RPM}(t\text{, total}) = \\ \frac{\sum_{}^{}{\ \left\lbrack \text{Renewal RPM}(t\text{, attachment}) * \text{New Business Limit}(t\text{, attachment}) \right\rbrack}}{\text{New Business Limit}(t\text{, total})}\text{.} \end{gathered} \tag{6.1i}$$

The adjusted RPM relativity is then the ratio of the total new business RPM to the notional total renewal business RPM:

$$\small \text{Adjusted Relativity}(t\text{, total}) = \frac{\text{New Business RPM}(t\text{, total})}{\text{Notional Renewal RPM}(t\text{, total})}\text{.} \tag{6.1j}$$

Table 10 illustrates this approach.

Column 14 of Table 10 calculates the notional total renewal business RPM as $5,925. This is lower than the total renewal business RPM of $6,970, because it accounts for the new business limit distribution that is more concentrated in higher attachments (that have lower RPMs in this example). The adjusted RPM relativity is then $7,429 / $5,925 = 1.25, as in the first approach.

Appendix B further explains how the two approaches are equivalent and achieve the goal of setting new business and renewal business on the same level of limit and attachment.

6.2. Weighting new business and renewal business rate levels

For the weightings of new business and renewal business rate levels, the goal is to derive the total portfolio rate level. As shown in Appendix A, this means the weights must be based on the level of exposure. To achieve this, we divide the current premium by the current rate level to normalize for varying rate levels between new and renewal business. Designate the premium for year t for renewal business as Renewal Premium(t) and for new business as New Business Premium(t).

We can derive the volume-based premiums for renewal business for year t as

$$\text{Renewal Volume}(t) = \frac{\text{Renewal Premium}(t)}{\text{Renewal Rate Level}(t)}\tag{6.2a}$$

and for new business as

$$\small \text{New Business Volume}(t) = \frac{\text{New Business Premium}(t)}{\text{New Business Rate Level}(t)}\text{.}\tag{6.2b}$$

Then the weight for renewal business for year t is

$$\begin{gathered}\text{Renewal Weight}(t) = \\\frac{\text{Renewal Volume}(t)}{\text{Renewal Volume}(t) + \text{New Business Volume}(t)}\text{.} \end{gathered}\tag{6.2c}$$

The weight for new business for year t is

$$\text{New Business Weight}(t) = 1 - \text{Renewal Weight}(t)\text{.}\tag{6.2d}$$

7. Proposed new method: Assumptions and limitations

The intention of the proposed new method is to provide a way to incorporate new business into rate change calculations that utilize commonly available data and are not too computationally burdensome. In achieving this intention, certain assumptions are made that result in limitations of this method.

Assumptions include the following: First, we are assuming that new business and renewal business are not fundamentally different in risk level, which is reasonable because both new and renewal policies are subject to underwriting guidelines. Second, because we are making an explicit adjustment only for limit and attachment, we are assuming that once adjusted for, new business and renewal business are at the same level of risk. Third, we need to assume that any difference between new business RPMs and renewal business RPMs arises from a difference in rate adequacy and not a difference in loss propensity. Using these assumptions, RPM works as a proxy for rate level. To the extent that any particular real-world situation diverges from these assumptions, the actuary could assign partial credibility of between 0% and 100% weight on the calculated rate changes that emerge from this method that includes new business; the complement of credibility could be applied to the conventional renewal-to-renewal rate changes.

Consider a limitation and a note of caution. If new business RPM is higher simply because of higher risk of loss rather than higher rate adequacy, then the proposed method will misfire: It will detect an increase in rate adequacy where there is none. Therefore, the actuary should use skill and judgment when applying the proposed new method.

8. Area for further research

The method proposed here allows for new business to be incorporated into rate change calculations. This is an improvement over a more traditional approach that includes only renewal business. However, the proposed method does not provide a mechanism to quantify the effect of nonrenewing policies, which drop out of the portfolio. This is an area for further research.

9. Conclusion

This method provides a straightforward algorithm for incorporating new business into rate change calculations, overcoming the frequent paucity of prior-year information for new business. Including new business can improve accuracy by allowing us to measure the rate change of a larger portion of the insurance portfolio rather than limiting our measurement to only the portion of the portfolio that comprises renewing policies. Ultimately, the proposed method can improve actuarial estimation of rate change, on-level factors, and reinsurance experience rating.

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