# Ratemaking

Ratemaking, or insurance pricing, is the determination of rates charged by insurance companies. The benefit of ratemaking is to ensure insurance companies are setting fair and adequate premiums given the competitive nature.

## Exposure

Exposure is a measure of potential risk. There are different ways of measuring exposure based on the type of insurance coverage provided. The amount charged to the insured will typically be quoted in terms of the relevant exposure unit. The measurement used to quantify exposure for a line of business is called an exposure base. Here are examples of common exposure bases:

Personal AutomobileEarned Car Year
HomeEarned House Year
Workers' CompensationPayroll
Commercial LiabilitySales Revenue, Square Footage, Number of Units

What makes a good exposure base? First, the expected loss should be proportional to the exposure level: if $L$ denotes loss for coverage of a risk with exposure base $e$, then ideally $\operatorname{E}[L | e = c] = c \operatorname{E}[L | e = 1]$. Second, the exposure base should be operational and practical -- it should be fairly easy and inexpensive to measure exposure levels and they shouldn't be easily manipulated by the parties involved.

The amount of money charged by the insurer to the policyholder for the coverage set forth in the insurance policy is called the premium.

There are two aggregation methods relevant for premium: calendar year and policy year aggregation. We consider aggregation of premium on policies with annual terms and with premium being defined in two different ways: written premium and earned premium.

Written premium is calculated by looking at the time when a policy is issued and paid for. Aggregation of written premium is the total premium collected for policies that were written or activated during a specified period.

The following table demonstrates how to aggregate written premium by calendar year:

Policy Effective Date Expiration Date Premium CY 2015 CY 2016 CY 2017
A5/01/154/30/16$175$175--
B3/01/162/28/17$225-$225-
C5/01/164/30/17$275-$275-
D8/01/167/31/17$300-$300-
E1/01/1712/31/17$250--$250
Total$1,225$175$800$250

The following table demonstrates how to aggregate written premium by policy year:

Policy Effective Date Expiration Date Premium PY 2015 PY 2016 PY 2017
A5/01/154/30/16$175$175--
B3/01/162/28/17$225-$225-
C5/01/164/30/17$275-$275-
D8/01/167/31/17$300-$300-
E1/01/1712/31/17$250--$250
Total$1,225$175$800$250

Earned premium is the portion of an insurance which is considered "earned" by the insurer, based on the part of the policy period that the insurance has been in effect, and during which the insurer has been exposed to loss. For instance, if a 365-day policy with a full premium payment at the beginning of the term has been in effect for 120 days, 120/365 of the premium is considered earned. Earned premium will not be returned to the insured if the policy is cancelled. Aggregation of earned premium is the total premium earned for policies during a specified period.

The following table demonstrates how to aggregate earned premium by calender year:

Policy Effective Date Expiration Date Premium CY 2015 CY 2016 CY 2017
A5/01/154/30/16$175$116.67$58.33- B3/01/162/28/17$225-$187.5$37.50
C5/01/164/30/17$275-$183.33$91.67 D8/01/167/31/17$300-$125$175
E1/01/1712/31/17$250--$250
Total$1,225$116.67$554.16$554.17

The following table demonstrates how to aggregate earned premium by policy year:

Policy Effective Date Expiration Date Premium PY 2015 PY 2016 PY 2017
A5/01/154/30/16$175$175--
B3/01/162/28/17$225-$225-
C5/01/164/30/17$275-$275-
D8/01/167/31/17$300-$300-
E1/01/1712/31/17$250--$250
Total$1,225$175$800$250

### Current rate level adjustment: parallelogram Method

When calculating loss ratios, the actuary needs to adjust earned premium for the experience period so that it takes into account all the rates changes that have occured. This adjustment is called a current level adjustment. The primary method to adjust earned premium is the parallelogram method.

The parallelogram method, also known as the geometric method, derives the on-level factor by dividing a cumulative rate level index by a weighted average rate level index with the weights representing geometric areas of a unit square.The key assumption underlying this method is that the distribution of written premium is uniform over time. We demonstrate the method through an example.

Suppose the following holds:

• Policies are annual
• Experience period is 3 calendar years CY1, CY2, and CY3
• Premium increased by 5% on July 1 of calendar year 1
• Premium increased by 10% on October 1 of calendar year 2
• Earned premium for calendar year 2 is $15,000,000 The situation can be described graphically: What is the on-level earned premium for calendar year 2? The cumulative rate index is 1.1 * 1.05 = 1.155 and the weighted average rate index is given by the formula [$] \textrm{weighted average rate index} = A1 + A2 \cdot 1.05 + A3 \cdot 1.05 \cdot 1.1 [$] with A1 denoting area 1 in CY2, A2 denoting area 2 in CY2 and A3 denoting area 3 in CY2. Basic geometry gives A1 = 0.25, A2 = 0.6875 and A3 = 0.0625. The on-level factor equals the cumulative rate index divided by the weighted average rate index: 1.155/(0.25 + 0.6875 * 1.05 + 0.0625 * 1.155) = 1.1063.  And finally the on-level earned premium for calendar year 2 equals the on-level factor multiplied by the earned premium: $15,000,000 * 1.1063 = $16,594,500.  ## Losses and LAE The largest and most important component of an insurer's expenses are losses and loss adjustments expenses (LAE). Broadly speaking, losses represent payments to claimants while loss adjustment expenses represent administrative and legal fees associated with settling claims. ### Loss Definitions Different loss amounts can be associated to a claim from the claim's initial reporting to its ultimate settlement and closure: paid loss, reported loss and ultimate loss. Type Description Paid losses Payments maid to claimants Reported losses The sum of paid losses and case outstanding. Case outstanding is an estimate, usually established by claims adjusters, of the remaining future payments to claimants. Ultimate losses The insurer's estimate of the sum of all payments ultimately made to claimants to settle past or future claims for a fixed group of policies. Ultimate losses is different than reported losses. First, the case outstanding established by the claims adjuster may not be a sufficiently accurate estimate of the remaining stream of payments necessary to settle existing claims. Second, reported losses doesn't include claims that have yet to be reported. ### Loss Data Aggregation There are four common aggregation methods: calendar year, accident year, policy year and report year. Method Description Calendar year Only looks at changes in paid or reported loss amounts during a calendar year. The aggregation amounts are fixed and aren't subject to adjustments or development based on information/activity occurring after the calendar year has expired. Accident year Only looks at paid or reported loss amounts related to the year when the incident (accident) that triggered a claim occurred. Unlike calendar year aggregation, these amounts are subject to change, adjustment and development until the claim is finally settled. Policy year Only looks at paid or reported loss amounts related to claims covered by policies written in a specific calendar year. It's similar to accident year aggregation except that the inception date of the policy covering the losses associated with the claim triggering incident (accident) is of relevance instead of the actual calendar year of the accident. The coverage period for policy year 2015 is two years (2015 and 2016), so policy year aggregation for 2015 may include incidents (accidents) that occurred during 2015 and 2016. Report year Only looks at paid or reported losses for claims that were reported in a specific calendar year. It is similar to accident year aggregation except that the reporting date of the claim is relevant instead of the date of the incident (accident) that triggered the claim. Assume we have the following two claims: Claims Summary ClaimPolicy periodDate of AccidentDate of Claim Report ASeptember 1, 2017 to August 31, 2018October 12, 2017 October 12, 2017 BMay 1, 2017 to April 30, 2018March 15, 2018 April 10, 2018 The following tables show complete historical claim transactions for claims A and B: Claim A claim transaction history DateIncremental PaymentCase Outstanding 10/12/17$0$8,000 02/22/18$3,000$4,500 05/17/18$2,000$2,250 01/11/19$1,750$0 Claim B claim transaction history DateIncremental PaymentCase Outstanding 04/10/18$0$7,000 05/12/18$1,500$5,500 12/15/18$4,000$1,250 03/18/19$2,000$0 We consider the aggregation of losses. Calendar year 2017 reported losses equal$8,000. Calendar year 2018 reported losses equal $6,000: the sum of the incremental payments ($10,500) minus the drop in case outstanding ($4,500). Calendar year 2019 reported losses equal$250: the sum of the incremental payments ($3,750) minus the drop in case outstanding ($3,500).

Accident year 2017 reported losses equal $6,750: the sum of all the incremental payments for claim A ($6,750) plus the remaining case outstanding ($0). Accident year 2018 reported losses equal$7,500: the sum of all incremental payments for claim B ($7,500) plus the remaining case outstanding ($0).

Policy year 2017 reported losses equal $14,250: both claims were written in 2017, so policy year reported losses equal the sum of the accident year losses ($6,750 + $7,500). Finally, report year reported losses are identical to accident year reported losses since the report dates are in the same calendar year as accident dates. ### Loss Development Insurance claims, especially in long-tailed lines such as liability insurance, are often not paid out immediately. Claims adjusters set initial case reserves for claims; however, it is often impossible to predict immediately what the final amount of an insurance claim will be, due to uncertainty around defense costs, settlement amounts, and trial outcomes (in addition to several other factors). Loss development refers to the evolution of the aggregation of losses through time. Historical losses need to be fully developed to ultimate losses in order to be used in ratemaking methods; however, the ratemaking portion of the exam don't require advanced loss development techniques since there is already a portion of the exam dedicated to loss development. ### Loss Trend Just like loss development, trending historical losses is another necessary adjustment for ratemaking purposes. More precisely, trending losses is simply inflating/deflating losses observed during the experience period so that such adjusted losses can be used to predict future losses. In order to trend losses for ratemaking purposes, one needs to establish the trend period which is usually a period in time delimited by the midpoint in the experience period and the midpoint in the forecast period. For example, assume the following: • The experience period is given by accident year 2016 • Policies are annual • A rate change effective January 1, 2019 to December 31, 2019 is proposed The midpoint of the experience period is July 01, 2016. The forecasting period is January 1 2019 to December 31, 2020. and the midpoint for the forecasting period is December 31,2019. Hence the trend period is July 01, 2016 to December 31, 2019 and the loss trend period, is 3.5 years. Assuming an annual inflation rate of 2%, the trend factor equals 1.023.5. Now suppose that the experience period is policy year 2016 instead of accident year 2016. In this scenario, the midpoint of the experience period is December 31, 2016, the trend period is 3 years and the trend factor is 1.023. ## Basic Ratemaking Methods We present two basic and important methods used in ratemaking. The first method, the pure premium method, gives a rate per unit of exposure whereas the second method, the loss ratio method, indicates how current rates should be modified. ### The fundamental insurance equation The starting point for the ratemaking methods explored here is the fundamental insurance equation [$] P = L + E_L + E_V + E_F + Q\cdot P [$] with $P$ denoting premium; $L$ denoting losses; $E_L$ denoting loss adjustments expenses; $E_V$ denoting variable underwriting expenses; $E_F$ fixed underwriting expenses; and $Q$ denoting the profit percentage. From the fundamental insurance equation, the ratemaking process must produce rates which produce a profit percentage $Q$ which equals or exceeds the targeted profit $Q_T$. ### Pure Premium Method To derive the pure premium method, we rewrite the fundamental insurance equation in terms of indicated premium and target profit percentage [$] $$\label{pp-1} P_I = L + E_L + (E_F + V \cdot P_I) + (Q_T \cdot P_I)$$ [$] with $P_I$ denoting indicated premium, the premium collected at the new (indicated) rates, and $Q_T$ denoting the target profit percentage. Simple algebraic manipulation of \ref{pp-1} gives [$] $$\label{pp-2} P_I = \frac{(L+ E_L) + E_F}{1 - V - Q_T}.$$ [$] To finally obtain the indicated pure premium formula, we simply divide both sides of \ref{pp-2} by the exposure level: [$] $$\label{pp} \overline{P_I} = \frac{\overline{L + E_L} + \overline{E_F}}{1 - V - Q_T}.$$ [$] Equation \ref{pp} underlies the pure premium method. For instance, suppose the actuary has the following assumptions and projections: • Projected pure premium including LAE:$250
• Projected fixed underwriting expense per exposure: $35 • Variable expense: 20% • Targeted underwriting profit: 10% Then, following \ref{pp}, the indicated average rate per exposure equals ($250 + $35)/(1 - 0.2 - 0.1) =$407.

### Loss Ratio Method

The loss ratio method is used to derive an indicated change factor which is the factor by which current rates should be adjusted. As with the pure premium method, we derive the formula by manipulating the fundamental insurance equation. More precisely, we begin with

[$] $$\label{lr-1} P_C = L + E_L + (E_F + V \cdot P_C) + Q_C \cdot P_C.$$ [$]

with $P_C$ denoting indicated current premium, the premium collected at the current rates, and $Q_C$ denoting the current profit percentage.

By simple manipulation of \ref{lr-1}, we can get an expression for the current profit %:

[$] $$\label{lr-2} Q_C = 1 - \frac{(L + E_L) + E_F}{P_C} - V.$$ [$]

We replace $Q_C$ in \ref{lr-2} with $Q_T$ and replace $P_C$ with $P_C$ multiplied by the indicated change factor (ICF):

[$] $$\label{lr-3} Q_T = 1 - \frac{(L + E_L) + E_F}{P_C \cdot \operatorname{ICF}} - V.$$ [$]

And a simple manipulation of \ref{lr-3} finally gives the desired indicated change factor:

[$] $$\label{lr} ICF = \frac{(L + E_L)/P_C + E_F/P_C}{1 - V - Q_T}.$$ [$]

Equation \ref{lr} is the final loss ratio formula. For instance, suppose the actuary has the following assumptions and projections:

• Projected ultimate loss and LAE ratio: 60%
• Projected fixed expense ratio: 5%
• Variable expense %: 30%
• Target underwriting profit %: 15%

Then the indicated change factor equals (60% + 5%)/(1 - 30% - 15%) = 1.18; consequently, according to the loss ratio method, rates should be increased by 18%.

## Increased Limit Factors

The lowest level of insurance available is usually referred to as the basic limit and higher limits are referred to as increased limits. Increased limit factors or ILFs are multiplicative factors that are applied to premiums for basic limits of coverage to determine premiums for increased limits.

Often, limited data is available to determine appropriate charges for high limits of insurance. In order to price policies with high limits of insurance adequately, actuaries may first determine a "basic limit" premium and then apply increased limits factors. The basic limit is a lower limit of liability under which there is a more credible amount of data.[1]

For example, basic limit loss costs or rates may be calculated for many territories and classes of business. At a relatively low limit of liability, such as $100,000, there may be a high volume of data that can be used to derive those rates. For higher limits, there may be a credible volume of data at the countrywide level but not much data available for individual territories or classes. Increased limit factors can be derived at the countrywide level (or some other broad grouping) and then applied to the basic limit rates to arrive at rates for higher limits of liability.[1] ### Formula Suppose $B$ is the base rate for a policy with the base limit and $H$ is an increased limit. We wish to compute the increased limit factor for limit $H$, denoted $\operatorname{ILF}(H)$. Assuming that underwriting expenses are variable and permissible profit provisions do not vary with respect to $H$, we can mimic the pure premium approach to set indicated increased limit factors: [$] $$\label{ilf-1} \operatorname{Indicated ILF}(H) = \frac{(\overline{L + E_L})_H}{(\overline{L + E_L})_B}.$$ [$] If we express the losses $(L + E_L)_H$ and $(L + E_L)_B$ as the product of severity and frequency, and assume that frequency of losses doesn't change in $H$ and is independent of severity, then \ref{ilf-1} can be simplified to: [$] $$\label{ilf-2} \operatorname{Indicated ILF}(H) = \frac{\operatorname{LAS}(H)}{\operatorname{LAS}(B)}$$ [$] with $\operatorname{LAS}(H)$ denoting the limited average severity for increased limit $H$. We use this notation because sample losses can be expressed as the number of claims (frequency) multiplied by the average claim severity. The question is how to determine $\operatorname{LAS}(H)$ given the data available. ### Uncensored Losses We first consider the case where loss data is uncensored i.e., claims are reported regardless of severity. For instance, suppose we have the following claims data: Size of Loss (X)Reported ClaimsReported Losses X <=$100,0002,100$136,500,000$100,000 < X <= $200,0001,700$265,200,000
$200,000 < X <=$400,000500$147,500,000$400,000 < X <= $800,00050$32,500,000
Total4,350$581,170,000 To compute LAS(100,000), we add reported losses below or equal to$100,000 to the number of losses above $100,000 multiplied by$100,000:

 (136,500,000 + 100,000 * (4,350-2,100))/4,350 = 83,103.


And to compute LAS(200,000), we add reported losses below or equal to $200,000 to$200,000 multiplied by the number of losses above $200,000: (265,200,000 + 136,500,000 + 200,000 * (4,350-3,800))/4,350 = 117,632.  Assuming the base limit is$100,000, the indicated increased limit factor for $200,000 equals 117,632/83,103 = 1.415. ### Censored Losses It is common for claim severity to be censored beyond the underlying policy limit: if $H$ is a policy limit, then every loss beyond the limit $H$ is recorded as $H$ in the claims data instead of the actual (higher) amount. For instance, if 2,000 of the 4,350 claims represented in the claims data above represented claims on policies with a limit of$100,000, then the censored loss data for those claims could be:

Size of Loss (X)Reported ClaimsReported Losses
X <= $100,0002,000$100,055,000

Now suppose that we have censored claims data for policies with limits $H$ equal to $200,000 and$400,000:

$200,000 policy limit Size of Loss (X)Reported ClaimsReported Losses X <=$100,000750$33,750,000$100,000 < X <= $200,000670$103,850,000
Total1,420$137,600,000$400,000 policy limit
Size of Loss (X)Reported ClaimsReported Losses
X <= $100,000775$112,375,000
$100,000 < X <=$200,000550$89,650,000$200,000 < X <= $400,000240$74,400,000
Total1,565$276,425,000 As for uncessored losses, we assume the base limit is$100,000. To compute LAS(100,000), we can combine all the claims data and then proceed like we did with uncessored losses:

(100,055,000 + 33,750,000 + 112,375,000 + 100,000*(670 + 550 + 240))/4,985 = 78,672.


To compute LAS(200,000), we make use of the following formula:

[$] $$\label{las-diff} \operatorname{E}[X \wedge b] - \operatorname{E}[X \wedge a] = \operatorname{E}[X \wedge b -a | X \gt a]\operatorname{P}(X \gt a)$$ [$]

To apply \ref{las-diff}, with a = 100,000 and b = 200,000, we first estimate $\operatorname{E}[X \wedge b -a | X \gt a]$ using claims data for policies with limits $200,000 and$400,000:

(103,850,000  + 89,650,000  + 100,000(240 - 670 - 550))/(670 + 550 + 240) = 65,411.


And the estimate for $\operatorname{P}(X \gt a)$ equals

(670 + 550 + 240)/(1,420 + 1,565) = 0.489.


This gives LAS(200,000) = 78,672 + 65,411 * 0.489 = 110,658. To compute LAS(400,000), we use \ref{las-diff} with b = 400,000 and a = 200,000. The estimate for $\operatorname{E}[X \wedge b -a | X \gt a]$ equals

(74,400,000 - 240 * 200,000)/240  = 110,000.


This gives LAS(400,000) = 110,658 + 110,000*0.489 = 164,448. Finally, we obtain the desired increased limit factors: $110,658/$78,672 = 1.407 for the increased limit $200,000 and$164,448/$78,672 = 2.09 for the increased limit$400,000.

Computing limited average severity for censored losses

Suppose an insurer has policies with limits $H_1,\ldots,H_n$ with base limit $H_1$ and $H_i \lt H_{i+1}$. To compute the limited average severity for increased limit $H_i$, we proceed as follows:

1. Calculate the limited average severity for the base limit $H_1$ by pooling all the (censored) historical claims data with policy limits $H_1,\ldots,H_n$.
2. Estimate $\triangle_i = \operatorname{E}[X \wedge H_{i} - H_{i-1} | X \gt H_{i-1}]\operatorname{P}(X \gt H_{i-1})$ using historical claims data for policies with limits $H_{i},\ldots,H_n$.
3. Set $\operatorname{LAS}(H_i) = \operatorname{LAS}(H_{i-1}) + \triangle_i$.

## Deductible Pricing

We consider the problem of computing the indicated relativity as the deductible changes. We use the same assumptions as for the increased limit factors: underwriting costs and profit provisions do not depend on the deductible and all undewriting costs are variable. If $B$ is the base rate, then the indicated relativity for a deductible of size $D$ equals

[$] $$\label{dp-1} \operatorname{Indicated Deductible Relativity}(D) = \frac{(\overline{L + E_L})_D}{(\overline{L + E_L})_B} = 1 - \operatorname{LER}(D)$$ [$]

with $\operatorname{LER}(D)$ denoting the loss elimintation ratio for deductible $D$ defined as follows:

[$] \operatorname{LER}(D) = \frac{(L + E_L)_B - (L + E_L)_D}{(L + E_L)_B}. [$]

### Uncensored Losses

As with policy limits, we consider both censored and uncessored losses. Reported ground up losses refer to losses that would have been incurred without a deductible. When reported ground up losses are available, the empirical distribution can be used to calculate the loss elimination ratio for any deductible $D$ (the base deductible is 0 and $x_i$ denotes the loss data):

[$] $$\label{ler} \operatorname{LER}(D) = 1 - \frac{\sum_{i=1}^n\operatorname{Max}(0,x_i -D)}{\sum_{i=1}^n x_i}$$ [$]

#### Example

To illustrate the method, suppose D = $200 and we have the following claims data: LER for$200 deductible
Size of Loss (X)Reported ClaimsGround-Up Reported Losses
X < $100 2,900194,300$100 <= X < $2001,411218,705$200 <= X < $4001,122350,064$400 <= X < $900 1,8501,295,000$900 < X 2,3209,558,400
Total7,52311,616,469

Then, using \ref{ler}, LER(200) equals

1 - (350,064 + 1,295,000 + 9,558,400 - 200*(1,122 + 1,850 + 2,320))/11,616,469 = 0.127.


And the indicated relativity for a deductible of $200 is 1 - 0.127 = 0.873. ### Censored Losses An insured may not report claims that are perceived to be less than the deductible on their policy or the historical claims data might only contain net losses i.e., the portion of losses that exceed the relevant deductible. The presence of censoring implies that one cannot pool all claims data when computing loss elimination ratios -- we cannot use claims data for policies with a deductible that is higher than the deductible being priced. #### Example Suppose we wish to calculate the loss elimination ratio for inreasing the deductible from$200 to $400 based on the following claims data: LER for$200 deductible to $400 deductible DeductibleReported ClaimsNet Reported LossesNet Reported Losses with$400 DedNet Reported Losses with $200 Ded Losses Eliminated 0490651,700501,809588,13486,325 1006901,244,123 1,033,768 1,153,890 120,122 2001,3852,876,430 2,663,121 2,876,430 246,241 4002,3155,399,786 5,399,786 NA NA 900275819,722NANANA Total4,95010,172,039 To calculate the relevant loss elimination ratio, we divide the loss eliminated by increasing the deductible to$400, which equals the sum of the entries in the last column of the table above, by the total reported losses for a deductible of \$200, which equals the sum of the entries in the second to last column:

(86,325 + 120,122 + 246,241)/(588,134 + 1,153,890 + 2,876,430) = 0.098.


## Notes

1. Palmer, Joseph (July 2006), Increased limits ratemaking for liability insurance (PDF)