# Pricing for Short-Term Insurance Coverages

Experience rating is the determination of rates charged by insurance companies based on historical loss data.

## 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:

Line of Business 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 calender 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: [] \begin{align*} \textrm{on-level factor for CY2} &= 1.155/(0.25 + 0.6875 \cdot 1.05 + 0.0625 \cdot 1.155) \\ &= 1.11. \end{align*} [] And finally the on-level earned premium for calendar year 2 equals the on-level factor multiplied by the earned premium:15,000,000 * 1.11 = $16,650,000. ## 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%.

## Classification Ratemaking

The actuary may decide to classify a portfolio of risks based on a set of rating variables. For instance, an actuary may use driver age, gender and model year as rating variables for personal automobile insurance. There are different criteria used to select appropriate rating variables: statistical, operational, social and legal.

The new rating variable needs to partition the existing portfolio of risks into classes of risks that demonstrate sufficient and credible intra-class statistical homogeneity and inter-class statistical heterogeneity. Furthermore, these statistical properties need to be sustained through time.

The rating variable needs to operationally feasible and practical. First, the relevant levels for the rating variable need to be objectively observable. Second, the levels for the rating variable needs to be measurable in a cost effective manner. If it's too expensive to measure levels associated with a new rating variable for every risk in a portfolio, then the cost of implementing the new rating variable might exceed the benefits of having a more granular risk segmentation. Third, the insured shouldn't be able to easily manipulate the level of a rating variable that is being used to price his insurance.

The rating variable needs be perceived as being socially acceptable on different dimensions. First, the new rating variable should preserve affordability of the insurance coverage across all levels of the new rating variable. Affordability is especially important for insurance coverage that is mandatory in specific locales. While actuaries may be more concerned with a statistical correlation (dependendcy) between a rating variable and losses, consumers must feel that there is a causal relationship between the rating variable and losses. Third, the insured should have a certain, through non-deceptive and/or legal actions, amount of control on the level of the rating variable that he is assigned to by the insurer. Finally, some rating variables cannot be used because they require sensitive information about the insureds and privacy concerns may arise. For instance, it might be interesting for insurers to use tracking information for rating purposes, but insurerds may object to invasive tracking by the insurer.

Finally, the new rating variable needs to be legal. Some rating variables may be prohibited in some states while being legal in other states; consequently, actuaries need to consult legal professionals to make sure that new rating variable is legal in a specific geographic region.

### Setting Indicated Rating Differentials

After setting levels for a rating variable, the actuary must determine the variation in rates for each level. The standard approach is to determine a base level for the rating variable and set the rates for each of the other levels as a multiple, usually referred to as a multiplicative indicated differential, of the base rate. We consider two approaches to determining the indicated differential: the pure premium approach and the loss ratio approach. Before introducing the two main approaches to setting differentials, we introduce some new notation:

• $\operatorname{Rate}_{I,i}$ denotes the indicated rate for level $i$
• $\operatorname{R}_{I,i}$ denotes the indicated differential for level $i$
• $B_I$ denotes the indicated rate for the base level

#### Pure Premium Approach

The pure premium approach uses the pure premium method while assuming that the fixed underwriting expense is negligible and that the profit provision $Q_T$ and the variable expense % $V$ is the same for every level $i$:

[$] \operatorname{R}_{I,i} = \frac{[\overline{L + E_L}]_i}{[\overline{L + E_L}]_B}. [$]

#### Loss Ratio Approach

The loss ratio approach uses the loss ratio method, and, like in the pure premium method, assumes that the fixed underwriting expense is negligible and that both $Q_T$ and $V$ are identical for every level $i$:

[] \begin{align*} \frac{\operatorname{R1}_{I,i}}{\operatorname{R1}_{C,i}} &= \frac{\operatorname{Rate}_{I,i}}{\operatorname{Rate}_{C,i}} \cdot \frac{B_C}{B_I} \\ &= \frac{(L + E_L)_i/P_{C,i}}{(L + E_L)_B/P_{C,B}}. \end{align*} []

In other words, the indicated differential change factor equals the loss ratio for level $i$ divided by the loss ratio for the base level.

### Balancing Back

Suppose that the insurer sets a target overall change factor, say $\operatorname{OCF}_T$, for a given portfolio of risks. Given a rating variable, how would we set the rate for the base level given a new set of indicated differentials for that rating variable?

Suppose we have $i=1,\ldots,n$ levels for a rating variable. If $w_i$ is the exposure weight for risks with rating variable level $i$, then we must have

[] \begin{align*} \operatorname{OCF}_T &= \frac{\sum_{i=1}^n w_i B_I R_{I,i}}{\sum_{i=1}^n w_i B_C R_{C,i}} \\ &= \frac{B_I}{B_C} \cdot \frac{\sum_{i=1}^n w_i R_{I,i}}{\sum_{i=1}^n w_i R_{C,i}} \end{align*} []

and, after re-arranging, we obtain the desired indicated base rate:

[$] B_I = B_C \cdot \operatorname{OCF}_T \cdot \frac{\sum_{i=1}^n w_i R_{C,i}}{\sum_{i=1}^n w_i R_{I,i}}. [$]

In other words, the new base rate equals the old base rate multiplied by the target overall change factor multiplied by adjustment factor which equals the old average rate differential divided by the new average rate differential.

## Wikipedia References

• Wikipedia contributors. "Loss development factor". Wikipedia. Wikipedia. Retrieved 28 August 2019.
• Werner, Geoff; Modlin, Claudine. "Basic Ratemaking" (PDF). Casualty Actuarial Society. Retrieved 28 August 2019.