View source for Werner04.Exposures

'''Reading''': BASIC RATEMAKING, Fifth Edition, May 2016, Geoff Werner, FCAS, MAAA & Claudine Modlin, FCAS, MAAA
Willis Towers Watson

'''Chapter 4''':  Exposures

==Pop Quiz==

Identify the 3 objectives of data aggregation for ratemaking. <span style="background-color: lightblue; border-radius: 5px;"> ''[[Werner04.Exposures#POP_QUIZ_ANSWERS |Click for Answer]]''&nbsp;</span>

==Study Tips==

There is a fair bit to take in here and although it is core content according to the ''[[BattleBriefings | Ranking Table]]'', it still isn't a Top 5 chapter. That means you shouldn't get too hung up on the details the first time you read it. You will get plenty of practice working with similar concepts in later chapters. There are just a few things you have to memorize so that part isn't a problem. You mainly have to learn how to aggregate WE, EE, UEE, and IFE from policy information ''(Written Exposures, Earned Exposures, Unearned Exposures, In-Force Exposures.)'' There is a web-based practice problem to help you learn this as well as lots of old exam problems.

'''Estimated study time''': 2 days ''(not including subsequent review time)''

==BattleTable==

Based on past exams, the '''main things''' you need to know ''(in rough order of importance)'' are:

* <span style="color: green;">'''exposure base criteria:'''</span> identifying desirable criteria, evaluating a potential exposure basis
* '''exposure aggregation''': <span style="color: blue;">'''CYEE'''</span>, <span style="color: brown;">'''CYWE'''</span>, inforce exposures, PY aggregation

: {| class='wikitable' style='width: 1105px;'

|-
! style='width: 175px;' | reference !! style='width: 175px;' | part (a) !! style='width: 175px;' | part (b) !! style='width: 175px;' | part (c) !! style='width: 175px;' | part (d) !! part (e)

|-
|| [https://www.battleacts5.ca/pdf/Exam_(2019_2-Fall)/(2019_2-Fall)_(01).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2019.Fall #1)'''</span>
|| '''PYEE:'''</span> <br> - calculate
|| '''in-force exposures:''' <br> - calculate
|| '''CY unearned exposures:''' <br> - calculate
|| <span style="color: blue;">'''CYEE'''</span> <br> - calculate
| style="background-color: lightgrey;" |

|-
|| [https://www.battleacts5.ca/pdf/Exam_(2019_1-Spring)/(2019_1-Spring)_(01).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2019.Spring #1)'''</span>
|| <span style="color: brown;">'''CYWE:'''</span> <br> - calculate
|| <span style="color: blue;">'''CYEE'''</span> <br> - calculate
|| '''PYEE:''' <br> - calculate
|| '''in-force exposures:''' <br> - calculate
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - miles driven

|-
|| [https://www.battleacts5.ca/pdf/Exam_(2018_1-Spring)/(2018_1-Spring)_(01).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2018.Spring #1)'''</span>
|| <span style="color: brown;">'''CYWE:'''</span> <br> - calculate
|| <span style="color: blue;">'''CYEE'''</span> <br> - calculate
|| '''PYEE:''' <br> - calculate
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - # vehicle occupants
| style="background-color: lightgrey;" |

|-
|| [https://www.battleacts5.ca/pdf/Exam_(2017_1-Spring)/(2017_1-Spring)_(01).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2017.Spring #1)'''</span>
|| <span style="color: brown;">'''CYWE'''</span> & <span style="color: blue;">'''CYEE:'''</span> <br> - calculate
|| <span style="color: blue;">'''CYEE'''</span> <br> - calculate
| style="background-color: lightgrey;" | 
| style="background-color: lightgrey;" | 
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|-
|| [https://www.battleacts5.ca/pdf/Exam_(2016_2-Fall)/(2016_2-Fall)_(01).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2016.Fall #1)'''</span>
|| <span style="color: brown;">'''CYWE:'''</span> <br> - calculate
|| '''in-force exposures:''' <br> - calculate
|| <span style="color: blue;">'''CYEE'''</span> <br> - calculate
| style="background-color: lightgrey;" | 
| style="background-color: lightgrey;" | 

|-
|| [https://www.battleacts5.ca/pdf/Exam_(2016_1-Spring)/(2016_1-Spring)_(03).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2016.Spring #3)'''</span>
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - evaluate hours driven
|| [[ASOP.12]]
| style="background-color: lightgrey;" | 
| style="background-color: lightgrey;" | 
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|-
|| [https://www.battleacts5.ca/pdf/Exam_(2015_2-Fall)/(2015_2-Fall)_(02).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2015.Fall #2)'''</span>
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - restaurant general liability
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - hospital prof. liability
| style="background-color: lightgrey;" |
| style="background-color: lightgrey;" |
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|-
|| [https://www.battleacts5.ca/pdf/Exam_(2015_1-Spring)/(2015_1-Spring)_(02).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2015.Spring #2)'''</span>
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - annual fuel expense
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - miles driven
| style="background-color: lightgrey;" | 
| style="background-color: lightgrey;" | 
| style="background-color: lightgrey;" | 

|-
|| [https://www.battleacts5.ca/pdf/Exam_(2015_1-Spring)/(2015_1-Spring)_(03).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2015.Spring #3)'''</span>
|| <span style="color: blue;">'''CYEE'''</span> <br> - calculate
|| <span style="color: blue;">'''CYEE:'''</span> <br> - calculate ''(using retention)''
|| '''assess assumption:''' <br> - uniform writings
| style="background-color: lightgrey;" | 
| style="background-color: lightgrey;" | 

|-
|| [https://www.battleacts5.ca/pdf/Exam_(2014_2-Fall)/(2014_2-Fall)_(01).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2014.Fall #1)'''</span>
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - identify
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - WC <u>change</u> to hrs worked
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - impact on WC frequency
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - impact on WC severity
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> -  impact on WC loss ratio

|-
|| [https://www.battleacts5.ca/pdf/Exam_(2014_1-Spring)/(2014_1-Spring)_(03).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2014.Spring #3)'''</span>
|| '''provide a date range:''' <br> -  for effective dates
|| '''provide a date range:''' <br> -  for cancellation dates
|| '''prove:''' <br> - inequality is invalid
| style="background-color: lightgrey;" | 
| style="background-color: lightgrey;" | 

|-
|| [https://www.battleacts5.ca/pdf/Exam_(2013_2-Fall)/(2013_2-Fall)_(01).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2013.Fall #1)'''</span>
|| <span style="color: green;">'''exposure base criteria:'''</span> <br> - WC <u>change</u> to hrs worked
| style="background-color: lightgrey;" | 
| style="background-color: lightgrey;" | 
| style="background-color: lightgrey;" | 
| style="background-color: lightgrey;" | 

|-
|| [https://www.battleacts5.ca/pdf/Exam_(2013_1-Spring)/(2013_1-Spring)_(01).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2013.Spring #1)'''</span>
|| <span style="color: blue;">'''CYEE'''</span> <br> - calculate
|| '''PYEE:''' <br> - calculate
|| '''PYWE:''' <br> - calculate after cancellation
|| <span style="color: brown;">'''CYWE:'''</span> <br> - calculate after cancellation
| style="background-color: lightgrey;" | 

|}

[https://www.battleacts5.ca/FC.php?selectString=**&filter=both&sortOrder=natural&colorFlag=allFlag&colorStatus=allStatus&priority=importance-high&subsetFlag=miniQuiz&prefix=Werner04&suffix=Exposures&section=all&subSection=all&examRep=all&examYear=all&examTerm=all&quizNum=all<span style="font-size: 20px; background-color: lightgreen; border: solid; border-width: 1px; border-radius: 10px; padding: 2px 10px 2px 10px; margin: 10px;">'''Full BattleQuiz]'''</span> <span style="color: red;">'''You must be <u>logged in</u> or this will not work.'''</span>

==In Plain English!==

===Criteria for Exposure Bases===

Alice has a riddle for you:

: ''You are an auto insurance agent and a 2 different customers come to you wanting to buy liability insurance for their car. Liability insurance provides financial protection for drivers who harm someone else or their property while operating a vehicle. The first customer's car is worth $25,000 and the second customer's car is worth $30,000. Using only this information, which customer represents the higher risk?''

The answer is there's no difference in liability risk between these 2 customers. If the $25,000 car gets into an accident and damages another car, it's the damage to the ''other car'' that determines the insurance payout. Same for the $30,000 car. The point here is that the <span style="color: red;">'''value'''</span> of an insured's car ''(within a reasonable range)'' <u>is not</u> a good measure of the insurer's exposure to risk.

But suppose you have a husband and wife, each with their own car wanting to buy liability insurance versus a single person with just one car. In this case, the family with 2 cars is the higher risk, double the risk of the single person. The reason is obvious: The husband and wife together will do roughly twice as much driving as the single person and therefore have twice as many chances of getting into an accident. The <span style="color: red;">'''number of cars'''</span> <u>is</u> a good measure of the insurer's exposure to risk.

The key idea in pricing an insurance policy is measuring the exposure to risk. Riskier policies are charged a higher price. The very first step in the pricing process for any line of insurance is coming up with an '''exposure base'''.

:{| class='wikitable'
|- 
|| '''Definition''': an <u>exposure base</u> is the basic unit that measures a policy's exposure to loss
|}

In the above example, the appropriate exposure base for auto insurance would be ''car-years'' where 1 car-year represents coverage for 1 vehicle for 1 year. In general, there are specific criteria that a good exposure base should satisfy.

:{| class='wikitable'
|- 
|| '''Question''': what criteria should a good <u>exposure base</u> satisfy [Hint: <span style="color: red;">'''Triple <u>P</u>lay'''</span>]
|}

:: <span style="color: red;">'''P'''</span>roportional to expected loss
:: <span style="color: red;">'''P'''</span>ractical
:: <span style="color: red;">'''P'''</span>recents ''(you should consider historical precedents)''

Let's take a closer look at what these criteria mean:

: '''Proportional to expected loss'''
:* According to Werner: ''all else being equal, the expected loss of a policy with two exposures should be twice the expected loss of a similar policy with one exposure.''
:* If proportionality is satisfied, it will be easier for the insured to understand and will be seen as more fair.

: '''Practical'''
:* An exposure based should be: objective, easy & cheap to obtain & verify
:* For example, the number of cars a family owns is objective. You might argue the ''# of miles driven'' is a better exposure base because it also is objective and proportional to expected loss ''(even more than car-years)''. A problem however is # of miles driven is not as easy to verify. Furthermore, if it is self-reported it's open to manipulation by the insured. This may become less of an issue however as telematics devices become more common. Telematics and Usage-Based-Insurance are discussed in more detail in Exam 6.

: '''Precedents'''
:* The criterion of historical precedents becomes important when a <u>change</u> in exposure base is under consideration.
:* For example if telematics devices become commonplace, insurers might consider switching from car-years to # of miles driven for auto insurance. While the latter may indeed be a better exposure there are also potential drawbacks to consider whenever an exposure based is changed:
::: &rarr; large premium swings for individual insureds
::: &rarr; rating algorithm may need to be changed ''(may be costly to update rating systems and manuals)''
::: &rarr; data adjustments may be required ''(because a pricing analysis uses several years of data)''

Here's a link to Werner with a table showing typical examples of exposure bases for different lines of business:

: [https://www.battleacts5.ca/pdf/Werner_(p063-063)_exposure_bases.pdf <span style="color: white; font-size: 12px; background-color: sandybrown; border: solid; border-width: 2px; border-radius: 10px; border-color: sandybrown; padding: 1px 3px 1px 3px; margin: 0px;">'''''Common exposure bases'''''</span>]

And below is a typical exam problem on this topic. You have to evaluate whether ''hours driven'' satisfies the <span style="color: red;">'''Triple <u>P</u>lay'''</span> criteria for a personal auto exposure base. This type of question is easy points on the exam so make sure you know it.

: [https://www.battleacts5.ca/pdf/Exam_(2016_1-Spring)/(2016_1-Spring)_(03).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2016.Spring #3)'''</span>

There are more problems of this type in the quiz...

[https://www.battleacts5.ca/FC.php?selectString=**&filter=both&sortOrder=natural&colorFlag=allFlag&colorStatus=allStatus&priority=importance-high&subsetFlag=miniQuiz&prefix=Werner04&suffix=Exposures&section=all&subSection=all&examRep=all&examYear=all&examTerm=all&quizNum=1<span style="font-size: 20px; background-color: aqua; border: solid; border-width: 1px; border-radius: 10px; padding: 2px 10px 2px 10px; margin: 10px;">'''mini BattleQuiz 1]'''</span> <span style="color: red;">'''You must be <u>logged in</u> or this will not work.'''</span>

===Exposures for Large Commercial Risks===

This topic is covered in ''[[Werner15.Commercial | Chapter 15 - Commercial Lines Rating Mechanisms]]''. Ratemaking for large commercial lines is different and is often done using composite rating or loss-composite rating. Loss-composite rating uses the insured's own historical loss experience to calculate premium.

===Aggregation of Exposures===

The type of problem we're going to look at in this section concerns aggregation of exposures. Given information about a set of policies, calculate the following metrics:

* Written Exposure ''(WE)''
: &rarr; total exposures arising from policies issued during a specified time period such as a calendar quarter or a calendar year
* Earned Exposure ''(EE)''
: &rarr; portion of the written exposures for which coverage '''has''' been provided <u>as of</u> a certain point in time
* Unearned Exposure ''(UEE)''
: &rarr; portion of the written exposures for which coverage '''has not''' been provided <u>as of</u> a certain point in time
* In-Force Exposure ''(IFE)''
: &rarr; number of insured units that are exposed to having a claim <u>at</u> a given point in time

Conceptually these calculations aren't hard but it the details can get messy and it's easy to make a mistake if you don't have a methodical approach. We're going to start with a very easy example then build up gradually.

You may already be aware of the geometric interpretation of policy writings but you don't need that for these simple examples. We'll cover the geometric interpretation further down. Aggregation can be done either on a CY or PY basis. ''(Calendar year aggregation here is sometimes also called calendar-accident year as explained in [[Werner03.Data]].)''

[https://www.battleacts5.ca/FC.php?selectString=**&filter=both&sortOrder=natural&colorFlag=allFlag&colorStatus=allStatus&priority=importance-high&subsetFlag=miniQuiz&prefix=Werner04&suffix=Exposures&section=all&subSection=all&examRep=all&examYear=all&examTerm=all&quizNum=2<span style="font-size: 20px; background-color: aqua; border: solid; border-width: 1px; border-radius: 10px; padding: 2px 10px 2px 10px; margin: 10px;">'''mini BattleQuiz 2]'''</span> <span style="color: red;">'''You must be <u>logged in</u> or this will not work.'''</span>

====Example A: Calendar Year - 1 Individual Policy====

<span style="color: brown;">'''Hint:'''</span> As you read through the bullet points demonstrating these calculations, check each one off as you understand it. There will be a quiz with these types of questions further down. This may take 5-10 minutes for each example.

Personal auto policy information:

:{| class='wikitable'
|-
! policy !! # of vehicles !! effective date !! expiration date
|-
|| A     || 1  || Jan 1, 2020 || Dec 31, 2020
|}

The solution below looks more complicated than it really is. If this is the insurer's only policy, then referring to the definitions provided above even ''Ian-the-Intern'' can do the calculations:

* CY<sub>20</sub>(WE) = <u>1.0</u>
:: ''(just count the policies that were written during CY 2020 - there was just 1 policy written during that time period)''

The value of EE changes over the course of the year and may depend on an "as of" date: ''(If you want the EE for a whole CY, then the "as of" date would just be the end of the year)''

* CY<sub>20</sub>(EE) as of Dec 31, 2019 = <u>0.0</u>
:: ''(exposures are earned over the coverage period and as of Dec 31, 2019, the coverage period has not yet started)''
* CY<sub>20</sub>(EE) as of Mar 31, 2019 = <u>0.25</u>
:: ''(by Mar 31, one-quarter of the written exposure of 1.0 has been earned, assuming uniform earnings of exposures)''
* CY<sub>20</sub>(EE) as of Dec 31, 2020 = <u>1.0</u>
:: ''(the coverage period for this policy falls entirely within CY 2020 so by the end of 2020 the entire written exposure of 1.0 is earned)''

For aggregate UEE values, you usually want the amount at the start and end of the CY but for an individual policy, UEE also depends on an "as of" date:

* CY<sub>20</sub>(UEE) as of Mar 31, 2020 = <u>0.75</u>
:: ''(if 0.25 has been earned, then obviously 0.75 is still unearned)''
* CY<sub>20</sub>(UEE) as of Dec 31, 2020 = <u>0.0</u>
:: ''(by year-end 2020, the policy has expired and there is no more exposure left to earn)''

For IFE, just count how many policies are "in-force" at the "as of" date. You '''must''' have an "as of" date for IFE.

* IFE as of Mar 31, 2020 = <u>1.0</u>
:: ''(only 1 policy is active on this date)''
* IFE as of Jul 12, 2020 = <u>1.0</u>
:: ''(only 1 policy is active on this date)''
* IFE as of Dec 31, 2020 = <u>0.0</u>
:: ''(in-force exposures drops to 0 at the end of the year when this policy expires and is no longer active)''

====Example B: Calendar Year - 3 Individual Policies====

Personal auto policy information:

:{| class='wikitable'
|-
! policy !! # of vehicles !! effective date !! expiration date
|-
|| A     || 1  || Jan 1, 2020 || Dec 31, 2020
|-
|| B     || 1  || Oct 1, 2020 || Sep 30, 2021
|-
|| C     || 1  || Jan 1, 2021 || Dec 31, 2021
|}

If the insurer now has 3 policies, we do the calculations for each policy individually and sum the results. Two policies, A and B, were written in 2020 and policy C was written in 2021 so we have:

* CY<sub>20</sub>(WE) = <u>2.0</u>
:: '' (due to policies A and B)''
* CY<sub>21</sub>(WE) = <u>1.0</u>
:: '' (due to policy C)''

Calculating EE at different "as of" dates is still pretty easy but you can see how it could quickly get messy. Every additional policy means there's more you have to keep track of. Below we calculate the CY totals for 2020 and 2021, so the "as of" dates are effectively Dec 31, 2020 and Dec 31, 2021.

: <span style="color: green;">'''Policy A'''</span>
:* CY<sub>20</sub>(EE) = <u>1.00</u> ''(because the entire policy term is in 2020)''
: <span style="color: green;">'''Policy B'''</span>
:* CY<sub>20</sub>(EE) = <u>0.25</u> ''(because first 3 months of the policy term is in 2020)''
:* CY<sub>21</sub>(EE) = <u>0.75</u> ''(because last 9 months of the policy term is in 2021)''
: <span style="color: green;">'''Policy C'''</span>
:* CY<sub>21</sub>(EE) = <u>1.00</u> ''(because the entire policy term is in 2021)''

: <span style="color: green;">'''Total'''</span>
:* CY<sub>20</sub>(EE) = 1.00 + 0.25 = <u>1.25</u>
:* CY<sub>21</sub>(EE) = 0.75 + 1.00 = <u>1.75</u>

For UEE, we'll calculate the '''starting''' and '''ending''' values for CY 2020 and 2021.

: <span style="color: green;">'''Policy A'''</span>
:* CY<sub>20</sub>(UEE) at Jan 1, 2020 = <u>0.00</u> ''(because the policy is not yet in effect, this is the starting UEE for CY 2020)''
:* CY<sub>20</sub>(UEE) at Dec 31, 2020 = <u>0.00</u> ''(ending UEE for CY 2020 because the policy has expired and the whole exposure has been earned)''
: <span style="color: green;">'''Policy B'''</span>
:* CY<sub>20</sub>(UEE) at Jan 1, 2020 = <u>0.00</u> ''(starting UEE for CY 2020)''
:* CY<sub>20</sub>(UEE) at Dec 31, 2020 = <u>0.75</u> ''(because the policy still has 9 months of earning remaining, this is both the ending UEE for CY 2020 and the starting UEE for CY 2021 )''
:* CY<sub>21</sub>(UEE) at Dec 31, 2021 = <u>0.00</u> ''(because the policy has expired and the whole exposure has been earned, this is the ending UEE for CY 2021)''

: <span style="color: green;">'''Policy C'''</span>
:* CY<sub>21</sub>(UEE) at Jan 1, 2021 = <u>0.00</u> ''(starting UEE for CY 2021, policy is not yet in effect)''
:* CY<sub>21</sub>(UEE) at Dec 31, 2021 = <u>0.00</u> ''(ending UEE for CY 2021, policy has expired so the whole exposure has been earned)''

: <span style="color: green;">'''Total'''</span>
:* CY<sub>19</sub>(UEE) at Dec 31, 2019 = <u>0.00</u> ''(ending UEE for CY 2019 = starting UEE for CY 2020)
:* CY<sub>20</sub>(UEE) at Dec 31, 2020 = 0.00 + 0.75 = <u>0.75</u> ''(ending UEE for CY 2020 = starting UEE for CY 2021)
:* CY<sub>21</sub>(UEE) at Dec 31, 2020 ''(ending UEE for CY 2021 = starting UEE for CY 2022)

For IFE, just count how many policies are "in-force" at the "as of" date. You '''must''' have an "as of" date for IFE.

* IFE as of Mar 31, 2020 = <u>1.0</u>
:: ''(only policy A is active on this date)''
* IFE as of Oct 15, 2020 = <u>2.0</u>
:: ''(policies A and B are active on this date)''
* IFE as of Mar 31, 2021 = <u>2.0</u>
:: ''(policies B and C are active on this date)''
* IFE as of Nov 30, 2021 = <u>1.0</u>
:: ''(only policy C is active on this date)''

[https://www.battleacts5.ca/FC.php?selectString=**&filter=both&sortOrder=natural&colorFlag=allFlag&colorStatus=allStatus&priority=importance-high&subsetFlag=miniQuiz&prefix=Werner04&suffix=Exposures&section=all&subSection=all&examRep=all&examYear=all&examTerm=all&quizNum=3<span style="font-size: 20px; background-color: aqua; border: solid; border-width: 1px; border-radius: 10px; padding: 2px 10px 2px 10px; margin: 10px;">'''mini BattleQuiz 3]'''</span> <span style="color: red;">'''You must be <u>logged in</u> or this will not work.'''</span>

====Example C: Policy Year - 3 Individual Policies====

We'll work with the same policies as in Example B:

:{| class='wikitable'
|-
! policy !! # of vehicles !! effective date !! expiration date
|-
|| A     || 1  || Jan 1, 2020 || Dec 31, 2020
|-
|| B     || 1  || Oct 1, 2020 || Sep 30, 2021
|-
|| C     || 1  || Jan 1, 2021 || Dec 31, 2021
|}

As long as there isn't a mid-term cancellation, there isn't any difference between CY and PY in calculating WE. Two policies, A and B, were written in CY 2020 and policy C was written in CY 2021. By definition, all policies written in CY 2020 are part of PY 2020, and all policies written in CY 2021 are part of PY 2020.

* PY<sub>20</sub>(WE) = <u>2.0</u>
:: '' (due to policies A and B)''
* PY<sub>21</sub>(WE) = <u>1.0</u>
:: '' (due to policy C)''

If there <u>is</u> a mid-term cancellation then the situation is slightly different at least for CY aggregation. Suppose policy A cancels on Oct 1 ,2020 and loses one-quarter of its original 1-year term. Then:
:* adjusted CY<sub>20</sub>(WE) = 1.0 - 0.25 = <u>0.75</u>
:* adjusted PY<sub>20</sub>(WE) = 1.0 - 0.25 = <u>0.75</u> ''(same as CY calculation)''

Now let's look at policy B. If policy B cancels Dec 31, 2020, we have:
:* adjusted CY<sub>20</sub>(WE) = 1.0 - 0.75 = <u>0.25</u>
:* adjusted PY<sub>20</sub>(WE) = 1.0 - 0.75 = <u>0.25</u> ''(same as CY calculation)''

But if policy B cancels on Jul 1, 2021, losing one-quarter of its term, the effect of the cancellation goes into CY 2021, not CY 2020:
:* adjusted CY<sub>20</sub>(WE) = 1.0 ''(no adjustment to WE for CY 2020 because the cancellation occurred in CY 2021)''
:* adjusted CY<sub>21</sub>(WE) = -0.25 ''(summing CY 2020 and CY 2021 does indeed give the correct total WE of 0.75 even though the effect is now spread across 2 years)

Note however that the effect of the late cancellation of the policy is <u>not</u> split between PYs &rarr; goes entirely into PY 2020. In general all PY transactions relate to a single PY.
:* adjusted PY<sub>20</sub>(WE) = 1.0 - 0.25 = <u>0.75</u>

Similarly, calculating EE for PY is actually easier than for CY. For CY aggregation, an annual policy written in the middle of 2020 would have EE in both CY 2020 and CY 2021. But for PY aggregation, the same annual policy would have all its EE in PY 2020. Any policy with an effective date between Jan 1, 2020 and Dec 31, 2020 has all its WE and EE in PY 2020. That's the advantage of PY aggregation. You can compare Example C to Example B line by line to see where the differences are.

: <span style="color: purple;">'''Policy A'''</span>
:* PY<sub>20</sub>(EE) = <u>1.00</u> ''(because the entire policy term is in PY 2020)''
: <span style="color: purple;">'''Policy B'''</span>
:* PY<sub>20</sub>(EE) = <u>1.00</u> ''(because the entire policy term is in PY 2020)''
:* PY<sub>21</sub>(EE) = <u>0.00</u> ''(because none of the policy term is in PY 2020 and makes no contribution to PY 2021)''
: <span style="color: purple;">'''Policy C'''</span>
:* PY<sub>21</sub>(EE) = <u>1.00</u> ''(because the entire policy term is in PY 2021)''

: <span style="color: purple;">'''Total'''</span>
:* PY<sub>20</sub>(EE) = 1.00 + 1.00 = <u>2.00</u>
:* PY<sub>21</sub>(EE) = 0.00 + 1.00 = <u>1.00</u>

For UEE, we'll calculate the '''starting''' and '''ending''' values for PY 2020 and 2021. But the ending date for PY 2020 is <span style="color: red;">'''Dec 31, 2021.'''</span> This is because a policy written on Dec 31, 2020, still in PY 2020, doesn't expire until the end of 2021. This is the disadvantage of PY aggregation. Policies can be written anytime between Jan 1 and Dec 31, 2020, and the PY doesn't end until the last policy expires.

: <span style="color: purple;">'''Policy A'''</span>
:* PY<sub>20</sub>(UEE) at Jan 1, 2020 = <u>0.00</u> ''(starting UEE for PY 2020 because the policy is not yet in effect)''
:* PY<sub>20</sub>(UEE) at Dec 31, 2022 = <u>0.00</u> ''(ending UEE for PY 2020 because the policy has expired and the whole exposure has been earned)''

: <span style="color: purple;">'''Policy B'''</span>
:* PY<sub>20</sub>(UEE) at Jan 1, 2020 = <u>0.00</u> ''(starting UEE for PY 2020 because the policy is not yet in effect)''
:* PY<sub>20</sub>(UEE) at Dec 31, 2022 = <u>0.00</u> ''(ending UEE for PY 2020 because the policy has expired and the whole exposure has been earned)''
:* PY<sub>21</sub>(UEE) at Dec 31, 2023 = <u>0.00</u> ''(because this policy is entirely within PY 2020 and makes no contribution to PY 2021)''

: <span style="color: purple;">'''Policy C'''</span>
:* PY<sub>20</sub>(UEE) at Dec 31, 2022 = <u>0.00</u> ''(this policy is from PY 2021 and makes no contribution to PY 2020)''
:* PY<sub>21</sub>(UEE) at Dec 31, 2023 = <u>0.00</u> ''(ending UEE for PY 2021 because the policy has expired and the whole exposure has been earned)''

: <span style="color: purple;">'''Total'''</span>
:* PY<sub>20</sub>(UEE) at Jan 1, 2020 = <u>0.00</u> ''(starting UEE for PY 2020)
:* PY<sub>20</sub>(UEE) at Dec 31, 2022 = 0.00 + 0.00 = <u>0.00</u> ''(ending UEE for PY 2020)
:* PY<sub>21</sub>(UEE) at Dec 31, 2020 = <u>0.00</u> ''(ending UEE for PY 2021)

For IFE, there is no difference between CY and PY. The IFE metric is a 'point-in-time' metric which means it doesn't depend on a time period aggregation. Just count how many policies are "in-force" at the "as of" date. You '''must''' have an "as of" date for IFE.

* IFE as of Mar 31, 2020 = <u>1.0</u>
:: ''(only policy A is active on this date)''
* IFE as of Oct 15, 2020 = <u>2.0</u>
:: ''(policies A and B are active on this date)''
* IFE as of Mar 31, 2021 = <u>2.0</u>
:: ''(policies B and C are active on this date)''
* IFE as of Nov 30, 2021 = <u>1.0</u>
:: ''(only policy C is active on this date)''

====Geometric Interpretation====

The source text explains the geometric interpretation of writing and earning exposures very well. This is extremely important for later chapters. I've linked directly to the relevant section of the Werner & Modlin text below. Begin reading on the lower part of page 51 where you see ''Aggregation of Exposures'' highlighted. Continue to the top of page 60 and stop at ''Calculation of Blocks of Expoures'' which is also highlighted.

: [https://www.battleacts5.ca/pdf/Werner_(p063-072)_exposure_aggregation.pdf <span style="color: white; font-size: 12px; background-color: sandybrown; border: solid; border-width: 2px; border-radius: 10px; border-color: sandybrown; padding: 1px 3px 1px 3px; margin: 0px;">'''''Geometric Interpretation of Exposure Writing'''''</span>] &nbsp;&nbsp; &larr; pay very close attention to the diagrams

Here are the key points for the CY version of the diagram:

* The CY diagram is divided into <span style="color: red;">'''squares'''</span> representing CYs. ''(Vertical lines separate the CYs.)''
* The diagonal lines represent <span style="color: green;">'''rate changes'''</span> and the parallelogram areas between diagonals are groups of policies with the same rates
* The vertical axis represents <span style="color: green;">'''% of policy term expired'''</span>

The only difference for the PY version is that PYs are represented by <span style="color: red;">'''parallelograms'''</span>, not squares. ''(Since the rate changes are parallel diagonals, calculations on PY data are easier than calculations on CY data. You'll see why in later chapters.)''

There are also some formulas you need to keep in mind:

:{| class='wikitable'
|-
|| '''CY & PY Aggregation for <u>individual</u> policies at a certain point in time:''' &nbsp;&nbsp; WE &nbsp; = &nbsp; EE + UEE
|}

This is easy to understand at the level of an individual policy. An annual policy has WE = 1.0. At policy inception, EE = 0.0 and UEE 1.0. After one month, EE = 1/12 and UEE = 11/12. After 2 months, EE = 2/12 and uUEE = 10/12. Continue this pattern to the end of the policy term where we then have EE = 1.0 and UEE = 0.0.

For '''groups''' of policies, the above formula holds for PY aggregation. For CY aggregation however, it works a little differently.

:{| class='wikitable'
|-
|| '''CY Aggregation for <u>groups</u> of policies:''' &nbsp;&nbsp; WE &nbsp; = &nbsp; EE &nbsp; + &nbsp; (UEE<sub>start</sub> &ndash; UEE<sub>end</sub>)
|}

The reason the formula is different is that individual policies within a group all have different effective dates and expiration dates. As a very simple example, suppose you want EE for CY 2020, CY<sub>20</sub>(EE), and you're given:

* CY<sub>20</sub>(WE) = 100
* CY<sub>20:start</sub>(UEE) = 90
* CY<sub>20:end</sub>(UEE) = 140

Then by rearranging the above formula:

* CY<sub>20</sub>(EE) = 100 - (90 -140) = 100 - (-50) = <u>150</u>

===Calculation of Blocks of Exposures===

====Example D: Blocks of Exposures====

The earlier examples of calculating WE, EE, UEE, and IFE were done at the level of an individual policy. If an insurer has millions of policies however, older computer systems may not have the capacity to perform calculations at this level of detail. In that case we can make a simplifying assumption and do the calculations on '''blocks of exposures''' rather than individual exposures.

For example, suppose an insurer writes 240 annual policies in January 2020. The effective dates for these policies would likely be spread over the whole month. With sufficient resources, we could calculate WE, EE, UEE, and IFE for each individual policy, for any time period or "as of" date and sum the results to get the total for the whole block. '''Alternately''', we can make the simplifying assumption that every policy was written in the middle of the month, Jan 15, and calculate WE, EE, UEE, IFE based on that assumption. This is reasonable if policies were written uniformly over the whole month.

* WE<sub>Jan</sub> = 240
: &rarr; This is the same for both the individual and block calculation. 
* EE<sub>Jan</sub> = 1/24 x 240 = <u>10</u>
: &rarr; An annual policy earns over 12 months. If earnings are assumed to start Jan 15, 2020, they will end Jan 15, 2021. Think of this one-year period as being divided into 24 half-month periods. This is the 24<sup>th</sup>s method, also called the 15<sup>th</sup> of the month rule.
* IFE<sub>Jan 1</sub> = 0
: &rarr; Some of the Jan policies may have been written on Jan 1 but we're assuming all policies were written on the 15<sup>th</sup> so no IFE for Jan 1
* IFE<sub>Jan 15</sub> = 240
: &rarr; This is also true for every day in Jan after the 15<sup>th</sup>
Taking this block of exposures forward to Feb, we have:
* WE<sub>Feb</sub> = 0
: &rarr; We would probably have 240 more exposures written in Feb but we're only following the Jan block for now
* EE<sub>Feb</sub> = 2/24 x 240 = <u>20</u>
: &rarr; We have a full month of earnings in Feb, which is 1/12 or 2/24 of the annual term.
* IFE<sub>an day of Feb</sub> = 240
: &rarr; The policies written in Jan are in force throughout all of Feb

Now just continue the pattern. The example from the text, which is linked below, assumes the insurer writes 240 exposures every month. Each row in their table shows the contribution from each month to the total CY amount for 2010, 2011, and 2012. 

Notice that under the 24<sup>th</sup>s method, the earnings of annual policies are spread over 13 months, the first and last portions being half-months. When Alice-the Actuary was an intern, that struck her as a needless complication. If it's an arbitrary assumption, why not instead assume policies are written on the 1<sup>st</sup> of the month. Then an annual policy would fully earn over a 12-month period, one month at a time. It probably goes back many years to when data was available only on an annual basis. And going forward, it may not matter because modern computer systems can perform the calculations exactly by using individual policies.

Take 5 or 10 minutes to make sure you understand all the numbers in these tables:

: [https://www.battleacts5.ca/pdf/Werner_(p072-073)_exposure_aggregation_blocks.pdf <span style="color: white; font-size: 12px; background-color: sandybrown; border: solid; border-width: 2px; border-radius: 10px; border-color: sandybrown; padding: 1px 3px 1px 3px; margin: 0px;">'''''Calculations of blocks of exposures'''''</span>]

====A Hard Exam Problem====

Here's a problem with 3 twists. Most old exam questions ask you to calculate CY values for individual policies assuming uniform earnings of exposures over the policy period. Part (a) of the problem below asks for <strong>PY</strong> values for a <strong>block</strong> of policies with <strong>non-uniform</strong> earnings.  And the solution to part (a) in the examiner's report is not very helpful. You should give it a try before looking at the solution, but don't worry too much if you have trouble solving it.

: [https://www.battleacts5.ca/pdf/Exam_(2019_2-Fall)/(2019_2-Fall)_(01).pdf <span style='font-size: 12px; background-color: yellow; border: solid; border-width: 1px; border-radius: 5px; padding: 2px 5px 2px 5px; margin: 5px;'>E</span>] <span style='color: red;'>'''(2019.Fall #1)'''</span>

Let's fill in a few of the details that are missing from the examiner's report solution. Aside from the table providing written and earned exposures by calendar quarter, you're given 6 pieces of information in bullet points underneath. There are 2 points that you absolutely '''must''' pay attention to:

* The quarterly earnings pattern was set by analyzing historical experience across the industry and is <u>not uniform</u>
* All policies are written on the first day of the quarter ''(rather than mid-period as for the 24<sup>th</sup>s rule for monthly blocks)''

You're asked to find PY<sub>17</sub>(EE) as of Mar 31, 2018. Note that you're essentially given CY<sub>17</sub>(EE) as of Mar 31, 2018:

* CY<sub>17</sub>(EE) as of Mar 31, 2018 &nbsp; = &nbsp; 5.00 + 247.50 + 427.50 + 52.50 &nbsp; = &nbsp; <u>732.5</u>

For CY aggregation, it doesn't matter what happens after the end of the year so the earnings for 2018 Q1 are irrelevant for CY total. Recall that CY data is fixed and doesn't change after the end of the year. That is <u>not</u> the case for the PY total however. That's because PYs extend <u>past</u> the end of the actual CY. Policy years take 2 years to close so the total for PY<sub>17</sub>(EE) will not be known until Dec 31, 2018. We don't have to go that far here because the "as of" date is Mar 31, 2018.

<span style="color: red;">'''Ok, so how do we actually solve the problem?'''</span> We have to back into the industry earnings pattern using the given written and earned exposures. What we actually need is IFE (in-force) exposures, but we can easily get that from WE:

: '''2017-Q1'''
:* CQ<sub>17-Q1</sub>(EE) / CQ<sub>17-Q1</sub>(IFE) &nbsp; = &nbsp; 5.00 / 100 &nbsp; = &nbsp; <u>0.05</u>
:: ''(this means that 5% of the in-force exposures were earned in 2017-Q1)''

Let's continue and calculate the industry pattern for the remaining 3 quarters:

: '''2017-Q2'''
:* CQ<sub>17-Q2</sub>(EE) / CQ<sub>17 Q1,2</sub>(IFE) &nbsp; = &nbsp; 247.5 / (100 + 450) &nbsp; = &nbsp; <u>0.45</u>
:: ''(45% of the in-force exposures were earned in 2017-Q2)''

: '''2017-Q3'''
:* CQ<sub>17-Q3</sub>(EE) / CQ<sub>17-Q1,2,3</sub>(IFE) &nbsp; = &nbsp; 427.5 / (100 + 450 + 400) &nbsp; = &nbsp; <u>0.45</u>
:: ''(45% of the in-force exposures were earned in 2017-Q3

: '''2017-Q4'''
:* CQ<sub>17-Q4</sub>(EE) / CQ<sub>17-Q1,2,3,4</sub>(IFE) &nbsp; = &nbsp; 52.5 / (100 + 450 + 400 + 100) &nbsp; = &nbsp; <u>0.05</u>
:: ''(45% of the in-force exposures were earned in 2017-Q4)''

So we now have the earnings pattern for all 4 calendar quarters: 0.05, 0.45, 0.45, 0.05. Observe:

* All WE from 2017-Q1 will be earned by Mar 31, 2018
* All WE from 2017-Q2 will be earned by Mar 31, 2018 ''(because we were told all policies were written on the first day of the quarter)''
* WE from 2017-Q3 will be earned during Q3, Q4, and 2018-Q1 according to the pattern: 0.45, 0.05, 0.05 which sums to <u>0.55</u>
* WE from 2017-Q4 will be earned during Q4, and 2018-Q1 according to the pattern: 0.05, 0.05 which sums to <u>0.10</u>

Putting this all together givesL

:* PY<sub>17</sub>(EE) at Mar 31, 2018
::: = &nbsp; CY<sub>17-Q1</sub>(WE) + CY<sub>17-Q2</sub>(WE) + (0.55 x CY<sub>17-Q3</sub>(WE)) + (0.10 x CY<sub>17-Q1</sub>(WE))
::: = &nbsp; 100 + 450 + (0.45 x 400) + (0.10 x 100)
::: = &nbsp; <u>780</u> <span style="color: red;">&nbsp;&nbsp; &larr; final answer</span>

Alice thinks that's a lot of work for 0.5 pts. And note this is question 1(a) on the exam. That's a very tough way to start the exam. :-(

[https://www.battleacts5.ca/FC.php?selectString=**&filter=both&sortOrder=natural&colorFlag=allFlag&colorStatus=allStatus&priority=importance-high&subsetFlag=miniQuiz&prefix=Werner04&suffix=Exposures&section=all&subSection=all&examRep=all&examYear=all&examTerm=all&quizNum=4<span style="font-size: 20px; background-color: aqua; border: solid; border-width: 1px; border-radius: 10px; padding: 2px 10px 2px 10px; margin: 10px;">'''mini BattleQuiz 4]'''</span> <span style="color: red;">'''You must be <u>logged in</u> or this will not work.'''</span>

===Exposure Trend===

The topic of trending is covered in detail in later chapters. A core component of the ratemaking process is projecting (trending) premiums, exposures, and losses to a date in the future when policies are predicted to be sold. The simple fact of inflation is one reason trending premiums and losses is necessary: $1 of premiums or losses today does not have the same value as $1 in the future. Exposures are treated a little differently however because not all exposure bases are dollar-values.

: '''Inflation-sensitive exposures bases'''
:* payroll ''(Worker's Compensation)''
:* sales revenue ''(commercial General Liability)''

: '''Non-Inflation-sensitive exposures bases'''
:* car-years ''(personal auto)''
:* house-years ''(homeowners)''

Exposure bases that are inflation-sensitive like payroll likely need to be trended whereas non-inflation-sensitive bases like car-years do not.

[https://www.battleacts5.ca/FC.php?selectString=**&filter=both&sortOrder=natural&colorFlag=allFlag&colorStatus=allStatus&priority=importance-high&subsetFlag=miniQuiz&prefix=Werner03&suffix=Exposures&section=all&subSection=all&examRep=all&examYear=all&examTerm=all&quizNum=all<span style="font-size: 20px; background-color: lightgreen; border: solid; border-width: 1px; border-radius: 10px; padding: 2px 10px 2px 10px; margin: 10px;">'''Full BattleQuiz]'''</span> <span style="color: red;">'''You must be <u>logged in</u> or this will not work.'''</span>

==POP QUIZ ANSWERS==

Objective of data aggregation for ratemaking:

: <span style="color: red;">'''M'''</span>atch losses and premiums as closely as possible
: <span style="color: red;">'''M'''</span>inimize the cost of collecting the data
: <span style="color: red;">'''R'''</span>ecent? (Use the most recent data available)

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