Exposure calculations

Census-level data refers to a data set wherein there is one row per policy. Exposure-level data expands census-level data such that there is one record per policy per observation period. Observation periods could be any meaningful period of time such as a policy year, policy month, calendar year, calendar quarter, calendar month, etc.

A common step in experience studies is converting census-level data into exposure-level data. The expose() family of functions assists with this task. Specifically, the expose() family:

If you already have exposure-level data available, the expose() functions are not necessary. However, we recommend converting your data to the exposed_df format using the function as_exposed_df().

Toy census data

To get started, we’re going to use a toy census data frame from the actxps package that contains 3 policies: one active, one that terminated due to death, and one that terminated due to surrender.

toy_census contains the 4 columns necessary to compute exposures:

pol_num: a unique identifier for individual policies
status: the policy status
issue_date: issue date
term_date: termination date, if any. Otherwise NA

library(actxps)
library(dplyr)

toy_census
#>   pol_num    status issue_date  term_date
#> 1       1    Active 2010-01-01       <NA>
#> 2       2     Death 2011-05-27 2020-09-14
#> 3       3 Surrender 2009-11-10 2022-02-25

Policy year exposures

Let’s assume we’re performing an experience study as of 2022-12-31 and we’re interested in policy year exposures. Here’s what we should expect for our 3 policies.

Policy 1 was issued on January 1, 2010 and has not terminated. Therefore we expect 13 exposure years.
Policy 2 was issued on May 27, 2011 and was terminated in 2020 due to death. The death occurred after the 9th policy anniversary, therefore we expect 9 fully exposed years and a partial exposure in the 10th year.
Policy 3 was issued on November 10, 2009 and was terminated in 2022 due to surrender. The surrender occurred after the 12th policy anniversary, therefore we expect 12 fully exposed years and a partial exposure in the 13th year.

To calculate exposures, we pass our data to the expose() function and we specify a study end_date.


exposed_data <- expose(toy_census, end_date = "2022-12-31")

This creates an exposed_df object, which is a type of data frame with some additional attributes related to the experience study.

is_exposed_df(exposed_data)
#> [1] TRUE

Let’s examine what happened to each policy.

Policy 1: As expected, there are 13 rows for this policy. New columns were added for the policy year (pol_yr), date ranges (pol_date_yr, pol_date_yr_end), and exposure. All exposures are 100% since this policy was active for all 13 years.

When the data is printed, additional attributes from the exposed_df class are displayed.

exposed_data |> filter(pol_num == 1)
#> Exposure data
#> 
#>  Exposure type: policy_year 
#>  Target status:  
#>  Study range: 1900-01-01 to 2022-12-31
#> 
#> # A tibble: 13 × 8
#>    pol_num status issue_date term_date pol_yr pol_date_yr pol_date_yr_end
#>      <int> <fct>  <date>     <date>     <int> <date>      <date>         
#>  1       1 Active 2010-01-01 NA             1 2010-01-01  2010-12-31     
#>  2       1 Active 2010-01-01 NA             2 2011-01-01  2011-12-31     
#>  3       1 Active 2010-01-01 NA             3 2012-01-01  2012-12-31     
#>  4       1 Active 2010-01-01 NA             4 2013-01-01  2013-12-31     
#>  5       1 Active 2010-01-01 NA             5 2014-01-01  2014-12-31     
#>  6       1 Active 2010-01-01 NA             6 2015-01-01  2015-12-31     
#>  7       1 Active 2010-01-01 NA             7 2016-01-01  2016-12-31     
#>  8       1 Active 2010-01-01 NA             8 2017-01-01  2017-12-31     
#>  9       1 Active 2010-01-01 NA             9 2018-01-01  2018-12-31     
#> 10       1 Active 2010-01-01 NA            10 2019-01-01  2019-12-31     
#> 11       1 Active 2010-01-01 NA            11 2020-01-01  2020-12-31     
#> 12       1 Active 2010-01-01 NA            12 2021-01-01  2021-12-31     
#> 13       1 Active 2010-01-01 NA            13 2022-01-01  2022-12-31     
#> # ℹ 1 more variable: exposure <dbl>

Policy 2: There are 10 rows for this policy. The first 9 periods show the policy in an active status and the termination date (term_date) is set to NA. The last period includes the final status of “Death” and the actual termination date. The last exposure is less than one because roughly a third of a year elapsed between the last anniversary date on 2020-05-27 and the termination date on 2020-09-14.

exposed_data |> filter(pol_num == 2)
#> Exposure data
#> 
#>  Exposure type: policy_year 
#>  Target status:  
#>  Study range: 1900-01-01 to 2022-12-31
#> 
#> # A tibble: 10 × 8
#>    pol_num status issue_date term_date  pol_yr pol_date_yr pol_date_yr_end
#>      <int> <fct>  <date>     <date>      <int> <date>      <date>         
#>  1       2 Active 2011-05-27 NA              1 2011-05-27  2012-05-26     
#>  2       2 Active 2011-05-27 NA              2 2012-05-27  2013-05-26     
#>  3       2 Active 2011-05-27 NA              3 2013-05-27  2014-05-26     
#>  4       2 Active 2011-05-27 NA              4 2014-05-27  2015-05-26     
#>  5       2 Active 2011-05-27 NA              5 2015-05-27  2016-05-26     
#>  6       2 Active 2011-05-27 NA              6 2016-05-27  2017-05-26     
#>  7       2 Active 2011-05-27 NA              7 2017-05-27  2018-05-26     
#>  8       2 Active 2011-05-27 NA              8 2018-05-27  2019-05-26     
#>  9       2 Active 2011-05-27 NA              9 2019-05-27  2020-05-26     
#> 10       2 Death  2011-05-27 2020-09-14     10 2020-05-27  2021-05-26     
#> # ℹ 1 more variable: exposure <dbl>

Policy 3: There are 13 rows for this policy. The first 12 periods show the policy in an active status and the termination date (term_date) is set to NA. The last period includes the final status of “Surrender” and the actual termination date. The last exposure is less than one because the roughly a third of a year elapsed between the last anniversary date on 2021-11-10 and the termination date on 2022-02-25.

exposed_data |> filter(pol_num == 3)
#> Exposure data
#> 
#>  Exposure type: policy_year 
#>  Target status:  
#>  Study range: 1900-01-01 to 2022-12-31
#> 
#> # A tibble: 13 × 8
#>    pol_num status    issue_date term_date  pol_yr pol_date_yr pol_date_yr_end
#>      <int> <fct>     <date>     <date>      <int> <date>      <date>         
#>  1       3 Active    2009-11-10 NA              1 2009-11-10  2010-11-09     
#>  2       3 Active    2009-11-10 NA              2 2010-11-10  2011-11-09     
#>  3       3 Active    2009-11-10 NA              3 2011-11-10  2012-11-09     
#>  4       3 Active    2009-11-10 NA              4 2012-11-10  2013-11-09     
#>  5       3 Active    2009-11-10 NA              5 2013-11-10  2014-11-09     
#>  6       3 Active    2009-11-10 NA              6 2014-11-10  2015-11-09     
#>  7       3 Active    2009-11-10 NA              7 2015-11-10  2016-11-09     
#>  8       3 Active    2009-11-10 NA              8 2016-11-10  2017-11-09     
#>  9       3 Active    2009-11-10 NA              9 2017-11-10  2018-11-09     
#> 10       3 Active    2009-11-10 NA             10 2018-11-10  2019-11-09     
#> 11       3 Active    2009-11-10 NA             11 2019-11-10  2020-11-09     
#> 12       3 Active    2009-11-10 NA             12 2020-11-10  2021-11-09     
#> 13       3 Surrender 2009-11-10 2022-02-25     13 2021-11-10  2022-11-09     
#> # ℹ 1 more variable: exposure <dbl>

Study start date

The previous section only supplied data and a study end_date to expose(). These are the minimum required arguments for the function. Optionally, a start_date can be supplied that will drop exposure periods that begin before a specified date.

expose(toy_census, end_date = "2022-12-31", start_date = "2019-12-31")
#> Exposure data
#> 
#>  Exposure type: policy_year 
#>  Target status:  
#>  Study range: 2019-12-31 to 2022-12-31
#> 
#> # A tibble: 6 × 8
#>   pol_num status    issue_date term_date  pol_yr pol_date_yr pol_date_yr_end
#>     <int> <fct>     <date>     <date>      <int> <date>      <date>         
#> 1       1 Active    2010-01-01 NA             11 2020-01-01  2020-12-31     
#> 2       1 Active    2010-01-01 NA             12 2021-01-01  2021-12-31     
#> 3       1 Active    2010-01-01 NA             13 2022-01-01  2022-12-31     
#> 4       2 Death     2011-05-27 2020-09-14     10 2020-05-27  2021-05-26     
#> 5       3 Active    2009-11-10 NA             12 2020-11-10  2021-11-09     
#> 6       3 Surrender 2009-11-10 2022-02-25     13 2021-11-10  2022-11-09     
#> # ℹ 1 more variable: exposure <dbl>

Target status

Most experience studies use the annual exposure method which allocates a full period of exposure for the particular termination event of interest in the scope of the study.

The intuition for this approach is simple: let’s assume we have an unrealistically small study with a single data point for one policy over the course of one year. Let’s assume that policy terminated due to surrender half way through the year.

If we don’t apply the annual exposure method, we would calculate a termination rate as:

\[ q^{surr} = \frac{claims}{exposures} = \frac{1}{0.5} = 200\% \]

A termination rate of 200% doesn’t make any sense. Under the annual exposure method we would see a rate of 100%, which is intuitive.

\[ q^{surr} = \frac{claims}{exposures} = \frac{1}{1} = 100\% \]

The annual exposure method can be applied by passing a character vector of target statuses to the expose() function.

Let’s assume we are performing a surrender study.

exposed_data2 <- expose(toy_census, end_date = "2022-12-31", 
                        target = "Surrender")

Now let’s verify that the exposure on the surrendered policy increased to 100% in the last exposure period.

exposed_data2 |> 
  group_by(pol_num) |> 
  slice_max(pol_yr)
#> Exposure data
#> 
#>  Exposure type: policy_year 
#>  Target status: Surrender 
#>  Study range: 1900-01-01 to 2022-12-31
#> 
#> # A tibble: 3 × 8
#> # Groups:   pol_num [3]
#>   pol_num status    issue_date term_date  pol_yr pol_date_yr pol_date_yr_end
#>     <int> <fct>     <date>     <date>      <int> <date>      <date>         
#> 1       1 Active    2010-01-01 NA             13 2022-01-01  2022-12-31     
#> 2       2 Death     2011-05-27 2020-09-14     10 2020-05-27  2021-05-26     
#> 3       3 Surrender 2009-11-10 2022-02-25     13 2021-11-10  2022-11-09     
#> # ℹ 1 more variable: exposure <dbl>

Other exposures periods

The default exposure basis used by expose() is policy years. Using the arguments cal_expo and expo_length other exposure periods can be used.

Calendar years

If cal_expo is set to TRUE, calendar year exposures will be calculated.

Looking at the second policy, we can see that the first year is left-censored because the policy was issued two-fifths of the way through the year, and the last period is right-censored because the policy terminated roughly seven-tenths of the way through the year.


toy_census[2, ] |> 
  expose(end_date = "2022-12-31", cal_expo = TRUE, target_status = "Surrender")
#> Exposure data
#> 
#>  Exposure type: calendar_year 
#>  Target status: Surrender 
#>  Study range: 1900-01-01 to 2022-12-31
#> 
#> # A tibble: 10 × 7
#>    pol_num status issue_date term_date  cal_yr     cal_yr_end exposure
#>      <int> <fct>  <date>     <date>     <date>     <date>        <dbl>
#>  1       2 Active 2011-05-27 NA         2011-01-01 2011-12-31    0.6  
#>  2       2 Active 2011-05-27 NA         2012-01-01 2012-12-31    1    
#>  3       2 Active 2011-05-27 NA         2013-01-01 2013-12-31    1    
#>  4       2 Active 2011-05-27 NA         2014-01-01 2014-12-31    1    
#>  5       2 Active 2011-05-27 NA         2015-01-01 2015-12-31    1    
#>  6       2 Active 2011-05-27 NA         2016-01-01 2016-12-31    1    
#>  7       2 Active 2011-05-27 NA         2017-01-01 2017-12-31    1    
#>  8       2 Active 2011-05-27 NA         2018-01-01 2018-12-31    1    
#>  9       2 Active 2011-05-27 NA         2019-01-01 2019-12-31    1    
#> 10       2 Death  2011-05-27 2020-09-14 2020-01-01 2020-12-31    0.705

Quarters, months, and weeks

The length of the exposure period can be decreased by passing "quarter", "month", or "week" to the expo_length argument. This can be used with policy or calendar-based exposures.

toy_census[2, ] |> 
  expose(end_date = "2022-12-31", 
         cal_expo = TRUE,
         expo_length = "quarter", 
         target_status = "Surrender")
#> Exposure data
#> 
#>  Exposure type: calendar_quarter 
#>  Target status: Surrender 
#>  Study range: 1900-01-01 to 2022-12-31
#> 
#> # A tibble: 38 × 7
#>    pol_num status issue_date term_date cal_qtr    cal_qtr_end exposure
#>      <int> <fct>  <date>     <date>    <date>     <date>         <dbl>
#>  1       2 Active 2011-05-27 NA        2011-04-01 2011-06-30     0.385
#>  2       2 Active 2011-05-27 NA        2011-07-01 2011-09-30     1    
#>  3       2 Active 2011-05-27 NA        2011-10-01 2011-12-31     1    
#>  4       2 Active 2011-05-27 NA        2012-01-01 2012-03-31     1    
#>  5       2 Active 2011-05-27 NA        2012-04-01 2012-06-30     1    
#>  6       2 Active 2011-05-27 NA        2012-07-01 2012-09-30     1    
#>  7       2 Active 2011-05-27 NA        2012-10-01 2012-12-31     1    
#>  8       2 Active 2011-05-27 NA        2013-01-01 2013-03-31     1    
#>  9       2 Active 2011-05-27 NA        2013-04-01 2013-06-30     1    
#> 10       2 Active 2011-05-27 NA        2013-07-01 2013-09-30     1    
#> # ℹ 28 more rows

Convenience functions

The following functions are convenience wrappers around expose() that target a specific exposure type without specifying cal_expo and expo_length.

expose_py = exposures by policy year
expose_pq = exposures by policy quarter
expose_pm = exposures by policy month
expose_pw = exposures by policy week
expose_cy = exposures by calendar year
expose_cq = exposures by calendar quarter
expose_cm = exposures by calendar month
expose_cw = exposures by calendar week

Tidymodels recipe step

For machine learning feature engineering, the actxps package contains a function called step_expose() that is compatible with the recipes package from tidymodels. This function applies the expose() function within a recipe.


library(recipes)
#> Warning: package 'recipes' was built under R version 4.2.3
#> 
#> Attaching package: 'recipes'
#> The following object is masked from 'package:stats':
#> 
#>     step

expo_rec <- recipe(status ~ ., toy_census) |>
  step_expose(end_date = "2022-12-31", target_status = "Surrender",
              options = list(expo_length = "month")) |>
  prep()

expo_rec
#> 
#> ── Recipe ──────────────────────────────────────────────────────────────────────
#> 
#> ── Inputs
#> Number of variables by role
#> outcome:   1
#> predictor: 3
#> 
#> ── Training information
#> Training data contained 3 data points and 1 incomplete row.
#> 
#> ── Operations
#> • Exposed data based on policy months for target status Surrender: <none> |
#>   Trained

tidy(expo_rec, number = 1)
#> # A tibble: 1 × 4
#>   exposure_type target_status start_date end_date  
#>   <chr>         <chr>         <date>     <chr>     
#> 1 policy_month  Surrender     1900-01-01 2022-12-31

bake(expo_rec, new_data = NULL)
#> # A tibble: 416 × 7
#>    issue_date term_date status pol_mth pol_date_mth pol_date_mth_end exposure
#>    <date>     <date>    <fct>    <int> <date>       <date>              <dbl>
#>  1 2010-01-01 NA        Active       1 2010-01-01   2010-01-31              1
#>  2 2010-01-01 NA        Active       2 2010-02-01   2010-02-28              1
#>  3 2010-01-01 NA        Active       3 2010-03-01   2010-03-31              1
#>  4 2010-01-01 NA        Active       4 2010-04-01   2010-04-30              1
#>  5 2010-01-01 NA        Active       5 2010-05-01   2010-05-31              1
#>  6 2010-01-01 NA        Active       6 2010-06-01   2010-06-30              1
#>  7 2010-01-01 NA        Active       7 2010-07-01   2010-07-31              1
#>  8 2010-01-01 NA        Active       8 2010-08-01   2010-08-31              1
#>  9 2010-01-01 NA        Active       9 2010-09-01   2010-09-30              1
#> 10 2010-01-01 NA        Active      10 2010-10-01   2010-10-31              1
#> # ℹ 406 more rows

Miscellaneous

Column names

As a default, the expose() functions assume the census data frame uses the following naming conventions:

The policy number column is called pol_num
The status column is called status
The issue date column is called issue_date
The termination date column is called term_date

These default names can be overridden using the col_pol_num, col_status, col_issue_date, and col_term_date arguments.

For example, if the policy number column was called id in our census-level data, we could write:

expose(toy_census, end_date = "2022-12-31", 
       target = "Surrender",
       col_pol_num = "id")

Treatment of additional columns in the census data

If the census-level data contains other policy attributes like plan type or policy values, they will be broadcast across all exposure periods. Depending on the nature of the data, this may or may not be desirable. Constant policy attributes like plan type make sense to broadcast, but numeric values may or may not depending on the circumstances.

toy_census2 <- toy_census |> 
  mutate(plan_type = c("X", "Y", "Z"), 
         policy_value = c(100, 125, 90))

expose(toy_census2, end_date = "2022-12-31", 
       target = "Surrender")
#> Exposure data
#> 
#>  Exposure type: policy_year 
#>  Target status: Surrender 
#>  Study range: 1900-01-01 to 2022-12-31
#> 
#> # A tibble: 36 × 10
#>    pol_num status issue_date term_date plan_type policy_value pol_yr
#>      <int> <fct>  <date>     <date>    <chr>            <dbl>  <int>
#>  1       1 Active 2010-01-01 NA        X                  100      1
#>  2       1 Active 2010-01-01 NA        X                  100      2
#>  3       1 Active 2010-01-01 NA        X                  100      3
#>  4       1 Active 2010-01-01 NA        X                  100      4
#>  5       1 Active 2010-01-01 NA        X                  100      5
#>  6       1 Active 2010-01-01 NA        X                  100      6
#>  7       1 Active 2010-01-01 NA        X                  100      7
#>  8       1 Active 2010-01-01 NA        X                  100      8
#>  9       1 Active 2010-01-01 NA        X                  100      9
#> 10       1 Active 2010-01-01 NA        X                  100     10
#> # ℹ 26 more rows
#> # ℹ 3 more variables: pol_date_yr <date>, pol_date_yr_end <date>,
#> #   exposure <dbl>

If your experience study requires a numeric feature that varies over time (ex: policy values, crediting rates, etc.), you can always attach it to an exposed_df object using a join function.


# Illustrative example - assume `values` is a data frame containing the columns pol_num and pol_yr.

exposed_data |> 
  left_join(values, by = c("pol_num", "pol_yr"))

Stacking `exposed_df` objects

If you need to stack two exposed_df objects, vctrs::vec_rbind() is recommended over rbind() or dplyr::bind_rows(). The advantage of vctrs::vec_rbind() is that it will combine attributes across all exposed_df objects passed to the function. The study end date will be updated to maximum study end date. Similarly, the study start date will be set to the earliest study start date. Target statuses and transactions types will become a super set of all observed values. The other two functions will retain attributes from only the first object passed to them.

For example, below exposed_data2 contains study start and end dates that are before and after the study range in exposed_data. In addition, this object contains a target status of “Surrender” whereas exposed_data has none.

When vctrs::vec_rbind() is used to combine exposed_data and exposed_data2, the result combines attributes across both objects.


exposed_data2 <- expose(toy_census, 
                        end_date = "2023-12-31", 
                        start_date = "1890-01-01",
                        target_status = "Surrender")

vctrs::vec_rbind(exposed_data, exposed_data2)
#> Exposure data
#> 
#>  Exposure type: policy_year 
#>  Target status: Surrender 
#>  Study range: 1890-01-01 to 2023-12-31
#> 
#> # A tibble: 73 × 8
#>    pol_num status issue_date term_date pol_yr pol_date_yr pol_date_yr_end
#>      <int> <fct>  <date>     <date>     <int> <date>      <date>         
#>  1       1 Active 2010-01-01 NA             1 2010-01-01  2010-12-31     
#>  2       1 Active 2010-01-01 NA             2 2011-01-01  2011-12-31     
#>  3       1 Active 2010-01-01 NA             3 2012-01-01  2012-12-31     
#>  4       1 Active 2010-01-01 NA             4 2013-01-01  2013-12-31     
#>  5       1 Active 2010-01-01 NA             5 2014-01-01  2014-12-31     
#>  6       1 Active 2010-01-01 NA             6 2015-01-01  2015-12-31     
#>  7       1 Active 2010-01-01 NA             7 2016-01-01  2016-12-31     
#>  8       1 Active 2010-01-01 NA             8 2017-01-01  2017-12-31     
#>  9       1 Active 2010-01-01 NA             9 2018-01-01  2018-12-31     
#> 10       1 Active 2010-01-01 NA            10 2019-01-01  2019-12-31     
#> # ℹ 63 more rows
#> # ℹ 1 more variable: exposure <dbl>

If dplyr::bind_rows() were used instead, the attributes of exposed_data only are shown, which is likely incorrect.

dplyr::bind_rows(exposed_data, exposed_data2)
#> Exposure data
#> 
#>  Exposure type: policy_year 
#>  Target status:  
#>  Study range: 1900-01-01 to 2022-12-31
#> 
#> # A tibble: 73 × 8
#>    pol_num status issue_date term_date pol_yr pol_date_yr pol_date_yr_end
#>      <int> <fct>  <date>     <date>     <int> <date>      <date>         
#>  1       1 Active 2010-01-01 NA             1 2010-01-01  2010-12-31     
#>  2       1 Active 2010-01-01 NA             2 2011-01-01  2011-12-31     
#>  3       1 Active 2010-01-01 NA             3 2012-01-01  2012-12-31     
#>  4       1 Active 2010-01-01 NA             4 2013-01-01  2013-12-31     
#>  5       1 Active 2010-01-01 NA             5 2014-01-01  2014-12-31     
#>  6       1 Active 2010-01-01 NA             6 2015-01-01  2015-12-31     
#>  7       1 Active 2010-01-01 NA             7 2016-01-01  2016-12-31     
#>  8       1 Active 2010-01-01 NA             8 2017-01-01  2017-12-31     
#>  9       1 Active 2010-01-01 NA             9 2018-01-01  2018-12-31     
#> 10       1 Active 2010-01-01 NA            10 2019-01-01  2019-12-31     
#> # ℹ 63 more rows
#> # ℹ 1 more variable: exposure <dbl>

In order to stack exposed_df objects, the exposure period types and lengths must match. If they do not, an error will be thrown. For example, policy year exposure records cannot be combined with calendar month records.

Ordinary data frames can be stacked with exposed_df objects using dplyr::bind_rows() and rbind() (assuming all column names match). If the exposed_df object is the first argument, the exposed_df class will be preserved with its original attributes.

dplyr verb methods and `exposed_df` class persistence

The actxps package includes exposed_df methods for the dplyr verbs listed below. These methods ensure that the functions below will always return an exposed_df object.

dplyr::select()
dplyr::mutate()
dplyr::filter()
dplyr::arrange()
dplyr::group_by()
dplyr::ungroup()
dplyr::slice()
dplyr::rename()
dplyr::relocate()
dplyr::left_join()
dplyr::right_join()
dplyr::inner_join()
dplyr::full_join()
dplyr::semi_join()
dplyr::anti_join()

Generally speaking, any dplyr verbs that aren’t listed that return data frames will preserve the exposed_df class as long as the data is not grouped. If the data is grouped, the exposed_df class may not persist. If this creates problems with your code, there are two options:

If groups don’t matter when the function is applied to the data, ungroup() the data, call the function, and restore the groups with group_by().
If groups do matter when the function is applied to the data, convert the data to an ordinary data frame or tibble, call the function, and convert the data to an exposed_df using as_exposed_df().

Limitations

The expose() family does not support:

Studies with multiple changes from an active to an inactive status. For example, these functions do not support changes from an active status to an impaired status, and back to an active status.