Experience summaries"

In actxps: Create Actuarial Experience Studies: Prepare Data, Summarize Results, and Create Reports

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

After experience data has been prepared for analysis, the next step is to summarize results. The actxps package's workhorse function for summarizing termination experience is exp_stats(). This function returns an exp_df object, which is a type of data frame containing additional attributes about the experience study.

At a minimum, an exp_df includes:

• The number of claims (termination events, n_claims)
• The amount of claims (weighted by a variable of the user's choice; more on this below, claims)
• The total exposure (exposure)
• The observed termination rate (q_obs)

Optionally, an exp_df can also include:

• Any grouping variables attached to the input data
• Expected termination rates and actual-to-expected (A/E) ratios (ae_*)
• Limited fluctuation credibility estimates (credibility) and credibility-adjusted expected termination rates (adj_*)

To demonstrate this function, we're going to use a data frame containing simulated census data for a theoretical deferred annuity product that has an optional guaranteed income rider. Before exp_stats() can be used, we must convert our census data into exposure records using the expose() function^[See vignette('exposures') for more information on creating exposure records.]. In addition, let's assume we're interested in studying surrender rates, so we'll pass the argument target_status = 'Surrender' to expose().

library(actxps)
library(dplyr)

exposed_data <- expose(census_dat, end_date = "2019-12-31",
                       target_status = "Surrender")

The exp_stats() function

To use exp_stats(), pass it a data frame of exposure-level records, ideally of type exposed_df (the object class returned by the expose() family of functions).

exp_stats(exposed_data)

The results show us that we specified no groups, which is why the output data is a single row. In addition, we can see that we're looking at surrender rates through the end of 2019, which exp_stats() inferred from exposed_data.

The number of claims (n_claims) is equal to the number of "Surrender" statuses in exposed_data. Since we didn't specify any weighting variable, the amount of claims (claims) equals the number of claims.

(amount <- sum(exposed_data$status == "Surrender"))

The total exposure (exposure) is equal to the sum of the exposures in exposed_data. Had we specified a weighting variable, this would be equal to the sum of weighted exposures.

(sum_expo <- sum(exposed_data$exposure))

Lastly, the observed termination rate (q_obs) equals the amount of claims divided by the exposures.

amount / sum_expo

Grouped data

If the data frame passed into exp_stats() is grouped using dplyr::group_by(), the resulting output will contain one record for each unique group.

In the following, exposed_data is grouped by policy year before being passed to exp_stats(). This results in one row per policy year found in the data.

exposed_data |> 
  group_by(pol_yr) |> 
  exp_stats()

Multiple grouping variables are allowed. Below, the presence of an income guarantee (inc_guar) is added as a second grouping variable.

exposed_data |> 
  group_by(inc_guar, pol_yr) |> 
  exp_stats()

Target status

The target_status argument of exp_stats() specifies which status levels count as claims in the experience study summary. If the data passed to exp_stats() is an exposed_df object that already has a specified target status (via a prior call to expose()), then this argument is not necessary because the target status is automatically inferred.

Even if the target status exists on the input data, it can be overridden. However care should be taken to ensure that exposure values in the data are appropriate for the new status.

Using the example data, a total termination rate can be estimated by including both death and surrender statuses in target_status. To ensure exposures are accurate, an adjustment is made to fully expose deaths prior to calling exp_stats()^[This adjustment is not necessary on surrenders because the expose() function previously did this for us.].

exposed_data |> 
  mutate(exposure = ifelse(status == "Death", 1, status)) |> 
  group_by(pol_yr) |> 
  exp_stats(target_status = c("Surrender", "Death"))

Weighted results

Experience studies often weight output by key policy values. Examples include account values, cash values, face amount, premiums, and more. Weighting can be accomplished by passing the name of a weighting column to the wt argument of exp_stats().

Our sample data contains a column called premium that we can weight by. When weights are supplied, the claims, exposure, and q_obs columns will be weighted. If expected termination rates are supplied (see below), these rates and A/E values will also be weighted.^[When weights are supplied, additional columns are created containing the sum of weights, the sum of squared weights, and the number of records. These columns are used for re-summarizing the data (see the "Summary method" section on this page).]

exposed_data |> 
  group_by(pol_yr) |> 
  exp_stats(wt = 'premium')

Expected values and A/E ratios

As common metric in experience studies is the actual-to-expected, or A/E ratio.

$$ A/E\ ratio = \frac{observed\ value}{expected\ value} $$

If the data passed to exp_stats() has one or more columns containing expected termination rates, A/E ratios can be calculated by passing the names of these columns to the expected argument.

Let's assume we have two sets of expected rates. The first set is a vector that varies by policy year. The second set is either 1.5% or 3.0% depending on whether the policy has a guaranteed income benefit. First, we need to attach these assumptions to our exposure data. We will use the names expected_1 and expected_2. Then we pass these names to the expected argument when we call exp_stats().

In the output, 4 new columns are created for expected rates and A/E ratios.

expected_table <- c(seq(0.005, 0.03, length.out = 10), 0.2, 0.15, rep(0.05, 3))

# using 2 different expected termination assumption sets
exposed_data <- exposed_data |>
  mutate(expected_1 = expected_table[pol_yr],
         expected_2 = ifelse(exposed_data$inc_guar, 0.015, 0.03))

exp_res <- exposed_data |>
  group_by(pol_yr, inc_guar) |>
  exp_stats(expected = c("expected_1", "expected_2"))


exp_res |> 
  select(pol_yr, inc_guar, q_obs, expected_1, expected_2, 
         ae_expected_1, ae_expected_2)

As noted above, if weights are passed to exp_stats() then A/E ratios will also be weighted.

exposed_data |>
  group_by(pol_yr, inc_guar) |>
  exp_stats(expected = c("expected_1", "expected_2"), 
            wt = "premium") |> 
  select(pol_yr, inc_guar, q_obs, expected_1, expected_2, 
         ae_expected_1, ae_expected_2)

Credibility

If the credibility argument is set to TRUE, exp_stats() will produce an estimate of partial credibility under the Limited Fluctuation credibility method (also known as Classical Credibility) assuming a binomial distribution of claims.^[See Herzog, Thomas (1999). Introduction to Credibility Theory for more information on Limited Fluctuation Credibility.]

exposed_data |> 
  group_by(pol_yr, inc_guar) |>
  exp_stats(credibility = TRUE) |> 
  select(pol_yr, inc_guar, claims, q_obs, credibility)

Under the default arguments, credibility calculations assume a 95% confidence of being within 5% of the true value. These parameters can be overridden using the conf_level and cred_r arguments, respectively.

exposed_data |> 
  group_by(pol_yr, inc_guar) |>
  exp_stats(credibility = TRUE, conf_level = 0.98, cred_r = 0.03) |> 
  select(pol_yr, inc_guar, claims, q_obs, credibility)

If expected values are passed to exp_stats() and credibility is set to TRUE, then the output will also contain credibility-weighted expected values:

$$ q^{adj} = Z^{cred} \times q^{obs} + (1-Z^{cred}) \times q^{exp} $$ where,

• $q^{adj}$ = credibility-weighted estimate
• $Z^{cred}$ = partial credibility factor
• $q^{obs}$ = observed termination rate
• $q^{exp}$ = expected termination rate

exposed_data |> 
  group_by(pol_yr, inc_guar) |>
  exp_stats(credibility = TRUE, expected = "expected_1") |> 
  select(pol_yr, inc_guar, claims, q_obs, credibility, adj_expected_1, 
         expected_1, ae_expected_1)

Confidence intervals

If conf_int is set to TRUE, exp_stats() will produce lower and upper confidence interval limits for the observed termination rate.

exposed_data |> 
  group_by(pol_yr, inc_guar) |>
  exp_stats(conf_int = TRUE) |> 
  select(pol_yr, inc_guar, q_obs, q_obs_lower, q_obs_upper)

If no weighting variable is passed to wt, confidence intervals will be constructed assuming a binomial distribution of claims. However, if a weighting variable is supplied, a normal distribution for aggregate claims will be assumed with a mean equal to observed claims and a variance equal to:

$$ Var(S) = E(N) \times Var(X) + E(X)^2 \times Var(N) $$

Where S is the aggregate claim random variable, X is the weighting variable assumed to follow a normal distribution, and N is a binomial random variable for the number of claims.

The default confidence level is 95%. This can be changed using the conf_level argument. Below, tighter confidence intervals are constructed by decreasing the confidence level to 90%.

exposed_data |> 
  group_by(pol_yr, inc_guar) |>
  exp_stats(conf_int = TRUE, conf_level = 0.9) |> 
  select(pol_yr, inc_guar, q_obs, q_obs_lower, q_obs_upper)

If expected values are passed to expected, the output will also contain confidence intervals around any actual-to-expected ratios.

exposed_data |> 
  group_by(pol_yr, inc_guar) |>
  exp_stats(conf_int = TRUE, expected = "expected_1") |> 
  select(pol_yr, inc_guar, starts_with("ae_"))

Lastly, if credibility is TRUE and expected values are passed to expected, confidence intervals will also be calculated for any credibility-weighted termination rates.

Miscellaneous

Summary method

As noted above, the result of exp_stats() is an exp_df object. If the summary() function is applied to an exp_df object, the data will be summarized again and return a higher level exp_df object.

If no additional arguments are passed, summary() returns a single row of aggregate results.

summary(exp_res)

If additional variable names are passed to the summary() function, then the output will group the data by those variables. In our example, if pol_yr is passed to summary(), the output will contain one row per policy year.

summary(exp_res, pol_yr)

Similarly, if inc_guar is passed to summary(), the output will contain a row for each unique value in inc_guar.

summary(exp_res, inc_guar)

Column names

As a default, exp_stats() assumes the input data frame uses the following naming conventions:

• The exposure column is called exposure
• The status column is called status

These default names can be overridden using the col_exposure and col_status arguments.

For example, if the status column was called curr_stat in our data, we could write:

exposed_data |> 
  exp_stats(col_status = "curr_stat")

Applying exp_stats to a non-exposed_df data frame

exp_stats() can still work when given a non-exposed_df data frame. However, it will be unable to infer certain attributes like the target status and the study dates. For target status, all statuses except the first level are assumed to be terminations. Since this may not be desirable, a warning message will appear informing what statuses were assumed to be terminated.

not_exposed_df <- data.frame(exposed_data)

exp_stats(not_exposed_df)

If target_status is provided, no warning message will appear.

exp_stats(not_exposed_df, target_status = "Surrender")

Limitations

The exp_stats() function only supports termination studies. It does not contain support for transaction studies or studies with multiple changes from an active to an inactive status. For information on transaction studies, see vignette("transactions").

actxps documentation built on June 26, 2024, 9:07 a.m.