# Experience summaries

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()` function1. 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)
    #> Experience study results
    #>
    #>  Groups:
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>
    #> # A tibble: 1 × 4
    #>   n_claims claims exposure  q_obs
    #>      <int>  <int>    <dbl>  <dbl>
    #> 1     2869   2869  132634. 0.0216

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"))
    #> [1] 2869

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))
    #> [1] 132634

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

    amount / sum_expo
    #> [1] 0.02163096

### 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()
    #> Experience study results
    #>
    #>  Groups: pol_yr
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>
    #> # A tibble: 15 × 5
    #>    pol_yr n_claims claims exposure   q_obs
    #>     <int>    <int>  <int>    <dbl>   <dbl>
    #>  1      1      102    102   19252. 0.00530
    #>  2      2      160    160   17715. 0.00903
    #>  3      3      124    124   16097. 0.00770
    #>  4      4      168    168   14536. 0.0116
    #>  5      5      164    164   12916. 0.0127
    #>  6      6      152    152   11376. 0.0134
    #>  7      7      164    164    9917. 0.0165
    #>  8      8      190    190    8448. 0.0225
    #>  9      9      181    181    6960. 0.0260
    #> 10     10      152    152    5604. 0.0271
    #> 11     11      804    804    4390. 0.183
    #> 12     12      330    330    2663. 0.124
    #> 13     13       99     99    1620. 0.0611
    #> 14     14       62     62     872. 0.0711
    #> 15     15       17     17     268. 0.0634

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()
    #> Experience study results
    #>
    #>  Groups: inc_guar, pol_yr
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>
    #> # A tibble: 30 × 6
    #>    inc_guar pol_yr n_claims claims exposure   q_obs
    #>    <lgl>     <int>    <int>  <int>    <dbl>   <dbl>
    #>  1 FALSE         1       56     56    7720. 0.00725
    #>  2 FALSE         2       92     92    7103. 0.0130
    #>  3 FALSE         3       67     67    6447. 0.0104
    #>  4 FALSE         4      123    123    5799. 0.0212
    #>  5 FALSE         5       97     97    5106. 0.0190
    #>  6 FALSE         6       96     96    4494. 0.0214
    #>  7 FALSE         7       92     92    3899. 0.0236
    #>  8 FALSE         8      103    103    3287. 0.0313
    #>  9 FALSE         9       87     87    2684. 0.0324
    #> 10 FALSE        10       60     60    2156. 0.0278
    #> # … with 20 more rows

### 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()`2.

    exposed_data |>
      mutate(exposure = ifelse(status == "Death", 1, status)) |>
      group_by(pol_yr) |>
      exp_stats(target_status = c("Surrender", "Death"))
    #> Experience study results
    #>
    #>  Groups: pol_yr
    #>  Target status: Surrender, Death
    #>  Study range: 1900-01-01 to 2019-12-31
    #>
    #> # A tibble: 15 × 5
    #>    pol_yr n_claims claims exposure  q_obs
    #>     <int>    <int>  <int>    <dbl>  <dbl>
    #>  1      1      290    290    20199 0.0144
    #>  2      2      325    325    18754 0.0173
    #>  3      3      292    292    17054 0.0171
    #>  4      4      329    329    15602 0.0211
    #>  5      5      329    329    13946 0.0236
    #>  6      6      334    334    12371 0.0270
    #>  7      7      297    297    10869 0.0273
    #>  8      8      340    340     9510 0.0358
    #>  9      9      308    308     7953 0.0387
    #> 10     10      260    260     6489 0.0401
    #> 11     11      894    894     6505 0.137
    #> 12     12      398    398     3753 0.106
    #> 13     13      131    131     2135 0.0614
    #> 14     14       89     89     1306 0.0681
    #> 15     15       23     23      544 0.0423

## 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.3

    exposed_data |>
      group_by(pol_yr) |>
      exp_stats(wt = 'premium')
    #> Experience study results
    #>
    #>  Groups: pol_yr
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>  Weighted by: premium
    #>
    #> # A tibble: 15 × 8
    #>    pol_yr n_claims claims  exposure   q_obs  .weight  .weight_sq .weight_n
    #>     <int>    <int>  <dbl>     <dbl>   <dbl>    <dbl>       <dbl>     <dbl>
    #>  1      1      102  83223 25312813. 0.00329 26301746 60742993234     19995
    #>  2      2      160 170058 23352482. 0.00728 24275265 56232848027     18434
    #>  3      3      124 123554 21246774. 0.00582 22201817 51746834383     16806
    #>  4      4      168 176751 19270854. 0.00917 20200019 47142441689     15266
    #>  5      5      164 173273 17228981. 0.0101  18134795 42887920479     13618
    #>  6      6      152 163034 15246521. 0.0107  16192950 38828949428     12067
    #>  7      7      164 153238 13328780. 0.0115  14159437 34291451913     10541
    #>  8      8      190 174200 11476432. 0.0152  12346124 30121310640      9130
    #>  9      9      181 187337  9546258. 0.0196  10420172 25781142118      7591
    #> 10     10      152 157603  7707064. 0.0204   8543150 20882643976      6185
    #> 11     11      804 856379  6093183. 0.141    6783273 16219955859      4897
    #> 12     12      330 383055  3883534. 0.0986   4525027 11191462577      3093
    #> 13     13       99 123357  2450266. 0.0503   2891573  7075934189      1937
    #> 14     14       62  75534  1339240. 0.0564   1821026  4655661820      1182
    #> 15     15       17  19168   401173. 0.0478    783146  1944755748       510

## 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_data2 <- exposed_data |>
      mutate(expected_1 = expected_table[pol_yr],
             expected_2 = ifelse(exposed_data$inc_guar, 0.015, 0.03))

    exp_res <- exposed_data2 |>
      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)
    #> Experience study results
    #>
    #>  Groups: pol_yr, inc_guar
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>  Expected values: expected_1, expected_2
    #>
    #> # A tibble: 30 × 7
    #>    pol_yr inc_guar   q_obs expected_1 expected_2 ae_expected_1 ae_expected_2
    #>     <int> <lgl>      <dbl>      <dbl>      <dbl>         <dbl>         <dbl>
    #>  1      1 FALSE    0.00725    0.005        0.03          1.45          0.242
    #>  2      1 TRUE     0.00399    0.005        0.015         0.798         0.266
    #>  3      2 FALSE    0.0130     0.00778      0.03          1.67          0.432
    #>  4      2 TRUE     0.00641    0.00778      0.015         0.824         0.427
    #>  5      3 FALSE    0.0104     0.0106       0.03          0.985         0.346
    #>  6      3 TRUE     0.00591    0.0106       0.015         0.560         0.394
    #>  7      4 FALSE    0.0212     0.0133       0.03          1.59          0.707
    #>  8      4 TRUE     0.00515    0.0133       0.015         0.386         0.343
    #>  9      5 FALSE    0.0190     0.0161       0.03          1.18          0.633
    #> 10      5 TRUE     0.00858    0.0161       0.015         0.532         0.572
    #> # … with 20 more rows

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

    exposed_data2 |>
      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)
    #> Experience study results
    #>
    #>  Groups: pol_yr, inc_guar
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>  Expected values: expected_1, expected_2
    #>  Weighted by: premium
    #>
    #> # A tibble: 30 × 7
    #>    pol_yr inc_guar   q_obs expected_1 expected_2 ae_expected_1 ae_expected_2
    #>     <int> <lgl>      <dbl>      <dbl>      <dbl>         <dbl>         <dbl>
    #>  1      1 FALSE    0.00471    0.005        0.03          0.942         0.157
    #>  2      1 TRUE     0.00235    0.005        0.015         0.470         0.157
    #>  3      2 FALSE    0.0105     0.00778      0.03          1.36          0.351
    #>  4      2 TRUE     0.00513    0.00778      0.015         0.660         0.342
    #>  5      3 FALSE    0.00737    0.0106       0.03          0.698         0.246
    #>  6      3 TRUE     0.00479    0.0106       0.015         0.453         0.319
    #>  7      4 FALSE    0.0174     0.0133       0.03          1.30          0.579
    #>  8      4 TRUE     0.00377    0.0133       0.015         0.283         0.252
    #>  9      5 FALSE    0.0146     0.0161       0.03          0.907         0.487
    #> 10      5 TRUE     0.00710    0.0161       0.015         0.441         0.473
    #> # … with 20 more rows

## 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.4

    exposed_data2 |>
      group_by(pol_yr, inc_guar) |>
      exp_stats(credibility = TRUE) |>
      select(pol_yr, inc_guar, claims, q_obs, credibility)
    #> Experience study results
    #>
    #>  Groups: pol_yr, inc_guar
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>
    #> # A tibble: 30 × 5
    #>    pol_yr inc_guar claims   q_obs credibility
    #>     <int> <lgl>     <int>   <dbl>       <dbl>
    #>  1      1 FALSE        56 0.00725       0.192
    #>  2      1 TRUE         46 0.00399       0.173
    #>  3      2 FALSE        92 0.0130        0.246
    #>  4      2 TRUE         68 0.00641       0.211
    #>  5      3 FALSE        67 0.0104        0.210
    #>  6      3 TRUE         57 0.00591       0.193
    #>  7      4 FALSE       123 0.0212        0.286
    #>  8      4 TRUE         45 0.00515       0.172
    #>  9      5 FALSE        97 0.0190        0.254
    #> 10      5 TRUE         67 0.00858       0.210
    #> # … with 20 more rows

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 `cred_p` and `cred_r` arguments, respectively.

    exposed_data2 |>
      group_by(pol_yr, inc_guar) |>
      exp_stats(credibility = TRUE, cred_p = 0.98, cred_r = 0.03) |>
      select(pol_yr, inc_guar, claims, q_obs, credibility)
    #> Experience study results
    #>
    #>  Groups: pol_yr, inc_guar
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>
    #> # A tibble: 30 × 5
    #>    pol_yr inc_guar claims   q_obs credibility
    #>     <int> <lgl>     <int>   <dbl>       <dbl>
    #>  1      1 FALSE        56 0.00725      0.0969
    #>  2      1 TRUE         46 0.00399      0.0876
    #>  3      2 FALSE        92 0.0130       0.125
    #>  4      2 TRUE         68 0.00641      0.107
    #>  5      3 FALSE        67 0.0104       0.106
    #>  6      3 TRUE         57 0.00591      0.0976
    #>  7      4 FALSE       123 0.0212       0.145
    #>  8      4 TRUE         45 0.00515      0.0867
    #>  9      5 FALSE        97 0.0190       0.128
    #> 10      5 TRUE         67 0.00858      0.106
    #> # … with 20 more rows

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_data2 |>
      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)
    #> Experience study results
    #>
    #>  Groups: pol_yr, inc_guar
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>  Expected values: expected_1
    #>
    #> # A tibble: 30 × 8
    #>    pol_yr inc_guar claims   q_obs credibility adj_expected_1 expected_1 ae_exp…¹
    #>     <int> <lgl>     <int>   <dbl>       <dbl>          <dbl>      <dbl>    <dbl>
    #>  1      1 FALSE        56 0.00725       0.192        0.00543    0.005      1.45
    #>  2      1 TRUE         46 0.00399       0.173        0.00482    0.005      0.798
    #>  3      2 FALSE        92 0.0130        0.246        0.00905    0.00778    1.67
    #>  4      2 TRUE         68 0.00641       0.211        0.00749    0.00778    0.824
    #>  5      3 FALSE        67 0.0104        0.210        0.0105     0.0106     0.985
    #>  6      3 TRUE         57 0.00591       0.193        0.00966    0.0106     0.560
    #>  7      4 FALSE       123 0.0212        0.286        0.0156     0.0133     1.59
    #>  8      4 TRUE         45 0.00515       0.172        0.0119     0.0133     0.386
    #>  9      5 FALSE        97 0.0190        0.254        0.0168     0.0161     1.18
    #> 10      5 TRUE         67 0.00858       0.210        0.0145     0.0161     0.532
    #> # … with 20 more rows, and abbreviated variable name ¹​ae_expected_1

## 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)
    #> Experience study results
    #>
    #>  Groups:
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>  Expected values: expected_1, expected_2
    #>
    #> # A tibble: 1 × 8
    #>   n_claims claims exposure  q_obs expected_1 expected_2 ae_expected_1 ae_expec…¹
    #>      <int>  <int>    <dbl>  <dbl>      <dbl>      <dbl>         <dbl>      <dbl>
    #> 1     2869   2869  132634. 0.0216     0.0242     0.0209         0.892       1.04
    #> # … with abbreviated variable name ¹​ae_expected_2

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)
    #> Experience study results
    #>
    #>  Groups: pol_yr
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>  Expected values: expected_1, expected_2
    #>
    #> # A tibble: 15 × 9
    #>    pol_yr n_claims claims exposure   q_obs expected_1 expected_2 ae_ex…¹ ae_ex…²
    #>     <int>    <int>  <int>    <dbl>   <dbl>      <dbl>      <dbl>   <dbl>   <dbl>
    #>  1      1      102    102   19252. 0.00530    0.005       0.0210   1.06    0.252
    #>  2      2      160    160   17715. 0.00903    0.00778     0.0210   1.16    0.430
    #>  3      3      124    124   16097. 0.00770    0.0106      0.0210   0.730   0.367
    #>  4      4      168    168   14536. 0.0116     0.0133      0.0210   0.867   0.551
    #>  5      5      164    164   12916. 0.0127     0.0161      0.0209   0.788   0.607
    #>  6      6      152    152   11376. 0.0134     0.0189      0.0209   0.707   0.639
    #>  7      7      164    164    9917. 0.0165     0.0217      0.0209   0.763   0.791
    #>  8      8      190    190    8448. 0.0225     0.0244      0.0208   0.920   1.08
    #>  9      9      181    181    6960. 0.0260     0.0272      0.0208   0.955   1.25
    #> 10     10      152    152    5604. 0.0271     0.03        0.0208   0.904   1.31
    #> 11     11      804    804    4390. 0.183      0.2         0.0208   0.916   8.81
    #> 12     12      330    330    2663. 0.124      0.15        0.0200   0.826   6.18
    #> 13     13       99     99    1620. 0.0611     0.05        0.0197   1.22    3.11
    #> 14     14       62     62     872. 0.0711     0.05        0.0195   1.42    3.64
    #> 15     15       17     17     268. 0.0634     0.05        0.0191   1.27    3.31
    #> # … with abbreviated variable names ¹​ae_expected_1, ²​ae_expected_2

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)
    #> Experience study results
    #>
    #>  Groups: inc_guar
    #>  Target status: Surrender
    #>  Study range: 1900-01-01 to 2019-12-31
    #>  Expected values: expected_1, expected_2
    #>
    #> # A tibble: 2 × 9
    #>   inc_guar n_claims claims exposure  q_obs expected_1 expected_2 ae_ex…¹ ae_ex…²
    #>   <lgl>       <int>  <int>    <dbl>  <dbl>      <dbl>      <dbl>   <dbl>   <dbl>
    #> 1 FALSE        1601   1601   52123. 0.0307     0.0235      0.03    1.31     1.02
    #> 2 TRUE         1268   1268   80511. 0.0157     0.0247      0.015   0.637    1.05
    #> # … with abbreviated variable names ¹​ae_expected_1, ²​ae_expected_2

### 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)
    #> Warning: ✖ No target status was provided.
    #> ℹ Death, Surrender was assumed.
    #> Experience study results
    #>
    #>  Groups:
    #>  Target status: Death, Surrender
    #>  Study range:  to
    #>
    #> # A tibble: 1 × 4
    #>   n_claims claims exposure  q_obs
    #>      <int>  <int>    <dbl>  <dbl>
    #> 1     4639   4639  132634. 0.0350

If `target_status` is provided, no warning message will appear.

    exp_stats(not_exposed_df, target_status = "Surrender")
    #> Experience study results
    #>
    #>  Groups:
    #>  Target status: Surrender
    #>  Study range:  to
    #>
    #> # A tibble: 1 × 4
    #>   n_claims claims exposure  q_obs
    #>      <int>  <int>    <dbl>  <dbl>
    #> 1     2869   2869  132634. 0.0216

### 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")`.

1. See `vignette('exposures')` for more information on creating exposure records.↩︎

2. This adjustment is not necessary on surrenders because the `expose()` function previously did this for us.↩︎

3. 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).↩︎

4. See Herzog, Thomas (1999). Introduction to Credibility Theory for more information on Limited Fluctuation Credibility.↩︎