SynthETIC
Synthetic Experience Tracking Insurance Claims

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Use of SynthETIC to Generate Individual Claims of Realistic Features

Use of SynthETIC to Generate Individual Claims of Realistic Features
In SynthETIC: Synthetic Experience Tracking Insurance Claims 

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

This vignette aims to illustrate how the SynthETIC package can be used to generate a general insurance claims history with realistic distributional assumptions consistent with the experience of a specific (but anonymous) Auto Liability portfolio. The simulator is composed of 8 modelling steps (or "modules"), each of which will build on (a selection of) the output from previous steps:

  1. Claim occurrence: claim frequency, claim occurrence times
  2. Claim size: claim size in constant dollar values i.e. without inflation
  3. Claim notification: notification delay (delay from occurrence to notification)
  4. Claim closure: settlement delay (delay from notification to closure)
  5. Claim payment count: number of partial payments
  6. Claim payment size: sizes of partial payments in constant dollar values i.e. without inflation
  7. Claim payment time: inter-partial-payment delays, partial payment times in calendar period
  8. Claim inflation: sizes of inflated partial payments

In particular, with this demo we will output

Description R Object

N, claim frequency n_vector = # claims for each accident period

U, claim occurrence time occurrence_times[[i]] = claim occurrence time for all claims that occurred in period i

S, claim size claim_sizes[[i]] = claim size for all claims that occurred in period i

V, notification delay notidel[[i]] = notification delay for all claims that occurred in period i

W, settlement delay setldel[[i]] = settlement delay for all claims that occurred in period i

M, number of partial payments no_payments[[i]] = number of partial payments for all claims that occurred in period i

size of partial payments payment_sizes[[i]][[j]] = $ partial payments for claim j of occurrence period i

inter-partial delays payment_delays[[i]][[j]] = inter partial delays for claim j of occurrence period i

payment times (continuous time) payment_times[[i]][[j]] = payment times (in continuous time) for claim j of occurrence period i

payment times (period) payment_periods[[i]][[j]] = payment times (in calendar periods) for claim j of occurrence period i

actual payments (inflated) payment_inflated[[i]][[j]] = $ partial payments (inflated) for claim j of occurrence period i

Reference

For a full description of SythETIC's structure and test parameters, readers should refer to:

Avanzi, B, Taylor, G, Wang, M, Wong, B (2021). SynthETIC: An individual insurance claim simulator with feature control. Insurance: Mathematics and Economics 100, 296–308. https://doi.org/10.1016/j.insmatheco.2021.06.004

The work can also be accessed via arXiv:2008.05693.

To cite this package in publications, please use:

citation("SynthETIC")

Set Up

library(SynthETIC)
set.seed(20200131)

Package-wise Global Parameters

We introduce the reference value ref_claim partly as a measure of the monetary unit and/or overall claims experience. The default distributional assumptions were set up with a specific (but anonymous) Auto Liability portfolio in mind. ref_claim then allows users to easily simulate a synthetic portfolio with similar claim pattern but in a different currency, for example. We also remark that users can alternatively choose to interpret ref_claim as a monetary unit. For example, one can set ref_claim <- 1000 and think of all amounts in terms of $1,000. However, in this case the default functions (as listed below) will not work and users will need to supply their own set of functions and set the values as multiples of ref_claim rather than fractions as in the default setting.

We also require the user to input a time_unit (which should be given as a fraction of year), so that the default input parameters apply to contexts where the time units are no longer in quarters. In the default setting we have a time_unit of 1/4.

The default input parameters will update automatically with the choice of the two global variables ref_claim and time_unit, which ensures that the simulator produce sensible results in contexts other than the default setting. We remark that both ref_claim and time_unit only affect the default simulation functions, and users can also choose to set up their own modelling assumptions for any of the modules to match their experiences even better. In the latter case, it is the responsibility of the user to ensure that their input parameters are compatible with their time units and claims experience. For example, if the time units are quarters, then claim occurrence rates must be quarterly.

set_parameters(ref_claim = 200000, time_unit = 1/4)
ref_claim <- return_parameters()[1]
time_unit <- return_parameters()[2]

The reference value, ref_claim will be used throughout the simulation process (as listed in the table below).

Module Details

  1. Claim Size At ref_claim = 200000, by default we simulate claim sizes from S^0.2 ~ Normal (9.5, sd = 3), left truncated at 30.
    When the reference value changes, we output the claim sizes scaled by a factor of ref_claim / 200000.

  2. Claim Notification By default we set the mean notification delay (in quarters) to be $$min(3, max(1, 2 - \frac{1}{3} \log(\frac{claim_size}{0.5~ref_claim}))$$ (which will be automatically converted to the relevant time_unit) i.e. the mean notification delay decreases logarithmically with claim size. It has maximum value 3 and equals 2 for a claim of size exactly at 0.5 * ref_claim.

  3. Claim Closure The default value for the mean settlement delay involves a term that defines the benchmark for a claim to be considered "small": 0.1 * ref_claim. The default mean settlement delay increases logarithmically with claim size and equals 6 exactly at this benchmark. Furthermore there was a legislative change, captured in the default mean function, that impacted the settlement delays of those "small" claims.

  4. Claim Payment Count For the default sampling distribution, we need two claim size benchmarks as we sample from different distributions for claims of different sizes. In general a small number of partial payments is required to settle small claims, and additional payments will be required to settle more extreme claims.
    It is assumed that claims below 0.0375 * ref_claim can be settled in 1 or 2 payments, claims between 0.075 * ref_claim in 2 or 3 payments, and claims beyond 0.075 * ref_claim in no less than 4 payments.

  5. Claim Payment Size We use the same proportion of ref_claim as in the Claim Closure module, namely 0.1 * ref_claim. This benchmark value is used when simulating the proportion of the last two payments in the default simulate_amt_pmt function.
    The mean proportion of claim paid in the last two payments increases logarithmically with claim size, and equals 75% exactly at this benchmark.

  6. Claim Inflation Two benchmarks values are required in this section, one each for the default SI occurrence and SI payment functions.
    1) A legislative change, captured by SI occurrence, reduced claim size by up to 40% for the smallest claims and impacted claims up to 0.25 * ref_claim in size.
    2) The default SI payment is set to be 30% p.a. for the smallest claims and zero for claims exceeding ref_claim in size, and varies linearly for claims between 0 and ref_claim.

The time_unit chosen will impact the time-related modules, specifically

1. Claim Occurrence {#occurrence}

Unless otherwise specified, claim_frequency() assumes the claim frequency follows a Poisson distribution with mean equal to the product of exposure E associated with period $i$ and expected claim frequency freq per unit exposure for that period. The exposure and expected frequency are allowed to vary across periods, but not within a period.

Given the claim frequency, claim_occurrence() samples the occurrence times of each claim from a uniform distribution. Together, the two functions assume by default that the arrival of claims follows a Poisson process, with potentially varying rates across different periods (see Example 1.2).

Alternative sampling processes are discussed in Example 1.3 and 1.4.

Example 1.1: Constant exposure and frequency {#ex1.1}

Input parameters

years <- 10
I <- years / time_unit
E <- c(rep(12000, I)) # effective annual exposure rates
lambda <- c(rep(0.03, I))

Implementation and Output

# Number of claims occurring for each period i
# shorter equivalent code:
# n_vector <- claim_frequency()
n_vector <- claim_frequency(I = I, E = E, freq = lambda)
n_vector

# Occurrence time of each claim r, for each period i
occurrence_times <- claim_occurrence(frequency_vector = n_vector)
occurrence_times[[1]]

Example 1.2: Increasing exposure, constant frequency per unit of exposure {#ex1.2}

Note that variables named with _tmp are for illustration purposes only and not used in the later simulation modules of this demo.

## input parameters
years_tmp <- 10
I_tmp <- years_tmp / time_unit
# set linearly increasing exposure, ...
E_tmp <- c(rep(12000, I)) + seq(from = 0, by = 100, length = I)
# and constant frequency per unit of exposure
lambda_tmp <- c(rep(0.03, I))

## output
# Number of claims occurring for each period i
n_vector_tmp <- claim_frequency(I = I_tmp, E = E_tmp, freq = lambda_tmp)
n_vector_tmp

# Occurrence time of each claim r, for each period i
occurrence_times_tmp <- claim_occurrence(frequency_vector = n_vector_tmp)
occurrence_times_tmp[[1]]

Example 1.3: Alternative claim frequency distribution {#ex1.3}

Users can choose to specify their own claim frequency distribution via simfun, which takes both random generation functions (type = "r", the default) and cumulative distribution functions (type = "p"). For example, we can use the negative binomial distribution in base R, or the zero-truncated Poisson distribution from the actuar package.

# simulate claim frequencies from negative binomial
# 1. using type-"r" specification (default)
claim_frequency(I = I, simfun = rnbinom, size = 100, mu = 100)
# 2. using type-"p" specification, equivalent to above
claim_frequency(I = I, simfun = pnbinom, type = "p", size = 100, mu = 100)

# simulate claim frequencies from zero-truncated Poisson
claim_frequency(I = I, simfun = actuar::rztpois, lambda = 90)
claim_frequency(I = I, simfun = actuar::pztpois, type = "p", lambda = 90)

Similar to Example 1.2, we can modify the frequency parameters to vary across periods:

claim_frequency(I = I, simfun = actuar::rztpois, lambda = time_unit * E_tmp * lambda_tmp)

If one wishes to code their own sampling function (either a direct random generating function, or a proper CDF), this can be achieved by:

# sampling from non-homogeneous Poisson process
rnhpp.count <- function(no_periods) {
  rate <- 3000
  intensity <- function(x) {
    # e.g. cyclical Poisson process
    0.03 * (sin(x * pi / 2) / 4 + 1)
  }
  lambda_max <- 0.03 * (1/4 + 1)
  target_num_events <- no_periods * rate * lambda_max

  # simulate a homogeneous Poisson process
  N <- stats::rpois(1, target_num_events)              # total number of events
  event_times <- sort(stats::runif(N, 0, no_periods))  # random times of occurrence

  # use a thinning step to turn this into a non-homogeneous process
  accept_probs <- intensity(event_times) / lambda_max
  is_accepted <- (stats::runif(N) < accept_probs)
  claim_times <- event_times[is_accepted]

  as.numeric(table(cut(claim_times, breaks = 0:no_periods)))
}

n_vector_tmp <- claim_frequency(I = I, simfun = rnhpp.count)
plot(x = 1:I, y = n_vector_tmp, type = "l",
     main = "Claim frequency simulated from a cyclical Poisson process",
     xlab = "Occurrence period", ylab = "# Claims")

Example 1.4: Alternative specification of the claim arrival process {#ex1.4}

We note that the claim_occurrence() function for simulating the claim times conditional on claim frequencies assumes a uniform distribution, and that this cannot be modified without changing the module. Indeed, the modular structure of SynthETIC ensures that one can easily unplug any one module and replace it with a version modified to his/her own purpose.

For example, if one wishes to replace this uniform distribution assumption and/or the whole Claim Occurrence module, they can simply supply their own vector of claim times and easily convert to the list format consistent with the SynthETIC framework for smooth integration with the later modules.

# Equivalent to a Poisson process
event_times_tmp <- sort(stats::runif(n = 4000, 0, I))
accept_probs_tmp <- (sin(event_times_tmp * pi / 2) + 1) / 2
is_accepted_tmp <- (stats::runif(length(event_times_tmp)) < accept_probs_tmp)
claim_times_tmp <- event_times_tmp[is_accepted_tmp]

# Number of claims occurring for each period i
# by counting the number of event times in each interval (i, i + 1)
n_vector_tmp <- as.numeric(table(cut(claim_times_tmp, breaks = 0:I)))
n_vector_tmp

# Occurrence time of each claim r, for each period i
occurrence_times_tmp <- to_SynthETIC(x = claim_times_tmp, 
                                     frequency_vector = n_vector_tmp)
occurrence_times_tmp[[1]]

2. Claim Size {#size}

Example 2.1: Default power normal {#ex2.1}

By default claim_size() assumes a left truncated power normal distribution: $S^{0.2} \sim \mathcal{N}(\mu = 9.5, \sigma = 3)$, left truncated at 30. There is no need to specify a sampling distribution if the user is happy with the default power normal. This example is mainly to demonstrate how the default function works.

Input parameters

We can specify the CDF to generate claim sizes from. The default distribution function can be coded as follows:

# use a power normal S^0.2 ~ N(9.5, 3), left truncated at 30
# this is the default distribution driving the claim_size() function
S_df <- function(s) {
  # truncate and rescale
  if (s < 30) {
    return(0)
  } else {
    p_trun <- pnorm(s^0.2, 9.5, 3) - pnorm(30^0.2, 9.5, 3)
    p_rescaled <- p_trun/(1 - pnorm(30^0.2, 9.5, 3))
    return(p_rescaled)
  }
}

Implementation and Output

# shorter equivalent: claim_sizes <- claim_size(frequency_vector = n_vector)
claim_sizes <- claim_size(frequency_vector = n_vector, 
                          simfun = S_df, type = "p", range = c(0, 1e24))
claim_sizes[[1]]

Example 2.2: Alternative claim size distribution {#ex2.2}

Users can also choose any other individual claim size distribution, e.g. Weibull from base R or inverse Gaussian from actuar:

## weibull
# estimate the weibull parameters to achieve the mean and cv matching that of
# the built-in test claim dataset
claim_size_mean <- mean(test_claim_dataset$claim_size)
claim_size_cv <- cv(test_claim_dataset$claim_size)
weibull_shape <- get_Weibull_parameters(target_mean = claim_size_mean, 
                                        target_cv = claim_size_cv)[1]
weibull_scale <- get_Weibull_parameters(target_mean = claim_size_mean, 
                                        target_cv = claim_size_cv)[2]
# simulate claim sizes with the estimated parameters
claim_sizes_weibull <- claim_size(frequency_vector = n_vector,
                                  simfun = rweibull, 
                                  shape = weibull_shape, scale = weibull_scale)
# plot empirical CDF
plot(ecdf(unlist(test_claim_dataset$claim_size)), xlim = c(0, 2000000),
     main = "Empirical distribution of simulated claim sizes",
     xlab = "Individual claim size")
plot(ecdf(unlist(claim_sizes_weibull)), add = TRUE, col = 2)


## inverse Gaussian
# modify actuar::rinvgauss (left truncate it @30 and right censor it @5,000,000)
rinvgauss_censored <- function(n) {
  s <- actuar::rinvgauss(n, mean = 180000, dispersion = 0.5e-5)
  while (any(s < 30 | s > 5000000)) {
    for (j in which(s < 30 | s > 5000000)) {
      # for rejected values, resample
      s[j] <- actuar::rinvgauss(1, mean = 180000, dispersion = 0.5e-5)
    }
  }
  s
}
# simulate from the modified inverse Gaussian distribution
claim_sizes_invgauss <- claim_size(frequency_vector = n_vector, simfun = rinvgauss_censored)

# plot empirical CDF
plot(ecdf(unlist(claim_sizes_invgauss)), add = TRUE, col = 3)
legend.text <- c("Power normal", "Weibull", "Inverse Gaussian")
legend("bottomright", legend.text, col = 1:3, lty = 1, bty = "n")

Example 2.3: Simulating claim sizes from covariates

The applications discussed above assume that the claim sizes are sampled from a single distribution for all policyholders (e.g. the default power normal, custom sampling distribution specified by simfun).

Suppose we instead want to simulate from a model which uses covariates to predict claim sizes. For example, consider a (theoretical) gamma GLM with log link:

[ \begin{align} E(S_i) =\mu_i &=\exp(\boldsymbol{x}_i^\top \boldsymbol\beta)\ &= \exp(\beta_0 + \beta_1 \times age_i + \beta_2 \times age_i^2)\ &= \exp(27 - 0.768 \times age_i + 0.008 \times age_i^2) \end{align} ]

# define the random generation function to simulate from the gamma GLM
sim_GLM <- function(n) {
  # simulate covariates
  age <- sample(20:70, size = n, replace = T)
  mu <- exp(27 - 0.768 * age + 0.008 * age^2)
  rgamma(n, shape = 10, scale = mu / 10)
}

claim_sizes_GLM <- claim_size(frequency_vector = n_vector, simfun = sim_GLM)
plot(ecdf(unlist(claim_sizes_GLM)), xlim = c(0, 2000000),
     main = "Empirical distribution of claim sizes simulated from GLM",
     xlab = "Individual claim size")

Example 2.4: Bootstrapping from given loss data {#ex2.4}

Suppose we have an existing dataset of claim costs at hand that we wish to simulate from, e.g. ausautoBI8999 (an automobile bodily injury claim dataset in Australia) from CASDatasets. We can take a bootstrap resample of the dataset and then convert to SynthETIC format with ease:

# install.packages("CASdatasets", repos = "http://cas.uqam.ca/pub/", type = "source")
library(CASdatasets)
data("ausautoBI8999")
boot <- sample(ausautoBI8999$AggClaim, size = sum(n_vector), replace = TRUE)
claim_sizes_bootstrap <- to_SynthETIC(boot, frequency_vector = n_vector)

Another way to code this would be to write a random generation function to perform bootstrapping, and then use claim_size as usual:

sim_boot <- function(n) {
  sample(ausautoBI8999$AggClaim, size = n, replace = TRUE)
}
claim_sizes_bootstrap <- claim_size(frequency_vector = n_vector, simfun = sim_boot)

Alternatively, one can easily fit a parametric distribution to an existing dataset with the help of the fitdistrplus package and then simulate from the fitted parametric distribution (Example 2.2).

3. Claim Notification {#notification}

SynthETIC assumes the (removable) dependence of notification delay on claim size and occurrence period of the claim, and thus requires the user to specify a paramfun (parameter function) with arguments claim_size and occurrence_period (and possibly more, see Example 3.2). The dependencies can be removed if the arguments are not referenced inside the function; e.g. the default notification delay function (shown below) is independent of the individual claim's occurrence_period.

Other than this pre-specified dependence structure, users are free to choose any distribution, whether it be a pre-defined distribution in R, or more advanced ones from packages, or a proper user-defined function, to better match their own claim experience.

Indeed, although not recommended, users are able to add further dependencies in their simulation. This is illustrated in Example 4.2 of the settlement delay module.

Example 3.1: Default Weibull {#ex3.1}

By default, SynthETIC samples notification delays from a Weibull distribution:

## input
# specify the Weibull parameters as a function of claim_size and occurrence_period
notidel_param <- function(claim_size, occurrence_period) {
  # NOTE: users may add to, but not remove these two arguments (claim_size, 
  # occurrence_period) as they are part of SynthETIC's internal structure

  # specify the target mean and target coefficient of variation
  target_mean <- min(3, max(1, 2-(log(claim_size/(0.50 * ref_claim)))/3))/4 / time_unit
  target_cv <- 0.70
  # convert to Weibull parameters
  shape <- get_Weibull_parameters(target_mean, target_cv)[1]
  scale <- get_Weibull_parameters(target_mean, target_cv)[2]

  c(shape = shape, scale = scale)
}

## output
notidel <- claim_notification(n_vector, claim_sizes, 
                              rfun = rweibull, paramfun = notidel_param)

Example 3.2: Alternative distribution for notification delay {#ex3.2}

SynthETIC does not restrict the choice of the sampling distribution. For example, we can use a transformed gamma distribution:

## input
# specify the transformed gamma parameters as a function of claim_size and occurrence_period
trgamma_param <- function(claim_size, occurrence_period, rate) {
  c(shape1 = max(1, claim_size / ref_claim),
    shape2 = 1 - occurrence_period / 200,
    rate = rate)
}

## output
# simulate notification delays from the transformed gamma
notidel_trgamma <- claim_notification(n_vector, claim_sizes, 
                                      rfun = actuar::rtrgamma, 
                                      paramfun = trgamma_param, rate = 2)

# graphically compare the result with the default Weibull distribution
plot(ecdf(unlist(notidel)), xlim = c(0, 15),
     main = "Empirical distribution of simulated notification delays",
     xlab = "Notification delay (in quarters)")
plot(ecdf(unlist(notidel_trgamma)), add = TRUE, col = 2)
legend.text <- c("Weibull (default)", "Transformed gamma")
legend("bottomright", legend.text, col = 1:2, lty = 1, bty = "n")

Clearly the transformed gamma with the parameters specified above accelerates the reporting of the simulated claims.

Example 3.3: User-defined sampling function for notification delay {#ex3.3}

One may wish to simulate from a more exotic sampling distribution that cannot be easily written as a nice pre-defined distribution function and its parameters. For example, consider a mixed distribution:

rmixed_notidel <- function(n, claim_size) {
  # consider a mixture distribution
  # equal probability of sampling from x (Weibull) or y (transformed gamma)
  x_selected <- sample(c(T, F), size = n, replace = TRUE)
  x <- rweibull(n, shape = 2, scale = 1)
  y <- actuar::rtrgamma(n, shape1 = min(1, claim_size / ref_claim), shape2 = 0.8, rate = 2)
  result <- length(n)
  result[x_selected] <- x[x_selected]; result[!x_selected] <- y[!x_selected]

  return(result)
}

In this case, we can consider claim_size as the "parameter" for the sampling distribution (just in the same way as shape and scale for gamma distribution). Then we can either define a parameter function like below:

rmixed_params <- function(claim_size, occurrence_period) {
  # claim_size is the only "parameter" required for rmixed_notidel
  c(claim_size = claim_size)
}

or simply run

notidel_mixed <- claim_notification(n_vector, claim_sizes, rfun = rmixed_notidel)

which would give the same result as

notidel_mixed <- claim_notification(n_vector, claim_sizes, 
                                    rfun = rmixed_notidel, paramfun = rmixed_params)

4. Claim Closure {#closure}

Claim settlement delay represents the delay from claim notification to closure. Like notification delay, SynthETIC assumes the (removable) dependence of settlement delay on claim size and occurrence period of the claim, and thus requires the user to specify a paramfun (parameter function) with arguments claim_size and occurrence_period (and possibly more, see Example 3.2).

Other than this pre-specified dependence structure, users are free to choose any distribution by specifying their own rfun and/or paramfun (see ?claim_closure).

Indeed, although not recommended, users are able to add further dependencies in their simulation. This is illustrated in Example 4.2.

Example 4.1: Default Weibull {#ex4.1}

Below we show the default implementation with a Weibull distribution.

## input
# specify the Weibull parameters as a function of claim_size and occurrence_period
setldel_param <- function(claim_size, occurrence_period) {
  # NOTE: users may add to, but not remove these two arguments (claim_size, 
  # occurrence_period) as they are part of SynthETIC's internal structure

  # specify the target Weibull mean
  if (claim_size < (0.10 * ref_claim) & occurrence_period >= 21) {
    a <- min(0.85, 0.65 + 0.02 * (occurrence_period - 21))
  } else {
    a <- max(0.85, 1 - 0.0075 * occurrence_period)
  }
  mean_quarter <- a * min(25, max(1, 6 + 4*log(claim_size/(0.10 * ref_claim))))
  target_mean <- mean_quarter / 4 / time_unit

  # specify the target Weibull coefficient of variation
  target_cv <- 0.60

  c(shape = get_Weibull_parameters(target_mean, target_cv)[1, ],
    scale = get_Weibull_parameters(target_mean, target_cv)[2, ])
}

## output
# simulate the settlement delays from the Weibull with parameters above
setldel <- claim_closure(n_vector, claim_sizes, rfun = rweibull, paramfun = setldel_param)
setldel[[1]]

There is no need to specify a sampling distribution if one is happy with the default Weibull specification. This example is just to demonstrate some of the behind-the-scenes work of the default implementation, and at the same time, to show how one may specify and input a random sampling distribution of their choosing.

Example 4.2: Additional dependencies {#ex4.2}

Suppose we would like to add the dependence of settlement delay on notification delay, which is not natively included in SynthETIC default setting. For example, let's consider the following parameter function:

## input
# an extended parameter function for the simulation of settlement delays
setldel_param_extd <- function(claim_size, occurrence_period, notidel) {

  # specify the target Weibull mean
  if (claim_size < (0.10 * ref_claim) & occurrence_period >= 21) {
    a <- min(0.85, 0.65 + 0.02 * (occurrence_period - 21))
  } else {
    a <- max(0.85, 1 - 0.0075 * occurrence_period)
  }
  mean_quarter <- a * min(25, max(1, 6 + 4*log(claim_size/(0.10 * ref_claim))))
  # suppose the setldel mean is linearly related to the notidel of the claim
  target_mean <- (mean_quarter + notidel) / 4 / time_unit

  # specify the target Weibull coefficient of variation
  target_cv <- 0.60

  c(shape = get_Weibull_parameters(target_mean, target_cv)[1, ],
    scale = get_Weibull_parameters(target_mean, target_cv)[2, ])
}

As this parameter function setldel_param_extd is dependent on notidel, it should not be surprising that we need to input the simulated notification delays when calling claim_closure. We need to make sure that the argument names are matched exactly (notidel in this example) and that the input is specified as a vector of simulated quantities (not a list).

## output
# simulate the settlement delays from the Weibull with parameters above
notidel_vect <- unlist(notidel) # convert to a vector
setldel_extd <- claim_closure(n_vector, claim_sizes, rfun = rweibull, 
                              paramfun = setldel_param_extd,
                              notidel = notidel_vect)
setldel_extd[[1]]

5. Claim Partial Payment - Number of Partial Payments {#payment-count}

claim_payment_no() generates the number of partial payments associated with a particular claim, from a user-defined random generation function which may depend on claim_size.

Example 5.1: Default mixture distribution {#ex5.1}

Below we spell out the default function in SynthETIC that simulates the number of partial payments (from a mixture distribution):

## input
# the default random generating function
rmixed_payment_no <- function(n, claim_size, claim_size_benchmark_1, claim_size_benchmark_2) {
  # construct the range indicators
  test_1 <- (claim_size_benchmark_1 < claim_size & claim_size <= claim_size_benchmark_2)
  test_2 <- (claim_size > claim_size_benchmark_2)

  # if claim_size <= claim_size_benchmark_1
  no_pmt <- sample(c(1, 2), size = n, replace = T, prob = c(1/2, 1/2))
  # if claim_size is between the two benchmark values
  no_pmt[test_1] <- sample(c(2, 3), size = sum(test_1), replace = T, prob = c(1/3, 2/3))
  # if claim_size > claim_size_benchmark_2
  no_pmt_mean <- pmin(8, 4 + log(claim_size/claim_size_benchmark_2))
  prob <- 1 / (no_pmt_mean - 3)
  no_pmt[test_2] <- stats::rgeom(n = sum(test_2), prob = prob[test_2]) + 4

  no_pmt
}

Since the random function directly takes claim_size as an input, no additional parameterisation is required (unlike in Example 3.1, where we first need a paramfun that turns the claim_size into Weibull parameters). We can simply run claim_payment_no() without inputting a paramfun.

## output
no_payments <- claim_payment_no(n_vector, claim_sizes, rfun = rmixed_payment_no,
                                claim_size_benchmark_1 = 0.0375 * ref_claim,
                                claim_size_benchmark_2 = 0.075 * ref_claim)
no_payments[[1]]

Note that the claim_size_benchmark_1 and claim_size_benchmark_2 are passed on to rmixed_payment_no and will not be required if we choose an alternative sampling distribution.

This mixture sampling distribution has been included as the default. There is no need to reproduce the above code if the user is happy with this default distribution. A simple equivalent to the above code is just

no_payments <- claim_payment_no(n_vector, claim_sizes)

This example is here only to demonstrate how the default function operates. If one would like to keep the structure of this function but modify the benchmark values, they may do so via

no_payments_tmp <- claim_payment_no(n_vector, claim_sizes,
                                    claim_size_benchmark_2 = 0.1 * ref_claim)

Example 5.2: Alternative distribution for number of partial payments {#ex5.2}

Suppose we want to use a zero truncated Poisson distribution instead, with the rate parameter as a function of claim_size:

## input
paymentNo_param <- function(claim_size) {
  no_pmt_mean <- pmax(4, pmin(8, 4 + log(claim_size / 15000)))
  c(lambda = no_pmt_mean - 3)
}

## output
no_payments_pois <- claim_payment_no(
  n_vector, claim_sizes, rfun = actuar::rztpois, paramfun = paymentNo_param)
table(unlist(no_payments_pois))

Interlude: Claims Dataset

We can use the following code to create a claims dataset containing all individual claims features that we have simulated so far:

claim_dataset <- generate_claim_dataset(
  frequency_vector = n_vector,
  occurrence_list = occurrence_times,
  claim_size_list = claim_sizes,
  notification_list = notidel,
  settlement_list = setldel,
  no_payments_list = no_payments
)
str(claim_dataset)

test_claim_dataset, included as part of the package, is an example dataset of individual claims features resulting from a specific run with the default assumptions.

str(test_claim_dataset)

6. Claim Partial Payment - Sizes of Partial Payments (without inflation) {#payment-size}

Example 6.1: Default Distribution {#ex6.1}

The default function samples the sizes of partial payments conditional on the number of partial payments, and the size of the claim:

## input
rmixed_payment_size <- function(n, claim_size) {
  # n = number of simulations, here n should be the number of partial payments
  if (n >= 4) {
    # 1) Simulate the "complement" of the proportion of total claim size 
    #    represented by the last two payments
    p_mean <- 1 - min(0.95, 0.75 + 0.04*log(claim_size/(0.10 * ref_claim)))
    p_CV <- 0.20
    p_parameters <- get_Beta_parameters(target_mean = p_mean, target_cv = p_CV)
    last_two_pmts_complement <- stats::rbeta(
      1, shape1 = p_parameters[1], shape2 = p_parameters[2])
    last_two_pmts <- 1 - last_two_pmts_complement

    # 2) Simulate the proportion of last_two_pmts paid in the second last payment
    q_mean <- 0.9
    q_CV <- 0.03
    q_parameters <- get_Beta_parameters(target_mean = q_mean, target_cv = q_CV)
    q <- stats::rbeta(1, shape1 = q_parameters[1], shape2 = q_parameters[2])

    # 3) Calculate the respective proportions of claim amount paid in the 
    #    last 2 payments
    p_second_last <- q * last_two_pmts
    p_last <- (1-q) * last_two_pmts

    # 4) Simulate the "unnormalised" proportions of claim amount paid 
    #    in the first (m - 2) payments
    p_unnorm_mean <- last_two_pmts_complement/(n - 2)
    p_unnorm_CV <- 0.10
    p_unnorm_parameters <- get_Beta_parameters(
      target_mean = p_unnorm_mean, target_cv = p_unnorm_CV)
    amt <- stats::rbeta(
      n - 2, shape1 = p_unnorm_parameters[1], shape2 = p_unnorm_parameters[2])

    # 5) Normalise the proportions simulated in step 4
    amt <- last_two_pmts_complement * (amt/sum(amt))
    # 6) Attach the last 2 proportions, p_second_last and p_last
    amt <- append(amt, c(p_second_last, p_last))
    # 7) Multiply by claim_size to obtain the actual payment amounts
    amt <- claim_size * amt

  } else if (n == 2 | n == 3) {
    p_unnorm_mean <- 1/n
    p_unnorm_CV <- 0.10
    p_unnorm_parameters <- get_Beta_parameters(
      target_mean = p_unnorm_mean, target_cv = p_unnorm_CV)
    amt <- stats::rbeta(
      n, shape1 = p_unnorm_parameters[1], shape2 = p_unnorm_parameters[2])
    # Normalise the proportions and multiply by claim_size to obtain the actual payment amounts
    amt <- claim_size * amt/sum(amt)

  } else {
    # when there is a single payment
    amt <- claim_size
  }
  return(amt)
}

## output
payment_sizes <- claim_payment_size(n_vector, claim_sizes, no_payments,
                                    rfun = rmixed_payment_size)
payment_sizes[[1]][[1]]

As this is the default random generation function that SynthETIC adopts, a shorter equivalent command would be to call claim_payment_no without specifying a rfun.

payment_sizes <- claim_payment_size(n_vector, claim_sizes, no_payments)

Example 6.2: Alternative payment size distribution {#ex6.2}

Let's consider a simplistic example where we assume the partial payment sizes are (stochastically) equal. This will result in the following simulation function:

## input
unif_payment_size <- function(n, claim_size) {
  prop <- runif(n)
  prop.normalised <- prop / sum(prop)

  return(claim_size * prop)
}

## output
# note that we don't need to specify a paramfun as rfun is directly a function
# of claim_size
payment_sizes_unif <- claim_payment_size(n_vector, claim_sizes, no_payments,
                                         rfun = unif_payment_size)
payment_sizes_unif[[1]][[1]]

7. Claim Payment Time {#payment-time}

The simulation of the inter-partial delays is almost identical to that of partial payment sizes, except that it also depends on the claim settlement delay - the inter-partial delays should add up to the settlement delay.

Other than this, the SynthETIC function implementation of claim_payment_delay() is almost the same as claim_payment_size(), but of course, with a different default simulation function:

## input
r_pmtdel <- function(n, claim_size, setldel, setldel_mean) {
  result <- c(rep(NA, n))

  # First simulate the unnormalised values of d, sampled from a Weibull distribution
  if (n >= 4) {
    # 1) Simulate the last payment delay
    unnorm_d_mean <- (1 / 4) / time_unit
    unnorm_d_cv <- 0.20
    parameters <- get_Weibull_parameters(target_mean = unnorm_d_mean, target_cv = unnorm_d_cv)
    result[n] <- stats::rweibull(1, shape = parameters[1], scale = parameters[2])

    # 2) Simulate all the other payment delays
    for (i in 1:(n - 1)) {
      unnorm_d_mean <- setldel_mean / n
      unnorm_d_cv <- 0.35
      parameters <- get_Weibull_parameters(target_mean = unnorm_d_mean, target_cv = unnorm_d_cv)
      result[i] <- stats::rweibull(1, shape = parameters[1], scale = parameters[2])
    }

  } else {
    for (i in 1:n) {
      unnorm_d_mean <- setldel_mean / n
      unnorm_d_cv <- 0.35
      parameters <- get_Weibull_parameters(target_mean = unnorm_d_mean, target_cv = unnorm_d_cv)
      result[i] <- stats::rweibull(1, shape = parameters[1], scale = parameters[2])
    }
  }

  # Normalise d such that sum(inter-partial delays) = settlement delay
  # To make sure that the pmtdels add up exactly to setldel, we treat the last one separately
  result[1:n-1] <- (setldel/sum(result)) * result[1:n-1]
  result[n] <- setldel - sum(result[1:n-1])

  return(result)
}

param_pmtdel <- function(claim_size, setldel, occurrence_period) {
  # mean settlement delay
  if (claim_size < (0.10 * ref_claim) & occurrence_period >= 21) {
    a <- min(0.85, 0.65 + 0.02 * (occurrence_period - 21))
  } else {
    a <- max(0.85, 1 - 0.0075 * occurrence_period)
  }
  mean_quarter <- a * min(25, max(1, 6 + 4*log(claim_size/(0.10 * ref_claim))))
  target_mean <- mean_quarter / 4 / time_unit

  c(claim_size = claim_size,
    setldel = setldel,
    setldel_mean = target_mean)
}

## output
payment_delays <- claim_payment_delay(
  n_vector, claim_sizes, no_payments, setldel,
  rfun = r_pmtdel, paramfun = param_pmtdel,
  occurrence_period = rep(1:I, times = n_vector))

# payment times on a continuous time scale
payment_times <- claim_payment_time(n_vector, occurrence_times, notidel, payment_delays)
# payment times in periods
payment_periods <- claim_payment_time(n_vector, occurrence_times, notidel, payment_delays,
                                      discrete = TRUE)
cbind(payment_delays[[1]][[1]], payment_times[[1]][[1]], payment_periods[[1]][[1]])

8. Claim Inflation {#inflation}

Input parameters

# Base inflation: a vector of quarterly rates
# In this demo we set base inflation to be at 2% p.a. constant for both past and future
# Users can choose to randominise the future rates if they wish
demo_rate <- (1 + 0.02)^(1/4) - 1
base_inflation_past <- rep(demo_rate, times = 40)
base_inflation_future <- rep(demo_rate, times = 40)
base_inflation_vector <- c(base_inflation_past, base_inflation_future)

# Superimposed inflation:
# 1) With respect to occurrence "time" (continuous scale)
SI_occurrence <- function(occurrence_time, claim_size) {
  if (occurrence_time <= 20 / 4 / time_unit) {1}
  else {1 - 0.4*max(0, 1 - claim_size/(0.25 * ref_claim))}
}
# 2) With respect to payment "time" (continuous scale)
# -> compounding by user-defined time unit
SI_payment <- function(payment_time, claim_size) {
  period_rate <- (1 + 0.30)^(time_unit) - 1
  beta <- period_rate * max(0, 1 - claim_size/ref_claim)
  (1 + beta)^payment_time
}

Implementation and Output

# shorter equivalent code:
# payment_inflated <- claim_payment_inflation(
#   n_vector, payment_sizes, payment_times, occurrence_times, claim_sizes, 
#   base_inflation_vector)
payment_inflated <- claim_payment_inflation(
  n_vector,
  payment_sizes,
  payment_times,
  occurrence_times,
  claim_sizes,
  base_inflation_vector,
  SI_occurrence,
  SI_payment
)
cbind(payment_sizes[[1]][[1]], payment_inflated[[1]][[1]])

Interlude: Transaction Dataset

Use the following code to create a transactions dataset containing full information of all the partial payments made.

# construct a "claims" object to store all the simulated quantities
all_claims <- claims(
  frequency_vector = n_vector,
  occurrence_list = occurrence_times,
  claim_size_list = claim_sizes,
  notification_list = notidel,
  settlement_list = setldel,
  no_payments_list = no_payments,
  payment_size_list = payment_sizes,
  payment_delay_list = payment_delays,
  payment_time_list = payment_times,
  payment_inflated_list = payment_inflated
)
transaction_dataset <- generate_transaction_dataset(
  all_claims,
  adjust = FALSE # to keep the original (potentially out-of-bound) simulated payment times
)
str(transaction_dataset)

test_transaction_dataset, included as part of the package, is an example dataset showing full information of the claims features at a transaction/payment level, generated by a specific SynthETIC run with the default assumptions.

str(test_transaction_dataset)

Output

SynthETIC includes an output function which summarises the claim payments by occurrence and development periods. The usage of the function takes the form

claim_output(
  frequency_vector = ,
  payment_time_list = ,
  payment_size_list = ,
  aggregate_level = 1,
  incremental = TRUE,
  future = TRUE,
  adjust = TRUE
)

Note that by default, we aggregate all out-of-bound transactions into the maximum development period. But if we set adjust = FALSE, then the function would produce a separate "tail" column to represent all payments beyond the maximum development period (see function documentation ?claim_output).

Examples:

# 1. Constant dollar value INCREMENTAL triangle
output <- claim_output(n_vector, payment_times, payment_sizes,
                       incremental = TRUE)

# 2. Constant dollar value CUMULATIVE triangle
output_cum <- claim_output(n_vector, payment_times, payment_sizes,
                           incremental = FALSE)

# 3. Actual (i.e. inflated) INCREMENTAL triangle
output_actual <- claim_output(n_vector, payment_times, payment_inflated,
                              incremental = TRUE)

# 4. Actual (i.e. inflated) CUMULATIVE triangle
output_actual_cum <- claim_output(n_vector, payment_times, payment_inflated,
                                  incremental = FALSE)

# Aggregate at a yearly level
claim_output(n_vector, payment_times, payment_sizes, aggregate_level = 4)

Note that by setting future = FALSE we can obtain the upper left part of the triangle (i.e. only the past claim payments). The past data can then be used to perform chain-ladder reserving analysis:

# output the past cumulative triangle
cumtri <- claim_output(n_vector, payment_times, payment_sizes,
                       aggregate_level = 4, incremental = FALSE, future = FALSE)
# calculate the age to age factors
selected <- vector()
J <- nrow(cumtri)
for (i in 1:(J - 1)) {
  # use volume weighted age to age factors
  selected[i] <- sum(cumtri[, (i + 1)], na.rm = TRUE) / sum(cumtri[1:(J - i), i], na.rm = TRUE)
}
# complete the triangle
CL_prediction <- cumtri
for (i in 2:J) {
  for (j in (J - i + 2):J) {
    CL_prediction[i, j] <- CL_prediction[i, j - 1] * selected[j - 1]
  }
}

CL_prediction

We observe that the chain-ladder analysis performs very poorly on this simulated claim dataset. This is perhaps unsurprising in view of the data features and the extent to which they breach chain ladder assumptions. Data sets such as this are useful for testing models that endeavour to represent data outside the scope of the chain-ladder.

Plot of Cumulative Claims Payments

Note that by default, similar to the case of claim_output and claim_payment_inflation, we will truncate the claims development such that payments that were projected to fall out of the maximum development period are forced to be paid at the exact end of the maximum development period allowed. This convention will cause some concentration of transactions at the end of development period $I$ (shown as a surge in claims in the $I$th period).

Users can set adjust = FALSE to see the "true" picture of claims development without such artificial adjustment. If the plots look significantly different, this indicates to the user that the user's selection of lag parameters (notification and/or settlement delays) is not well matched to the maximum number of development periods allowed, and consideration might be given to changing one or the other.

plot(test_claims_object)
# compare with the "full complete picture"
plot(test_claims_object, adjust = FALSE)

# plot by occurrence and development years
plot(test_claims_object, by_year = TRUE)

Multiple Simulation Runs

Once all the input parameters have been set up, we can repeat the simulation process as many times as desired through a for loop. The code below saves the transaction dataset generated by each simulation run as a component of results_all.

times <- 100
results_all <- vector("list")
for (i in 1:times) {
  # Module 1: Claim occurrence
  n_vector <- claim_frequency(I, E, lambda)
  occurrence_times <- claim_occurrence(n_vector)
  # Module 2: Claim size
  claim_sizes <- claim_size(n_vector, S_df, type = "p", range = c(0, 1e24))
  # Module 3: Claim notification
  notidel <- claim_notification(n_vector, claim_sizes, paramfun = notidel_param)
  # Module 4: Claim settlement
  setldel <- claim_closure(n_vector, claim_sizes, paramfun = setldel_param)
  # Module 5: Claim payment count
  no_payments <- claim_payment_no(n_vector, claim_sizes, rfun = rmixed_payment_no,
                                  claim_size_benchmark_1 = 0.0375 * ref_claim,
                                  claim_size_benchmark_2 = 0.075 * ref_claim)
  # Module 6: Claim payment size
  payment_sizes <- claim_payment_size(n_vector, claim_sizes, no_payments, 
                                      rfun = rmixed_payment_size)
  # Module 7: Claim payment time
  payment_delays <- claim_payment_delay(n_vector, claim_sizes, no_payments, setldel,
                                        rfun = r_pmtdel, paramfun = param_pmtdel,
                                        occurrence_period = rep(1:I, times = n_vector))
  payment_times <- claim_payment_time(n_vector, occurrence_times, notidel, payment_delays)
  # Module 8: Claim inflation
  payment_inflated <- claim_payment_inflation(
    n_vector, payment_sizes, payment_times, occurrence_times,
    claim_sizes, base_inflation_vector, SI_occurrence, SI_payment)

  results_all[[i]] <- generate_transaction_dataset(
    claims(
      frequency_vector = n_vector,
      occurrence_list = occurrence_times,
      claim_size_list = claim_sizes,
      notification_list = notidel,
      settlement_list = setldel,
      no_payments_list = no_payments,
      payment_size_list = payment_sizes,
      payment_delay_list = payment_delays,
      payment_time_list = payment_times,
      payment_inflated_list = payment_inflated),
    # adjust = FALSE to retain the original simulated times
    adjust = FALSE)
}

What if we are interested in seeing the average claims development over a large number of simulation runs? The plot.claims function in this package at present only works for a single claims object so we need to come up with a way to combine the claims objects generated by each run. A much simpler alternative would be to just increase the exposure rates and plot the resulting claims object. This has the same effect as averaging over a large number of simulation runs.

This long-run average of claims development offers insights into the effects of the distributional assumptions that users have made throughout the way, and hence the reasonableness of such choices.

The code below runs only for 10 simulations and we can already see the trend emerging, which matches with the result of our single simulation run above. Increasing times to run simulation will show a smoother trend, which we refrain from producing here because running simulation on this amount of data takes some time (100 simulations take around 10 minutes on a quad-core machine). We remark that the major simulation lags are caused by the claim_payment_delay and (less severely) claim_payment_size functions.

start.time <- proc.time()
times <- 10

# increase exposure to E*times to get the same results as the aggregation of
# multiple simulation runs
n_vector <- claim_frequency(I, E = E * times, lambda)
occurrence_times <- claim_occurrence(n_vector)
claim_sizes <- claim_size(n_vector)
notidel <- claim_notification(n_vector, claim_sizes, paramfun = notidel_param)
setldel <- claim_closure(n_vector, claim_sizes, paramfun = setldel_param)
no_payments <- claim_payment_no(n_vector, claim_sizes, rfun = rmixed_payment_no,
                                claim_size_benchmark_1 = 0.0375 * ref_claim,
                                claim_size_benchmark_2 = 0.075 * ref_claim)
payment_sizes <- claim_payment_size(n_vector, claim_sizes, no_payments, rmixed_payment_size)
payment_delays <- claim_payment_delay(n_vector, claim_sizes, no_payments, setldel,
                                      rfun = r_pmtdel, paramfun = param_pmtdel,
                                      occurrence_period = rep(1:I, times = n_vector))
payment_times <- claim_payment_time(n_vector, occurrence_times, notidel, payment_delays)
payment_inflated <- claim_payment_inflation(
  n_vector, payment_sizes, payment_times, occurrence_times,
  claim_sizes, base_inflation_vector, SI_occurrence, SI_payment)

all_claims <- claims(
  frequency_vector = n_vector,
  occurrence_list = occurrence_times,
  claim_size_list = claim_sizes,
  notification_list = notidel,
  settlement_list = setldel,
  no_payments_list = no_payments,
  payment_size_list = payment_sizes,
  payment_delay_list = payment_delays,
  payment_time_list = payment_times,
  payment_inflated_list = payment_inflated
)
plot(all_claims, adjust = FALSE) +
  ggplot2::labs(subtitle = paste("With", times, "simulations"))
proc.time() - start.time

Users can also choose to plot by occurrence year, or remove the inflation by altering the arguments by_year and inflated in

plot(claims, by_year = , inflated = , adjust = )


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SynthETIC documentation built on Sept. 3, 2023, 5:06 p.m.

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