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:
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
payment_inflated[[i]][[j]]
= $ partial payments (inflated) for claim j of occurrence period iFor 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")
library(SynthETIC) set.seed(20200131)
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
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
.
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
.
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.
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.
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.
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
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.
years
= number of years consideredI
= number of claims development periods considered (which equals the number of years divided by the time_unit
)E[i]
= exposure associated with each period ilambda[i]
= expected claim frequency per unit exposure for period iyears <- 10 I <- years / time_unit E <- c(rep(12000, I)) # effective annual exposure rates lambda <- c(rep(0.03, I))
# 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]]
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]]
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")
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]]
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.
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) } }
# 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]]
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")
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")
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).
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.
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)
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.
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)
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.
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.
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]]
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
.
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)
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))
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)
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)
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.normalised) } ## 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]]
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]])
base_inflation_past
= vector of historic quarterly inflation rates for the past $I$ periods, base_inflation_future
= vector of expected quarterly base inflation rates for the next $I$ periods (users may also choose to simulate the future inflation rates); the lengths of the vector might differ from $I$ when a time_unit
different from calendar quarter is usedclaim_payment_inflation
)SI_occurrence
= function of occurrence_time
and claim_size
that outputs the superimposed inflation index with respect to the occurrence time of the claimSI_payment
= function of payment_time
and claim_size
that outputs the superimposed inflation index with respect to payment time of the claim# 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 }
# 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]])
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)
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.
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)
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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