Chapter 5 Aggregate Loss Models | Loss Data Analytics: Short Course

5.1 Introduction

In this section, we learn how to:

Record aggregate losses from an insurance system.
Identify actuarial applications of aggregate loss models.

Video: Introduction to Aggregate Loss Models

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5.2 Individual Risk Model

In this section, we learn how to:

Build an individual risk model for a portfolio of insurance contracts.
Apply the individual risk model to life and nonlife insurance.
Compute the distribution of aggregate losses from an individual risk model.

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5.2.1 Exercise. Simulating Claim Frequency

Assignment Text

In this assignment, we revisit Example 5.2.1 of Loss Data Analytics using simulation in R. Even though we “officially” start to work with simulation beginning in Chapter 6 in Loss Data Analytics, this exercise provides you with an opportunity to taste a flavor of simulation.

Instructions

Use the function rbinom() to simulate the number of claims that follow binomial distribution. For reproducibility of results set the initial ‘seed’ as 1000 and then generate 10,000 observations of claim frequencies for each group of policies.
Specify the size and probability when you simulate the number of claims with binomial distribution in each group of policies.

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5.2.2 Exercise. Individual Risk Model

Assignment Text

Continuing our work in Exercise 5.2.1, we now reproduce the theoretical results for calculation of moments in individual risk models using simulation. We use the simulated frequencies for each group of policies from Exercise 5.2.1 and continue to simulate compound losses in an individual risk model as elaborated in Section 5.4.2 on Loss Data Analytics.

Instructions

Use the function runif() to claim amounts that follow uniform distribution for each group of policies.
Specify the parameters of uniform random variables, minimum and maximum values of claim amounts from each type of policies.
Initialize an empty vector of length \(R\) to store \(R\) observations of simulated \(S\). In each simulation, calculate the \(j\)th simulated aggregate loss and store it in \(j\)th component of \(S\) for \(j=1, \ldots, R\).
Compute the mean and variance of \(S\) by simulation using mean() and var(). Then, compare these values to their corresponding theoretical values, 2,800 and 600,467, respectively.
Produce a visual summary of the distribution of \(S\) using the histogram function, hist() .

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5.3 Fitting a Collective Risk Model in Two Parts

In this section, we learn how to:

Build a collective risk model for a portfolio of insurance contracts.
Calculate the mean and variance of the aggregate loss.
Fit frequency and severity components in a collective risk model.

Video: Fitting a Two Part Collective Risk Model

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5.3.1 Exercise. Estimating the Frequency Component of the Collective Risk Model

Assignment Text

In this exercise and the next, you will use data to calibrate the frequency and severity components of the collective risk model. The basis for this is the LGPIF data, introduced in Section 1.3 and used in many exercises.

In this assignment, we fit a negative binomial distribution to the frequency component. You may wish to review Exercise 2.3.5 where we fit a negative binomial distribution to a small synthetic data set.

Instructions

Restrict the sample to the subset of towns and identify the claims counts as a global variable \(N\).
Construct a function to determine the negative log-likelihood. Do this based on the probability mass function for the negative binomial distribution, dnbinom(). Note that we use the same parametrization as in the Section 2.2 of Loss Data Analytics so that \(\text{size}=r\) and \(\text{prob}=1/(1+\beta)\).
Determine initial, or starting, values for the algorithm using the method of moments.
Use the function optim() to find the maximum likelihood estimates of the frequency distribution.
Provide a table to compare the empirical distribution to the fitted negative binomial distribution.

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5.3.2 Exercise. Estimating the Severity Component of the Collective Risk Model

Assignment Text

This is a continuation of Exercise 5.3.1. In this assignment, we fit a Pareto distribution to the severity component. You may wish to review Exercise 3.5.1 where we fit a Pareto distribution to a different subset of the LGPIF data.

Instructions

Restrict the sample to the subset of towns and identify the average claim severity as a global variable \(Xbar\).
Construct a function to determine the negative log-likelihood. Do this based on the probability density function dlomax(). Note that we use the same parametrization as in the Section 3.2 of Loss Data Analytics so that \(\text{shape}=\alpha\) and \(\text{scale}=\theta\).
Determine initial, or starting, values for the algorithm using the method of moments.
Use the function optim() to find the maximum likelihood estimates of the severity distribution.
Provide a \(qq\) plots of the empirical versus fitted quantiles to assess the adequacy of the model fit.

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

5.4 Computation Strategies for a Collective Risk Model

In this section, we learn how to:

Compute the aggregate loss distribution.
Implement numerical strategies in R.

Video: Computing a Collective Risk Model Distribution

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xx ### Exercise. Collective Risk Model II

Assignment Text

In this assignment, we analyze the aggregate loss distribution by combining fitted frequency and severity components. Because it is usually very hard to obtain a closed form expression for the distribution function of the compound loss in a collective risk model, we rely on simulations to compute the aggregate loss distribution. We use the fitted parameters from the exercises in Section 5.3. Specifically, fitted shape and scale negative binomial parameters have already been recorded as the global variables r.est and beta.est, respectively. In the same way, fitted shape and scale Pareto parameters have already been recorded as the global variables alpha.est and theta.est, respectively.

Instructions

As an input to calculating the aggregate distribution, we need the expression function. This function allows to call functions without actually evaluating them.
Also, to generate random samples from the fitted frequency and severity models, we use rnbinom() and rlomax(). Note that these generated random samples are also used as inputs to compute the aggregate claim amount distribution.
We use aggregateDist() to simulate compound losses that match the observed data upon the fitted frequency and severity models.
With the aggregate distribution, plot its distribution function. For comparison, superimpose on this plot the empiricial (cumulative) distribution function, created using the ecdf() function.

{"language":"r","pre_exercise_code":"Insample <- read.csv(\"https://raw.githubusercontent.com/OpenActTexts/LDACourse1/main/Data/Insample.csv\", header=T, na.strings=c(\".\"), stringsAsFactors=FALSE)\nlibrary(lme4)\n\naggregate.portfolio <- function(x, by = names(x$nodes), FUN = sum,\n                                classification = TRUE, prefix = NULL, ...)\n{\n    level.names <- names(x$nodes)       # level names\n    nlevels <- length(level.names)      # number of levels\n    years <- level.names[nlevels]       # name of last level\n\n    ## Match level names in 'by' to those in the model\n    by <- match.arg(by, level.names, several.ok = TRUE)\n\n    ## Version of FUN able to work on lists\n    fun <- function(x, ...) FUN(unlist(x), ...)\n\n    ## The most common case should be to aggregate claim amounts by\n    ## node. This case being very simple, it is treated separately.\n    if (identical(by, level.names))\n        return(cbind(if (classification) x$classification,\n                     array(sapply(x$data, FUN, ...), dim(x$data),\n                           dimnames = list(NULL, paste(prefix, colnames(x$data), sep = \"\")))))\n\n    ## Summaries only by last level (years) are also simple to handle.\n    if (identical(by, years))\n    {\n        res <- apply(x$data, 2, fun, ...)\n        names(res) <- paste(prefix, colnames(x$data), sep = \"\")\n        return(res)\n    }\n\n    ## The other possibilities require to split the data in groups as\n    ## specified in argument 'by'. If the last level (years) is in\n    ## 'by', then the matrix structure must be retained to make the\n    ## summaries. Otherwise, it can just be dropped since summaries\n    ## will span the years of observation.\n    ##\n    ## Convert the sequence of subscripts into factors by pasting the\n    ## digits together. It is important *not* to sort the levels in\n    ## case the levels in 'by' are not in the same order as in\n    ## 'level.names'.\n    rows <- setdiff(by, years)          # groups other than years\n    s <- x$classification[, rows, drop = FALSE] # subscripts\n    f <- apply(s, 1, paste, collapse = \"\")      # grouping IDs\n    f <- factor(f, levels = unique(f))          # factors\n    s <- s[match(levels(f), f), , drop = FALSE] # unique subscripts\n    xx <- split(x$data, f)                      # split data\n\n    ## Make summaries\n    if (years %in% by)\n    {\n        xx <- lapply(xx, matrix, ncol = ncol(x$data))\n        res <- t(sapply(xx, function(x, ...) apply(x, 2, fun, ...), ...))\n        cols <- colnames(x$data)\n    }\n    else\n    {\n        res <- sapply(xx, fun, ...)\n        cols <- deparse(substitute(FUN))\n    }\n\n    ## Return results as a matrix\n    structure(cbind(if (classification) s, res),\n              dimnames = list(NULL, c(if (classification) rows, paste(prefix, cols, sep = \"\"))))\n}\n\nsimS <- function(n, model.freq, model.sev)\n{\n  \n    n=1000\n    model.freq = freq\n    model.sev = sev\n    \n    ## Prepare the call to simul() by building up 'nodes'\n    level.names <- names(if (is.null(model.freq)) model.sev else model.freq)\n    nlevels <- length(level.names)\n    nodes <- as.list(c(rep(1, nlevels - 1), n))\n    names(nodes) <- level.names\n\n    ## Get sample\n    x <- aggregate(simul(nodes = nodes,\n                         model.freq = model.freq,\n                         model.sev = model.sev))[-1]\n\n    ## Compute the empirical cdf of the sample. Done manually instead\n    ## of calling stats:::ecdf() to keep a copy of the empirical pmf\n    ## in the environment without computing it twice.\n    x <- sort(x)\n    vals <- unique(x)\n    fs <- tabulate(match(x, vals))/length(x)\n    FUN <- approxfun(vals, pmin(cumsum(fs), 1), method = \"constant\",\n                     yleft = 0, yright = 1, f = 0, ties = \"ordered\")\n    class(FUN) <- c(\"ecdf\", \"stepfun\", class(FUN))\n    assign(\"fs\", fs, envir = environment(FUN))\n    FUN\n}\n\naggregateDist <-\n    function(method = c(\"recursive\", \"convolution\", \"normal\", \"npower\", \"simulation\"),\n             model.freq = NULL, model.sev = NULL, p0 = NULL, x.scale = 1,\n             convolve = 0, moments, nb.simul, ...,\n             tol = 1e-06, maxit = 500, echo = FALSE)\n{\n    Call <- match.call()\n\n    ## The method used essentially tells which function should be\n    ## called for the calculation of the aggregate claims\n    ## distribution.\n    method <- match.arg(method)\n\n    if (method == \"normal\")\n    {\n        ## An error message is issued if the number of moments listed\n        ## is not appropriate for the method. However it is the user's\n        ## responsibility to list the moments in the correct order\n        ## since the vector is not required to be named.\n        if (missing(moments) || length(moments) < 2)\n            stop(\"'moments' must supply the mean and variance of the distribution\")\n        FUN <- normal(moments[1], moments[2])\n        comment(FUN) <- \"Normal approximation\"\n    }\n\n    else if (method == \"npower\")\n    {\n        if (missing(moments) || length(moments) < 3)\n            stop(\"'moments' must supply the mean, variance and skewness of the distribution\")\n        FUN <- npower(moments[1], moments[2], moments[3])\n        comment(FUN) <- \"Normal Power approximation\"\n    }\n    else if (method == \"simulation\")\n    {\n        if (missing(nb.simul))\n            stop(\"'nb.simul' must supply the number of simulations\")\n        if (is.null(names(model.freq)) && is.null(names(model.sev)))\n            stop(\"expressions in 'model.freq' and 'model.sev' must be named\")\n        FUN <- simS(nb.simul, model.freq = model.freq, model.sev = model.sev)\n        comment(FUN) <- \"Approximation by simulation\"\n    }\n    else\n    {\n        ## \"recursive\" and \"convolution\" cases. Both require a\n        ## discrete distribution of claim amounts, that is a vector of\n        ## probabilities in argument 'model.sev'.\n        if (!is.numeric(model.sev))\n            stop(\"'model.sev' must be a vector of probabilities\")\n\n        ## Recursive method uses a model for the frequency distribution.\n        if (method == \"recursive\")\n        {\n            if (is.null(model.freq) || !is.character(model.freq))\n                stop(\"frequency distribution must be supplied as a character string\")\n            dist <- match.arg(tolower(model.freq),\n                              c(\"poisson\",\n                                \"geometric\",\n                                \"negative binomial\",\n                                \"binomial\",\n                                \"logarithmic\",\n                                \"zero-truncated poisson\",\n                                \"zero-truncated geometric\",\n                                \"zero-truncated negative binomial\",\n                                \"zero-truncated binomial\",\n                                \"zero-modified logarithmic\",\n                                \"zero-modified poisson\",\n                                \"zero-modified geometric\",\n                                \"zero-modified negative binomial\",\n                                \"zero-modified binomial\"))\n            FUN <- panjer(fx = model.sev, dist = dist, p0 = p0,\n                          x.scale = x.scale, ..., convolve = convolve,\n                          tol = tol, maxit = maxit, echo = echo)\n            comment(FUN) <- \"Recursive method approximation\"\n        }\n\n        ## Convolution method requires a vector of probabilities in\n        ## argument 'model.freq'.\n        else if (method == \"convolution\")\n        {\n            if (!is.numeric(model.freq))\n                stop(\"'model.freq' must be a vector of probabilities\")\n            FUN <- exact(fx = model.sev, pn = model.freq, x.scale = x.scale)\n            comment(FUN) <- \"Exact calculation (convolutions)\"\n        }\n        else\n            stop(\"internal error\")\n    }\n\n    ## Return cumulative distribution function\n    class(FUN) <- c(\"aggregateDist\", class(FUN))\n    attr(FUN, \"call\") <- Call\n    FUN\n    }\n\n\nplot.aggregateDist <- function(x, xlim,\n                               ylab = expression(F[S](x)),\n                               main = \"Aggregate Claim Amount Distribution\",\n                               sub = comment(x), ...)\n{\n    ## Function plot() is used for the step cdfs and function curve()\n    ## in the continuous cases.\n    if (\"stepfun\" %in% class(x))\n    {\n        ## Method for class 'ecdf' will most probably be used.\n        NextMethod(main = main, ylab = ylab, ...)\n    }\n    else\n    {\n        ## Limits for the x-axis are supplied if none are given\n        ## in argument.\n        if (missing(xlim))\n        {\n            mean <- get(\"mean\", envir = environment(x))\n            sd <- sqrt(get(\"variance\", envir = environment(x)))\n            xlim <- c(mean - 3 * sd, mean + 3 * sd)\n        }\n        curve(x, main = main, ylab = ylab, xlim = xlim, ylim = c(0, 1), ...)\n    }\n    mtext(sub, line = 0.5)\n}\n\nsimul <- function(nodes, model.freq = NULL, model.sev = NULL, weights = NULL)\n{\n    ## Get level names. Each could be NULL.\n    level.names <- names(nodes)\n    freq.names <- names(model.freq)\n    sev.names <- names(model.sev)\n\n    ## 'nodes' must be a named list. One exception is allowed: there\n    ## is only one level. In this case, add a predetermined name if\n    ## there isn't one already and make sure 'nodes' is a list.\n    if (length(nodes) == 1L)\n    {\n        if (is.null(level.names))\n            names(nodes) <- \"X\"\n        nodes <- as.list(nodes)\n    }\n    else\n    {\n        if (!is.list(nodes) || is.null(level.names))\n            stop(\"'nodes' must be a named list\")\n    }\n\n    ## Determine if frequency and severity models are present. Keep\n    ## for future use.\n    has.freq <- !all(sapply(model.freq, is.null))\n    has.sev  <- !all(sapply(model.sev, is.null))\n\n    ## Check that at least one of 'model.freq' or 'model.sev' is\n    ## present and that the level names match with those of 'nodes'.\n    ## Perhaps is there a fancier way to do all these tests, but the\n    ## version below is at least easy to follow.\n    if (has.freq)\n    {\n        if (has.sev)\n        {\n            if (! (identical(level.names, freq.names) &&\n                   identical(level.names, sev.names)))\n                stop(\"level names different in 'nodes', 'model.freq' and 'model.sev'\")\n        }\n        else\n        {\n            if (!identical(level.names, freq.names))\n                stop(\"level names different in 'nodes', 'model.freq' and 'model.sev'\")\n        }\n    }\n    else\n    {\n        if (has.sev)\n        {\n            if (!identical(level.names, sev.names))\n                stop(\"level names different in 'nodes', 'model.freq' and 'model.sev'\")\n        }\n        else\n            stop(\"one of 'model.freq' or 'model.sev' must be non-NULL\")\n    }\n\n    ## The function is written for models with at least two levels\n    ## (entity and year). If there is only one, add a dummy level to\n    ## avoid scattering the code with conditions.\n    if (length(nodes) < 2L)\n    {\n        nodes <- c(node = 1, nodes)\n        model.freq <-\n            if (has.freq) c(expression(node = NULL), model.freq) else NULL\n        model.sev <-\n            if (has.sev) c(expression(node = NULL), model.sev) else NULL\n    }\n\n    ## Frequently used quantities\n    level.names <- names(nodes)         # need to reset!\n    nlevels <- length(nodes)            # number of levels\n\n    ## Recycling of the number of nodes (if needed) must be done\n    ## \"manually\". We do it here once and for all since in any case\n    ## below we will need to know the total number of nodes in the\n    ## portfolio. Furthermore, the recycled list 'nodes' will be\n    ## returned by the function.\n    for (i in 2L:nlevels)       # first node doesn't need recycling\n        nodes[[i]] <- rep(nodes[[i]], length = sum(nodes[[i - 1L]]))\n\n    ## Simulation of the frequency mixing parameters for each level\n    ## (e.g. class, contract) and, at the last level, the actual\n    ## frequencies. If 'model.freq' is NULL, this is equivalent to\n    ## having one claim per node.\n    if (has.freq)\n    {\n        ## Normally, only the immediately above mixing parameter will\n        ## be used in the model for a level, but the code here allows\n        ## for more general schemes. For this to work, all mixing\n        ## parameters have to be correctly recycled at each level\n        ## where they *could* be used. Since the model at any level\n        ## could be NULL, 'params' will keep track of the mixing\n        ## parameters that were simulated in previous iteration of the\n        ## forthcoming loop.\n        params <- character(0)\n\n        for (i in seq_len(nlevels))\n        {\n            ## Number of nodes at the current level\n            n.current <- nodes[[i]]\n\n            ## Extract simulation model for the level.\n            Call <- model.freq[[i]]\n\n            ## Repeat the mixing parameters of all levels above the\n            ## current one that were simulated in the past.\n            for (j in seq_along(params))\n                eval(substitute(x <- rep.int(x, n.current),\n                                list(x = as.name(params[[j]]))))\n\n            ## Simulate data only if there is a model at the current\n            ## level.\n            if (!is.null(Call))\n            {\n                ## Add the number of variates to the call.\n                Call$n <- sum(n.current)\n\n                ## Simulation of the mixing parameters or the data. In\n                ## the latter case, store the results in a fixed\n                ## variable name.\n                if (i < nlevels)\n                {\n                    assign(level.names[[i]], eval(Call))\n                    params[i] <- level.names[[i]] # remember the parameter\n                }\n                else\n                    frequencies <- eval(Call)\n            }\n        }\n    }\n    else\n        frequencies <- rep.int(1, sum(nodes[[nlevels]]))\n\n    ## Simulation of the claim amounts. If 'model.sev' is NULL, this\n    ## is equivalent to simulating frequencies only.\n    if (has.sev)\n    {\n        ## Repeat the same procedure as for the frequency model, with\n        ## one difference: when reaching the last level (claim\n        ## amounts), the number of variates to simulate is not given\n        ## by the number of nodes but rather by the number of claims\n        ## as found in 'frequencies'.\n        params <- character(0)\n\n        for (i in seq_len(nlevels))\n        {\n            n.current <- nodes[[i]]\n            Call <- model.sev[[i]]\n\n            for (j in seq_along(params))\n                eval(substitute(x <- rep.int(x, n.current),\n                                list(x = as.name(params[[j]]))))\n\n            if (!is.null(Call))\n            {\n                ## The rest of the procedure differs depending if we\n                ## are still simulating mixing parameters or claim\n                ## amounts.\n                if (i < nlevels)\n                {\n                    ## Simulation of mixing parameters is identical to the\n                    ## simulation of frequencies.\n                    Call$n <- sum(n.current)\n                    assign(level.names[[i]], eval(Call))\n                    params[i] <- level.names[[i]]\n                }\n                else\n                {\n                    ## For the simulation of claim amounts, the number\n                    ## of variates is rather given by the\n                    ## 'frequencies' object. Furthermore, the mixing\n                    ## parameters must be recycled once more to match\n                    ## the vector of frequencies.\n                    for (p in intersect(all.vars(Call), params))\n                        eval(substitute(x <- rep.int(x, frequencies),\n                                        list(x = as.name(p))))\n                    Call$n <- sum(frequencies)\n                    severities <-eval(Call)\n                }\n            }\n        }\n    }\n    else\n        severities <- rep.int(1, sum(frequencies))\n\n    ## We must now distribute the claim amounts in vector 'severities'\n    ## to the appropriate nodes. This is complicated by the\n    ## possibility to have different number of nodes (years of\n    ## observation) for each entity. The result must be a matrix\n    ## with the number of columns equal to the maximum number of last\n    ## level nodes.\n    ##\n    ## The number of nodes (years of observation) per entity is\n    ## given by 'n.current' since we reached the last level in (either\n    ## one of) the above loops.\n    ##\n    ## Assign a unique ID to each node, leaving gaps for nodes without\n    ## observations.\n    ind <- unlist(mapply(seq,\n                         from = seq(by = max(n.current), along = n.current),\n                         length = n.current))\n\n    ## Repeating the vector of IDs according to the frequencies\n    ## effectively assigns a node ID to each claim amount. The vector\n    ## of claim amounts is then split by node, yielding a list where\n    ## each element corresponds to a node with claims.\n    f <- rep.int(ind, frequencies)\n    severities <- split(severities, f)\n\n    ## Identify nodes with frequency equal to 0, which is different\n    ## from having no observation (NA).\n    freq0 <- ind[which(frequencies == 0)]\n\n    ## Rearrange the list of claim amounts in a matrix;\n    ##\n    ##      number of rows: number of nodes at the penultimate level\n    ##                      (number of entities)\n    ##   number of columns: maximum number of nodes at the last level\n    ##                      (number of years of observation).\n    ##\n    ## Moreover, assign a value of 'numeric(0)' to nodes with a\n    ## frequency of 0.\n    nrow <- length(n.current)           # number of entities\n    ncol <- max(n.current)              # number of years\n    res <- as.list(rep.int(NA, nrow * ncol))\n    res[unique(f)] <- severities\n    res[freq0] <- lapply(rep.int(0, length(freq0)), numeric)\n    res <- matrix(res, nrow, ncol, byrow = TRUE,\n                  dimnames = list(NULL,\n                                  paste(level.names[nlevels],\n                                        seq_len(ncol), sep = \".\")))\n\n    ## Reshape weights as a matrix, if necessary.\n    weights <- if (is.null(weights))\n        NULL\n    else\n    {\n        ## Integrate NAs into the weights matrix as appropriate.\n        w <- rep.int(NA, nrow * ncol)\n        w[ind] <- weights\n        matrix(w, nrow = nrow, byrow = TRUE, dimnames = dimnames(res))\n    }\n\n    ## Finally, create a matrix where each row contains the series of\n    ## identifiers for an entity in the portfolio, e.g. if the data\n    ## is denoted X_{ijkt}, one line of the matrix will contain\n    ## subscripts i, j and k. As we move from right to left in the\n    ## columns of 'm', the subscripts are increasingly repeated.\n    ncol <- nlevels - 1L\n    m <- matrix(1, nrow, ncol,\n                dimnames = list(NULL, head(level.names, ncol)))\n    for (i in seq_len(ncol - 1L))    # all but the last column\n    {\n        ## Vector 'x' will originally contain all subscripts for one\n        ## level. These subscripts are then repeated as needed to give\n        ## the desired result. To avoid another explicit loop, I use a\n        ## 'lapply' with a direct assignment in the current\n        ## frame. Somewhat unusual, but this is the simplest procedure\n        ## I managed to come up with.\n        x <- unlist(lapply(nodes[[i]], seq))\n        lapply(nodes[(i + 1L):(nlevels - 1L)],\n               function(v) assign(\"x\", rep.int(x, v), envir = parent.frame(2)))\n        m[, i] <- x\n    }\n    m[, ncol] <- unlist(lapply(nodes[[ncol]], seq)) # last column\n\n    ## Return object of class 'portfolio'\n    structure(list(data = res,\n                   weights = weights,\n                   classification = m,\n                   nodes = nodes,\n                   model.freq = model.freq,\n                   model.sev = model.sev),\n              class = \"portfolio\")\n}\n\ndlomax <- function(x, shape=1, scale=1, log=FALSE) {\n  if (log==FALSE) result <- shape*scale^shape / (scale + x)^(shape+1)\n  if (log==TRUE ) result <- log(shape)+shape*log(scale)-(shape+1)*log(scale + x)\n  return(result) }\n\nrlomax <- function(n, shape=1, scale=1) {\n  u <- runif(n)\n  result <- scale*((1 - u)^(-1/shape)-1)\n  return(result) }\n\nInsampleTown <- subset(Insample, TypeTown==1)\n\n# The negative binomial frequency model: N\nN <- InsampleTown$Freq\nfreq_lik <- function(parm) {\n  r    <- parm[1]\n  beta <- parm[2]\n  lik  <- -sum(dnbinom(N, size=r, prob=1/(1+beta), log=TRUE))\n  return(lik)\n}\ninit.parm.freq <- c(mean(N)/(var(N)/mean(N)-1), var(N)/mean(N)-1) # initial estimates by method of moments\nfreq_mod       <- optim(par=init.parm.freq, fn=freq_lik) # Maximum likelihood estimation for the frequency model\n\nr.est    <- freq_mod$par[1]\nbeta.est <- freq_mod$par[2]\n\n# The Pareto severity model: Xbar\nXbar <- InsampleTown$yAvg[which(InsampleTown$Freq>0)]\nsev_lik <- function(parm) {\n  alpha <-  parm[1]\n  theta <-  parm[2]\n  lik   <- -sum(dlomax(Xbar, shape=alpha, scale=theta, log=TRUE))\n  return(lik)\n}\ninit.parm.sev <- c( 2/(1-mean(Xbar)^2/var(Xbar))  , mean(Xbar)*(2/(1-mean(Xbar)^2/var(Xbar))-1) ) # initial estimates by method of moments\nsev_mod       <- optim(par=init.parm.sev, fn=sev_lik, method=\"L-BFGS-B\") # Maximum likelihood estimation for the severity model\nalpha.est     <- sev_mod$par[1]\ntheta.est     <- sev_mod$par[2]","sample":"# Parameter estimates from the frequency model: alpha.est, theta.est\n# Parameter estimates from the severity  model:     r.est,  beta.est\n\n# Random samples from fitted frequency and severity models\nfreq <- expression(data =  ??(size=??, prob=1/(1+??)))\nsev  <- expression(data =  ??(shape=??, scale=??))\n# The aggregate distribution \nFs   <- aggregateDist(\"simulation\", nb.simul = 1000, model.freq = ??, model.sev = ??)\nplot(Fs)\nlines(ecdf(InsampleTown$y),  col=\"blue\")\nlegend(\"bottomright\", legend = c(\"Simulation\", \"Empirical\"), col=c(\"black\",\"blue\"), lty=c(1,2), pch=c(1, 19))","solution":"# Parameter estimates from the frequency model: alpha.est, theta.est\n# Parameter estimates from the severity  model:     r.est,  beta.est\n\n# Random samples from fitted frequency and severity models\nfreq <- expression(data =  rnbinom(size=r.est, prob=1/(1+beta.est)))\nsev  <- expression(data =  rlomax(shape=alpha.est, scale=theta.est))\n# The aggregate distribution \nFs   <- aggregateDist(\"simulation\", nb.simul = 1000, model.freq = freq, model.sev = sev)\nplot(Fs)\nlines(ecdf(InsampleTown$y),  col=\"blue\")\nlegend(\"bottomright\", legend = c(\"Simulation\", \"Empirical\"), col=c(\"black\",\"blue\"), lty=c(1,2), pch=c(1, 19))","sct":"Fsmsg <- \"Did you correctly specify the object `Fs`?\"\nex() %>% check_object(\"Fs\", undefined_msg = \"Make sure to not remove `Fs`!\") %>% check_equal(incorrect_msg=Fsmsg)\nfreqmsg <- \"Did you correctly specify the object `freq`?\"\nex() %>% check_object(\"freq\", undefined_msg = \"Make sure to not remove `freq`!\") %>% check_equal(incorrect_msg=freqmsg)\nsevmsg <- \"Did you correctly specify the object `sev`?\"\nex() %>% check_object(\"sev\", undefined_msg = \"Make sure to not remove `sev`!\") %>% check_equal(incorrect_msg=sevmsg)\nsuccess_msg(\"Good job! This example demonstrates how to use simulation to calculate a complicated (distribution) function. It is an important task in order to visualize the distribution of compound losses and make subsequent decisions.\")","hint":"We should use the estimated parameters from the frequency and severity distributions in calculation of the aggregate claim amount distribution."}

5.5 Tweedie Distribution

In this section, we learn how to:

Construct the Tweedie distribution from a collective risk model.
Establish the Tweedie distribution as a member of the exponential family of distributions.
Fit Tweedie distribution as a generalized linear model.

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5.5.1 Exercise. Fitting a Tweedie Distribution

Assignment Text

In this assignment, we fit a Tweedie model for a collective risk model using observed compound losses in LGPIF data with R. Unlike the previous tutorials that required separate model fitting for the frequency and severity components, use of the Tweedie distribution enables us to fit a collective risk model in a single step.

Instructions

Restrict the sample to the subset of towns and identify the total claim severity as a global variable \(S\).
Construct a function to determine the negative log-likelihood. Do this based on the hybrid probability mass/density function dtweedie(). This is a built-in function from R library tweedie .
From the fitted Tweedie model via optim(), extract the estimated mu, phi, and p.
After estimating the parameters of the Tweedie model, plot the empirical quantiles against the quantiles from the fitted model. Note that we generate the fitted quantiles via qtweedie() and the estimated Tweedie parameters.

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5.6 Effects of Coverage Modifications

In this section, we learn how to:

Examine the impact of aggregate deductible on the aggregate loss.
Examine the effect of per-occurrence deductible on frequency and severity components in the aggregate loss.

Video: Effects of Coverage Modifications

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Contributors

Authors. Himchan Jeong, Simon Fraser University, and Peng Shi, University of Wisconsin-Madison, are the principal authors of the initial version of this chapter.
Chapter Maintainers. Please contact Himchan himchan_jeong@sfu.ca and/or Jed at jfrees@bus.wisc.edu for chapter comments and suggested improvements.