## Clark Cape Cod method

### Description

Analyze loss triangle using Clark's Cape Cod method.

### Usage

```ClarkCapeCod(Triangle, Premium, cumulative = TRUE, maxage = Inf,
G = "loglogistic")
```

### Arguments

 `Triangle` A loss triangle in the form of a matrix. The number of columns must be at least four; the number of rows may be as few as 1. The column names of the matrix should be able to be interpreted as the "age" of the losses in that column. The row names of the matrix should uniquely define the year of origin of the losses in that row. Losses may be inception-to-date or incremental. `Premium` The vector of premium to use in the method. If a scalar (vector of length 1) is given, that value will be used for all origin periods. (See "Examples" below.) If the length is greater than 1 but does not equal the number of rows of `Triangle` the `Premium` values will be "recycled" with a warning. `cumulative` If `TRUE` (the default), values in `Triangle` are inception to date. If `FALSE`, `Triangle` holds incremental losses. `maxage` The "ultimate" age to which losses should be projected. `adol` If `TRUE` (the default), the growth function should be applied to the length of time from the average date of loss ("adol") of losses in the origin year. If `FALSE`, the growth function should be applied to the length of time since the beginning of the origin year. `adol.age` Only pertinent if `adol` is `TRUE`. The age of the average date of losses within an origin period in the same units as the "ages" of the `Triangle` matrix. If `NULL` (the default) it will be assumed to be half the width of an origin period (which would be the case if losses can be assumed to occur uniformly over an origin period). `origin.width` Only pertinent if `adol` is `TRUE`. The width of an origin period in the same units as the "ages" of the `Triangle` matrix. If `NULL` (the default) it will be assumed to be the mean difference in the "ages" of the triangle, with a warning if not all differences are equal. `G` A `character` scalar identifying the "growth function." The two growth functions defined at this time are "loglogistic" (the default) and "weibull".

### Details

Clark's "Cape Cod" method assumes that the incremental losses across development periods in a loss triangle are independent. He assumes that the expected value of an incremental loss is equal to the theoretical expected loss ratio (ELR) times the on-level premium for the origin year times the change in the theoretical underlying growth function over the development period. Clark models the growth function, also called the percent of ultimate, by either the loglogistic function (a.k.a., "the inverse power curve") or the weibull function. Clark completes his incremental loss model by wrapping the expected values within an overdispersed poisson (ODP) process where the "scale factor" sigma^2 is assumed to be a known constant for all development periods.

The parameters of Clark's "Cape Cod" method are therefore: ELR, and omega and theta (the parameters of the loglogistic and weibull growth functions). Finally, Clark uses maximum likelihood to parameterize his model, uses the ODP process to estimate process risk, and uses the Cramer-Rao theorem and the "delta method" to estimate parameter risk.

Clark recommends inspecting the residuals to help assess the reasonableness of the model relative to the actual data (see `plot.clark` below).

### Value

A `list` of class "ClarkLDF" with the components listed below. ("Key" to naming convention: all caps represent parameters; mixed case represent origin-level amounts; all-lower-case represent observation-level (origin, development age) results.)

 `method` "CapeCod" `growthFunction` name of the growth function `Origin` names of the rows of the triangle `Premium` Premium amount for each origin year `CurrentValue` the most mature value for each row `CurrentAge` the most mature "age" for each row `CurrentAge.used` the most mature age used; differs from "CurrentAge" when adol=TRUE `MAXAGE` same as 'maxage' argument `MAXAGE.USED` the maximum age for development from the average date of loss; differs from MAXAGE when adol=TRUE `FutureValue` the projected loss amounts ("Reserves" in Clark's paper) `ProcessSE` the process standard error of the FutureValue `ParameterSE` the parameter standard error of the FutureValue `StdError` the total standard error (process + parameter) of the FutureValue `Total` a `list` with amounts that appear on the "Total" row for components "Origin" (="Total"), "CurrentValue", "FutureValue", "ProcessSE", "ParameterSE", and "StdError" `PAR` the estimated parameters `ELR` the estimated loss ratio parameter `THETAG` the estimated parameters of the growth function `GrowthFunction` value of the growth function as of the CurrentAge.used `GrowthFunctionMAXAGE` value of the growth function as of the MAXAGE.used `FutureGrowthFactor` the ("unreported" or "unpaid") percent of ultimate loss that has yet to be recorded `SIGMA2` the estimate of the sigma^2 parameter `Ldf` the "to-ultimate" loss development factor (sometimes called the "cumulative development factor") as defined in Clark's paper for each origin year `LdfMAXAGE` the "to-ultimate" loss development factor as of the maximum age used in the model `TruncatedLdf` the "truncated" loss development factor for developing the current diagonal to the maximum age used in the model `FutureValueGradient` the gradient of the FutureValue function `origin` the origin year corresponding to each observed value of incremental loss `age` the age of each observed value of incremental loss `fitted` the expected value of each observed value of incremental loss (the "mu's" of Clark's paper) `residuals` the actual minus fitted value for each observed incremental loss `stdresid` the standardized residuals for each observed incremental loss (= residuals/sqrt(sigma2*fitted), referred to as "normalized residuals" in Clark's paper; see p. 62) `FI` the "Fisher Information" matrix as defined in Clark's paper (i.e., without the sigma^2 value) `value` the value of the loglikelihood function at the solution point `counts` the number of calls to the loglikelihood function and its gradient function when numerical convergence was achieved

Daniel Murphy

### References

Clark, David R., "LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach", Casualty Actuarial Society Forum, Fall, 2003 https://www.casact.org/sites/default/files/database/forum_03fforum_03ff041.pdf

`ClarkLDF`

### Examples

```X <- GenIns
colnames(X) <- 12*as.numeric(colnames(X))
CC.loglogistic

# Clark's "CapeCod method" also works with triangles that have
# more development periods than origin periods. The Premium
# is a contrived match to the "made up" 'qincurred' Triangle.

# Method also works for a "triangle" with only one row:
# 1st row of GenIns; need "drop=FALSE" to avoid becoming a vector.

# If one value of Premium is appropriate for all origin years
# (e.g., losses are on-level and adjusted for exposure)
# then only a single value for Premium need be provided.

# Use of the weibull function generates a warning that the parameter risk
# approximation results in some negative variances. This may be of small
# concern since it happens only for older years with near-zero
# estimated reserves, but the warning should not be disregarded
# if it occurs with real data.

# The plot of the standardized residuals by age indicates that the more
# mature observations are more loosely grouped than the less mature, just
# the opposite of the behavior under the loglogistic curve.
# This suggests that the model might be improved by analyzing the Triangle
# in two different "blocks": less mature vs. more mature.
# The QQ-plot shows that the tails of the empirical distribution of
# standardized residuals are "fatter" than a standard normal.
# The fact that the p-value is essentially zero says that there is
# virtually no chance that the standardized residuals could be
# considered draws from a standard normal random variable.
# The overall conclusion is that Clark's ODP-based CapeCod model with
# the weibull growth function does not match up well with the qincurred