LDF Interpolation Based on Recent Boor Paper in Variance

[rajesh06]

Could you consider adding a function with the captioned. Here is a version that we wrote:

interpolate_ldfs <- function(observed_ldf_df, interp_age){
  # observed_ldf_df <- sel_data
  # interp_age <- 9

  ## At some age ('ldf_2_one') all selected 'ldfs' = 1 for all 'ages' >= ldf_2_one 
  ## Hence our 'pct_ibnr' -> inf for all 'ages' >= 'ldf_2_one', 
  ## and recieve error when fit linear model
  ## Test if 'interp_age' >= 'ldf_2_one' then return 1. Else proceed to interpolation 

  ldf_2_one <- min(observed_ldf_df$age[observed_ldf_df$ldf == 1]) 
  #the first age which the ldf is 1

  if (interp_age >= ldf_2_one) {
    return(1)
  } else {

    ## Exclude rows from 'observed_ldf_df' where ldf == 1  
    observed_ldf_df <- observed_ldf_df[observed_ldf_df$ldf != 1,]

    observed_ldf_df <- observed_ldf_df %>% 
      dplyr::mutate(pct_ibnr = 1 - (1 / ldf)) 

    ## Fit weibull model
    weibul_model <- lm(log(-log(observed_ldf_df$pct_ibnr)) ~ 
        log(observed_ldf_df$age)) # Boor Eq (8)

    ## Define the age of the ldfs above and below the interpulated age  
    age_below <- interp_age - (interp_age %% 12) 
    age_above <- interp_age + (12 - (interp_age %% 12))

    fit_below <- exp(-exp(weibul_model$coefficients[1] + 
        weibul_model$coefficients[2] * log(age_below))) 
    fit_above <- exp(-exp(weibul_model$coefficients[1] + 
        weibul_model$coefficients[2] * log(age_above)))
    fit_at <- exp(-exp(weibul_model$coefficients[1] + 
        weibul_model$coefficients[2] * log(interp_age))) 

    ## Selected ldfs at age_below and age_above
    observed_below <- observed_ldf_df$pct_ibnr[observed_ldf_df$age == age_below]
    observed_above <- observed_ldf_df$pct_ibnr[observed_ldf_df$age == age_above]

    ## observed_below is na when age_below < 12. Set equal to 1
    if(interp_age < 12) observed_below = 1

    ## variables to make extrapolation easier
    max_obs_age <- max(observed_ldf_df$age)

    if(interp_age < max_obs_age){   # interpolate
      interp_along_curve <- observed_below + (((fit_at - fit_below) / 
          (fit_above - fit_below)) * 
          (observed_above - observed_below))
    }  else{                           # extrapolate

      fit_at_max_age <- exp(-exp(weibul_model$coefficients[1] + 
          weibul_model$coefficients[2] * 
          log(max_obs_age)))

      obs_at_max_age <- observed_ldf_df$pct_ibnr[observed_ldf_df$age == 
          max_obs_age]

      interp_along_curve <- fit_at * obs_at_max_age / fit_at_max_age
    }
    ## Calculate ldf
    implied_ldf <- 1 / (1 - interp_along_curve)
    ## Adjust for age < 12 months 
    implied_full_ay_ldf <- ifelse(interp_age >= 12, implied_ldf, 
      implied_ldf * 12 / interp_age) 

    return(implied_full_ay_ldf)

  }}

[trinostics]

Hi Raj,

Will you or others be willing to maintain that function if and when anyone raises any "Issues" for it?

Including new functions in a package can be a bit of a chore. May I lean on you for some requests?

1. A link to the Variance article. That will help me understand what's going on and can be included as a reference in the help.
2. Speaking of help .. that's probably the most taxing aspect of this exercise .. we will need wording for the following (I'm sure you're familiar with all of this .. see ?ata as an example): Title, Description Definitions for each of the arguments Details (if any) Value (not always found in everyone's help, but I always find useful) Author(s) Examples (I will volunteer to create an Runit test for the function with the examples if you can give me the expected outcome! thanks)

I think that's about it. I can cobble together the Rd file with your input.

Feel free to submit that information to my email. Look forward to hearing from you.

Dan

[jimbrig]

I have some interpolation functions in my lossrx package as well here:

• Source: lossrx/actuary-interpolation.R · jimbrig/lossrx
• Docs: interp - Interpolate — interp • lossrx (jimbrig.com))

#' `interp` - Actuarial Interpolation
#'
#' Interpolate Cumulative Loss Development Factors (CDFs).
#'
#'
#' This generic function comes with three possible `method`s:
#'  1. Linear Interpolation
#'  2. Exponential Interpolation
#'  3. Double Exponential Interpolation
#'
#' @details
#' Actuaries often have to *interpolate* values in-between the selected Loss
#' Development Factors (LDFs) / Cumulative Loss Development Factors (CDFs)
#' in order to derive development factors at a variety of possible ages of
#' maturity, outside the scope of the selected factors by the actuary.
#'
#' For example, an actuary will select factors by maturity or development age
#' in months using actuarial triangles. Due to the fact the actuarial selections
#' are limited to the maturities present in the triangle (i.e. ages 12, 24, etc.),
#' the factors for ages before, after, and in-between the selection ages must be
#' *interpolated*.
#'
#' A comprehensive approach to deriving the interpolated values would follow a
#' pattern similar to the following:
#'
#' - For ages of maturity `<= First Selected Age of Maturity` (i.e. `<= 12`),
#'   factors are derived using *persistencies*. A persistency is simply a
#'   percentage value representing the percent paid/reported at a given age
#'   compared to the age's ceiling and floor. For example, a persistency as of
#'   age 3 would represent the percent paid/reported at 3 months of development
#'   out of the total percent paid/reported at 12 months of development. The
#'   persistency as of age 15 would represent the percent paid/reported at age
#'   15 compared to the total percent paid/reported between ages 12 and 24.
#'
#' - For ages of maturity `Selected Age of Maturity Floor <= x <= Selected Age of Maturity Ceiling`,
#'   i.e. *in-between* ages, the factors are derived using *double-exponential*
#'   interpolation using the selections at the floor and ceiling ages.
#'
#' - For ages of maturity `>= Last Selected Age of Maturity` (i.e. `>= 106`),
#'   a *decay factor approach* is used to *decay* the final selected factor
#'   across the ages beyond that final age.
#'
#'
#' @param new_age integer - value of the new age whose CDF is to be interpolated
#' @param cdf_array numeric vector of CDFs (usually representative of the selected factors)
#' @param age_array numeric vector of ages corresponding to the supplied `cdf_array`
#' @param cutoff the largest possible age, after which, no interpolation is performed
#' @param method integer - must be 1, 2, or 3 where 1 represents linear, 2 represents exponential,
#'   and 3 represents double exponential. Defaults to 3, but falls back onto 1 if necessary.
#'
#' @return derived numeric value for the supplied `new_age`'s CDF
#' @export
#'
#' @name interp
#'
#' @examples
#' cdfs <- c(3.579, 2.866, 2.489, 2.121, 1.876, 1.543, 1.222, 1.150, 1.109, 1.005, 1.0025)
#' ages <- seq(from = 12, to = (length(cdfs) * 12), by = 12)
#'
#' interp(14, cdfs, ages)
#' interp(12, cdfs, ages) == cdfs[[1]]
#' interp(27, cdfs, ages, method = 2)
interp <- function(new_age, cdf_array, age_array, cutoff = 450, method = 3) {

  stopifnot(
    is.numeric(new_age),
    is.numeric(cdf_array),
    is.numeric(age_array),
    length(cdf_array) == length(age_array),
    length(cdf_array[cdf_array < 0]) == 0,
    length(cdf_array[age_array < 0]) == 0,
    is.numeric(cutoff),
    is.numeric(method),
    method %in% c(1:3),
    new_age <= cutoff
  )

  age_high <- min(age_array[age_array >= new_age])
  age_low <- max(age_array[age_array <= new_age])
  cdf_high <- cdf_array[match(age_high, age_array)]
  cdf_low <- cdf_array[match(age_low, age_array)]

  if (any(age_high == age_low, cdf_high == cdf_low)) return(cdf_high)

  if (any(method == 1, new_age < min(age_array), cdf_high < 1, cdf_low < 1)) {
    out <- interp.linear(new_age, age_high, age_low, cdf_high, cdf_low)
    return(out)
  }

  if (method == 2) {
    out <- interp.exp(new_age, age_high, age_low, cdf_high, cdf_low)
    return(out)
  }

  out <- interp.dblexp(new_age, age_high, age_low, cdf_high, cdf_low)
  return(out)

}

#' @describeIn interp Double Exponential Interpolation
#' @param age_low,age_high Low and High ages
#' @param cdf_low,cdf_high Low and High CDFs
interp.dblexp <- function(new_age, age_high, age_low, cdf_high, cdf_low) {

  new_cdf <- exp(exp(((age_high - new_age) * log(log(1 / (1 / cdf_low))) +
                        (new_age - age_low) * log(log(1 / (1 / cdf_high)))) /
                       (age_high - age_low)))

  new_cdf

}

#' @describeIn interp Exponential Interpolation
#' @param age_low,age_high Low and High ages
#' @param cdf_low,cdf_high Low and High CDFs
interp.exp <- function(new_age, age_high, age_low, cdf_high, cdf_low) {

  new_cdf <- 1 / (1 - exp(((age_high - new_age) * log(1 - (1 / cdf_low)) +
                             (new_age - age_low) * log(1 - (1 / cdf_high))) /
                            (age_high - age_low)))
  new_cdf

}

#' @describeIn interp Linear Interpolation
#' @param age_low,age_high Low and High ages
#' @param cdf_low,cdf_high Low and High CDFs
interp.linear <- function(new_age, age_high, age_low, cdf_high, cdf_low) {

  new_cdf <- 1 / ((( age_high - new_age) * (1 / cdf_low) + (new_age - age_low) * (1 / cdf_high)) /
                    (age_high - age_low))

  new_cdf

}