IBNR.Rmd

---
title: "IBNR"
author: "Dan Murphy"
date: "May 26, 2018"
output:
  html_document: default
  pdf_document: default
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

Prediction is hard. Especially of the future.  
\~ Old Danish Proverb

It is in the public interest for insurance companies 
to have sufficient assets to make all claim payments 
owed in the future on policies purchased in the past.
The estimated total future value owed on behalf of policyholders 
as of any given financial statement date is called the company’s 
“Liability for Claims” 
(technically that’s not the NAIC’s word for it, 
but close enough). 
Estimating this liability, usually the largest on a company's balance sheet, 
falls under the purview of two company departments: 
Claims and Actuarial.

When a policyholder files a claim, 
it is the responsibility of the Claims Department 
to estimate the total amount likely to be paid out. 
These claim-level estimates are called "case reserves." 
Case reserves reflect the value of a claim from the company’s standpoint. 
But the total value of case reserves usually falls short of the company’s 
true Liability for Claims for two reasons:

A. Updated claims: as claims stay open and 
new facts are discovered, 
values tend to rise

B. Delayed claims: new, as-yet-unknown 
claims are often filed after a financial statement date

The estimate of the total shortfall (A+B) is called the 
“Incurred But Not Reported” (IBNR) reserve. 
It is the responsibility of the Actuarial Department to estimate IBNR.

Most actuarial techniques for estimating IBNR actually 
estimate the ultimate value of claims first, 
then subtract paid-to-date to get total reserves,
and finally subtract case reserves to get IBNR.

```{r, echo = FALSE}
plot(c(-3,10), c(-3,10), type = "n", xlab = "", ylab = "", 
     bty = "n", axes = FALSE)
rect(0,0,2,5, col = "blue")
rect(0,5,2,8, col = "green")
rect(0,8,2,10, col = "yellow")
text(1, 2.5, "Paid", pos = 3, col = "white")
text(1, 6.5, "Case\nReserves", col = "black")
text(1, 9, "IBNR", col = "black")
library(pBrackets)
brackets(2, 8, 2, 0, h = NULL, ticks = 5/8, curvature =.2, type = 2,
         col = 1, lwd = 1, lty = 1, xpd = TRUE)
text(x=2.5, y=3,"Case Incurred = Paid + Case Reserves", pos = 4)
brackets(2.5, 10, 2.5, 5, ticks = .4, curvature = .2, type = 2,
         col = 1, lwd = 1, lty = 1)
text(x=3, y=8,"Total Reserves = Case Reserves + IBNR", pos = 4)
brackets(0, 0, 0, 10, ticks = .5, curvature = .2, type = 2,
         col = 1, lwd = 1, lty = 1, h=1)
text(x=-1, y=5,"Ultimate", pos = 2)

```

Actuaries have many techniques for estimating the ultimate value of claims,
and thus IBNR.
One of the simplest techniques is called the Loss Development Method.

# The Loss Development Method

In a nutshell,
the Loss Development Method multiplies the total current value of 
a cohort of claims by a factor.

called a "development factor," 
the product of which is called the "ultimate" value of 
total Liability for Claims

the Chain Ladder method looks at ratios of
total claim values as of 
The purpose of this paper is 
to investigate the mathematics behind the Chain Ladder Method.

We start this investigation with summarized claim data 
as displayed in triangular format
in Schedule P of the U.S. *Annual Statement*.

## Annual Statement: Schedule P

For the benefit of US regulators, 
as of every December 31st every insurance company must file a 
set of reports called the *Annual Statement*.
This report, a.k.a the "Yellow Book," 
is a thick compendium 
of statistics describing all aspects of an insurance company's business:
from premium to claims to investments and well beyond
deep into the bowels of an insurance company.
One of the key aspects monitored in the yellow book is 
the company's Liability for Claims.
Not only is the size of that liability important, 
but changes in that liability are important too because 
changes can directly impact the 
company's bottom line.
We will see how that works with two tables in 
**Schedule P** of the yellow book,
**Part2** and **Part3**.

Changes in that liability are closely scrutinized because they directly 
flow through to the bottom line.

year over year directly impact the
company's net income that year. over time is illuminated from various perspectives. 
The data that we will poke and prod in this paper is called GenIns.

## GenIns

The ChainLadder package describes GenIns as a
"run off triangle of accumulated general insurance claims data."
GenIns is a real, not simulated, cumulative paid loss  was created by the Australian actuary Greg Taylor.


egins with its representation in Schedule P - Part 3, meticulously named “Cumulative Paid Net Loss and Defense and Cost Containment Expenses Reported at Year End ($000 omitted)”:

```{r, echo = FALSE}
#setwd("C:/Users/Dan/Desktop/GenIns")
library(knitr)
library(kableExtra)
suppressPackageStartupMessages(library(ChainLadder))
suppressPackageStartupMessages(library(mondate))
df <- as.data.frame(GenIns, na.rm = TRUE)
df$Accident_Year <- as.numeric(df$origin) + 2007 # give labels to the origin's
df$Evaluation_YearEnd <- df$Accident_Year + df$dev - 1
df[["Evaluation_Date"]] <- mondate.ymd(df$Evaluation_YearEnd)
tri <- as.triangle(df, origin = "Accident_Year", 
                   dev = "Evaluation_YearEnd", value = "value")
#tri
#round(tri)
#noquote(format(round(tri / 1000), big.mark = ","))

class(tri) <- "matrix"
options(knitr.kable.NA = '')
kable(round(tri / 1000), align = 'r', row.names = TRUE, 
      format.args = list(big.mark = ','),
      caption = "GenIns in Sch P part 3 format") %>% 
  kable_styling(font_size = 8, full_width = FALSE)

###
## ignore below
###
tri2 <- as.triangle(df, origin = "Accident_Year", 
                    dev = "Evaluation_Date", value = "value")
#tri2
#round(tri2)
#noquote(format(round(tri2/1000), big.mark = ","))

m <- round(tri2/1000)
names(dimnames(m))[2L] <- "Evaluation_Date ($000 omitted)"
#noquote(format(m, big.mark = ","))
mf <- format(m, big.mark = ",")
nams <- names(dimnames(m))
dimnms <- dimnames(m)
mfp <- paste("   ", mf)
dim(mfp) <- dim(mf)
dimnames(mfp) <- dimnms
#noquote(mfp)
mfp[is.na(m)] <- "     XXX"
#noquote(mfp)

mfp2 <- rbind(Prior = c("     000", rep("", 9)), mfp)
#print(noquote(mfp2))
```

Then plain ol' GenIns by kable:

```{r}
G <- GenIns
class(G) <- "matrix"

kable(G, row.names = TRUE, 
      format.args = list(big.mark = ','),
      caption = "GenIns in triangle format")%>% 
  kable_styling(font_size = 8, full_width = FALSE)
```