TheStory.Rmd

---
title: "The Liability for Claims"
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

In the Fall of 2016 you cut down some trees 
along your back property line
that impeded your view
of the beautiful valley beyond.
Your friends said "Wow."
Your neighbor complained about his loss of privacy.
Empathetically you offered your side:
"They blocked my view and, in any case, they were my trees."
Your neighbor wasn't so sure and after
researching county records discovered
the trees were on his property.
"Oh, gee, sorry,"
you say.
"Honest mistake."
He demanded restitution.
You clarified "Honest mistake!"
He filed suit.

Now you find yourself facing
hundreds of thousands of dollars in lost property value,
aledgedly,
with additional damages TBD.
You file a claim with your insurance company
to engage lawyers,
defend you, and,
heaven forbid,
pay whatever damages are levied against you
(within certain limits, of course).[^2]

[^2]:http://www.insuranceobserver.com/PDF/2003/071603.pdf

How do you know your insurance company will be around to the bitter end?

It is in every policyholder's interest 
that their insurance company stay in business
long enough to pay all benefits owed under contract.
This is usually a very large number for every insurance company.

588B = 2/3*Industry

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*[^1]
and is usually the largest liability on an insurer's balance sheet.
After estimating those liabilities and other costs,
the company takes stock of its assets to determine
if it's bankrupt or still a going concern.
Estimating the *Liability for Claims* is a very important function
of an insurance company. 
This function
falls under the purview of two key company departments: 
Claims and Actuarial.

[^1]:The NAIC 
(National Association of Insurance Commissioners)
breaks *Liability for Claims* into two categories:
Losses and Loss Adjustment Expenses
(costs to defend and manage the claim).
Liability for Claims is my term for the sum of the two.
Furthermore, in my opinion the term "loss"
connotes something detrimental from the policyholder's standpoint,
whereas those amounts are actually policyholder benefits
to which they are entitled by virtue of their 
premium payment every year.

### Claims Department

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 "claim reserves." 
A claim reserve reflects the value of that claim 
from the insurance company’s standpoint. 
But the total value of claim reserves usually falls short of a 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.
For example,
if property values in your area increase significantly,
so could the value of your claim.

B. Delayed claims: new, as-yet-unknown 
claims are often filed after a financial statement date.
For example,
as of 12/31/2016 Voter Insurance did not know
that your claim had yet to be filed.
But you paid your premium so Voter
must have enough money set aside to respond to your claim,
whatever the outcome.

### Actuarial Department

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 claim reserves to get IBNR.
Figure 1 below illustrates these concepts.

```{r, echo = FALSE}
plot(c(-3.5,10), c(-3.5,10), type = "n", xlab = "", ylab = "", 
     bty = "n", axes = FALSE)
rect(-1,0,1,5, col = "blue")
rect(-1,5,1,8, col = "green")
rect(-1,8,1,10, col = "yellow")
text(0, 2.5, "Claims\nPaid", pos = 3, col = "white")
text(0, 6.5, "Claims\nReserves", col = "black")
text(0, 9, "IBNR", col = "black")
library(pBrackets)
brackets(1, 8, 1, 0, h = NULL, ticks = 5/8, curvature =.2, type = 2,
         col = 1, lwd = 1, lty = 1, xpd = TRUE)
text(x=1.5, y=3,"Case Incurred = Claims Paid + Claims Reserves", pos = 4)
brackets(1.5, 10, 1.5, 5, ticks = .4, curvature = .2, type = 2,
         col = 1, lwd = 1, lty = 1)
text(x=2, y=8,"Total Reserves = Claims Reserves + IBNR", pos = 4)
brackets(-1, 0, -1, 10, ticks = .5, curvature = .2, type = 2,
         col = 1, lwd = 1, lty = 1, h=1)
text(x=-2, y=5,"Ultimate", pos = 2)

```

### Why is IBNR important?



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 remainder of this paper will study this method and suggested variants.

### The Claims Development Method

In a nutshell,
the Claims Development Method multiplies the current
aggregated value of 
a cohort of claims by a factor to get the ultimate value 
for that cohort.
The factor is called the Claims Development Factor (CDF).

$$ClaimsUltimate_{asof} = ClaimsValue_{asof} * CDF$$

IBNR is the difference between that estimated ultimate value
and the claims incurred value as of the financial statement date:

$$IBNR_{asof} = ClaimsUltimate_{asof} - ClaimsIncurred_{asof}$$

The substript "asof" is important. 
It is the date on which all claims values,
paids and reserves,
are pulled for entry into the company's financial statements
and into their actuarial models.[^2]

Since all reserves, claims reserves and IBNR, are estimates,
any changes in these estimates from one asof date to the next
will reflect
positively or negatively on the company's income statement
over the calendar period between those dates.

[^2]:This is the most succinct situation.
As you might imagine, 
accountants have come up with a number of different dates
that can have bearing on a company's financial statements 
filed with regulatory authorities.
We can ignore those complications here.

Let's look at an example.

### Votre Insurance Company

Your carrier,
Votre Insurance Company,
has been around since 2011.
This is what the company had on its financial statements
as of 12/31/2016:

```{r, echo = FALSE}
suppressPackageStartupMessages(library(ChainLadder))
suppressPackageStartupMessages(library(mondate))
# repeatedly run MCL method on MCL data for smaller and smaller triangles
row.names(MCLincurred) <- row.names(MCLpaid) <- 2011:2017
m <- mondate::mondate.ymd(2011:2017)

L <- list()

f <- function(i){
  M <- MunichChainLadder(MCLpaid[1:i, 1:i], MCLincurred[1:i, 1:i], .1, .1)
  rptd.diag <- c(getLatestCumulative(MCLincurred[1:i, 1:i]))
  paid.diag <- c(getLatestCumulative(MCLpaid[1:i, 1:i]))
  S <- summary(M)$ByOrigin
  paid.ulti <- S$`Ult. Paid`
  rptd.ulti <- S$`Ult. Incurred`
  pmean <- function(u, v) {apply(cbind(u, v), 1, mean)}
  ulti <- pmean(paid.ulti, rptd.ulti)
  ulti <- structure(
    sapply(seq_along(ulti), function(i) {
      if (ulti[i] <= rptd.diag[i]) ulti[i] <- max(paid.ulti[i], rptd.ulti[i], rptd.diag[i])
      ulti[i]}),
    names = names(rptd.diag)
  )
  cbind(paid.diag = paid.diag, rptd.diag = rptd.diag,
        case.diag = rptd.diag - paid.diag,
        ulti = ulti,
               paid.ulti = paid.ulti, rptd.ulti = rptd.ulti,
               ibnr = ulti - paid.diag,
        `Liability for Claims` = ulti - paid.diag)
  }
for (i in 7:3) L[[i]] <- f(i)
d <- round(L[[6]][,c("paid.diag", "case.diag", 
                     "ibnr", "Liability for Claims", "ulti")])
rbind(d, total = colSums(d))
```

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)
```