# Freakonometrics

## Tag - insurance

Saturday, June 23 2012

## Actuarial models with R, Meielisalp

I will be giving a short course in Switzerland next week, at the 6th R/Rmetrics Meielisalp Workshop & Summer School on Computational Finance and Financial Engineering organized by ETH Zürich, https://www.rmetrics.org/. The long version of the slides for the course on Actuarial models with R can be found online with the #Rmetrics tag, and the short version will be uploaded soon. There will be some practicals, based on Swiss mortality table for the part on demography. The datasets can be uploaded using the following code,

```DEATH=read.table(
"http://freakonometrics.free.fr/DeathsSwitzerland.txt",
"http://freakonometrics.free.fr/ExposuresSwitzerland.txt",
DEATH\$Age=as.numeric(as.character(DEATH\$Age))
DEATH=DEATH[-which(is.na(DEATH\$Age)),]
EXPOSURE\$Age=as.numeric(as.character(EXPOSURE\$Age))
EXPOSURE=EXPOSURE[-which(is.na(EXPOSURE\$Age)),]```
• based on those datasets, plot the log mortality rates for male and female,

• for those two datasets, plot the log mortality rates in 1900 and 1950, respectively
• for those two datasets, plot the log mortality rates for the cohort born on 1900 and 1950, respectively
• on the total dataset (male and female), fit a Lee-Carter model. Plot the age coefficients

• plot the time coefficients and propose a forecast for that series of estimators.

• plot the residuals obtained from the regression

• using those estimates, and the forecasts, project the log-mortality rates

• extrapolate the survival function of an insured aged 40 in 2000, and compare it with the one obtained on the longitudinal dataset.

• based on those survival functions, compute actuarial present values for several quantities, e.g. whole life annuities for some insured aged 40, and whole life insurances, and compare those values from 1900 till 2040 (on the graphs below, titles were inverted).

Then, we will briefly mention payment triangles. We will work on the triangle used on http://rworkingparty.wikidot.com/ that can be downloaded below,
```OthLiabData=read.csv(
"http://www.casact.org/research/reserve_data/othliab_pos.csv",
library(plyr)
OL=SumData=ddply(OthLiabData,.(AccidentYear,DevelopmentYear,
DevelopmentLag),summarise,IncurLoss=sum(IncurLoss_H1-BulkLoss_H1),
CumPaidLoss=sum(CumPaidLoss_H1), EarnedPremDIR=
sum(EarnedPremDIR_H1))
LossTri <- as.triangle(OL, origin="AccidentYear",
dev="DevelopmentLag", value="IncurLoss")
Year=as.triangle(OL, origin="AccidentYear",
dev="DevelopmentLag", value="DevelopmentYear")
TRIANGLE=LossTri
TRIANGLE[Year>1997]=NA```
Here, the triangle looks like that
```> TRIANGLE
dev
origin      1      2      3      4      5      6      7      8      9     10
1988 128747 195938 241180 283447 297402 308815 314126 317027 319135 319559
1989 135147 208767 270979 304488 330066 339871 344742 347800 353245     NA
1990 152400 238665 297495 348826 359413 364865 372436 372163     NA     NA
1991 151812 266245 357430 400405 423172 442329 460713     NA     NA     NA
1992 163737 269170 347469 381251 424810 451221     NA     NA     NA     NA
1993 187756 358573 431410 476674 504667     NA     NA     NA     NA     NA
1994 210590 351270 486947 581599     NA     NA     NA     NA     NA     NA
1995 213141 351363 444272     NA     NA     NA     NA     NA     NA     NA
1996 237162 378987     NA     NA     NA     NA     NA     NA     NA     NA
1997 220509     NA     NA     NA     NA     NA     NA     NA     NA     NA```
• suggest an estimation for the amount of reserves, all years.
• using a Poisson regression, propose a VaR with level 99.5% for future payments, for all claims that already occurred.

Monday, June 4 2012

## Longevity and mortality dynamics with R

Following the previous post on life contingencies and actuarial models in life insurance, I upload additional material for the short course at the 6th R/Rmetrics Meielisalp Workshop & Summer School on Computational Finance and Financial Engineering organized by ETH Zürich, https://www.rmetrics.org/. The second part of the talk (on Actuarial models with R) will be dedicated to longevity and mortality. A complete set of slides can be downloaded from the blog, but again, only some part will be presented.

As mentioned earlier, the codes are from a book on actuarial science in R, written with Christophe Dutang and Vincent Goulet (so far in French) that should appear, some day... The code used in the slides above can be downloaded from here, and datasets are the following,

```> DEATH <- read.table(
+ "http://freakonometrics.free.fr/Deces-France.txt",
+ "http://freakonometrics.free.fr/Exposures-France.txt",

For additional resources, I will use Rob Hyndman's package on demography, Heather Turner and David Firth's package on generalized nonlinear models (e.g. the slides of the short course Heather gave in Rennes at the UseR! conference in 2009), as well as functions developed by JPMorgan's LifeMetrics (functions are  fully documented in the LifeMetrics Technical Document). All those functions can be obtained using

```> library(demography)
> library(gnm)
> source("http://freakonometrics.free.fr/fitModels.R")```

Friday, June 1 2012

## Life contingencies with R

I will be giving in less than four weeks a short course at the 6th R/Rmetrics Meielisalp Workshop & Summer School on Computational Finance and Financial Engineering organized by ETH Zürich, https://www.rmetrics.org/. The talk will be on Actuarial models with R, and first part will be dedicated to life insurance. A complete set of slides can be downloaded from the blog, but in the talk, only some part will be presented.

The codes are from a book on actuarial science in R, written with Christophe Dutang and Vincent Goulet (so far in French) that should appear, some day... The code used in the slides can be downloaded from here, and datasets are the following,

```> TD <- read.table(
+ "http://perso.univ-rennes1.fr/arthur.charpentier/TD8890.csv",
+ "http://perso.univ-rennes1.fr/arthur.charpentier/TV8890.csv",

For additional resources, I recommend Emiliano's website, http://www.math.uconn.edu/, with great lectures on life insurance mathematics, and the (new) lifecontinfencies vignette on http://cran.r-project.org/,

`> library(lifecontingencies)`

Wednesday, December 14 2011

## Natural Catastrophe Insurance: How Should the Government Intervene?

An updated version of the joint paper with Benoit Le Maux is online on http://papers.ssrn.com/

"The present paper develops a new theoretical framework for analyzing the decision to provide or buy insurance against the risk of natural catastrophes. In contrast with conventional models of insurance, the insurer has a non-zero probability of insolvency that depends on the distribution of the risks, the premium rate, and the amount of capital in the company. Among several results, we show that risk-averse policyholders will accept to pay higher rates for a government-provided insurance with unlimited guarantee. However, depending on the correlation between and within the regional risks, a government program can be more attractive to high-correlation than to low correlation
areas, which may lead to inefficiencies if the insurance ratings are not appropriately chosen.
"

Friday, December 10 2010

## Insurance for a warming planet

I read yesterday an interesting review of the book written by Bjørn Lomborg, which proposes an economic analysis (the subtitle being "comparing costs and benefits") of climate change. Actually, it is a collective book written by the Copenhagen Business School. Martin L. Weitzman (here) wrote a nice review in Nature, a few week ago, precisely untitled 'Insurance for a warming planet'. As Martin Weitzman wrote it, surprisingly, there is no real discussion about uncertain in the book: "key parameters
are approximated by firm values, such as the median or mean, rather t
han a probability distribution. The modelling thus becomes a knobt-widdling exercise in optimizing outcomes". And to go further, "The economics of climate change is mainly about decision-making under extreme uncertainty [...] A striking feature of the economics of climate change is that rare but catastrophic events may have unfathomable costs. Deep uncertainty about the unknown unknowns of what might go wrong is therefore coupled with essentially unlimited liability. The resulting battle between declining probabilities and increasing damages is difficult to resolve. Alas, this uncertainty can figure prominently in evaluations of climate change policies. Its absence in a book dealing with economic comparisons of smart solutions is a serious omission [...] When confronted with the possibility of extreme damages at low probabilities, most people do not look to averages. Instead they think about how much insurance they need, and can afford to buy, to survive those events. Climate policy is better viewed as buying insurance for the planet against extreme outcomes than as the solution to a multivariate problem over which we have control".

Wednesday, November 17 2010

## Course on risk, insurance, and uncertainty

The course on risk and insurance in Luminy, starts at 10.30 on Friday (here) instead of Thursday (I switched with Patrice Bertail). The slides can be found here,

Tuesday, November 9 2010

## Natural Catastrophe Insurance: When Should the Government Intervene?

Drought risk is something extremely difficult to insure, mainly because most of the assumptions we usually have in insurance are no longer valid. For instance, there is a strong correlation in space and time: if your neighbor claims a loss due to drought, so will you, probably; and if you claimed a loss last year, this year you are very likely to claim a loss, again. In France, droughts are covered by the cat nat mechanism. On the map below, we can see how many times drought insurance has been used, per village (red points means that 5 years (over the past 20) there has been a drought... well, at least 5 years, the maximum being 16),

The insurance works in France only because of the cat nat system. We can see on that picture that there are two different regions in France: the one in white, i.e. no drought over the past 20 years, and the one with small dots, i.e. droughts every 5 or 10 years. This heterogeneity might probably end the market, in the case the cat nat system does not work any longer. In several regions, no one will buy such insurance (why getting insurance if there is no risk ?), while in other, everyone will. And then, we might face some bankruptcies since risks become too correlated in those small regions... Anyway, if you're interested in droughts, you can look here, or there, or here if you want (in French) a description of the cat nat system.
This was the starting point of a paper we started a few years ago with Benoît, untitled "Natural Catastrophe Insurance: When Should the Government Intervene? ". The paper is now on-line, and can be downloaded from here.
The abstract is the following "The present research relaxes three of the usual assumptions made in the insurance literature. It is assumed that (1) there is a finite number of risks, (2) the risks are not statistically independent and (3) the structure of the market is monopolistic. In this context, the article analyses two models of natural catastrophe insurance: a model of insurance with limited liability and a model with unlimited guarantee. Among others, the results confirm the idea that the natural catastrophe insurance industry is characterized by economies of scale. The government should consequently encourage the emergence of a monopoly and discipline the industry through regulated premiums. It is also shown that government intervention of last resort is not needed when the risks are highly correlated. Lastly, the results point out that when the risks between two regions are not sufficiently independent, the pooling of the risks can lead to a Pareto improvement only if the regions face similar magnitude of damage. If not, then the region with low-damage events needs the premium to decrease to accept the pooling of the risks.".

Tuesday, November 2 2010

## Lecture notes on risk and insurance

I just finished some lectures notes on risk and insurance The notes, that can be downloaded here, are in French, and will be used at the JES (Journées d'Etudes Statistiques), organised at the CIRM (mentioned here). Previous notes on risk measures (here) and copulas (there). Again, all comments are welcome...

Thursday, April 10 2008

## Econométrie de l'assurance, un rapide survol

Rapide exposé dans le cadre d'une rencontre entre les économistes et les statisticiens rennais: glm, gam, gnm et données de panels.