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GK Errata.tex
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\documentclass[]{article}
\usepackage{color}
\usepackage[normalem]{ulem}
\usepackage{array}
\usepackage{booktabs}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{soul}
\newcommand{\sti}{
{\mbox{\st{
$\displaystyle\prod_{j = 1}^{\color{red}i\color{black}}$
}}}
}
%opening
\title{Clarification and Errata to \\\textit{Catastrophe Modeling: \\A New Approach to Managing Risk} \\(Grossi, P. and Kunreuther, H., Editors)}
\author{Casualty Actuarial Society Syllabus Committee \thanks{This note was originally prepared by Rajesh Sahasrabuddhe, FCAS, MAAA, CAS Syllabus Committee Chairperson in 2013. It has been revised based on similar comments provided contemporaneously by Josh Taub, FCAS and Matthew M. Iseler, FCAS.} } %\\with assistance from Dave Clark, FCAS, MAAA}
\begin{document}
\maketitle
\begin{abstract}
\end{abstract}
This notes presents an errata and clarifying remarks to Section 2.4 Derivation and Use of an Exceedance Probability Curve of \textit{Catastrophe Modeling: A New Approach to Managing Risk}.
\section{Clarification}
The use of the phrase ``exceedance probability" in Section 2.4 is ambiguous. Specifically, “exceedance probability” can be used in one of three ways:
\begin{description}
\item[Occurrence Exceedance Probability (OEP)] The OEP is the probability that at least one loss exceeds the specified loss amount.
\item[Aggregate Exceedance Probability (AEP)] The AEP is the probability \\that the sum of all losses during a given period exceeds some amount.
\item[Conditional Exceedance Probability (CEP)] The CEP is the probability that the amount on a single event exceeds a specified loss amount; this is equal to 1-CDF of the severity curve as used by actuaries in other contexts.
\end{description}
For actuaries who have not worked with catastrophe models, the OEP may be a new concept. Actuaries usually think of severity distributions, which correspond to the CEP - not the OEP. In Section 2.4, the term ``exceedance probability" refers to the \textbf{Occurrence Exceedance Probability (OEP)}. The $OEP$ is the distribution of the largest loss in the period and is based on the theory of order statistics.
\section{Errata}
\begin{itemize}
\item The end continued paragraph at the top of page 30 is corrected as follows:\\
\\
A list of \color{red} \sout{15} 14\color{black}\footnote{Editor's note: The definition of $E_i$ includes events that ``\emph{could} damage a portfolio of structures'' (emphasis added). We assume that event~\#15 in the original Table 2.1 would have met this standard (e.g. a hurricane that turns away from land). We have removed event~\#15 in order to emphasize that the probabilities need not sum to 1.000.} such events is listed in Table 2.1 ranked in descending order of the amount of loss. \color{red} \sout{In order to keep the example simple and the calculations straightforward, these events were chosen so the set is exhaustive (i.e., sum of probabilities for all events equals one)}.\color{black}
\item The first complete paragraph on page 30 is corrected as follows:\\
\\
The events listed in Table 2.1 are assumed to be independent Bernoulli random variables \color{red} .\sout{, each with a}. It is assumed that each event only occurs at most once with \color{red} the \color{black} probability mass function defined as: \ldots
\item The second complete paragraph on page 30 is corrected as follows:\\
\\
If an event $E_i$ does not occur, the loss \color{red} for that event \color{black} is 0. \ldots
%\item The third complete paragraph on page 30 is corrected as follows:\\
%\\
%The \color{red} \sout{overall expected loss} average size of an individual loss \sout{for the entire series of events}\color{black}, denoted as the \color{red} \sout{average annual loss (AAL)}Average Loss Event ($ALE$) \color{black} in Table 2.1, is the sum of expected losses \color{red} \sout{of} for \color{black} each of the individual events for a given year and is given by:\\
%\begin{equation*}
% \color{red}ALE \color{black} = \sum_i p_i \times L_i \\
%\end{equation*}
%\color{red} Note: Candidates should review the definition of average annual loss (AAL) in the Glossary as this term is used in other sections of this material.
\color{black}
\item The fourth complete paragraph on page 30 is corrected as follows:\\
\\
Assuming that during a given year, at most only one \color{red} of each \color{black} disaster occurs, the \color{red} $OEP$ \sout{exceedance probability} \color{black} for a given level of loss, $\color{red}O \color{black}EP(L_i)$, can be determined by calculating: \ldots
\item The first sentence of the fifth complete paragraph on page 30 is corrected as follows:\\
\\
The resulting \color{red} $OEP$ is the probability that at least one loss exceeds a given value \sout{exceedance probability is the annual probability that the loss exceeds a given value}\color{black}.
\item The upper limit of the product in the last equation on page 30 is corrected from \color{red}$i$\color{black}~to \color{red}$i-1$\color{black}~as follows:
\begin{equation*}
OEP(L_i) = 1 - \sti\prod_{j = 1}^{\color{red}i-1\color{black}}(1 - p_i)
\end{equation*}
\newpage
\item Table 2.1 is replaced with the following:
% Table generated by Excel2LaTeX from sheet 'Sheet1'
\numberwithin{table}{section}
\begin{table}[h]
\centering
\caption{Events, Losses and Probabilities}
\begin{tabular}{crrrr}
\toprule
%Line 1
& \multicolumn{1}{c}{Annual} & & \multicolumn{1}{c}{Occurrence}
& \\
%Line 2
& \multicolumn{1}{c}{Probability of } &
& \multicolumn{1}{c}{Exceedance} & \\
%Line 3
Event & \multicolumn{1}{c}{Occurrence} & \multicolumn{1}{c}{Loss}
& \multicolumn{1}{c}{Probability} & \multicolumn{1}{c}{$E[L] =$ } \\
%Line 4
\multicolumn{1}{c}{$(E_i)$} & \multicolumn{1}{c}{$(p_i)$} & \multicolumn{1}{c}{$(L_i)$}
& \multicolumn{1}{c}{$[OEP(L_i)]$} & \multicolumn{1}{c}{$p_i \times L_i$ } \\
\midrule
1 & 0.002 & \$25,000,000 & 0.0000 & \$50,000 \\
2 & 0.005 & 15,000,000 & 0.0020 & 75,000 \\
3 & 0.010 & 10,000,000 & 0.0070 & 100,000 \\
4 & 0.020 & 5,000,000 & 0.0169 & 100,000 \\
5 & 0.030 & 3,000,000 & 0.0366 & 90,000 \\
6 & 0.040 & 2,000,000 & 0.0655 & 80,000 \\
7 & 0.050 & 1,000,000 & 0.1029 & 50,000 \\
8 & 0.050 & 800,000 & 0.1477 & 40,000 \\
9 & 0.050 & 700,000 & 0.1903 & 35,000 \\
10 & 0.070 & 500,000 & 0.2308 & 35,000 \\
11 & 0.090 & 500,000 & 0.2847 & 45,000 \\
12 & 0.100 & 300,000 & 0.3490 & 30,000 \\
13 & 0.100 & 200,000 & 0.4141 & 20,000 \\
14 & 0.100 & 100,000 & 0.4727 & 10,000 \\
%\sout{15} & \sout{0.283} & \sout{0} & \sout{0.5255} & \sout{0} \\
\midrule
Total & & \multicolumn{2}{r}{Average Annual Loss (AAL)} & 760,000 \\
\bottomrule
\end{tabular}%
\label{tab:addlabel}%
\end{table}%
\end{itemize}
\end{document}