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\title{Ratemaking in a Changing Environment, ASTIN Bulletin  }
 



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\hypertarget{online-appendices}{%
\section{Online Appendices}\label{online-appendices}}

\hypertarget{appendix-a-summary-statistics-at-the-policy-and-claim-level}{%
\subsection{Appendix A: Summary statistics at the policy and claim
level}\label{appendix-a-summary-statistics-at-the-policy-and-claim-level}}

Table \ref{DistContC6} summarizes the distribution of the claim
frequency and severity, which are the two continuous outcomes of
interest at the policy level. The Spearman correlation coefficient shows
that the two outcomes are correlated. The table also shows that the
coverage and deductible distributions are right-skewed, and we deal with
the skewness by taking logarithmic transformations on these variables.
The high correlation coefficient for coverage suggests it will be a
significant predictor in the claim frequency and severity model. In
addition, the table summarizes the transaction frequency to the
settlement of a reported claim and the average payment amounts; the
Spearman correlation coefficient shows that they are correlated.
Moreover, the correlation coefficient between deductible and the
transaction frequency and severity indicates it may be used as a
predictor.

\begin{table}[h]
\centering
\caption{Summary statistics for outcomes at the policy level (claim frequency and severity) and claim level (transaction frequency and severity), and continuous covariates (deductibles, and coverages).}
\label{DistContC6}
\resizebox{0.7\textheight}{!}{
    \begin{tabular}{lrrrrrr}
    \hline
     \multicolumn{7}{c}{Policy Level} \\
          & \multicolumn{1}{r}{Min.} & \multicolumn{1}{r}{Median} & \multicolumn{1}{r}{Average} & \multicolumn{1}{r}{Max.} & \multicolumn{1}{r}{Claim}& \multicolumn{1}{r}{Claim} \\
          &  &  &  &  & \multicolumn{1}{r}{Frequency }& \multicolumn{1}{r}{ Severity} \\
                    &  &  &  &  & \multicolumn{1}{r}{ ($\rho_S$)}& \multicolumn{1}{r}{ ($\rho_S$)} \\
    \hline
    Claim Frequency & 0     & 0   & 0.893 & 231  &  -     & 0.964\\
    Claim Severity & 0   & 0   & 9,649  & 12,922,218    & 0.964  & -\\
    Deductible & 500   & 1,000  & 3,407 & 100,000       & 0.051  & 0.090 \\
    Coverage(000'S) & 0.4     & 11,493  & 11,493  & 2,444,797   & 0.404  & 0.396 \\
    \hline
    \multicolumn{7}{c}{Claim Level} \\
              & \multicolumn{1}{r}{Min.} & \multicolumn{1}{r}{Median} & \multicolumn{1}{r}{Average} & \multicolumn{1}{r}{Max.} & \multicolumn{1}{r}{Transaction}& \multicolumn{1}{r}{Transaction} \\
          &  &  &  &  & \multicolumn{1}{r}{Frequency}& \multicolumn{1}{r}{ Payment} \\
                    &  &  &  &  & \multicolumn{1}{r}{ ($\rho_S$)}& \multicolumn{1}{r}{ ($\rho_S$)} \\
    \hline
    Transaction Frequency & 1     & 1   & 1.221 & 11           &  -     & 0.301\\
    Transaction Payment (Average per claim) & 8.2   & 2,609   & 11,757 & 1,174,293    & 0.301   & -\\
    Deductible & 500   & 1,000  & 6,739 & 100,000       & 0.095  & 0.132\\
    Coverage(000'S) & 102.3     & 58,046  & 295,660  & 2,444,797   & -0.019 & -0.079 \\
    \hline
    \end{tabular}
    }
\end{table}

Table \ref{DistCart} shows the summary statistics of the claim frequency
and severity at the policy level and the transaction frequency and
severity at the claim level for the categorical variables in the
dataset. The table suggests a high variation in the claim frequency and
average severity of the claims across the categorical variables at the
policy level. At the claim level, the transaction frequency does not
vary much, but there is a substantial variation with the average
severity.

\begin{table}[htbp]
\centering
\caption{Summary statistics at the policy and claim level by categorical variables.}
\label{DistCart}
\resizebox{0.55\textheight}{!}{
    \begin{tabular}{lrrrrrr}
    \hline
    \multicolumn{1}{r}{} & &\multicolumn{2}{c}{Policy Level} & &  \multicolumn{2}{c}{Claim Level} \\
  \cmidrule{1-1} \cmidrule{3-4} \cmidrule{6-7}
           \multicolumn{1}{l}{Variable} & &  \multicolumn{1}{r}{Average} & \multicolumn{1}{r}{Average} & &  \multicolumn{1}{r}{Average} & \multicolumn{1}{r}{Average}\\
    \multicolumn{1}{r}{} & & \multicolumn{1}{r}{Frequency} & \multicolumn{1}{r}{Severity} &  &\multicolumn{1}{r}{Frequency} & \multicolumn{1}{r}{Severity}\\        
 \hline
          \multicolumn{1}{l}{$\textit{Entity Type}$ }& &\multicolumn{1}{l}{} & &\multicolumn{1}{l}{}&\multicolumn{2}{r}{}\\
    Village     & & 0.349   & 2,977 & & 1.196   & 7,776\\
    City        & & 1.607   & 12,015 & & 1.222   &9,276\\
    County       &  & 3.330   & 17,453 & & 1.169   &13,089 \\
    Misc        & & 0.172   & 4,110  & & 1.127   &14,328\\
    School      & & 0.931   & 25,963 & &  1.231  &15,604 \\
    Town        &  & 0.092   & 1,204  & &  1.126  & 6,351\\
\hline
       \multicolumn{1}{l}{$\textit{Region}$}& &\multicolumn{1}{l}{}&\multicolumn{1}{l}{} & &\multicolumn{2}{r}{}\\
    Northeastern    & & 0.561   & 7,835 & & 1.189   & 12,840 \\
    Northern       &  & 0.410   &10,682 & & 1.229   & 13,879  \\
    Southeastern   & & 1.389   & 30,123& & 1.206   & 15,605 \\
    Southern       & &  1.196  & 5,820 & & 1.263   & 7,921\\
    Western        & & 0.483   &5,210 & & 1.216   &  7,989  \\
\hline
     \multicolumn{1}{l}{$\textit{Alarm Credit}$}& &\multicolumn{1}{l}{} &\multicolumn{1}{l}{}& &\multicolumn{2}{r}{}\\
    No Alarm Credit   &             & 0.226   & 2,427 & &  1.237  & 7,873\\
    Alarm Credit 5\%  &               & 0.290   & 3,508 & & 1.183   & 10,566 \\
    Alarm Credit 10\% &              & 0.275   & 3,016 & & 1.165   &10,250\\
    Alarm Credit 15\%  &            & 1.059   & 15,869  & &  1.212  &11,326\\
    Alarm Credit (Combination)  &    & 2.212   & 29,923 & & 1.233   &14,476\\
    \hline
    \end{tabular}
    }
\end{table}

\clearpage

\clearpage

\hypertarget{appendix-b-algorithm-to-construct-the-simulation-data.}{%
\subsection{Appendix B: Algorithm to construct the simulation
data.}\label{appendix-b-algorithm-to-construct-the-simulation-data.}}

We employ Algorithm \ref{MPPsim} to construct the simulation data.

\begin{algorithm}
\tiny
\caption{Data-generating process for MPP.}\label{MPPsim}

 \hspace*{\algorithmicindent} \textbf{Input:}  \begin{tabularx}{\textwidth}{l>{$}c<{$}X} 
 
 Number of policyholders $J$, parameters for claim frequency $\left\{\alpha,\beta_{11},\beta_{12}\right\}$,  parameters for reporting delay\\ 
  distribution $\{\gamma_{10},\gamma_{11}.\gamma_{12},\kappa\}$, parameters for transaction frequency $\left\{b,\pi_{11},\pi_{12}\right\}$,  parameters for transaction \\
payment $\left\{\phi_{10},\phi_{11},\phi_{12}, \sigma\right\}$, probability of settlement $\eta$, and ratemaking time $\tau$. \\
   \end{tabularx}
   
  \hspace*{\algorithmicindent} \textbf{Output:}  \begin{tabularx}{\textwidth}{l>{$}c<{$}X} 

Complete reported data - policy-level ($PL=\left\{(S_j,N_j,x_{j1},x_{j2});j=1,...,J \right\}$), \\
claim-level ($CL=\left\{(P^{ULT}_{ji},x_{j1},x_{j2},\delta_{ji},M_{ji}(\tau),M_{ji});N_j>0,i=1,...,N_j \right\}$), \\ 
and transaction-level ($TL=\left\{(P_{jik},x_{j1},x_{j2},M_{ji}(\tau));M_{ji}>0,k=1,...,M_{ji} \right\}$)\\
Datasets at ratemaking date - policy-level ($PL^R=\left\{PL; k \leq M_{ji}(\tau)\right\}$), \\
claim-level ($CL^R=\left\{CL; k \leq M_{ji}(\tau)\right\}$), \\
and transaction-level ($TL^R=\left\{TL; k \leq M_{ji}(\tau)\right\}$)

  \end{tabularx}
  
\begin{algorithmic}[1]
\For{ Policyholder $j=1,..., J$}
        \State 
       Generate $x_{j1}\sim \mbox{Bernouli}(0.3)$, $x_{j2}\sim \mbox{Normal}(0,1)$;
       \State 
       Generate  $N_j^T \sim \mbox{Poisson}(\psi_{j})$; $\psi_{j}=\exp(\log\alpha +x'_{j1}\beta_{11}+x'_{j2}\beta_{12})$;
          \State 
        Generate  $U_i \sim \mbox{Weibull}(N_j^T,\theta_{i},\kappa)$; $\theta_{i}=\exp(\gamma_{10} +x'_{j1}\gamma_{11}+x'_{j2}\gamma_{12})$;
         \State 
        Generate  $v_i \sim \mbox{Uniform}(N_j^T,0,\tau)$; 
          \State 
        Generate  $N_j \sim \mbox{Count}(U_i+v_i\leq \tau)$; 
      \If {$N_j > 0$}
      \For{$i=1,...,N_j$}
         \State 
       Generate  $M_{ji} \sim \mbox{Poisson}(\lambda_{ji})$; $\lambda_{ji}=\exp(\log b+x'_{j1}\pi_{11}+x'_{j2}\pi_{12})$ ;
        \If {$M_{ji} > 0$}
         \For{$k=1,...,M_{ji}$}
         \State
         Generate $P_{jik}\sim \mbox{Gamma} \left(\frac{\exp(\mu_{jik})}{\sigma},\sigma\right)$;
         \State where $\mu_{jik}=\exp(\phi_{10}+x'_{j1}\phi_{11}+x'_{j2}\phi_{12})$ ;
         \EndFor
         \State \textbf{return} $\left\{P^{ULT}_{ji}=P_{ji1}+\ldots+P_{jiM_{ji}}\right\}$;
         \State  Generate $\delta_{ji}\sim \mbox{Bernouli}(\eta)$
          \For{$\delta_{ji}=0$}
         \State $M_{ji}(\tau) \sim \mbox{ Discrete Uniform}(0,M_{ji})$
          \EndFor
         \Else
         \State $P_{jik}=0$,$P^{ULT}_{ji}=0$, $M_{ji}(\tau)=0$ and $\delta_{ji}=1$
         \EndIf
      \EndFor
      \State
      \textbf{return} $\left\{S_j=P^{ULT}_{j1}+\ldots+P^{ULT}_{jN_j}\right\}$; 
      \Else
      \State $S_j=0$
      \EndIf
  \EndFor
     \State 
        \textbf{return}
        \begin{tabularx}{\textwidth}{l>{$}c<{$}X}
         $PL=\left\{(S_j,N_j,x_{j1},x_{j2})\right\}$; $PL^R=\left\{PL; k \leq M_{ji}(\tau)\right\}$;\\
         $CL=\left\{(P^{ULT}_{ji},x_{j1},x_{j2},\delta_{ji},M_{ji}(\tau),M_{ji})\right\}$; $CL^R=\left\{CL; k \leq M_{ji}(\tau)\right\}$\\
         $TL=\left\{(P_{jik},x_{j1},x_{j2},M_{ji}(\tau))\right\}$; $TL^R=\left\{TL; k \leq M_{ji}(\tau)\right\}$
         \end{tabularx}
\end{algorithmic}
\end{algorithm}

\newpage

\hypertarget{appendix-c-loss-cost-prediction-using-the-frequency-severity-model}{%
\subsection{Appendix C: Loss Cost Prediction Using the
Frequency-Severity
Model}\label{appendix-c-loss-cost-prediction-using-the-frequency-severity-model}}

A rating formula based on the Frequency-Severity model will be achieved
by the product of exponentiated estimates from the claim frequency, and
severity models. The following rating formula calculates the predicted
loss cost: \begin{equation}
\label{PredForm}
\begin{array}{rl}
\text{Loss Cost}&= e_{j} \exp(\ln\hat{\alpha}_T+\bm{x}'_{j}\hat{\bm{\beta}})\times \exp(\bm{x}'_{j}\hat{\bm{\phi}})\\
&= \text{Exposure}\\
&\times \text{Expected number of claims}\\
& \times \text{Expected payment per claim}.\\
\end{array}
\end{equation}

\noindent Where \(e_{j}\) is the exposure variable and \(\bm{x}_{j}\)
are rating factors for the new contract. \(\{\hat{\alpha}_T\}\) are the
fitted trend parameters from the most recent policy year for the
reported claim. Also, \(\{\hat{\bm{\beta}},\hat{\bm{\phi}}\}\) are the
fitted parameters for rating variables from the claim frequency model,
and the severity model building blocks. Here, the Poisson model is used
to model the claim frequency, and a gamma GLM with a logarithmic link is
used to model the loss amounts from claims.




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\end{document}
